Download CS 296L - Won J. You

A Mathematical Model
of the Dual Immune Response
Won You
CS 296L
June 15, 2001
Abstract:
This paper will propose a system of nonlinear differential equations, built upon systems theory,
which models the human body’s immune response to disease and cancer. To accomplish this
task, two models of the human immune system will be presented, one derived by Kirschner and
Panetta and the other by Rundell et al. These two models provide the foundation on which the
dual immune system response model is built. Next, this paper will discuss the computer
implementation of the models that was used to test and validate their accuracy. Also, a proposal
is made on how the cancer vaccine Theratope can be incorporated with a set of equations into the
dual immune response model. Lastly, an overview of the major immune processes will be
presented in order to illustrate some of the more fundamental traits of the model.
Motivation:
According to the World Health Organization, more than 1.2 million people, around the world,
will be diagnosed with breast cancer in this year alone (2001). Unfortunately, in spite of the
staggering number of people afflicted with this deadly disease, there is still little progress made
in preventing and curing the victims. In light of this dire situation, it is useful to investigate from
a control systems view the mechanisms and relationships between the human immune response
and breast cancer. A mathematical model allows the simulation and in-depth analysis of the
immune system processes on removing cancer and the possible reasons why it fails to clear all of
the tumor cells. But in order to introduce such a model of the immune system dynamics, a
better understanding of immunology must be obtained.
An Introduction to the Immunology:
All vertebrates are protected by a dual immune system, featuring cellular and humoral immunity.
When cancer cells proliferate to a detectable threshold number in a given physiological space of
the human anatomy, the body’s own natural immune system is triggered into a search-anddestroy mode. However, this immune response is only possible if the cancer cells possess
distinctive surface markers known as tumor-specific antigens. In other words, the immune
system must, first, be able to identify the cancer cells as foreign. The immune response begins
when a white blood cell called a macrophage encounters a virus and consumes it. Next, the
macrophage digests the virus and displays pieces of the virus called antigens on its surface. Once
this happens, helper T cells recognizes the antigen displayed and binds to the macrophage. This
union stimulates the production of chemical substances—such as interleukin-1 (IL-1) and tumor
necrosis factor (TNF) by the macrophage, and interleukin-2 (IL-2) and gamma interferon (IFNy) by the T cell—that allow intercellular communication. Continuing this immune response, IL2 instructs other helper T cells and the killer T cells to multiply. In this manner, helper T cells
are responsible for organizing the immune response. On the other hand, killer T cells, also
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known as CTLs, attack virus-infected cells. The proliferating helper T cells also release
substances that cause B cells to multiply and produce antibodies. B cells make antibodies, which
are Y-shaped molecules that attach to and neutralize viruses floating free in the bloodstream,
thereby preventing the viruses from infecting other cells. As this process is taking place, the B
cells continue to clone and differentiate into plasma cells and memory cells. Plasma cells are
what secrete the antibodies, and the memory cells identify and record the information about the
virus for future attacks. Once the virus or foreign substance is brought under control, suppressor
T cells causes the activated T and B cells to turn off. If, in the future, the body is re-invaded by
the same virus or antigen, the memory cells will be reactivated and respond faster and more
powerfully to destroy the virus. This is the principle behind the vaccinations that are given to
children against the measles or mumps. The steps involved in the deployment of the immune
response are illustrated in the following diagrams:
Figure 1: The broad graphical look at the two immune responses taken from
http://www.people.virginia.edu/~rjh9u/imresp.html
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Figure 2: This diagram is taken from the following website:
http://defiant.wbc.edu/wbc/jjohnson/Pages/InfDis/ImmuneResponse.html
This figure shows the humoral immune response that occurs after a B cell encounters an antigen
or binds to an epitope. The B cells begin to clone themselves and start to differentiate into shortlived and long-lived plasma cells and memory cells.
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Figure: Gives a flow chart diagram of the various phases of both the humoral and cellular
immune response
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Now research shows that some degree of immune response against cancer exists in animals and
humans. Moreover, elements of the immune system are capable of recognizing cancer cells and
have even been identified in patients with certain cancers through various research. Although
progress is being made in this field, scientists still do not completely understand precisely how
the immune system works to suppress cancer and why it sometimes fails to do so. While there
are several techniques and methodologies being researched to enhance the natural immune
response, this paper will focus primarily on specific active cancer immunotherapy.
The Purpose of Theratope
With active specific immunotherapy (ASI), the subject is actively immunized against a defined
cancer by using a specific cancer-associated antigenic determinant or epitope in a germane
formulation. In order to produce this formulation, however, an epitope had to be first identified;
one such epitope was found in mucins.
The cells in the human body produce mucins, which are large molecular weight glycoproteins.
However, the mucins on cancer cells have been found to be underglycosylated. This aberration
provides an epitope by which cancer cells can be distinguished from normal cells. One of these
epitopes associated with adenocarcinomas is the carbohydrate, Sialyl-Tn (STn). Biomira’s
vaccine Theratope uses a synthetized mimic of the cancer-associated antigen STn conjugated to
the protein carrier Keyhole Limpet Hemocyanin (KLH).
For the first few treatments Theratope is administered with the immune adjuvant Detox-B Stable
Emulsion, to augment the immune response to the vaccine. Also, preceding the first treatment of
Theratope is a single low-dose administration of cyclophosphamide. This is a kind of
chemotherapy done to help overcome the activity of suppressor T cells. It is hypothesized that
when a cancer mucin sheds in the body the mucin causes the stimulation of suppressor T cells,
which can impede the effectiveness of the cancer vaccine.
Some Findings by Biomira
In a paper published in 1993, Biomira reported the promising findings of their experiments.
Using STn-conjugated human serum albumin in a solid-phase enzyme-linked immunosorbent
assay, Biomira’s scientists found that all patients treated with the vaccine had an increase in the
production of IgM and IgG antibodies reactive with natural STn determinants. Unfortunately,
this paper only published brief tabulations of the results from their experiments. In their
conclusion, they write that their study demonstrates the specificity of the humoral anti-hapten
response. Because of the lack of data available to simulate their findings, the scope of this paper
will be limited to proposing a possible model for simulating the dynamics induced or stimulated
by ASI. Since the company’s experiments give evidence of both a humoral and cellular
response, this suggests that the model should incorporate both of these responses, first. This
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merging of the two immune responses will be the first to be done for mathematical models. In
the appendix, an alternative method of modeling the dual immune response is discussed, one
which uses cellular automata theory. Be that as it may, there are several papers that do address
the human immune response. Unfortunately, these papers concentrate on either the humoral
response or the cellular response, each in isolation of the other; no attempt is made to merge the
two. Therefore, to formulate the dual immune response model, the cell-mediated response model
given in a paper by Kirschner and Panetta will be combined with the humoral immune response
model, written by Rundell et al. To begin, this paper will examine the Kirschner-Panetta model.
Existing Models
1. Kirschner-Panetta Model:
While there have been no models developed specifically for Theratope or other specific active
cancer immunotherapy (ACI), there has been work done on specific passive cancer
immunotherapy. Vaccines are considered active immunotherapy agents because the body is
stimulated to make its own antibodies, whereas with passive immunotherapy the subject is
administered a synthesized injection of effector cell stimulating serum. One model which
presents the interaction of cancer cells with a type of specific passive cancer therapy is the
Kirschner-Panetta model. This model explores cancer treatment by adoptive cellular
immunotherapy. This type of treatment serves to boost the immune system’s capacity to fight
the cancer. The immunotherapy here attempts to use cytokines, the communication/stimulation
proteins produced, released, and used by cells, to enhance cellular activity. The cytokine used in
the Kirschner-Panetta model is interleukin-2 (IL-2). Interleukin-2 is the main cytokine
responsible for lymphocyte activation, growth and differentiation. In general, adoptive cellular
immunotherapy refers to the injection of cultured immune cells that have anti-tumor reactivity
into tumor bearing hosts. Therefore, with adoptive cellular immunotherapy, the patient is
injected with cells derived from lymphocytes recovered from the patient’s tumors. Before
injection, these recovered cells are incubated with high concentrations of IL-2 in vitro along with
natural killer cells and cytotoxic T cells. More importantly, this injection of antigen-sensitized T
cells provides a growth stimulus for other lymphocytes to proliferate into a high enough cell
number capable of mounting an effective attack against cancer. The processes involved in the
immune response are described by cellular and molecular kinetics, along with the principles of
conservation and mass-action. From these principles, a set of differential equations are written
with the state variables, representing the different cellular populations as concentrations. The
equations which encapsulate the adoptive cellular immunotherapy dynamic is presented below:
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k EI
dE
 k1T  k 2 E  3 L  s1
dt
k4  I L
k ET
dT
 k 6 (1  k11T )T  7
dt
k5  T
k ET
dI L
 8
 k10 I L  s 2
dt
k9  T
Initial Conditions:
E ( 0)  E 0
T (0)  T0
I L ( 0)  I L 0
Parameter Values:
Kirschner-Panetta
Parameter Names
c
This Paper’s
Parameter Names
k1
Values
0  c  0.05
g1
g2
g3
μ2
k4
k5
k9
k2
2e7 (1/ml)
1e5 (1/ml)
1e3 (1/ml)
0.03 (1/day)
μ3
k10
10 (1/day)
p1
k3
0.1245
(1/day)
p2
k8
5 (1/day)
a
k7
1 (1/day)
b
k11
1e-9 (1/ml)
8
Description
The antigenicity of the tumor, the
ability of a substance to trigger an
immune response in a particular
organism.
Half-life for effector cells
Half-life for tumor cells
Half-life for Interleukin-2
The removal rate constant for
effector cells
The removal rate constant for
Interleukin
The rate constant for the selflimiting production of effector
cells
The rate constant for the selflimiting production of IL-2
The rate constant that represents
the strength of the immune
response, modeled by MichaelisMenten kinetics to indicate the
limited immune response to the
tumor.
The growth limiting constant
r2
k6
0.18 (1/day)
The logistic growth constant
E: The concentration of the effector cells, such as cytotoxic T-cells, macrophages, and natural
killer cells.
T: The concentration of the tumor cells.
IL: The concentration of IL-2 in the single tumor-site compartment being modeled.
s1,s2: The external input of LAK or IL-2 to the site, respectively.
r2(T): The logistic growth function of the tumor cells.
In their model, cancer treatment by immunotherapy is presented as a competition between
normal and cancer cells, the classic predator-prey model. The anti-cancer cells i.e. the effector
cells are thought of as predators to the cancer cells. The tumor-immune dynamics represented
here surround three primary concentrations: the effector cells, tumor cells, and the cytokine (IL2). Several terms are of the Michaelis-Menton form to indicate the plateau that results from the
saturation that occurs. For example, in equation (1), the third term indicates the saturated effects
of the immune response. Effector cells have a natural lifespan of a few days.
2. The Rundell-DeCarlo-HogenEsch-Doerschuk Model:
This model examines the humoral immune response of the human body to Haemophilus
influenzae Type b. While this model certainly does not address the production of antibodies
against cancer cells, the dynamics involved are very similar and worth investigating.
Specifically, this model explicitly incorporates the interaction between the memory cells, T-zone
and germinal center (GC) B cell dynamics, IgM and IgG antibodies, avidity maturation, and IC
presentation by FDCs.
3.1 An Overview and Elaboration of the Humoral Immune Response:
The humoral response can be summarized by three major phases: the primary response, the late
follicular response, and the secondary response. In the primary response, the immune system is
just beginning to mount a response to the foreign agent, which, in this case, happens to be
influenza. The secondary response describes the immune response when the body re-encounters
the bacteria after having developed immunity to it.
Primary Response summary:
1) Antigen activates naive helper T-cells and B-cells
2) Proliferation of B-cells
3) Some B-cells become B-blasts
4) B-blasts differentiate into short lived plasma cells which produce low avidity antibodies
5) Remaining B-blasts either die or migrate to the primary follicles
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6) Migrating B-blasts form germinal centers (GCs), the factory that supports B-cell proliferation
7) Mutations of the immunoglobulin genes cause avidity maturation and an increase in the
strength of antibody-antigen bond
8) Majority of the GC B-blasts undergo apoptosis
9) Remaining higher avidity cells become memory B-cells or long lived plasma cells that
produce antibodies
Late Follicular Response:
1) Memory B-cell activation by immune complex presenting follicular dendritic cells creates
pockets of low level B-cell proliferation
2)These late-B-blasts differentiate into long lived plasma or memory B-cells
Secondary Response:
All subsequent encounters with the foreign substance or agent will be met by earlier antibody
production, higher antibody titers and avidity. Also, the IgG isotope antibodies tend to
characterize the secondary response.
To review, when a vertebrate first encounters an antigen, it exhibits a primary humoral immune
response in which the B-cells begin to proliferate and differentiate. The progeny lymphocytes
include not only effector cells but also clone of memory cells, which retain the capacity to
produce both effector and memory cells upon subsequent stimulation by the original antigen.
The effector cells live for a few days; therefore, the antibody titer increases and decreases within
a month. However, the memory cells live for a lifetime and can be reactivated with a secondary
response. Thus when an antigen is encountered a second time, its memory cells quickly produce
effector cells which can rapidly produce massive quantities of antibodies. The primary response
begins with IgG, and then switches to IgM.
Based on the summary of the triggering events described above, the Rundell et al paper creates
the an 11th order system of nonlinear ordinary differential equations, listed on the following
page:
Note: Appendix A shows the entire set of equations, which include auxiliary functions not shown
here.
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dAg
 k1 Ag  k 2 K M Ag AbM  k 3 K G ( PL , PS , K ) Ag AbG
dt
dA
(2) bM  k 4 I SF ( BG ) Ps  k 6 k 2 K M Ag AbM  k 7 AbM
dt
dAbG
(3)
 k 8 [1  I SF ( BG )]Ps  k 8 PL  k 9 k 3 K G ( PL , Ps , K )TAbG  k10 AbG
dt
dP
(4) S  k11k12 {1  k13 H Z (t  k14 )}xBZ (t  k14 )1 (t  k14 )  k16 PS
dt
dP
(5) L  k15 ( S , K )k17 {1  k13 S (t  k14 ) H g (t  k14 )}xBg (t  k14 )1 (t  k14 )  k18 H g (t  2k14 )
dt
xM (t  2k14 )1 (t  2k14 )  k19 PL
(1)
dBZ
 k 20 H Z (t  k14 )U NB ( B g (t  k14 ) B z (t  k14 )) x1 (t  k14 )  k 21k13 H Z BZ  k12 [1  k13 H Z ]BZ
dt
dB g
(7 )
 {k 22 ( B g )k12 [1  k13 H Z (t  k14 )]BZ (t  k14 ) x1 (t  k14 )  k 23 H Zm (t  k14 ) M (t  k14 )
dt
Bg
x1 (t  k14 )  k 24 ( K )k13 H g SB g }2 x(1 
)  k17 [1  k13 SH g ]B g
B g max
( 6)
dM
 k 25 ( B g )k17 [1  k13 S (t  k14 ) H g (t  k14 )}xBg (t  k14 )1 (t  k14 )  k 26 H g (t  2k14 )
dt
xM (t  2k14 )1 (t  2k14 )  k 23 H Zm M  k 27 M
(8)
dI c
 k 28 K M Ag AbM  k 29 K G ( PL , PS , K ) Ag AbG  k 30 H g M  k 31 I c
dt
dK
K
(10)
 k 32 AMR ( B g , I C ) SK (1 
)
dt
K max
(9)
(11)
dS
 RUA ( A g )k 33 S  k 33 S
dt
Ag = the antigen concentration; AbM = IgM antibody concentration; AbG = IgG antibody
concentration; Ps = short-lived plasma cell concentration; PL = long-lived plasma cell
concentration; M = memory cell concentration; Bz = T-zone B-blasts concentration; Bg = GC Bblasts concentration; Ic = immune complex concentration; K = avidity; S = stimulation factor
11
The following is a simpler and more visual representation of the main terms of the differential
equations.
The rate of change of bacteria = dAg/dt
Replication
Rate
=
Removal by
complexing with
IgM
Removal by
complexing with
IgG
The antibodies produced by the humoral immune system destroys the bacteria upon antigenantibody complexing.
The rate of change of the IgM antibody = dAbM/dt =
Production Rate
Removal by
complexing with
Ag
Half-Life
Removal
The rate of change of IgG antibody = dAbG/dt =
Production
Rate
Removal by
complexing with
Ag
Half-Life Removal
The rate of change in the stimulation factor = dS/dt =
Antigen induced
Stimulation
Decay of
Stimulation
The rate of change of the short lived plasma cells = dPs/dt =
Differentiation from Tzone
B-blasts
Half-Life
Removal
The rate of change of the long lived plasma cells = dPL/dt =
Differentiation from
GC B-blasts
Differentiation from
Ic activated memory
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HalfLife
removal
The rate of change immune complex presentation = dIc/dt =
Uptake of IgM
complexes
Uptake of IgG
complexes
Activation of
memory cells
Half-Life
removal
The rate of change in the GC B-blast avidity maturation rate = dK/dt =
Avidity
Maturation
Physiological constraint on
avidity maximum
Computer Simulation:
To reproduce the findings of both the Kirschner-Panetta and the Rundell et al model, a computer
simulation program called VisSim was used. VisSim is a software program for the modeling and
simulation of complex dynamic systems that allows the user to build a system model with block
diagrams. This program was chosen for its ease of use and its robust simulation engine that
provides fast and accurate solutions for linear, nonlinear, continuous and discrete time, and timevarying designs.
All of the runs were conducted on Pentium-powered PCs under the Windows operating system.
The integration method employed was the Bulirsh-Stoer method for stiff ordinary differential
equations (ODE). This method is an adaptive numerical analysis method which compensates for
the vast variations in the rate of change between the numerous state variables. Other integration
methods, such as the Euler and Runge-Kutta 5th order method, were also used to test the models,
with equal success. The figures shown below illustrate the results.
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Figure: This screenshot shows a portion of the VisSim configuration for the Rundell et al model.
60000
40000
20000
0
0
6
IgM (microgram/ml)
Antigen (cfu/ml)
Antigen Concentration following primary response
80000
100
200 300
Time (hour)
400
500
Primary Antibody Response
5
4
3
2
1
0
0
14
100
200
300
Time (hour)
400
500
8
6
4
2
500
80000
70000
60000
50000
40000
30000
20000
10000
0
0
Primary Plasma Cell Response
400000
GC B-Blast Concentration (cells/ml)
2000
125 250 375
Time (hour)
Primary Plasma Cell Response
90000
1750
1500
1250
1000
750
500
250
0
0
100
30000000
200
300
Time (sec)
400
Primary GC B-Blast Response
150000
100000
50000
100
Primary IC response
15000000
10000000
5000000
200
300
Time (hour)
400
500
15
200
300
Time (hour)
400
500
400
500
Primary Avidity Maturation
9
8
7
6
5
0
100
500
200000
0
0
20000000
400
250000
500
25000000
200
300
Time (hour)
300000
10
0
0
100
350000
Avidity (ml/microgram)
Plasma Cell Concentration (cells/ml)
0
0
IC Concentration (Complexes/ml)
Plasma Cell Concentration (cells/ml)
IgG (microgram/ml)
Primary Antibody Response
10
100
200
300
Time (hour)
.8
.6
.4
.2
T-zone B-blast Concentration (cells/ml)
0
0
90000
100
200
300
Time (hour)
400
500
Memory B-cell Concentration (cells/ml)
Stimulation Factor (unitless)
Primary Simulation Factor
1.0
70000
60000
50000
40000
30000
20000
10000
100
200
300
Time (sec)
400
500
16
Primary Memory Cell Response
50000
40000
30000
20000
10000
0
0
Primary T-zone B-blast response
80000
0
0
60000
100
200
300
Time (hour)
400
500
The following are the results for the simulation of the Kirschner-Panetta model with c = k1 =
0.01, c = 0.02, and c = 0.035:
Effector Cell Concentration
40000000
200000000
IL Concentration
20000000
Tumor Cell Concentration
175000000
20000000
10000000
Volume
Volume
Volume
15000000
150000000
30000000
125000000
100000000
75000000
10000000
5000000
50000000
25000000
0
0
500
1000 1500
Time (sec)
2000
0
0
Effector Cell Concentration
200000
400000
500
1000
1500
Time (sec)
0
0
2000
Tumor Cell Concentration
50000
Volume
Volume
Volume
50000
300000
100000
250000
200000
150000
10000
50000
500
1000
1500
Time (sec)
2000
0
0
500
Effector Cell Concentration
200000
50000
40000
Volume
Volume
30000
30000
1000
Time (sec)
1500
0
0
2000
Tumor Cell Concentration
30000
175000
25000
150000
20000
Volume
0
0
40000
20000
100000
60000
IL Concentration
60000
350000
150000
500 1000 1500 2000
Time (sec)
125000
100000
75000
10000
50000
5000
10000
25000
0
0
500
1000
1500
Time (sec)
0
0
500
2000
17
1000
Time (sec)
1500
2000
1000 1500
Time (sec)
2000
IL Concentration
15000
20000
0
0
500
500
1000 1500
Time (sec)
2000
The Proposal
Before offering a proposal of the active specific immunotherapy model, it is important to present
the integration of the Kirschner-Panetta (KP) and Rundell et al model for covering the dual
immune system response to antigens. Now, the three equations of the Kirschner-Panetta model
will be added to the Rundell et al model in the following manner:
1) The rate of change of the tumor cell concentration from the KP model will be combined with
the rate of change of the antigen concentration, since in this case, the tumor cells are the
antigens. Therefore, the notation of Ag has been replaced by T.
2) The exogenous input terms s1 and s2 are set equal to 0, because the dual immune response
model will only represent the immune response to some initial cancer cell or antigen
population when there is no treatment.
3) Continuing this set of differential equations are the equations derived from the Rundell et al
model.
(1)
k EI
dE
 k1T  k 2 E  3 L
dt
k4  I L
( 2)
k ET
dI L
 8
 k10 I L
dt
k9  T
(3)
k ET
dT
 k1T  49
  k 2 K M TAbM  k 3 K G ( PL , PS , K )TAbG
dt
k 50  T
dAbM
 k 4 I SF ( BG ) Ps  k 6 k 2 K M TAbM  k 7 AbM
dt
dA
(5) bG  k 8 [1  I SF ( BG )]Ps  k 8 PL  k 9 k 3 K G ( PL , Ps , K )TAbG  k10 AbG
dt
dP
(6) S  k11k12 {1  k13 H Z (t  k14 )}xBZ (t  k14 )1 (t  k14 )  k16 PS
dt
dP
(7) L  k15 ( S , K )k17 {1  k13 S (t  k14 ) H g (t  k14 )}xBg (t  k14 )1 (t  k14 )  k18 H g (t  2k14 )
dt
xM (t  2k14 )1 (t  2k14 )  k19 PL
( 4)
(8)
dBZ
 k 20 H Z (t  k14 )U NB ( B g (t  k14 ) B z (t  k14 )) x1 (t  k14 )  k 21k13 H Z BZ  k12 [1  k13 H Z ]BZ
dt
18
(9)
dB g
dt
 {k 22 ( B g )k12 [1  k13 H Z (t  k14 )]BZ (t  k14 ) x1 (t  k14 )  k 23 H Zm (t  k14 ) M (t  k14 )
x1 (t  k14 )  k 24 ( K )k13 H g SB g }2 x(1 
Bg
B g max
)  k17 [1  k13 SH g ]B g
dM
 k 25 ( B g )k17 [1  k13 S (t  k14 ) H g (t  k14 )}xBg (t  k14 )1 (t  k14 )  k 26 H g (t  2k14 )
dt
xM (t  2k14 )1 (t  2k14 )  k 23 H Zm M  k 27 M
(10)
dI c
 k 28 K M TAbM  k 29 K G ( PL , PS , K )TAbG  k 30 H g M  k 31 I c
dt
dK
K
(12)
 k 32 AMR ( B g , I C ) SK (1 
)
dt
K max
(11)
(13)
dS
 RUA (T )k 33 S  k 33 S
dt
For the purposes of this paper, the pharmacokinetics of the cancer vaccine, Theratope will be
limited to a certain extent. By this, I mean that some of the interactions derived by the
introduction of Theratope into the body will be represented as simply and naively as possible.
This may seem a little unwarranted, but it is a result of the scarcity of information available on
the manner in which the drug interacts with the body. As a result, this paper will offer a
beginning look into the complex manner by which active specific immunotherapy affects the
growth of cancer cells. More specifically, this paper will attempt to unravel the dynamics
between the immune system, the cancer cells, and the normal cell population. This will help to
reveal whether or not the dual response of the immune response will be successful in eliminating
the cancer cells before the tumor cells overtake the normal cells. This will be done by integrating
the different equations provided in the preceding models. Also, it is also worthy to note that the
use of cyclophosphamide will also be excluded from this study. Since cyclophosphamide has the
effect of destroying both cancer cells and normal cells, I will treat its overall influence as being
negated in the long term dynamics between the immune system and cancer cells. Incidentally,
cyclophosphamide is used to abrogate the activity of suppressor T-cells which are known to be
activated by shed mucin cells. But this model can be easily implemented in later versions of the
model. For instance, the single dose treatment of cyclophosphamide can be represented as a two
or three compartment model, with one exogenous input, u1. The initial compartment can be the
blood pool, which is directly attached to the body, with a leak.
Some of the possible additions to the model are:
1) Now that the pharmacokinetics are being examined, I introduced a u1 to the effector cell
concentration. Also, several additional inputs may be required due to the existence of several
19
substances in the Theratope treatment. The first injection is the cyclophosphamide; the next
is four treatments of Detox-B, and the last treatment is the cancer vaccine itself.
2) An equation may have to be introduced to handle the adoptive stimulation of the immune
response. This may be taken care of either by making the antigenicity of the cellular
response a function of the input of ASI and/or by adding a term in the avidity maturation
equation which dependent on the presence of the vaccine. The motivation behind these
changes is that the cancer vaccine increases the ability of the B cells and T cells to identify
the otherwise stealthy tumor cells. One way of interpreting this may be to say that the
vaccine increases the antigenicity of the tumor cells and/or the avidity of the binding to the
tumor cells. Investigation into the published papers of Biomira, the vaccine’s manufacturer
showed that little is known about the actual mechanics, governing the vaccine’s elicitation of
the immune response. Moreover, correspondence with one of the company’s scientists
revealed that no formal studies were done on the vaccine’s pharmacokinetics, at least none
that she was aware of. What is known is that this vaccine is primarily aimed at metastatic
breast cancer, although there are several other adaptations of the vaccine in development for
other diseases. The drug is predicated on the belief that the immune system fails to identify
and thus attack cancer cells, because of their secretion of mucins. More specifically, the
glycoprotein, MUC-1 gene has been discovered to be particularly overexpressed in breast
tumors, making it a good candidate for immunotherapy. By synthesizing cancer antigens,
Biomira’s scientists have found an effective way of tricking the body into recognizing the
cancerous cells as foreign. To accomplish this, the vaccine is formulated with a protein
marker that is carried on the surfaces of breast cancer cells, along with another protein that
stimulates a general immune response. In essence, the aim here is to stimulate T-cells to
become active killer T-cells and eradicate tumors in patients.
3) A final consideration for the implementation of the Theratope vaccine on the immune system
is to carefully decide whether or not another equation is needed to represent the interaction
between the T cells and the B cells. The T cells and B cells, as explained earlier in the
immunology review, communicate and in some ways, regulate one another. This may
require some further rumination. More auxiliary functions may be incorporated to fit the
interaction, or the influence the two responses have on each other may be embedded in a
black-box model or function. 2) As far as the humoral immune response is concerned, the
notation of Ag has been replaced by T. Originally, Ag was used to represent the bacterial
concentration. And to reflect the enhanced sensitivity to the STn epitope, there is also a k
constant added to the stimulation factor equation (Equation 10) No other changes have been
made to the other 11 equations.
With equation 1, the rate of change for the effector-cell population is determined by two factors,
besides its current population: the recruitment of effector cells, which is determined by the
amount of tumors multiplied by the tumor’s antigenicity (the constant c) and the growth source
from the stimulation by lymphokines, represented by the third term in the equation. The third
20
term is of the Michaelis-Menton form, in order to reflect the saturation effects of the immune
response. Equation 2 marks the rate of change of the tumor cells.
The rate constant a represents the strength of the immune response and is modeled by MichaelisMenton kinetics to indicate the limited immune response to the tumor. The third equation
represents the rate of change of the normal cells or non-cancerous cells. The remaining equations
display the complicated manner by which the B-cells are activated in the human body in reaction
to an antigen. The final refinements give the following:
21
(1)
k EI
dE
 k1T  k 2 E  3 L  u1
dt
k4  I L
( 2)
k ET
dI L
 8
 k10 I L
dt
k9  T
k ET
dT
 k1T  49
 k 2 K M TAbM  k 3 K G ( PL , PS , K )TAbG
dt
k 50  T
dA
(4) bM  k 4 I SF ( BG ) Ps  k 6 k 2 K M TAbM  k 7 AbM
dt
dA
(5) bG  k 8 [1  I SF ( BG )]Ps  k 8 PL  k 9 k 3 K G ( PL , Ps , K )TAbG  k10 AbG
dt
dP
(6) S  k11k12 {1  k13 H Z (t  k14 )}xBZ (t  k14 )1 (t  k14 )  k16 PS
dt
dP
(7) L  k15 ( S , K )k17 {1  k13 S (t  k14 ) H g (t  k14 )}xBg (t  k14 )1 (t  k14 )  k18 H g (t  2k14 )
dt
xM (t  2k14 )1 (t  2k14 )  k19 PL
(3)
dBZ
 k 20 H Z (t  k14 )U NB ( B g (t  k14 ) B z (t  k14 )) x1 (t  k14 )  k 21k13 H Z BZ  k12 [1  k13 H Z ]BZ
dt
dB g
(9)
 {k 22 ( B g )k12 [1  k13 H Z (t  k14 )]BZ (t  k14 ) x1 (t  k14 )  k 23 H Zm (t  k14 ) M (t  k14 )
dt
Bg
x1 (t  k14 )  k 24 ( K )k13 H g SB g }2 x(1 
)  k17 [1  k13 SH g ]B g
B g max
(8)
dM
 k 25 ( B g )k17 [1  k13 S (t  k14 ) H g (t  k14 )}xBg (t  k14 )1 (t  k14 )  k 26 H g (t  2k14 )
dt
xM (t  2k14 )1 (t  2k14 )  k 23 H Zm M  k 27 M
(10)
dI c
 k 28 K M TAbM  k 29 K G ( PL , PS , K )TAbG  k 30 H g M  k 31 I c
dt
dK
K
(12)
 k 32 AMR ( B g , I C ) SK (1 
)
dt
K max
(11)
(13)
dS
 RUA (T )k 33 S  k 33 S
dt
22
(14)
dB g
dt
 { g ( B g ) dZ [1   2 H Z (t   d )]BZ (t   d ) x1 (t   d )   n 2 H Zm (t   d ) M (t   d )
x1 (t   d )   pg ( K ) 2 H g SB g }2 x(1 
Bg
B g max
)   dg [1   2 SH g ]B g
dS
 kRUA (T )  e S   e S
dt
dM
(16)
  gZ ( B g ) dZ [1   2 S (t   d ) H g (t   d )}xBg (t   d )1 (t   d )  p Mic H g (t  2 d )
dt
xM (t  2 d )1 (t  2 d )   n 2 H Zm M   M M
(15)
dI c
  i M K M TAbM   i G K G ( PL , PS , K )TAbG   Mic2 H g M   ic I c
dt
dK
K
(18)
  k AMR ( B g , I c ) SK (1 
)
dt
K max
(17)
(19) xi (t 0 )  xio  0, i  1,2
(20) I SF ( B g ) 
1
1   1 Bg
(21)U NB ( B g , BZ ) 
1
1   2 ( B g  BZ )
T
Tgth  T
(22) RUA 
(23)1 (t   d )  0 : t   d
(24)1 (t   d )  1 : t   d
(25) H Z 
KMT
1 KMT
KT
1  KT
KI c
(27) H g 
1  KI c
(26) H Zm 
23
Parameter Estimation:
Although one of the goals of this paper was to construct a model which accurately captures the
essence of the dynamics of Theratope and to put this model through the rigors of parameter
estimation, it became clear after an extensive search of the available medical database that the
necessary experimental data did not exist. Despite the best efforts to find experimental data on
other diseases that may help to validate the model of the dual immune response, no success was
made. The list of parameters, which require further investigation and experimental data are:
1) k1 = antigen growth rate for breast cancer; the current k1 value for Haemophilus influenzae is
0.15/hr
2) k2 = removal rate constant of IgM bound breast cancer cells; the current k2 value for the
bacteria is 0.10/hr
3) k3 = removal rate constant of IgG bound breast cancer cells; the current k3 value for the
bacteria is 0.10/hr
4) k13 = the differentiation rate constant of T-zone B cells to short lived plasma cells
5) k11 = Maximal percentage of germinal center B-blasts that continue to proliferate
6) k25 = Differentiation rate of germinal center B-blasts to memory cells
Some other necessary modifications is the conversion of parameters’ units found in the
Kirschner-Panetta model. The units must be made uniform.
Conclusions:
A mathematical model of the dual immune response is offered in this paper, including a method
for studying active specific immunotherapy. This paper helps to establish the basis for future
work by presenting the prerequisite background in immunology, as well as investigating the
dynamics shown in the immune response from several papers. The results in the published
papers were duplicated using a computer visualization program called VisSim. The difficult task
of evaluating and testing the accuracy of the models’ mechanisms and behaviors was not made,
because of the lack of supporting data.
Future Work And Considerations:
The greatest challenge that remains in this project is the elicitation of the parameter values
related to the human immune response to breast cancer. In the process of conducting the
research necessary to develop a mathematical model of the pharmacokinetics of the breast cancer
vaccine Theratope, it was discovered that no experimental data was available for the immune
response to breast cancer—in the form that is needed, anyway— in the medical literature and
research; this prevents the completion and full analysis of the quality of the model. One
24
alternative to this problem is to recruit parameter values for another well-documented disease for
which experimental data is plentiful, such as HIV, the adeno virus, or other bacteria. So far, this
too has not led anywhere, as there is still no information for the immune system parameter
values. Moreover, many of the papers published by Biomira on Theratope as well as other papers
found in the medical literature measure the cellular response through a bioassay, say enzymelinked immunosorbent assay (ELISA) for example. Perhaps, in the future, more data may
become available, but until that time, the formulation and validation of the model cannot be
completed.
Also, another approach to the problem may be to identify the possible values or range of values
for the numerous parameters in my model and test the effects on the model, to, at least, limit the
possibilities.
Lastly, future work will also be needed in the areas of parameter identification, parameter
sensitivity, and parameter estimation.
25
References:
Evidence of Cellular Immune Response Against Sialyl-Tn in Breast and Ovarian Cancer
Patients After High-Dose Chemotherapy, Stem Cell Rescue, and Immunization with
Theratope STn-KLH Cancer Vaccine. Journal of Immunotherapy. 22(1), 1999: 54-66.
Asachenkov, A., Marchuk, G., Mohler, R. & Zuev, S. (1994b). Immunology and Disease
Control: A Systems Approach. IEEE Trans. Biomed. Eng. 41(10), 943-953.
Homberg, B. & Oparin, D., & Gooley, T., & Lilleby. K., & Bensiner W., & Reddish, M., &
MacLean, G., & Longenecker, B., & Sandmaier, B. Clinical Outcome of Breast and Ovarian Cancer
Patients Treated with High-Dose Chemotherapy, Autologous Stem Cell Rescue and Theratope STnKLH Cancer Vaccine. Bone Marrow Transplantation (2000) 25, 1233-1241.
Kirschner, D & Panetta, JC. Modeling Immunotherapy Of The Tumor-immune Interaction.
Journal of Mathematical Biology, 1998 Sep, 37(3): 235-52.
Kohler, B., Puzone, R., Seiden, P., & Celada, F. A Systematic Approach to Vaccine Complexity Using an
Automaton Model of the Cellular And Humoral Immune System I. Viral characteristics and polarized
responses. Vaccine (2000)19, 862-876.
Longenecker, B. Active Specific Immunotherapy (ASI) of Carcinomas Using Synthetic
Cancer Antigen ‘Vaccines’. Clinical Chemistry, 1992 ,Vol. 38(6): 933-934.
MacLean, G., & Miles, D., & Rubens, R., & Reddish, M., & Longenecker, B. Enhancing the
Effect of Theratope STn-KLH Cancer Vaccine in Patients with Metastatic Breast
Cancer By Pretreatment with Low-Dose Intravenous Cyclophosphamide. Journal of Immunotherapy,
(1996), 19(4), pp. 309-316.
MacLean, G. & Reddish, M.& Koganty & T., Wong, T & Gandhi, S. & Smolenski, M. &
Samuel, J. & Nabholtz, J & Longenecker, B. Immunization of Breast Cancer Patients
Using a Synthetic Sialyl-Tn Glycoconjugate Plus Detox Adjuvant. Cancer
Immunology Immunotherapy, 1993, 36:215-222.
Nani, F & Freedman, HI. A Mathematical Model Of Cancer Treatment By Immunotherapy.
Mathematical Biosciences, 2000 Feb, 163(2):159-99.
Rundell, A. & DeCarlo, R. & Hogenesch, H. & Doerschuk, P. (1998). The Humoral Immune
Response to Haemophilus influenzae Type b: A Mathematical Model Based on Tzone and Germinal Center B-cell Dynamics. Journal of Theoretical Biology, 1998 Oct
7, 194(3):341- 81.
Rundell, A. & DeCarlo, R. & Hogenesch, H. & Ventkataramanan, B. (1998). Systematic
Method for Determining Intravenous Drug Treatment Strategies Aiding the
Humoral Immune Response. IEEE Transactions on Biomed. Engineering, 1998 April,
45(4): 429- 39.
Visual Solutions, http://www.vissim.com
26
Appendix A: Main Model Equations In Rundell et. al’s Humoral Immune Response Model
dAg
 k1 Ag  k 2 K M Ag AbM  k 3 K G ( PL , PS , K ) Ag AbG
dt
dA
(2) bM  k 4 I SF ( BG ) Ps  k 6 k 2 K M Ag AbM  k 7 AbM
dt
dA
(3) bG  k 8 [1  I SF ( BG )]Ps  k 8 PL  k 9 k 3 K G ( PL , Ps , K )TAbG  k10 AbG
dt
dP
(4) S  k11k12 {1  k13 H Z (t  k14 )}xBZ (t  k14 )1 (t  k14 )  k16 PS
dt
dP
(5) L  k15 ( S , K )k17 {1  k13 S (t  k14 ) H g (t  k14 )}xBg (t  k14 )1 (t  k14 )  k18 H g (t  2k14 )
dt
xM (t  2k14 )1 (t  2k14 )  k19 PL
(1)
dBZ
 k 20 H Z (t  k14 )U NB ( B g (t  k14 ) B z (t  k14 )) x1 (t  k14 )  k 21k13 H Z BZ  k12 [1  k13 H Z ]BZ
dt
dB g
(7 )
 {k 22 ( B g )k12 [1  k13 H Z (t  k14 )]BZ (t  k14 ) x1 (t  k14 )  k 23 H Zm (t  k14 ) M (t  k14 )
dt
Bg
x1 (t  k14 )  k 24 ( K )k13 H g SB g }2 x(1 
)  k17 [1  k13 SH g ]B g
B g max
( 6)
dM
 k 25 ( B g )k17 [1  k13 S (t  k14 ) H g (t  k14 )}xBg (t  k14 )1 (t  k14 )  k 26 H g (t  2k14 )
dt
xM (t  2k14 )1 (t  2k14 )  k 23 H Zm M  k 27 M
(8)
dI c
 k 28 K M Ag AbM  k 29 K G ( PL , PS , K ) Ag AbG  k 30 H g M  k 31 I c
dt
dK
K
(10)
 k 32 AMR ( B g , I C ) SK (1 
)
dt
K max
(9)
(11)
dS
 RUA ( A g )k 33 S  k 33 S
dt
27
Appendix B: Auxiliary Functions for the Humoral Immune Response Model By Rundell et. al
Hz 
K M Ag
1  K M Ag
H Zm 
Hg 
KAg
1  KAg
KI C
1  KI C
KP  KP
PL  PS  1
K G ( PL , PS , K ) 
1
1  k 34 B g
I SF ( B g ) 
2S 1
 )( k 41 tan 1 (k 42 ( K  k 43 ))  k 44 )
3 3
1
U NB ( B g , BZ ) 
1  k 35 ( B g  BZ )
k15 ( S , K )  k15 (
k 22 ( B g ) 
k 22
1  k 36 B g
k 24 ( K )  k 24 (k 38 tan 1 ( K  k 39 )  k 40 )
k 25 ( S )  k 25 (1 
AMR ( B g , I C ) 
RUA ( Ag ) 
2S
)
3
Bg
k 37 I C  1
Ag
Agth  Ag
28
Appendix C: Rundell et al. Model Parameters
Rundell
Parameter
Names
λ1
This Paper’s
Parameter
Name
k1
Primary Value
αbM
k2
0.1 (1/hr)
αbG
k3
0.1 (1/hr)
ρM
k4
1.7e-6 (microgram/hr)
(1/cell)
χ1
k5
1.7e-5 (ml/cells)
ηM
k6
1.6e-8 (microgram/cfu)
α2M
k7
ρG
k8
ηG
k9
2.4e-9 (microgram/cfu)
α2G
k10
0.002 (1/hr)
γSZ
k11
0.80
κdZ
k12
0.06
γ2
k13
0.70
τd
k14
0.15 (1/hr)
0.0042
(microgram/cfu)
1.72e-6
(microgram/hr)(1/cell)
6 hr
γ1g
k15
0.001
αS
k16
0.014 (1/hr)
κdg
k17
0.009 (1/hr)
ρMg
k18
1.04e-5 (1/hr)
29
Description
Hib growth rate constant
Removal rate constant of IgM bound
bacteria
Removal rate constant of IgG bound
bacteria
IgM production rate constant
Normalizing Isotope switching
parameter
Conversion factor for multivalency and
unit conversions
IgM half-life removal rate constant
IgG production rate constant
Conversion factor for multivalency and
unit conversions
IgG half-life removal rate constant
Differentiation rate constant of Bz to
short lived plasma cells
T-zone B-blast cell
differentiation/migration/death rate
constant
Maximal percentage of blast for
continued proliferation
Cell Differentiation Delay
Differentiation rate constant of low
avidity Bg to long lived plasma cells
Short lived plasma cells life-span
removal rate constant
GC B-blast cell differentiation/death rate
constant
Plasma cell maintenance rate constant
αL
k19
0.0014 (1/hr)
ρn1
k20
1157 (cell/ml)(1/hr)
κpZ
k21
0.01 (1/hr)
γgZ
k22
0.0098
ρn2
k23
0.048 (1/hr)
κpG
k24
0.0624 (1/hr)
γmg
k25
0.10
ρMic
k26
0.000828 (1/hr)
αM
k27
0.00085 (1/hr)
ρiM
k28
10 (complex/cfu)(1/hr)
ρiG
k29
10 (complex/cfu)(1/hr)
ρMic2
k30
αic
ρk
k31
k32
1.04e-5
(complex/cell)(1/hr)
0.000463 (1/hr)
6.85e-8 (1/hr)
βe
k33
0.014 (1/hr)
χ1
k34
1.7e-5 (ml/cell)
χ2
k35
1e-6 (ml/cell)
χ3
k36
0.0001 (ml/cell)
χ4
γk
ηK
βK
k37
k38
k39
k40
1e-10 (cells/complex)
0.5351
10.27 (ml/microgram)
1.7375
γ1
k41
106.088
η11
k42
3
η12
k43
12 (ml/microgram)
30
Long lived plasma cells life-span
removal rate constant
Naive cell stimulation factor
Proliferation adjustment parameter for
B-blast cell cycle in T-zone
Migration rate constant
Stimulation rate constant of memory Bcells by free antigen
Proliferation adjustment parameter for
low avidity B-blast cell cycle in GC
Differentiation rate parameter of Bg to
memory B-cells
Memory maintenance rate constant
Memory cells effective average life-span
removal rate constant
Uptake rate constant of IgM ICs by
FDCs
Uptake rate constant of IgG ICs by
FDCs
IC removal rate constant by activating
Memory
IC half-life removal rate constant
Avidity maturation rate constant
Maximum stimulation decay rate
constant
Normalizing isotype switching
parameter
Normalizing naive recruitment factor
Normalizing migration exclusion
parameter
IC normalization rate constant
Arctan parameter for Bg cell cycle
Arctan parameter for Bg cell cycle
Arctan parameter for Bg cell cycle
Arctan parameter for Bg to long lived
plasma cells
Arctan parameter for Bg to long lived
plasma cells
Arctan parameter for Bg to long lived
plasma cells
β1
k44
162.492
Km
Kmin
Kmax
Agth
Bgmax
Km
Kmin
Kmax
Agth
Bgmax
0.77
5.14 (ml/microgram)
20.55 (ml/microgram)
0.000333333 (cfu/ml)
5e7 (cells/ml)
31
Arctan parameter for Bg to long lived
plasma cells
IgM constant avidity
IgG low avidity
IgG maximum avidity
Threshold of antigen elimination
GC limiting cell concentration
Appendix D: Cellular Automata: An Alternative Method of Modeling the Dual-Immune
Response
Introduction:
First conceived in the 1940’s by Stanislaw Ulam and John von Neumann, cellular automata (CA)
theory was introduced as a means for simulating complex dynamical systems at a time when
computer computational power was limited. To begin, a cellular automaton consists of an ndimensional grid of cells, which can be in one of k possible states. Each cell is identical and is
updated synchronously in discrete time steps according to local interaction and behavioral rules.
Hence, by its very nature, CAs are limited to a strictly defined world. However, although
cellular automatons are discrete dynamical systems that are confined by discrete time and space,
it can still be designed to model complex continuous dynamic behaviors and interactions. For
instance, an early computer implementation of CA theory was a program called the Game of Life
(GOL), developed by John Horton Conway. This game was invented to represent the
microscopic cycles that exist in biological population growth, patterns such as an exponential
increase from plentiful resources and a decrease related to overpopulation and competition.
The Game of Life is played on a finite two-dimensional grid or lattice that is in one of two states:
alive, represented by the number 1, or dead, represented by 0. The set of rules governing the
cells’ states are as follows:
 For each time step t, a dead cell becomes alive at time t+1 if exactly three of the eight
neighboring cells at time t were alive.
 For each time step t, a live cell becomes dead when at time t less than two or greater than
three neighboring cells are alive.
While these rules may seem deceptively simple, it belies the rich tapestry of patterns that can
emerge, reflecting, too, the colorful variations seen in the real world. CA provides an inside look
at the microscopic events of a system while preserving its general trends, such as growth or
diffusion and the like. As a result, CA theory can be applied to modeling other phenomena, such
as basic heat and wave equations from physics and predator-prey dynamics from biology, to
name a few. In recent years, CA theory has resurfaced from its humble beginnings to be applied
to more complicated problems. The most interesting application with respect to this paper is the
problem of modeling the dual-immune response to various diseases.
IMMSIM and its Computer Implementation:
In 1992, F. Celada and P.E. Seiden from Princeton University proposed a simulation program by
the name of IMMSIM to model the human immune response. The first version of this program
was implemented under Windows 95, using the programming language APL2. Since then,
IMMSIM has undergone numerous modifications along with an extension of its features. Some
more recent developments have been in rewriting the software in both C and C++. However,
32
the most significant enhancement has been the addition of the cell-mediated immune response.
Prior to this year (2001), the program was only designed to simulate the humoral immune
response.
At its core, IMMSIM applies the fundamentals of CA theory to the major players of the immune
system. For example, IMMSIM tracks six cellular entities: B cells, T cells, antigen-presenting
cells (APC), antibodies, antigens, and immune complexes. Each of these entities follows a strict
set of probabilistic interaction rules for behavior, described within modules. As an analogy, the
entities can be thought of as finite state machines (FSM) that are allowed to travel in physical
space, which, in this case, is the human body. Also, modules are logical units that allow the
program to decompose the simulation into several smaller self-contained set of processes or a
piece of physical space. For instance, the modules are the lymph node, thyroid, bone marrow,
and the environment.
The program allows a user to optionally specify the initial parameter values of the system—the
half-life of the plasma cells for example. Then IMMSIM displays the population of the B-cells,
antigen, immune complexes, etc, with respect to the number of time steps (shown in the figure
below).
Figure: This is screenshot of the type of results that can be found using IMMSIM++
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The current version of IMMSIM utilizes a triangular lattice, which is intended to represent a
small portion of the human body. A population of six entities inhabit each site, distributed in a
random manner. When simulation begins, the interaction rules are applied to each entity. These
rules are probabilistic rather than the usual deterministic variety. Once all of the rules are
applied, the birth of new cells is determined. In regards to the implementation of this program,
listed below is the half-lives used by IMMSIM:
Entity
APC
B cell
Th cell
Tc cell
Epithelial cell
Plasma cell
Active state of an APC
Active state of a Th cell
Active state of a Tc cell
Anergy state
Antibody
Damage
Time steps
50
50
50
50
100
10
50
50
50
100
10
3
Bound MHC/peptide complexes and antigen half-lives are given the exorbitantly high value of
10,000, to allow these cells to be long-lasting. Other important traits of the computer
implementation of this model are that:



The size of the body array is 16x15 sites
The diffusion rate is chosen so that entities spread uniformly over the site and its six
neighbors on one time step. Epithelial cells do not diffuse.
All runs are terminated at 2000 time steps (and called chronic) if they have not previously
terminated due to cure or death. Cure results when all antigen is eliminated. Death results
when 50% of the epithelial cells are infected or the viral load is greater than 200,000.
As mentioned before, there are four different modules that comprise the IMMSIM model:
Module
boneMarrow
thymus
Description
produces a steady stream of new B-Cells, T-Cells and APCs which flow to the
lymphNode and thymus modules
simulates positive and negative selection of T-Cells which eventually flow to
the lymphNode module
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lymphNode
simulates the interactions between the cells and molecules of the IMMSIM
model
handles Antigen injections to the lymphNode module
environment
An example of the interaction rules for APCs and antigens are given below:
APC
Event
DEATH
Description
 ln 2
STEP
BIND_ANTIGEN
BIND_T_CELL
BIND_IMMUNE_COMPLEX
Use Pdie  e  where , the half-life, is tauAPC, to determine
if the APC will die.
Update internal state information such as age. If an MPC is
being presented, remove it with probability pRemoveAg. If an
Antigen has been bound then internalize it, process its peptides
and present an MPC.
After binding to an Antigen, an APC will subsume it. Any
Antigen that had previously been internalized is removed.
Internalized Antigen is processed and presented during the
STEP event.
APCs that are exposing MPC can interact with T-Cells. After
an APC binds a T-Cell, it will remove any MPCs which were
exposed on its surface.
This is exactly the same as BIND_ANTIGEN. In this case the
Antigen, is the FC of the Immune Complex.
Antigen
Event
DEATH
DIVIDE
STEP
Description
 ln 2
Use Pdie  e  where , the half-life, is tauAg, to determine if the Antigen will die
(i.e., be removed from the simulation).
Each time step, Antigens divide at a rate specified by agMultRate. This provides a
means of simulating the growth of live antigens such as bacteria.
Update internal state information such as age.
Discussion:
The findings of Kleinstein and Seiden demonstrate that IMMSIM simulates the general behavior
of a real immune response. This was accomplished by running a simulation of the system for
100 time steps after the injection of an antigen. As expected, after the first injection of the
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antigen, the antibody production was slow, a result of the time-consuming cloning process of the
relatively small naive B cell population. But after a second injection, a far more rapid production
of antibodies was witnessed, a familiar secondary response found in nature. Overall, Kleinstein
and Seiden were able to show that their program is a useful research and educational tool for
studying aspects of clonal selection, mutation, and affinity maturation. Other advantages of the
IMMSIM program are that it is easily expandable and adaptive. New interactions can be added
to program very easily, and behaviors can be modified simply by changing the parameter values.
Future Work:
In the future, it may be worthwhile to make a comparison between the accuracy and quality of
the results produced using a mathematical model versus a CA model. But there are no
immediate plans to try to develop a hybrid model that unifies CA theory as well as mathematical
modeling theory.
References:
http://www.ifs.tuwien.ac.at/~aschatt/info/ca/ca.html
http://lslwww.epfl.ch/~moshes/ca_main.html
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