a short course in modeling chemotaxis - ritter

8/13/2019 A Short Course in Modeling Chemotaxis - Ritter

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A SHORT COURSE IN

THE MODELING OF CHEMOTAXIS

By

L. R. Ritter1

As part of the Research Experience for Undergraduates

Summer Course 2004

Texas A & M University

College Station Texas

1This is a work in progress. Please disregard some redundancy in the last part of chap-

ter 4, and the incomplete results in that section. Please email any critical comments to lrit-

[email protected].


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cCopyright by L. Rylie Ritter 2004

All Rights Reserved

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Contents

List of Figures iii

1 Introduction 1

1.1 What isChemotaxis? . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 The Story of Dicty . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Beyond Dicty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Modeling: The Macroscopic Approach 7

2.1 The first derivation by Keller and Segel . . . . . . . . . . . . . . . . . 7

2.2 The Keller-Segel Model of Chemotaxis . . . . . . . . . . . . . . . . . 14

2.3 Traveling bands of chemotactic bacteria . . . . . . . . . . . . . . . . . 15

2.4 Some conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 Modeling: The Microscopic Approach 22

3.1 A Local Information Model . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 A Barrier Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.3 A Nearest Neighbor Model . . . . . . . . . . . . . . . . . . . . . . . . 30

3.4 A Gradient-Based Model . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.5 Chapter Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

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4 Applications of the Keller-Segel Model 36

4.1 Angiogenesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.1.1 What is Angiogenesis? . . . . . . . . . . . . . . . . . . . . . . 36

4.1.2 A Mathematical Model of Angiogenesis . . . . . . . . . . . . . 38

4.2 Atherogenesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.2.1 What is Atherosclerosis? . . . . . . . . . . . . . . . . . . . . . 46

4.2.2 A Detailed Look at Atherogenesis . . . . . . . . . . . . . . . . 52

4.2.3 A Mathematical Model of Atherogenesis . . . . . . . . . . . . 61

References and Suggested Reading 72

Appendix A

Linear Chemical Reactions 76

Appendix B

Bachmann-Landau Order Symbols 80

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List of Figures

1.1 Life Cycle of Dictyostelium Discoideum . . . . . . . . . . . . . . . . . 5

2.1 The experimental configuration . . . . . . . . . . . . . . . . . . . . . 16

4.1 Cartoon of the process of tumor induced angiogenesis. . . . . . . . . 394.2 Time evolution of the onset of angiogenesis for the system (4.12)

(4.15). Evolution of fibronectin decay showing an opening of about 6microns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.3 Time evolution of the endothelial cell movement. Note the bimodalityfor each time slice. This figure fails to correctly capture the events onthe time interval [0, 0.05] where the growth factor is changing mostrapidly. The numerical solution breaks down sometime after 0.2 hours

when the two maxima coalesce (the authors do not provide a plot ofthis phenomenon). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.4 This plot shows the evolution of the EC as in 4.3 on the time interval[0, 0.05]. Bimodality appears almost immediately. . . . . . . . . . . . 49

4.5 Evolution of the growth factor decay showing that growth factor isgone within 0.025 hours. . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.6 Evolution of the protease showing that the concentration comes to anear steady state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.7 Cartoon picture of the thick walled, multi-layered artery. . . . . . . . 554.8 Lesion with Smooth Muscle Cell Cap . . . . . . . . . . . . . . . . . . 71

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Chapter 1

Introduction

1.1 What is Chemotaxis?

In biological systems, living species sense their environment and respond to it.

Consider this example: A male tiger may enter into a new territory and detect the

scent left by a female tiger. Hoping for an opportunity to breed, our tiger is drawn

further into the new area. On the other hand, suppose the incoming male instead

senses the scent left by another male tiger. He interprets the scent as a message that

this territory belongs to another. If he is not up to fighting for hunting and breeding

rights, he will retreat. His response to the stimulus, a scent marking, whether he

moves toward or away from it is called taxis(from the Greek taxisto arrange).

Typically the word taxis is preceded a prefix that is determined by the type of

stimulus that organisms in a given system respond to. Several types of taxis are

well known; a few of these are outlined in table 1.1. The type of taxis that we

are currently interested in is chemotaxis. According to Horstmann [9], chemotaxis

is ...the influence of chemical substances in the environment on the movement of

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mobile species. Referring back to the story of our male tiger we note that the

influence the chemical signal in the scent marking might be to draw him nearer to

the source of the signal; we typically call this positive chemotaxis, and refer to

the chemical as a chemoattractant. When an organism is driven away from the

source of a chemical signal we refer to this as negative chemotaxis and in this

case we call the chemical species a chemorepellentor achemoinhibitor.

Table 1.1: Types of Taxis

Type of Taxis Stimulus Example Species

geotaxis gravitational force flies, birds

aerotaxis air (oxygen) bacteria

phototaxis light moths, mollusks

haptotaxis adhesive red blood cells in mammals

chemotaxis chemical signal bacteria, amoebae, red blood cells

Our focus will be on the mathematical modeling of chemotaxis and some of the

applications of these models. In particular, we will explore the derivation (from

different perspectives) of a model typically referred to as the Classical Keller-

Segel Model of Chemotaxis. The model takes its name from Evelyn F. Keller

and Lee A. Segel, mathematicians who introduced the model equations as a model

of chemotaxis in 1970. The most distiguishing characteristic of this model (which

will become clear in later sections) actually precedes the work of Keller and Segel. In

1953, Clifford S. Patlak derived essentially the first equation of the Keller-Segel

model. Patlak was motived in part by recent (at that time) studies of biological

systems such as modeling of migration of mosquitoes, mollusk, and homing pigeon.

He was also interested in modeling other phenomena such as predicting the length


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of polymer chains. Though seemingly unrelated, these topics can both be thought

of in terms of random walks in which various forces come into play. We will see that

this probabilistic approach has been used extensively in the modeling and study

of chemotaxis. Despite Patlaks publication, the name Keller-Segel Model is the

one commonly associated with a particular model of chemotaxis. Keller and Segels

early work was inspired by a tiny but impressive little character called dictyostelium

discoideumknown as Dicty to admirers (of which there are many, just try a

google search of Dicty amoebae to see how big of a following these little guys

have).

1.2 The Story of Dicty

Dictyostelium discoidium is a unicellular organism, in particular he is what is

known as a myxamoeba or more commonly slime mold. He is typically found

living in cultivated soil as well as in the sticky humus layers of woodland soils where

he feeds on bacteria. Dicty was first discovered by K. B. Raper in 1935, and it

became quite clear that he was part of a very strange family of amoebae. Most

notably, his life cycle depended on him being a social creature (see figure 1.1).

Given sufficient food supplies, a population of dictys move very little and simply

engulf the bacteria they need to survive. Once food resources are depleted, however,

the dictys will spread out over the available region. Eventually one of them will begin

to emit a chemical signal called cyclic Adenosine Monophosphate (cAMP). Other

myxamoebae will be attracted to this founder cell. They will migrate towards

the founder and will amplify the signal by emitting cAMP as well. Over time, the

myxamoebae will aggregate and differentiate to form a multicellular organism. In


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this new form, as a plasmoid, the individual dictys actually maintain their individual

integrity. This plasmoid or slug will move on slime toward light. Eventually, it will

sprout a fruiting body, release spores, and the life cycle starts again.

1.3 Beyond Dicty

The aggregation of myxamoebae and myxobacteria 1 has been of great interest to

those researchers who have developed models of chemotaxis [11, 12, 13, 20, 19, 26].

In fact the movement of myxobacteria is far less understood, the interested reader

is referred to [19] for an analysis of modeling of two mechanisms of aggregation

(reinforced slime trails and chemical signaling). However, the techniques used to

understand the behavior of dictyostelium discoidium are currently being used to

understand seemingly unrelated phenomena such as angiogenesis 2 (see section 4.1)

and more recently atherogenesis 3

(see section 4.2).

The next two chapters of this report will be dedicated to the mathematical

modeling of chemotaxis. There are two approaches to modeling to consider. These

are the macroscopic approach in which the behavior of a population is considered as a

whole and the microscopic approach where the focus is on the irregular movements of

a single member of a population. While the microscopic approach seems to predate

1Myxobacteria are a gram negative bacteria that exhibit gliding movement on solid substrates.

They were first isolated by Roland Thaxter in 1892. The prefix myx is from the Greek muxa

meaning slime or mucus.2Angiogenesis is the process by which tumors induce the growth of their own capillary network.3Atherogenesis is the onset of the disease atherosclerosis. This disease, commonly called hard-

ening of the arteries, involves the build up of lipids and debris in the arterial wall and is the leading

cause of human mortality throughout the developed world.


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Figure 1.1: Life Cycle of Dictyostelium Discoideum


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the macroscopic (see Patlak [20]), it was the macroscopic approach that was taken

by Keller and Segel and so will be presented here first. The modeling in chapter

2 will be based on the notions of Ficks and Fouriers law, and mass balance to

derive the equations governing a given system. We will find that this approach gives

rise to a system of (typically) parabolic partial differential equations (PDEs). In

chapter 3, the microscopic approach is presented. The tools used in this section are

primarily probability theory and the notion of a discrete space random walk. This

results in a continuous time, discrete space master equation for the probability of

finding an organism at a given place at a given time. However, we will see that from

the discrete space equations of the microscopic models one can formally recover the

reaction-diffusion PDEs of the macroscopic approach.


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Chapter 2

Modeling: The Macroscopic

Approach

2.1 The first derivation by Keller and Segel

Perhaps the best approach to studying this topic is to go through the work of

Keller and Segel as presented in their ground breaking paper of 1970 [11]. This

predates the formal Keller-Segel model, but is actually its first appearance. In

[11], the authors were interested in describing the aggregation behavior of cellular

slime mold which they proposed was the result of an instability. To this end, they

identified the following four species significant to the process:

a(x, t) The density of myxamoebae at the pointx (in R, R2, or R3) and at the time

t.

(x, t) The concentration of the chemical attractant acrasin. Here, you can think of

acrasin as being the chemical cAMP described in section 1.2.

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(x, t) The concentration of acrasinase, an enzyme that degrades the chemoattractant

acrasin.

c(x, t) The concentration of a complex that forms when acrasin and acrasinase react.

Next, Keller and Segel outline the assumptions that their model will be based on.

These are

(i) Acrasin is produced by the amoebae at a rate off() per amoebae.

(ii) Acrasinase is produced by the amoebae at a rate ofg(, ) per amoebae.

(iii) Acrasin and acrasinase react to form a complex that dissociates into a free

enzyme (acrasinase) and a degraded product vis.

+

k1

k1

c k2

+ degraded product1

(iv) , , andc diffuse by Ficks Law.

(v) The amoebaea move in the direction of increasing gradient of acrasin () and

by diffusion. The total number of amoebae remains relatively fixed.

In order to derive the equations of motion we consider an arbitrary volume

element of our system and balance the mass of each species. Let Vbe an arbitrary

1This compact notation really denotes three reactions

+ k1 c c

k1

+ and c k2 + degraded product

The values k with a subscript are the reaction rates.


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volume element with boundaryV. Balance of mass requires that the myxamoebae

density, for example, satisfy

Change in mass inV =

t

V

a(x, t) dx

Flux out of the boundary ofV = V

J(a) n dS (2.1)

+

Mass created (birth)

or deleted (death) in V=

V

Q(a) dx

whereJ(a) is the flux vector associated with speciesa,n is an outward unit normal

to V, and Q(a) is the net mass of amoebae created (birth - death) per unit time

per unit volume. We can write similar equations for the three other species giving

rise to the analogous terms J(), Q(), J(), Q(), J(c), and Q(c) all of which will

require clarification. However, before we determine the proper forms (based on the

modeling assumptions) of these terms, let us note that we can use the divergence

theorem to obtain2 V

J(a) n dS=V

J(a) dx.

Hence the equation (2.1) can be written

V

t

a(x, t) +

J(a)

Q(a) dx= 0.

2Unless otherwise stated, we will assume throughout this report that various functions and

vector fields are sufficiently smooth to satisfy the hypotheses of any theorem invoked and to write

as many derivatives as are necessary. Necessary conditions on the volume Vare similarly assumed

to hold.


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The only way for this result to hold for all possible choices ofV is if the integrand

is identically zero. So, from the integral mass balance (2.1), we arrive at the PDE

ta(x, t) = J(a) +Q(a).

Similar equations for the other species of course follow. But it remains to specify

the termsJ() and Q() for=a, , ,and c.

To characterize the flux and growth terms we turn back to the assumptions of

the model. The flux terms , , andc are determined by assumption (iv) as

J() = D for =, , c.

(This is a classical Fickian diffusion flux.) The parameterD with a subscript is the

diffusion coefficient which we can take here to be constants (although in general a

diffusion coefficient could depend on space or on the diffusing species themselves).

We use assumption (v) to determine a reasonable form for J(a) as follows:

J(a) = D2a+D1.

This flux is the defining characteristic of the model at hand. We can think of it

as capturing two important aspects of the movement of the species. Each of these

aspects corresponds to one of the two terms in the flux. Namely, the term

D2a D2> 0 (Fickian term)

says that the organisms avoid increasing concentrations of their own kind. We

can think of this loosely as capturing the spreading out to avoid overcrowding

phenomenon (recall that the vectorais the direction of steepest descent of thedensity a). The second term3

D1 (Fourier type term)3The numberingD1 andD2 are in keeping with Keller and Segels original notation


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captures the movement of the species in response to the chemical . This is the

chemotaxis phenomenon. WhenD1 >0 we can interpret this terms as saying that

amoebae move from low concentrations of towards higher concentrations. This

is positive chemotaxis indicating that is a chemoattractant. IfD1


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Equation (2.2) is the identifying equation of the Classical Keller-Segel model of

chemotaxis. Ill mention this again shortly. For now, let us return to Keller and

Segels analysis of this system just derived.

Let us note that the system (2.2)(2.5) is a coupled system of nonlinear parabolic

PDEs. A solution would require that we first impose initial conditions of each of

the species as well as identify the relevant geometry (where the variable x lives) and

what boundary conditions each species should satisfy. Dealing with four equations

of this form is certainly not impossible, and in fact is much easier to do today (at

least numerically) than it was in 1970. To a great extent, how one approaches a

system is influenced by what features one is trying to investigate. In this early work

[11], the authors proposed that the onset of aggregation of the myxamoebae was

the result of an instability. So, at this stage their intent was to perform a stability

analysis of (2.2)(2.5). To this end, they first simplified the model so as to highlight

the central characteristics.5 This is done by assuming that the complex c is in a

steady state (tc 0) and that the total concentration of enzymes (c+) is constant.This yields

k1 k1c k2c= 0, and c+= 0= const.

Substituting this into the system, we obtain the system of two equations

a

t = (D2aD1) (2.6)

t

= (D) +af() k() (2.7)

where

k() =0k2K/(1 +K), and K=k1/(k1+k2).

5In their own words In such a preliminary investigation as this it is useful for the sake of clarity

to employ the simplest reasonable model.


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The stability analysis consists of assuming that there is a steady state for the system

(a, ) = (a0, 0) and considering a perturbation off of this state by setting

a= a0+et cos(1x1+2x2), = 0+e

t cos(1x1+2x2).

(Herex= (x1, x2) R2.) Stability is determined by the sign of since positive indicates that the perturbation term grows exponentially and negative indicates

that the perturbation decays back toward the steady state. After some computationsKeller and Segel arrived at the following condition for instability:

D10D2a0

+a0f

(0)

k >1, where k= k(0) +0k

(0).

The motivation for writing the condition in this form is that we can see an expected

result by analyzing the first term. Since only positive solutions (a, ) are physically

feasible the size of the left hand side of this inequality is determined primarily by

the value of the ratioD1D2

=chemotactic effects

diffusive effects .

Instability, and consequently aggregation, is expected whenever the movement of

the amoebae is influenced primarily by the chemical signal. This is what is observed

in experiments and so this criterion seems to support both the hypothesis that an

instability is at work in the aggregation process as well as the proposed model of

chemotaxis. For further discussion of this analysis, the reader is encouraged to see[11].


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2.2 The Keller-Segel Model of Chemotaxis

The purpose of the current section is to clarify much of the language and notation

that will be used through this report. We will call any system of the following form

u

t = (u) ((u, v)v) +G(u, v) (2.8)

v

t = F(u, v) (2.9)

the Keller-Segel model of chemotaxis. In this formulation, the variable u represents

the density of some motile living species, and the variable v represents the concen-

tration of some chemical species. The coefficient is called the motility coefficient

and is the analog of the diffusion coefficients for nonliving species. In general,

may depend on space, u, v or some combination of these variables. Typically, how-

ever, 0 whether it is constant or not. The function is called the chemotacticsensitivity function. Most often, and throughout the present text, the chemotactic

sensitivity function is assumed to be linear in the species u [12, 9, 17] which is in

keeping with the notion that the flux of a species should be proportional to the

density of that species. Hence, we can write

(u, v) =u0(v).

Keller and Segel call 0 the chemotactic coefficient6. As we saw in section 2.1, if

0 >0 then (2.8) corresponds to positive chemotaxis and v is a chemoattractor. If

0< 0 then (2.8) corresponds to negative chemotaxis and v is a chemoinhibitor. In

the case that 0 equation (2.8) is a pure diffusion equation and no chemotaxisis accounted for in the model.

6Some authors refer to 0 as the chemotactic sensitivity function based on an apriori notion

that only linear dependence on u can occur in the chemotactic term.


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2.3 Traveling bands of chemotactic bacteria

If you ever encounter a traveling band of chemotactic bacteria, dont make di-

rect eye contact; they hate that.7 Joking aside, this section is based on the results

presented by Keller and Segel in 1971 [13] in which the authors consider an exper-

imental setup that results in visible bands of bacteria traveling up a capillary tube

filled with a mixture of oxygen and nutrient rich substrate. This example of the use

the Keller-Segel model is particularly interesting because it raises the question of

existence of a traveling wave solution to the system (2.8)(2.9).

The experiment that is being modeled consists of placing a population of Es-

cherichia coli (E. coli) bacteria in a petri dish and placing a capillary tube lined

with an oxygen and nutrient substrate into the center of the dish (figure 2.3). Af-

ter some time, a visible band of bacteria can be seen propagating up the capillary

tube as it consumes the nutrient substrate. In the case that all of the substrate is

not completely consumed by this band, a second band of bacteria will follow and

also propagate up the tube. A question that one might seek to answer is: what is

required of the system for traveling bands to be observed?

The assumptions of the model are

(i) There is a single substrate the concentration of which is denoted bys(x, t).

(ii) Bacteria, whose density is given by b(x, t), move by chemotaxis toward highconcentrations of the substrate and by diffusion.

(iii) Diffusion of the substrate is negligible

(iv) The bacteria consume the substrate at a rate ofk(s) per bacteria

7I was trying my best to be funny. Ill stop; I promise.


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Figure 2.1: The experimental configuration


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(v) The system is closed so that neither bacteria nor substrate enter (exit) through

the boundaries.

We can write the system of equations8 (in the same fashion as done in section 2.1)

as

b

t =

x

(s)

b

x b(s) s

x

(2.10)

s

t = k(s)b (2.11)

fort >0 and 0< x < L (L=length of capillary tube), and with the conditions

s(x, 0) =s0(x), b(x, 0) =b0(x), b

x=

s

x= 0 when x= 0, L.

The no flux conditions are because of assumption (v), ands0 and b0are given initial

distributions of substrate and bacteria, respectively. Since the focus of this study is

to determine if traveling waves can be observed, and since the length of the capillary

tube is considered to be very large in comparison to a bacterial band, the analysis

is facilitated by introducing a traveling wave coordinate and extending the domain

of interest to the whole real line. That is, let

z= x ct < z < , b(x, t) =b(z), and s(x, t) =s(z)

Further assume that k, , and s0 are constant and that in the far field s= s0. In

the variable z the system (2.10)(2.11) becomes9

cb = b + (b(s)s) (2.12)

cs = kb (2.13)8Observe that in this one space dimension setting the operator = x .9

ddz


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exercise for the reader11 The solution is given by

b(z) = c2s0k( )

s

s0

/ecz/ (2.15)

s(z) = s0

1 +ecz/ (2.16)

From this solution we see something similar to what was observed for the sys-

tem (2.6)(2.7). Note that the solution (b, s) above is only bounded if

>0 i.e if > .

That is to say the chemotactic influence must prevail over the diffusive influence

on bacteria for the solution (2.15)(2.16) to be meaningfulfor traveling waves to

exist. The model here is further supported by matching the wave speed c with

reported data. This is done by the authors in [13] where they showed that the wave

speed determined by (2.15)(2.16) is in reasonable agreement with available data.

The speed c can be determined by substituting (2.16) into (2.15) and integrating

over all z(since the number of bacteria is assumed constant). One finds that

c=N k

as0where N=a

b(z) dz

andais the cross sectional area of the capillary tube.

11HINT: Integrate (2.12) and use the condition at infinity to deal with the constant of integration.

Then divide through by b and integrate again. The solution will look like

b= Qs/ exp(cz/)

with Q arbitrary (corresponding to horizontal translation in z). To get the solution to look

like (2.15)(2.16) choose

Q= c2s0k( )

s/0 .


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2.4 Some conclusions

In this chapter we have seen the modeling of chemotaxis approached from the

consideration of an entire population of organisms and how they interact with their

environment. The most notable result is the equation (2.8) which captures in a

mathematical language the phenomenon of taxis of a living species in response to

a nonliving speciessome chemical in the environment. The form of the second

equation in the Classical Keller-Segel model as stated in (2.9) is decidedly vague.

The influences that drive the evolution of the chemical species can vary from sys-

tem to system and must be specified in correspondence with the assumptions of a

particular model. Hopefully the model examples discussed in sections 2.1 and 2.3

have given you some idea of what types of factors might go into determining the

appropriate right hand side of equation (2.9).

Does the organism produce the chemical?

Does the organism deplete the chemical, for example by eating it?

Are there other sources of the chemical, perhaps it is produce by some chemicalreaction?

Does the chemical naturally degrade?

Is there appreciable diffusion of the chemical?

These are just a few questions the modeler might want to consider. We will see

more examples of equations governing chemical species in the fourth chapter.

Now, we turn our attention to the modeling of chemotaxis from the microscopic

perspective. Given the uncertainty encountered when attempting to quantify the


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behavior of a single organism we will find that the appropriate setting for modeling

is that of probability theory.


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Chapter 3

Modeling: The Microscopic

Approach

I briefly stated at the end of chapter 1 that the microscopic approach to modeling

chemotaxis actually appears to predate the macroscopic. In his 1953 paperRandom

Walk with Persistence and External Bias1 [20], C. Patlak derives what is essentially

the first equation of the Keller-Segel model (equation (40) in [20]). Patlak was

himself motivated by studies of the motion of animals and the study of variability of

genes in populations. The primary method he used, and the method to be discussed

in this chapter, is that of treating a motion as a random walkin which forces that

influence the walker are accounted for. The majority of the discussion contained

in this chapter as well as the notation used herein is taken from the 1997 paper by

1Patlak defines persistence of direction as meaning that the probability a particle (organism)

travels in a given direction is not the same in all directions. External bias means that the

probability that a particle travels in a given direction depends on external forces acting on the

particle.

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Hans Othmer and Angela Stevens [19]. Even though the formal Classical Model

of chemotaxis came eighteen years after Patlaks paper, and the paper of Othmer

and Stevens another twenty-six years after that, Patlak stated in 1953

[I]t may be pointed out that the motion of organisms under the influ-

ence of various stimuli...may, to a good approximation, be treated by this

method.

Now we are interested in characterizing the motion of a single organism walking

on a lattice who can, in the interest of simplicity, move from one lattice point to any

adjacent lattice point at any instant in time. To further simplify the exposition, we

can begin by assuming that the lattice is in one spatial dimension with the points

on the lattice indexed by the integers i Z. Thus, at any time t, a walker ati maymove to the point i 1, i+ 1, or remain at i. Rather than introducing a variableto represent our walker, we seek to determine a probability density function (pdf)

corresponding to the movement. To that end, define the continuous time, conditional

pdf

pi(t) =P(the walker is at i at time t|the walker started at i = 0 at time t = 0).

We are interested in chemotaxis, so we would also like to represent the control

speciesi.e. chemoattractor or repellent. In keeping with [19], we will assume that

the control species is distributed on an embedded lattice of half step sizes. Tocharacterize this, we introduce the sequence

W = (...,wi1/2, wi, wi+1/2,...,w1/2, w0, w1/2,...)

of weights associated with the control species on the embedded lattice (you can think

ofwq as the amount of chemical at the point qfor half integer q). To construct a


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master equation of our pdf we need to represent the transitional probability functions

that describe the probability that our walker takes a step. These functions should

depend on Wsince our walker is under the influence of the control species. Define

T+i (W) = transitional probability per unit time of a jump from i to i+ 1,

and

Ti (W) = transitional probability per unit time of a jump from i toi 1.

We can write the master equation [19]

pit

=Ti+1(W)pi+1(t) +T+i1(W)pi1(t) (T+i (W) +Ti (W))pi(t) (3.1)

In plain English the above can be stated (roughly) as:

The change in probability of finding our walker at i

(Probability the walker moves from i + 1 to i) (the probability the walker was at i + 1)

+

(Probability the walker moves from i 1 toi) (the probability the walker was at i 1)

|

(Probability the walker moves from i) (the probability the walker was at i).

Moreover, we can say that the quantity

(T+i (W) +Ti (W))

1


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is the mean waiting time at the ith site. Equation (3.1) appears at first like a

Markov process (the current state being independent of the previous state). How-

ever, this is in general not the case since the walker can influence the weights. We

saw this in chapter 2 in the models described there. That is, the organisms may

secrete, eat, or otherwise influence the state of the control species. So, even though

the transitional probabilities are not shown to depend explicitly on the p is, there is

implicit dependence as Wcan, and usually does, depend on the pis.

In the rest of this chapter, we will consider four different models that determine

the form of the transitional probabilities. These are (1) a local information model,

(2) a barrier model, (3) a nearest neighbor model, and (5) a gradient-based model.

We will also see how these relate, at least formally, to the macroscopic model de-

scribed in the previous chapter. Equations that govern the control species will be

ignored here.

3.1 A Local Information Model

Here, the transitional probability depends only on the weight at the node of

interesti.e.

Tn(W) =T(wn) n2.

In this assignment we are also separating the units by letting the parameter have

units of 1/time so that T is dimensionless. Substitution into the master equa-

tion (3.1) yields

pit

=T(wi+1)pi+1(t) +T(wi1)pi1(t) 2T(wi)pi(t) (3.2)2Note that this means thatT+n =T

n.


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This equation assumes that our walker can only sense and hence respond to the

measure of the chemical right where he is standing. We can analyze the features

of this model in more depth by relating it to a continuous time, continuous space

model.

To do this, let us assume that the mesh consists of equally spaced nodes a distance

ofhapart and define x= nh, n Z. Next, suppose that we can expand the termson the right hand side of (3.2) in Taylor series3 for small h (for ease of notation I

will write T(wi) as Ti and denote derivatives with respect to x by a subscript)

Ti+1pi+1 = Tipi+h

Tipix

+h2

2

Tipi

xx

+h3

3!

Tipi

xxx

+O(h4) (3.3)

Ti1pi1 = Tipi h

Tipix

+h2

2

Tipi

xxh

3

3!

Tipi

xxx

+ O(h4) (3.4)

The term O(h4) means that all remaining terms are at most of order h4 as h 0.An explanation of this order notation can be found in Appendix B. It should be

noted here these expansions are not rigorous because we have not shown that these

derivatives are indeed bounded as h 0. It is not a trivial matter to establishboundedness of the derivatives and in fact for some relevant models (choices ofT)

it turns out that it can not be done4. Nevertheless, we proceed by substituting (3.3)

and (3.4) back into the right hand side of (3.2) and dropping the subscript i. Note

3To clarify the computations, for x = ih Fi1 denotes F((i 1)h) =F(x h). And for small

h we can write the Taylor expansion about x as

F(x h) = F(x) + (h)dF

dx(x) +

(h)2

2!

d2F

dx2(x) +

(h)3

3!

d3F

dx3(x) +

4Many interesting systems exhibit blow-up in finite time or Dirac concentration solutions. These

solutions can have relevant physical meaning.


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that the odd terms and the zero-order terms vanish to leave

p

t =h2

T(w)p

xx

+O(h4).

We (formally) set

limh0

lim

h2 =:D (0,).

When and h both go to zero this leads to

pt

= D 2

x2

T(w)p

= D

x

T(w)

p

x+ pT(w)

w

x

.

It is not difficult to show that in higher dimensions the above generalizes to

p

t =D T(w)p +pT

(w)w .This is an equation of the form (2.8). We can rewrite this equation as

p

t =D (p (p, w)w) (3.5)

and see that we have terms similar to the motility coefficient5

= DT(w)

and the chemotactic sensitivity function

(p, w) = pDT(w)

with the prime here indicating differentiation with respect to w.

5I say similar to since p is a probability density function as opposed to number or mass.


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A number of observations can be made about equation (3.5). Most obvious is that

the chemotactic sensitivity function is linear in our measure of the mobile speciesp.

This is not surprising and is in keeping with the notion that the chemotactic flux

be proportional to species density. The dependence of on Talso yields interesting

information about how the form of the transitional probability function affects the

anticipated behavior of the system. In particular, if we consider the case that T

is constant in w, that is the probability of a movement to an adjacent node is

independent of the control species, then T and hence is zero. Then (3.5) is a pure

diffusion equation and the system does not exhibit chemotaxis. IfTis a decreasing

function ofw, then >0 which we expect to indicate positive chemotaxis. To check

the reasonableness of this observation, we recall that by our definition ofTand the

mean waiting time:

T(w)1 = 2 times the mean waiting time where the weight is w.So, ifT(w)< 0, then

w

T(w)

1>0

and the mean waiting time is increasing in w. If follows that accumulation of the

organisms is more likely to occur where the concentration of the control species is

large. This is consistent with positive chemotaxis. It is similarly clear that ifT is

an increasing function, then the system exhibits negative chemotaxis. Finally, I willmention that (3.5) admits a nonconstant steady state solution T p harmonic.


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3.2 A Barrier Model

The model is again determined by what form the transitional probabilities take

and how they depend on the control species. In the current section we want the

movement of our organism from one node an adjacent node to depend on barrier

between the nodes. Here is where it is particularly useful that we prescribe the

weights of the control species be on an embedded lattice of half step size. As an

example, suppose that

Ti (W) =T(wi1/2)

so that a jump from i to i+ 1, for example, depends on a barrier at i+ 1/2. The

coefficient 1/time is again introduced so that Tis dimensionless. For this choiceof transitional probability the master equation is

pi

t

=T(wi+1/2)pi+1+T(wi1/2)pi1

T(wi+1/2) + T(wi1/2)pi (3.6)

As in the previous section we define x = ih whereh is the size of the grid mesh and

assume that

D= limh0

lim

h2

is a positive, finite number. Upon expanding the right hand side of (3.6) to fourth

order and letting , h 0 we arrive at

pt =D x

T(w) px

(3.7)

I leave the computation as an exercise for the reader. The higher dimensional analog

isp

t =D

T(w)p

(3.8)


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The notable characteristic of this equation is that there is no dependence on the

second spatial derivatives of the control species. That is to say that this is not an

equation of the form of the Keller-Segel model. Rather it is a convection diffusion

equation. This type of equation has been studied by Lapidus in [15] who showed

that solutions can exhibit transient aggregation but that this does not last. He

terms this phenomenon pseudochemotaxis. From the form of the transitional

probabilities there is no dependence on the weights that are on the walkers lattice.

Moreover, there is no correlation between left and right jump probabilities. One

possible adjustment to this is to insist that the mean waiting time be constant on

the lattice. An example of this would be to define the transitional probabilities as

Ti (W) =T(wi1/2)

T(wi+1/2) + T(wi1/2)(3.9)

so that

T+i (W) +Ti (W) = i.

It can be shown that this choice ofT does lead to an equation of the Keller-Segel

type. But, rather than consider this example now, I will conclude this section and

reintroduce essentially this same transitional probability form for a nearest neighbor

model.

3.3 A Nearest Neighbor Model

As the name suggests, a nearest neighbor model is one in which the transitional

probability function at a given site depends on the weights of the control species at

one or both of the adjacent sites. The nearest neighbor model that will be considered

as an example in this section is similar to the last one described in which the mean


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waiting time at all sites is constant. We define the transitional probabilities by

T+n(W) = wn+1

wn+1+ wn1, and Tn(W) =

wn1wn+1+wn1

.

Note that this is of the same general form as (3.9) except for dependence on the

integer valued grid points, but it corresponds to the special form ofT(w) = w.

This particular choice was given by Davis [3] to describe a reinforced random walk.

In Davis model, he considered the value of a weight at a given spatial node as a

function of the number of times a walker occupied that node. This type of model is

analyzed by Othmer and Stevens [19] in relation to the movement of myxobacteria.

This is because a myxobacterium can excrete a slime trail that may be used by other

myxobacteria to facilitate their movement. Having the weight at a node increase

each time it is occupied reflects the bacterias preference for moving along existing

slime trails.

Let us introduce the new function Ndefined by

Tn+1(W) = wn

wn+2+wn=:N(wn, wn+2).

Then

T+n1(W) =N(wn, wn2),

and the master equation for this case can be written

pnt

= (pn+1N(wn, wn+2) +pn1N(wn, wn2)pn) . (3.10)

To evaluate this model we again expand the the right hand side about the point

x = nh for small h and formally take to infinity and h to zero. (The reader is

encouraged to work out the details)6 Defining the parameter D in the same way as

6PutN(u, v) = uu+v and use that

N(u, u+ u) = N(u, u) +

vN(u, u)u+O(u2) =

1

2

1

4uu+O(u2).


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before, the corresponding continuous space equation to (3.10) is

p

t =D

x

1

2

p

x p

w

w

x

.

The analog in two or more spatial dimensions is

p

t =D

1

2p p

ww

. (3.11)

First, we see that (3.11) is an equation of the type of the first equation of theKeller-Segel model of chemotaxis. Moreover, we can identify the motility coefficient

and chemotactic sensitivity function as

=D

2 (p, w) =D

p

w=Dp

d

dwln(w).

We see that the chemotactic sensitivity function is again linear in the measure of

the motile species p. Also, this form of corresponds to the Weber-Fechner Law

used by Keller and Segel in [13]. This may also reflect the slightly larger range of

lattice that influences the conditionally probability at a given node. That is, the

probability at the nth lattice point depends on the weights at the (n 2)nd, thenth,and the (n+ 2)nd nodes.

We can consider a more general form of the transitional probabilities

Tn(W) =T(wn1)

T(wn+1) + T(wn1)

of which Tis the identity is just one example. The analysis leads to the continuous

equation

p

t =D

1

2ppT

(w)

T(w)w

. (3.12)


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The motility coefficient is the same as in (3.11) and the chemotactic sensitivity

function is

(p, w) =DpT(w)

T(w)=Dp

d

dwln(T(w)).

The chemotaxis is there for positive whenever Tis strictly increasing and negative

when T is strictly decreasing. This is quite different than the local information

model of section 3.1.

3.4 A Gradient-Based Model

We conclude this chapter with one last random walk model. A gradient based

model is intended to capture the phenomena of organism that probe the envi-

ronment before making a move and having the decision to move be based on a

consideration of weights at the current point and the next point to be occupied.

To keep things simple we consider a model in which the transitional probability

function depends linearly on the difference of the weight at a given point and the

next adjacent point. For example

Tn(W) = (+((wn1) (wn))) (3.13)

where is a known continuous function7 and and are nonnegative constants.

We again derive an equation of a continuous space variable. For this model, we

obtainp

t =D

x

p

x 2p(w) w

x

, (3.14)

7The form of T in (3.13) could be prescribed to depend on barrier weights (e.g. at

wn1/2); it could also be extended to higher order dependence for example by adding a term

((wn1) (wn))2

and so forth.


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or in two or more dimensions

p

t =D (p 2p(w)w) . (3.15)

If both and are positive, then this equation is of the Keller-Segel type with

motility coefficient = D and chemotactic sensitivity function

(p, w) = 2p(w).

The chemotaxis represented would be positive ifis increasing and negative if is

decreasing. If= 0 (equivalently ifis constant) then there is no taxis and (3.15)

is a pure diffusion equation. This is identical to the case considered in section 3.1 in

which the probability of movement is independence of the control species. If there

is no basal transition rate, = 0, then there is no diffusion and the only transport

is due to chemotaxis.

3.5 Chapter Conclusions

We have seen that the defining characteristic of the Keller-Segel model can be

derived directly as in chapter 2 from mass balance considerations and, at least

formally, by considering irregular movements of a single member of a population

from a probabilistic perspective. But, as was stated earlier, we have given little

consideration as to the evolution of the control species other than to point out atthe end of chapter 2 what some of the phenomena might need to be accounted for.

Some issues of interest at this point may be

What are equations for the stimulus?

What types of solutions to these equations might arise?


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What are some applications of the Keller-Segel model?

In the next chapter, I will focus on the last question which will include some examples

of the previous two.


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Chapter 4

Applications of the Keller-Segel

Model

The model as first derived by Keller and Segel was done so in the context of a

specific example of chemotaxisi.e. aggregation of slime mold [11]. In this final

chapter, we will explore two other examples of problems for which the Keller-Segel

model has been used to describe the evolution of a system in which organisms (in

these cases human cells) respond to chemicals in their environment.

4.1 Angiogenesis

4.1.1 What is Angiogenesis?

When a tumor first appears in the human body it is in a stage that would be

termed avascularmeaning that it does not have its own blood supply but rather

depends on diffusion of oxygen and other nutrients from nearby tissues for its survival

36


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and growth. Since nutrients are unable to reach all the cells in a new tumor some

of the cells begin to die (become necrotic). Small avascular tumors are typically

characterized by a necrotic core surrounded by living cells. In the absence of its own

blood supply the young tumor will obtain a limited size of 1-3 mm [17]. Continued

growth can only occur if the tumor can somehow induce the growth of its own

capillary network. A hypoxic tumor may be able to do just this by sending chemical

signals into the surrounding tissue which are responded to by cells in the vasculature

in such a way that these responding cells migrate and build capillaries connecting

the tumor to the rest of the blood stream. The process of inducing new vasculature

is called angiogenesis. Angiogenesis is not unique to tumor growth; it occurs for

example during prenatal development. However, angiogenesis has become, in some

circles, essentially synonymous with tumor development [17].

In order to understand the process, it is necessary to have some basic idea of

the structure of vessels in the human body. There are several vessel types in the

body that vary in size, structure and function. These include large arteries such

as the aorta that delivers fresh blood from the heart, veins and the vena cava

that return blood to the heart, and tiny little capillaries throughout the body1

that deliver nutrient rich blood to all of tissues of the body including some of the

larger vessels. The process of tumor induced angiogenesis2 primarily involves the

small veins and capillaries3. These can be considered as small cylinders that are

lined with a monolayer of cells called endothelial cells(ECs) resting on a thin basal

membrane and perhaps supported on the outer side by layered smooth muscle cells

1There are on the order of 1010 capillaries in a human being.2Throughout the rest of this section, the word angiogenesis will always be used to denote tumor

induced angiogenesis.3Most cells in the body are no more than 50m away from a capillary ([10] pg. 250).


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(SMCs). The ECs are the main cellular species outside of the tumor responsible for

angiogenesis.

Angiogenesis occurs in three stages (see figure 4.1): (1) activation of ECs, (2)

migration and proliferation of ECs, and (3) maturation of the new vessels. The

first stage is triggered when a hypoxic tumor emits a chemical signal in the form

of any of several specific chemicals known collectively as TAFs (tumor angiogenic

factors). The chemical signal is heard by an endothelial cell when the chemical

binds to a receptor on the EC surface. This results in gene expression within the

EC causing it to change phenotypes from a quiescent to an activated type. The EC,

which are lining a vessel near to the tumor, then begin to excrete MMPs (matrix

metalloproteinases) which degrades the basal membrane and any cells comprising

the outer wall of the vessel. In this way the ECs can begin to migrate from the

vessel wall toward the signaling tumor. This movement is facilitated by the TAFs

(chemotaxis) as well as by adhesive fibronectin secreted by the ECs (haptotaxis).

It is believed that the migrating ECs begin to attach to each other to form tube

like structures [17]. Finally, SMCs may migrate and attach to the new structure to

enhance its stability. A loop may form that runs from the vasculature to the tumor

and back. This is a particularly dangerous development from a medical perspective

as it can result in metastases.

4.1.2 A Mathematical Model of Angiogenesis

During the past decade, mathematics has been used to attempt to better under-

stand the process of angiogenesis as well as develop methods of predicting the effects

of various interventions. Numerous models and studies have been documented (c.f.


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the review [17] and the references therein). In this section, I will discuss the model

of Levine, Sleeman, and Hamilton given in [16]. In their 2001 paper Mathematical

Modeling of the Onset of Capillary Formation Initiating Angiogenesis, they offer a

model of the first two stages of angiogenesis that captures the bimodal migration of

endothelial cells (the formation of a tube structure).

Levine et al. base their model on the interactions between one cellular species

(ECs), four chemical species, and the receptors on the ECs. These are denoted by

v The concentration of TAFs.

c The concentration of proteolytic enzymes (MMPs).

r The density of receptors to TAFs on the ECs.

l The concentration of an intermediate receptor complex.

The density of ECs. and

f The density of fibronectin.

The model is then based on the following assumptions:

(i) A TAF (v) bonds with a receptor (r) to form the intermediate complex (l).

v+r

k1

k1

l

(ii) The complex l induces the production of a new receptor and an MMP.

l k2 c+r


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(iii) The MMP degrades fibronectin to produce an intermediate product that in

turn produces an MMP plus a degraded product.

c+f k3 q k4 c+ degraded product

(The intermediate product q is not of interest.) This occurs by Michaelis-

Menten kinetics4 (as opposed to linear reactions).

(iv) The ECs produce fibronectin at a rate off(fM f) per EC; wherefM is theamount of fibronectin in a normal capillary.

(v) The decay of fibronectin and MMP is negligible.

(vi) The ECs follow a reinforced random walk using a nearest neighbor model with

transitional probabilities T(c, f) = T1(c)T2(f).

(vii) The kinetics of the reactions in assumptions (i) and (ii) are in steady stateprior to EC migration.

(viii) The system is closed with no flux of EC into or out of the domain.

(ix) The system is on a 1D lattice x= 0 to x = 1

4For a linear reaction, we might interpret this schematic as (see appendix A)

df

dt = k3cf.

The assumption that the kinetics are of Michaelis-Menten type means that we modify this equation

so that iff is small, the equation is essentially the same linear kinetics, but when f is large the

equation is of zero orderdf/dt= constant. This can be achieved by writing

df

dt = k3cf

1 +2f.


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The first two assumptions immediately give us preliminary differential equations

forv, r, l, and c based on the condition of linear reactions. These are

v

t = k1rv+k1l (4.1)

r

t = k1rv+ (k1+k2)l (4.2)

l

t = k1rv (k1+ k2)l (4.3)

c

t = k2l (4.4)

Assumptions (iii) through (v) allow us to write the equation for fibronectin

f

t =2cf1 +2f

+f(fM f) (4.5)

where the 2= k3, and 2= k4/k3. Finally, we can obtain the equation

t

=D

x

x

x

(ln ) (4.6)which is an equation consistent with (3.12) and assumption (vi), namely follows a

reinforced random walk of a nearest neighbor model5. At this point, an equation for

each of the species has been specified. But rather than consider the system (4.1)

(4.6) the authors of [16] note that while EC move on a time scale of days, the

chemical kinetics described in the first two assumptions occur on the time scale of

seconds. They thus argue that species r and l are in a pseudo steady state during

EC movement. The phrase pseudo steady state is used because the solution r,

5The term in equation (4.6) is equivalent to the term Tin (3.12) and should not be confused

with the use of the character used in chapter 3 to discuss the gradient based model. While this

notational change may be confusing, the use of the character in chapter 3 was consistent with

the notation of Othmer and Stevens whose work chapter 3 was based on, and here the use of is

consistent with the notation of Levine et al. the authors of the current model.


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l is obtained by mechanically setting the left hand side of (4.3) to zero. However,

the solution obtained by this method can not hold for small t unless r(x, 0) = 0 or

v(x, 0) = 0 which is physically unfeasible. To see how this argument proceeds, note

that since there is no complex l prior to the reactions we can sum equations (4.2)

and (4.3) and integrate to obtain

r(x, t) + l(x, t) =r(x, 0). (4.7)

Setting the left hand side of (4.3) to zero and using the above relation we can write

expressions for r,l, ct , and vt

r(x, t) = r(x, 0)

1 +1v(x, t) (4.8)

l(x, t) = 1r(x, 0)v(x, t)

1 +1v(x, t) (4.9)

c

t =

k21r(x, 0)v(x, t)

1 +1v(x, t) (4.10)

v

t =

(k11 k1)r(x, 0)v(x, t)1 +1v(x, t)

(4.11)

Finally, by employing the observation that the number of receptors on an EC remains

essentially constant

r(x, t)(x, t)

r(x, 0)(x, t)

= const,

assuming that the parameters k21 = k1 k11 :=k, and by setting 1 =k we


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arrive at the system of interest

t = D

x

xln

(4.12)

v

t =

1v1 +1v

(4.13)

c

t =

1v

1 +1v (4.14)

f

t =

2cf1 +2f

+f(fM f). (4.15)

Completing the statement is the no flux boundary condition (assumption (viii))

xln

= 0 when x= 0, 1

and given initial distributions

(x, 0) =0(x) 0, v(x, 0) =v0(x) 0,c(x, 0) =c0(x) 0, f(x, 0) =fM(x) 0.

We conclude this subsection with the results of a numerical experiment conducted

by the authors of [16] that shows some of the expected outcomes such as the decay of

fibronectin in an interval consistent in length with a typical capillary, the bimodal

concentration of ECs in this interval indicating a capillary sprout, and the rapiduptake of angiogenic growth factor by the ECs.

Figures 4.24.6 show the solutions of the system (4.12)(4.15) obtained numeri-

cally. The form of the transitional probability function used is

(c, f) =1(c)2(f) =

1+c

2+c

1 1+f+f

2


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45

which is intended to account for both chemotaxis in response to protease c and

haptotaxis in response to fibronectin f. This is based on an ad hoc postulate that

the mechanics obey a power law (1(c) = c1 for example) which is then mollified

(that isi is bounded away from both zero and infinity). The problem is scaled with

initial conditions

(x, 0) = 1, c(x, 0) = 0, f(x, 0) = 1, and v(x, 0) = 15100(1 cos(2x))100.

The value of100 = 0.933 1030 is chosen so that10

v(x, 0) dx = 1. This choice

of initial condition on the TAFs corresponds essentially to a unit pulse 6 of TAF at

the center of the lattice at the onset of the numerical experiment. The remaining

parameter values used are given in table 4.1.

Table 4.1: Parameter Values [16]

D= 3.5e 5 1 = 0.001 2 = 1.0 1 = 1.2 1 = 1.0 2 = 0.001

2 = 1.2 1 = 73.0 1 = 0.007 = 0.222 2 = 19.0 2 = 1.28

Figure 4.2 shows the decay of fibronectin in an interval x (0.44, 0.56) aboutwhich the TAFs are originally concentrated. The width of the channel corresponds

to about 6 microns which is in the range of diameters of a typical capillary. Figures

4.3 and 4.4 show the corresponding bimodal concentration of endothelial cells at

the ends of this interval which is indicative of the tube structure growth of a new

capillary. The remaining figures, 4.5 and 4.6, show the phenomenon of rapid uptake

of TAFs by the endothelial cellsthe TAF is nearly completely depleted within

6The sequenceFm(x) = 15m(1cos(2x))m tends to the Dirac function (x1/2) asm

in the sense of distributions.


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1/40th of an hour, and during this same short period the density of protease attains

a near steady profile. For an analysis of the system with respect to changes in some

of the parameter values please see [16].

4.2 Atherogenesis

A more recent application of the Keller-Segel model of chemotaxis is the current

study of the disease atherosclerosis. Before discussing the modeling approach, let

me familiarize the reader with the this disease.

4.2.1 What is Atherosclerosis?

Atherosclerosis is a chronic disease involving the accumulation of lipid laden

cells in the arterial wall. The precursors of the disease can begin in humans during

early childhood with the appearance of fatty streaks which are purely inflamma-

tory lesions [23]. These early lesions, which consist primarily of monocyte7 derived

macrophages8, T-lymphocytes9, and lipids10, may be found throughout the arterial

7A large white blood cell with finely granulated chromatin dispersed throughout the nucleus

that is formed in the bone marrow, enters the blood, and migrates into the connective tissue where

it differentiates into a macrophage.8A phagocytic tissue cell of the mononuclear phagocyte system that may be fixed or freely

motile, is derived from a monocyte, and functions in the protection of the body against infection

and noxious substances9Any of several lymphocytes (as a helper T cell) that differentiate in the thymus, possess highly

specific cell-surface antigen receptors, and include some that control the initiation or suppression

of cell-mediated and humoral immunity.10Any of various substances that are soluble in nonpolar organic solvents (as chloroform and

ether), that with proteins and carbohydrates constitute the principal structural components of


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Figure 4.2: Time evolution of the onset of angiogenesis for the system (4.12)(4.15).

Evolution of fibronectin decay showing an opening of about 6 microns.


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Figure 4.4: This plot shows the evolution of the EC as in 4.3 on the time interval

[0, 0.05]. Bimodality appears almost immediately.


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Figure 4.5: Evolution of the growth factor decay showing that growth factor is gone

within 0.025 hours.


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Figure 4.6: Evolution of the protease showing that the concentration comes to a

near steady state.


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tree. More advanced atherosclerotic lesions, known as fibrofatty and fibrous plaques,

are principally found in the larger muscular arteries. An advanced lesion, in partic-

ular a fibrous plaque, is characterized by a core of lipid laden macrophages (called

foam cells), leukocytes, debris (potentially necrotic) and a fibrous cap composed of

smooth muscle cells intermingled with connective tissue proteins. Initially, as the

fibrous plaque forms, and the wall of the artery thickens, there is a compensatory

dilation and luminal preservation, a process called remodeling. With continued

plaque growth, there is eventual luminal encroachment restricting the flow of blood

through the artery. Rupture of the fibrous cap can result in sudden, complete occlu-

sion of an artery leading to ischemia of the heart or brain i.e. heart attack or stroke.

Atherosclerosis, with its potential for causing these catastrophic medical events, is

the leading cause of human mortality in the western world and much of Asia [23].

Through mathematical modeling and numerical simulation, we seek to provide a

deeper understanding of the disease process and to develop methods of predicting

the influence of various risk factors such as blood pressure and cholesterol as well as

changes in the structural and mechanical properties of the artery, aspects that are

critical to developing strategies for combating this prolific disease.

4.2.2 A Detailed Look at Atherogenesis

Recently, mathematical approaches to studying some of the physical-chemicalevents necessary for the development of atherosclerotic lesions have been offered in-

cluding models of lipoprotein transport and accumulation in the arterial wall [18, 24].

The presence of low density lipoprotein (LDL), more precisely of chemically mod-

living cells, and that include fats, waxes, phospholipids, cerebrosides, and related and derived

compounds.


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ified or oxidized LDL (oxLDL), appears to be a key trigger of the initial immune

response that characterizes atherogenesis. Cobbold, Sherratt, and Maxwell [2] pro-

posed a mathematical model ofin vitroLDL oxidation. Their model takes the form

of a system of time dependent ordinary differential equations (ODEs) governing the

oxidation process. In the work described herein, we have adapted their approach to

allow for spatial variation in our model of atherogenesis.

We propose a mathematical model that describes the early stages of atheroscle-

rosis including formation of a fibrofatty plaque. Our model focuses on the inflam-

matory component that is key to the pathogenesis of CVD. In particular, we model

the secretion of and response to chemotactic stimuli by various cellular species and

LDL in the blood and arterial wall. Because this study centers on the interplay

between chemical and cellular species, we employ a classical model of chemotaxis

first proposed by Keller and Segel in 1970 [11]. This approach is inspired inpart by

recent models of angiogenesis such as that described in section 4.1.2. Our model is

expressed as a nonlinear system of coupled parabolic partial differential equations.

Disease Description

Although the fatty streaks that can occur even in infancy may be found through-

out the arterial network, the advanced lesions of the disease of atherosclerosis are

found in the larger arteries such as the abdominal aorta, coronary arteries, cerebral

arteries and others [10]. These various arteries, while of different sizes and loca-

tions in the body, have a similar structure. They can be thought of as thick walled,

multilayered tubes whose innermost surface is exposed to flowing blood with the

outermost surface bounded by surrounding tissue (see figure 4.7). The radius of the

lumen is roughly one half of the radius of the artery itself. The arterial wall contains


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three distinctive layersthe intima, media, and adventitia, whose constituents are

suited to the functions of the given layer. The intima is the innermost layer of the

arterial wall. It is characterized by a monolayer of endothelial cells that form the

crucial interface between the arterial wall and the blood flow, a thin basal membrane

on which the endothelial cells rest, and a subendothelial layer of smooth muscle cells

and fine collagen fibrils whose thickness varies with age and health. The intima ini-

tially accounts for < 10 % of the total wall thickness. However, the inflammation

that characterizes the atherogenic process takes place primarily within this layer

which is considered extensively in our proposed model. The endothelial cells serve

several purposes that include providing a smooth surface for fluid flow, secretion

of anticoagulants to maintain the fluid state of the blood, and chemical signaling

of immune cells. This barrier assumes great significance in the development of and

protection against atherosclerosis by regulating the passage of chemical and cellular

species between the artery wall the and the adjacent blood stream. The media is

the largest layer of the artery and consists of multiple layers of smooth muscle cells

aligned nearly in the circumferential direction in an extra cellular matrix of collagen

and elastin. The muscular media is high in strength and is the most resistant layer

to loads both in the axial and circumferential directions [8]. The outermost layer,

the adventitia, consists of fibroblasts, fibrocytes (collagen and elastin producing

cells), and thick bundles of collagen fibers. The fibers are aligned nearly in the axial

direction. They provide reinforcement of the arterial wall and, at high pressures,

prevent overstretch and rupture of the vessel.

According to recent theory [23], the lesions of atherosclerosis that form the in-

timal layer constitute a chronic inflammatory response to injury. The first step

involves,endothelial dysfunction. Although poorly understood, this process appears


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Figure 4.7: Cartoon picture of the thick walled, multi-layered artery.


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to be characterized by a change in the permeability of the endothelial layer that al-

lows lipids to migrate into the subendothelial layer followed by an influx of the cells

that comprise the immune response. The change in permeability is also accompanied

by an increase in the adhesiveness of the endothelial layer and a change from an-

ticoagulant to procoagulant properties. A number of factors have been considered

as possible causes of endothelial dysfunction. These include multiple typical and

atypical CVD risk factors such as cigarette smoking, diabetes mellitus and hyper-

tension (all of which generate free radicals), elevated LDL cholesterol blood levels,

and possibly even infection by microorganisms (e.g. herpes viruses or chlamydia

pneumonae) [23]. It is observed that certain sites, in particular the opposing wall

at an arterial bifurcation, are especially vulnerable to endothelial injury and subse-

quent dysfunction resulting in plaque formation. This pattern suggests that blood

flow changes including increased turbulence, decreases in shear stress, flow reversal,

and stagnation zones occurring at these sites may also contribute to the dysfunction.

Endothelial cells are known to be sensitive to shear stress, and recirculation due to

flow separation can deprive the vessel wall of fresh, nutrient rich blood resulting in

dysfunctional cells.

Lipoprotein Oxidation

Lipoproteins are micellar particles produced by the liver and intestines which

contain regulatory proteins that direct the blood trafficking of cholesterol and other

lipids to various cells in the body. There are four major classes of lipoprotein

very low-density lipoprotein (VLDL), intermediate-density lipoprotein (IDL), low-

density lipoprotein (LDL, the bad cholesterol containing particle), and high-

density lipoprotein (HDL, the good cholesterol containing particle). The bulk


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of cholesterol is contained within the latter two particles. The lipoprotein structure

consists of a lipid core containing cholesterol esters and triglicerides, and a coat

that is composed of regulatory surface proteins, unesterified cholesterol, phospho-

lipids and a variety of other minor components that may include molecules and

proteins associated with antioxidant defenses. LDL particles transport cholesterol

that is needed for various cellular functions such as cell membrane formation and

hormone synthesis. About 60% to 70% of the total body cholesterol is contained in

the LDL particles. HDL particles account for most of the remaining cholesterol. The

function of the HDL particles appears to be involved with the return of excess lipids

from tissues to the liver for subsequent processing (a process referred to as reverse

transport). Many studies have unequivocably shown that elevated blood levels of

LDL cholesterol confer a higher risk of developing CVD. Although LDL particles

are not found in atherosclerotic plaques, oxidatively modified LDL particles are.

A mathematical model of the in vitro oxidation process of LDL is described by

Cobbold, Sherratt and Maxwell [2]. As described above, the LDL particle contains

on its surface a number of antioxidant defenses including-tocopherol (vitamin E).

In the plasma where the concentration of free radicals is low and other antioxidant

particles are present, LDL usually remain in its native, unoxidized form. As the LDL

particle migrates into the intima, it may expend all of its innate defenses against

oxidation. According to the model in [2], each contact with a free radical depletes

the LDL particle of one of its vitamin E molecules. Once these are exhausted,

the next contact with a free radical initiates peroxidation of the lipid core. In [2],

the authors consider a typical LDL particle with six -tocopherol molecules on its

surface exposed to free radical attack. Letting L6(t) denote the concentration of

LDL particles with six antioxidant defenses at time t, the authors assume a linear


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reaction with a free radical R at a rate ke (L6+R ke L5), to obtain the variable

L5(t), the concentration of LDL particles with five antioxidant defenses remaining.

In this fashion, they obtain a system of nine ordinary differential equations governing

the concentration of free radicals R(t), the concentration of LDL particles with i

antioxidant defenses Li(t), i= 0..6, and the concentration of oxidized LDL Lox(t).

The time rate of change of each species of LDL is given by the linear reactions,

and the change in radical concentration is increased by a production term (that isestimated since it is not measurable) and decreased by reaction with LDL species.

In the model of atherogenesis proposed in this paper, we adopt Cobbold Sherratt

and Maxwells model of LDL oxidation. We modify their equations only by adding

spatial dependence in the form of a diffusion term. LDL in both its native and

oxidized form are only slightly mobile within the arterial tissue. We account for this

by a small diffusion coefficient. This process is obligatory in the formation of the

atherosclerotic plaque. In 1977, Goldstein et al. [5] discovered that certain immune

cells, in particular macrophages, have a high affinity for oxidatively-modified LDL

but not native LDL. This results in trapping of cholesterol within the arterial wall.

Macrophages engorged with lipids are referred to as foam cells. Unable to perform

their normal duty of degrading debris, these lipid-laden cells accumulate and signal

other immune cells to the site instigating plaque growth.

Corruption of the normal immune process

Circulating immune cells in the blood, such as monocytes and T-lymphocytes,

migrate into tissues in response to chemical signals secreted by various cells including

endothelial cells and other immune cells. Under normal healthy conditions, these

immune cells aid in the degradation of apoptic cells as well as the removal of foreign


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Lesion progression

Smooth muscle cells (SMCs) also respond to chemical signals produced during

the accumulation of oxLDL, foam cells, and debris. SMCs migrate around the lesion

to form a fibromuscular cap overlaying the plaque. This process is also mediated

by chemo-attractants which entice SMCs into the region as well as chemo-inhibitors

that keep the SMCs outside of the lesion core. The fibrofatty atherosclerotic plaque

is characterized by a fibrous cap of SMCs, poorly formed connective tissue and, in

some cases, lipid filled macrophages due to continued influx of leukocytes. The cap

covers the core which contains dead cells, foam cells, and potentially nectrotic tissue.

As described in the introduction, remodeling occurs that initially results in the

abluminal expansion of the arterial wall followed by eventual luminal encroachment

as the cells, cell matrix, and debris accumulate in the plaque. The overlaying surface

becomes thrombogenic resulting in platelet adherence due to increase in expression

of platelet-endothelial-cell adhesion molecule 1. The thrombus can further diminish

or even completely occlude blood flow at this site [23].

Continued disease progression results in an advanced lesion described as a fi-

brous plaque. These types of plaques are characterized by a dense cap composed of

SMCs, collagen, elastin, and basement membrane fibers. These plaques often cause

moderate (40%50%) to severe (> 90%) arterial occlusion. However, the danger

of clinically significant ischemia imposed by such a lesion has more to do with thestability of the plaque (which is primarily determined by the composition of the

cap and the lipid core) than the degree of occlusion caused by the plaque [14, 4].

The cap may be uniformly thick which provides stability to the lesion; the SMCs

are the main arbiters of the integrity of the cap. A nonuniform cap, primarily one


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with a region that is thin (commonly found at the shoulders of the plaque), is me-

chanically unstable. Factors such as rheological forces or the production of various

proteases influence the stability of the cap [4]. Rupture of an unstable by moderate

sized plaque can result in complete occlusion of an artery with catastrophic medical

consequences such as stroke or myocardial infarction. Whereas a lesion that may

occlude the lumin to a much greater extent but which has a stable cap amy be less

of an immediate threat to a person because neovascular development can compen-

sate for the chronic and slowly-forming occlusion. One of the primary goals of the

present study is to model and simulate the formation of the atherosclerotic plaque

covered by a fibrous cap. Success in this endeavor is expected to lead developing a

much better understanding of the pathogenesis of CVD with the ultimate hope of

minimizing the morbidity and mortality attributable to CVD.

4.2.3 A Mathematical Model of Atherogenesis

Our model is based on a continuum mechanical, i.e. macroscopic, view of the

process. Hence the equations governing the evolution of various species are derived

by considering a balance of mass of each of these. To this end, we identify the cellular

and chemical species most significant to the process and consider their movement,

production, and death in an arbitrary control volume. In so doing, we account

for the highly interactive nature of disease developmentthe secretion of chemicalspecies by cells, movement of cells in response to chemical signals and so forth.


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Modeling Assumptions

We begin by identifying and labeling the following six generalized cellular and

chemical species that are to be considered responsible for disease in this model:

S1. n1 (number density)Immune cells: These are primarily monocyte derived

macrophages but may also include monocytes, T-lymphocytes, and other im-

mune response cells.

S2. n2 (number density)Smooth muscle cells: Also included in n2 are any cells

that are inherent in the production of the extra cellular matrix.

S3. n3 (number density)Debris: Debris is defined as dead cells, apoptic cells,

deposited lipids, and foam cells (corrupted macrophages).

S4. c1 (concentration e.g. M/L)Chemo-attractant: Here, we make no distinc-

tion between various types of chemo-attractants such as macrophage colonystimulating factor, monocyte chemotactic protein, interleukin-1, and others.

Rather,c1 refers to any chemical that induces positive chemotaxis.

S5. c2 (concentration)Native LDL: The value ofc2 is the concentration of LDL

molecules that are not oxidized regardless of how many innate defenses any

given molecule has remaining.

S6. c3 (concentration)Oxidized LDL

The change in concentration of each species in any volume element is subject to the

following assumed conditions:

A1. Endothelial dysfunction occurs by an unspecified mechanism allowing LDL to

enter the intima.


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A2. All species are subject to some random flux through the boundary of a volume.

For species n3,c2, andc3 this flux is very small as these are nearly immobile.

A3. Immune cells exhibit positive chemotaxis in response to chemo-attractant and

oxidized LDL. Smooth muscle cells exhibit positive chemotaxis in response to

chemo-attractant and negative chemotaxis in response to high concentrations

of debris. Cells are sensitive to the relative gradiente.g. cc

in respect

to chemotaxis. This is consistent with the observation that cells are highly

sensitive to changes in chemical concentration at very low concentrations, and

it also accounts for saturation effects at high concentrations.

A4. Chemo-attractant is produced in an unspecified way at a rate off1(n3) per

unitn3. It is decreased when an immune cell or SMC neutralizes it on contact.

A5. Native LDL enters the system through the boundary where it may become

oxidized (produce c3) following the general scheme as described in [2].

A6. Immune cells and SMCs die or become corrupted to produce debris.

A7. The concentration of debris increases due to death or corruption of other

cellular species, and it decreases by degradation by immune cells at a rate f2

per unit debris per unit immune cells. The rate of decrease depends on the

concentration of oxidized LDL.

The governing equations

We formulate the flux, production, and consumption terms for each species in

accordance with the above assumptions. To satisfy assumption A1. we include

in the assumed flux of each species a classical Fickian dependence on the species


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gradient. For the immune cells and the SMCs, we add to this a term that describes

the chemotaxis of these cells. Using the model of Keller and Segel [11], we define the

flux, Jni of species ni that exhibits chemotactic movement in response to chemical

species cj by

Jni = ini+ij(cj, ni)cj.

The coefficientij is the chemotactic sensitivity function. In general, it can be either

positive or negative, with positive ij corresponding to attractive chemotaxis, and

negative ij to repellent chemotaxis. This sensitivity function is typi

a short course in modeling chemotaxis - ritter

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