ANALYSIS OF TRANSPORT IN THE BRAIN by Lori Ann Ray A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Chemical Engineering MONTANA STATE UNIVERSITY Bozeman, Montana April 2021 ©COPYRIGHT by Lori Ann Ray 2021 All Rights Reserved ii DEDICATION This work is dedicated to all those living with our lack of knowledge about the inner workings of the brain. To my son, Ian, who lost his Junior and Senior years of high school and first year of college to what we understand in retrospect was post-concussive syndrome. To his continuing to advocate that things were “not right” in his brain despite being told repeatedly by doctors that he was fine, just needed to rest, and would adapt to the changes (meaning new limitations) in his brain. To his openness to try the novel therapies I found in my desperate search to help him and understand myself what was wrong. To his patience, determination, and unbelievable discipline in making the lifestyle changes to care for his brain as it slowly healed, part of which required “reigning in” an incredibly goal-oriented athlete and student. Ian did all this (and still graduated high school and got accepted at his college of choice) despite being misunderstood by many of the people he was close to and feeling pretty awful through much of the process. He remains dedicated to all the ways in which he cares for his brain and body five years after his symptoms began. Ian is now a thriving Chemical Engineering and Pre-Medicine student at MSU. He will be an even more phenomenal doctor due to his experience being marginalized due to our society’s lack of understanding at the time of his injury and his personal journey to regaining his health. To my granddaughter, Bailie, who through a 1 in 100 billion fluke accident or illness in utero, was born with her cognitive functions intact and trapped in a body over which she has little control. She has cerebral palsy, but more specifically bilateral schizencephaly. Bailie will have to rely on others to care for her for the entirety of her life. Her condition is so rare that each time we ended up in the hospital in her early years they were reticent to release her. To my family working through how to care for a child that no one knew how to care for. To the blessed family that has walked this with us, the Jarvis’ (who Bailie calls Memzie, Pa, and Auntie), carrying us when we could not carry ourselves. And to Bailie’s mom who could only have been completely overwhelmed with this diagnosis at the age of 19 when Bailie was born but has stayed with it and continues to have a very special relationship with Bailie. Bailie is our small, but sassy, beacon of joy and unconditional love. To my mother-in-law, Nancy, who died from amyotrophic lateral sclerosis (ALS). A disease that slowly robs you of your motor skills, such as speaking, swallowing and breathing, and is diagnosed by the frustrating process of elimination. Nancy faced this challenge bravely and openly. She involved her church community in her daily care, such as trips to the grocery store, in order to ease the burden on her family and to educate others. Nancy was a class act. I imagine a future in which there are medicines for neurological diseases like ALS, guidelines for proactive healing of brain injuries, and technology that enables those with iii disabilities to enjoy the power of moving their bodies and speaking their minds. Having “met” several amazing scientists, engineers, and philanthropists through the course of my studies, either in person or through their work, I believe this future is not far away. This work is also dedicated to my husband, Rod, who was an important source of encouragement, the person with whom I could debate my ideas and understandings, and my editor. This could not have been easy as a husband--sometimes ours exchanges were stimulating and others they were incredibly frustrating for him. I deeply appreciate his support throughout my doctoral adventure. My work, especially the written part, was made better by his involvement and I am eternally grateful. iv TABLE OF CONTENTS 1. INTRODUCTION ........................................................................................................... 1 Neurodegeneration and Waste Clearance ........................................................................ 2 Waster Clearance in the Brain differs from the Body ..................................................... 4 Mechanisms of Transport ................................................................................................ 6 Theories of Transport in the Brain .................................................................................. 7 Summary of Research ...................................................................................................... 8 References Cited ............................................................................................................ 10 2. FLUID FLOW AND MASS TRANSPORT IN BRAIN TISSUE: BACKGROUND AND LITERATURE REVIEW ....................................................... 12 Contribution of Authors and Co-Authors ...................................................................... 12 Manuscript Information Page ........................................................................................ 13 Abstract .......................................................................................................................... 14 Introduction ................................................................................................................... 14 Background .................................................................................................................... 15 Relevant Physiology .............................................................................................. 15 Relevant Transport ................................................................................................ 18 Diffusion and Advection ........................................................................... 19 Osmotics .................................................................................................... 19 Transport Equations for Interstitial Space ................................................. 20 Equations of Flow for Perivascular Space ................................................. 21 Transport Parameters ................................................................................. 21 Evolution of the Field ................................................................................................... 22 Key Experimental Work ........................................................................................ 22 Glymphatic Hypothesis ......................................................................................... 24 Transport in the Whole Brain: Dynamic Contrast-Enhanced MRI ...................... 25 Efflux Routes and Meningeal Lymphatic Vessels ................................................ 26 Efflux Routes ............................................................................................. 26 Meningeal Lymphatic Vessels .................................................................. 27 Sleep Enhances Glymphatic Function ................................................................... 27 Unanswered Questions .......................................................................................... 28 Perivascular Flow: Influx ......................................................................... 28 Interstitial Flow ......................................................................................... 31 Transport between Perivascular Space and Interstitium ............................ 34 Perivascular Flow: Efflux ......................................................................... 35 Glymphatic Debate is Focusing ............................................................................. 36 Early Human Results ............................................................................................. 36 Discussion ..................................................................................................................... 37 Transport Time-Scale Analysis ............................................................................. 37 Mass/Volumetric Flow Balance ............................................................................ 39 v TABLE OF CONTENTS CONTINUED Discussion Summary ............................................................................................. 40 Conclusions and Areas of Future Work ........................................................................ 41 Acknowledgements ....................................................................................................... 42 References Cited ............................................................................................................ 42 3. ANALYSIS OF CONVECTIVE AND DIFFUSIVE TRANSPORT IN THE BRAIN INTERSTITIUM ....................................................... 48 Contribution of Authors and Co-Authors ...................................................................... 48 Manuscript Information Page ........................................................................................ 49 Abstract .......................................................................................................................... 50 Background .................................................................................................................... 50 Physiology of the Brain Interstitium ..................................................................... 50 Transport in Biological Tissues ............................................................................. 51 Experimental Techniques for Investigating Brain Transport and Their Findings ................................................................................................. 52 Estimating Interstitial Bulk Flow .......................................................................... 54 Published Simulations ........................................................................................... 54 RTI Experiments in the Context of Interstitial Bulk Flow .................................... 55 Methods ......................................................................................................................... 55 Model Parameters and Variables ........................................................................... 58 Results ........................................................................................................................... 58 Interstitial Bulk-Flow Simulations ........................................................................ 58 Simulations of Real-Time Iontophoresis Experiments .......................................... 58 Implications for Large-Molecule Transport .......................................................... 62 Clearance Simulations ........................................................................................... 63 Discussion ...................................................................................................................... 64 Conclusions ................................................................................................................... 65 Acknowledgements ....................................................................................................... 66 References Cited ............................................................................................................ 66 4. QUANTIFICATION OF TRANSPORT IN THE WHOLE MOUSE BRAIN ............................................................................................ 68 Contribution of Authors and Co-Authors ...................................................................... 68 Manuscript Information Page ........................................................................................ 69 Abstract .......................................................................................................................... 70 Introduction ................................................................................................................... 72 Improved Understanding of the Glymphatic System and Current Questions ........................................................................................... 75 Literature Review of DCE MRI for Study of Brain Transport ............................. 81 Literature Review of Whole-Brain Transport Modelling ...................................... 84 vi TABLE OF CONTENTS CONTINUED Approach ....................................................................................................................... 86 Results ........................................................................................................................... 91 Tracer Concentration from Dynamic Contrast-Enhanced MRI ............................ 91 Background for Interpreting DCE MRI .................................................... 92 Cerebral Vasculature & Cerebrospinal Fluid Circulation Anatomy .............................................................. 92 Tracer Injection Site and Ventricular Accumulation ....................... 94 T2 Interference ................................................................................ 95 Tracer Concentration from DCE MRI for Wild-type Mice ....................... 96 Tracer Concentration from DCE MRI for Transgenic Mice ..................... 98 Transport Model Simulation Results ................................................................... 101 Anatomical Subdomains defined by MRI Data ....................................... 102 Preferential Transport Routes correspond to Periarterial Space ........................................................................... 103 Brain Tissue Subdomain combines Several Mechanisms of Transport .............................................................. 105 Effective Diffusivities Exceed Apparent Diffusivity .............................. 107 Wild-type Mice .............................................................................. 107 Transgenic Mice ............................................................................ 110 Comparison of Data to “Best Fit” Simulation ......................................... 112 Sources of Error ................................................................................................... 114 Sources of Error from DCE MRI ............................................................ 114 Effect of Injection on Results ........................................................ 116 Sources of Error from Transport Modelling Assumptions ...................... 123 Discussion .................................................................................................................... 126 Conclusions ................................................................................................................. 131 Future Work ......................................................................................................... 133 Methods ....................................................................................................................... 135 Calculation of Tracer Concentration and Identification of Anatomical Details from DCE MRI Data ....................................................... 135 Transport Model .................................................................................................. 138 Acknowledgements ..................................................................................................... 139 References Cited .......................................................................................................... 140 5. CONCLUSIONS AND CONTRIBUTIONS .............................................................. 145 Key Contributions ....................................................................................................... 147 Future Work ................................................................................................................. 148 References Cited .......................................................................................................... 151 REFERENCES CITED ................................................................................................... 177 vii TABLE OF CONTENTS CONTINUED APPENDICES ................................................................................................................. 153 APPENDIX A: Varying Perivascular Astroglial Endfoot Dimensions Along the Vascular Tree Maintain Perivascular-Interstitial Flux through the Cortical Mantle .................................. 153 APPENDIX B: Supplemental Material for Chapter Four ................................... 170 viii LIST OF TABLES Table Page 2.1. Acronyms ....................................................................................................... 15 2.2. Key Parameters describing Transport in the Brain and their Values .............................................................................................. 22 2.3. Estimated interstitial velocity for awake state and proposed interstitial velocity in the anesthetized state for mice .................................................................................................. 27 2.4. Summary of Computational Models and Measurements of Periarterial Flow ............................................................................................. 29 2.5. Summary of Computational Models of Interstitial Flow and Mass Transport ........................................................................................ 32 2.5. Revised Summary of Computational Models of Interstitial Flow and Mass Transport ............................................................................... 33 2.6. Interstitial Superficial-Velocity Simulation Results for Various Authors .............................................................................................. 33 2.7. Characteristic Transport Times in the Mouse Brain ....................................... 39 2.8. Volumetric Flow Balance Comparisons for Periarterial and Interstitial Flow in the Mouse Brain ........................................................ 40 3.1. Summary of Extracellular Space (ECS) Structural Parameters determined by TMA Real-time Iontophoresis (RTI) Experiments on Neocortex oof Healthy Anesthetized Adult Rats and Mice ................................................................. 52 3.2. Model Parameters and Variables .................................................................... 57 3.3. Simulation Results for Bulk-Flow Superficial-Velocity in the Brain Interstitium .................................................................................. 59 3.4. Summary oof Simulations and Sensitivity Analysis Performed .................... 59 3.5. Summary of Boundary Condition Sensitivity Analysis ................................. 59 ix LIST OF TABLES Table Page 4.1. Symbols and Acronyms .................................................................................. 70 4.2. Quantitative Analysis of Brain Transport Parameters for each Anatomical Subdomain .................................................................. 128 x LIST OF FIGURES Figure Page 1.1. Physiology of Molecular Transport in the Body and the Brain ........................ 5 2.1. Production and Circulation of Cerebrospinal Fluid ........................................ 17 2.2. Illustration of Anatomical Details within Brain Tissue .................................. 18 2.3. Characteristic Transport Time in the Interstitial Space of Tissues as a Function of Molecular Size .................................................... 19 2.4. Hypothetical perivascular and paravascular flow pathways around an artery ..................................................................... 23 2.5. Comparison of RTI Experimental Data and Finite-element Simulation for Asleep and Awake States ....................................................... 28 2.6. Illustrations for Characteristic Time and Mass Balance Calculations ...................................................................................... 38 3.1. Illustration of Movement of Fluid and Solutes in Brain Tissue between Interstitial and Perivascular Space surrounding Penetrating Vasculature ................................................... 51 3.2. TMA Concentration Curves for each Replicate of Young Adult Mice from Kress ....................................................................... 53 3.3. Finite-element Domain Illustrating Physiology Incorporated into Model ................................................................................. 56 3.4. Superficial Velocity Streamlines and Velocity Profile for 𝑣 = 50 µm/min .............................................................................. 58 3.5. Range in TMA Concentration versus Time Curves for Experimental Data compared with Diffusion-only with Perivascular Exchange Simulations ............................... 60 3.6. Range in TMA Concentration versus Time Curves for Experimental Data compared with Diffusion and Convection Simulations with Perivascular Exchange .................................... 61 xi LIST OF FIGURES CONTINUED Figure Page 3.7. Peclet Number versus Apparent Diffusivity for Molecules of Interest in Brain Transport ........................................................ 62 3.8. Ab Clearance from Interstitial Injection, experimental Data Compared with Simulations ................................................................... 63 4.1. Physiology of Molecular Transport in the Body versus the Brain ................. 73 4.2. Mouse Anatomy: Cerebral Vasculature and Cerebral Spinal Fluid (CSF) Circulation ........................................................ 93 4.3A. Wild-type Mouse Tracer (Gad) concentration contours calculated from DCE MRI Data ................................................... 97 4.3B. Transgenic Mouse Tracer (Gad) concentration contours calculated from DCE MRI Data ................................................... 99 4.4. Anatomical Details extracted from MRI data define Transport Model Subdomains ...................................................................... 103 4.5. Effective Diffusivity for Different Brain Regions of Wild-type Mice (A) and a Transgenic Mouse (B). ....................................... 109 4.6. Comparison of Wild-type “Best Fit” Simulation to DCE MRI concentration data for Representative Mouse ............................. 113 4.7. Effective Diffusivities estimated for the Surface Periarterial Space and Brain Tissue regions during and after the Injection for a Representative Wild-type Mouse ........................... 120 4.8. Schematic of Brain Transport Analysis using Finite-element Modeling with DCE MRI Data ............................................ 135 xii NOMENCLATURE Symbol Description a Void volume, porous media characteristic a flip angle, in MRI parameter of pulse sequence l tortuosity, porous media characteristic f Void fraction, in porous media theory. Same as void volume. ADCw Apparent diffusivity of water, as measured by MRI 𝑐 tracer concentration 𝐷 Free Diffusivity, unhindered as in not through porous media 𝐷∗ Apparent Diffusivity, Diffusivity of a molecule through a porous medium 𝐷"## Apparent Diffusivity, Diffusivity of a molecule through a porous medium 𝐷$%&# Dispersion Coefficient 𝐷'(( Effective Diffusivity, Lumped Mass Transport Parameter 𝐷'((,*+ Effective Diffusivity, Brain Tissue Region in Model 𝐷'((,*,-. Effective Diffusivity, Branching Periarterial Space in Model 𝐷'((,.,-. Effective Diffusivity, Surface Periarterial Space in Model 𝑓(𝑐) Cellular uptake, in mass transport equation 𝑘′ Hydraulic conductivity 𝐿 Characteristic length 𝑃 Pressure 𝑃𝑒 Peclet Number, ratio of convective to diffusive transport 𝑟/ Tracer relaxivity 𝑅𝑒 Reynolds number, ratio of inertial to viscous forces 𝑟𝑚𝑠 Root-mean square error 𝑠 Source term, in mass transport equation 𝑆 MRI signal 𝑆0 Pre-contrast MRI signal, baseline 𝑆1$ Post-contrast MRI signal 𝑇/,0 Pre-contrast relaxation time 𝑇𝑅 Repetition time 𝑣 Superficial velocity 𝑣% Intrinsic velocity = 𝑣/𝛼 𝑣2. velocity in interstitial space 𝑣,-. velocity in periarterial space xiii NOMENCLATURE CONTINUED Acronym Description Ab Beta-amyloid, protein aggregate implicated in Alzheimer’s Disease Acer Anterior cerebral artery AQP Aquaporin, cell membrane protein channel for water transport aquaporin-4, concentrated on astrocytes at border of PVS & interstitial AQP4 space Bas Basilar artery BBB Blood-brain barrier BT Anatomical region in FEM Model, Brain Tissue CoW Circle of Willis, system of arteries on ventral surface of brain CSF Cerebrospinal fluid DCE MRI Dynamic contrast-enhanced MRI DTI MRI Diffusion tensor imaging MRI ECM Extracellular matrix ECS Extracellular space EM Electron microscopy or micrograph FEM Finite-element model EMG Electromyography Gad Gaditeridol (Prohance), DCE MRI contrast agent (tracer) GAG Glycosaminoglycans, primary component of ECM in the brain ICA Internal carotid artery ICP Intracranial pressure iNHP Idiopathic normal-pressure hydrocephalus Integrative Optical Imaging, experimental method for measuring interstitial IOI transport in the brain ISF Interstitial fluid IVIM MRI Intravoxel Incoherent Motion MRI KO Knock-out, genetically modified animal model, also called transgenic Mcer Middle cerebral artery MRI Magnetic resonance imaging NREM Non-rapid eye-movement sleep Olfac Olfactory artery PAS Periarterial space Pcer Posterior cerebral artery PET Positron emission tomography xiv NOMENCLATURE CONTINUED Acronym Description BPAS Branching periarterial space PVS Perivascular space PVW Perivascular “wall” rOMT regularized Optimal mass transport model Real-time iontophoresis, experimental method for measuring interstitial RTI transport in the brain, analysis assumes diffusion-only transport SAS Subarachnoid space, CSF-filled space surrounding the brain SBA Anatomical region in FEM Model, Surrounding Branching Arteries SPAS Surface periarterial space SSA Anatomical region in FEM Model, Surrounding Surface Arteries TBI Traumatic brain injury TMA Tetramethylammonium, tracer in RTI experiments WT Wild-type, unaltered animal model xv ABSTRACT Neurodegeneration is one of the most significant medical challenges facing our time, yet the gap between therapies and understanding of the inner workings of the brain is great. Impairment of waste clearance has been identified as one key underlying factor in the vulnerability of the brain to neurodegeneration, stimulating research towards understanding transport of molecules in the brain. Based on experimental findings, a unique-to-the-brain circulation has been proposed, the glymphatic system, where cerebrospinal fluid surrounding the brain moves into the brain along the periarterial space that surrounds cerebral arteries, flows through the interstitial space between brain cells, where cellular wastes reside, and carries waste out of the brain tissue along perivenous routes. However, current gaps in knowledge about the driving force for fluid flow have generated scientific skepticism, and an independent method for quantifying transport and demonstrating the presence or absence of convection is desirable. In this work, computational transport models are developed and used to analyze published experimental data to determine fundamental transport parameters for different aspects of the glymphatic circulation. Calculated transport parameters are compared to the known diffusivity of tracers through brain tissue to draw conclusions about the presence and significance of bulk flow, or convection. Based on these analyses, transport in the periarterial spaces surrounding major arteries is over 10,000 times faster than diffusion and in brain tissue, containing both periarterial and interstitial space, transport is around 10 times faster than diffusion alone (for characteristic transport lengths around 1 mm). Interstitial velocity is determined to be on the order of 0.01 mm/min, making convection in the interstitial spaces of the brain critical to the transport of large, slow-to- diffuse molecules implicated in neurodegeneration. Convection is demonstrated to be a significant mechanism of transport throughout the brain. Observations and analyses from this work contribute further evidence to a circulatory-like system in the brain with relatively rapid convection along periarterial space, branching throughout the brain tissue and slower convection across that tissue, in the interstitial spaces of the brain. Transport models developed in this work are demonstrated to be useful tools for gleaning further information from experimental data. 1 CHAPTER ONE INTRODUCTION The brain is wider than the sky and deeper than the sea. --Emily Dickenson It is the purpose of this work to investigate molecular transport in the brain. The brain has been called the last frontier in physiology and medicine--unique among organs and long under-explored due to concerns of breaching the cranial vault. Molecular transport is an essential link in many physiological and pathological processes of the brain. In particular, the waste-clearance aspect of molecular transport has gained notoriety in recent years due to its established connection to neurodegeneration. Due to the blood brain barrier, which protects the brain by restricting movement of molecules across cerebral blood vessel walls, the brain’s system for molecular transport is known to differ from the rest of the body. Therefore, understanding the movement of molecules within the brain is an important aspect of expanding our knowledge about the brain. A brain-specific circulation has been proposed, the glymphatic system [1], in which cerebrospinal fluid surrounding the brain moves into the brain along an annular space surrounding cerebral arteries, called the periarterial space; crosses the interstitial space between brain cells, dissolving and entraining waste molecules; and moves out of the brain along perivenous routes. However, the driver of fluid flow for this proposed circulation, such as the heart pumping the blood, has not been firmly established. In addition, although experimental observations are plentiful, little work has been done to 2 quantify transport within the structure of the glymphatic system, tying observations to fundamental transport parameters. In the work presented here, fundamental transport parameters are determined for interstitial and perivascular space by analyses utilizing finite-element transport models and published experimental data. Transport parameters are compared to the known diffusivity of tracer molecules to draw conclusions about the presence and magnitude of convection in the interstitial and perivascular compartments. In addition, these connected compartments are analyzed as a circulatory system at both the brain-tissue and whole- brain scale. A major goal of this work is the development and demonstration of transport models as useful tools for analyzing data and expanding knowledge as new discoveries are made. Neurodegeneration and Waste Clearance Neurodegeneration is one of the most significant medical challenges facing our time, yet the gap between therapies and understanding of the inner workings of the brain is great—referred to by neuroscientists as the black valley of death [2]. Alzheimer’s, the most prevalent of the neurodegenerative diseases, is the fastest growing cause of death in the U.S. (excepting COVID-19)--projected to grow from 6 million affected Americans in 2020 to 13 million in 2050 [3]. Yet no effective treatments exist, and several recent clinical studies of new drug therapies based on the latest scientific theories have failed. With the initial harmful events that bring later neurodegenerative decline known to begin 20 years before symptoms become apparent, the improved understanding of brain 3 physiology that fuels prevention and early intervention is essential to curbing the growth of neurodegenerative disease. A hallmark of neurodegenerative disease is the build-up of aggregated proteins in the brain, called plaques. In fact, most neurodegenerative diseases are characterized, and often diagnosed, by their “signature” protein aggregates. Since aggregates form at elevated protein concentrations, neurodegeneration has been linked to deficiencies in the system that removes waste products from the brain, thereby maintaining homeostatic (optimal) concentrations. Impairment of waste clearance of mis-aggregating proteins from brain tissue is believed to underlie the vulnerability of the aging and injured brain to neurodegeneration [4, 5]. Waste clearance is a considerable task in the brain. The brain consumes about 25% of the body’s energy, although it makes up only 2% of its mass. In the process of making and using all this energy, approximately 3 lbs. per year of potentially toxic protein wastes and biological debris are generated [6]. This molecular waste must be “cleared” from the brain. Lack of sufficient clearance and associated increases in concentration can cause cellular and/or systemic damage--experienced as problems with cognition, memory, or other activities of the brain. At elevated concentrations, proteins aggregate together eventually becoming too large to move through brain tissue—thereby getting “trapped” and causing irreversible damage. In fact, the pathology of Alzheimer’s, Parkinson’s and other neurodegenerative diseases can be reproduced in animal models by the forced over- production of protein aggregates [6]. 4 Waste Clearance in the Brain differs from the Body The body’s primary tools for the transport of molecules, including waste clearance, are the circulatory system, comprised of the heart and 100,000 km of blood vessels, and the lymph system. The heart pumps blood through the vasculature, carrying nutrients and wastes between specialized tissues and organs over relatively vast distances. Facile exchange of molecules between the blood and the surrounding tissue occurs through “loose” junctions between cells at the blood vessel wall. These loose junctions also allow leakage of fluid from arteries that creates interstitial flow through tissue, delivering nutrients and sweeping away waste, which is collected in veins and lymph vessels (Figure 1A). Circulatory physiology in the brain differs markedly from the rest of the body due to the blood brain barrier (BBB), which protects the brain from pathogens and molecules that could upset the unique biochemical environment required for cerebral activities. The BBB is comprised of tight junctions between cells at the blood vessel wall, with transport of molecules across the BBB strictly controlled and occurring through protein channels optimized for specific molecules. As a consequence of the BBB, the brain must employ uniquely evolved methods for transport of molecules, different from the rest of the body. Many of the physiological details of this brain-specific transport system are only now being discovered. 5 A. Molecular Transport in the Body B. Molecular Transport in the Brain: Glymphatic Hypothesis Artery Vein Blood-Brain Periarterial Perivenous Artery Vein Barrier influx efflux Interstitial Solute !'( $ ! %&&"# Interstitial Flow Lymph Vessel Perivascular Brain Astrocytic Perivascular Space Interstitium Endfoot Space Figure 1. Physiology of Molecular Transport in the Body versus the Brain. A. Illustration of physiological system of molecular transport in tissues outside the brain. Transport in the body occurs via the circulatory system, interstitial flow through tissues, and the lymphatic system. Fluid (white arrows) leaks from arteries into the interstitial space between cells, generating interstitial flow that delivers nutrients (blue dots) and carries away waste (blue dots) to veins or lymphatic vessels to be processed by the liver and kidneys [7]. B. Illustration of hypothesized physiological system of molecular transport in brain tissue. Unique to the brain are annular fluid-filled spaces surrounding cerebral blood vessels, called perivascular space (PVS), that are connected to the cerebrospinal fluid that surrounds the brain. The PVS is bounded by the vascular wall on the inside and by the “endfeet” of astrocyte cells (yellow cells with long processes) on the outside. It is hypothesized that fluid (green arrows) moves from the cerebrospinal fluid in the subarachnoid space surrounding the brain along periarterial space following penetrating arteries deep into the brain. The fluid enters the interstitium, where interstitial solutes (purple) and fluid are transported to the perivenous space and carried to extracranial efflux routes. Unique to the brain is an annular space surrounding cerebral blood vessels, called the perivascular space (PVS) (Figure 1B), which resides on the brain-tissue side of the BBB and is connected to the cerebrospinal fluid (CSF) that surrounds the brain. This perivascular space follows the vasculature as it branches deep into the brain and has been identified as a potential route of molecular transport and a source of fluid for flow through the interstitial regions of the brain. However, this brain-specific route lacks an 6 obvious driver of flow, like the heart for blood circulation, leaving many questions about transport mechanisms in the brain. Mechanisms of Transport Within the physiological systems described above, molecular transport is governed by fundamental and established principles according to the mass transport equation: 34 = 𝐷𝛻6𝑐 − 𝑣 ∙ 𝛻𝑐 + 𝑠(𝑐) − 𝑓(𝑐) (1) 35 where: 𝑐 = concentration 𝐷 = diffusion coefficient 𝑣 = velocity (a vector field) 𝑠(𝑐) = source term (i.e., cellular excretion) 𝑓(𝑐) = sink term (i.e., cellular uptake) The right-hand-side of equation 1 contains a diffusion term, a convection term, and source and sink terms. Diffusion occurs through the random (Brownian) motion of molecules from areas of high to low concentration. Diffusion is always occurring when a concentration gradient is present. The rate of diffusive transport is strongly related to molecular size and can be extremely slow for large molecules. Convection is the transport of a molecule by bulk flow. In a free medium, convective transport is independent of molecular size; all solutes move in the direction and with the velocity of the bulk flow. Convection can be much faster than diffusion, but requires an external driving force, such as a pressure gradient. Diffusion is sufficient for moving molecules 7 over very short distances and in delicate environments, such as within a cell. However, the body typically employs convection for transit over larger distances, such as blood flow and breathing. Theories for Transport in the Brain Because the BBB prevents facile exchange of molecules with the blood and traditionally understood interstitial flow, transport in the brain was historically believed to occur by diffusion alone. However, in 1974, Cserr et al. observed dye injected into the brain to move faster than could be explained by diffusion and to follow preferential pathways along the vasculature [8]. Dyes of markedly different molecular size moved at similar rates, offering evidence of convection [9]. Iliff et al. used specialized in vivo microscopic imaging to demonstrate movement of tracers from CSF surrounding the brain into the brain along the perivascular space of penetrating arteries and then into the interstitium [1] (Figure 1B). Ex vivo microscopy showed the tracers leaving the interstitial space via venous perivascular space [1]. Based on the observations described above, the perivascular space, filled with CSF, appears to behave like a circulatory system in the brain providing a pathway for rapid movement of fluid (and molecules) into deeper parts of the brain, and a source of fluid for interstitial flow through brain tissue. This brain-specific transport system has been named the glymphatic hypothesis [6, 10]. However, unlike the blood that is pumped by the heart, no clear pressure gradient exists to generate these flows, causing several researchers to reject the idea of convection in the perivascular or interstitial spaces of the brain [11, 12]. 8 Iliff et al. and others hypothesize that fluid in the perivascular space moves by a peristaltic-like flow created by the pumping action of the arterial wall that makes the internal boundary of the perivascular space [13, 14]. Some argue the movement of the arterial wall does not generate a net flow, but accelerates transport by dispersion [15]. (As used here, dispersion occurs when flows with no net bulk flow, such as eddies, facilitate mixing and accelerate transport relative to diffusion.) Scientific consensus regarding convection in the brain remains elusive. In particular, the following questions have not been satisfactorily answered: 1) Is convection a significant transport mechanism in the brain? 2) What is the magnitude of its contribution to the overall transport rate in different regions of the brain? 3) Does the experimental data, analyzed using the fundamentals of transport phenomena, support the circulatory flow proposed in the glymphatic hypothesis? Summary of Research In the work reported in this Doctoral Thesis, the nature of molecular transport in the brain is investigated and quantified. Contemporary understanding of brain physiology is combined with principles of transport phenomena to build quantitative models of molecular transport in the brain. Transport across short (µm) and long (mm) length scales is investigated, which physiologically corresponds to transport through interstitial tissue (short distance) and across the whole brain (long distance), constituting a system of perivascular and interstitial space. Published experimental data is analyzed within the structure of the transport models to calculate fundamental transport parameters such as 9 convective velocity (𝑣) and effective diffusivity (𝐷'((), a lumped parameter including all mechanisms of transport in a given region (perivascular convection, interstitial convection, and diffusion). The magnitudes of transport parameters are compared with known values of molecular diffusivity in brain tissue (𝐷"##) to draw conclusions about the presence and magnitude of convection, therefore elucidating its importance to overall transport. It is the hope of the author that the validated models, quantified transport parameters, and improved understanding of brain transport will make a useful contribution to disease diagnosis and treatment, delivery of therapeutic molecules to the brain, and brain health. The literature for transport in the brain spans several disciplines, from neuroscience and physiology to electrochemistry and cell biology. Chapter II reviews this broad literature from an engineering perspective, with the goal of understanding the physiology with respect to its impact on transport phenomena and presenting current theories and knowledge about transport in the brain. In addition, engineering analyses are performed on published data. For example, Reynolds numbers are calculated to characterize observed and predicted flows. Additionally, the effect of molecular size on transport mechanisms, a critical factor impacting movement of large proteins implicated in neurodegeneration, is investigated through analysis of characteristic times and Peclet number. Chapter II also introduces novel engineering analysis based on constructal theory [16] that seeks to develop insights into the optimality of the system of transport in the brain. 10 In Chapters III and IV, molecular transport is investigated by focusing on different scales and physiological aspects of the overall transport system in the brain. Experimental data is from published studies in mice and rats. Interstitial transport, which occurs at the µm scale, is investigated in Chapter III. Chapter IV analyzes transport across the whole mouse brain, utilizing dynamic contrast-enhanced (DCE) magnetic- resonance imaging (MRI) data and examining all scales of transport. Overall conclusions are presented in Chapter V. References Cited 1. Iliff, J.J., et al., A Paravascular Pathway Facilitates CSF Flow Through the Brain Parenchyma and the Clearance of Interstitial Solutes, Including Amyloid beta. Science Translational Medicine, 2012. 4(147). 2. Gupta, S., Keep Sharp. 2021: Simon and Schuster. 3. Alzheimer's, A., 2021 Alzheimer's disease facts and figures. Alzheimers & Dementia, 2021. 17(3): p. 19-27. 4. Selkoe, D.J. and J. Hardy, The amyloid hypothesis of Alzheimer's disease at 25years. Embo Molecular Medicine, 2016. 8(6): p. 595-608. 5. Iliff, J.J., et al., Impairment of Glymphatic Pathway Function Promotes Tau Pathology after Traumatic Brain Injury. Journal of Neuroscience, 2014. 34(49): p. 16180-16193. 6. Nedergaard, M. and S.A. Goldman, Brain Drain (vol 314, pg 44, 2016). Scientific American, 2016. 315(1): p. 6-6. 7. Bailey, R. Lymphatic Vessels. 2018 [cited 2021 March 10]; Interstitial Fluid Image]. Available from: https://www.thoughtco.com/lymphatic-vessels-anatomy- 373245. 8. Cserr, H.F. and L.H. Ostrach, BULK FLOW OF INTERSTITIAL FLUID AFTER INTRACRANIAL INJECTION OF BLUE DEXTRAN 2000. Experimental Neurology, 1974. 45(1): p. 50-60. 11 9. Cserr, H.F., et al., EFFLUX OF RADIOLABELED POLYETHYLENE GLYCOLS AND ALBUMIN FROM RAT-BRAIN. American Journal of Physiology, 1981. 240(4): p. F319-F328. 10. Nedergaard, M., Garbage Truck of the Brain. Science, 2013. 340(6140): p. 1529- 1530. 11. Hladky, S.B. and M.A. Barrand, Mechanisms of fluid movement into, through and out of the brain: evaluation of the evidence. Fluids and Barriers of the Cns, 2014. 11. 12. Carare, R.O., et al., Solutes, but not cells, drain from the brain parenchyma along basement membranes of capillaries and arteries: significance for cerebral amyloid angiopathy and neuroimmunology. Neuropathology and Applied Neurobiology, 2008. 34(2): p. 131-144. 13. Iliff, J.J., et al., Cerebral Arterial Pulsation Drives Paravascular CSF-Interstitial Fluid Exchange in the Murine Brain. Journal of Neuroscience, 2013. 33(46): p. 18190-18199. 14. Mestre, H., et al., Flow of cerebrospinal fluid is driven by arterial pulsations and is reduced in hypertension. Nature Communications, 2018. 9. 15. Asgari, M., D. de Zelicourt, and V. Kurtcuoglu, Glymphatic solute transport does not require bulk flow. Scientific Reports, 2016. 6. 16. Bejan, A., Shape and structure, from engineering to nature. 2000, New York: New York : Cambridge University Press. 12 CHAPTER TWO FLUID FLOW AND MASS TRANSPORT IN BRAIN TISSUE: A LITERATURE REVIEW Contribution of Authors and Co-Authors Manuscript in Chapter 2 Author: Lori A. Ray Contributions: Researched and wrote the article. Performed transport analysis of literature data included in the article. Co-Author: Jeffrey J. Heys Contributions: Reviewed and edited the article. 13 Manuscript Information Lori A. Ray and Jeffrey J. Heys Fluids Status of Manuscript: ____ Prepared for submission to a peer-reviewed journal ____ Officially submitted to a peer-reviewed journal ____ Accepted by a peer-reviewed journal __X__ Published in a peer-reviewed journal MDPI 2019, Volume 4, Issue 4 DOI: 10.3390/fluids4040196 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 CHAPTER THREE ANALYSIS OF CONVECTIVE AND DIFFUSIVE TRANSPORT IN THE BRAIN INTERSTITIUM Contribution of Authors and Co-Authors Manuscript in Chapter 3 Author: Lori A. Ray Contributions: Conceived and designed study with Jeffrey J. Heys. Developed the model, performed the simulations, and analyzed the results. Wrote the article with input from Jeffrey J. Heys and Jeffrey J. Iliff. Co-Author: Jeffrey J. Iliff Contributions: Provided valuable knowledge regarding neuro-physiology. Provided input to and editing of the article. Co-Author: Jeffrey J. Heys Contributions: Conceived and designed study with Lori A. Ray. Provided input to and editing of the article. 49 Manuscript Information Lori A. Ray, Jeffrey J. Iliff, and Jeffrey J. Heys Fluids and Barriers of the CNS Status of Manuscript: ____ Prepared for submission to a peer-reviewed journal ____ Officially submitted to a peer-reviewed journal ____ Accepted by a peer-reviewed journal __X__ Published in a peer-reviewed journal BMC 2019, Volume 16, Issue 6 DOI: 10.1186/s12987-019-0126-9 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 CHAPTER FOUR QUANTIFICATION OF TRANSPORT IN THE WHOLE MOUSE BRAIN Contribution of Authors and Co-Authors Manuscript in Chapter 4 Author: Lori A. Ray Contributions: Conceived and designed study with Jeffrey J. Heys. Developed the model, performed the simulations, and analyzed the results. Wrote the article with input from Jeffrey J. Heys. Co-Author: Martin Pike Contributions: Provided expertise on dynamic contrast-enhanced (DCE) MRI experiments and analysis. Provided processed DCE MRI signal data for the transport analysis performed in the article. Co-Author: Jeffrey J. Iliff Contributions: Provided valuable knowledge regarding neuro-physiology. Provided input to the article. Co-Author: Jeffrey J. Heys Contributions: Conceived and designed study with Lori A. Ray. Provided input to and editing of the article. 69 Manuscript Information Lori A. Ray, Martin Pike, Jeffrey J. Iliff, and Jeffrey J. Heys Status of Manuscript: __X__ Prepared for submission to a peer-reviewed journal ____ Officially submitted to a peer-reviewed journal ____ Accepted by a peer-reviewed journal ____ Published in a peer-reviewed journal 70 CHAPTER FOUR QUANTIFICATION OF TRANSPORT IN THE WHOLE MOUSE BRAIN Authors: Lori A Ray, Martin Pike, Jeffrey J Iliff, Jeffrey J Heys Affiliations: Department of Chemical and Biological Engineering, Montana State University; Advanced Imaging Research Center, Oregon Health and Sciences University; Department of Psychiatry and Behavioral Sciences, University of Washington School of Medicine; Department of Neurology, University of Washington School of Medicine Abstract Understanding molecular transport in the brain is critical to care and prevention of neurological disease and the healing of brain injuries. A critical question is whether transport in the brain occurs primarily by diffusion or faster mechanisms are also at work, such as convection or dispersion. The glymphatic system of fluid transport in the brain has been proposed, which hypothesizes a circulation from the subarachnoid cerebrospinal fluid that surrounds the brain, along periarterial spaces that follow the branching cerebral vasculature into the brain, across the interstitial space between cells, and out of the brain along perivenous routes. The majority of findings for the glymphatic system have been at the microscopic level, demonstrating, and occasionally quantifying, elements within the system. Recently, dynamic contrast-enhanced (DCE) MRI has been established for study of the glymphatic system, offering a broad view of transport across the whole brain. DCE MRI offers the potential for quantitative analysis to fundamental transport and kinetic parameters, due to the ability to convert signal to tracer concentration. However, 71 few studies have utilized this potential, instead analyzing signal and reporting parameters with arbitrary units dependent upon the experimental situation and disconnected from fundamental transport processes. In this work, DCE MRI experimental data is combined with subject-specific finite-element models of the whole mouse brain to quantify transport parameters in different regions across the brain. Effective diffusivity (𝐷'((), a lumped transport parameter combining all mechanisms of transport, is determined for each anatomical region and compared to apparent diffusivity (𝐷"##), the actual diffusivity of the contrast agent through brain tissue, to draw conclusions about dominant transport mechanisms. In the periarterial regions of major arteries 𝐷'(( was found to be 20,000 times greater than 𝐷"##; a result that can only be explained by convection. In the brain tissue, which in the model combines interstitial space and the periarterial space of smaller arteries, 𝐷'(( was 10-25 times greater than 𝐷"##, indicating convection is a significant mechanism of transport throughout the brain. Importantly, this work demonstrates the continuation of periarterial convection along penetrating and smaller arteries, an aspect of the glymphatic system that has been in question. Key Words: biotransport, brain transport, glymphatic, perivascular transport, interstitial transport, finite-element modelling, dynamic contrast-enhanced MRI 1. Introduction 72 Neurodegeneration is one of the most significant medical challenges of our time, yet the gap between therapies and our understanding of the inner workings of the brain is great—referred to by neuroscientists as the “black valley of death” [2]. A hallmark of neurodegenerative diseases is the presence of aggregated proteins in the brain tissue. In fact, the pathology of Alzheimer’s, Parkinson’s and other neurodegenerative diseases can be reproduced in animal models by the forced over-production of protein aggregates specific to each disease [6]. Since aggregates form when concentrations of these proteins rise, impairment of protein removal from brain tissue, called waste clearance, is believed to underlie the vulnerability of the aging and injured brain to neurodegeneration [4, 5]. In the rest of the body (i.e., outside the brain), fluid “leaks” from arteries into the interstitial space between cells, delivering nutrients and carrying waste to lymphatic vessels or veins to be processed by the liver and kidneys (Figure 1A). Fluid flows from high-pressure arteries through the interstitium to low-pressure veins and lymph vessels. Circulatory physiology in the brain differs markedly from the rest of the body due to the blood brain barrier (BBB), which protects the brain from pathogens and molecules that could upset the unique biochemical environment required for cerebral activities. The BBB is comprised of tight junctions between cells at the blood vessel wall, with transport of molecules across the BBB strictly controlled and occurring through protein channels optimized for specific molecules. In addition, lymph vessels appear to be absent from brain tissue. These constraints specific to brain anatomy raise a key question for this work: “ how are wastes cleared from the brain?” 73 A. Molecular Transport in the Body B. Molecular Transport in the Brain: Glymphatic Hypothesis Blood-Brain Periarterial Perivenous Barrier efflux Artery influxVein Interstitial Solute !'( $ ! %&&"# Interstitial Flow Lymph Perivascular Brain Astrocytic Perivascular Vessel Space Interstitium Endfoot Space Figure 1. Physiology of Molecular Transport in the Body versus the Brain. A. Illustration of physiological system of molecular transport in tissues outside the brain. Transport in the body occurs via the circulatory system, interstitial flow through tissues, and the lymphatic system. Fluid (white arrows) leaks from arteries into the interstitial space between cells, generating interstitial flow that delivers nutrients (blue dots) and carries away waste (blue dots) to veins or lymphatic vessels to be processed by the liver and kidneys [7]. B. Illustration of hypothesized physiological system of molecular transport in brain tissue. Unique to the brain are annular fluid-filled spaces surrounding cerebral blood vessels, called perivascular space (PVS), that are connected to the cerebrospinal fluid that surrounds the brain. The PVS is bounded by the vascular wall on the inside and by the “endfeet” of astrocyte cells (yellow cells with long processes) on the outside. It is hypothesized that fluid (green arrows) moves from the cerebrospinal fluid in the subarachnoid space surrounding the brain along periarterial space following penetrating arteries deep into the brain. The fluid enters the interstitium, where interstitial solutes (purple) and fluid are transported to the perivenous space and carried to extracranial efflux routes. Fluid flow from the periarterial space into the interstitial space is facilitated by aquapoprin-4 channels (AQP4) (green dots) concentrated on the astrocytic endfeet. (Aquaporin channels are protein channels optimized for the transport of water across cell walls and regulation of water between compartments.) While the brain’s system for transporting molecules originally seemed well explained by diffusion [17], Cserr et al. and Rennels et al. observed tracers to move along preferential paths that looked to follow the vasculature and tracer progress was much faster than could be explained by diffusion [18, 19]. These preferential paths were 74 postulated to be the perivascular space (PVS), an annular space surrounding cerebral vessels that is connected to the fluid surrounding the brain, cerebrospinal fluid (CSF). The PVS is bounded by the vascular wall on the inside and by the “endfeet” of astrocyte cells on the outside. (Astrocyte cells are brain cells with many functions; their endfeet are cellular processes extending from the cell body.) Astrocytic endfeet have a high concentration of aquaporin-4 protein channels (AQP4) that specifically transport and regulate water. Recently, Iliff et al. reported evidence of the PVS as a mechanism for convective transport in the brain [20] and a system for molecular transport in the brain was proposed (Figure 1B), where: 1) CSF surrounding the brain moves into the brain along periarterial routes, in the same direction as blood flow (antegrade flow), possibly driven by the peristaltic pumping action of arterial pulsation [13]; 2) from the periarterial space, fluid moves into the interstitium, facilitated by AQP4; 3) fluid flows across the interstitium, driven by a small pressure difference between the periarterial and perivenous space, dissolving or entraining interstitial solutes (e.g., waste products); 4) then flows (with the solutes) to the perivenous space to be transported outside the brain via primary perivenous drainage pathways (e.g., meningeal (dural) lymphatic vessels). 75 This potential flow circuit, named the glymphatic system, provides a source of interstitial fluid, the CSF, and a pathway out of the brain, an efflux route, for interstitial waste. PVS surrounds the vasculature as it branches from large vessels to ever-smaller penetrating vessels. Therefore, the PVS is a conduit for distributing fluid from the surface of the brain deep into its interior. To summarize, the perivascular spaces of the brain may play a similar role to “leaky” arteries and lymph vessels in the peripheral body, providing fluid to sweep away waste products from the interstitial space between cells and a route for waste to exit the brain to be processed by the liver and kidneys. Interstitial flow may be similar in both environments, although the interstitial space in the brain may be more sensitive to flow due to brain-specific functions like synaptic activity. Quantifying transport rates in this glymphatic system of perivascular and interstitial spaces of the brain is the fundamental objective of the work presented here. 1.1.Improved Understanding of the Glymphatic System and Current Questions Amid continued observations of the glymphatic system for molecular transport in the brain, research over the last decade has expanded our knowledge of this system and much work remains to understand the mechanisms underlying transport. 76 Table 1. Symbols and Acronyms Symbol Description Acronym Description a MRI flip angle Acer Anterior cerebral artery tortuosity, porous media aquaporin-4, water transporting l characteristic AQP4 cell protein channel f void fraction Bas Basilar artery 𝑐 tracer concentration BBB Blood-brain barrier Anatomical region in FEM Model, 𝐷 Diffusivity BT Brain Tissue Apparent Diffusivity, Diffusivity of a 𝐷!"" molecule through a porous medium CoW Circle of Willis 𝐷#$%" Dispersion Coefficient CSF Cerebrospinal fluid Effective Diffusivity, Lumped Mass 𝐷&'' Transport Parameter DCE MRI Dynamic contrast-enhanced MRI Effective Diffusivity, Brain Tissue 𝐷&'',)* Region in Model DTI MRI Diffusion tensor imaging Effective Diffusivity, Branching 𝐷&'',)+,- Periarterial Space in Model ECM Extracellular matrix Effective Diffusivity, Surface 𝐷&'',-+,- Periarterial Space in Model FEM Finite Element Model Peclet Number, ratio of convective to Gaditeridol (Prohance), DCE MRI 𝑃𝑒 diffusive transport Gad contrast agent 𝑟. Tracer relaxivity ICA Internal carotid artery 𝑆 MRI signal ICP Intracranial pressure 𝑆/ Pre-contrast MRI signal, baseline IOI Integrative Optical Imaging 𝑆0# Post-contrast MRI signal IVIM MRI Intravoxel Incoherent Motion MRI 𝑇.,/ Pre-contrast relaxation time Mcer Middle cerebral artery 𝑇𝑅 Repetition time MRI Magnetic resonance imaging 𝑣 superficial velocity Olfac Olfactory artery 𝑣1- velocity in interstitial space PAS Periarterial space 𝑣+,- velocity in periarterial space Pcer Posterior cerebral artery BPAS Branching periarterial space PVS Perivascular space rms Root-mean square error regularized Optimal mass transport rOMT model SAS Subarachnoid space Anatomical region in FEM Model, SBA Surrounding Branching Arteries SPAS Surface periarterial space Anatomical region in FEM Model, SSA Surrounding Surface Arteries 77 With respect to the biology, the role of AQP4 and the structure of the perivascular space have been examined. AQP4 has been broadly demonstrated to support transport from perivascular space to interstitial space and to facilitate tracer transport through the brain [21], but its specific mechanism is unknown. The perivascular structure surrounding the arteries as they penetrate into the brain is significantly more complicated than the hollow channel depicted in Figure 1B. The structure of this complex space surrounding the cerebral vasculature has been probed and described at a cellular and molecular level [20, 22, 23], spawning debate and ongoing work regarding the specific location of perivascular flow within that structure. It is also unknown whether the perivascular space, through which the CSF flows, is open (empty) or filled with a network of proteins and other macro-molecules called the extracellular matrix (ECM). With respect to fluid flow, perivascular flow has been measured experimentally and shown to correlate with arterial pulsation, but the physics of how this flow is generated have yet to be proven. No known pressure gradient exist in the brain sufficient to drive flow through the periarterial space [24]. The correlation of periarterial flow with arterial pulsation [13, 14, 25] implies peristaltic flow, generated by pulsation of the arterial (inner) wall against a mostly rigid, but also porous, outer “wall” comprised of astrocytes. However, models to date of peristaltic flow in the periarterial space do not support the flows observed [15, 26, 27], leaving questions about both the appropriate model assumptions and the mechanism of peristaltic flow under the unique properties of the arterial pulse wave. Dispersion, which occurs when flows with no net directional 78 flow (such as eddies) facilitate mixing and accelerate transport relative to diffusion, has been hypothesized to enhance periarterial transport over diffusion alone, with models predicting enhancement factors of 2-220 depending on molecular diffusivity and assumptions [15, 28]. In the interstitial space, transport models validated with experimental data have estimated interstitial flows ranging from no flow to 0.1 mm/min [29-31]. Differences between models originate from varying assumptions about the ease with which fluid can move through porous interstitial space (hydraulic conductivity) and pressure gradients between periarterial and perivenous space. Pressures in the periarterial and perivenous space are not known. Known static pressure differences in the brain are not sufficient for significant interstitial flow [29], but the necessary pressure difference may be generated by periarterial flow. It has also been argued that significant flow may disrupt the sensitive environment required for neuronal function [32], but this has not been demonstrated and diffusion of small molecules is faster than the interstitial flows quoted above [27]. For comparison, measurements of interstitial flow outside the brain (in the peripheral body) report interstitial velocities of 𝑣2. = 0.006-0.12 mm/min [33]. Recent publications of particle tracking velocimetry and DCE MRI experiments demonstrate convective transport in periarterial space, but these demonstrations are limited and questions remain. Bedussi et al. and Mestre et al. independently measured an average periarterial velocity of 17-19 µm/s in the direction of blood flow, by tracking the 79 movement of fluorescent microspheres [14, 34]. The University of Rochester group further demonstrated the characteristics of the periarterial flow to be indicative of flow in an open channel and not a porous one [14, 25]. Open channels present a much lower hydraulic resistance, or resistance to fluid flow, than a porous periarterial space filled with ECM, requiring less driving force for the measured periarterial flow. However, both groups also observed the 1-µm diameter microspheres were excluded from penetrating arteries [25, 34]. Flow was measured in the perivascular space of the middle cerebral artery (Mcer) and some of its immediate branches. The Mcer is a pial artery, meaning it resides on the surface of the brain, as did the immediate branches investigated. Microspheres were excluded from the periarterial space of advancing generations of arterial branches of the Mcer that entered into the brain tissue, penetrating arteries. It was hypothesized the penetrating periarterial space may be “somewhat porous”, excluding the relatively large microspheres [25, 34]. Bedussi et al. alternatively proposed periarterial flow may not continue into the brain, but only occur around surface arteries [34]. Dynamic contrast-enhanced (DCE) MRI experiments, where a contrast agent is injected into the CSF, have shown that tracer follows preferential routes following arteries along the surface of the brain and penetrating into the brain at a rate too rapid to be described by diffusion [35-41]. However, DCE MRI results thus far have been semi- quantitative, analyzing signal instead of tracer concentration to determine direction of 80 movement and penetration into different tissues. Results have not been utilized to calculate fundamental transport parameters, such as velocity or effective diffusivity. An exception, discussed below, is the computational modeling of Valnes et al. and Croci et al. simulating DCE MRI in the human brain [42, 43]. However, the complexity of the highly-folded surface of the human brain and the nature of the DCE MRI data, in which the subjects’ behavior was not as tightly controlled as mice and rats, resulted in a broad range for calculated transport parameters. Opportunities exist for using DCE MRI to quantify transport in the brain. Additionally, DCE MRI offers a brain-wide, systemic view of transport, complimentary to the microscopic view of flow surrounding a single artery. Quantitative analysis utilizing DCE MRI may help to answer contemporary questions about glymphatic transport, such as whether periarterial flow penetrates into the brain. The goal of this work is to quantify transport across the whole brain, utilizing tracer concentration calculated from DCE MRI data and computational transport models to calculate fundamental transport parameters. “DCE MRI is minimally invasive and quantitative, and therefore can be useful in validating results of theoretical models and making parameter calculations.” [44] The brain is divided into anatomical regions relevant to the system of brain-wide transport and transport parameters estimated for each region. Estimated transport parameters are analyzed to establish the significance of transport mechanisms “of a magnitude greater than diffusion”, such as dispersion and 81 convection. Quantification of fundamental transport parameters is essential for accurately comparing and predicting transport in the brain. This work will be particularly useful to the field in its ability to: 1) quantify DCE MRI observations using fundamental parameters, 2) corroborate convection in the surface periarterial space by a different method, and 3) demonstrate the presence or absence of periarterial convection penetrating into the brain and interstitial flow through the brain. Below, the current literature is reviewed for these two research areas of DCE MRI for brain transport and whole-brain transport modelling. 1.2.Literature Review of DCE MRI for Study of Brain Transport Dynamic contrast-enhanced MRI is a tool for investigating transport at the scale of the whole brain first applied to glymphatic research by Iliff et al. [35]. Several disease states have now been investigated using DCE MRI in the brain: stroke [39, 45], diabetes [40], neurodegeneration [41]. In DCE MRI, a Gadolinium contrast agent is used that greatly increases the normal T1-weighted signal in proportion to the local concentration of the contrast agent. Signal is measured for each “voxel” in the 3-dimensional domain at various time points. (In MRI, the measurement domain is divided into a grid of small volumes, called voxels. An average signal is measured for each voxel to create a 3-dimensional image.) The 82 propagation of the contrast agent (from here on called “tracer”) is observed through this change in signal relative to a pre-contrast (baseline) image. Tracer concentration can be calculated from the change in signal relative to the baseline. It is a more complex calculation (see Methods) than the direct proportionality that is normally assumed when discussing experimental results. The ability to convert signal, an arbitrary variable, into concentration, a true physical variable, enables calculation of fundamental transport parameters and kinetic variables. An additional advantage of MRI is different anatomical tissues can be identified by their unique pre-contrast signals, describing anatomical regions such as arteries and ventricles. To study brain transport using DCE MRI, tracer is injected into the CSF, usually into a subarachnoid cistern at the back of the brain. Reported studies in mice and rats give the same general observations [36, 40, 46, 47] of rapid movement of tracer along the ventral (bottom) surface of the brain, often extending the full length of the brain (12-19 mm) in the first 5-15 minutes, then tracer penetrates superiorly (upwards) into the brain, with highest concentrations near the major arteries. The experimentalists in this area most often use cluster analysis to understand the overwhelming amount of data generated by DCE MRI; a precedent set by Iliff et al. [35]. In cluster analysis, regions of the brain are grouped together, either by anatomy (cortex, subcortex, hippocampus, etc.) or spatial distribution (voxels with similar signal kinetics). The averaged signal for each region is plotted over time, looking for different rates of 83 uptake [21, 23, 35, 40, 41, 47, 48]. The percent of total volume that displays a signal is also reported [40], which gives an indication of tracer penetration by region or in total. This type of analysis provides qualitative information about the direction and the relative rate of penetration of the tracer, enabling comparisons between anatomical region, disease states and genetic modifications. It does not result in quantitative transport parameters that take into consideration distance from injection site, transport routes and processes for the tracer arriving at these different regions, or efflux routes for exiting the regions. Bojd et al. took a step towards quantifying kinetic parameters by performing a classic-to-MRI two-compartment kinetic-model perfusion analysis on DCE MRI in rats [46]. The analysis was performed after applying cluster analysis to consolidate voxels that demonstrate similar kinetic behavior with respect to Gadolinium concentration. Three parameters were calculated: 1) 𝜏$'4"7 based on an exponential fit of the signal decay from Cmax, which should be related to clearance rate; 2) retention, an algebraic combination of fitted kinetic parameters; and 3) loss, also an algebraic combination of fitted kinetic parameters. Using the above values, a clear difference was demonstrated between healthy rats and those with diabetes. 84 1.3.Literature Review of Whole-Brain Transport Modelling Several recent papers have pointed out the need for quantitative modelling to aid in the understanding of molecular transport in the brain. Abbott et al. write “more sophisticated quantitative modelling of (brain) transport is urgently needed and will likely play a larger role in the future to better clarify some of the discrepancies (between seemingly opposing observations) and to suggest new experiments that may help resolve them” [32]. In a recent review of MRI and glymphatic modeling, Kaur et al. state “future sophisticated modelling techniques hold the potential to generate quantitative maps for glymphatic system parameters that could contribute to diagnosis, monitoring, and prognosis of neurological disease.” [49] Such modelling is just beginning to show itself in the literature; several methods and their results are summarized below. Koundal et al. developed a regularized optimal mass transport model (rOMT), interpolating “particle pathlines” useful for visualizing glymphatic flow [47]. (rOMT is a computational method predicting an optimal interpolation path that conserves mass and minimizes transport energy.) They concluded that both convection and diffusion were required for the model to resemble transport patterns in the live brain and observed 2-fold regional differences in transport rate in brain tissue. (This analysis was performed using DCE MRI signal.) In addition, less convection was observed as one moves away from the larger pial arteries with diffusion becoming more dominant in deeper portions of the brain tissue. An advantage of OMT is that it requires very little in the way of assumptions or prior knowledge about the boundaries or tissue properties. However, the 85 OMT method is not quantitative, calculating relative ‘speeds’ using arbitrary units instead of fundamental parameters. rOMT is useful for visualization and making relative comparisons. Croci et al. [43] and Valnes et al. [42] have both reported results of finite-element modeling of the human brain, taking very different approaches and using the results of recently published human DCE MRI data [50, 51] to draw conclusions. Croci et al. use stochastic models, which predict the probability of outcomes (or results), and uncertainty quantification. It was concluded diffusion alone is sufficient to describe tracer concentrations observed in grey matter, but diffusion is not sufficient to explain transport of tracer deep into the white matter. Valnes et al. built a finite-element transport model, solving for the apparent diffusion coefficient (ADC) that best fits the experimental data. The ADC is a lumped parameter that includes all mechanisms of transport. The model included separate regions for grey and white matter. Notably, Valnes et al. are the first to use calculated concentration instead of signal in a DCE MRI-based model. Folding on the surface of the human brain presents a significant obstacle to applying boundary conditions from noisy MRI data and obtaining a good fit between the model and the data, as a result several smoothing techniques are applied that resulted in a range of ADC values. They conclude ADC for grey matter fitted from the simulations is somewhat larger than the actual apparent diffusivity of the tracer, leaving potential for additional transport mechanisms at work in the brain. It should be noted that the time span for the 86 above models are 24-48 hours and neither account for changes in transport during sleep, which has been demonstrated in mice [52, 53] and humans [54]. Review of the literature shows promise in combining the quantitative potential of DCE MRI experiments for brain transport with a computational model of the whole brain. Such brain transport analysis could be useful in interpreting and understanding experimental observations, predicting transport in different situations, such as injury or disease state, and suggesting future experiments. Here, we focus on a model of the mouse brain, which is less complex than the human brain and about which more is understood from the perspective of waste clearance and brain transport, including numerous published DCE MRI experiments in mice and rats verifying results. 2. Approach In this work, tracer concentration calculated from DCE MRI data and simulations from computational transport models are used to quantify transport parameters in different regions of the brain. It is the purpose of this work to describe broad-scale transport mechanisms for various anatomical regions of import to molecular transport in the brain. Transport of molecules (in this case the Gadolinium tracer) in brain tissue may occur by a combination of mechanisms according to: 34 = 𝐷 6"##𝛻 𝑐 + 𝐷 6$%&#𝛻 𝑐 − 𝑣 ∙ 𝛻𝑐 + 𝑠(𝑐) − 𝑓(𝑐) (1) 35 87 where: 𝑐 = concentration, 𝐷"## = apparent diffusivity, 𝐷$%&# = dispersion coefficient, 𝑣 = superficial velocity (a vector field), 𝑠(𝑐) = source term (e.g., injection), and 𝑓(𝑐) = sink term (e.g., efflux route). The mass transport equation (Eqn. 1) contains a diffusion term, a dispersion term, a convection term, and source or sink terms (given in order). Diffusion occurs by the random motion of molecules from high to low concentration. Diffusion is slow, but always occurring (when there is a concentration difference), and its rate depends on the size of the molecule. Convection is the transport of molecules by bulk flow. In a free medium, convection is molecular-size independent; all molecules move at the velocity of the bulk flow. Superficial velocity, 𝑣, is a non-physical variable peculiar to porous media flow, as in through biological tissues. It is a hypothetical flow velocity calculated as if the mobile phase, the interstitial fluid in the brain tissue, were the only phase present in a given cross-sectional area. Dispersion occurs when there is convection that does not result in net bulk flow (e.g., an eddy), but the flow facilitates mixing that increases the transport rate relative to diffusion alone. In the situation of transport in the brain, we lack the information for such a complex model of brain-wide transport, like dispersion coefficients or the pressure gradients required to calculate interstitial or perivascular flow. In addition, the full model would require a great deal of complexity, include several adjustable parameters, and consume an enormous amount of computational resource. On the other hand, tracer concentration as a function of time is known in every DCE MRI voxel of the brain. Therefore, a simple transport model is utilized: 88 34 = 𝐷 6'((𝛻 𝑐 (2) 35 where all transport mechanisms are “lumped” into a single parameter, effective diffusivity (𝐷'((). (𝐷'(( is equivalent to ADC used by Valnes et al. discussed in Literature Review of Whole-Brain Transport Modelling.) Since the MRI voxels measure about 60 µm on a side, major vasculature can be identified, but smaller vasculature and perivascular space cannot be separated from the brain interstitium. Therefore, 𝐷'(( represents a combination of diffusion, perivascular dispersion, perivascular convection, and interstitial convection in a single transport parameter. This simplification from the full to the simplified mass-transport model also forces the mathematical structure of diffusion, which has a quadratic relationship between concentration and distance (Eqn. 1), consistent with dispersion but differing from convection. The goal of this work is to quantify parameters that describe transport over the broad systemic scale, not the small details, and draw conclusions about the prevalence and magnitude of convection or dispersion in moving molecules across the brain. As such, the simplifications applied, including anatomical simplifications discussed below, aid in developing useful transport parameter estimates. An additional advantage of this simplified transport model is that it is unbiased about the mechanism of transport, which is advantageous due to the broad range in proposed theories and evidence about transport in the brain--ranging from diffusion only to dispersion to convection [27]. Modelling transport with 𝐷'(( where the dominant mechanisms are uncertain is a general approach taken by others [42, 55] to quantify and compare transport rates. 89 Subject-specific, finite-element models of the whole mouse brain are developed using Equation 2 and the MRI data (see Methods). Within the models, the brain is separated into different anatomical regions, or subdomains (discussed further in Simulation Results), defined by the MRI data and representing broad systems of transport across the brain. The “best fit” effective diffusivity for each anatomical region in each mouse is determined by minimizing the difference, or error, between the simulation of tracer concentration and tracer concentrations calculated from DCE MRI experimental data. A cumulative error is calculated, summing over each vertex and time point (see Methods for details of concentration calculations and error calculations). The goal is not to exactly simulate the DCE MRI data, but to use the data to quantify transport in each region, at least to its order-of-magnitude. Because diffusion is always occurring (given a concentration gradient) and the other transport mechanisms require an external driver, diffusivity is a useful benchmark with which to compare the overall transport parameters determined in this work. The effective diffusivities calculated from model simulations are compared to the known diffusivity of the DCE MRI tracer through brain tissue, apparent diffusivity (𝐷"##). From this comparison, conclusions are drawn about the prevalence and magnitude of dispersion and convection for each region identified in the transport model. Since 𝐷"## is such a critical parameter to our analysis, details of its calculation are described below. If diffusion is the dominant mode of transport, the effective diffusivity will be very 90 similar to the apparent diffusivity. If dispersion is present or convection is of a similar rate as diffusion, the effective diffusivity may be an order of magnitude (ten times) greater than the apparent diffusivity. If significant convection exists, the effective diffusivity will be several orders of magnitude greater than the apparent diffusivity. Apparent diffusivity describes the diffusive rate of a molecule through a porous media, such as brain tissue. It is defined by: 𝐷 6"## = 𝐷⁄𝜆 (3) where 𝐷 is the free diffusivity, the diffusivity in an open volume and outside the porous, and l is tortuosity, which represents the degree to which molecular transport is slowed by the porous medium. The Gadolinium contrast agent for the DCE MRI experiments reported here is gaditerol a small molecule of 559 Da. Rattanakijsuntorn et al. measured the free diffusivity of gaditerol (trade name Prohance) in the vitreous humor of bovine eyes, which is similar to cerebrospinal fluid, of 𝐷 = 0.016 mm2/min [56]. The tortuosity of brain tissue for molecules of this size has been reported as l=1.6 [57] and l=1.85 [31]. Therefore, 𝐷"## = 0.016/(1.73)2 = 0.005 mm2/min. The above analysis is performed for both wild-type (normal) mice and mice that have been genetically modified (transgenic mice) to study the importance of aquaporin-4 (AQP4) channels to interstitial transport--examining this aspect of the glymphatic hypothesis. Recall that AQP4 channels are concentrated at the astrocytic endfeet comprising the interface between the perivascular and interstitial space of the brain. 91 AQP4 channels are hypothesized to facilitate transport of fluid from the periarterial space into the interstitium, creating interstitial convection that aids in the transport of molecules (particularly large molecules) across brain tissue. In the transgenic mice, the concentration of AQP4 channels on the astrocytic endfeet has been significantly reduced. According to the glymphatic hypothesis, the calculated transport parameter associated with interstitial transport should be lower in the transgenic than the wild-type mice. 3. Results 3.1 Tracer Concentration from Dynamic Contrast-Enhanced (DCE) MRI The work reported in this paper uses previously published dynamic contrast-enhanced (DCE) MRI data investigating transport in the brain [21]. The published work reported signal, Figure 3 reports concentration calculated from DCE MRI signal for a representative wild-type and a representative transgenic mouse. (See Methods for calculation of concentration from DCE MRI signal data.) The contrast agent is gadoteridol (Gad) with a molecular size of 559 Da. A Gad solution was injected continuously for the first 20 minutes of the experiment and its progress followed through an additional 30 to 60 minutes, for 50 to 80 total minutes of tracer progress. The injection site was a CSF aqueduct at the back of the brain, leading from the fourth ventricle to the subarachnoid space surrounding the brain (brain anatomy and the injection site are discussed below). 92 It is useful to understand details and implications of the injection site, which effects tracer movement, and the MRI pulse sequence, which effects measured DCE MRI signal, prior to interpreting the tracer concentration images. A deeper understanding of the vascular and CSF circulation anatomy of the mouse brain is also helpful for understanding both the DCE MRI data and the transport model. These details are discussed below, prior to reporting the tracer concentration results. 3.1.1. Background for Interpreting DCE MRI 3.1.1.1 Cerebral Vasculature and Cerebrospinal Fluid (CSF) Circulation Anatomy Figure 2A illustrates the cerebral vascular anatomy of a mouse, where arteries are red and veins are blue [58]. The major arteries feeding the brain run along its ventral (bottom) surface; an image of this ventral arterial system, comprised of the basilar artery (Bas), the Circle of Willis (CoW), and the anterior cerebral artery (Acer), is depicted in Figure 2B [59]. From the ventral system of arteries, other major arteries branch upwards, including the posterior cerebral arteries (Pcer), middle cerebral arteries (Mcer), and olfactory arteries (Olfac). The branching of the Pcer is shownin the blue box of Figure 2A. The Mcer is shown branching from the CoW in Figure 2B. The group of Olfac arteries branching from the Acer are shown in the green box of Figure 2 A. (In Figure 2A (green box), Olfac arteries include Olo, Cof, and Lofr branching at the Azac.) Also note from Figure 2A that in the mouse brain, large arteries are concentrated in the inferior (lower part) of the brain and large veins in the superior (top part) of the brain. 93 A B Acer MCer CoW Bas D Injection site C Injection Subarachnoid site Space Figure 2. Mouse Anatomy: Cerebral Vasculature and Cerebral Spinal Fluid (CSF) Circulation. A. Atlas of mouse cerebral arteries (red) and veins (blue), sagittal view of surface. Blue box shows detail at posterior cerebral artery (Pcer). Green box shows detail for anterior cerebral artery (Acer) and olfactory bulb arteries (Olo, Cof, Lofr). [58] B. Image of the major arterial system on the ventral surface of the brain, made up of basilar artery (Bas), Circle of Willis (CoW), and anterior cerebral artery (Acer). Also noted is middle cerebral artery (Mcer) branching out from the CoW. [58] C. 3- dimensional illustration of ventricular system inside the brain including the lateral ventricles (LV), the third ventricle (3V), and the fourth ventricle (4V) [60]. D. Illustration of CSF circulation [61], modified to include injection site. CSF is produced in the third and lateral ventricles, where it flows to the fourth ventricle and then into the subarachnoid space (SAS) surrounding the brain and spinal cord. CSF exits along spinal nerves, the cribiform plate, cranial nerves, arachnoid villa in the SAS and through meningeal lymphatic vessels. In the DCE MRI experiments, tracer was injected into the aqueduct between the fourth ventricle and the SAS (Red arrow labelled “Injection Site”). 94 Figure 2D illustrates the cerebrospinal fluid circulation in the brain. CSF is produced in the third and lateral ventricles, where it flows to the fourth ventricle and then into the subarachnoid space (SAS) surrounding the brain and spinal cord [62]. CSF exits along spinal nerves, the cribiform plate, cranial nerves, arachnoid villa in the SAS and through meningeal (dural) lymphatic vessels [27, 62]. A 3-dimensional illustration locating the ventricles within the brain is shown in Figure 2C. 3.1.1.2. Tracer Injection Site and Ventricular Accumulation The intended goal of the injection was to deliver tracer to the subarachnoid space CSF, since this is the source of fluid for glymphatic transport. However, tracer also enters the fourth ventricle and spreads through the ventricular system. The expected injection site (achieved by surgery following specific anatomical markers) was the cisterna magna, a subarachnoid cistern at the back of the brain, that would have delivered the tracer directly into the SAS CSF. Based on the DCE MRI data, tracer looks to have been injected at the aqueduct leading from the fourth ventricle to the SAS (see Figure 2D “Injection Site”). Since the CSF circulation flows from the fourth ventricle into the SAS, much of the tracer achieved the goal of entering into the SAS CSF from where it moves along the described preferential routes into the brain. However, tracer also moved into the fourth ventricle and spread by dispersion and diffusion throughout the ventricular system over the course of the experiment (see Figure 2 C&D for images of CSF ventricles in a mouse). This phenomenon, referred to as ventricular accumulation, is 95 identified in Figure 3A. Tracers have been observed to pass into the brain tissue from the SAS [1, 63], but not across the ventricular walls. The DCE MRI data support that observation. Although ventricular accumulation complicates interpretation of the concentration images, transport through the ventricular system is effectively independent of transport into and through the brain tissue. Ventricular transport is well-delineated and can be easily separated for our analysis. (Ventricular accumulation simply “robs” some of the tracer from the experiment.) 3.1.1.3. T2 Interference The DCE MRI experiments used a pulse sequence and an injection strategy optimized for measuring low tracer concentrations (seeking to identify efflux routes far from the injection site). A result of that strategy was interference between T1 and T2* MRI signals for voxels of high tracer concentration that compromised the T1-weighted signal values used to calculate concentration. This phenomenon is called “T2 interference” and renders the data for some voxels (for some time points) useless. In MRI, T1 relaxation is related to the speed of proton realignment with the magnetic field and T2* relaxation is related to the speed of proton-spin dephasing [64]. Both of these proton relaxations are occurring during a DCE MRI experiment, but at different speeds and the pulse sequence is normally adjusted to “weight” the signal towards one type of relaxation. However, complete separation of T1 and T2* signals is difficult. The presence of Gad generates a very high T1 signal. The presence of Gad decreases the T2* signal (relative to baseline), but its interaction with Gad is weak and produces very little signal except in places of 96 high concentration. In voxels of high tracer concentration, the T2* signal overlaps, or interferes with, the T1 signal resulting in useless data. An average of 0.2-0.5% of the voxels in the brain domain were affected by T2 interference, with the greatest percentage at the 20 or 30-minute time points. Based on examination of the data, T2 interference is estimated to occur above tracer concentrations of about 1.7 mmol. In Figure 3 (and the figures that follow), voxels with T2 interference have been assigned a concentration value of 1.7 mmol, which is shown as the darkest red in the concentration images. 3.1.2. Tracer Concentration from DCE MRI for Wild-type Mice Tracer concentrations images for a representative wild-type mouse are shown in Figure 3A for digital “slices” with relevant anatomy, such as major surface and branching arteries. (See Supplemental Material for movies of tracer concentration data.) Upon careful examination of the concentration data (Figure 3A), regions of distinctly different concentration dynamics are observed. These regions correlate with anatomical features of the brain and are observed in all experimental subjects. From its injection site at the posterior (back) of the brain, tracer moves rapidly anterior (forward) along the ventral (bottom) surface of the brain following the major arterial system made up of the basilar artery (Bas), the Circle of Willis (CoW), and the anterior cerebral artery (Acer) (see Figure 2B for image of arterial system on ventral surface). The tracer path along the surface arteries is best illustrated in the cross section of the Circle of Willis (Figure 3a), where the tracer path clearly splits into two branches, following the branched arteries. 97 Along this route, tracer moves from the posterior to the anterior of the brain (approximately 12 mm) by the first time point (10 min)—a rate of 1.2 mm/min or faster. T2 interference Pcer Branch Circle of Willis Acer Gad Concentration (mmol) Ventricular Accumulation 1 mm Figure 3A. Wild-type Mouse Tracer (Gad) concentration contours calculated from DCE MRI Data. Scale is c=0 (blue) to c=0.5 mmol (red). Concentrations exceeding 0.5 mmol are dark red (top of the color bar). The first column shows the center sagittal slice. The next three columns show coronal slices at different locations in the brain associated with major arteries and preferential transport routes. From its injection point in the CSF aqueduct at the posterior of the brain (see Figure 2), tracer moves along preferential routes first along the system of arteries on the ventral surface of the brain, then along major branching arteries in the superior direction. From these preferential routes, tracer moves outward into the brain. Coronal slices illustrate this behavior at major arterial features. T2 Interference: Occurs at high tracer concentrations ([Gad]>1.7 mmol) obscuring the T1 signal used to calculate tracer concentration. Ventricular Accumulation: In addition to moving into the SAS, tracer also moves into the fourth ventricle and spreads by dispersion and diffusion throughout the ventricular system over the course of the experiment. Since tracer transport across the ventricular walls and into the brain tissue is negligible, transport through the ventricular system is effectively independent of transport into and through the brain tissue. See Additional Information for animation of tracer concentration data. 10 min 20 min 30 min 50 min Gad Injection (0-20 min) 98 As tracer reaches the length of the brain along this ventral route, it also moves superior (upwards) through the brain at an equally rapid or even faster pace. High tracer concentration is observed around large arteries that branch from the surface arteries in the superior direction—the posterior and middle cerebral arteries (Pcer and Mcer), and the olfactory arteries (Olfac). Tracer penetrates into the brain tissue from these primary routes. These observations are consistent with those from previously published DCE MRI experiments [35, 36, 40, 41]. Tracer is also observed to move outwards directly from the injection site into the brain tissue. Tracer transport across blood vessel walls is negligible due to the BBB, therefore, the arteries are obvious from their absence of tracer, amid high concentrations in the brain tissue. (Seen most clearly at intermediate and later time points in the olfactory bulb and on the ventral surface, Figure 3A.) 3.1.3. Tracer Concentration from DCE MRI for Transgenic Mice The transgenic mice exhibit mis-localization of aquaporin-4 (AQP4), through Snta 1 gene deletion. Aquaporin-4 is a protein in the cell wall (or membrane) that serves as a “channel” to transfer water across the cell membrane. Recall that AQP4 channels are normally localized on the astrocytic endfeet that make up the interface between the perivascular space (PVS) and the brain tissue. The deletion of the Snta 1 gene impairs APQ4 localization, meaning the AQP4 is distributed around the entire cell wall of the astrocyte, instead of being concentrated at the endfeet. AQP4 on the astrocytic endfeet 99 are believed to serve a role in the transport of fluid from the PVS into the interstitial space of the brain tissue [1, 21], creating interstitial flow. Therefore, it is hypothesized that a lower concentration of AQP4 on the endfeet will impede interstitial transport. T2 interference Pcer Branch Circle of Willis Acer Mcer Gad Concentration (mmol) 1 mm Figure 3B. Transgenic Mouse Tracer (Gad) concentration contours calculated from DCE MRI Data exhibits significantly different behavior than wild-type mice. The high concentration, T2 interference region extends the length of the brain along the arteries of the ventral surface. (The T2-interference regions may also include arterial space, as the two are difficult to separate.) Tracer concentrations within the tissue are lower than the wild-type mice, but those concentrations also extend further from the preferential routes—the concentration gradient is not as steep. The high concentrations surrounding the surface and olfactory arteries indicate rapid movement along a path following the arteries and slow movement out of this preferential path and into the brain tissue. In addition, the lack of steep concentration gradients in the brain tissue is also indicative of slow transport there. Transport into and through the brain tissue is inhibited in the transgenic mice relative to the wild-type mice, due to the mis-localization of AQP4, which impedes transport from periarterial to interstitial spaces of the brain. 10 min 20 min 30 min 50 min Gad Injection (0-20 min) 100 Figure 3B shows tracer concentration contours for a typical transgenic mouse, which exhibits significantly different behavior than the wild-type mice. First, wide T2 interference regions extend along the entire route surrounding the arterial system on the ventral surface and the olfactory arteries into the 30-minute time point. Second, tracer concentrations within the tissue are lower, but those concentrations also extend further from the preferential routes—the concentration gradient is not as steep--compared to wild-type mice. Blood vessels and T2 interference can present similarly, as the blood flowing through vessels constantly “refreshes” protons leading to low MRI signal or signal lower than baseline. Therefore, it is possible that either: 1) the surface and olfactory arteries in the transgenic mice are much larger than in the wild-type, or 2) T2 interference is occurring across the entire ventral length of the transgenic mouse brain, indicating very high tracer concentrations. Arteries can be identified prior to injection by their proton density. Proton density calculations (see Methods) show the artery dimensions to be much less than the negative concentration regions observed in the DCE MRI data. In addition, concentration contours at later time points (see Figure 3B, t=50 min), when tracer has dispersed and the “lower than baseline signal” regions have shrunk, reveal the artery dimensions to be less than the low signal dimensions at earlier time points. This leaves T2 interference as the cause of low MRI signal. Therefore, the concentration of tracer is very high and remains high for a significant period of time throughout the region surrounding the arterial system on the ventral surface and the olfactory arteries. (T2- 101 interference regions of high concentration depicted in the tracer concentration images (Figure 3B) may also include arterial space, as the two are difficult to separate.) The high concentrations surrounding the surface and olfactory arteries indicate rapid movement along a path following the arteries, and slow movement out of this preferential path and into the brain tissue relative to wild-type mice. In addition, the lack of steep concentration gradients in the brain tissue is also indicative of slow transport there. Based on comparison of transgenic and wild-type mice observations in the DCE MRI concentration data, transport into and through the brain tissue is inhibited in the transgenic mice. Therefore, the reduced concentration of AQP4 channels at the interface between the periarterial and interstitial space impedes transport both across the interface and through the interstitial space. This observation is exactly as expected if AQP4 plays a significant role in the transport of fluid from periarterial space to interstitial space. 3.2. Transport Model Simulation Results The goal of this work is to quantify transport in different, physiologically relevant, regions of the brain using subject-specific, finite-element models as discussed in the Approach section. Recall that the simplified transport model utilizes effective diffusivity, 𝐷'((, a lumped transport parameter that combines diffusion, perivascular dispersion, perivascular convection, and interstitial convection. The effective diffusivity for each anatomical region in each mouse is determined by minimizing the difference, the root 102 mean square error (rms), between the simulation of tracer concentration and tracer concentration calculated from DCE MRI experimental data. (The T2 interference region, the ventricles, and the blood vessels are excluded from the error calculation.) Comparison of calculated effective diffusivities to apparent diffusivity, the actual diffusivity of the tracer through brain tissue, is used to draw conclusions about dispersion and convection. 3.2.1. Anatomical Subdomains defined by MRI Data Recall from the calculated tracer concentration images (Figure 3A), regions of distinctly different concentration dynamics are observed that correlate with anatomical features of the brain. The MRI data is used to define these regions and create subdomains within the finite-element model where different effective diffusivities are applied. Within the FEM, the brain is divided into five subdomains: 1) ventricles, 2) arteries, 3) the region surrounding the surface arteries (SSA), 4) the region surrounding the major branching arteries (SBA), and 5) the rest of the brain, which we call brain tissue (BT). These subdomains are depicted in Figure 4 for a representative mouse and agree well with the mouse anatomy illustrated in Figure 2. As tracer transport into the arteries and across the ventricular walls (between the fluid inside the ventricles and the brain tissue) is negligible, these regions are effectively excluded from the model by assigning extremely low effective diffusivities. Therefore, modelling is focused on quantifying transport in the two regions surrounding major arteries, the SSA and the SBA, and the bulk of the brain tissue, BT. 103 A. Arteries C. Slices with all material types shown Arteries Acer BPAS Mcer CoW SPAS Bas Brain Tissue Circle of Willis Ventricles B. Ventricles Pcer Branch LV 4V LV 3V CoW Figure 4. Anatomical Details extracted from MRI data define Transport Model Subdomains. A. 3D “crystal brain” image of arterial subdomain (red) extracted from the MRI data viewed from the ventral surface. Opaque white shows the surface of the brain. Extracted arteries (Bas, CoW, and Acer) correspond well with the ventral surface arteries of anatomical images shown in Figure 2B. Lateral ventricle also visible (blue). B. 3D “crystal brain” image of the ventricular subdomain (blue) extracted from the MRI data from a sagittal view. The ventricles defined by MRI data compare well with the anatomical illustration in Figure 2C. Arteries also shown (red) C. 2D “slices” showing all anatomical subdomains incorporated in the transport model as noted in the color bar. The SSA and SBA anatomical regions are shown in grey and orange, and labelled as SPAS and BPAS for surface and branching periarterial space. 3.2.1.1. Preferential Transport Routes correspond to Periarterial Space The regions surrounding the arteries (the SSA and SBA) were defined not by anatomy, but by their unique tracer concentration and transport behaviors. To ensure these regions were defined by the data, a sensitivity analysis was performed on the width of the SSA that clearly showed a wider SSA (up to around 400 µm) led to better 104 agreement between the simulation and the data. Therefore, the DCE MRI data indicate that the determined SSA and SBA boundaries are significant to transport. It seems obvious from the brain anatomy and the glymphatic hypothesis (Figure 1) discussed in the Introduction that these regions, which follow and surround the major arteries of each subject, are periarterial space (PAS). However, the widths of the SSA and the SBA regions in the model are wider than expected given in vivo measurements of the PAS. Mestre et al. found the width of the PAS to be similar to the diameter of the vessel it surrounds [14]--observed PAS widths were 0.75 to 1.5 times the artery diameter (Figure 1e in Mestre et al.). Tithof et al. measured PAS dimensions, determining the PAS width for surface arteries (at its widest point) to be 1 to 2.5 times the artery diameter [65]. For example, the internal carotid artery (ICA), which is part of the Circle of Willis has a diameter of around 140 µm (measured ex vivo, which may cause shrinkage) [66]. Therefore, one would estimate a PAS width at 105 – 350 µm. The width of the SSA in the transport model (measured close to the ICA) averages 400 + 100 µm, larger than the highest estimate. The greater widths of the SSA and SBA, relative to the PAS, can be explained by artifacts inherent to MRI and to interpolation of data onto a finite-element mesh. The MRI signal for each voxel is the average of the materials inside the voxel. So, if an anatomical feature exists within two neighboring voxels, it will have an impact on the signal for both. Voxel width for the experiments reported here is 60 µm. Thus, a feature 105 of dimension 60 µm that “straddles” two voxels will have an apparent width of 120 µm, a stretch factor of 2 or greater. Near the vasculature, this phenomenon is amplified by the pulsing of the arteries, which causes movement of the artery and the surrounding tissue (the periarterial space) and blurs neighboring voxels. For example, the large branching arteries (Olfac, Pcer, and Mcer) are known to have a diameter around 40-90 µm [58]. Within the MRI data, these arteries have a diameter of about 3 voxels or 180 µm, a stretch factor of 2 to 4. In the finite-element model, their diameter increases to about 220 µm, a 20% increase from the MRI data, due to interpolation of the MRI data onto the less-refined finite-element mesh. Therefore, both spread and blur attributed to the experimental and modelling techniques account for a stretching of the PAS by 2 to 4 times, corresponding to a PAS width of 280 – 560 µm for the ICA and bringing the width of the SSA and surface periarterial space into agreement. In conclusion, strong evidence supports the subdomains defined by the model surrounding the arteries (SSA and SBA) are periarterial space, which from now on will be referred to as the surface periarterial space (SPAS) and the branching periarterial space (BPAS) respectively. Artifacts inherent to MRI and interpolation cause the periarterial space to be stretched relative to its actual dimensions. 3.2.1.2. Brain Tissue Subdomain combines Several Mechanisms of Transport The brain tissue subdomain represents the bulk of the brain, what is left after the other anatomical regions have been separated out. Within this region, the brain interstitium is 106 combined with the perivascular space of smaller penetrating vessels. Therefore, several potential transport mechanisms are present and contribute to effective diffusivity: diffusion, periarterial convection, periarterial dispersion, and interstitial convection. In addition, the tracer used in the DCE MRI experiments (Gad) is a relatively small molecule with a fast apparent diffusivity. Ray et al. predicted that for a molecule around 600 Da interstitial convection and diffusion would have similar rates [31]. Therefore, within the brain tissue region of the model, interstitial convection is convoluted with both periarterial convection and diffusion and is hypothesized to be of similar magnitude to diffusion. This convolution of transport mechanisms in the region containing the interstitial space has implications for the study of the transgenic mice. AQP4 mislocation is hypothesized to decrease interstitial convection. In AQP4 knockout mice, mice genetically modified to lack AQP4 channels completely, Iliff et al. found periarterial flow to be unaffected relative to wild-type mice and tracer movement into the interstitium to be “abolished” [1]. Since both periarterial convection and interstitial convection are combined in the effective diffusivity of the brain tissue region and interstitial convection is believed to be of a similar rate to diffusion, periarterial convection and/or dispersion (in smaller penetrating arteries) would need to be small for a change in interstitial convection to be apparent in the 𝐷'((,*+ results. 107 3.2.2. Effective Diffusivities Exceed Apparent Diffusivity Effective diffusivity for each anatomical subdomain is greater than 𝐷"## (Figure 5), indicating transport is faster than diffusion alone throughout the brain. 𝐷'(( is greatest for the regions surrounding arterial space, SPAS and BPAS, and a similar order of magnitude for both of those regions. In fact, the 𝐷'(( in the periarterial regions is so much faster than 𝐷"## (10,000+ times) that transport can only be explained by convection. In the brain tissue region, which combines interstitial and periarterial space, the enhancement of overall transport over 𝐷"## is smaller (10-25 times), but still supports the presence of convection. Even in the brain tissue region, periarterial convection is found to be the primary contributor to 𝐷'((, concealing differences in interstitial convection that may be present between the wild-type and transgenic mice, and resulting in no statistical difference between their effective diffusivity results. 3.2.2.1. Wild-type Mice Figure 5A reports the “best fit” 𝐷'(( parameters for the wild-type mice for each of the three transport regions, both for each subject and the mean of all four subjects. (See Supplemental Material for example contour plots of rms error versus 𝐷'((.) Overall, good agreement was found for the effective diffusivities of each region between different wild-type subjects. The agreement between simulation and data was most highly dependent on the effective diffusivity in the brain tissue (which constitutes the majority of the brain volume in the model), 𝐷'((,*+ = 0.10 + 0.04 mm2/min, dependent on the 108 SPAS to a lesser degree, 𝐷 2'((,.,-. = 95 + 40 mm /min, and least dependent on BPAS effective diffusivity, 𝐷 2'((,*,-. = 140 + 40 mm /min. A clear minimum was usually exhibited for 𝐷'((,*+ and 𝐷'((,.,-.. A minimum was rarely found with BPAS effective diffusivity, where error decreased continuously with increasing 𝐷'((,*,-.. However, a significant change in slope was observed, a point at which an extremely small change in error results from a very large change in BPAS effective diffusivity, and 𝐷'((,*,-. was determined at this slope change. Wild-type Effective Diffusivities by Transgenic Effective Diffusivities by Subdomain Subdomain 1000 1000 100 100 Mouse ID 10 WT031417 10 WT021517 Mouse ID 1 1 WT032217 KO062217 0.1 WT020217 0.1 Mean 0.01 0.01 !!"" = 0.005 ''#⁄'() !!"" = 0.005 ''#⁄'() 0.001 0.001 BT SPAS BPAS BT SPAS BPAS Figure 5. Effective diffusivity for different brain regions of wild-type mice (A) and a transgenic mouse (B). 𝐷'(( is calculated by minimizing the difference between simulations and concentration calculated from DCE MRI data. Error bars give one standard deviation. Effective diffusivity for each region is greater than 𝐷"## indicating transport is faster than diffusion alone throughout the brain. 𝐷'(( is greatest for the regions surrounding arterial space, SPAS and BPAS, and so much faster than 𝐷"## (10,000+ times) that transport can only be explained by convection. 𝐷'(( values for the transgenic mouse are effectively the same as the wild-type mice. This result is likely due to the small effect of interstitial convection on the lumped parameter of 𝐷'((. Effective Diffusivity (mm2/min) Effective Diffusivity (mm2/min) 109 The branching periarterial space (BPAS) was more difficult to define than the SPAS. Although the BPAS was clearly a preferential route, it was not as delineated by a steep concentration difference as the SPAS. Arterial pulsation causes significant movement of the arteries in the BPAS relative to the small size of their periarterial space, blurring signal and shifting the location of the anatomical feature between time points. This leads to more noise and variability in the data near these pulsing branching arteries. Such concentration variability likely contributed to the lack of a clear minimum and lowers confidence in 𝐷'((,*,-. relative to the other two regions. Effective diffusivity in the perivascular spaces (𝐷'((,.,-. 𝑎𝑛𝑑 𝐷'((,*,-.) is about 20,000 times greater than the apparent diffusivity of Gad, which indicates the presence of periarterial convection surrounding major arteries, and possibly also dispersion. Dispersion has been proposed as a mechanism that might increase periarterial transport relative to diffusion. Asgari et al. predicted that dispersion in a porous media-filled periarterial space, induced by the pulsation of the arterial wall, could increase the rate of transport over diffusion by a factor of as much as two [15], depending on the size of the molecule. Faghih et al. calculated a larger dispersion enhancement of 220 for an open periarterial channel [28]. However, their analytical model made the assumption of a small gap relative to the vessel radius approximating the annular channel as fluid between flat plates, which differs from experimental observations of the periarterial space [14, 65]. Therefore, dispersion may be present, but it does not account for the large effective diffusivities in the regions surrounding the arteries, which only convection can explain. 110 Effective diffusivity in the bulk of the brain tissue, 𝐷'((,*+, is much slower than the periarterial regions, but is still 10-25 times faster than 𝐷"##. As discussed above, 𝐷'((,*+ represents a compilation of diffusion, periarterial convection, periarterial dispersion, and interstitial convection. In addition, for a small molecule like Gad interstitial convection and diffusion are predicted to have similar rates [31]. Dispersion could also enhance transport in the smaller periarterial spaces, but, as discussed above, is unlikely to have a contribution high enough to account for the remaining enhancement of 𝐷'((,*+ over 𝐷"##. Therefore, the majority of the increase over 𝐷"## observed in the brain tissue anatomical region may be primarily attributable to periarterial convection surrounding smaller penetrating arteries. 3.2.2.2.Transgenic Mice Recall from the Approach section, if AQP4 plays a significant role in the transport of water between the perivascular and interstitial space, then a decrease in the transport parameter associated with interstitial space is expected in the transgenic mice. The periarterial subdomains of the model do not include interstitial space and, therefore, 𝐷'((,.,-. 𝑎𝑛𝑑 𝐷'((,*,-. are not expected to differ between the wild-type and transgenic mice. The brain tissue subdomain is the anatomical region in the model that contains interstitial space. However, the brain tissue region also includes the periarterial space of smaller penetrating arteries and, therefore, combines several mechanisms of transport. As discussed above, for a small molecule like Gad, interstitial convection has a similar 111 rate to diffusion and interstitial convection will only be apparent in 𝐷'((,*+ if penetrating periarterial flow is slow. Penetrating periarterial flow was found to be the primary contributor to 𝐷'((,*+, therefore a significant change in 𝐷'((,*+ for transgenic mice compared to wild-type mice is not expected. Figure 5B reports the “best fit” 𝐷'(( parameters for the transgenic mouse, which are effectively the same as the wild-type mice. This result is more likely due to the small effect of interstitial convection on the lumped transport parameter for the brain tissue, 𝐷'((,*+, than a lack of difference in interstitial transport between the mouse types, particularly as the concentration dynamics were markedly different in the transgenic mice. It should also be noted that the significant T2 interference observed in the periarterial space regions with the transgenic mice was likely to effect accurate calculation of effective diffusivities in all the regions, but the SPAS in particular. An experimental technique that can isolate or emphasize transport in the interstitial space, such as integrative optical imaging (IOI), may provide a better setting in which to test the transgenic mice and the AQP4 hypothesis. 3.2.3. Comparison of Data to “Best Fit” Simulation Figure 6 shows simulated concentration contours compared to MRI DCE data for an example wild-type mouse. Simulations visually match the data well at early time points and deviate more at later time points. At later time points the data exhibits continued heterogeneity of concentration (distinct areas of high and low), while in the simulation, 112 the tracer smoothly disperses through the tissue. This smooth dispersal is the expected outcome of a diffusive model (Eqn. 2). Therefore, given the model chosen and the anatomical simplifications made in this work, a different outcome is not possible. The heterogeneity of the tracer concentration data at later time points tells us the tissue is more heterogeneous than modelled, as is the transport underlying the anatomical differences. Certainly, the local agreement between the simulations and the data can be improved by including more anatomical details. However, the addition of these details adds complexity and requires more adjustable parameters, which dilutes the usefulness of the quantified parameters, adds new sources of error, and requires extremely large computational resources. It is the purpose of this work to describe broad-scale transport mechanisms in the brain. Although the simulated concentration contours are “smoothed” by the model assumptions compared to the data, the transport parameters remain valid estimates for each broad region of fluid movement in the brain. A closer look at the comparison between simulation and data shows additional evidence of periarterial convection surrounding the surface arteries (SPAS). The simulated concentration is higher than the data at the posterior (back) of the brain and lower towards the anterior (front) (see Figure 6, center sagittal slice). Recall that the transport model used in the simulation (Eqn. 2) was simplified from the full mass transport equation (Eqn. 1). This simplification also forced the mathematical form of diffusion on the simulation. Note that in the simplified equation (Eqn. 2) the relationship between concentration and distance is quadratic, or second order. In the full mass 113 transport equation (Eqn. 1), convection (the third term on the right-hand side of the equation including velocity, 𝑣) has a linear, or first order, relationship between concentration and distance. A quadratic model, such as that used in the simulation, will have a higher concentration near the source (posterior of the brain) and fall off more steeply moving forward, while a linear model, such as that of convection in the full transport equation, will demonstrate a constant slope. The pattern of deviation between simulation and data, from the posterior to the anterior of the brain, supports better agreement with a concentration gradient resulting from a linear relationship between concentration and distance, consistent with convective transport. Simulation Data Simulation Data Figure 6. Comparison of wild-type “best fit” simulation (𝐷'((,*+ = 0.08 mm2/min, 𝐷'((,.,-. = 130 mm2/min, 𝐷'((,*,-. = 80 mm2/min) to DCE MRI concentration data for representative mouse. Simulations visually match the data well at early time points. At later time points, the data exhibits continued heterogeneity of concentration (distinct areas of high and low), while, in the simulation, the tracer smoothly disperses through the tissue. This smooth dispersal is the expected outcome of a diffusive model (Eqn. 2). The intention of the model was not to exactly match the data, but to calculate transport parameters that are useful estimates for each broad region of fluid movement in the brain. See Additional Information for simulation animation. Gad Concentration (mmol) 10 min 20 min 30 min 50 min Gad Injection (0-20 min) 114 As discussed above, no significant differences were found in the effective diffusivity results between transgenic and wild-type mice. This finding was due to the lack of sensitivity of the analysis, and possibly the DCE MRI experiment, to the changes in interstitial flow expected due to a lower concentration of AQP4 on the astrocyte endfeet forming the boundary between perivascular and interstitial space. The negative outcome provides useful information, confirming the role of AQP4 in interstitial, and not perivascular, transport and suggesting a different experimental technique for study of the transgenic mice, one focused on interstitial transport. 3.3. Sources of Error Several sources of error exist both in the physics of the DCE MRI experiments and in the assumptions made to simplify the transport model. The sources and their potential impact on the results are discussed below. Although the results are affected by these sources of variability and bias that may adjust effective diffusivities up or down, none of the sources are significant enough to impact the overall finding for convective transport in the brain. 3.3.1. Sources of Error from DCE MRI Even though it is a powerful experimental tool, MRI is an inherently noisy technique with a high signal-to-noise ratio. In live biological subjects, this noise is exacerbated by small movements in the tissue, such as blood pumping through vessels, or movement of the entire subject. MRI data is often filtered to reduce noise and smooth the data, but at 115 the cost of averaging the data over regions considered large relative to anatomical details. In order to investigate the interaction of transport with specific anatomical features. the decision was made to leave the data unfiltered. (Some minimal Gaussian filtering occurred as a result of interpolating the MRI data onto the finite-element mesh.) A consequence of using the unfiltered data is greater variability between points in space and time, leading to greater error between the simulation and the data. Error between the data and the simulation is calculated by summing the magnitude of the difference between the simulation and the data at every vertex for every time point. Noise, whether above or below the simulation value, adds to the absolute error. Filtering data would produce an average of the noisy “ups and downs”, effectively leading to an average magnitude of error among vertices instead of one that is summed, and a lower absolute error. However, all parameter combinations experience this same noise, and the transport parameters giving the minimum difference between simulation and data, which is relative, should not be greatly affected. T2 interference contributes to error by obscuring important, desirable data. The DCE MRI experimental parameters (i.e., pulse sequence) were optimized for measuring low concentrations of the tracer in order to follow the Gad deep into the tissue and possibly elucidate efflux routes. As a result, T2 interference was great and the high concentration regions were sacrificed. T2 interference patterns varied from mouse to mouse, which added to variability in results as different regions had to be excluded from the calculation for each mouse. Small differences in the location of the injection relative to the anatomy 116 of the subject were the likely cause of some differences in T2 interference patterns. In particular, T2 interference obscured the rapid transport path from the injection site to the tissue surrounding the ventral surface arteries, adding uncertainty to the SPAS geometry used in the model. T2 interference was more likely to increase variability than to result in higher or lower values for 𝐷'((. 3.3.1.1. Effect of Injection on Results Does the injection induce convection? It is a valid question and a difficult variable to balance experimentally, as too slow of an infusion rate results in low tracer concentration, and therefore a low signal, and too fast can upset intracranial pressure and CSF flow. The modelling tools developed in this Thesis can be used to study this effect. The injection rate and site were designed to be minimally disruptive to the experimental objectives, while delivering sufficient tracer into the subarachnoid space (SAS) CSF. The infusion rate for the DCE MRI experiments presented here was 0.5 µl/min for 20 min. To put that in context, CSF is produced in the ventricles at 0.32-0.41 µl/min [67, 68]. After circulating through the ventricles and around the brain and spine, CSF exits to the peripheral body by various routes (see Figure 2D illustrating the anatomy of the CSF circulation). The mouse brain holds about 40 µl of CSF [62, 69]. While the infusion volume may seem like a significant portion of the total CSF volume delivered over a short time, raising concerns about increasing pressure and 117 inducing convection, several mitigating factors are at work. First, the entire volume of CSF in the mouse is replaced every 100-125 minutes, or 12-15 times per day. Second, we are not investigating transport in just the CSF, but all the periarterial and interstitial fluid as well, called the extracellular fluid. The extracellular fluid volume in the brain is estimated at 100 µl (based on a 500 mg mouse brain with a 20% void fraction [57]). Therefore, the injection volume is only 7% of the total volume of the extracellular plus CSF fluid (140 µl). Third, it is well known that mouse physiology will make adjustments in response to the additional fluid in order to maintain its desired intracranial pressure (ICP). Increased pressure will naturally increase CSF drainage along the exit paths described in Cerebral Vasculature and CSF Circulation Anatomy, which are essentially microscopic orifices throughout the barrier, and a physiological feedback mechanism controls ICP by reducing CSF production or cerebral blood volume [70]. Lastly, looking at the data (Figure 3), the tracer is observed to move “upstream” into the fourth ventricle as rapidly as it moved “downstream” into the cranial SAS, which indicates infusion flow was not directed towards periarterial routes on the ventral surface. Although the tracer injection will inevitably have some impact on fluid dynamics in the brain, and therefore the experiment, several aspects of mouse-brain physiology and the location of the injection mitigate that impact. Raghunandan et al. addressed the question of injection inducing convection directly, comparing a tracer study with traditional injection techniques, where tracer solution is slowly infused into the CSF, to a study where CSF was removed from the cranium at the 118 same rate the tracer solution was injected, for no net addition of fluid [71]. The experimental results, where periarterial velocity in mice was measured using the same methods as Mestre et al. [14], were statistically identical for the two cases. ICP was maintained at near-physiological levels for the experiment where CSF was removed. For the experiment using traditional injection techniques, and an injection rate of 2 µl/min, ICP was observed to increase to a maximum of just over 2 mmHg over the course of the injection and recover to near-physiological pressure about 4 minutes after the injection ceased [71]. (Normal mouse ICP is around 4 mmHg and varies 1 mmHg between mice [72].) The injection rate used by Raghunandan et al. is four times higher than the injection rate for the DCE MRI experiments reported here. Raghunandan et al. directly demonstrated that fluid flows observed in the periarterial spaces of the middle cerebral artery and its immediate branches are not an artifact of tracer injection. The whole-brain transport model developed in this work is an appropriate and useful tool for investigating the effect of the injection on transport parameters during a DCE MRI experiment. First, the change in effective diffusivity over time (based on the best fit to the cumulative data at each time point) is investigated. Effective diffusivities are expected to decrease over time due to: 1) losses of tracer across the boundaries, which are not accounted for in the model (discussed further in Sources of Error from Transport Model Assumptions), and 2) the injection and its associated increase in pressure, a potential driving force for flow, at the beginning of the experiment. The decrease in 𝐷'(( associated with loss of tracer across boundaries is anticipated to be a steady loss 119 throughout the experiment, while a decrease in 𝐷'(( associated with the injection is expected to be more abrupt and correlate with the end of the injection (around 24 min into the experiment). Figure 7A shows that effective diffusivity decreases with cumulative experimental time. For the surface periarterial space, effective diffusivity remains fairly constant over the first 50 minutes of the experiment, then decreases by almost 30% by the last time point. The change in 𝐷'((,.,-. over time does not correlate with the injection timing and is better explained by tracer losses discussed above. For the brain tissue region, effective diffusivity increases slightly from 20-30 min, then decreases significantly from 30-50 min, falling to a third of the initial value by 80 min. The change in 𝐷'((,*+ over time does look to correlate with the injection. The error calculation used to determine the best fit between data and the simulation is biased towards regions of high concentration. For the brain tissue region, the highest concentrations occur near the injection site. Therefore, it follows the brain tissue effective diffusivities are influenced more by the injection than those of the surface periarterial space. Differences in the dominant mechanisms of transport and the effect of small pressure changes on that mechanism, such as the pumping action of peristaltic-like flow versus flow from pressure gradient, may also play a role in the effect of the injection. Next, simulations were performed to determine the best-fit effective diffusivities during the injection and after the injection for each anatomical subdomain. (The post- 120 A Brain Tissue Region Effective Diffusitivty B Brain Tissue Region Effective Diffusitivty vs. Cumulative Experimental Time fit for Injection Dynamics Cumulative During Injection Post Injection 0.14 0.14 0.12 0.12 0.1 0.1 0.08 0.08 0.06 0.06 0.04 0.04 0.02 Injection 0.02 0 Injection0 0 20 40 60 80 0 20 40 60 80 Time (min) Time (min) Surface PAS Region Effective Diffusitivty Surface PAS Region Effective Diffusitivty vs. Cumulative Experimental Time fit for Injection Dynamics Cumulative During Injection Post Injection 160 160 140 140 120 120 100 100 80 80 60 60 40 40 20 20 Injection Injection 0 0 0 20 40 60 80 0 20 40 60 80 Time (min) Time (min) Figure 7. Investigation of Potential Effect of Injection on Effective Diffusivities. The tracer injection is a possible source of error in the experiments, potentially accelerating transport at the beginning of the experiment. The transport model is used to investigate this hypothesis. A. Best-fit effective diffusivities versus cumulative experimental time for a representative wild-type mouse. Effective diffusivity in the surface periarterial space remains fairly constant over the first 50 minutes of the experiment, then decreases by 30% by the last time point. Changes in 𝐷'((,.,-. correlate better with tracer losses across the boundary than the injection. Changes in brain tissue effective diffusivity correlate with the injection. 𝐷'((,*+ increases slightly during the injection, then decreases significantly post injection, falling to a third of the initial value by 80 min. B. Separate effective diffusivities calculated during and after the injection for the SPAS and brain tissue regions (𝐷'((, ,-. = 55 mm2/min during injection and 50 mm2/min post injection). The results of the simulations, which are expected to underestimate post- injection diffusivities, estimate 𝐷'((,.,-. and 𝐷'((,*+ are almost 6 times faster during than after the injection. The results demonstrate the injection rate used for the experimental data reported here (0.5 µl/min for 24 min) has an effect on transport observed by DCE MRI, with higher effective diffusivities estimated during the injection than following the injection. However, the overall findings of 𝐷'(( > 𝐷"## hold. SPAS Effective Diffusivity (mm2/min) BT Effective Diffusivity (mm2/min) SPAS Effective Diffusivity (mm2/min) BT Effetcive Diffusivity (mm2/min) 121 injection set of effective diffusivities are expected to be the “physiological” effective diffusivities.) The total injection duration is 24 minutes (20 min of tracer solution followed by 4 min of saline solution). Raghunandan et al. found the ICP returned to near normal levels about 4 minutes after the injection. Therefore, separate transport parameter sets were fitted for the first 31 minutes, the first three data sets that occurred during the injection, and 31-83 minutes, the last data sets that occurred after the injection. Both the cumulative losses over boundaries not accounted for in the model and the high degree of T2 interference near the injection site add uncertainty to the calculation and have a greater overall downward effect on post-injection effective diffusivities. The best fit effective diffusivities for the surface periarterial space (SPAS) and brain tissue regions during and after the injection for a representative wild-type mouse are reported in Figure 7B, compared with the previous results reported in Figure 3A (fitted over the first 52 minutes of the experiment). For both the SPAS and the brain tissue, 𝐷'((,.,-. and 𝐷'((,*+ are six times faster during the injection than after the injection. Although the post-injection effective diffusivities (calculated in this way) are a low estimate of physiological transport, 𝐷'(( corrected for injection effects remains significantly greater than the apparent diffusivity of the tracer for all anatomical regions. If this same trend follows for the other mice, 𝐷'((,.,-. is more than 1,000 times greater than 𝐷"## and 𝐷'((,*+ is about 5 times greater than 𝐷"##. 122 The combined results of the injection analysis reported in Figure 7 indicate that the injection rate of 0.5 µl/min for 24 min has some effect on transport in the brain observed by DCE MRI, with higher effective diffusivities estimated during the injection than following the injection. Errors inherent to the experimental data and the simulation assumptions add uncertainty to the calculation and force underestimation of post- injection transport. In particular, significant T2 interference near the injection site limits the reliability of the analysis reported above. Additional work is warranted to determine the conditions (particularly injection rate) under which the injection impacts transport measurements by DCE MRI, through new experimentation and refined transport modelling. However, even for these underestimated effective diffusivities, the results of the injection simulation do not change the outcome of 𝐷'(( >> 𝐷"## in the periarterial space and 𝐷'(( > 𝐷"## in the brain tissue region. It is also worth noting that the most prominent engineering argument against flow resulting from the injection is that convection requires a pressure gradient, while pressure within a closed volume, such as the skull, equilibrates quickly. Therefore, pressure would be constantly equilibrating during the injection and one would expect only a very small pressure gradient (significantly less than the full increase in ICP) across the brain. Since the contents within the skull are significantly more complex than a container filled with fluid and it is not truly closed but has some efflux, a sophisticated calculation is necessary to determine the true rate at which pressure will equilibrate in response to the 123 injection, greatly reducing pressure gradients within the brain. Such a calculation is beyond the scope of this work but an interesting area of future work. 3.3.2. Sources of Error from Transport Model Assumptions Several assumptions were required to develop a computationally feasible model: 1) the anatomical features of the brain were grossly simplified, 2) a no flux boundary condition was applied and 3) efflux routes were not included, and 4) the form of the transport equation was simplified. In order to develop a reasonable first-generation model, it was necessary to simplify the complex details of the brain into five manageable tissue regions. However, the brain tissue, even within a mouse, is highly varied and complex with many specialized regions. For example, the cortex makes up the bulk of the brain tissue and is primarily homogenous grey matter, the hippocampus contains white matter tracts that make the tissue heterogenous with respect to transport, and glands, such as the pituitary and pineal glands, are comprised of dense tissue. Each of these specialized tissues has different properties, which could be included in the model in an attempt to “match the data”. However, the outcome of that work would be a long list of parameters that could be adjusted to fit any data set and a model of great complexity adding further sources of error. The goal of this work is to quantify parameters that describe transport over the broad systemic scale, not the small details, and draw conclusions about the prevalence 124 and magnitude of convection or dispersion in moving molecules across the brain. Here, the simplification aids in developing useful transport parameter estimates. The transport model assumed a no-flux boundary condition, meaning no material leaves the brain. It would have been ideal to use the concentration data on the boundary, but accurate concentration data for the full boundary were not available due to T2 interference. In addition, significant noise is observed on the boundaries of the majority of the subjects that may make the concentration at the boundary inherently elusive from DCE MRI data. If a no-flux boundary condition accurately represents the physical situation, then the total amount of Gad in the brain would increase during the injection, over the first 20 minutes, and then remain constant. Due to T2 interference, the concentration is unknown in the most concentrated volumes near the injection plume. However, for all of the mice (with the exception of one wild-type mouse) the amount of Gad in the brain increases or plateaus over the full course of the experiment (up to 84 minutes). The increasing amount of Gad over the course of the experiment is due to tracer being “released” from the T2 interference region—as the tracer disperses, regions where the concentration was high enough to be obscured by T2 interference, become low enough to be measured. The fact that total amount of Gad remains constant or increases over time supports the appropriateness of a no-flux boundary condition for the time scale investigated. 125 In the glymphatic hypothesis, fluid and molecules exit the brain tissue through the perivenous space, ultimately emptying to extracranial lymphatic vessels outside the blood-brain barrier [1]. Perivenous routes for efflux from the brain tissue were not included in the model. If efflux was significant to transport over the time scale of the experiments, the total amount of tracer in the brain would be expected to decrease steadily after the injection was complete. However, calculations show increasing total amounts of Gad over the course of the experiment, which supports that efflux is not a significant contributor to transport over the time scale investigated. Further, in order to minimize any effect of excluding efflux routes, error was calculated over the first 52 minutes of DCE MRI experiment (later time points were omitted). As discussed previously, the equation describing transport in the brain was also simplified for this analysis, lumping all transport mechanisms into a single parameter for each region and forcing the mathematical structure of diffusive transport. Differences observed between the simulation and data due to this simplification were discussed above for concentration in the surface periarterial space (SPAS) where the importance of convection is supported by the estimated 𝐷'((,.,-.. The model could be greatly improved by specifically including all mechanisms of transport as shown in Equation 1. However, to calculate convection, a driving force, such as pressure, must be known throughout the brain, which it is not. In addition, convection is a vector, which has a magnitude and a direction, adding a great deal of complexity. Alternatively, a convective field could be measured experimentally, possibly by Intravoxel Incoherent Motion 126 (IVIM) MRI [73]. The velocity field could be applied to the finite-element mesh directly and the transport model could be used to answer more subtle questions about brain transport, i.e. modelling disease states. 4. Discussion The effective diffusivities reported in the Results section indicate convection is significant to transport in the brain and prevalent throughout the brain. Optimal agreement between the simulation and MRI data was obtained with 𝐷'(( > 𝐷"## for all modelled brain regions and 𝐷'(( >> 𝐷"## in the periarterial spaces. In this Discussion section, the effective diffusivity results are further analyzed using concepts from transport phenomena and the results of that analysis compared to transport parameters reported in literature. Dimensionless groups are an important engineering tool for assessing the relative importance of different transport mechanisms. Péclet number (Pe) is the ratio of convective to diffusive transport rates: 𝑃𝑒 = 8"5' :( 4:;<'45%:; = >< = ?!""@ ?#$$@?%&'$ (4) 8"5' :( $%((=&%:; ? ?#$$ If transport is predominantly by diffusion, then 𝑃𝑒 ≪ 1, and if transport is predominantly by convection, Pe ≫ 1. Given an effective diffusivity, as estimated in this work, the 𝑃𝑒 calculation is straightforward. Table 2 reports 𝑃𝑒 for each anatomical region using 𝐷"## = 0.005 mm2/min for Gad in typical brain tissue and 𝐷$%&# = 𝐷"## (based on Asgari et al. [15]). In both periarterial spaces (SPAS and BPAS), 𝑃𝑒 exceeds 10,000 and 127 convection is the dominant mechanism of transport. The brain tissue exhibits an intermediate 𝑃𝑒 = 18, where convection and diffusion are both relevant, but convective rates may be greater. In our model, the brain tissue represents a combination of interstitial tissue and periarterial space for smaller penetrating vessels. From the Péclet analysis, we can conclude convection in the broad brain tissue region is present and significant to transport, but the contribution of each of its physiological components is not known. The convection observed in this work is impactful for molecules of interest in neurodegeneration, which are generally larger than Gad. Gad, the tracer used in this DCE MRI study, is small (559 Da) compared to molecules of interest in neurodegeneration, such as beta amyloid (4.5 kDa) and tau (45 kDa). Recall from the Approach section, diffusivity, 𝐷"##, and dispersion, 𝐷$%&#, decrease with molecular size, while the rate of convective transport is independent of molecular size. Therefore, for large molecules relevant to neurodegeneration, 𝑃𝑒 will be greater and convection will have an even greater impact on transport, particularly in the brain tissue region where convective rates are nearer in magnitude to diffusive rates. Using the definition of Péclet number (Eqn. 4) and a characteristic length for transport in each region, 𝐷'(( can be used to estimate fluid velocity (𝑣), which is reported for each region in Table 2. Periarterial convection is estimated to have an average 128 Table 2. Quantitative Analysis of Brain Transport Parameters for each Anatomical Subdomain. Table reports: 1) best fit 𝐷'(( for wild-type mice, 2) 𝑃𝑒𝑐𝑙𝑒𝑡 number giving the ratio of convective to diffusive transport, 3) dimensionless rate and velocity calculated from 𝐷'(( using a characteristic length for the anatomical region, and 4) measured (or estimated from a model) brain transport parameters from the literature for comparison. Sub 𝐷'(( from 𝑃𝑒𝑐𝑙𝑒𝑡 Charac. Charac. Estimated Published Domain Simulation Number Length Rate Velocity Transport (𝑚𝑚6/ (𝑚𝑚) (𝑚𝑖𝑛@/) (𝑚𝑚⁄𝑚𝑖𝑛) Parameter 𝑚𝑖𝑛) Branching 60 + 20 12,000 5 2 + 1 12 + 4 𝑣!"# Periarterial = 1.2𝑚𝑚⁄𝑚𝑖𝑛 Space [14, 34] Surface 95 + 40 19,000 12 0.7 + 8 + 3 𝑣!"# Periarterial 0.3 = 1.2𝑚𝑚⁄𝑚𝑖𝑛 Space [14, 34] Brain 0.1 + 0.04 18 0.5 0.4 + 0.2 + 0.1 𝐷$%% = Tissue: 0.2 0.005 𝑚𝑚&/ Interstitial 𝑚𝑖𝑛 [56, 57] Space & 𝑣'# = Periarterial 0.01𝑚𝑚⁄𝑚𝑖𝑛 space of [31] small 𝑣!"# = not vessels known velocity of 𝑣,-. = 8 mm/min in the SPAS1 and 𝑣,-. = 12 mm/min in the BPAS, which agrees with observations from the DCE MRI data of 𝑣 > 1.2 mm/min. The choice of characteristic length has a significant impact on the velocity calculated in this way. The characteristic lengths reported in Table 2 for the SPAS and BPAS were chosen based on 1 If the lower values of effective diffusivity calculated from the post injection simulation are used, then 𝑣()* = 2.1 mm/min. 129 the length dimension of the brain in each direction. As the vasculature that the perivascular space follows takes a tortuous, branching path and is connected from the major arteries down to the capillaries, the appropriate characteristic length for each of these regions may be longer than estimated. Perivascular convection in a surface artery was measured to be 1.2 mm/min by both Mestre et al. and Bedussi et al. [14, 34]. The periarterial velocity values estimated from simulations are greater than the literature value, but they are of similar orders of magnitude. As discussed above, the brain tissue region represents a combination of interstitial and periarterial transport surrounding smaller penetrating arteries (illustrated in Figure 1B). Therefore, literature values are investigated for each and compared to the estimated average velocity of 𝑣*+ = 0.1-0.3 mm/min (Table 2)2. In previous work, Ray et al. estimated interstitial flow in the brain of 𝑣2. = 0.01 mm/min [31]. Measurements of interstitial flow outside the brain (in the peripheral body), which likely represent an upper limit on brain interstitial transport, report 𝑣2. = 0.006-0.12 mm/min [33]. Both estimates of interstitial velocity are lower by an order of magnitude than the average velocity estimated from the effective diffusivity in the lumped brain tissue region, indicating a significant contribution from penetrating periarterial flow. 2 If the lower values of effective diffusivity calculated from the post-injection simulation are used, then 𝑣+, = 0.02-0.1 mm/min, which still suggests a contribution from penetrating periarterial flow.) 130 Another way to think about velocity in the brain tissue region is as the velocity of a transport “front”. The smaller penetrating arteries branch out in many directions, so there is no clear direction of flow on the scale of the brain tissue region either in the penetrating periarterial space or for the interstitial flow that runs from periarterial to perivenous space. Instead, tracer progresses more rapidly than by diffusion alone, but along a tortuous convective path. Therefore, the velocity calculated in Table 2, 𝑣*+ = 0.1-0.3 mm/min, is essentially the velocity of a “front” progressing deeper into the brain, as opposed to the velocity of the actual interstitial or periarterial flow. Plog et al. measured front velocities on the order of 0.1-1 mm/min in mice in the space between large vessels (analogous to the brain tissue region in this work), using a novel technique called transcranial fluorescence mesoscopic imaging [74]. This front velocity, or an enhancement factor or effective diffusivity, may be a better representation of transport for the combined effects of periarterial convection, interstitial convection, dispersion, and diffusion in the brain tissue. The results presented above for both the periarterial space of major branching arteries (BPAS) and the periarterial space of smaller penetrating arteries (included in the brain tissue region) clearly support that periarterial flow continues into the brain along penetrating arteries. Recall from the introduction, 1-µm microspheres tracking fluid movement in the periarterial space of the middle cerebral artery (Mcer) were excluded from branches of that artery penetrating into the brain [14, 34]. This observation brought 131 into question whether periarterial flow continues into the brain, and if it does, whether the penetrating periarterial space is filled with protein networks that behave like a porous media and impede flow. In the model, the BPAS is comprised of the MCer, which is a pial artery residing primarily on the surface of the brain, but also the posterior cerebral artery (Pcer) and the olfactory arteries (Olfac), which penetrate into the brain tissue. Convection was apparent along the large caliber arteries of the BPAS at a magnitude that suggests open channel flow, not flow through porous media. Convection is also supported for the periarterial space of smaller (and higher branch order) penetrating arteries. Although the actual flow velocity in the periarterial space of these smaller arteries cannot be determined from the analysis performed here at the scale of the whole- brain, it is clear that such flow significantly enhances transport over diffusion alone. Therefore, the results reported here support convection along the periarterial space of large and small penetrating arteries, where open-channel flow is demonstrated along large caliber arteries and it cannot be determined whether flow along smaller arteries is through open channels or channels filled with protein networks that impede flow. 5. Conclusions Through quantitative analysis of DCE MRI data for the whole mouse brain using a finite-element transport model, convection is demonstrated throughout the brain. Convection is shown to be dominant in the periarterial space of both surface and major branching arteries, such as the posterior cerebral and olfactory arteries, where Pe > 10,000. Periarterial velocities are estimated at around 10 mm/min, exceeding but 132 comparable to previous experimental measurements of convection in the periarterial space. Importantly, convection was predicted to continue into the brain tissue, demonstrating it is more than a surface phenomenon. Periarterial convection for smaller penetrating arteries could not be separated from interstitial convection as the data and model were not sufficiently refined to make this separation; these mechanisms are combined in the “brain tissue” region of the model. However, comparison to estimated and upper bound values for interstitial flow suggests appreciable convection in the periarterial space throughout the brain. As the penetrating arteries branch in many directions, it is difficult to estimate and actual velocity in the periarterial and interstitial space of smaller arteries from the whole brain viewpoint of this work. An enhancement factor over diffusion for the progressing concentration front as fluid flows through the branching network of the periarterial space is estimated at 10-25. Such enhancement can only be achieved by significant convection in the periarterial space of smaller penetrating arteries (𝑣,-. > 0.2 mm/min). The convection observed here, following the transport of the small Gad tracer, is even more significant to the transport of larger molecules (with slower diffusivities) implicated in neurodegeneration. These results, quantifying transport in the brain, will continue to be refined as further work is completed to understand and mitigate the factors discussed in Sources of Error. 133 Tracer concentration dynamics were markedly different for transgenic mice, supporting the hypothesis of an AQP4 role in the transport of fluid into the interstitial space. However, these differences could not be quantified using the current model, due to lumping in the DCE MRI data, and therefore the model, of interstitial space with the perivascular space of small penetrating arteries. Testing the transgenic mice with an experimental method focused on interstitial transport, such as integrated optical imaging (IOI), may better elucidate differences and enable quantification of interstitial transport. 5.1. Future Work The transport model described in this work is structured for continuous improvement as new information becomes available. If fluid velocity across the whole brain is measured, such as by IVIM MRI, convection can be directly incorporated using Equation 2. As efflux routes are better understood, efflux can be added either through specific routes or an efflux rate can be distributed over an entire region. As the current transport parameters become firmed through further study, additional anatomical regions can be added, such as separating the olfactory bulb from the bulk of the brain tissue. More ambitiously, a multi-scale transport model can be developed by appropriately connecting the whole-brain model reported here with a small-scale model, such as the model for interstitial transport described in Ray et al. [31] or models for transport in branching periarterial space. 134 In the world of computational modelling, one is elated to have access to high-quality experimental data such as the DCE MRI data presented here, but the experiments are often not designed specifically for the analysis being performed. The DCE MRI data used here were collected using a method designed to detect very low concentrations deep in the brain tissue, and therefore information valuable to our analysis was lost to T2 interference. An injection and MRI method designed for measuring higher tracer concentration and avoiding T2 interference, especially if combined with this work, would greatly improve the reliability of the transport parameters calculated here. Future experimental work aimed at quantifying transport in the brain would benefit from investigating molecules of different sizes, i.e. 500 Da and 10,000 Da. Diffusion and dispersion are dependent on molecular size, while convection is independent of molecular size (until the molecule is so large as to be excluded from regions or to have its progress slowed by interaction with surfaces). Comparison of transport data and parameters from tracers of significantly different molecular size would provide further evidence towards convective versus diffusive or dispersive transport. In addition, as DCE MRI equipment and methods improve, data collected over a more refined grid of smaller dimensions and more rapidly to achieve a greater number of data sets over a given time will improve the analysis presented here. 135 5. Methods A graphical depiction of the methods is presented in Figure 8. Calculate T1,0 & Tracer Concentration Solution minimizing rms error Data rms Interpolate Simulation DCE Concen-tration Finite-Element Model: Iterate:MRI D Signal Build Anatomical Masks Transport Equation & eff, BT Data & Anatomical Subdomains Deff, SPASData Masks !" D# eff, PPAS Build 3D Finite- onto !# = &!""' " Extract Element Mesh Mesh Brain Surface Figure 8. Schematic of Brain Transport Analysis using Finite-element Modeling with DCE MRI Data. DCE MRI signal data is used to calculate tracer concentration and 𝑇/,0, a parameter that is used to identify different types of tissues. Concentration and 𝑇/,0 are used to define anatomical masks for regions relevant to brain-wide transport. DCE MRI data is also utilized to extract the surface of the brain. The brain surface and anatomical masks are unique to each experimental subject. A 3-dimensional tetrahedral mesh is built from the brain surface. The concentration data and anatomical masks are interpolated onto the mesh. A finite-element model is developed utilizing the simplified mass transport equation and subdomains defined by the anatomical masks. Unique effective diffusivities are applied to each subdomain. Simulations are performed for varying effective diffusivities to find the combination of effective diffusivities that minimizes the difference between the concentrations data and the simulation. 5.1. Calculation of Tracer Concentration and Identification of Anatomical Details from DCE MRI Data The majority of DCE MRI data in the literature are reported and analyzed as signal, often assuming tracer concentration is directly proportional to MRI signal. T1-weighted signal is increased by the presence of the tracer, but the relationship between signal and concentration is more complex. In order to quantify transport parameters, concentration, a 136 fundamental physical variable, is required. The process for calculating concentration from DCE MRI signal follows. Tracer concentration is related to MRI signal by [75]: [𝐺𝑎𝑑] ≅ / I.0%@ ./J (5) 8-∙+-,/ ./ where: 𝑟/ = tracer relaxivity, 3.2 x 10-3 L/mmol-ms for gadoteridol (Prohance) [76] 𝑇/,0 = pre-contrast relaxation time (ms) 𝑆1$ = signal intensity after contrast injection (as a function of time) 𝑆0 = baseline signal intensity prior to contrast agent injection The proportionality assumption between concentration and signal is correct if 𝑇/,0 is constant across the sample. However, 𝑇/,0 is dependent on the density of protons and varies a great deal (500-4,500) across different tissues in a biological subject, especially one as complex as the brain. 𝑇/,0 can be estimated from baseline MRI signals collected at different flip angles according to the following equation [77]: 12342-,/ 𝑆(𝑀0, 𝛼) = 𝑀0 𝑠𝑖𝑛(𝛼) /@' 123 (6) /@4:&(C) ' 42-,/ where: 𝑇𝑅 = repetition time (16 ms for the experiments reported here), and 𝛼 = flip angle 137 For each voxel, 𝑇/,0 is calculated from 𝑆0 (baseline) images at 𝛼 = 3° and 15° using a curve fit to Equation 6. Knowing 𝑇/,0, Equation 5 is used to calculate Gad concentration from DCE MRI signal for each voxel and time point. The calculated concentration, [Gad], is superficial concentration, 𝑐̅, used in porous media theory, where 𝑐̅ = 𝑐 ∙ 𝜙 and 𝜙 is void fraction or porosity. It is assumed the void fraction is consistent across the brain; 𝜙 in brain tissue has a well-known value of 20% [57]. Throughout the text the accent on superficial concentration has been dropped and it is denoted simply as 𝑐. Since 𝑇/,0 varies markedly between different types of tissue, it can be used to identify anatomical features, such as blood vessels. 𝑇/,0 and concentration thresholds were used to develop anatomical masks, voxels assigned to a specific anatomical feature, for regions of interest. The nibabel Python library [78] was used to import the MRI files into Python arrays, where all calculations were performed. The analyzed data was visualized using a Python code [79]. 5.2. Transport Model Subject-specific, finite-element models (FEM) of transport in the mouse brain were developed based on the diffusion equation (Eqn 2). The models are 3-dimensional models of the whole brain (skull and subarachnoid space excluded). Subdomains were developed for each relevant anatomical region using the anatomical masks described above. Different 138 effective diffusivities were applied to each anatomical subdomain to determine the combination of parameters that best fit the DCE MRI concentration data. Brain surfaces were extracted from the MRI data using MRIcroS [80]. The resulting surface was remeshed for FEM using Meshlab [81]. The 3-dimensional, tetrahedral mesh of 800,000-900,000 cells was generated with gmsh [82]. The dynamic (changing with each time point) tracer concentration data, dynamic T2 interference boundaries, and the static anatomical masks were interpolated onto the mesh using an inverse-distance weighted (IDW) method. The IDW method is essentially the application of a Gaussian filter (using 27 nearest neighbors), and therefore had the effect of also smoothing the data. Approximately 2,500,000 data locations were interpolated onto around 150,000 mesh vertices, resulting in 850,000 mesh cells. The simple model of the diffusion equation was solved within the complex geometry of the whole mouse brain, including five anatomical subdomains where different material properties were applied, notably 𝐷'((. The tracer injection was modelled as a point source in space and a rectangular function in time. Due to T2 interference, the concentration at the boundary was not complete and could not be used as the boundary condition. Therefore, a no-flux boundary condition was applied, based on the fact that the skull represents an impenetrable barrier. The brain is however surrounded by CSF which could carry tracer out of the domain. The use of a no-flux boundary condition may overestimate the simulated concentration of tracer in the brain. 139 Effective diffusivities were varied and mapped (See Supplemental Material) to determine the combination resulting in the minimum root mean square (rms) error between the data and the simulation. 𝑟𝑚𝑠 = O∑+ (4%#5#@ 4'&678#5&9:) ; 4E/ (7) + The T2 interference region, the ventricles, and the blood vessels are excluded from the error calculation. Time points up to 52 minutes were used in the error calculation, later time points were excluded to minimize the error due to assumptions of no efflux routes and the no-flux boundary condition. The finite-element problem was solved using FEniCS [83, 84], an open-source solver of partial differential equations by the finite-element method. The time derivative was discretized using a backward difference (i.e., an implicit method). Post processing of simulation and DCE MRI concentration data was carried out using Python [85], Paraview [86], and Excel. Every effort was made to use open-source software. Acknowledgements: Ian Ray for creating the program to generate error contours using his newly acquired Python skills. Rod Ray for fruitful discussions about transport phenomena and editorial input to this manuscript. 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If diffusion is dominant, then therapies can focus on maintaining strong concentration gradients or reducing barriers to diffusion, such as maintaining or increasing the width of the interstitial space between cells that can impede diffusion of large molecules. In the work presented in this Thesis, convection, along with diffusion, is demonstrated to be an important mechanism of transport throughout the brain, and across transport scales. Analyses were performed at two scales: 1) interstitial transport, utilizing Real-time Iontophoresis (RTI) data, and 2) whole-brain transport divided into pertinent anatomical regions, utilizing dynamic-contrast enhanced (DCE) MRI data. The greatest convective transport is found in the periarterial space surrounding major arteries, and convection is also found to be significant in the periarterial space surrounding smaller, branching arteries. In the interstitial space, convection is a critical augmentation to diffusion for large, slow-to-diffuse molecules. It follows that observations and analyses from this work contribute further evidence to a circulatory-like system in the brain with rapid transport of cerebrospinal fluid (CSF) into the brain along preferential 147 routes, periarterial space, branching throughout the brain tissue and slower transport across that tissue, in the interstitial spaces of the brain. Transport in the interstitial space is slower overall and the role of convection is highly dependent on molecular size. From the body of work focused on interstitial space, the interstitial flow velocity is demonstrated to be on the order of 0.01 mm/min. This value falls within the range of interstitial velocity in the peripheral body of 0.006-0.12 mm/min [33]. For small molecules, like the ions essential to neuronal activity (around 30 Da), diffusion is relatively fast and convection contributes little to overall transport rate. For large molecules (>3 kDa), such as proteins implicated in neurodegeneration, convection is demonstrated to be an important mechanism of transport in the interstitium. These molecules have small free diffusivities, made even smaller when moving through narrow and tortuous space between brain cells. Periarterial transport surrounding major arteries is so rapid (𝐷'((/𝐷"## > 10,000), it can only be explained by convection. Periarterial flow velocity estimated using results from the whole brain transport model agrees with limited experimental measurements [14, 34]. The periarterial space of smaller penetrating arteries could not be isolated in this work, as it is combined with interstitial space in the DCE MRI data. However, enhancement for combined interstitial and smaller periarterial transport over diffusion alone is 𝐷'((/𝐷"## = 10-25. The DCE MRI tracer is an intermediate molecular size, where, as described above, interstitial convection is expected to be similar in rate to diffusion. Therefore, diffusion and interstitial convection account for up to 𝐷'((/𝐷"## = 2, and the additional enhancement to 𝐷'((/𝐷"## = 10-25 supports significant periarterial 148 convection surrounding small, branching arteries. Importantly, periarterial convection is demonstrated within the brain tissue, both surrounding large arteries that penetrate the brain and smaller arteries that branch within the brain tissue, confirming periarterial flow is more than a surface-artery phenomenon. The finite-element transport models described in this work, of the whole brain and the brain interstitium, are demonstrated to be useful tools for the study of brain transport and structured for continuous improvement as new information is discovered. In fact, the whole-brain model was applied to investigate and quantify the potential effect of DCE MRI tracer injection on transport parameters. Key Contributions • Useful tools for quantifying transport in the brain at different length scales, including finite-element transport models and unique methods of analyses. • Quantification of fundamental transport parameters from published experimental data in the murine (mouse or rat) brain. • Demonstration of the presence and magnitude of periarterial convection within brain tissue. • Demonstration of the presence and magnitude of potential interstitial flow within brain tissue. • Corroboration of the presence and magnitude of periarterial convective velocity surrounding surface arteries. 149 • Further evidence supporting the overarching hypothesis of a brain-specific circulatory system for transport of molecules. Future Work Key areas of future work for understanding transport in the brain include: Brain-Transport Modelling • Development of a multi-scale transport model through the appropriate connection of the whole-brain model with small-scale models, such as the model for interstitial transport developed for this Thesis or models for fluid flow and transport in branching periarterial space. • The whole-brain transport model can be improved by: o Inclusion of convective velocity obtained through measurement of velocity fields, such as by Intravoxel Incoherent Motion (IVIM) MRI. o Addition of efflux routes, either specific routes as they are identified or distributed over an entire region. o Further refinement of anatomical regions, particularly as transport parameters become firmed for current regions. • The interstitial transport model can be improved as new information comes to light such as: vascular arrangement and separation, anatomical and physiological details about the structure of the boundary between the perivascular space and the interstitial space, or details about pressure within the spaces of the brain. 150 Experiments for Study of Transport in the Brain • DCE MRI o Experimental work investigating molecules of different sizes, i.e. 500 Da and 10,000 Da, would provide further evidence towards convective versus diffusive or dispersive transport. Diffusion and dispersion are dependent on molecular size, while bulk convection is independent of molecular size (until the molecule is so large as to be excluded from regions or to have its progress slowed by interaction with surfaces). Tracking the movement of different sized molecules experimentally would enable further delineation of the mechanisms contributing to overall transport. o As equipment and methods improve, more rapid collection of data over a more refined grid will result in more data sets for a given experimental time and data over smaller dimensions, elucidating further anatomical details. These refinements will improve the analyses presented here. o The transport analyses reported in this Thesis can be applied to new and existing experimental data to optimize tracer injection methods (site, rate, etc.) either minimizing or separating the effects of the injection on transport through the brain. In particular, it would be interesting to apply this analysis to DCE MRI data collected in rats, which have a larger brain volume relative to the normal injection volume and transport may be less affected by the injection protocol. 151 • Real-time Iontophoresis (RTI) is an excellent method for studying interstitial transport in the brain, however, it is limited to a few molecules that are quite small relative to the molecules of interest in neurodegeneration. Quantitative experimental methods for direct measurement of interstitial flow, or of the transport of larger molecules, would greatly expand knowledge towards understanding of neurodegeneration. The modelling tools developed here could be applied to such experimental data for facile transport analysis. Drivers of flow in the Glymphatic System One of the greatest opportunities for improvement of general understanding of convection in the brain is knowledge of the driving forces for bulk flow along the glymphatic circulation. A significant piece of understanding this circulation would be to demonstrate the mechanisms of causality between arterial pulsation and periarterial flow, hypothesized to be a peristaltic motion. Experimentation on a model system, outside the body and properly scaled to reproduce similar physics, and computational fluid dynamics modelling would certainly provide useful additions to recent in vivo measurements characterizing periarterial flow. Knowledge of the pressure cascade through the different compartments of the glymphatic system would demonstrate the potential for convection and provide useful information for modelling and calculating bulk flow velocities. Pressure in the periarterial space may come from models of peristaltic-like periarterial flow or sophisticated measurements in this region of µm dimensions. Pressure in the perivenous space may come from better understanding of the perivenous efflux routes and their routes out of the brain. 152 References Cited 14. 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Journal of Neuroscience Research, 2018. 96(12): p. 1876-1886. 168 APPENDICES 169 APPENDIX A VARYING PERIVASCULAR ASTROGLIAL ENDFOOT DIMENSIONS ALONG THE VASCULAR TREE MAINTAIN PERIVASCULAR-INTERSTITIAL FLUX THROUGH THE CORTICAL MANTLE 170 APPENDIX A VARYING PERIVASCULAR ASTROGLIAL ENDFOOT DIMENSIONS ALONG THE VASCULAR TREE MAINTAIN PERIVASCULAR-INTERSTITIAL FLUX THROUGH THE CORTICAL MANTLE Contribution of Authors and Co-Authors Manuscript in Appendix A Author: Marie X. Wang Contributions: Performed and analyzed experiments. Wrote article. Co-Author: Lori A. Ray Contributions: Conceived of computational model with Jeffrey J. Heys. Performed calculations associated with image projection. Reviewed and edited the article. Co-Author: Kenji F. Tanaka Contributions: Provided expertise regarding immunochemistry. Reviewed and edited the article. Co-Author: Jeffrey J. Iliff Contributions: Conceived of experimental study with Marie X. Wang. Reviewed and edited the article. Co-Author: Jeffrey J. Heys Contributions: Conceived of computational model with Lori A. Ray. Developed model and conducted simulations. Wrote simulation sections of article. Reviewed and edited the article. 171 Manuscript Information Marie X. Wang, Lori A. Ray, Kenji F. Tanaka, Jeffrey J. Iliff, and Jeffrey J. Heys GLIA Status of Manuscript: ____ Prepared for submission to a peer-reviewed journal ____ Officially submitted to a peer-reviewed journal ____ Accepted by a peer-reviewed journal __X__ Published in a peer-reviewed journal Wiley 2021, Volume 69, Issue 3 DOI: 10.1002/glia.23923 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 APPENDIX B SUPPLEMENTAL MATERIAL FOR CHAPTER FOUR QUNATIFICATION OF TRANSPORT IN THE WHOLE MOUSE BRAIN 187 Supplemental Material I: 𝑇/,0 Example Image and Histogram 𝑇/,0, pre-contrast relaxation time is related to the proton density of the material, and is therefore a fundamental property of biological tissues. As discussed in Methods, 𝑇/,0 can be estimated from baseline MRI signals collected at different flip angles according to the following equation: 12342-,/ 𝑆(𝑀0, 𝛼) = 𝑀0 𝑠𝑖𝑛(𝛼) /@' 123 (6) /@4:&(C) ' 42-,/ where: 𝑇𝑅 = repetition time (16 ms for the experiments reported here), and 𝛼 = flip angle Shown below are a center sagittal slice of 𝑇/,0 for a representative mouse (Figure I.A.), which illustrates different tissues in the mouse brain, and a histogram of 𝑇/,0 for the same mouse (Figure I.B.), which shows different groupings of 𝑇/,0 values. Vasculature has a low 𝑇/,0, because protons are “refreshed” by the flow of blood, and appears bright in the T1-weighted image. Cerebrospinal fluid in the ventricles has a high 𝑇/,0 and is dark. 188 Figure I.A. Center sagittal image of 𝑇/,0 for mouse KO062217. 𝑇/,0values correspond to different types of biological tissue. Therefore, different anatomical features can be identified in the MRI images given their calculated 𝑇/,0. For example, vasculature has a low 𝑇/,0, because protons are “refreshed” by the flow of blood, and appears bright. Cerebrospinal fluid in the ventricles has a high 𝑇/,0 and is dark. Figure I.B. Histogram of 𝑇/,0 for mouse KO062217. 𝑇/,0values correspond to different types of biological tissue. For example, 𝑇/,0>3000 corresponds to CSF in the ventricles. 189 Supplemental Material II: Example Error Contours Effective diffusivities were varied and mapped to determine the combination resulting in the minimum root mean square (rms) error between the simulation and the data. Root mean square error was summed over all mesh vertices, excluding ventricles, arteries, and T2 interference regions. Error Contours for a representative wild-type mouse are reported in Figures II.A and B. A clear minimum is exhibited for 𝐷'((,.,-. and 𝐷'((,*+ (Figure II.A), while 𝐷'((, ,-. exhibits a significant change in slope, which is deemed the “best fit.” Figure II.A.. Root mean square error times 10 for 𝐷'((,.,-. vs. 𝐷'((,*+ at 𝐷 2'((, ,-.=100 mm /min. A clear minimum is exhibited at 𝐷'((,*+ = 0.08 and 𝐷'((,.,-. = 120-140 mm2/min. 190 Figure II.B. Root mean square error times 10 for 𝐷'((, ,-. vs. 𝐷'((,.,-. at 𝐷'((,*+ = 0.08 mm2/min. 𝐷 2'((,.,-. = 120-140 mm /min as demonstrated in Figure IIA. Surface flattens significantly around 𝐷'((, ,-. = 50 mm2/min. 191 In Sources or Error, the model was used to simulate the effect of the injection, by fitting different sets of effective diffusivities during (Figures II.C and D) and after (Figure II.E) the tracer solution injection. Figure II.C. Root mean square error times 10 during injection for 𝐷'((,.,-. vs. 𝐷'((,*+ with 𝐷'((, ,-.=50 mm2/min. A significant change in slope is noted around 𝐷'((,*+ = 0.13 mm2/min and 𝐷'((,.,-. = 150 mm2/min. 192 Figure II.D. Root mean square error times 10 during injection for 𝐷'((, ,-. vs. 𝐷'((,.,-. with 𝐷'((,*+ =0.12 mm2/min. Slope change estimated at 𝐷'((,.,-. = 140 mm2/min and 𝐷 2'((,.,-. = 55 mm /min. Figure II.E. Root mean square error times 10 calculated after injection for 𝐷'((,.,-. vs. 𝐷 2'((,*+ with 𝐷'((, ,-.=50 mm /min. Transport parameters during injection were: 𝐷'((,*+=0.12, 𝐷'((,.,-.=140 and 𝐷'((, ,-.=55 mm2/min. A clear minimum is exhibited at 𝐷'((,*+ = 0.02 & 𝐷'((,.,-. = 25 mm2/min. No change in the contours was observed when 𝐷'((, ,-. was changed to 10 mm2/min. 193 Supplemental Material III: Periarterial Space Sensitivity Analysis To define a surface periarterial space (SPAS) subdomain that best represents the concentration data, a sensitivity analysis was performed to determine the SPAS width that resulted in the minimum rms error, or difference between the simulation and the data. The analysis was performed at 𝐷'((,*+=0.2 and 𝐷'((,.,-.=72 mm2/min. The results, shown in Figure III.A., show a minimum around 5,500 vertices, which corresponds to a SPAS width of around 400 𝜇m at the Circle of Willis. SPAS Sensitivity 0.15 0.14 0.13 0.12 0.11 0.1 0.09 0.08 3,000 3,500 4,000 4,500 5,000 5,500 6,000 6,500 7,000 Number of SPAS Vertices Figure III.A. Results of sensitivity analysis for rms error vs. total number of vertices in the surface periarterial space (SPAS) subdomain. The number of SPAS vertices is directly proportional to the average width of the periarterial space. A minimum is exhibited around 5,500 vertices which corresponds to a SPAS width of around 400 𝜇m at the Circle of Willis. rms 194 Supplemental Material IV: Example FEniCS Code for Whole-Brain FE Model """ # The purpose of this code is to model transport in the mouse brain. # In particular effective diffusivities for specific regions in the brain # are varied and compared to DCE MRI data to determine the set of # effective diffusivities that minimizes the difference, or rms error. # The domain surface was extracted from MRI data using MRIcrOS # The domain was meshed using gmsh and this mesh is imported in this code. # Anatomical subdomains were defined using thresholds for T1,0 or conc # and were then interpolated onto the mesh. Each anatomical mask is imported # and then used to create subdomains with unque effective diffusivities. # the data has been inerpolated onto the mesh so rms can be directly calculated. # FEM problem is solved using FEniCS project (dolfin and mshr). from dolfin import * from mshr import * import numpy as np #----Inputs-------------------------------------------------------------------- # Enter Effective Diffusivities to be Tested DBT = np.zeros(3) DSPAS = np.zeros(4) DPPAS = np.zeros(1) len_DBT = DBT.shape[0] len_DSPAS = DSPAS.shape[0] len_DPPAS = DPPAS.shape[0] DBT[0] = 0.08 DBT[1] = 0.04 DBT[2] = 0.12 DSPAS[0] = 160 DSPAS[1] = 140 DSPAS[2] = 120 DSPAS[3] = 100 DPPAS[0] = 100 print(DBT, DSPAS, DPPAS) # Define Parameters time_pts = 8 data_dt = 10.4 #min Q = 0.00335 # source rate = 66.7 mM * 0.5 e-6 l/min = 0.0000335x100 src = Point(-0.09, -4.89, -0.78) # source location dt = data_dt / 10. 195 T = (time_pts) * data_dt #----Various Error Functions--------------------------------------------------- def sim_rms(original, sim): rms = np.sqrt(np.mean((original-sim)**2)) rmspe = np.sqrt(np.mean(np.square((original-sim)/original))) return (rms, rmspe) def mape(original, sim): mpe = np.mean((original-sim)/original) mape = np.mean(np.abs((original-sim)/original)) return (mape, mpe) #----Import Data--------------------------------------------------------------- # Load concentration data conc_0 = np.load('mouse_interpconct0ext_ondomain_t1.npy') conc_1 = np.load('mouse_interpconct0ext_ondomain_t2.npy') conc_2 = np.load('mouse_interpconct0ext_ondomain_t3.npy') conc_3 = np.load('mouse_interpconct0ext_ondomain_t4.npy') conc_4 = np.load('mouse_interpconct0ext_ondomain_t5.npy') conc_5 = np.load('mouse_interpconct0ext_ondomain_t6.npy') conc_6 = np.load('mouse_interpconct0ext_ondomain_t7.npy') conc_7 = np.load('mouse_interpconct0ext_ondomain_t8.npy') # combine into one array conc_m = np.concatenate((conc_0, conc_1, conc_2, conc_3, conc_4, conc_5, conc_6, conc_7), axis=1) # Load T2 mask data # T2 interference has rendered data in these voxels useless and they are #excluded from the error calculations. T2 interference region changes in time T2mask_0 = np.load('mouse_interpT2ext2mask_t1.npy') T2mask_1 = np.load('mouse_interpT2ext2mask_t2.npy') T2mask_2 = np.load('mouse_interpT2ext2mask_t3.npy') T2mask_3 = np.load('mouse_interpT2ext2mask_t4.npy') T2mask_4 = np.load('mouse_interpT2ext2mask_t5.npy') T2mask_5 = np.load('mouse_interpT2ext2mask_t6.npy') T2mask_6 = np.load('mouse_interpT2ext2mask_t7.npy') T2mask_7 = np.load('mouse_interpT2ext2mask_t8.npy') # combine into one array T2_mask = np.concatenate((T2mask_0, T2mask_1, T2mask_2, T2mask_3, T2mask_4, T2mask_5, T2mask_6, T2mask_7), axis=1) print(T2_mask.shape) # load anatomical mask data 196 # ventricles vent_mask = np.load('Mouse_vent_interp_00706.npy') # arteries Acer_mask = np.load('Mouse_interp_acer_nofilt.npy') aCoW_mask = np.load('Mouse_interp_surfart_nofilt.npy') Bas_mask = np.load('Mouse_interp_bas_nofilt.npy') Pcer_mask = np.load('Mouse_interp_pcermask_nofilt.npy') Mcer_mask = np.load('Mouse_interp_mcermask_nofilt.npy') Olfac_mask = np.load('Mouse_interp_olfacmask_nofilt.npy') # periarterial space CoW_mask = np.load('Mouse_interpt0_SPASrev0924b_01.npy') # surface arteries penperi_mask = np.load('Mouse_interp_PPAS_rev1011cp_01.npy') # branching arteries # Load the Mesh mesh = Mesh('mouse_mesh_00706.xml.gz') coor = mesh.coordinates() # save mesh mfilename = 'mouse_mesh_00706.pvd' meshfile = File(mfilename) meshfile << mesh #----exclude certain vertices from error calculation--------------------------- # ventricle points nv = mesh.num_vertices() exclude = np.zeros((nv, time_pts)) for i in range(nv): if (vent_mask[i] > 0 or aCoW_mask[i] > 0 or Bas_mask[i] > 0): for j in range(time_pts): exclude[i, j] = 1 # conc_m=0 points for i in range(nv): for j in range(time_pts): if conc_m[i,j] == 0: exclude[i, j] = 1 # T2 interference points for i in range(nv): for j in range(time_pts): if T2_mask[i, j] > 0: exclude[i, j] = 1 #---Map concentration data onto mesh for erro calc----------------------------- # create Function space C = FunctionSpace(mesh, 'Lagrange', 2) 197 comm = MPI.comm_world cncdatafile = HDF5File(comm, 'mouse_concdata_T2ext.h5', 'r') # map concentration data onto function space CD = FunctionSpace(mesh, 'CG', 1) cd1 = Function(CD) cncdatafile.read(cd1, '/cd1') cd2 = Function(CD) cncdatafile.read(cd2, '/cd2') cd3 = Function(CD) cncdatafile.read(cd3, '/cd3') cd4 = Function(CD) cncdatafile.read(cd4, '/cd4') cd5 = Function(CD) cncdatafile.read(cd5, '/cd5') cd6 = Function(CD) cncdatafile.read(cd6, '/cd6') cd7 = Function(CD) cncdatafile.read(cd7, '/cd7') cd8 = Function(CD) cncdatafile.read(cd8, '/cd8') #----Create anatomical subdomains---------------------------------------------- # create anatomical arrays for use in marking subdomains vertex_values = np.zeros(mesh.num_vertices()) vent_vertex_values = np.zeros(mesh.num_vertices()) Acer_vertex_values = np.zeros(mesh.num_vertices()) SA_vertex_values = np.zeros(mesh.num_vertices()) Bas_vertex_values = np.zeros(mesh.num_vertices()) Mcer_vertex_values = np.zeros(mesh.num_vertices()) Pcer_vertex_values = np.zeros(mesh.num_vertices()) Olf_vertex_values = np.zeros(mesh.num_vertices()) CoW_vertex_values = np.zeros(mesh.num_vertices()) PP_vertex_values = np.zeros(mesh.num_vertices()) # map anatomical masks onto mesh using a Function # vertex to dof_map approach AN = FunctionSpace(mesh, 'CG', 1) # so that dofs are only in mesh vertices # ventricles vm = Function(AN) for vertex in vertices(mesh): vent_vertex_values[vertex.index()] = vent_mask[vertex.index()] 198 vm.vector()[:] = vent_vertex_values[dof_to_vertex_map(AN)] # arteries SAm = Function(AN) SAm.set_allow_extrapolation(True) #work around--rounding error causes crash for vertex in vertices(mesh): SA_vertex_values[vertex.index()] = aCoW_mask[vertex.index()] SAm.vector()[:] = SA_vertex_values[dof_to_vertex_map(AN)] Acerm = Function(AN) Acerm.set_allow_extrapolation(True) #work around--rounding error causes crash for vertex in vertices(mesh): Acer_vertex_values[vertex.index()] = Acer_mask[vertex.index()] Acerm.vector()[:] = Acer_vertex_values[dof_to_vertex_map(AN)] Basm = Function(AN) Basm.set_allow_extrapolation(True) #work around--rounding error causes crash for vertex in vertices(mesh): Bas_vertex_values[vertex.index()] = Bas_mask[vertex.index()] Basm.vector()[:] = Bas_vertex_values[dof_to_vertex_map(AN)] # periarterial space CoWm = Function(AN) CoWm.set_allow_extrapolation(True) #work around--rounding error causes crash for vertex in vertices(mesh): CoW_vertex_values[vertex.index()] = CoW_mask[vertex.index()] CoWm.vector()[:] = CoW_vertex_values[dof_to_vertex_map(AN)] PPm = Function(AN) PPm.set_allow_extrapolation(True) #work around--rounding error causes crash for vertex in vertices(mesh): PP_vertex_values[vertex.index()] = penperi_mask[vertex.index()] PPm.vector()[:] = PP_vertex_values[dof_to_vertex_map(AN)] # more arteries Mcerm = Function(AN) Mcerm.set_allow_extrapolation(True) #work around--rounding error causes crash for vertex in vertices(mesh): Mcer_vertex_values[vertex.index()] = Mcer_mask[vertex.index()] 199 Mcerm.vector()[:] = Mcer_vertex_values[dof_to_vertex_map(AN)] Pcerm = Function(AN) Pcerm.set_allow_extrapolation(True) #work around--rounding error causes crash for vertex in vertices(mesh): Pcer_vertex_values[vertex.index()] = Pcer_mask[vertex.index()] Pcerm.vector()[:] = Pcer_vertex_values[dof_to_vertex_map(AN)] Olfm = Function(AN) Olfm.set_allow_extrapolation(True) #work around--rounding error causes crash for vertex in vertices(mesh): Olf_vertex_values[vertex.index()] = Olfac_mask[vertex.index()] Olfm.vector()[:] = Olf_vertex_values[dof_to_vertex_map(AN)] # mark anatomical subdomaons materials = MeshFunction("size_t", mesh, mesh.topology().dim()) materials.set_all(0) class Ventricle(SubDomain): def inside(self, x, on_boundary): if vm(x[0], x[1], x[2]) > 0: return True else: return False subdomain1 = Ventricle() subdomain1.mark(materials, 1) class SA(SubDomain): def inside(self, x, on_boundary): if SAm(x[0], x[1], x[2]) > 0: return True else: return False subdomain4 = SA() subdomain4.mark(materials, 1) class Acer(SubDomain): def inside(self, x, on_boundary): if Acerm(x[0], x[1], x[2]) > 0: return True else: 200 return False subdomain5 = Acer() subdomain5.mark(materials, 1) class Bas(SubDomain): def inside(self, x, on_boundary): if Basm(x[0], x[1], x[2]) > 0: return True else: return False subdomain6 = Bas() subdomain6.mark(materials, 1) class PenPeri(SubDomain): def inside(self, x, on_boundary): if PPm(x[0], x[1], x[2]) > 0: return True else: return False subdomain3 = PenPeri() subdomain3.mark(materials, 3) class Mcer(SubDomain): def inside(self, x, on_boundary): if Mcerm(x[0], x[1], x[2]) > 0: return True else: return False subdomain7 = Mcer() subdomain7.mark(materials, 1) class Pcer(SubDomain): def inside(self, x, on_boundary): if Pcerm(x[0], x[1], x[2]) > 0: return True else: return False subdomain8 = Pcer() subdomain8.mark(materials, 1) class Olf(SubDomain): def inside(self, x, on_boundary): if Olfm(x[0], x[1], x[2]) > 0: return True 201 else: return False subdomain9 = Olf() subdomain9.mark(materials, 1) class CoW(SubDomain): def inside(self, x, on_boundary): if CoWm(x[0], x[1], x[2]) > 0: return True else: return False subdomain2 = CoW() subdomain2.mark(materials, 2) # save subdomains to file mfilename = 'sim_data_WT031417/mouse_matls_comp2_rev0924b01_rev1011c01_ppnv.pvd' matlsfile = File(mfilename) matlsfile << materials #----Begin loop generating conc simulations for various effective diffusivities err=np.zeros((len_DBT*len_DSPAS*len_DPPAS, 15)) err_ct = 0 for k in range(len_DBT): for l in range(len_DSPAS): for m in range(len_DPPAS): d_0 = DBT[k] d_1 = d_0 * 0.00000001 d_2 = DSPAS[l] d_3 = DPPAS[m] print('Diffusivities: ', d_0, d_2, d_3) # create files to save simulation concentration cfilename = 'Map/WT031417_concl_D'+str(d_0)+'Q100_SPAS'+str(d_2)+'_PPAS'+str(d_3)+'.pvd' cncfile = File(cfilename) # and difference between simulation and data for visualization delfilename = 'Map/WT031417_deltal_D'+str(d_0)+'Q100_SPAS'+str(d_2)+'_PPAS'+str(d_3)+'..pvd' delfile = File(delfilename) #----define diffusivity on subdomains------------------------------------------ class Diff(UserExpression): def __init__(self, materials, d_0, d_1, d_2, d_3, **kwargs): super().__init__(**kwargs) 202 self.materials = materials self.d_0 = d_0 self.d_1 = d_1 self.d_2 = d_2 self.d_3 = d_3 def eval_cell(self, values, x, cell): if self.materials[cell.index] == 1: values[0] = self.d_1 elif self.materials[cell.index] == 2: values[0] = self.d_2 elif self.materials[cell.index] == 3: values[0] = self.d_3 else: values[0] = self.d_0 def value_shape(self): return () AD = Diff(materials, d_0, d_1, d_2, d_3, degree=0) # mark boundaries def boundary(x, on_boundary): return on_boundary # define boundary condition # no flux boundary condition is the natural boundary condition 0*ds # define initial condition c_1 = interpolate(Constant(0.0), C) #----define finite element variational problem--------------------------------- c = TrialFunction(C) phi = TestFunction(C) a = phi*c*dx + dt*AD*inner(nabla_grad(phi), nabla_grad(c))*dx L = c_1*phi*dx A = assemble(a) b = assemble(L) # solve concentration problem # define functions c = Function(C) # Transfer a function in CD to a function in C using a transfer matrix, M: # used for calculating difference between simulation and data for visualization M = PETScDMCollection.create_transfer_matrix(CD,C) 203 cdc = Function(C) delta = Function(C) t = dt time_pt = 0 conc = np.zeros((nv, time_pts)) #---time stepping loop, solve dynamic mass transport PDE----------------------- while t <=T+.0000000001: b = assemble(L, tensor=b) # add or remove point source source_add = PointSource(C, src, Q) source_add.apply(b) if t >= 20.: source_remove = PointSource(C, src, -Q) source_remove.apply(b) # bc.apply(A, b) solve(A, c.vector(), b, "gmres", "ilu") # save concentration sim to array and to file at data time points if (t<10.5 and t>10.3): #save to file cncfile << c, t cdc.vector()[:] = M*cd1.vector() delta.vector()[:] = cdc.vector()[:] delta.vector()[:] = delta.vector()[:] - c.vector()[:] delfile << delta, t # save to array for error cacluation at end vertex_values_c = c.compute_vertex_values(mesh) conc[:, time_pt] = vertex_values_c[:] time_pt += 1 # new_bc = 'cd' + str(time_pt) # cb.assign(new_bc) if (t<20.9 and t>20.7): cncfile << c, t cdc.vector()[:] = M*cd2.vector() delta.vector()[:] = cdc.vector()[:] delta.vector()[:] = delta.vector()[:] - c.vector()[:] delfile << delta, t vertex_values_c = c.compute_vertex_values(mesh) conc[:, time_pt] = vertex_values_c[:] time_pt += 1 if (t<31.3 and t>31.1): cncfile << c, t cdc.vector()[:] = M*cd3.vector() 204 delta.vector()[:] = cdc.vector()[:] delta.vector()[:] = delta.vector()[:] - c.vector()[:] delfile << delta, t vertex_values_c = c.compute_vertex_values(mesh) conc[:, time_pt] = vertex_values_c[:] time_pt += 1 if (t<41.7 and t>41.5): cncfile << c, t cdc.vector()[:] = M*cd4.vector() delta.vector()[:] = cdc.vector()[:] delta.vector()[:] = delta.vector()[:] - c.vector()[:] delfile << delta, t vertex_values_c = c.compute_vertex_values(mesh) conc[:, time_pt] = vertex_values_c[:] time_pt += 1 if (t<52.1 and t>51.9): cncfile << c, t cdc.vector()[:] = M*cd5.vector() delta.vector()[:] = cdc.vector()[:] delta.vector()[:] = delta.vector()[:] - c.vector()[:] delfile << delta, t vertex_values_c = c.compute_vertex_values(mesh) conc[:, time_pt] = vertex_values_c[:] time_pt += 1 if (t<62.5 and t>62.3): cncfile << c, t cdc.vector()[:] = M*cd6.vector() delta.vector()[:] = cdc.vector()[:] delta.vector()[:] = delta.vector()[:] - c.vector()[:] delfile << delta, t vertex_values_c = c.compute_vertex_values(mesh) conc[:, time_pt] = vertex_values_c[:] time_pt += 1 if (t<72.9 and t>72.7): cncfile << c, t cdc.vector()[:] = M*cd7.vector() delta.vector()[:] = cdc.vector()[:] delta.vector()[:] = delta.vector()[:] - c.vector()[:] delfile << delta, t vertex_values_c = c.compute_vertex_values(mesh) conc[:, time_pt] = vertex_values_c[:] time_pt += 1 if (t<83.3 and t>83.1): cncfile << c, t 205 cdc.vector()[:] = M*cd8.vector() delta.vector()[:] = cdc.vector()[:] delta.vector()[:] = delta.vector()[:] - c.vector()[:] delfile << delta, t vertex_values_c = c.compute_vertex_values(mesh) conc[:, time_pt] = vertex_values_c[:] time_pt += 1 c_1.assign(c) t += dt #---calculate fit-------------------------------------------------------------- for i in range(nv): for j in range(time_pts): if exclude[i, j] > 0: conc[i, j] = 1. conc_m[i, j] = 1. err[err_ct, 0] = d_0 err[err_ct, 1] = d_2 err[err_ct, 2] = d_3 print('Diffusivities: ', d_0, d_2, d_3) err_mape, err_mpe = mape(conc_m[:,0:4], conc[:,0:4]) err_rms, err_rmspe = sim_rms(conc_m[:,0:4], conc[:,0:4]) err[err_ct, 3] = err_rms err[err_ct, 4] = err_rmspe err[err_ct, 5] = err_mape print('total exclude T2 error--t=10-50 min') print('exclude T2 mape = ', err_mape) print('exclude T2 rms = ', err_rms) print('exclude T2 rmspe = ', err_rmspe) err_mape, err_mpe = mape(conc_m[:,0:3], conc[:,0:3]) err_rms, err_rmspe = sim_rms(conc_m[:,0:3], conc[:,0:3]) err[err_ct, 6] = err_rms err[err_ct, 7] = err_rmspe err[err_ct, 8] = err_mape print('total exclude T2 error--t=10-40 min') print('exclude T2 mape = ', err_mape) print('exclude T2 rms = ', err_rms) print('exclude T2 rmspe = ', err_rmspe) err_mape, err_mpe = mape(conc_m[:,0:2], conc[:,0:2]) err_rms, err_rmspe = sim_rms(conc_m[:,0:2], conc[:,0:2]) 206 err[err_ct, 9] = err_rms err[err_ct, 10] = err_rmspe err[err_ct, 11] = err_mape print('total exclude T2 error--t=10-30 min') print('exclude T2 mape = ', err_mape) print('exclude T2 rms = ', err_rms) print('exclude T2 rmspe = ', err_rmspe) err_mape, err_mpe = mape(conc_m[:,0:7], conc[:,0:7]) err_rms, err_rmspe = sim_rms(conc_m[:,0:7], conc[:,0:7]) err[err_ct, 12] = err_rms err[err_ct, 13] = err_rmspe err[err_ct, 14] = err_mape print('total exclude T2 error--t=10-80 min') print('exclude T2 mape = ', err_mape) print('exclude T2 rms = ', err_rms) print('exclude T2 rmspe = ', err_rmspe) err_ct += 1 np.save('Map/WT031417_errcont_030521.npy', err)