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\begin{document}

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\title{\ifnum\TeacherEdition=1 TEACHER VERSION \else STUDENT VERSION \fi\\Should Cancer Therapy Start Before or After Surgery?}




\markboth{Surgery and Therapy}{Surgery and Therapy}

\author{

\hspace*{.5 true cm} \begin{minipage}[t]{7.5 true cm}
Maila Brucal-Hallare\\
Norfolk State University\\
Norfolk  VA 23504 USA  \\
mcbrucal-hallare@nsu.edu
\end{minipage}
\ \quad \
\begin{minipage}[t]{7.5 true cm}
Iordanka Panayotova\\
Christopher Newport University\\
Newport News VA 23606 USA\\
iordanka.panayotova@cnu.edu
\end{minipage}
}


\makeStitlePDFLaTex

\begin{abstract}
This activity builds upon elementary models on population growth. In particular, we compare two different treatment models of cancer therapy where in one, surgery happens before therapy and in the other, surgery happens after therapy. 
Activities will help students appreciate the importance of initial values in the existence of solutions.
\end{abstract}


\section*{SCENARIO DESCRIPTION}
Heart disease, according to the Center for Disease Control and Prevention, is the leading cause of mortality in the USA.  Next in the list is cancer.  There are many types of cancer but all of them have the same definition, according to the National Cancer Institute \cite{cancer.gov}:

\begin{quote}
\textit{Cancer is a disease in which some of the body’s cells grow uncontrollably and spread to other parts of the body.}
\end{quote}

This activity is inspired by some current research on having surgery before therapy \cite{surgery before therapy} and having surgery after therapy \cite{surgery after therapy}.    

As a student of differential equations, there are two verbs in this definition that should engage you: grow (uncontrollably) and spread (to other parts of the body).  These two dynamic verbs provide rationale for why many mathematical oncologists use differential equations to investigate cancer  growth, spread, and treatments.

Currently, cancer treatments include surgery and various kinds of therapy: radiotherapy, chemotherapy, immunotherapy, hormonal therapy, and even viro-therapy \cite{cancer.net}.  Refer to another SIMIODE Modeling Activity by the authors on viro-therapy \cite{Panayotova2021}.  In this activity, you will explore a basic question: does the order of treatment matter? In particular, suppose that the oncologist has recommended surgery and chemotherapy to treat a cancer.  Should the surgery happen before chemotherapy, or after chemotherapy?  We shall use a highly simplified differential equation to explore this question.  Please remember that this activity is for classroom exploration only and the results of the exploration should not be used in making medical decisions.

Let $x = x(t)$ represent the cancer cell population at any time $t$.  Suppose $G(x)$ describes the growth rate of cancer cells and $D(x)$ describes the death rate of $x$ due to a certain kind therapy.  Let us assume that both growth and death rates are modeled by the linear functions
$$
G(x) = ax, \:\:\: D(x) = b x,
$$
for some growth parameter $0 < a < 1$ and some treatment parameter $0 < b  < 1$ that depend on the specific cancer under investigation.
\begin{figure}
\centering
\includegraphics[width=14cm]{Treatment Case 1.PNG}
\caption{Observation timeline for treatment model 1.}
\end{figure}

\begin{figure}
\centering
\includegraphics[width=14cm]{Treatment Case 2.PNG}
\caption{Observation timeline for treatment model 2.}
\end{figure}

To analyze the exploration question, we shall assume the following:
\begin{enumerate}
\item the initial population of cancer cells is $x(0) = X_{0}$;
\item surgery happens instantaneously;
\item therapy happens once and over an interval of time of positive length; and
\item after surgery, 99$\%$ of the cancer cells have been removed.
\end{enumerate}

We shall explore two treatment models with stages as described below.  Both treatment models will have three stages, where $t = 0$ is the initial time, $t=t_{S}$ is the surgery time, $t = t_{T}$ is the start of therapy time, and $t = t_{F}$ is the final time: 
\begin{itemize}
\item Treatment model 1 considers the treatment scenario when surgery is applied after therapy.  The time intervals are $0 < t_{T} < t_{S} = t_{F}.$  See Figure 1.

\item Treatment model 2 considers the treatment scenario when surgery is applied before therapy.  The time intervals are $0 < t_{S} = t_{T} < t_{F}.$  See Figure 2.
\end{itemize}

In which of these two treatment scenarios do we end up with less cancer cells?  The exercises below will guide you towards a concrete result.


\section*{ACTIVITY}


\subsection*{Treatment model 1: surgery after therapy.
}
\begin{enumerate}
\item The initial-value problem for stage 1 is
$$
\left\{
\begin{array}{llll}
\dfrac{dx}{dt} & = & ax, & 0 \leq t < t_{T},\\
x(0) & = & X_{0}. & \\ 
\end{array}
\right.
$$
Using separation of variables and then plugging-in the initial value $x(0) = X_{0}$ yields
$$
\dfrac{dx}{x} = a \: dt \implies x(t) = \rule{4cm}{0.15mm}.
$$

\item What is the initial-value problem for stage 2?
$$
\left\{
\begin{array}{llll}
\dfrac{dx}{dt} & = & ax-bx, \:\:\:t_{T} \leq t < t_{S},\\
x(t_{T}) & = & \rule{4cm}{0.15mm}, & \\ 
\end{array}
\right.
$$
Using separation of variables and then plugging-in the initial value yields $x(t_{T}) = X_{0}e^{at_{T}}$, 
$$
\dfrac{dx}{x} = (a-b) \: dt \implies x(t) = \rule{4cm}{0.15mm}.
$$

\item In stage 3, surgery happens instantaneously.  Using your answers in the previous item, what is the number of cancer cells at the end of treatment model 1, assuming that surgery removes $99\%$ of the cancer cells from the previous stage?
\begin{equation}\label{EndofTreatmentModel1}
x(t_{S}) = x(t_{F}) = \rule{4cm}{0.15mm}.
\end{equation}
\end{enumerate}

\subsection*{Treatment model 2: surgery before therapy.
}
\begin{enumerate}
\item The initial-value problem for stage 1 is
$$
\left\{
\begin{array}{llll}
\dfrac{dx}{dt} & = & ax, & 0 \leq t < t_{S},\\
x(0) & = & X_{0}. & \\ 
\end{array}
\right.
$$
Using separation of variables and then plugging-in the initial value we find
$$
\dfrac{dx}{x} = a \: dt \implies x(t) = \rule{4cm}{0.15mm}.
$$

\item In stage 2, surgery happens instantaneously at time $t = t_{S}$.  Using your answers in the previous item, what is the number of cancer cells at the end of stage 2, assuming that surgery removes $99\%$ of the cancer cells from the previous stage?
$$
x(t_{S}) = \rule{4cm}{0.15mm}.
$$

\item What is the initial-value problem for stage 3?
$$
\left\{
\begin{array}{llll}
\dfrac{dx}{dt} & = & ax-bx,\:\:\: t_{T} \leq t < t_{F},\\
x(t_{T}) & = & \rule{4cm}{0.15mm}. & \\ 
\end{array}
\right.
$$
Using separation of variables and then plugging-in the initial value, we find
$$
\dfrac{dx}{x} = (a-b) \: dt \implies x(t) = \rule{4cm}{0.15mm}.
$$

\item How many cancer cells are there at the end of treatment model 2?
\begin{equation}\label{EndofTreatmentModel2}
x(t_{F}) = \rule{4cm}{0.15mm}.
\end{equation}

\item Having completed your computations in the previous exercise, the next step is to compare the numbers in (\ref{EndofTreatmentModel1}) and (\ref{EndofTreatmentModel2}).  Did you get equal quantities? Explain why.

\end{enumerate}


\subsection*{Treatment Model 3: Modified Treatment Model 2}
\begin{enumerate}

\item Treatment models 1 and 2 show us that commuting surgery with therapy, with the assumptions above, did not give any difference in the final counts of the cancer cells.  Let us look at a modified version of Treatment model 2.  
\begin{figure}
\centering
\includegraphics[width=14cm]{Treatment Case 3.PNG}
\caption{Observation timeline for treatment model 2.}
\end{figure}
In Treatment model 3, we assume that after surgery, the growth of cancer cells is now modeled by $f(x) = a^2 x$, where $0 <a < 1.$  See Figure 3.

\item The initial-value problem for stage 1 is
$$
\left\{
\begin{array}{llcl}
\dfrac{dx}{dt} & = & ax, & 0 \leq t < t_{S},\\
x(0) & = & X_{0}. & \\ 
\end{array}
\right.
$$
Using separation of variables and then plugging-in the initial value we have,
$$
\dfrac{dx}{x} = a \: dt \implies x(t) = \rule{4cm}{0.15mm}.
$$

\item In stage 2, surgery happens instantaneously at time $t = t_{S}$.  Using your answers in the previous item, what is the number of cancer cells at the end of stage 2, assuming that surgery removes $99\%$ of the cancer cells from the previous stage?
$$
x(t_{S}) = \rule{4cm}{0.15mm}.
$$

\item What is the initial-value problem for stage 3?
$$
\left\{
\begin{array}{llll}
\dfrac{dx}{dt} & = & a^2 x-bx, \:\:\: t_{T} \leq t < t_{F},\\
x(t_{T}) & = & \rule{4cm}{0.15mm}. & \\ 
\end{array}
\right.
$$
Using separation of variables and then plugging-in the initial value we find 
$$
\dfrac{dx}{x} = (a^2-b) \: dt \implies x(t) = \rule{4cm}{0.15mm}.
$$

\item How many cancer cells are there at the end of treatment model 3?
\begin{equation}\label{EndofTreatmentModel3}
x(t_{F}) = \rule{4cm}{0.15mm}.
\end{equation}

\item Having completed your computations in the previous exercise, the next step is to compare the numbers in (\ref{EndofTreatmentModel2}) and (\ref{EndofTreatmentModel3}), for example, by computing the magnitude of their ratios. Observe that many factors will cancel out.  

\item Since $0 < a < 1$, conclude your investigation by writing a mathematical prescription on whether surgery should be applied before therapy, given the above assumptions.  Make sure that you capture all the assumptions and the main result. 

\end{enumerate}

\subsection*{Investigating the assumptions of the models}
\begin{enumerate}
\item The models that you see above assume that the growth and death rates are both proportional to their population.  These assumptions are highly idealized. Depending on the kind of cancer, these rates may vary considerably.  Use the internet to find information on the growth rates that mathematical oncologists have been using in their models.  A good starting point is \cite{growth models}.      

\item The models assume that the surgery is successful and that the population is reduced by a constant factor (in the treatment models above, we assumed $99\%$).  A variant of this is a constant factor $K$ where $|K| \leq e^{-L}$ for some $L > 0$.  However, some cancer cells cannot be removed by surgery.  In the models, we assumed that the surgery happens instantaneously.  How would you modify the model if two non-instantaneous therapies are applied, for example, radiotherapy and chemotherapy?       

\item How would you modify the equations in case the oncologist recommends three treatments, one after the other?  

\item Out of mathematical curiosity and not out of realistic assumptions, mathematical oncologists analyze equations like $\dfrac{dx}{dt} = ax - bx$ for $a = b, a< b, a > b$.  Reflect on the implications of each of these three cases on the two models.  Make an intelligent guess on how the magnitudes of $a$ and $b$ affect the efficacy of either treatment models.


\end{enumerate}


\teacher{\section*{NOTES FOR TEACHERS }

There is a lot of research effort on the efficacy of various cancer treatments and on developing different treatment modes.  This activity is inspired by some current research on having surgery before therapy \cite{surgery before therapy} and having surgery after therapy \cite{surgery after therapy}.    

In our experience, students appreciate it when they are exposed to current cancer research as early as possible.  We believe that presenting meaningful real-life applications, such as studying cancer treatments, supplements the learning of the mechanics of any of the methods that they will learn in class.  We recommend that this straightforward activity be presented during the first few sessions of a one-semester course in ordinary differential equations to engage students' interests on using differential equations to model real-life phenomena.  

The equations are linear and separable.  Moreover, this activity brings to life the importance of initial values in the existence of solutions.  Figures 1, 2, and 3 are diagrams that illustrate the various treatment models covered in this activity. 

These are the formal learning objectives for this activity:
\begin{enumerate}
\item Create a mathematical model of a treatment scenario where therapy occurs before surgery and therapy occurs after surgery. 
\item Solve the resulting mathematical models using separation technique and compare results.
\item Provide opportunities to learn how mathematics is used in cancer research.
\end{enumerate}
}
%\subsection*{Numerical Investigation: %Does the order of the treatment %matter?
%}
%\begin{enumerate}
%\item Assume the following values for %the parameters $\alpha, \beta$ and %time $t_{S}, t_{T}, t_{F}$ when %surgery happens after therapy:
%$$
%\alpha = 1.2, \beta = 0.8, X_0 = 100, %t_{S} = 10, t_{T} = 12, t_{F} = 20. 
%$$
%Compute $x(t)$ as given in %(\ref{surgerythentherapy}).

%\item Assume the following values for %the parameters 
%$\alpha, \beta$ and time $t_{T}, %t_{S}$ when surgery happens before %therapy:
%$$
%\alpha = 1.2, \beta = 0.8, X_0 = 100, %t_{T} = 12, t_{S} = t_{F} = 20. 
%$$
%Compute $x(t)$ as given in %(\ref{therapythensurgery}).

%\item Compare the values that you %obtained from 2(a) and 2(b).  

%\end{enumerate}

\begin{thebibliography}{98}

\bibitem{Panayotova2021} Panayotova, I.~N.  and M.~Brucal-Hallare.  2021. The Good Kind of Virus: Oncolytic Viruses vs Cancer Cells. 6-017-S-OncolyticViruses.

\bibitem{cancer.net} Breast Cancer: Types of Treatment. 2020.  Cancer.net. \href{https://www.cancer.net/cancer-types/breast-cancer/types-treatment}{https://www.cancer.net/cancer-types/breast-cancer/types-treatment}.

\bibitem{cancer.gov} National Cancer Institute. 2021. What is cancer? \href{https://www.cancer.gov/about-cancer/understanding/what-is-cancer}{https://www.cancer.gov/about-cancer/understanding/what-is-cancer}.

\bibitem{growth models}
Gerlee,  P. 2013. The model muddle: in search of tumor growth laws. {\it Cancer Res\/}. 73(8): 2407-2411. doi: 10.1158/0008-5472.CAN-12-4355. Epub 2013 Feb 7. PMID: 23393201.


\bibitem{surgery after therapy}
Hennon, Mark, and Sai Yendamuri. 2019. For radiation therapy before surgery in esophageal cancer, dose matters, and with each answer comes more questions.  {\it Journal of Thoracic Disease\/}. 11,12 (2019): 5662-5663. doi:10.21037/jtd.2019.12.62

\bibitem{surgery before therapy}
Sekiguchi, Atsushi {\it et al}. 2016. Postoperative hormonal therapy prevents recovery of neurological damage after surgery in patients with breast cancer. {\it Scientific reports}.  Vol. 6 34671. 6 Oct. 2016. doi:10.1038/srep34671.

\end{thebibliography}



\end{document}
