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\title{\ifnum\TeacherEdition=1 TEACHER VERSION  \else STUDENT VERSION \fi\\Exploring Tumor Growth Modeling}




\markboth{Tumor Growth }{Tumor Growth }

\author{Randy Boucher and Ryan Miller\\
United States Military Academy\\
West Point NY 10996 USA
}

\makeStitlePDFLaTex


\begin{abstract}
Students will transform, solve, and interpret a tumor growth scenario using non-linear differential equation models. Two population growth models (Gompertz and logistic) are applied to model tumor growth. Students use technology to solve the Gompertz model and answer a series of questions designed to further their understanding of growth models and to refine their ability to analyze and compare mathematical models.  
\end{abstract}


\section*{SCENARIO DESCRIPTION}

This is an INDIVIDUAL assignment. Your primary submission is a hard copy report that should \textbf{not exceed three (3) pages}. To support your primary report, you should include annotated and referenced appendices. 

You should include any additional supporting graphs, equations, and computations in the appendices. If you think that your code is important, you may include it in an appendix, but you should ensure that it includes annotations describing what you are doing. If you think a computation or equation is important for the reader, then ensure  it is in the report or appendices. 

Ensure all work is logical, neat, and organized. Properly document any sources or assistance you receive.  Doing the mathematics correctly is important, but it is also critical to be able to analyze and effectively communicate your mathematical results, as well as reflect on the relevance of your results in the real world.
\pagebreak{}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% GOALS
\section*{Goals}
\begin{enumerate}
\item Clearly communicate your mathematical model and solution methods in a written form.
\item Explore aspects of the problem from a different disciplinary perspective to supplement mathematical modeling.
\item Design and solve a mathematical model for a real world problem using differential equations.
\item Confront the ambiguity of problem solving and understand the necessity of making assumptions in formulating a mathematical model. Explore the impact these assumptions have on solutions.
\item Apply technology learned in the course to analyze and to solve a differential equations model. Develop an appreciation for how technology enhances your problem solving capabilities.
\end{enumerate}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% TOOLS
\section*{Helpful tools}
\begin{enumerate}
\item Mathematical computation software.
\item Differential equations textbook.
\end{enumerate}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% PROJECT
\section*{BACKGROUND}

Tumor growth has been investigated extensively for more than 60 years~\cite{Benzekry2014}. Normally, human cell repair and reproduction is achieved through a regulated cell division process. Lumps or tumors develop when cell division continues at an uncontrolled rate (Figure $1$). There are many factors that contribute to tumor growth and development. This complex system of influencing factors can be simplified and represented as a mathematical model. Designing models helps researchers understand many issues surrounding tumors including evaluation of growth hypotheses, predicting patient prognosis, and  determining the effectiveness of different therapies applied to combating tumor growth~\cite{Laird1965,Colin2010,Bernard2012}.  \\ 

 \begin{figure}[h!]
 \centering
  \includegraphics[width=.75\textwidth]{cancer.pdf}
   \label{tumor}
 \caption{The development of a tumor~\cite{Tidy2018}. }
 \end{figure}
 
Experimental observation has revealed that the relative growth rate of a tumor decreases with time~\cite{Benzekry2014}. This observation implies a connection between the principles of tumor growth and general growth laws, which can be expressed as ordinary differential equations~\cite{Gerlee2013}. Many mathematical models have been studied to predict tumor growth, however, the most widely used are the \textbf{logistic model} and \textbf{Gompertz model} described below. \\
\vspace{.1in}
The \textbf{logistic model} is given by 
\begin{equation}\label{logistic}
\frac{dV_L}{dt}=r_L(1-\frac{V_L}{K_L})V_L,\quad V_L(0)=V_{L0},
\end{equation}  
where $V_L(t)$ is the tumor volume at time $t$, $r_L$ is the growth rate constant of the tumor, $K_L$ is the carrying capacity, and $V_{L0}$ is the volume of the tumor at the time of initial observation, $t=0$. The general logistic model solution is given by
\begin{equation*}
%\label{log_sol}
V_L=\frac{V_{L0}K_L}{V_{L0}+(K_L-V_{L0})e^{-r_Lt}}.
\end{equation*} 
The \textbf{Gompertz model} is given by 
\begin{equation}\label{gompertz}
\frac{dV_G}{dt}=r_G\ln{(\frac{K_G}{V_G})}V_G,\quad V_G(0)=V_{G0},
\end{equation} 
where $V_G(t)$ is the tumor volume at time $t$, $r_G$ is the growth rate constant of the tumor, $K_G$ is the carrying capacity, and $V_{G0}$ is the tumor volume at, $t=0$. \\
\vspace{.1in}
Notice that both models are \textbf{autonomous} differential equations and so concepts introduced in your textbook may be useful to analyze solutions of each model.  Table 1 displays data for a typical spheroid tumor growth over a 60 day period.
\vspace{.1in}

\begin {table}[h]
% \label{data}

\begin{center}
\scalebox{0.9}{
    \begin{tabular}{ | c | c | c | c | c | c | c | c |}
    \hline
    time (days) & 0 & 10 & 20 & 30 & 40 & 50 & 60\\ \hline
    volume ($\mu$m$^3$) & $2.90\times 10^3$ & $1.20\times 10^8$ & $3.00\times 10^9$ & $6.00\times 10^9$ & $6.60\times 10^9$ & $7.20\times 10^9$ & $7.30\times 10^9$ \\ \hline
        \end{tabular}}
\end{center}
\vspace{-12pt}
\caption {Typical spheroid tumor growth over a 60 day period.}
\end{table}

\vspace{-.03in}
Additionally, best fit curves were obtained for models (\ref{logistic}) \& (\ref{gompertz}) resulting in the optimal parameter values: $r_L=.721$ per day, $K_L=6.77\times 10^9$ $\mu$m$^3$, $r_G=.141$ per day and $K_G=7.23\times 10^9$ $\mu$m$^3$. 

Figure 2 illustrates the spheroid growth process for a tumor for the first 20 days. 


 \begin{figure}[h!]
 \centering
  \includegraphics[width=.75\textwidth]{spheroid.jpg}
   \label{tumor}
 \caption{Example of Spheroid Growth~\cite{Wong2017}. }
 \end{figure}
\newpage

\section*{TASK: TUMOR GROWTH MODEL EVALUATION AND COMPARISON}

Conduct an analysis of the Gompertz model (\ref{gompertz}) and provide a comparison of it to the logistic model using the data provided in Table 1. Address the following questions in your report: 
\begin{enumerate}
	\item Find the solution to the Gompertz model.

	\item Discuss at least two assumptions that are both reasonable and necessary to model tumor growth with the Gompertz model.

	\item Plot the graph of $V_L(t)$ \& $V_G(t)$ verses $t$ for $0\leq t\leq 60$ (with the given parameter values) along with the data provided in Table 1. Properly label the graph. Qualitatively comment on the similarities and differences of the two solution curves.
	\item Plot the graph of $f(V_G)=r_G\ln{(K_G/V_G)}V_G$ versus $V_G$, find the critical point(s) and label each as \textit{asymptotically stable} (attractor), \textit{unstable} (repeller), or \textit{semi-stable}. What does it mean to be asymptotically \textit{stable}, \textit{unstable}, or \textit{semi-stable} within the context of tumor growth?

	\item For $0\leq V_G\leq K_G$, determine where $V_G$ versus $t$ is \textit{concave up} and where it is \textit{concave down}. Comment on the dynamics of the tumor growth associated within each interval.

	\item Conduct a goodness-of-fit test of the logistic and Gompertz  models using the data provided in Table 1. One method is to complete a sum of square errors (SSE) analysis. A SSE can be computed by summing the square of each error in the model's predicted values at $t=0,\,10,\,20,\,30,\,40,\,50\,\textrm{and}\,60$ days. Which is the better model? Why?

	\item Suppose $V(\tau)=3\times 10^9$ $\mu$m$^3$ is the tumor size for which cancer treatments (such as immunotherapy and chemotherapy) are unlikely to bring about a  successful outcome.  Use the logistic \textit{and} Gompertz models to solve for time $\tau$. Which value of $\tau$ do you think is more accurate? Why? 
	
	\item The logistic model is typically used to to model population growth. Per (unit) population growth rate (PPGR) measures the growth one person contributes. By re-arranging the logistic equation we can express (\ref{logistic}) as the per (unit) tumor volume growth rate (PVGR) as a function of $V_L$, which is
	\begin{equation}
	PVGR(V_L) =\frac{1}{V_L}\frac{dV_L}{dt}=r_L(1-\frac{V_L}{K_L}).\quad
	\end{equation} 
    What does PVGR measure as it applies to our tumor growth logistic model?

	\item Rearrange (\ref{gompertz}) to find $PVGR(V_G)$. For the logistic and Gompertz models respectively, find the following:
	 $$\text{(i)} \lim_{V_L\to 0}PVGR(V_L) \hspace{1in}\text{     (ii)} \lim_{V_L\to K_L}PVGR(V_L)$$
	
	 $$\text{(iii)} \lim_{V_G\to 0}PVGR(V_G)  \hspace{.9in}\text{     (iv)} \lim_{V_G\to K_G}PVGR(V_G) $$
	
	%\vspace{3in}
	\item Compare and contrast the behavior of the PVGR of these models. Provide a final recommendation as to which model is better for modeling tumor growth. 
   \end{enumerate}


\begin{thebibliography}{98}

  \bibitem{Benzekry2014} Benzekry, Sébastien {\it et al\/}. 2014. Classical mathematical models for description and prediction of experimental tumor growth. {\it PLOS computational biology\/}. 10.8: 1-19. 
  
  \bibitem{Bernard2012} Bernard, Apexa {\it et al\/}. 2012. Mathematical modeling of tumor growth and tumor growth inhibition in oncology drug development. {\it Expert opinion on drug metabolism \& toxicology\/}. 8.9: 1057-1069. 

  \bibitem{Colin2010} Colin, Thierry {\it et al\/}. 2010. Prediction of the Evolution of Thyroidal Lung Nodules Using a Mathematical Model. {\it ERCIM News\/}.  2010.82: 37-38.

  \bibitem{Gerlee2013} Gerlee, Philip. 2013. The model muddle: in search of tumor growth laws. {\it Cancer research\/}. 73.8: 2407-2411. 
  
  \bibitem{Laird1965}  Laird, Anna Kane. 1965. Dynamics of tumour growth: comparison of growth rates and extrapolation of growth curve to one cell. {\it British Journal of Cancer\/}. 19.2: 278. 
  
  \bibitem{Tidy2018} Tidy, Colin. 2018. Stages of Cancer. {\it Patient\/}.  \href{https://patient.info/health/cancer/stages-of-cancer\#}{https://patient.info/health/cancer/stages-of-cancer\#}.
  
  \bibitem{Wong2017} Wong, Chung {\it et al\/}. 2012. Human neuroendocrine tumor cell lines as a three-dimensional model for the study of human neuroendocrine tumor therapy. {\it Journal of visualized experiments: JoVE\/}. 66: 1-7. Web. Accessed 18 August 2017. \href{https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3486771/}{https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3486771/}.

\end{thebibliography}

\teacher{
\section*{NOTES FOR TEACHERS}

This project was introduced to first semester freshmen that were enrolled in an introductory differential equations course for only four weeks. Note that the robustness of the activities and specifications on the tasks are not intended to be challenging. This was done to allow students to focus on the writing component of the project requirements. This application was initially developed as an in-class exercise, and could easily be retooled to serve this purpose again. Students spent between six and eight hours on the project. Students employed their basic familiarity with Mathematica  to solve the Gompertz model, however, any mathematical computation software can be used. 

We include a Mathematic notebook solution file and a pdf version for those who do not use Mathematica. We further, offer the introductory PowerPoint presentation, 1-081-S-TumorGrowth - Class Project Intro, as well as a pdf version,  we used with our class.


}

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