Main

Mathematical modelling of blood perfusion and

oxygenation in microvascular networks with

applications in stroke research

Paul Sweeney

September 2014

University College London

Department of Mechanical Engineering

Supervised by Dr Rebecca Shipley

& Dr Simon Walker-Samuel

Submitted in part fulfilment of the requirements for the degree of

Master of Research in Mechanical Engineering of University College London

Abstract

Stroke is the sudden onset of paralysis caused by either a bleed or blockage within

a brain’s vasculature, which prevents transportation of oxygen to brain tissue (brain

hypoxia). Stroke victims can recover with no or few disabilities if they are treated

promptly. Understanding the processes of fluid and mass transport behind this illness

could allow for the development of more effective methods of treatment and improve

our fundamental understanding of the disease state. Diagnosis of stroke itself is clinical,

with assistance from medical imaging. However, there is no commonly used blood test

for diagnosis. Control of blood pressure has conclusively shown to prevent ischaemic

stroke [39], although once a ischaemic stroke is diagnosed, it is not clear whether lower

blood pressure is helpful towards treatment. Currently two options are available for

therapy, breaking down the blockage or by removing it mechanically.

The aim of this report is to use and develop a mathematical model which simulates

blood perfusion and oxygenation in microvascular networks. Simulations are run using

the vascular data of a mouse brain gathered by CABI† using medical imaging. A com-

parison of network fluid pressure and tissue oxygen distributions in healthy networks

versus one containing an ischaemic stroke at the microvascular level will be presented.

Imaging is a ever developing research area, but current techniques can already extract

blood vessels down to a resolution of 5 µm. With access to these incredibly detailed

data sets on vascular structure, modelling can provide a tool to extract functional in-

formation.

Chapter 1 gives a brief introduction to brain anatomy, with focusing turning towards

current models of hemodynamics based on in vivo and in vitro blood rheology descrip-

tions [31, 33, 34, 38]. Chapter 2 focuses on the development of an algorithm to retrieve

the largest subnetwork from an incomplete network dataset (a method required due to

an incomplete vascular casting), with later discussion turning to the implementation of a

geometry-based algorithm developed by Smith [49]. Chapter 3 presents the framework

behind mathematically modelling discrete (i.e. individual vessel) microvascular fluid

transport. Work by Fry et al. [16] is then presented which allows for an adaption of the

discrete model in the presence of incomplete boundary data. A method which can be

used on microvascular networks containing a large portion of unknown boundary con-

ditions, although this correlates to an increased run-time of the algorithm. Chapter 4

introduces the mathematics behind modelling oxygen transport to tissue using a Green’s

function method [45]. Chapter 5 presents the results of simulations on healthy and is-

†Centre for Advanced Biomedical Imaging, UCL.

2

chaemic stroke induced networks - the discrete model uses C++ code built in-house

(developing upon work by Rebecca Shipley‡), while the oxygen transport model uses

externally sourced code [43]. Later chapters discuss the results which shows a pressure

drop of ∼ 10 mmHg behind the ischaemic stroke and an increase of ∼ 17% of hypoxic

tissue in this region. Future work for this report will also be discussed.

‡Department of Mechanical Engineering, UCL.

3

Acknowledgments

I would like to thank my supervisors, Dr Rebecca Shipley and Dr Simon Walker-Samuel,

for their time, expertise, patience and supply of Starbucks tea/coffee, without which this

project would not be complete.

I would like to acknowledge and thank the funding provided by EPSRC.

I would like to thank my family for their constant encouragement and guidance

throughout my education.

Finally, I would like to thank Andy, Chris, Ed, Charly, Andrew and Ryan for their

unwavering support and humour throughout the past year.

4

Contents

1 Introduction 13

1.1 Motivations & Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2 Brain Anatomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.2.1 Arterial cerebral circulation . . . . . . . . . . . . . . . . . . . . . 15

1.2.2 Circle of Willis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.2.3 Brain drainage . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.2.4 Morphological features of intracortical arteries and veins . . . . . 17

1.3 Hemodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.3.1 Erythrocytes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.3.2 The Fahræus Effect . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.3.3 The Fahræus-Lindqvist Effect . . . . . . . . . . . . . . . . . . . . 21

1.3.4 Microvascular blood viscosity in vivo and the endothelial surface

layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.3.5 Phase Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.3.6 Blood flow measurements . . . . . . . . . . . . . . . . . . . . . . 25

2 Network Extraction & Analysis 26

2.1 Largest Subnetwork Algorithm . . . . . . . . . . . . . . . . . . . . . . . 27

2.2 Geometrical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3 Discrete Modelling of Microvascular Fluid Transport 33

3.1 Basic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2 Incorporating Variable Hematocrit . . . . . . . . . . . . . . . . . . . . . 35

3.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.4 Blood Flow Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4 Analysis of Oxygen Transport to Tissue 41

4.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2 Green’s Function Approach to Oxygen Transport . . . . . . . . . . . . . 44

4.2.1 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.2.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . 47

5 Mathematical Simulation of Fluid & Mass Transport 48

5.1 Flow Estimation Method of Fry et al. . . . . . . . . . . . . . . . . . . . 49

5.1.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5

5.2 Green’s Function Method for Oxygen Delivery . . . . . . . . . . . . . . 51

5.2.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

6 Discussion 54

7 Conclusion 58

Bibliography 59

6

Nomenclature

αp Oxygen solubility in blood plasma

αeft Effective solubility of oxygen in blood

αRBC Oxygen solubility in RBCs

δ Delta function

∅ Empty set

ε Aspect ratio

λ Langrange multiplier

µabs Effective blood viscosity

µvitro In vitro relative apparent blood viscosity

φ Sink strength of tissue

τj Wall shear stress in segment j

τ0j Target shear stress at segment j

µm Micrometres

cP Centipoise

dyn Dyne

oC Celsius

mmHg Millimetre of mercury

α Solubility of oxygen in tissue

µvivo In vivo effective blood viscosity

φ Concentration

θ RBC concentration field

A Bifurcation law variable

7

B Bifurcation law variable

C Curvature relationship between relative apparent viscosity and hematocrit

c Shear stress variable

D Diffusive coefficient of oxygen in tissue

D Vessel diameter

d Vessel diameter or daughter vessel

Dr Critical parent-daughter diameter ratio

d50 Parameter from the optimisation procedure for modelling ESL

dcrit Parameter from the optimisation procedure for modelling ESL

doff Parameter from the optimisation procedure for modelling ESL

Dph Diameter of the region in which the blood can flow freely in a vessel

dA Capillary cross-section increment

Eamp Parameter from the optimisation procedure for modelling ESL

EHD Parameter from the optimisation procedure for modelling ESL

Epeak Parameter from the optimisation procedure for modelling ESL

Ewidth Parameter from the optimisation procedure for modelling ESL

f Rate of convective oxygen transport

G Green’s function

Gtv Matrix of tissue PO2 at the midpoint of tissue region resulting from a unit source

at the midpoint of another segment

Gvt Matrix of tissue PO2 at the midpoint of a segment resulting from a unit source

at the midpoint of a tissue region

Gvv Matrix of average tissue PO2 at the surface of a segment as a result of a unit

source distributed on the surface of another segment

H Flow estimation matrix

HD Discharge hematocrit

8

Hi Discharge hematocrit from vessel i

HT Tube hematocrit

hChange Change in hematocrit between old and new values

hInit Initial hematocrit value

hOld Old hematocrit value

hTol Hematocrit tolerance

iterMAX Maximum number of convergence loop iterations

J Diffusive flux

K Matrix forming the sparse linear system or a constant representing intravascular

resistance to radial oxygen transport

kτ Weighting factor

kp Weighting factor

kτ,final Final value of kτ

kτ,init Initial value of kτ

L Start/end node segment matrix

l Vessel length

M Fluid conductance or oxygen consumption rate

M0 Uniform oxygen demand in tissue given a non-limiting supply of oxygen

MI Matrix which stores all subnetworks

N Set of all vessel nodes

nD Number of daughter vessels

ni End node of daughter i

nS Total number of segments

nBC Total number of boundary nodes

P Set of segments which have been parents or PO2

p Parent vessel or fluid pressure

9

P0 PO2 at half-maximal consumption

Pb Blood PO2 level

Pv Average tissue PO2

p0k Target pressure at node k

P50 PO2 at 50% saturation

Q Flux

q(x) Source strength at x

qv Rate of diffusive oxygen efflux

Q0i Boundary node vector

qChange Change in flux between old and new values

qTol Flux tolerance

r Radial component

RN Set of reached nodes

RS Set of reached segments

Re Reynolds number

S Set of all vessel segments or oxyhemoglobin saturation

s Arc length down the central axis of a vessel

Si Segment with ith boundary node

T Set of tissue points

u Horizontal velocity component

v Blood velocity field

wk Vessel length associated with node k

Was Function in the modelling of the ESL

Weft Function describing the effect thickness of the ESL

Wmax Maximal ESL thickness

10

Wpeak Function in the modelling of the ESL

Wph Function describing the effect thickness of the ESL

X0 Bifurcation law variable

q Flux

11

List of Figures

1.1 Superficial dissection of the right side of the neck & the Circle of Willis 15

1.2 Superior sagittal sinus & sagittal section of the skull . . . . . . . . . . . 16

1.3 Morphological features of intracortical arteries and veins . . . . . . . . . 17

1.4 General view of cortical vessels & coiling of an artery . . . . . . . . . . . 18

1.5 Blood constituents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.6 Micrographs of RBCs flowing through small-bore glass tubes & mesenteric

microvessels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.7 Possible causes for high resistance in small microvessels . . . . . . . . . 22

2.1 A simple discrete network . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2 Frequency distributions of network diameters and lengths . . . . . . . . 26

2.3 3D visualisation of the collected mouse brain network . . . . . . . . . . 27

2.4 Initial stages of the geometry-based algorithm of Smith [49] . . . . . . . 28

2.5 Later stages of the geometry-based algorithm of Smith [49] . . . . . . . 30

2.6 Geometrical analysis of the rate mesentery network . . . . . . . . . . . . 32

2.7 Network classification comparison . . . . . . . . . . . . . . . . . . . . . . 32

5.1 Largest subnetwork, further network extraction & network reduction . . 49

5.2 2D representations of discrete flow solutions . . . . . . . . . . . . . . . . 50

5.3 Pressure distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.4 2D planar slice representing PO2 . . . . . . . . . . . . . . . . . . . . . . 52

5.5 PO2 frequency distributions . . . . . . . . . . . . . . . . . . . . . . . . . 53

6.1 Tumour morphology and tumour vascularisation . . . . . . . . . . . . . 56

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1 Introduction

1.1 Motivations & Objectives

The subject of stroke is prevalent in today’s society where 1 in 6 people around the world

will suffer from a stroke in their lifetime [1]. In the UK alone, there are approximately

152,000 strokes every year, with 1 in 5 being fatal [1]. Stroke is a sudden and devastat-

ing illness but by combining mathematical modelling with medical imaging, we aim to

better identify and treat damaged sections of the brain and improve our fundamental

understanding of the mechanical and transport processes at play.†

The aim of this thesis is to use and combine existing mathematical models which

simulate blood perfusion and oxygenation in microvascular networks. Simulations will

be run on a healthy network and a ischaemic stroke induced network - both are identical

with the exception of reducing the diameter of a microvessel to simulate a ischaemic

stroke. A comparison in the spatial distribution of hypoxia and blood pressures between

the two networks will be made in order to try and understand the changes in network

environment caused by stroke.

A novelty of this report’s work is the use of brand new datasets. Structural data of

the mouse brain was acquired using a vascular casting technique with the resulting cast

scanned using Micro-CT. Current techniques can already extract blood vessels down to a

resolution of 5 µm, whereas brain capillaries can be as low as 3 µm [12]. Therefore, using

this method, an incomplete vascular network was collected - hence the use of the flow

estimation algorithm of Fry et al. [16] in the modelling. The modelling also makes use

of MRI fluid distribution data. A key motivation behind this report is that imaging can

only get flow information on a coarse scale, whereas the mathematical model can predict

fluid flow in individual vessel. Detailed information on oxygenation is also not currently

available using imaging - data which could prove vital in the treatment of stroke. Thus,

the modelling provides a tool to take the structural and fluid flow information to predict

blood flow and oxygenation.

The mathematical modelling process uses these data to parameterise the models and

also to compare against the model predictions of fluid distribution. A 1D Poiseuille flow

model, which describes a linear relationship between the flux and the pressure gradi-

ent along a vessel segment, is used, combined with in vivo blood rheology descriptions

such as the Fahræus-Lindqvist effect [34]. The oxygen transport model includes oxy-

gen bound to erythrocytes, dissolved within blood plasma, diffusion through the tissue

†Extract taken from Sweeney et al. [51].

13

and uptake by the cellular population [45]. The flows in major feeding and draining

vessels can be measured by MRI or estimated from literature values of brain perfusion;

this information parameterises boundary conditions for the larger vessels. Boundary

conditions for terminal micro vessels remain unknown but are required for a fully de-

termined linear system for the networks fluid pressures. An algorithm by Fry et al. [16]

which minimises the deviation of pressures and wall shear stresses from specified target

values, resolves this indeterminacy. Oxygen partial pressure levels are needed on the in-

flow boundary nodes in the oxygen transport model, these figures are based on current

literature. The resulting models and boundary conditions form a system of ordinary

differential equations representing conservation of mass and momentum, which are then

solved using numerical techniques in C++.

An ischaemic stroke is caused by a sudden blockage of blood flow to the central nervous

system. The most common causes are by thrombosis (a blood clot within a vessel) or

embolism (a foreign object such as plaque or a broken off section of a blood clot which

has been carried in the bloodstream). A loss of blood behind the stroke creates a hypoxic

environment which can lead to necrosis (cell death). The location of the ischaemic stroke

can have drastic effects, however, the size of the hypoxic region does not necessarily

correlate to the magnitude of the neurological damage. For example, a very small area

of hypoxia in the brainstem can be more devastating compared to relatively large damage

to the cerebellum or cerebral hemispheres [42]. Current therapies for stroke are designed

to restore blood flow to the hypoxic region as soon as possible after the blockage event.

However, even if the blockage is removed, the capillaries may be too damaged for efficient

perfusion to the surrounding tissue, or muscle cells may have undergone necrosis due to

a lack of oxygen. This is known as the no-reflow phenomenon. Hence, time is a crucial

factor when treating stroke.

In order to simulate an ischaemic stroke, a healthy simulation is taken and then an

individual microvessel is blocked. This is simply done by reducing the chosen microvessel

diameter to ∼ 0µm. A comparison in the spatial distribution of hypoxia and blood

pressures between the healthy and stroke induced networks will be made. With access to

incredibly detailed data sets on vascular structure collected through imaging techniques,

functional information can be extracted from these simulations which can be used for

stroke identification and treatment.

14

1.2 Brain Anatomy

(a) Superficial dissection of theright side of the neck, showingthe carotid and subclavian arter-ies.

(b) Diagram of the Circle ofWillis. This is regarded as thenormal configuration, however,fewer than half have this shape.

Figure 1.1: Reproducedfrom Gray [19].

The brain can be divided lengthwise down its middle

into what are known as the cerebral hemispheres. Each

hemisphere is subdivided into four lobes (frontal, oc-

cipital, parietal and temporal) by sulci and gryi (the

grooves and bumps on the surface of the brain). Within

the fluid-filled ventricles (the cavities between the lobes)

is the cerebrospinal fluid (CSF), whose function is to act

as a buffer, reduce pressure at the base of the brain, ex-

crete waste products and act as a endocrine medium for

the brain. Absorption of CSF into the blood stream can

occur in the superior sagittal sinus when CSF pressure is

greater than venous pressure, however, diffusion is only

unidirectional.

Metabolically, the brain is very active. A stable and

copious blood supply is require as the brain has no ef-

fective means of storing oxygen or glucose - the brain

accounts for 25% of the body’s oxygen consumption

[42]. The overall flow rate is maintained at a con-

stant level but may vary in certain regions depending

on neural activity. Currently the mechanisms control-

ling cerebral blood flow are not fully understood, but

there is general agreement that the following methods

are used. The first process is called autoregulation,

in which blood vessel themselves act to maintain con-

stant flow by constricting (therefore increasing flow re-

sistance) in response to increased blood pressure and

relax due to a decrease in pressure (decreasing flow re-

sistance). The second mechanism is a local response by

neurones, astrocytes and cerebral vessels to increased

neural activity.

1.2.1 Arterial cerebral circulation

Arterial cerebral circulation can be divided into two

categories, anterior and posterior cerebral circulation.

Anterior cerebral circulation is supplied by the internal

carotid artery (80% of supply) which bifurcates into the

15

anterior and middle cerebral arteries. Before bifurcation occurs, the internal carotid

artery gives rise to two smaller branches the anterior charoidal and posterior communi-

cating arteries. The posterior cerebral circulation is supplied by the vertebral arteries

(remaining 20% supply to the brain), which consists of the basilar, posterior cerebral

and posterior communicating arteries.

1.2.2 Circle of Willis

The Circle of Willis is an arterial polygon formed by the connection of the posterior

cerebral and the internal carotid arteries (see Figure 1.1b). Normally little blood flows

through the Circle of Willis since the fluid pressures in the internal carotid arteries are

approximately the same as that in the posterior cerebral arteries, leading to little blood

flow through the posterior communicating arteries. However, if a major vessel becomes

occluded over a sustained period of time, it can provide an alternative route to prevent

neurological damage. However, since the anterior and posterior communicating arteries

vary largely in size, in the case of an abrupt blockage, blood vessels my rupture.

1.2.3 Brain drainage

The principle route for venous drainage of the brain is through a system of cerebral

veins that empty into the dural venous sinuses and ultimately into the internal jugular

veins. This is often considered the only significant route through which venous blood

can leave the head, however, only in a supine position. Otherwise, the internal jugulars

collapse and blood leaves through the vertebral venous plexus instead.

Cerebral veins are conventionally divided into two group, superficial veins which lie on

the surface of the cerebral hemispheres (emptying into the superior sagittal sinus), and

(a) Superior sagittal sinus laid openafter removal of the skull cap. (b) Sagittal section of the skull.

Figure 1.2: Reproduced from Gray [19].

16

deep veins which drain internal structures (emptying into the straight sinus). Superficial

veins are quite variable and only three veins are reasonably constant from one brain

to another - the superficial middle cerebral vein, the superior anastomotic vein and

the inferior anastomotic vein. On the other hand, deep veins are more constant in

configuration.

Note, in addition to superficial and deep veins, there is a separate and complex col-

lection of veins that serves the cerebellum and brainstem.

1.2.4 Morphological features of intracortical arteries and veins

The vascular network of a brain contains complex and dense system of branch and

mesh-like blood vessels. Duvernoy et al. [12] categorised intracortical vessels into six

groups:

1. Group 1 - composed of vessels which reach the molecular layer (layer I) with the

possibility of extending into the external granular layer (layer II).

2. Group 2 - continues into the superficial part of the pyramidal layer (layer III a &

b).

3. Group 3 - are the most numerous vessels, which penetrate as far as the high

vascularised middle region of the cortex (centred in the internal granular layer IV

while overlapping in the pyramidal, III c and ganglionic, V a, layers).

Figure 1.3: Morphological features of intracortical arteries and veins. Arteries and veinsare divided into six groups, Ai and Vi where i = 1, . . . , 6, respectively. Reproduced fromDuvernoy et al. [12].

17

(a) General view of cortical vessels.(1) Cortical arteries. (2) Cortical vein.(3) Medullary artery (type A6). (4)Subcortical white matter. (b) Coiling of an artery of type A3.

Figure 1.4: Reproduced from Duvernoy et al. [12].

4. Group 4 - these vessels reach the multiform layer and inner limit of the subcortical

white matter.

5. Group 5 - vessels are comprised of both arteries and veins which pass through the

cortex and adjacent white matter.

6. Group 6 - a special group of large arteries which pass through the grey matter

without branching and supply blood only to the white matter.

Current literature conforms with morphological analysis in that vasculature takes

the form of tree (arteries/veins) and mesh (capillaries) like structures [26]. Chapter

2.2 uses these branching and looping formations to create an algorithm that analyses

a microvascular network’s topology and distinguishes between arterioles, venules and

capillaries. Our aim is to apply this algorithm to a dense brain network in order to

distinguish between arteries, veins and capillaries.

Work by Duvernoy et al. [12] only serves to show the complexities of aiming to simulate

blood perfusion in an entire brain network. Not only will the process be computationally

demanding due to the shear vascular density of brain networks (see Figure 1.4a) but

potentially challenging in terms of imaging due to various topological structures such as

coiling of arterial branches (see Figure 1.4b).

1.3 Hemodynamics

A vital component of modelling blood perfusion through vascular networks is using bio-

logically realistic mathematical descriptions of blood rheology. Throughout the human

18

Figure 1.5: Blood constituents. Reproduced and altered from [7].

body blood flow displays different characteristics, from inertia driven flow in the aorta

to the highly non-Newtonian, viscous driven flow at the capillary scale. Mathematical

modelling of these properties is currently on going. Presently, In vitro and in vivo mea-

surements are used to further our understanding of these mechanisms governing fluid

transport.

The following sections will discuss the fundamental biological factors in the study

of hemodynamics which become particularly important at the non-Newtonian capillary

scale. The mathematical descriptions will also be presented in order to implement them

in the discrete fluid transport model presented in Chapter 3.

1.3.1 Erythrocytes

Blood is a fluid vital in sustaining life in animals. The bodily fluid delivers integral

nutrients and oxygen to cells and transports metabolic waste products away from said

cells. Blood consists of erythrocytes (red blood cells or RBCs), thrombocytes (platelets),

leukocytes (white blood cells) and plasma. Blood plasma constitutes ∼ 55% of blood

volume and is essentially∼ 92% water, with the remaining∼ 8% being dissolved proteins

and trace amounts of other minerals. RBCs take up a much higher volume within the

blood when compared to other cells - normal human blood has a hematocrit (the volume

percentage of RBCs contained in blood - also known as packed cell volume or PVC) in

the range of 37− 48% for women and 42− 52% for men [4].

In vertebrates, RBCs are the principle means of delivery oxygen to tissue. Uptake

of oxygen happens within the lungs or gills and is released into surrounding tissue

while squeezing through capillaries. RBCs are biconcave disks approximately 8µm in

diameter. The disk structure is very flexible, so much so that when entering into very

thin capillaries, they deform into bullet-like shapes with approximately a minimum

19

Figure 1.6: Micrographs of RBCs flowing through small-bore glass tubes (those on theleft) and mesenteric microvessels (those on the right). In the vessel diameter range of∼ 6 − 13µm, cells may flow in single file at low hematocrits, HD. With increasinghematocrit, more cells are present in a vessel cross-section, leading to a transition tomulti-file flow and increased interaction of cells with the wall. Reproduced from Priesand Secomb [34].

diameter of 2.8µm [46]. Hence, the size of a RBC is relative to the capillary diameter

and has a significant impact on the apparent viscosity of the blood, and therefore, its

non-Newtonian characteristics. The non-Newtonian nature of blood is thought to be

important in vessels of diameters less than 300µm [37].

In large vessel or a high hematocrit, the interactions between cells, deformation of

said cells, combined with plasma and vessel wall interactions are all integral parts of

fluid transport. Whereas, in capillaries of less than 6µm, RBCs are forced into single-

file flow and so assumptions can be made, such as Lubrication Theory to model blood

plasma between the cells and cell wall, or assuming an axisymmetric erythrocyte shape.

Attempts have been made to model individual cells by assuming axisymmetric cells [46]

or by focusing attention to membrane deformation [13]. However, three-dimensional

simulations of more than a few cells is very computationally expensive and so currently

is not feasible. Alternatively, continuum descriptions can be used.

1.3.2 The Fahræus Effect

The Fahræus effect is the apparent decrease of hematocrit with decreasing capillary di-

ameter [14]. The velocity profile of Poiseuille flow (see Chapter 3 for details) is parabolic,

with no-slip boundary conditions at the walls (zero velocity condition) and has maximal

velocity down the vessel centre. Hence, due to the biological properties of blood, RBCs

travel along the centre of the vessel faster than the lubricating plasma layer adjacent to

20

the vessel walls.

A parametric description of the reduction of tube hematocrit, HT , relative to discharge

hematocrit, HD, captures the phenomenon of the Fahræus effect. The tube hematocrit

is the cross-sectionally averaged RBC concentration and is defined as

HT =1

A

∫θdA, (1.1)

where θ is the RBC concentration field and A is the capillary cross-section. The discharge

hematocrit is defined as the cross-sectionally averaged RBC concentration weighted by

blood velocity and thus defined as

HD =

∫vθdA∫vdA

, (1.2)

where v is the blood velocity field.

Based on experiments in which human RBC suspensions were perfused through glass

tubes with varied diameters, Pries et al. [38] derived an empirical relationship between

tube and discharge hematocrit. Mathematically the relationship can be described as

HT

HD= HD + (1−HD) · (1 + 1.7e−0.415d − 0.6e−0.011d), (1.3)

where d is the tube diameter in µm. Assuming this formula can be applied in vivo,

Equation (1.3) can be used to calculate discharge hematocrit, which is needed for flow

simulations.

1.3.3 The Fahræus-Lindqvist Effect

Another biological characteristic of blood flow is the Fahræus-Lindqvist effect, which

is the apparent change of the viscosity of blood with the diameter of the blood vessel

it is travelling through. Through in vitro experiments, Fahræus and Lindqvist [15]

demonstrated the decrease of apparent blood viscosity with decreasing (thin) glass tube

diameters, ranging from 300µm to ∼ 10µm.

Pries et al. [33] developed an in vitro experimentally-based empirical formulae to

describe this non-Newtonian behaviour. This empirical law predicts relative apparent

viscosity as a function of tube diameter d (µm) and discharge hematocrit, HD:

µvitro(d,HD) = 1 + (µ0.45(d)− 1) · (1−HD)C − 1

(1− 0.45)C − 1, (1.4)

where

µ0.45(d) = 220e−1.3d + 3.2− 2.44e−0.06d0.645

, (1.5)

21

Figure 1.7: Schematic drawings showing possible reasons for high resistance in smallmicrovessels. Reproduced from Pries et al. [38].

is the apparent viscosity when HD = 0.45 and C is given by

C(d) = (0.8 + e−0.075d) ·(

1

1 + 10−11 · d12− 1

)+

1

1 + 10−11 · d12. (1.6)

Here, C describes the curvature of relationship between relative apparent viscosity and

hematocrit with tube diameter.

1.3.4 Microvascular blood viscosity in vivo and the endothelial

surface layer

The Fahræus-Lindqvist effect exhibits a distinctive minimum at 6 − 7µm. However,

in vivo flow resistance in small vessels is substantially higher than predicted in vitro

viscosity data. The disparity between in vitro and in vivo data is unclear, although the

endothelial surface layer (ESL) is currently thought to be the most likely factor causing

this discrepancy.

The ESL is the interface between the endothelium (vessel wall) and the blood flowing

through the vessel lumen. It is estimated the ESL has a thickness ranging from 0.5µm

to over 1µm and is composed of soluble plasma components lined with a glycocalyx, a

layer of membrane-bound macromolecules. It is hypothesised that the presence of ESL

blocks RBCs and impedes plasma flow, which accounts for the increase in flow resistance

in vivo compared to in vitro.

A physical model, which fits in vivo data more closely compared to the previous in

vitro model, was proposed by Pries and Secomb [34] using data gathered from the rat

mesentery. Flow resistance is described as a function of vessel diameter and hematocrit

in vivo by combining in vitro blood viscosity with effects of diameter dependent ESL.

22

The effective viscosity was defined as

µvivo(d,HD) = µvitro(Dph, HD) ·[

d

d− 2Weff (d,HD)

]4, (1.7)

where Dph describes the diameter of the region in which the blood can flow freely and

is defined as

Dph(d) = d− 2Wph(d). (1.8)

The functions Weff and Wph describe the effect thickness of the ESL (dependent on HD)

and the physical thickness of the ESL, respectively. Mathematically, they are defined as

Weff (d,HD) = Was(d) +Wpeak(d) · (1 +HD · EHD) (1.9)

and

Wph(d) = Was(d) +Wpeak(d) · Epeak. (1.10)

With the following defined variables,

Was(d) =

{0, if d ≤ doff ,

d−doffd+d50−2doff ·Wmax, if doff < d,

(1.11)

and

Wpeak(d) =

0, if d ≤ doff ,

Eamp ·d−doff

dcrit−doff , if doff < d ≤ dcrit,Eamp · e−Ewidth(d−dcrit), if dcrit < d,

(1.12)

where Wmax = 2.6µm (the maximal layer thickness).

The following parameters were obtained using an optimisation procedure which min-

imised the deviation between measured and modelled flow velocities,

doff = 2.4µm, d50 = 100µm, dcrit = 10.5µm,

Epeak = 0.6, Eamp =1.1, Ewidth = 0.03, EHD = 1.18(1.13)

The effect viscosity is then found using the following

µabs(d,HD) = µvivo(d,HD)× µp, (1.14)

where µp = 1.05 cP is the viscosity of blood plasma for a rat at 37oC [35].

23

1.3.5 Phase Separation

The distribution of RBCs and plasma at microvascular bifurcations is a significant phe-

nomenon to be included in modelling of blood flow. At a branching point, the direction

a RBC takes is highly dependent upon red cell screening, the process in which the

balance of fluid forces determines the destination of the RBC. The effect of red cell

screening increases as the vessel diameter decreases. Through experimental simulation,

phase separation has been shown to be most effected in vessels of diameter lower than

30µm and that branching angles at bifurcations is not thought to be important [31].

Daughter branches with lower fractional flow have lower hematocrit than the feeding

vessel. Those with very low flow may experience plasma skimming, the process in which

small branching vessels skim off the cell-depleted layer along the wall of the parent vessel

and so these daughter vessels only contain plasma.

To simulate phase separation, conservation of RBC flux needs to be enforced at each

vessel node in addition to conservation of total flux. Pries et al. [31] derived a non-linear

empirical formula with modifications to render predictions more robust for extreme

combinations of input hematocrit and vessel diameter [34]. The fractional flow of RBCs

into one daughter branch, is calculated from the respective fractional blood flow. The

following expression is used to calculate Hα, the discharge hematocrit in daughter branch

α,

ln

(HαQα

HfQf −HαQα

)= A+B · ln

(Qα/Qf −X0

Qβ/Qf −X0

), (1.15)

where Q is the flux - subscripts f , α and β represent the feeding parent vessel and its

two daughter vessels. The bifurcation law’s (Equation (1.15)) variables are defined as,

A = −13.29 ·[

(dα/dβ)2 − 1

(dα/dβ)2 + 1

]·

1−Hf

df, (1.16)

B = 1 + 6.98 ·1−Hf

df, (1.17)

and

X0 = 0.964 ·1−Hf

df, (1.18)

where dk (µm) where k = f, α, β, is either the diameter of the parent or a daughter

vessel. The discharge hematocrit in daughter branch β is calculated using conservation

of RBC flux and is given by,

Hβ =HfQf −HαQα

Qβ. (1.19)

To conclude, empirical laws have been introduced which describe non-Newtonian

24

capillary-scale blood flow characteristics - the in vivo viscosity law, Equation (1.14),

and the bifurcation law given by Equation (1.15). These laws will be later combined to

derive discrete (i.e. individual vessel) flow solutions in brain networks.

Schmid et al. [41] researched the impact of RBCs on the flow in cortical microvascular

vasculature and the influence of capillary dilation on the distribution of these erythro-

cytes. Numerical simulations were executed on both artificial and real microvascular

networks, with focus towards analysing flow separation at divergent bifurcations. Based

on their results, RBCs strongly influence the ratio of outflow rates at divergent bifur-

cations, such that in ideal cases the flow rates are approximately equal in both outflow

branches. Comparatively, the distribution of erythrocytes due to capillary dilation does

not significantly change flow rates, but results in an accumulation of erythrocytes and

a reduced flow velocity in the dilated capillary. Whereas in an ideal network (well-

balanced network), flow in in the surrounding vessels is hardly affected by capillary

dilation. Thus, they concluded that optimal capillary networks are well-balanced, since

this increases their ability to adapt to local changes.

1.3.6 Blood flow measurements

The blood rheological phenomena previously discussed were discovered via the study of

microvascular networks within the rat mesentery. Application of these empirical laws

for modelling brain blood flow requires confirmation using imaging. Detre et al. [9]

formed a study with a purpose to use non-invasive MRI to evaluate cerebral blood flow

(CBF) in a cohort of patients suffering from a cerebrovascular disease (either a transient

ischaemic attack or a severe carotid stenosis). Non-invasive assessment of CBF was

done by MRI using magnetic labelling of arterial water as a diffusible flow tracer (CASL

technique). However, resolution does not allow for individual blood flow measurements

at the microvascular scale.

Dunn et al. [11] studied dynamic imaging of CBF using laser speckle. The method

was applied to CBF imaging during pathophysiologic events such as focal ischaemia and

cortical spreading depression. The technique can be used for accurate, high-resolution

imaging of CBF, thus allowing for individual measurement of blood flow at the capillary

scale. However, due to the density of brain networks and the scale of simulations we

intend to produce, using just one imaging technique to confirm simulations at both

macro and micro scales may not be effective and so multiple techniques may be used.

25

2 Network Extraction & Analysis

In this chapter, network extraction is briefly discussed along with the development of

an algorithm for extracting subnetworks (a process needed due to incomplete casting

perfusion and resolution limitations of Micro-CT) and the implementation of a geometry-

based algorithm by Smith [49], which classifies blood vessels in terms of arterioles,

capillaries and venules.

Figure 2.1: Asimple discretenetwork contain-ing three vesselsconnected byfour nodes. Rednodes representboundary nodes.

Structural data of the vasculature of the mouse brain was ob-

tained by Angela D’Esposito† through a novel method called vascu-

lar casting, a process which involves an invasive method of injecting

a casting agent into the vascular system of the mouse. The brain

was then extracted and scanned using Micro-CT. The resolution of

Micro-CT is 5 µm, given capillaries can be 3 µm in diameter, a full

dataset of the mouse brain is not expected. Analysis of vessel diam-

eters (see Figure 2.2a) shows a large number of diameters close to

the mean of 6.61 µm, but two diameters were found to be 2.5 µm,

suggesting a potential error the segmentation and skeletonisation

step.

The process of segmentation and skeletonisation of the resulting

dataset was done by Simon Walker-Samuel†. Segmentation of the

network allows for application of the discrete fluid transport model.

During this process the vascular network is sectioned into a series

(a) (b)

Figure 2.2: Plots of the frequency (logarithmically scaled) distributions of diametersand lengths (µm).

†Member of CABI, UCL.

26

(a) (b)(c)

Figure 2.3: 3D visualisation of the collected mouse brain network. (a) The full dataset.(b) & (c) the largest subnetwork. Vessels are to scale and colour is diameter dependent.

of vessel segments (sections of blood vessels that contain no bifurcations) which are

connected by by nodes (junctions between vessels - these internal nodes can connect

multiple segments) or in the case of a boundary vessel, a segment has a boundary node

(no connection to another segment at a boundary node i.e. a node connected to only

one segment) - see Figure 2.1. Frequency distributions of vessel diameters and lengths

can be seen in Figure 2.2. Diameters ranged from 2.5 to 67.4 µm with a mean of 6.61

µm. Whereas lengths ranged from 2 to 154 µm with a mean of 12.7 µm.

In order to parametrise the discrete fluid transport model in terms of boundary condi-

tions, phase contrast MRI was used to gain average blood flow velocities for the internal

carotid arteries (obtained by Rajiv Ramasawmy†). The boundary conditions are dis-

cussed in further detail in Section 3.3, but currently the mathematical modelling of these

networks is hindered by the lack of detailed flow information.

2.1 Largest Subnetwork Algorithm

Due to resolution limitations of Micro-CT and potential incomplete perfusion of the

casting agent around the mouse brain, analysis showed a scattering of disconnected

subnetworks. Perfusion occurred in arteries, some arterioles and very few capillaries,

therefore, none occurred in the venous side of the brain (confirmed by vessel diameters

and morphological comparisons). Hence, the next step was to design an algorithm to

extract the largest subnetwork (i.e. the subnetwork with the largest number of vessel

segments) in the dataset (see Algorithm 1) in order to have a coherent structure to

run discrete fluid flow simulations on. To reduce the run time of Algorithm 1 isolated

(single) vessels segments were removed from the dataset along with duplicate vessel

nodes (which connect vessel segments) - an artefact from segmentation.

The largest subnetwork algorithm starts by iterating through all boundary nodes. For

the ith boundary node, the segment connected to said node, Si is found and added to the

27

set of reached segments, RS . The algorithm then checks to see if Si has been connected

to a previously found subnetwork, if it has, the algorithm iterates to the next boundary

node. Otherwise, the convergence loop (see Algorithm 1) starts to iterate for a preset

number of iterations. Inside the loop, all daughter branches, Sk, of segments contained

within RS but are not members themselves are found and then added to RS . The loop

continues to find each generation of daughter branches until no vessel segments are being

added to RS - the algorithm has converged on the entire subnetwork. At this point the

subnetwork is stored in the matrix MI . Once all boundary nodes have been iterated

through, the largest subnetwork is then extracted from MI . MI itself is a matrix of

size nBC ×nS , where nBC and nS are the total number of network boundary nodes and

segments, respectively. The matrix acts as a ”flagging” system for identifying which

segments are assigned to the subnetwork associated with a boundary node. From here,

network data such as diameter and length distributions can easily be extracted.

Note, the largest subnetwork algorithm was tested on the mesentery networks of Pries

et al. [32]. For this report the largest subnetwork was wanted, however, this algorithm

can be easily adapted to find, say, the longest subnetwork or the subnetwork containing

the highest volume of blood.

2.2 Geometrical Analysis

When analysing the vascular network of a brain it is useful to be able to categorise the

vessels in terms of arterioles, venules and capillaries as in future work (see Discussion),

each will be treated using a different mathematical model. A novel geometry-based al-

gorithm by Smith [49] characterises the distinction between branching arterioles/venules

and mesh-like capillaries, the transition from the different topologies is marked by iden-

tifying loops within the network. The algorithm can be applied to an arbitrary network

nN nodes and nS segments, each with a given vessel diameter, Dj .

The pseudo code for the geometry-based algorithm is presented in Algorithm 2. The

algorithm starts by selecting the main feeding arteriole, which, if pressures are known, is

(a) (b)

Figure 2.4: Initial stages of the geometry-based algorithm applied to rat mesentery databy Pries et al. [32]. (a) the 1st and (b) the 3rd iteration. Figure has been reproducedfrom Smith [49].

28

Algorithm 1 Largest subnetwork algorithm.

Set iterMAX . . iterMAX - maximum number of iterations for convergence loop.Initialise MI = ∅. . MI - a matrix which stores a record of all subnetworks.for i = 1 : nBC do . nBC - number of boundary nodes.

Initialise RS = {Si}. . RS - the set of reached segments.. Si - segment who’s start node is the ith boundary node.

if Si /∈MI then . If Si has not been recorded in a previous subnetwork.ConvergeCounter = 0.for j = 1 : iterMAX do . Convergence loop.

Counter = 0.for k = 1 : nS do

if Sk is a daughter of segment in RS and Sk /∈ RS . thenAdd Sk to RS .

end ifend forSet Counter. . Counter - the number of segments in RS .if Counter = ConvergeCounter then

Terminate j loop.else

ConvergeCounter = Counter.end if

end forend ifStore subnetwork in MI .Set RS = ∅.

end forExtract largest subnetwork from MI .

the highest nodal pressure. This vessel segment is then classed as a parent and added to

the set of segments that have been classed as parent vessel, P , and the reached segment

set RS (likewise its corresponding nodes are added to the set of reached nodes, RN ).

On the ith iteration, parent segment pi is found to be the maximal diameter vessel

available (promoting the characteristic that arterioles or venules have larger diameters

when compared to capillaries) that is contained in RS but has not been previously classed

as a parent - pi and its corresponding end node is added to P and RN respectively. Its

corresponding daughter vessels, dj , are then found and added to RS .

The parent and daughter vessels are then classified by the following conditions:

1. If the end node, nj , of the daughter vessel, dj , is contained within the set of reached

nodes, RN , dj is a capillary.

2. If the above condition is satisfied and the diameter of the parent vessel, pi, is

smaller than the critical parent-daughter diameter ratio multiplied by the diameter

of the daughter vessel (i.e. Dr ×D(dj)), pi is a capillary.

29

3. If pi is classified as a capillary, then all of its daughter vessels are capillaries.

4. Else if the above conditions are not satisfied, pi is classified as an arteriole/venule.

The geometry-based algorithm by Smith [49] is run from both the arteriolar and

venular trees i.e. when classifying venules, the first parent segment is that which has

the lowest nodal pressure. The two results are compared, those vessels not classified as

either arterioles of venules are labelled as capillaries. The choice of Dr is a crucial factor

when running this algorithm as it can cause an overlap of the arteriole and venular trees,

which biologically is not correct as each tree has distinct pressure profiles. Smith [49]’s

choice of Dr was based on a comparison of results by Roy et al. [40] on rat mesentery

networks obtained by Pries et al. [32]. The method by Roy et al. [40] relies on predefined

flow directions to identify connected diverging or converging tree structures.

Work on implementing the geometry-based algorithm by Smith [49] is currently ongo-

ing. The mesentery network (from the Pries et al. [32] collection) is being used a testing

bed during implementation as it allows for a comparison with the work of Smith [49]

and Roy et al. [40]. The current result is shown in Figure 2.6c (Dr = 2.675), visually

it displays similarities to both Figures 2.6a and 2.6b. However, the algorithm currently

does not work when applied to the brain subnetwork extracted in Section 2.1.

The objective is to apply the geometry-based algorithm to our brain networks to

(a)

(b)

(c)

Figure 2.5: (a) Later stages of the geometry-based algorithm on the mesentery networkof Pries et al. [32]. (b) Magnification of the region outlined in (a). (c) the capillaryclassification of daughter branch, d2, one step later. Figure has been reproduced fromSmith [49].

30

study macro and microangiopathy (see Discussion for further details). Difficulty may

arise when selecting Dr (currently, sensitivity analysis for the brain networks has not

been done) for our networks and so the method by Roy et al. [40] or an alternative

geometry-based method by Cassot et al. [6] may need to be considered.

Algorithm 2 Geometry-based algorithm by Smith [49].

Select first parent segment p1. . p1 : the main feeding or draining vessel.Initialise P = ∅. . P : the set of segments which have been parents.Initialise RS = {p1}. . RS : the set of reached segments.Initialise RN = {start node of p1} ∪ {endnode of p1}.

. RN : the set of reached nodes.for i = 1 : nS do . Loop through parent segments.

pi ← maxD(s ∈ {RS\P}).. pi : the max diameter segment ∈ RS but /∈ P .

Add pi to P .Find daughter vessels dj (j = 1, . . . , nD) of parent pi.

. nD : the number of daughter vessels of pi.Add start/endnode of pi to RNAdd dj (j = 1, . . . , nD) to RS .for j = 1 : nD do

Find nj . . nj : end node of dj .if nj ∈ RN then

dj is a capillary.if D(pi) < Dr ×D(dj) then

. Dr : critical parent-daughter diameter ratio.pi is a capillary.

end ifelse

Add nj to RN .end if

end forif pi is a capillary then

dj (j = 1, . . . , nD) are also capillaries.else

pi is an arteriole/venule.end if

end for

31

(a) (b)

(c)

Figure 2.6: Geometrical analysis of the rat mesentery network. (a) Roy et al. [40].(b) Reproduced from Smith [49]. (c) Attempt at implementation of the geometry-basedalgorithm. Red - arterioles, green - capillaries and blue - venules.

Figure 2.7: Comparison of the number of arterioles, capillaries and venules presentedby Smith [49] and implementation for this report. Roy et al. [40] data not available.

32

3 Discrete Modelling of Microvascular

Fluid Transport

The objectives of this chapter is to introduce the concept and modelling of discrete

microvascular fluid transport. A basic model is presented which can be used when all

boundary conditions are known. An extension of this basic model is given, which uses

a flow estimation algorithm by Fry et al. [16] for the case when there are unknown

boundary conditions.

The key components when modelling blood flow within a discrete microvascular net-

work are network geometry and topology, blood rheology within vessels, in addition to

flow or pressure boundary conditions at all boundaries of the network. Blood flow in

microvascular networks is viscous dominated, along with the assumptions of Lubrica-

tion Theory the Navier-Stokes equations can be reduced. Application of Lubrication

Theory assumes blood is modelled as a continuum, and blood vessels are long and thin.

Mathematically, the assumptions can be defined as,

ε =D

L� 1 and ε2Re� 1, (3.1)

where ε is the aspect ratio between a vessel’s diameter, D, and length, L. Re is the

Reynolds number (a ratio between inertial and viscous forces) of the fluid. Under these

assumptions, Poiseuille’s Law, a linear relationship between a vessel (represented by an

axisymmetric tube) flux, Q, and its pressure gradient, can be used

Q = − πd4

128µ(d,HD)

dp

ds, (3.2)

where d is vessel diameter, µ is the dynamic viscosity, HD is blood discharge hematocrit,

p is the fluid pressure and s is the arc length down the central axis of the vessel.

A discrete microvascular network is a series of individual blood vessels, where Equa-

tion (3.2) can be applied to each individual vessel (along with network conservation

laws). The following section describes the process of modelling multiply connected ves-

sels combined with the previously seen in vivo model by Pries and Secomb [34], to

describe the flow resistance in microvessels.

33

3.1 Basic Model

As previously described in Chapter 2, to use a discrete model, the microvascular network

is segmented and represented by a series of vessel segments which are connected by

nodes and/or contain boundary nodes. For each vessel segment a positive flow direction

is defined from its start node to end node (this is arbitrarily chosen, although care needs

to be taken when defining flow at the boundaries i.e. positive or negative values for

inflow or outflow, respectively). Under the assumption of Poiseuille flow, the relationship

between the nodal pressures pk and the segment fluxes, Qj , can be expressed in the form

Qj =∑k∈N

Mjkpk, j ∈ S, (3.3)

where N and S are the set of all nodes and vessel segments in the microvascular network

respectively, and

Mjk =

+πd4j/ (128µjlj) , if k is the start node of segment j,

−πd4j/ (128µjlj) , if k is the end node of segment j,

0, otherwise,

(3.4)

where lj denotes the length of segment j.

As a consequence of the conservation of mass, the sum of the flow at each interior

node is zero. This can be combined with the boundary nodes to give∑j∈S

LijQj +Q0i = 0 for i ∈ N, (3.5)

where

Lij =

−1, if i is the start node of segment j,

+1, if i is the end node of segment j,

0, otherwise.

(3.6)

For all interior nodes, Q0i = 0. If i is a boundary node, Q0i is the inflow (or outflow

if negative). Combining (3.3) with (3.5) yields∑k∈N

Kikpk = −Q0i for i ∈ N, (3.7)

where

Kik =∑j∈S

LijMjk. (3.8)

Equation (3.7) forms a sparse linear system of nN (total number of nodes) equations.

If the pressure or flow is known at every boundary node (at least one pressure condi-

34

tion is needed), the system can be solved for nodal pressures using standard methods.

Substitution of resulting pressure solutions into equation (3.3) solves for segment fluxes.

3.2 Incorporating Variable Hematocrit

As detailed in Section 1.3.2, the modelling of variable hematocrit at microvascular bi-

furcations accounts for the non-Newtonian behaviour of blood perfusion at the capillary

scale. In this thesis, the empirical laws given by Pries et al. [38] are used to model this

process.

Absolute effective viscosity given by Equation (1.14), is a function of hematocrit and

since the fluxes, Qj , are needed to calculate vessel hematocrits, an iterative procedure

is used. To summarise the pseudo code presented in Algorithm 3, hematocrit is set to

an initial value (0.4) which is then used to calculated the absolute effective viscosity

on the first iteration. Equation (3.7) is then used to calculate vessel fluid pressures

and consequently segment fluxes from Equation (3.3). The nodes connecting vessel

segments are then listed in order of flow direction. From the list, a hematocrit boundary

condition is applied to a node if it is located at the network boundary. If the node

corresponds to a converging bifurcation, conservation of RBC flux is used. Finally, if

the node is a diverging bifurcation, the empirical law given by Equation (1.15) is used.

A relaxation factor (multiplied by a factor of 0.8 every fifth iteration) is included when

updating hematocrit values to ensure hematocrit convergence. Consequently, the new

and old values of vessel hematocrit and flux are compared and if the differences are

below previously defined tolerances, the algorithm terminates.

The algorithm for the phase separation iterative procedure is presented in Algorithm

3, however, implementation is ongoing and so constant hematocrit (with a value of 0.4)

is used for this report. Note, with constant hematocrit, Algorithm 3 terminates once

segment fluxes, Qj , are calculated.

3.3 Boundary Conditions

Rapid development for imaging of microvascular networks allows us to model detailed

three-dimensional networks. The distribution of blood flow rates fundamentally influ-

ences perfusion, solute transport, flow regulation, and growth and adaption in the mi-

crovascular networks but current available methods for observation of blood flow within

individual vessels is limiting. Experimental measurements of the main feeding arte-

riole(s) and draining venules(s) can be taken, but measurements of blood flow within

individual capillaries, in a capillary dense network such as the brain, would be extremely

time consuming, if not impossible using current methods. Thus, the mathematical mod-

35

Algorithm 3 Phase separation algorithm.

hOldj = hInit. . Defining initial hematocrit.relaxFactor = 1. . Set initial relaxation factor.hTol = 10−3 , qTol = 10−3. . Tolerances for hematocrit and flux.IterMax = 100. . Set the maximum number of iterations.for iter = 1 : iterMax do

if iter mod 5 ≡ 0 thenrelaxFactor ← 0.8× relaxFactor.

end ifCalculate µabs,j(dj , hOldj) using in vivo viscosity law - Equation (1.14).Solve Equation (3.7) for pressures, pk.From Equation (3.3), calculate fluxes, Qj .{nsort,inod} ← Generate list of nodes in order of flow direction.for iseg = 1 : nN do . nN - Total number of nodes.

if nsort,inod is a boundary node thenAssign hematocrit boundary condition.

else if nsort,inod is converging thenUse conservation of RBC flux.

else nsort,inod is divergingApply bifurcation law given by Equation (1.15).

end ifend forhChange = hNewj − hOldj .hNewj = hOldj + relaxFactor × hChange.qChange = qNewj − qOldj .hOld← hNew. . Update hematocrit values.if |hChange| < hTol and |qChange| < qTol then

Nonlinear iterations have converged. . Algorithm terminates.end if

end for

elling of these networks is hindered by the lack of detailed flow information.

Lorthois et al. [27] researched the effect of boundary conditions on Poiseuille flow

simulations (incorporating the in vivo viscosity and bifurcation laws of Pries et al. [38])

relating to brain blood flow regulation by intra-cortical arterioles in a large human vascu-

lar network. The human brain samples came from the Duvernoy et al. [12] collection and

the data set was collected using confocal laser microscopy by Cassot et al. [6]. Bound-

ary nodes within the network were categorised as either arteriolar, venular or capillary

nodes. If the arterial nodes were connected to an arteriolar tree, a pressure condition

of 74 mmHg was set at the base of this tree and a zero flow condition was assigned

to all other interrupted segments of the tree. Likewise, for venular boundary nodes, if

belonging to a venular tree, a pressure condition of 15 mmHg was set at its base and zero

flow was assigned to all other interrupted segments. For capillary boundary nodes three

36

boundary conditions were tested. The first being a zero flow condition, which was found

to minimise total blood flow. The second case assigned a uniform pressure, which was

chosen so that the net capillary flux was zero - this condition minimised the total inflow.

The final case was to assign a power law which defined the flow rate as a function of

capillary radius (the power law, Qcap = Kγrcap, where K and γ are coefficients obtained

from adjustment of the results of the first two case). However, further study of the third

condition was stopped as it soon became clear that assigning this boundary condition

was inappropriate for capturing the hemodynamics of the intra-cortical network under

study. Hematocrit at the boundary was set to 0.4 for all cases.

The study by Lorthois et al. [26] illustrates the strong variability of network flow

properties when assigning boundary conditions to capillary vessels. The following section

illustrates another option given by Fry et al. [16] in estimating blood flow boundary

conditions.

3.4 Blood Flow Estimation

The model described in Section 3.1 requires boundary conditions in the form of flow or

pressure values at all boundary nodes within a given network. The method of Fry et al.

[16] offers an alternative approach to assigning boundary conditions at the potentially

numerous, smaller micro vessels forming the network boundary. The approach is based

on the idea that microcirculation is regulated in response to signals relating to flow and

shear stresses by Pries et al. [36]. Therefore, the method seeks to minimise the deviation

from specified target wall shear stresses and pressures values derived from independent

information about typical network hemodynamic properties.

If not all boundary conditions are known, the system, (3.7), reduces to∑k∈N

Kikpk = −Q0i, i ∈ I ∪B, (3.9)

where I is the set of all interior nodes and B is the set of all boundary nodes with

known boundary conditions. The resulting system has multiple solutions and so further

assumptions are needed to obtain a unique solution. The method proposed by Fry et al.

[16] involves solving a constrained optimisation problem, which can be formulated in

terms of a Lagrangian objective function, which is defined as

L =1

2kp∑k∈N

wk(pk − p0k)2+1

2kτ∑j∈S

`j(τj − τ0j)

+∑i∈I∪B

λi

(∑k∈N

Kikpk +Q0i

),

(3.10)

37

where p0k is the target pressure at node k, τj is the wall shear stress in segment j, τ0j

is the corresponding target shear stress, kp and kτ are weighting factors associated with

the pressure and shear deviations from the target values, λi is the Lagrange multiplier

associated with node i and wk is the vessel length associated with node k, and defined

as

wk =1

2

∑`j , (3.11)

where `j is the vessel length associated with segment j.

Note, shear stress is defined as

τj = µj∂uj∂r

∣∣∣∣r=rj

, (3.12)

where uj is the horizontal velocity component and r is the vessels’ radial component.

This can be calculated using the definition of the fluid velocity field in Poiseuille flow

uj(r) = −(r2 − r2j )

4µj`j4pj , (3.13)

where 4pj is the pressure change in segment j. Substition of (3.13) into (3.12) leads

to

τj = µjrj

2µj`)j4pj =

4µjπr3j×

πd4j128µj`j

4pj . (3.14)

Using Poiseuille’s law, (3.3), the shear stress can be written in the form

τj = cjQj = cj∑k∈N

Mjkpk, (3.15)

where

cj =4µjπr3j

(3.16)

Hence, (3.15) is used in the Lagrange optimisation problem given by (3.10).

Setting ∂L/∂pi = 0 gives

kτ∑k∈N

Hikpk + kpwi(pi − p0i) +∑

k∈I∪BKkiλk

= kτ∑j∈S

τ0jcjMji`j , i ∈ N,(3.17)

where

Hik =∑j∈S

c2jMjiMjk`j . (3.18)

Setting ∂L/∂λi = 0 simply recovers equation (3.9).

38

Combining equations (3.9) and (3.17) we gain the following sparse linear system[K 0

kτH + kpW KT

]×

[p

λ

]=

[−Q0

kpW × p0 + kτMT × (τ0 · c · `)

](3.19)

where K, H, W and M denote the matrix forms of Kik, Hik, wk and Mij (note, W is

a diagonal matrix with entries wk) respectively, whereas p, λ, Q0, p0, τ0, c and ` are

the vector forms of pk, λk, Q0i, p0k, τ0j , cj and `j respectively. The result is a square

linear system of nI + nB + n linear equations, where nI , nB and n are the number of

interior nodes, known boundary nodes and the total number of nodes. This system can

be solved for unknowns pi and λi using standard methods.

In the approach presented here, the target pressures at each node and target shear

stresses in each segment are assumed to be constants - the choice of which will be

discussed in further detail later. However, through observations of hemodynamic char-

acteristics in sub-regions of a network, the target pressures and shear stresses can be

represented by functions of variables. The sign of the shear stress, τ0j , in each segment

is equal to the corresponding flow sign, but this cannot be known until a solution is ob-

tained. Therefore, the system of equations must be solved iteratively with target shear

stresses (initially set with random flow signs) being used to estimate the flow distribu-

tion and then updating the signs in each segment for the next iterative step. During

optimisation (with application to the rate mesentery networks of Pries et al. [32]), Fry

et al. [16] initially set kτ = 10−4 and on each iteration the value of kτ was doubled. This

process was repeated until two consecutive iterations gave the same flow directions in

all segments. The constant kp is used to maintain pressures in a physiologically-realistic

range and was arbitrarily set to 0.1 in order to bias the nodal pressures towards the

specified target pressure. This iterative procedure, with variable hematocrit, is shown

in Algorithm 4. Note, work on implementing the process of calculating the final value of

kτ is ongoing. The final value of kτ given by Fry et al. [16] was used for the simulations

in this report. Note, these simplifications means kτ does not iterate in Algorithm 4, and

the algorithm terminates once segment fluxes, Qj , are calculated.

Testing of the flow estimation algorithm was done on the rat mesentery networks of

Pries et al. [32] and in order to replicate the results of Fry et al. [16] and Smith [49].

39

Algorithm 4 Flow estimation algorithm.

Set p0j . . Assign initial segment pressure value(s).Set τ0j = rand(−1, 1)× τ0.

. Assign initial value, τ0, for target shear stress, with randomly assigned signs.kp = 0.1.kτ = kτ,init. . Set initial value for kτ .Set kτ,final. . Assign final value for kτ .hTol = 10−3 , qTol = 10−3. . Tolerances for hematocrit and flux.while kτ ≤ kτ,final do

while flowSign changes dohOldj = hInit. . Defining initial hematocrit.relaxFactor = 1. . Set initial relaxation factor.for iter = 1 : iterMax do

if iter mod 5 ≡ 0 thenrelaxFactor ← 0.8× relaxFactor.

end ifCalculate µabs,j(dj , hOldj) using Equation (1.14).Solve Equation (3.7) for pressures, pk.From Equation (3.3), calculate fluxes, Qj .{nsort,inod} ← Generate list of nodes in order of flow direction.for iseg = 1 : nN do . nN - Total number of nodes.

if nsort,inod is a boundary node thenAssign hematocrit boundary condition.

else if nsort,inod is converging thenUse conservation of RBC flux.

else nsort,inod is divergingApply bifurcation law given by Equation (1.15).

end ifend forhChange = hNewj − hOldj .hNewj = hOldj + relaxFactor × hChange.qChange = qNewj − qOldj .hOld← hNew. . Update hematocrit values.if |hChange| < hTol and |qChange| < qTol then

Nonlinear iterations have converged. . Algorithm terminates.end if

end forflowSignj = sign(qj). . Record flow directions.τ0j = flowSignj × τ0. . Update shear stress signs.

end whilekτ ← 2× kτ .

end while

40

4 Analysis of Oxygen Transport to

Tissue

In this chapter, the governing equations describing oxygen transport are presented,

along with a Green’s function approach to this model. In this report oxygen transport

is simulated using the Green’s function method of Secomb et al. [45].

An essential function of the circulatory system is to delivery oxygen to tissue. The

distance that oxygen can diffuse into oxygen-consuming tissue is small, and so tissue

oxygenation is critically dependent on the spatial arrangement of microvessels in the

tissue. Thus, an ischeamic stroke at both the macro and micro scale can have a dras-

tic affect on oxygen consumption within tissue, eventually leading to the formation of

hypoxic regions within a circulatory network. Due to unavailable experimental infor-

mation, theoretical methods have been developed to simulate the spatial distribution of

oxygen levels in tissue surrounding a network of microvessels. Often blood is treated as

a continuum or computations are performed in the frame of reference of erythrocytes.

The first method struggles to capture the sharp O2 gradients near erythrocyte mem-

branes, and the second is problematic when assigning boundary conditions and so does

not allow for detailed information at the individual capillary scale.

Recent work by Lucker et al. [28] provided a new method to modelling oxygen trans-

port. Their model relies on moving meshes following individual erythrocytes, which are

then mapped over a fixed mesh serving for the calculation of oxygen transport in the

capillaries and surrounding tissue. The approach can account for variational hematocrit,

RBC velocity and metabolism. Lucker et al. [28] have run both steady-state and tran-

sient simulations to observe the influences of these parameters of PO2 (partial oxygen

pressure - the hypothetical pressure of O2 if it alone occupied the volume of the blood

at the same temperature) levels and have validated against PO2 measurements in the

rodent brain given by Parpaleix et al. [29]. Early simulations have been confined to a

single capillary network due to high computational costs, but work is currently being

done to expand simulations to larger networks.

Current work by Boas et al. [5] into capillary transit time heterogeneity, offers po-

tential when assigning PO2 boundary conditions. Boas et al. [5] made detailed mea-

surements of PO2 distribution from pail arterioles through a capillary network to pial

veins using a two photon microscope combined with an oxygen quenched phosphorescent

dye. This techniques allows to measure the speed and flux of individual RBC through

an individual vessel within a vascular network. Application of this method to vascular

41

networks of increasing size is time consuming, however, results could be used for average

PO2 levels in various regions of the brain. Boas et al. [5] also created a mathematical

model to calculate a PO2 distribution for a small subsection of the brain using a Krogh

cylinder model. The Krogh cylinder model, models solute transport from capillary to

surrounding tissues without the need of blood flow data. The structure of the model

implies that each capillary vessel is responsible for oxygen supply to the corresponding

surrounding cylindrical tissue section. However, there are multiple limitations to using

this model, for example, it assumes cylindrical angular symmetry around the central

axis while neglecting longitudinal diffusion of gas within the tissue, implying blood flow

is the sole mechanism for oxygen diffusion. The Krogh model also assumes there is

no diffusion of oxygen between surrounding tissue cylinders and as such, may not be

biologically realistic.

Finite-difference methods have been used for three-dimensional oxygen transport

problems by Goldman and Popel [17] & [18] and by Beard and Bassingthwaighte [2]

& [3], and for two-dimensional problems by Lo et al. [25]. For example, the following

equation is a two-dimensional time-dependent diffusion equation for the distribution of

PO2, P (x, y, t),∂P

∂t= Dα

(∂2P

∂x2+∂2P

∂y2

)− M0P

P + P0, (4.1)

where D is the diffusion coefficient of oxygen in tissue, α is the solubility, M0 is a

uniform oxygen demand in the tissue given a non-limiting supply of oxygen and P0

represents the PO2 at half-maximal consumption. A finite-difference method discretises

Equation (4.1) on to a grid and solves it using a numerical scheme. However, relative to

a Green’s function approach, the finite-difference method generally has a larger number

of unknowns in the numerical formulation and does not allow for rapid computations

for complex vascular geometries.

In this thesis, a Green’s function approach by Secomb et al. [45] is used to model

oxygen transport - a method derived from observations of skeletal muscle, brain, and

tumour tissues. In comparison to finite difference methods, the Green’s function ap-

proach reduces the number of unknowns in the numerical analysis and allows for faster

computations for complex networks.

4.1 Governing Equations

The following standard mathematical model describes oxygen transport through tissue

which is represented as a homogeneous medium with oxygen diffusivity D and solubility

α. Here, vessel walls are treated as part of tissue space and steady-state conditions are

42

assumed. Using Fick’s law of diffusion,

J = −D∇φ, (4.2)

which relates the diffusive flux, J , to the concentration, φ, combined with the conser-

vation of mass, leads to the governing equation

Dα∇2P = M(P ), (4.3)

where P is the tissue PO2 and M(P ) is the consumption rate. The dependence of

oxygen consumption on PO2 is represented by a Michaelis-Menten relationship

M(P ) =M0P

P0 + P, (4.4)

where M0 is a uniform oxygen demand in the tissue given a non-limiting supply of

oxygen and P0 represents the PO2 at half-maximal consumption.

The rate of convective oxygen transport along a vessel segment is given by

f(Pb) = Q [HDC0S(Pb) + αeffPb] , (4.5)

where Pb is the blood PO2 level (considered as an average value over the cross-section

of the blood vessel), Q is the blood flow rate, HD is the discharge hematocrit (the

volume flux of erythrocytes as a faction of the total volume flux in a vessel), C0 is the

concentration of haemoglobin-bound oxygen in a fully saturated erythrocyte, S is the

oxyhemoglobin saturation and αeff is the effective solubility of oxygen in blood. Both

Q and HD can be considered constant within a segment but may change from one vessel

segment to the next using the previously seen empirical bifurcation law given by Pries

et al. [38], whereas Pb varies with the position along a segment. The oxyhemoglobin

saturation, S, is represented by the Hill equation

S(Pb) =Pnb

Pnb + Pn50, (4.6)

where n is a constant and P50 is the PO2 at 50% saturation. The effective solubility of

oxygen in blood is given by

αeff = (1−HD)αp +HDαRBC , (4.7)

where αp and αRBC are the solubilities in blood plasma and RBCs, respectively. These

values are similar [22] and so αeff slightly depends on hematocrit and so can be approx-

imated by a constant value.

43

Conservation of oxygen infers

df(Pb)

ds= −qv(s) (4.8)

in each vessel segment, where s is variable defining segment length and qv is the rate

of diffusive oxygen efflux per unit of s.

Continuity of oxygen flux and PO2 must be enforced over the interface between blood

and tissue. Assuming a cylindrical segment leads to

qv(s) = −Dα∫ 2π

0

∂P

∂rrvdθ, (4.9)

where rv is the vessel radius and integration is performed over the circumference of

the vessel, denoted by the azimuthal angle θ. The blood vessel delivering the oxygen

generally exceeds local PO2 levels at the interface with the surrounding tissue. This

behaviour can be described by the following equation by Hellums [21],

Pv(s) = Pb(s)−Kqv(s), (4.10)

where Pv(s) is the average tissue PO2 and K represents intravascular resistance to

radial oxygen transport - a complex process, but here it is assumed K is a constant

dependent on each vessel’s diameter.

4.2 Green’s Function Approach to Oxygen Transport

In the following Green’s function method, blood vessels are represented as a set of

discrete oxygen sources. Using superposition principles, the resulting fields from these

sources represent the PO2 field in the tissue. As in equation (4.3), the tissue region is

represented by a set of discrete oxygen sinks. If the rate of oxygen uptake within the

tissue is known, the only unknowns in the problem are the strengths of both the sources

and sinks. The following mathematical model is presented by Secomb et al. [45] and is

used for this thesis.

Modelling oxygen transport, the Green’s function, G(x;x′), for a given tissue domain

may be defined as the PO2 at a point x resulting from a unit point source at x′. That

is to say, the Green’s function would be the solution to

Dα∇2G = −δ3(x− x′), (4.11)

44

where δ3 is the delta function in three-dimensions. The PO2 is thus given by

P (x) =

∫Sources

G(x;x′)q(x′)dx′, (4.12)

where q(x) represents the distribution of source strengths. The oxygen field resulting

down a long narrow blood vessel can be represented by a series of sources down its centre.

However, if it is assumed the distribution of these source strengths are equal to the

diffusive flux, qv(s), as defined by equation (4.9), the oxygen field close to the vessel may

not be accurately represented. In an attempt to improve this method, Hsu and Secomb

[23] assumed that the oxygen sources along the vessel axis must be uniformly distributed

around the circumference of the vessel on the blood-tissue boundary. However, Pozrikidis

and Farrow [30] noted that in order to satisfy a flux boundary condition, along with the

distribution of sources, a distribution of dipoles on the surface is generally required to

cancel the inwards flux into the blood vessel as a result of the distribution of sources.

Thus, implying the method of Hsu and Secomb [23] is an approximation. However, in

the case of oxygen transport, the continuous connectivity of vessel segments, gives rise

to smoothly varying oxygen sources and so the errors resulting from Hsu and Secomb

[23]’s approximation are small. Therefore, the approach of Hsu and Secomb [23] shall

be used in this approach.

4.2.1 Numerical Analysis

The following describes the numerical methods used to solve for oxygen transport using

the Green’s function approach. In a infinite domain, the solution is the singular function

G =1

4πDα|x− x′|. (4.13)

In a finite domain, additional nonsingular terms may be needed in order to satisfy

boundary conditions for the domain. If the microvascular network is segmented into nS

segments and the tissue is discretised into nT approximately cubic regions, a nS × nSmatrix of coefficients is defined in which Gvvij is the average tissue PO2 at the surface

of segment i as a result of a unit source distributed on the surface of segment j. Gvvij is

derived by integrating the Green’s function G(x;x′) with respect to both x and x′ over

the surface of segments i and j. An nS × nT matrix is formed using the coefficients of

Gvtij , which is defined as the tissue PO2 at the midpoint of segment i resulting from a

unit source at the midpoint of tissue region j. Likewise, a nT × nS is formed from the

coefficients of Gtvij , which is defined as the tissue PO2 at the midpoint of tissue region i

resulting from a unit source at the midpoint of segment j.

Use of the above definitions combined with equation (4.12), the average PO2 at both

45

the blood tissue interface of segment i and the tissue region i are given respectively by

Pv,i =

nS∑j=1

Gvvij qj +

nT∑j=1

Gvtijφj +G0, i ∈ S, (4.14a)

Pt,i =

nS∑j=1

Gtvij qj +

nT∑j=1

Gttijφj +G0, i ∈ T, (4.14b)

where qj is the source strength of segment j, φj is the sink strength of tissue region j

and T is the set of tissue points.

From Equation (4.10) we have

Pv,i = Pb,i −Kqi, i ∈ S, (4.15)

where Pb,i is the PO2 of blood at the midpoint of segment i.

From Equation (4.4) we have

φj = −M(Pt,j) = − M0Pt,jP0 + Pt,j

, i ∈ T, (4.16)

where Pt,j represents the PO2 in tissue region j as given by (4.14b). When calculating

the tissue PO2, in the case of no-flux boundary conditions on the tissue domain, or if

no boundary conditions are imposed, the constant G0 is required. G0 is determined by

the following conditionns∑j=1

qj +

nT∑j=1

φj = 0, (4.17)

which states that the net oxygen delivery is equal to oxygen consumption.

By integrating Equation (4.8) with respect to segment length, the rate of convective

oxygen transport along vessel segment i at its midpoint, f(Pbi), can be expressed as a

linear function of the source strengths qj ,

f(Pb,i) = f(P 0b,i)−

ns∑j=1

αijqj for i ∈ S; (4.18)

where P 0bi is the intravascular PO2 of segment i in the absence of diffusive oxygen

exchange. In this case, qj = 0 for all j and P 0bi is typically given by an assumed PO2

by an upstream vessel entering the tissue domain. For a series of segments forming an

unbranched vessel, αij is equal to 1 if segment j is upstream of segment i, 12 if i = j and

0 otherwise. If the segments form a branching network, αij depends on the partition of

oxygen fluxes at converging bifurcations.

46

Combining Equations (4.14a), (4.15) and (4.18) we get the following equation

ns∑j=1

Gvvij qj +G0 − f−1f(P 0

b,i)−nS∑j=1

αijqj

+Kqi

= −nT∑j=1

Gvtijφj , i ∈ S,

(4.19)

which is a set of ns equations to be solved together with Equation (4.17) for qj and G0.

C++ code generated by Secomb [43] is used to simulate the described Green’s function

model. The inputted network file is contained with a cuboid which must include all

oxygen-consuming regions. The midpoint of each vessel segment represents a oxygen

source, and so a segments efflux is a product of source strength and vessel length. The

tissue domain is split into subregions each of which are centred on a tissue node point.

In each of these subregions, the oxygen consumption rate is uniform and dependent on

the PO2 at the nodal point according to the Michaelis-Menten kinetics (Equation 4.4).

Next, inside the cuboid, the smallest convex region containing all tissue nodes within a

pre-defined distance of the nearest vessel is calculated. This convex region is equivalent

to the simulated tissue domain.

4.2.2 Boundary Conditions

In general, the boundary conditions on the exterior surface of the tissue domain cannot

be determined by available experimental information, in which case, additional assump-

tions must be made to form a well-posed problem. Two choices are imposing no-flux

(pointwise no-flux or net no-flux) or periodic boundary conditions. Imposing a no-flux

boundary condition is not realistic when applied to vascular networks, which tend to

have irregular geometries and therefore local oxygen fluxes can occur across any ar-

bitrarily chosen boundary in the tissue. Use of this boundary condition can result in

exaggeration of the amount of hypoxia in the tissue. However, using the Green’s function

method, the tissue volume can be considered as being embedded in an infinite domain

with the same diffusivity, where no oxygen sources or sinks are located externally to

the given tissue region. Hence, a well-posed problem is formed without imposing ex-

plicit boundary conditions on the outer surface of the tissue region. When compared

to finite-difference methods, the approach minimises artefacts associated with assigning

boundary conditions.

47

5 Mathematical Simulation of Fluid &

Mass Transport

In Chapter 3, the mathematics behind discrete fluid transport in blood vessel and blood

flow estimation of Fry et al. [16] was shown. In this Chapter, simulations of discrete

fluid transport on a brain subnetwork are run for both healthy and ischaemic stroke

induced networks. The subsequent flow solutions are then use in order to partially

parametrise the Green’s function approach to an oxygen transport model. The run-time

of simulations using these models greatly increases with respect to the number of vessel

segments contained within a network. Due to time limitations, a further network from

the largest subnetwork (see Figure 2.3b) was acquired.

Since averaged blood flow velocity data of the left and right carotid arteries was

obtained by MRI, the conditions for Lubrication Theory (see Equation 3.1) at the carotid

vessels where tested. For example, for one of the carotid arteries, the aspect ratio

was calculated as ε ≈ 0.07997 (where the diameter and length of the vessel are given

by 22.29µm and 278.747µm respectively) and the Reynolds number as Re ≈ 0.6045

(where the blood velocity, viscosity and density are given by 12.6 cm/s, 4.879 cP [52]

and 1050×10−18 kgµm−3 [24] respectively). Thus, the conditions for Lubrication Theory,

ε ≈ 0.07997 � 1 and ε2Re ≈ 0.0028 � 1, were thought to be within tolerances. Since

at the internal carotid arteries diameters are relatively large and the blood velocities

are the highest in the entire network, the discrete blood flow model is applicable to

the entire mouse brain data. Subsequently, a network was extracted in the vicinity of a

carotid artery. The network was retrieved by simply selecting a blood vessel and running

a simple algorithm to keep all vessels contained within a predefined box (4000× 5000×4000µm). The largest subnetwork algorithm was then run again to obtain the largest

network within the acquired data (see Figure 5.1b).

The region of the mouse brain in Figure 5.1b was selected due to the characteristics

of the Circle of Willis. A network with a looping structure was thought to possess more

desirable results when simulating an ischaemic stroke. For example, if a network was

chosen that has two branching regions connected by a a single blood vessel, simulating

an ischaemic stroke in this vessel could (dependent on boundary conditions) result in

an entire region of the network having zero blood flow. Comparatively, if an ischaemic

stroke occurred in a looping structure, more interesting results would be produced in

terms of how the network compensates blood flow through the remaining unblocked

vessel(s).

48

(a) (b) (c)

Figure 5.1: (a) The largest subnetwork obtained from the original mouse data. (b)Extracted section of the Circle of Willis, with the carotid artery circled. (c) Furtherreductions for the oxygen transport model. Colouring of vessels is diameter dependent.

The run-time of the oxygen transport solver varies drastically depending on the num-

ber of vessel segments within a network. Due to time limitations, reductions to the

number of vessel segments within the network used for this model had to be made (see

Figure 5.1c). To give a understanding of the sheer quantity of blood vessel segments

within the network and subsequent subnetworks - the full brain dataset shown in Fig-

ure 2.3a contains 186, 245 segments, the largest subnetwork (see Figure 5.1a) contains

72, 429 segments, the section of the Circle of Willis (see Figure 5.1b) contains 4, 718

segments and its reduction (see Figure 5.1c) only contains 515 segments.

5.1 Flow Estimation Method of Fry et al.

In this section, discrete flow solutions are derived for the sub-section in Figure 5.1b.

Limited physiological data on flow and pressure distributions in the vascular networks

are available, meaning the determining boundary conditions remains problematic when

simulating flows in discrete microvascular networks.

In this thesis, discrete flow solutions were obtained via the flow estimation method of

Fry et al. [16] introduced in Chapter 3. The optimisation procedure was developed and

tested on the mesentery networks of Pries et al. [32]. Parameter values were assigned

for the target pressure, the target shear stress and weighting of shear stress to pressure

terms. Based on diastolic pressures given by Czosnyka [8], target pressure was set at a

constant value of p0 = 40 mmHg. No data was found to allow accurate assignment of

a target shear stress value. Fry et al. [16] assumed the magnitude of the target shear

stress to be |τ0j | = 15 dyn/cm2, the average shear stress corresponding to the median

49

segment pressure across all four mesentery networks - to avoid obscure results, the same

constant value was used.

As per Fry et al. [16], kp was arbitrarily set to 0.1. The same value as Fry et al. [16]

was also used for kτ = 2.147×105. Uniform hematocrit boundary conditions of 0.4 were

applied at all inflow boundary nodes. Two pressure boundary conditions, based on data

by Czosnyka [8], were set to 60 mmHg on two vessels located on the Circle of Willis

(see Figure 5.2a). Using the MRI data collected for the internal carotid arteries, a flow

boundary condition at the carotid vessel was set to 12.6 cm/s.

5.1.1 Results

(a) (b)

Figure 5.2: 2D representations of the discrete flow solutions for pressures (mmHg). (a)The healthy network - flow boundary condition given by the solid circle and the twopressure conditions shown by the dashed circles. (b) Ischaemic stroke induced network- blocked vessel has been circled.

Blood fluid pressure values for both the healthy and ischaemic stroke induced networks

ranged from 48.16 to 66.85 mmHg (consistent with physiological results by Czosnyka [8])

and 45.86 to 61.01 mmHg respectively. Inducing an ischaemic stroke reduced the mean

pressure from 54.75 mmHg in the healthy network to 51.71 mmHg. Standard deviation

increased from 3.61 to 3.87 mmHg, indicating a slight increase in the range of pressure

values for the stroke induced network. This shift can easily be seen in Figure 5.3, where

blood vessel pressures fluctuate from 55 to 57 mmHg, towards the 46 to 49 mmHg range.

Visually this can be seen when comparing Figures 5.2a and 5.2b - the region north of

50

Figure 5.3: A plot comparing network pressures against the % of blood vessels withina network. The distributions of the healthy and stroke induced networks are overlaidfor ease of comparison.

the stroke experiences a pressure drop of ∼ 10 mmHg suggesting an increased likelihood

of hypoxia in this region.

5.2 Green’s Function Method for Oxygen Delivery

In this section, the Green’s function method see in Chapter 4 is used to simulated

oxygen transport from the extracted subnetwork of blood vessels to the finite volume of

tissue representing a portion of the mouse brain. As previously described, the numerical

method used treated vessels as a distribution of oxygen sources and the tissue as a

distribution of oxygen sinks.

As previously described, the discrete flow solutions parametrise the Green’s function

model. Since the number of vessel segments were reduced (see Figures 5.1b & 5.1c)

in our chosen network, the flow solutions are effected. Thus, the discrete flow model

was run again on the simpler network shown in Figure 5.1c. The resulting flow solu-

tions combined with hematocrit values for all vessel segments and the PO2 and solute

concentrations of boundary nodes (inflow nodes), were used to parametrise the Green’s

function model. Solute transport and reaction parameters are predefined as per the

C + + code provided by Secomb [43] - oxygen is handled differently compared to other

solutes to account for binding by haemoglobin. Intravascular resistance to oxygen is also

parameterised as per Secomb [43].

The cuboidal box containing the network was reduce as compactly as possible - the

size was reduced to 3580×4430×3910µm. The solver subdivides this domain, with each

subdivision centred on a tissue node point. The simulations were parametrised so that

51

there were 50 tissue points in all dimensional directions. The algorithm approximates

the smallest convex region inside the cuboid in order to simulate the tissue domain. The

smallest convex region includes all tissue node points within a distance of the nearest

vessel - this distance was specified as 30µm.

5.2.1 Results

Comparing Figures 5.4a and 5.4b, it is clear the ischaemic stroke induces oxygen depri-

vation in both the surrounding blood vessels and brain tissue in the chosen PO2 planar

slice. Inducing an ischaemic stroke reduced the mean tissue PO2 level from 33.76 mmHg

to 13.5 mmHg, while decreasing the standard deviation from 39.92 to 33.38 mmHg -

indicating a shift of PO2 distributions more towards the mean in the stroke network.

This distribution shift is apparent in Figure 5.5b, suggesting an increase in potential

oxygen deprivation. Note, in Figure 5.5a there is a significant increase (∼ 17%) in tis-

sue with a PO2 of zero, implying a large increase in hypoxia within the stroke induced

brain tissue. This would pose a problem to administering drugs to the patient as drug

transport would be affected in the same way as oxygen transport.

(a) (b)

Figure 5.4: A 2D planar slice representing PO2 (mmHg) in the blood vessels andsurrounding tissue for both the (a) healthy and (b) stroke (circled) induced network.

52

(a) (b)

Figure 5.5: Frequency distributions for PO2 levels in the surrounding tissue for boththe healthy and stroke networks. (b) is a zoomed in region for the 1 to 40 mmHg ofPO2 range of (a).

53

6 Discussion

The brain demands high quantities of oxygen which is distributed through its densely

packed vascular network. Under normal conditions, steep gradients of PO2 are found

in the vicinity of microvessels but low levels may be found in tissue [44]. Hence, any

decrease in perfusion as a result of a stroke can lead to tissue hypoxia and potential

necrosis - affecting the treatment and recovery of a patient suffering from stroke.

This report has presented a combination of mathematical models which simulate blood

perfusion and oxygenation in a microvascular network. These models were applied to a

network containing a section of the Circle of Willis. Simulations were run to model fluid

transport in a healthy and ischaemic stroke induced network in order to give us a insight

into the mechanics behind blood and O2 transport. Work focused on an ischaemic stroke

occurring in just one vessel, but future study would aim to understand the variability

of stroke in different areas of the brain.

Chapter 1’s main focus was introducing current mathematical models which describe

non-Newtonian blood rheology phenomena. The bifurcation law of Pries and Secomb

[34] attempts to resolve the disparity between in vitro and in vivo data by combining

in vitro blood viscosity effects with effects of a diameter dependent endothelial surface

layer. However, the cause of the discrepancy remains to be fully known and could be a

combination of factors, including ESL. A limitation of this model is that it was developed

for bifurcations with three vessel segments connected to one node, whereas within the

capillary network of a brain there are often multiple vessels attached to a node. The

empirical phase separation model of Pries et al. [31] was also presented. Future work

following on from this report would be to complete the integration of this empirical

formula into the discrete fluid transport model. However, work by Guibert et al. [20],

who simulated cerebral blood flow in primate cortex, found that phase separation had

little effect on network-wide hemodynamic properties when imposing uniform hematocrit

boundary conditions. Thus, neglecting phase separation in a large brain network may

be a reasonable assumption, but obviously more work is needed.

Chapter 2 presented a geometry-based algorithm by Smith [49]. During the process of

running fluid transport simulations on the subnetwork of the Circle of Willis, it became

clear that the algorithm of Smith [49] would not work on an entire brain network. The

geometry-based algorithm uses the topological structures of arteriolar/venular trees and

capillary meshes to classify blood vessels. It was hoped that this could be used on a brain

network to distinguish between arteries, capillaries and veins. However, considering the

Circle of Willis is a looping structure composed of arteries, the algorithm by Smith

54

[49] would classify it as a capillary network. Therefore, a modification of the algorithm

or use of perhaps the flow-based algorithm by Roy et al. [40] may be preferential for

application to an entire brain network.

The application of a algorithm to classify the blood vessels within a brain network

is not just interesting from a statistical point of view, it can be used as a tool into the

study of angiopathy. Angiopathy is a generic term for a disease of the blood vessels,

which comes in two forms, macroangiopathy and microangiopathy. Macroangiopathy is

the occurrence of atherosclerosis (the thickening of an artery wall as a result of invasion

and accumulation of white blood cells) - the result being the formation of a blood clot

preventing the flow of blood i.e. stroke. Microangiopathy is where the walls of small

blood vessels become so thick and weak that they bleed and leak protein, increasing

flow resistance and therefore slowing the flow of blood through the body. In this case,

the discrete fluid transport would be need to be modified to include Darcy’s Law (a

constitutive equation that describes the flow of a fluid through a porous medium), how-

ever, this will be discussed later. Application of a blood vessel classification algorithm

could lead to automatic simulation of either macro or microangiopathy (or selective use

of the discrete or continuum fluid transport models - discussed later) in the appropriate

vessels.

Chapter 3 presented the formulation of a discrete fluid transport model, along with

an adaption which incorporates the flow estimation algorithm of Fry et al. [16]. The al-

gorithm attempts to address the lack of experimental microcirculatory flow distribution

information by minimising the deviation from target wall shear stresses and pressure

values - a useful method when not all flow or pressure conditions are known at a net-

work’s boundaries. However, this approach has significant limitations as it is trying

to solve a system that is mathematically underdetermined when constrained by avail-

able data. Inevitably results are an estimate, with the number of known boundary

conditions greatly affecting the accuracy of the solution [16]. With increasing known

boundary conditions, the flow estimation errors are reduced. Considering the network

that was used for this report (see Figure 5.1b), only 3% of boundary conditions were

”known”, suggests the potential for a large amount of flow errors. Further investigation

is needed in order to derive the optimal balance of shear stress and pressure terms in the

optimisation procedure for a brain network as these parameters have a large weighting

on the discrete flow solutions. Nonetheless, Fry et al. [16]’s method could lead to new

insights into functional properties of microcirculation.

As mentioned in Chapter 2, vascular casting of a mouse brain is a complicated proce-

dure that can result in incomplete perfusion of the casting agent, and therefore retrieval

of a system of disconnected subnetworks that form a part of the mouse brain. Recent

work by Dobosz et al. [10] looks to overcome the limitations of conventional imag-

55

Figure 6.1: Visualisation of tumour morphology and tumour vascularisation. Scalebar, 250 µm and 100 µm (blowup). Reproduced from Dobosz et al. [10].

ing techniques. Dobosz et al. [10] applied multispectral fluorescence (UM) to the field

of tumour analysis and thus was able to create unprecedented three-dimensional and

quantitative insights into whole tumours with cellular resolution (see Figure 6.1). This

technique provides an exciting opportunity to procure an entire vascular dataset of a

mouse brain. The blood to the brain is supplied by the carotid and vertebral arteries,

and drained through the cerebral veins. Flow information for these blood vessels can

be acquired through MRI. Hence, with a full dataset, all boundary conditions would be

known and thus the flow estimation algorithm of Fry et al. [16] would not be needed in

the discrete model. In turn, this would lead to a significant reduction in the run-time

of simulations of discrete fluid transport, although considering the potential number of

vessel segments in such a dense dataset, the run-time would still be large.

Recent developments in mathematical modelling of microvascular networks presents

an exciting option for modelling blood flow in large incomplete networks. Shipley and

Chapman [47] developed continuum models for fluid transport through leaky neovas-

culature and porous interstitium of a solid tumour using mathematical homogenisation

methods. The multi-scale method exploits the separation of length scales between indi-

vidual capillaries and the tissue by assuming a periodic microstructure. This continuum

model has already been applied by Smith et al. [50] on coronary microcirculation. Fur-

ther development by Shipley et al. [48] has resulted in a coupled discrete-continuum

model for microcirculatory blood flow. Here, Poiseuille’s law is explicitly used to de-

scribe fluid transport through arteriolar vessels and coupled via point sources of flux

to a Darcy model for transport in the capillary bed. Considering current drawbacks of

56

network acquisition, the density of a brain network and limitations of the flow estima-

tions of Fry et al. [16], the discrete-continuum model provides an exciting option for the

mathematical modelling of fluid transport in large portions of the mouse brain.

Chapter 4 introduced the concept of a Green’s function approach by Secomb et al. [45]

to the transport of oxygen through blood vessels and into the surrounding tissue. The

novel approach allowed for faster computation of PO2 levels compared to finite difference

methods due to the lower number of unknowns. However, there is no experimental

data available giving detailed data of PO2 distributions in the mouse brain. Therefore,

the accuracy of this model of the microvascular networks of a brain cannot be fully

confirmed. Simulations of this model were run using external software provided by

Secomb [43]. Further work needs to be done to further understand the C + + coding

of this package to allow for a accurate adaption to the mouse brain, combined with an

increase in computational power to allow for simulations on more detailed networks.

57

7 Conclusion

This report has presented a novel combination of mathematical models for fluid and

mass transport in microvascular networks with applications to stroke research. Using

a discrete fluid transport model, experimental limitations are overcome by allowing for

detailed flow information at the individual blood vessel scale. The resulting information

was used to gain an insight into oxygen transport around the brain using a innovative

Green’s function model which allowed for a comprehensive study of oxygen diffusion at

the micro-scale. Understanding the processes of fluid and mass transport that occur

during an ischaemic stroke, could allow for the development of more effective methods

of treatment and improve our fundamental understanding of the disease state. Although

confirmation of these results are hard to come by due to experimental drawbacks, the

merging of these two models allows for insight into the deeply complex and relatively

unknown mechanisms occurring in the brain during this devastating illness. Recent

and future developments in both medical imaging and mathematical modelling offer

extremely exciting opportunities to build upon this report’s work, and to further our

understanding into the brain’s vascular environment during stroke, which can hopefully

help in the treatment and recovery of stroke patients.

58

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