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\begin{document}
\begin{center}\textbf{ {\Large 
Cancer Treatment Using Multiple Chemotheraputic Agents \\[1ex] 
Subject to Drug Resistance } \\[2ex]
J. J. Westman \\
Department of Mathematics \\
University of California \\
Box 951555 \\
Los Angeles, CA  90095-1555 \\ USA  \\[2ex]
B. R. Fabijonas \\ 
Department of Mathematics \\
Southern Methodist University \\
P.O. Box 750156 \\
Dallas, TX  75275-0156 \\ USA \\[2ex]
D. L. Kern \\ 
Institute for Mathematics and Its Applications \\
University of Minnesota \\ 
Minneapolis, MN 55404 \\ USA\\[2ex]
F. B. Hanson \\ 
Laboratory for Advanced Computing \\
M/C 249 \\
University of Illinois at Chicago \\
Chicago, IL  60607-7045 \\ USA \\
}
\end{center}
%}
%\makeatletter
%\author{
%\parbox[c]{6.5in}{\centering
%J. J. Westman$^a$, B. R. Fabijonas$^b$, D. Kern$^c$, and F. B. Hanson$^d$ \thanks{Work supported in part by the National Science Foundation Grant DMS-99-73231.} \\
%\vspace{.5em}
%{\small {\it $^a$
%Department of Mathematics,
%University of California,
%Box 951555,
%Los Angeles, CA  90095-1555, USA \\
%%
%$^b$Department of Mathematics,
%Southern Methodist University,
%P.O. Box 750156,
%Dallas, TX  75275-0156, USA \\
%%
%$^c$Institute for Mathematics and Its Applications, University of
%Minnesota, Minneapolis, MN 55404, USA\\
%%
%$^d$Laboratory for Advanced Computing,
%M/C 249,
%University of Illinois at Chicago,
%Chicago, IL  60607-7045, USA \\}}}\\
%\\*[.5em]
%MTNS 2002}
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%\makeatother
%\begin{document}
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%\maketitle
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%\noindent\hrulefill
\begin{abstract}
A compartment model for the evolution of cancer subject to
multiple chemotherapeutic agents is presented.  The formulation 
accounts for the heterogeneous nature of cancer and drug resistance, leading
to a system that can be used as a tool for optimizing treatments.
Chemotherapeutic or cytotoxic agents have limited effectiveness due to  
of toxicity and the development of drug resistance.  
Toxicity limits the amount 
of the agent that can be used and acts as an upper bound on the dosage.  Drug 
resistance causes the clinical value of the cytotoxic agent to become nominal 
after a small number of treatments.  Therefore, several non-cross resistant 
cytotoxic agents are used to extend the number of treatments that can be administered.
\end{abstract}
%\hrulefill
%
\section{Introduction}
\setcounter{equation}{0}
One of the most challenging aspects of treating 
cancer with chemotherapy is the 
development of resistance to the cytotoxic agents.  Drug resistance, whether 
intrinsic or acquired, ultimately leads to the 
ineffectiveness of the agent where
there is nominal clinical benefit compared to the
 side effects of the treatment.
To counter the development of the resistance, other treatment 
modes or options are used that destroy the resistant and sensitive cells.  
The use of multimode treatments, for example chemotherapy 
and radiotherapy, leads to a more effective treatment plan.
One of these multimode options is to use another 
cytotoxic agent that utilizes a different mechanism to destroy the cells and therefore should 
not be cross resistant, i.e., cells should not share all of the same
drug resistance mechanisms.  
Treatment therapies utilizing multiple cytotoxic agents are commonly used in 
practice, see for example \cite{lippincotts_98,cridland_78,cch_97}.

This paper extends the four compartment model presented 
in \cite{mbs01} for the evolution of cancer subjected to chemotherapy 
to account for the use of multiple cytotoxic agents.  For concreteness, 
we examine the case of two non-cross resistant cytotoxic agents (say agents 
$A$ and $B$) which yields 
an eight compartment model.  This can be extended rather easily 
to $M$ non-cross resistant cytotoxic agents, where the number of 
compartments would be given by
\eq\nonumber
2\sum_{i=0}^M  \left ( \begin{array}{c}
  M \\
  i
\end{array} \right ) = 2^{M+1}.
\nq
The resulting set of ordinary differential equations is viewed as a system 
which describes the evolution of the compartment populations.  
For the sake of brevity, 
the reader is referred to \cite{mbs01} for model details.  

\section{Compartmental Model}
Cancer can be viewed as a system consisting of three types of cells:
proliferating, clonogenic and end cells.  The division is based 
on the cells' function.  
The first type is the {\it proliferating fraction}, which 
consists of cells that are currently undergoing cell division and represents
the mechanism by which the cancer grows and spreads.  During the 
cell life
cycle, a number of checkpoints exist to ensure that the 
resulting daughter cells 
will be viable after mitosis.  For each viable daughter cell, there exists a 
positive probability for the daughter cell to become any 
one of the three types 
of cells. 
The quiescent or 
{\it clonogenic fraction} consists of {\it stem cells} 
that are capable of propagating
the cancer but are currently inactive.  Given a proper stimulus, the cells in
the clonogenic fraction can be recruited into the other two groups.  The final 
type are the {\it end cells} which consists of necrotic tissue and terminally 
differentiated somatic tissues such as vascular structures which can 
support but not propagate
the cancer.  The goal of most treatments 
is to move all of the cells from the growth and 
clonogenic fractions into the end compartment.  
The end cells are not accounted for in this model 
since they cannot generate new cancerous cells.

Drug resistance can either be intrinsic
or acquired.  Intrinsic resistance is resistance that exists prior to 
treatment. 
Acquired resistance to a cytotoxic agent evolves as a result of
treatment with the cytotoxic agent and develops after mitosis,
{\it randomly} providing the cell with mechanisms to defeat the actions
of cytotoxic agents via mutations.  Mutations can only occur in
cells that are actively dividing, therefore only cells in the
proliferating fraction can mutate.  
Either kind of drug resistance is assumed to be inherited by the
daughter cells and their progeny.  Note that resistance to a
particular drug normally implies that resistance should exist for
all drugs in the same category that act with the same mechanism.
Thus, two drugs which use the same mechanism to attack the cancerous
cell are referred to as cross resistant.

A treatment consists of administering both agents at 
specified times.  For example, a typical 
treatment period is 28 days with agent $A$ 
administered on day $1$ and agent $B$ on day $14$.  
A treatment cycle consists of $n$ treatments, and at most $m$ cycles can be 
administered.  Therefore a total of $N=n \times m$ treatments may be given.  
Regimens for treatment normally start with a single treatment 
cycle and continue
until the patient goes into remission or all of the available treatments have 
been administered.  Upon recurrence of the cancer, a 
new treatment cycle begins 
provided a treatment is available.
Figure~\ref{fig:8comptreat} represents the eight compartment model for
the evolution of heterogeneous cancer with treatment.
In the figure, the solid lines represent the evolutionary dynamics of 
the cancer and the the dashed lines show the effects of treatments.
\begin{figure}%[htp]
~\\*[-13em]
\begin{center}{%
\includegraphics
[width=6.0in]
{chemo8.eps}
}
\caption{Schematic representation of the eight compartmental model
subject to chemotherapy with cytotoxic agents $A$ and $B$.   
The dashed lines are chemotherapy terms.
Compartments to the left of the figure's center denote the 
proliferating compartments $P_i$,
and similarly, those on the right denote the 
clonogenic compartments $C_i$, for $i=0,~ 1,~ 2,~ 3$.
Compartments aligned in rows are susceptible to neither agent ($P_0,C_0$), 
to agent $A$ only ($P_1,C_1$), to agent $B$ only ($P_2,C_2$), or both 
cytotoxic agents ($P_3,C_3$).  
} \label{fig:8comptreat}
\end{center}
\end{figure}

\subsection{Mathematical Model}
Cancer is a stochastic evolutionary process for the various 
types of cells.  
The physical process for the development of cancer is an
intractable problem since all cells and their evolution 
would need to be followed.
Therefore, the mathematical model presented here is based on 
macroscopic or bulk 
actions and is not necessarily related to the biological processes 
of the individual cancerous cells. However, the bulk dynamics may be 
observed through means such as clinical {\it in vitro} trials 
or {\it in vivo} methods such as imaging.

Define the unit dose impulse function, $\Delta_{j,k}(t)$ 
for the time $t=t_{j,k}$ at which a 
treatment $j$ of cytotoxic agent $X_k$ is administered by
\eq
\Delta_{j,k}(t) = \left \{
\begin{array}{ll}
1 ~~ & ~~ t=t_{j,k} \\
0 ~~ & ~~ {\rm otherwise}
\end{array}
\right \},
\nq 
where $X_1 = A$ and $X_2 = B$.
Define the indicator set
\eq
{\cal I} \equiv \left \{ ~~ 0, ~~ 1, ~~ 2, ~~ 3 \right \}
\nq
to represent the classification of the cells and their resistance to 
the cytotoxic agents, where $P_i(t)$ and $C_i(t)$ for $i\in{\cal I}$ denote 
the number of cells in the proliferating and clonogenic fractions, 
respectively, 
where the index $i$ denotes no resistance if $i=0$, resistance to 
agent $A$ if $i=1$,
resistance to agent $B$ if $i=2$, resistance to both agents if $i=3$. 
%
The model that is
generated predicts the evolution of the cancer starting from
a single 
cell. Let time $t=0$ be the time of the appearance
of the first malignant cell from which a colony will form.  This
first cell must be in the growth fraction, which yields the
initial conditions for the model as
%%
\eq \label{eq:initdata}
\begin{array}{ccc}
 P_0(0)=1, ~~ & ~~ P_i(0) = 0 ~~{\rm for}~~ i=1,~2,~3, 
 ~~ & ~~ C_i(0) = 0 ~~{\rm for}~~ i \in {\mathcal I}. %BRF3%i=0,~1,~2,~3.
\end{array}
\nq

For the remainder of this section let $i,s \in {\cal I}$ be the indicator
for the drug resistance, 
$X_k\in \{A,B\}$ be the cytotoxic agent administered, $Y\in\{P,~C\}$
represent proliferating or clonogenic fraction, and $j = 1,~2,~\ldots,~N$ 
denote the treatment number.  The overall growth dynamics for the untreated
cancer is assumed 
to be Gompertzian in nature with relative rate of change form
%%
\eq \label{eq:growthrate4}
f(P_0,P_1,P_2,P_3,C_0,C_1,C_2,C_3) = \lambda \log \Big (
\frac{K}{P_0 + P_1 + P_2 + P_3 + C_0 + C_1 + C_2 + C_3} \Big ),
\nq
%%
where the time dependence of $P_i(t)$ and $C_i(t)$ %BRF3 %for $i\in{\cal I}$ 
has been suppressed.
The evolutionary rates for the growth of the cancer are 
$\alpha_{i}$ denoting natural migration of daughter cells after mitosis 
from the proliferating to the clonogenic fraction,
$\beta_{i}$ the natural back migration of clonogenic cells which 
restart the cell life cycle,
and $\delta_{\sP,i}$ or $\delta_{\sC,i}$ the natural loss of cells 
either from the proliferating or clonogenic compartments to the end cell 
compartment.
All of these rates are, in general, functions of
$(P_0,P_1,P_2,P_3,C_0,C_1,C_2,C_3,t)$.  The probabilistic
mutation rates for the viable mutated cells are denoted by
$\mu_{{\scriptscriptstyle Y_i},{\scriptscriptstyle Y_s}}$,
where $Y_i$ is the compartment for the
parent cell and $Y_s$ is the compartment of the daughter cell.
All possible mutations are allowed,  
except for those mutations that have zero probability of occurring.  
Transitions from mutations are not allowed 
to stay in the same compartment, therefore 
$\mu_{{\scriptscriptstyle Y_i},{\scriptscriptstyle Y_i}}\equiv 0$, 
and mutations are not allowed where resistance to one agent is lost 
and the resistance to the other is gained, 
therefore $\mu_{{\scriptscriptstyle Y_1},{\scriptscriptstyle Y_2}}\equiv 0$ 
and $\mu_{{\scriptscriptstyle Y_2},{\scriptscriptstyle Y_1}}\equiv 0$.

Let $\Delta_{j,k}\kappa_{{\scriptscriptstyle Y_i},j,k}$ denote the loss of cells in the $Y_i$ 
compartment due to treatment $j$ of agent $X_k$. 
A feature of this model is that only susceptible cells to a given
agent $X_k$ are killed by the administration of the treatment, which
is assumed to be uniform.  However, 
the kill rates
can vary to account for semi-cross resistance of cytotoxic agents.  
For the model in Figure~\ref{fig:8comptreat}, the nonzero kill rates are 
$\Delta_{j,k}\kappa_{{\scriptscriptstyle P_0},j,k}$, 
$\Delta_{j,2}\kappa_{{\scriptscriptstyle P_1},j,2}$, and 
$\Delta_{j,1}\kappa_{{\scriptscriptstyle P_2},j,1}$. 
As a result of the large numbers of cells
killed by the administration of cytotoxic agent $X_k$, a stimulus is
generated that recruits cells form the clonogenic fraction into
the proliferating fraction to ensure the viability of the cancer
at rate $\Delta_{j,k}\gamma_{{\scriptscriptstyle C_i},{\scriptscriptstyle Y_i},j,k}$.  
Side effects of treatment
include the development acquired resistance at a rate 
$\Delta_{j,k}\mu_{{\scriptscriptstyle Y_i},{\scriptscriptstyle Y_s},j,k}$.  
It is assumed, 
without 
loss of generality, that those cells that acquire resistance
stay in the proliferating fraction. The only nonzero rates for the
development of acquired resistance are 
$\Delta_{j,1}\mu_{{\scriptscriptstyle P_0},{\scriptscriptstyle P_1},j,1}$,
$\Delta_{j,1}\mu_{{\scriptscriptstyle P_2},{\scriptscriptstyle P_3},j,1}$, 
$\Delta_{j,2}\mu_{{\scriptscriptstyle P_0},{\scriptscriptstyle P_2},j,2}$, and
$\Delta_{j,2}\mu_{{\scriptscriptstyle P_1},{\scriptscriptstyle P_3},j,2}$.

The equations for the proliferating compartments of the
compartment model are
%%
\eq
\label{eq:8comptreatprol}
\frac{dP_i}{dt} &=& \Bigg [
        \Big \{ 1-\alpha_{i}-\sum_{s=1}^4
    (\mu_{{\scriptscriptstyle P_i},{\scriptscriptstyle P_s}}
     + \mu_{{\scriptscriptstyle P_i},{\scriptscriptstyle C_s}}) \Big \} P_i 
     + \sum_{s=1}^4 
    \mu_{{\scriptscriptstyle P_s},{\scriptscriptstyle P_i}}P_s  \Bigg ] f 
    + \beta_{i} C_i - \delta_{\sP, i}P_i \nonumber \\
    && +
         \sum_{j=1}^N \sum_{k=1}^2\Bigg \{ 
      \Big (-\Delta_{j,k}\kappa_{{\scriptscriptstyle P_i},j,k} -
        \sum_{s=i}^4\Delta_{j,k}
        \mu_{{\scriptscriptstyle P_i},{\scriptscriptstyle P_s},j,k}  \Big ) P_i \\
        && \hspace{10em} + \sum_{s=1}^{i} \Delta_{j,k}
          \mu_{{\scriptscriptstyle P_s},{\scriptscriptstyle P_i},j,k}P_s 
        + \gamma_{{\scriptscriptstyle C_i},{\scriptscriptstyle P_i},j,k} C_i \Bigg \},
  \nonumber
\nq
%%
where $i \in {\mathcal I}$.
The corresponding equations for the clonogenic compartments are
%%
\eq\label{eq:8comptreatclon}
\frac{dC_i}{dt} &=& \Bigg [ \alpha_{i}P_i 
   + \sum_{s=1}^4 \mu_{{\scriptscriptstyle P_s},{\scriptscriptstyle C_i}}P_s  
          \Bigg ] f 
    - \beta_{i} C_i - \delta_{\sC, i}C_i 
        - \sum_{j=1}^N \sum_{k=1}^2\Delta_{j,k} \Big ( \kappa_{{\scriptscriptstyle C_i},j,k} 
           + \gamma_{{\scriptscriptstyle C_i},{\scriptscriptstyle P_i},j,k} \Big )C_i,
\nq
%%
where $i \in {\mathcal I}$.
Therefore, the system of equations which represents the dynamics as illustrated in Figure~\ref{fig:8comptreat} are given by \eqref{eq:8comptreatprol} and 
\eqref{eq:8comptreatclon}, which are subject to the initial condition \eqref{eq:initdata}.

\section{Discussion}
Ultimately, the majority of cells reside
in the $P_3$ and $C_3$ compartments so that if any cytotoxic agent
is administered the change in the number of cells is algebraic at
best.  This suggests that alternate treatment modes should be sought
to reduce the drug resistant populations and the number of
cancerous cells.  Such treatment modalities as the use of surgery,
radiotherapy, and immunotherapies using cytokines, angio-genesis
inhibitors, and anti-metastatic agents could be employed 
in a comprehensive treatment plan for the patient.  

The formulation of the cancer evolution and the subsequent treatment
focuses on the heterogenous nature of the cancer and the effects
drug resistance.  Thus, this provides a system that can be utilized
by clinicians to determine the effectiveness of treatment regimens
employing multiple cytotoxic agents to predict the response of the
cancer to a given strategy.  Additionally, the model presents a
framework in which alternate treatment strategies can be evaluated.
A future direction is to solve numerically the optimal control problem 
for the treatment schedule in order
to maximize the life expectancy and quality of life of the
patient.

Modeling the treatment of cancer has difficulties since  
a number of the 
underlying biological mechanisms are not clearly understood, for example, 
the development of resistance to a given treatment.  
Typically, tumors are not composed of a single strain of 
cancer cell but 
consist of several prominent strains, due to mutations, 
which will react differently to the various cytotoxic agents 
utilized for treatment.  These mutations greatly complicate 
the model and need to be included for a better understanding of 
the effects of a given treatment plan.
Future research directions include the addition of more realism 
in the model to account for mutations in the cancer cells and a 
more comprehensive treatment plan, including considerations of 
pharmacokinetics and toxicity of treatments,   to include all 
possible modes, in particular radiotherapy and angiogenesis 
inhibitors, for combatting the cancer and retarding its development.
Additionally, an optimal control problem needs to be developed, 
tailored to every patient, that determines the most effective 
treatment plan for the patient that accounts for the disease and 
the patients' health and mental state.
We believe that the use of a tailored individual comprehensive treatment plan using multimode treatments should lead to a better quality of life for the patient as well as a longer life expectancy.
%

\vspace{.1in}
 \noindent \textbf{Acknowledgement}: FBH was supported in part by 
the National Science Foundation Grant DMS-99-73231.

\begin{thebibliography}{10}
\bibitem{lippincotts_98}
D. C. Baquiran and J. Gallagher, 
   {\it Lippincott's Cancer Chemotherapy Handbook}, 
   Lippincott-Raven Publishers--Philadelphia, 1998.

\bibitem{cridland_78}
M. D. Cridland, 
   {\it Fundamentals of Cancer Chemotherapy,} 
   University Park Press--Baltimore, 1978.

\bibitem{cch_97}
D. S. Fischer, M. T. Knobf, and H. J. Durivage, 
    {\it The Cancer Chemotherapy Handbook, 5th ed.,} 
    Mosby--St Louis, 1997.

\bibitem{mbs01}
J. J. Westman, B. R. Fabijonas, D. L. Kern, and F. B. Hanson, 
    ``Compartmental 
    Model for Cancer Evolution: Chemotherapy and Drug Resistance,'' 
    {\it Math. Biosci.} (submitted 2001).
\end{thebibliography}
%
\end{document}
