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\begin{document}

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\title{\ifnum\TeacherEdition=1 TEACHER VERSION  \else STUDENT VERSION \fi\\Introduction to Systems of ODEs: Tumor Growth}





\markboth{Tumor Growth}{Tumor Growth}

\author{Allison Lewis\\
Lafayette College\\
Easton  PA 18042 USA\\
lewisall@lafayette.edu
}

\makeStitlePDFLaTex

\begin{abstract}
This modeling scenario guides a student familiar with single ordinary differential equation (ODE) models towards the development of a more complex system of two ODEs for describing the evolution of tumor growth over time.  Students should have prior experience with solving ODEs using the separable method, and may have prior exposure to the logistic growth model (though this is not required).  By the end of the activity. students will have been exposed to using compartment diagrams to build systems of ODEs, and will be able to discuss how including an additional layer of biological complexity may better enable the modeler to fit the model to experimental data.  This activity is designed to take place over 1-2 class periods and requires no supplies beyond a simple worksheet and accessibility to software for visually fitting a model to data (i.e. Desmos or similar).  
\end{abstract}


\section*{SCENARIO DESCRIPTION}

How do cancerous tumors evolve in size over time?  In general, tumor cells are characterized by their fast division (and hence quick proliferation). In this way, tumor volume may exhibit exponential growth early in its evolution.  However, tumors cannot grow forever without bound.  There are many factors that may limit a tumor's ability to grow beyond a certain point, including limited resources in terms of nutrients and space.  When a tumor reaches this point where it can grow no further without additional resources, we say that it has reached its \textit{carrying capacity}.  

In a clinical setting, it is of utmost importance that clinicians understand how a cancerous tumor may change over time, so that they can design an appropriate treatment regimen.  Mathematical modeling can play a role here; we can use our knowledge of how tumors evolve biologically to express their growth dynamics mathematically and make predictions about what a tumor may look like at some point in the future. Models for tumor growth in the literature range in complexity from a single differential equation to systems of tens of equations with hundreds of parameters \cite{Yin2019}.  In this activity, we'll investigate several models for describing the evolution of a solid tumor over time, using what we've learned about ordinary differential equations. \\


\noindent \textbf{Part 1: A Single ODE Model} \\

\noindent Our first goal is to build a simple model that will describe how the volume of a tumor changes over time in the absence of any form of treatment.  We'll begin with a simplifying assumption: assume that the tumor is a homogeneous mass comprised only of viable (i.e. proliferating) cells.  

\begin{enumerate}
\item[(a)] Using your intuition and our description of basic tumor mechanics above, sketch a graph depicting the general shape of the curve of tumor size over time you would expect to observe, with time on the $x$-axis and volume on the $y$-axis. Identify any equilibria that you would expect to be present.  Discuss your graph with a partner, and justify your choices using biological reasoning. 
\item[(b)] The following table gives a set of simulated data describing tumor volume, $V$ (mm$^3$), as a function of time, $t$ (days), for a tumor growing in vitro (i.e. in a culture dish) in a laboratory setting.  

\begin{table}[h]
\begin{center}
\begin{tabular}{|p{0.7cm}|p{0.7cm}|p{0.7cm}|p{0.7cm}|p{0.7cm}|p{0.7cm}|p{0.7cm}|p{0.7cm}|p{0.7cm}|p{0.7cm}|p{0.7cm}|p{0.7cm}|p{0.7cm}|}
\hline
\textbf{Day} & 0 & 4 & 8 & 12 & 16 & 20 & 24 & 28 & 32 & 36 & 40 & 44\\
\hline
\textbf{Vol.} & 0.020 & 0.118 & 0.350 & 0.540 & 0.599 & 0.661 & 0.662 & 0.662 & 0.662 & 0.662 & 0.662 & 0.662\\
\hline
\end{tabular}
\caption{Tumor volume (mm$^3$) reported over a period of 44 days for a tumor growing in vitro in a laboratory setting.  Data is simulated using the cellular automaton model described in \cite{Kannan2019}.}
\end{center}
\end{table}


\noindent Plot the data and compare the shape of the resulting curve to your sketch from part (a).  Develop a differential equation to describe the change in tumor volume over time, $dV/dt$.  Make sure to check that your chosen ODE contains the equilibria that you identified in part (a).  

\item[(c)]  Solve your ODE using an appropriate method from class.  Use your data and/or a graphing tool like Desmos to estimate the values of any unknowns so that your model provides a reasonable estimate of the data.

\item[(d)]  Using your model solution and estimated parameter values from (c), evaluate your model at each day for which data is available, and complete Table \ref{tab:sse} to evaluate the overall error in your model. (Note: the SSE, or \textit{error sum of squares}, is a natural method for quantifying model error.  We measure the deviations of the model from the data at each point, square them all so that they all contribute positively to the final error calculation, and sum them all together to obtain a final quantity that summarizes the accuracy of the fit.) 

\begin{table}[h]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
\textbf{Day} & \textbf{Actual Volume} & \textbf{Estimated Volume} & \textbf{(Actual - Estimated)$^2$}\\
\hline
0 & 0.020 & & \\
\hline
4 & 0.118 & & \\
\hline
8 & 0.350 & & \\
\hline
12 & 0.540 & & \\
\hline
16 & 0.599 & & \\
\hline
20 & 0.661 & & \\
\hline
24 & 0.662 & & \\
\hline
28 & 0.662 & & \\
\hline
32 & 0.662 & & \\
\hline
36 & 0.662 & & \\
\hline
40 & 0.662 & & \\
\hline
44 & 0.662 & & \\
\hline
\multicolumn{3}{|r|}{\textbf{Sum of Squared Errors (SSE):}} & \\
\hline
\end{tabular}
\end{center}
\caption{Sum of squares computation for assessing model fit to provided data.}
\label{tab:sse}
\end{table} 

\end{enumerate}


\noindent \textbf{Part 2: A More Complex Model} \\

\noindent In Part 1, we made a very large assumption; namely, that all cells in the tumor are viable and proliferating.  In actuality, there is a great deal of heterogeneity in a tumor mass.  For one, as a tumor grows, cells in the center of the mass are unable to receive nutrients as effectively as those on the outside due to a limited vasculature system (the system of blood vessels that connects a tumor to its outside environment).  These cells may undergo natural cell death as a result, in essence forming a necrotic (dead) core at the center of the tumor.  This core still contributes to the overall volume of the tumor, but these cells are no longer proliferating.  Our simple model from Part 1 may no longer be adequate to represent the biological complexity of our situation!  Our goal now becomes to develop a more complex model that can track both the viable and necrotic cell populations.

\begin{enumerate}
\item[(a)]  In the following table, data is provided from a simulation where a significant portion of the tumor is necrotic tissue.  Using this data set, try fitting your model from Part 1 to this new data set by adjusting the values of any unknowns accordingly.  

\begin{table}[h]
\begin{center}
\begin{tabular}{|p{0.7cm}|p{0.7cm}|p{0.7cm}|p{0.7cm}|p{0.7cm}|p{0.7cm}|p{0.7cm}|p{0.7cm}|p{0.7cm}|p{0.7cm}|p{0.7cm}|p{0.7cm}|p{0.7cm}|}
\hline
\textbf{Day} & 0 & 4 & 8 & 12 & 16 & 20 & 24 & 28 & 32 & 36 & 40 & 44\\
\hline
\textbf{Vol.} & 0.020 & 0.129 & 0.402 & 0.729 & 1.134 & 1.512 & 1.833 & 2.149 & 2.396 & 2.526 & 2.661 & 2.689\\
\hline
\end{tabular}
\caption{Tumor volume (mm$^3$) reported over a period of 44 days for a tumor growing in vitro in a laboratory setting with a large necrotic core.  Data is simulated using the cellular automaton model described in \cite{Kannan2019}.}
\end{center}
\end{table}


\item[(b)] Assess the fit of your model to this new data set both visually and computationally by analyzing the graphical fit and computing the SSE as done in Part 1.  How does this fit to this second data set compare to your fit in Part 1?  Write a few sentences explaining why we are unable to achieve such a close fit to this second data set.

\item[(c)]  In order to account for both viable and necrotic cells, we need to create a \textit{system of ODEs} capable of tracking both cell types.  We can do this using \textit{compartment modeling}.  Below, we diagram our system by including a compartment for each of our cell populations, $V$ for viable and $N$ for necrotic.  Complete the diagram by drawing arrows between compartments that describe how cells move through the system. (Hint: When we work with compartment diagrams, we think of ``input-output".  What is entering each compartment?  What is exiting?  When something exits one compartment, where does it go?  How does each compartment depend on the other?  Consider the mechanisms by which cells may both enter and exit each compartment, and label each of your arrows with the corresponding terms that you'll use in your model.)  

\hspace{100pt}\includegraphics[width=2.5in]{CompartmentModel.png}

\item[(d)]  Use your compartment diagram from (c) to develop a system of two ODEs: one for $dV/dt$ (representing viable cells)  and one for $dN/dt$ (representing necrotic cells).  Discuss your system with a partner and assign biological meaning to each of your parameter values.  Prepare a short explanation of your model to share out to the full class.  

\end{enumerate}


\begin{thebibliography}{98}



\bibitem{Kannan2019} Kannan, P.,  M.~Paczkowski, A.~Miar, {\it et al\/}. 2019. Radiation resistant cancer cells enhance the survival and resistance of sensitive cells in prostate spheroids. {\it bioRxiv}. doi: 10.1101/564724. 


\bibitem{Yin2019} Yin, A., D.J.A.R.~Moes, J.G.C.~van Hasselt, J.J.~Swen, H.-J.~Guchelaar.  2019. A Review of Mathematical Models for Tumor Dynamics and Treatment Resistance Evolution of Solid Tumors. {\it CPT: Pharmacometrics Syst. Pharmacol\/}. 8(10): 720-737.

\end{thebibliography}

\teacher{
\section*{NOTES FOR TEACHERS}

This activity is inspired by the investigation in \cite{Cho2021}, which examines the tradeoff between model complexity and data availability when trying to parametrize models for describing tumor growth.  The activity provides a valuable opportunity for a class discussion of such a tradeoff; while more complex models may provide additional insight into the biological underpinnings of the scenario, the additional complexity may lead to difficulty in obtaining analytic solutions or finding unique parameter sets that yield a best fit to the data \cite{Cho2021, Yin2019}.  It is worth exposing students to this contradiction, and allowing them to explore scenarios under which they must decide what level of model complexity is appropriate for a given situation.  The following comments provide suggestions for implementation, descriptions of solutions that students may develop, and an explanation of all supporting materials.  


\subsection*{Prerequisites and Implementation}

This activity is designed to be used as an in-class exploration over 1-2 class periods where students undertake with a partner or in groups of three (though certainly it could be adapted so that a portion of the activity is completed by students outside of class).  Students are expected to have working knowledge of solving separable differential equations, and may have already seen the logistic equation used in another context.  This activity is intended to serve as an motivation to introduce systems of ODEs; thus, we assume no prior knowledge of the development of or analysis for systems of ODEs.  

\subsection*{Solution Strategies and Hints}

The following walk-through provides example solutions and suggestions for guiding students through the activity.

\subsubsection*{Part 1: A Single ODE Model}

As a first task in (a), students are encouraged to think about how a tumor might evolve over time subject to limiting constraints.  Students should sketch a sample graph that mimics the logistic growth curve, and connect it back to the underlying biology (tumor cells grow exponentially at the start of the time period, but eventually division is limited by space and nutrient constraints, forcing the tumor volume to level off at its carrying capacity).  In (b), students can verify (or correct) their sketch by plotting the given data set and observing how it follows an S-shape.  

At this point, students are asked to develop a differential equation to describe the observed dynamics.  Those students who have had exposure to the logistic growth function will make the jump quickly, but others can be guided to the correct equation by asking them to first identify the equilibrium states (corresponding to zero volume and the volume at carrying capacity), and ``back-hack" an equation that contains those two equilibria.  At this point, students should have an equation similar to the following:  
\begin{eqnarray}
\frac{dV}{dt} &=& \lambda V\left(1-\frac{V}{K}\right), \nonumber 
\end{eqnarray}
where $\lambda$ describes the growth rate (units of 1/time) and $K$ represents the carrying capacity (units of volume).  Some possible variations include:
\begin{enumerate}
\item[$\bullet$] Students may choose to parametrize their equation as \begin{eqnarray}\frac{dV}{dt} = \lambda V \left(V-K\right),\nonumber \end{eqnarray} such that while $K$ still has units of volume, $\lambda$ is now expressed in terms of 1/(volume $\times$ time).
\item[$\bullet$] Students with a strong background in biology may recognize the need for a death term for the tumor cells.  They may append an extra term to their equation such as \begin{eqnarray} \frac{dV}{dt} = \lambda V \left(1-\frac{V}{K}\right) - \eta V,\nonumber \end{eqnarray} where the $-\eta V$ term suggests that tumor cells die at a rate proportional to their volume.  Students who choose to go this route may now go down one of two paths: (1) they can solve their equation as is (this can still be solved using the separable method), or (2) they may recognize that this system can be reduced back to a two-parameter system and re-parametrize as $\frac{dV}{dt} = AV-BV^2$, where $A = \lambda - \eta$ (net growth rate), and $B = \frac{\lambda}{K}$.  Alternatively, it is not imperative that a death term be included for this activity to work, so the teacher may instead guide students to ignore such a mechanism for the time being.
\end{enumerate}

Regardless of their choice of parametrization, students should be encouraged to justify their model biologically, ensuring that it contains all of the equilibria that were present in their original sketch.  This would be a good point to check-in with each group and ensure that they have a working model that can be solved via an analytic technique they have learned in class.

If students have chosen any of the above three variations, they will now be able to solve their equation in (c) using the separable method (or another appropriate method chosen by the instructor).  If using the first parametrization given above, the solution should look like \begin{eqnarray} V(t) = \frac{CKe^{\lambda t}}{1+Ce^{\lambda t}},\nonumber \end{eqnarray} where $C$ is a constant arising from the integration process. Students should utilize the $V(0) = 0.020$ initial condition to solve for their constant of integration.  The carrying capacity parameter, $K$, can either be inferred from the data, or estimated along with $\lambda$ using a tool like Desmos \cite{Desmos} to estimate both values. Figure \ref{fig:desmos_ocm} shows a possible fit that students might obtain; here we estimate $K = 0.67$ based on the given data, use $K$ along with the initial condition to solve for $C = 0.0308$ as a constant of integration, and then utilize Desmos to estimate $\lambda \approx 0.4$ by fitting the graph to the data visually.  (Note: instructors looking to learn how to use the graphing functionality in Desmos can see \cite{Desmos2} for additional resources.)

\begin{figure}[!bth]
\begin{centering}
\includegraphics[width=5in]{desmos_ocm.png}
\caption{Fitting the single ODE model to Part 1 data in Desmos using an estimated carrying capacity of $K=0.67$ and an integration constant of $C = 0.0308.$}
\label{fig:desmos_ocm}
\end{centering}
\end{figure}

Having obtained their parameter estimates, students can now complete part (d), by plugging the time points into their model solutions and computing the error sum of squares.  They should have a very nice fit to the data at this point, with a relatively small SSE (the SSE value obtained using the parameter estimates given above is 0.0065).  

Note: As an alternative to visual fitting in Desmos, more advanced students may wish to minimize the SSE function---written as a function of the unknown parameters---using either a computer algebra system or the Solver function in EXCEL, in order to obtain an exact optimal solution.


\subsubsection*{Part 2: A More Complex Model}

Students are now presented with a second set of data, generated from a tumor spheroid with a much larger necrotic core.  Using the same model solution as in Part 1, i.e. $V(t) = \frac{CKe^{\lambda t}}{1+Ce^{\lambda t}}$, they should attempt to fit the data as best they can using Desmos or a similar tool to adjust their parameter estimates from Part 1.  For instance, by estimating $K = 2.7$ based on the new data set and using that value together with the initial condition to obtain $C = 0.0075$, one can estimate $\lambda \approx 0.26$ as in Figure \ref{fig:desmos_tcm}.  At this point, students should recognize that they are unable to achieve as close a fit visually to the data as they were able to do in Part 1.  They can further confirm this by computing the SSE again in part (b); with the values used in Figure \ref{fig:desmos_tcm}, our SSE value is 0.5195, several orders of magnitude larger than our value from Part 1.  

\begin{figure}[!bth]
\begin{centering}
\includegraphics[width=5in]{desmos_tcm.png}
\caption{Fitting the single ODE model to Part 2 data using an estimated carrying capacity of $K=2.70$ and an integration constant of $C = 0.0075.$}
\label{fig:desmos_tcm}
\end{centering}
\end{figure}

Students are now asked in part (c) to think about how to develop a more complex model that is capable of tracking both viable and necrotic cells.  They are given two compartments, $V$ and $N$, and asked to draw arrows between the two representing the different mechanisms for entering and exiting each compartment.  One such compartment drawing might look like Figure \ref{fig:TCM_Demo}, where students show that (1) viable (proliferating) cells produce yet more viable cells with the $\lambda$ arrow, (2) viable cells die naturally and become necrotic cells with the $\eta$ arrow, and (3) necrotic cells decay naturally over time and exit the system with the $\zeta$ arrow. (Note: students do not have to incorporate the decay mechanism for the necrotic cells for this activity to work, but those same students that recognized the need for a death mechanism for viable cells above may also realize that necrotic cells won't stick around forever!)  Students who are stuck on this step can be encouraged to think of this as an input-output model.  What should enter the compartment?  What should exit?  When it exits, where does it go?  How does each compartment depend on what is currently in the other?  Such guiding questions can be used to help students get a sketch down on paper.


    \begin{figure}[!ht]\centering
    \tikzstyle{startstop} = [rectangle, rounded corners, minimum width=1.5cm, minimum height=1.5cm,text centered, draw=black, fill=cyan!30]
    \tikzstyle{arrow}=[thick, ->, >=stealth]
    \tikzstyle{stem}=[thick]
    \begin{tikzpicture}[node distance= 2 cm]
    %Nodes
    \node (V) [startstop] {\Large V};
    \node (N) [startstop, right of= V, xshift = 1cm] {\Large N};
    \node [draw=none, above of= V] (lambda) {\Large $\lambda$};
    \node [draw=none, right of= V, xshift = -.5cm, yshift = .5cm] (eta) {\Large $\eta$};
    \node [draw=none, right of= N, xshift = -.6cm, yshift = .5cm] (zeta) {\Large $\zeta$};
    %Arrows
    \draw[ultra thick, ->] (V) to[loop above] node[auto] {} (V);
    \draw[ultra thick, ->] (V.east) -- (N.west);
    \draw[ultra thick, ->] (N) to[right] node[auto] {} ++ (2,0);
    \end{tikzpicture}
    \caption{Compartment diagram for the double ODE model.}
    \label{fig:TCM_Demo} 
    \end{figure}


In part (d), students are asked to convert their compartment diagram from part (c) into a system of ODEs.  One possibility is 
\begin{eqnarray}
\frac{dV}{dt} &=& \lambda V\left(1-\frac{V}{K}\right) -\eta V, \nonumber \\
\frac{dN}{dt} &=& \eta V, \nonumber
\end{eqnarray} for those students who did not incorporate a decay mechanism for the necrotic cells.  If such a mechanism \textit{was} incorporated, the system will look like 
\begin{eqnarray}
\frac{dV}{dt} &=& \lambda V\left(1-\frac{V}{K}\right) -\eta V, \nonumber \\
\frac{dN}{dt} &=& \eta V - \zeta N. \nonumber
\end{eqnarray}

Students should be prepared to justify each of the terms in their model/diagram, and share their model with the full class.  

\subsection*{Optional Class Discussion Following Part 2}

Since students do not yet know how to solve or analyze systems of differential equations, asking them to provide a new fit to the data at this point is likely beyond their capabilities.  Instructors may choose to leave it here for the time being, having given students a reason to believe that systems of ODEs may be useful in some scenarios.  They can then return to this example after covering graphical analysis of systems, and further explore this example using those tools. 

If, instead, an instructor would prefer to show students at this point that their new model is capable of fitting the Part 2 data set better than the original single ODE model, they can use the provided Matlab demonstration (requiring access to a Matlab license) to compare the two models.  It estimates the parameters for both models using an optimization method (\texttt{fmincon} with a provided SSE function), and plots both models against the given data set, while printing the SSE value for each.  Since measuring viable vs. necrotic composition is inherently invasive and expensive, in the case of the two-compartment model we rely on fitting $V(t)+N(t)$ to total tumor volume only, to align with realistic clinical data expectations. The model comparison plots for the two data sets are shown below in Figure \ref{fig:comparison}; using this demo can help to convince students of the necessity of using a more complex model for the second data set, whereas the first data set is homogeneous enough in composition to allow for use of the single ODE model.

\begin{figure}[!bth]
\begin{centering}
\centerline{
\includegraphics[width=3in]{comparison_Data1.jpg} 
\includegraphics[width=3in]{comparison_Data2.jpg} }
\caption{Comparing model fits with optimized parameter values for both Part 1 (left) and Part 2 (right) data sets.  The graph on the right shows that the additional complexity of the two-compartment model is required to achieve an accurate fit to the data.  On the left, we see that either model is able to produce a reasonable fit to the data.}
\label{fig:comparison}
\end{centering}
\end{figure}

If the instructor chooses to implement this activity as a class-wide exploration (as opposed to having students work in small groups), they may also use this Matlab demo to allow students to explore how changing the parameters affects the fit of the model to the data.  The ``params" variable obtained from the \texttt{fmincon} function can be overridden to contain whatever parameter values the students would like to try, and the resulting graph will reflect their choices.  The instructor may allow them to play around with trying to fit the model manually, and then finish with the optimization algorithm to let students see how close they got to the optimal parameters.

\subsection*{Assessment}

I use this activity as an exploration device for motivating the use of systems of ODEs.  As such, all of the work is performed during class time in small groups, and followed up with a class-wide discussion using the Matlab demonstration provided.  I generally do not use this as a graded exercise, though it would certainly be reasonable to ask students to submit a writeup of their answers to each of the prompts.  


\begin{thebibliography}{98}

\bibitem{Kannan2019} Kannan, P.,  M.~Paczkowski, A.~Miar, {\it et al\/}. 2019. Radiation resistant cancer cells enhance the survival and resistance of sensitive cells in prostate spheroids. {\it bioRxiv}. doi: 10.1101/564724. 

\bibitem{Yin2019} Yin, A., D.J.A.R.~Moes, J.G.C.~van Hasselt, J.J.~Swen, H.-J.~Guchelaar.  2019. A Review of Mathematical Models for Tumor Dynamics and Treatment Resistance Evolution of Solid Tumors. {\it CPT: Pharmacometrics Syst. Pharmacol\/}. 8(10): 720-737.


\bibitem{Cho2021} Cho, H., A.L.~Lewis, K.M.~Storey, R.~Jennings, B.~Shtylla, A.M.~Reynolds, and H.M.~Byrne. 2021.  A framework for performing data-driven modeling of tumor growth with radiotherapy treatment. Springer Special Issue: Using Mathematics to Understand Biological Complexity. {\it Association for Women in Mathematics\/}. Series 22.

\bibitem{Desmos}
Desmos Graphing Calculator. 2015. {\it Desmos Graphing Calculator\/}.  Available at: \href{https://www.desmos.com/calculator}{https://www.desmos.com/calculator}. Retrieved on 3 August  2021.

\bibitem{Desmos2}
Desmos: Learn Graphing. {\it Desmos Graphing Calculator\/}. Available at: 
\href{https://learn.desmos.com/graphing}{https://learn.desmos.com/graphing}.  Retrieved on 3 August 2021.


\end{thebibliography}




}
\end{document}