## 2017 Macklin Lab speaking schedule

Members of Paul Macklin’s lab are speaking at the following events:

**Feb. 28, 2017:**Paul Macklin, at the NCI PSON-CSBC Mathematical Oncology Meeting- Open source tools and resources for reproducible 3-D multicellular cancer systems biology

**Mar. 3, 2017:**Edwin F. Jarez, at the USC Department of Electrical Engineering- PhD Dissertation defense

**Mar. 17, 2017, 4:00 pm CST:**Paul Macklin, at the University of Nebraska-Lincoln Dept. of Mathematics Colloquium- New open source tools for computational modeling of cancer and multicellular systems
- Abstract and more information: Abstract

**Mar. 27, 2017:**Paul Macklin, at the MBI Emphasis Workshop: Hybrid Multi-Scale Modelling and Validation- From Single Models to Community Advances: open source codes and data standards
- Abstract and more information: Workshop Schedule

**Apr. 26-8, 2017:**Paul Macklin, Keynote Speaker at the Frontiers in Mathematical Oncology: Young Investigators Conference- Advances towards open source 3-D multicellular cancer systems biology

**Jun. 12, 2017:**Paul Macklin, at the Gordon Research Conference in Mammary Gland Biology- 3-D Simulations of Multicellular Systems Biology in Ductal Carcinoma
*In Situ*

- 3-D Simulations of Multicellular Systems Biology in Ductal Carcinoma
**Jul. 17-20, 2017:**Paul Macklin, at the Annual Meeting of the Society for Mathematical Biology- Title TBD
- Minisymposium: Liver as a model system for mechanics, flow, and multiscale mathematical biology

## Coarse-graining discrete cell cycle models

### Introduction

One observation that often goes underappreciated in computational biology discussions is that a computational model is often * a model of a model of a model of biology*: that is, it’s a numerical approximation (a model) of a mathematical model of an experimental model of a real-life biological system. Thus, there are three big places where a computational investigation can fall flat:

- The experimental model may be a bad choice for the disease or process (not our fault).
- Second, the mathematical model of the experimental system may have flawed assumptions (something we have to evaluate).
- The numerical implementation may have bugs or otherwise be mathematically inconsistent with the mathematical model.

Critically, you can’t use simulations to evaluate the experimental model or the mathematical model until you verify that the numerical implementation is consistent with the mathematical model, and that the numerical solution converges as \( \Delta t\) and \( \Delta x \) shrink to zero.

There are numerous ways to accomplish this, but ideally, it boils down to having some analytical solutions to the mathematical model, and comparing numerical solutions to these analytical or theoretical results. In this post, we’re going to walk through the math of analyzing a typical type of discrete cell cycle model.

### Discrete model

Suppose we have a cell cycle model consisting of phases \(P_1, P_2, \ldots P_n \), where cells in the \(P_i\) phase progress to the \(P_{i+1}\) phase after a mean waiting time of \(T_i\), and cells leaving the \(P_n\) phase divide into two cells in the \(P_1\) phase. Assign each cell agent \(k\) a current phenotypic phase \( S_k(t) \). Suppose also that each phase \( i \) has a death rate \( d_i \), and that cells persist for on average \( T_\mathrm{A} \) time in the dead state before they are removed from the simulation.

The mean waiting times \( T_i \) are equivalent to transition rates \( r_i = 1 / T_i \) (Macklin et al. 2012). Moreover, for any time interval \( [t,t+\Delta t] \), both are equivalent to a transition probability of

\[ \mathrm{Prob}\Bigl( S_k(t+\Delta t) = P_{i+1} | S(t) = P_i \Bigr) = 1 – e^{ -r_i \Delta t } \approx r_i \Delta t = \frac{ \Delta t}{ T_i}. \] In many discrete models (especially cellular automaton models) with fixed step sizes \( \Delta t \), models are stated in terms of transition probabilities \( p_{i,i+1} \), which we see are equivalent to the work above with \( p_{i,i+1} = r_i \Delta t = \Delta t / T_i \), allowing us to tie mathematical model forms to biological, measurable parameters. We note that each \(T_i\) is the average duration of the \( P_i \) phase.

#### Concrete example: a Ki67 Model

Ki-67 is a nuclear protein that is expressed through much of the cell cycle, including S, G_{2}, M, and part of G_{1} after division. It is used very commonly in pathology to assess proliferation, particularly in cancer. See the references and discussion in (Macklin et al. 2012). In Macklin et al. (2012), we came up with a discrete cell cycle model to match Ki-67 data (along with cleaved Caspase-3 stains for apoptotic cells). Let’s summarize the key parts here.

Each cell agent \(i\) has a phase \(S_i(t)\). Ki67- cells are quiescent (phase \(Q\), mean duration \( T_\mathrm{Q} \)), and they can enter the Ki67+ \(K_1\) phase (mean duration \(T_1\)). When \( K_1 \) cells leave their phase, they divide into two Ki67+ daughter cells in the \( K_2 \) phase with mean duration \( T_2 \). When cells exit \( K_2 \), they return to \( Q \). Cells in any phase can become apoptotic (enter the \( A \) phase with mean duration \( T_\mathrm{A} \)), with death rate \( r_\mathrm{A} \).

### Coarse-graining to an ODE model

If each phase \(i\) has a death rate \(d_i\), if \( N_i(t) \) denotes the number of cells in the \( P_i \) phase at time \( t\), and if \( A(t) \) is the number of dead (apoptotic) cells at time \( t\), then on average, the number of cells in the \( P_i \) phase at the next time step is given by

\[ N_i(t+\Delta t) = N_i(t) + N_{i-1}(t) \cdot \left[ \textrm{prob. of } P_{i-1} \rightarrow P_i \textrm{ transition} \right] – N_i(t) \cdot \left[ \textrm{prob. of } P_{i} \rightarrow P_{i+1} \textrm{ transition} \right] \] \[ – N_i(t) \cdot \left[ \textrm{probability of death} \right] \] By the work above, this is:

\[ N_i(t+\Delta t) \approx N_i(t) + N_{i-1}(t) r_{i-1} \Delta t – N_i(t) r_i \Delta t – N_i(t) d_i \Delta t , \] or after shuffling terms and taking the limit as \( \Delta t \downarrow 0\), \[ \frac{d}{dt} N_i(t) = r_{i-1} N_{i-1}(t) – \left( r_i + d_i \right) N_i(t). \] Continuing this analysis, we obtain a linear system:

\[ \frac{d}{dt}{ \vec{N} } = \begin{bmatrix} -(r_1+d_1) & 0 & \cdots & 0 & 2r_n & 0 \\ r_1 & -(r_2+d_2) & 0 & \cdots & 0 & 0 \\ 0 & r_2 & -(r_3+d_3) & 0 & \cdots & 0 \\ & & \ddots & & \\0&\cdots&0 &r_{n-1} & -(r_n+d_n) & 0 \\ d_1 & d_2 & \cdots & d_{n-1} & d_n & -\frac{1}{T_\mathrm{A}} \end{bmatrix}\vec{N} = M \vec{N}, \] where \( \vec{N}(t) = [ N_1(t), N_2(t) , \ldots , N_n(t) , A(t) ] \).

For the Ki67 model above, let \(\vec{N} = [K_1, K_2, Q, A]\). Then the linear system is

\[ \frac{d}{dt} \vec{N} = \begin{bmatrix} -\left( \frac{1}{T_1} + r_\mathrm{A} \right) & 0 & \frac{1}{T_\mathrm{Q}} & 0 \\ \frac{2}{T_1} & -\left( \frac{1}{T_2} + r_\mathrm{A} \right) & 0 & 0 \\ 0 & \frac{1}{T_2} & -\left( \frac{1}{T_\mathrm{Q}} + r_\mathrm{A} \right) & 0 \\ r_\mathrm{A} & r_\mathrm{A} & r_\mathrm{A} & -\frac{1}{T_\mathrm{A}} \end{bmatrix} \vec{N} .\]

(If we had written \( \vec{N} = [Q, K_1, K_2 , A] \), then the matrix above would have matched the general form.)

### Some theoretical results

If \( M\) has eigenvalues \( \lambda_1 , \ldots \lambda_{n+1} \) and corresponding eigenvectors \( \vec{v}_1, \ldots , \vec{v}_{n+1} \), then the general solution is given by

\[ \vec{N}(t) = \sum_{i=1}^{n+1} c_i e^{ \lambda_i t } \vec{v}_i ,\] and if the initial cell counts are given by \( \vec{N}(0) \) and we write \( \vec{c} = [c_1, \ldots c_{n+1} ] \), we can obtain the coefficients by solving \[ \vec{N}(0) = [ \vec{v}_1 | \cdots | \vec{v}_{n+1} ]\vec{c} .\] In many cases, it turns out that all but one of the eigenvalues (say \( \lambda \) with corresponding eigenvector \(\vec{v}\)) are negative. In this case, all the other components of the solution decay away, and for long times, we have \[ \vec{N}(t) \approx c e^{ \lambda t } \vec{v} .\] This is incredibly useful, because it says that over long times, the fraction of cells in the \( i^\textrm{th} \) phase is given by \[ v_{i} / \sum_{j=1}^{n+1} v_{j}. \]

#### Matlab implementation (with the Ki67 model)

First, let’s set some parameters, to make this a little easier and reusable.

parameters.dt = 0.1; % 6 min = 0.1 hours parameters.time_units = 'hour'; parameters.t_max = 3*24; % 3 days parameters.K1.duration = 13; parameters.K1.death_rate = 1.05e-3; parameters.K1.initial = 0; parameters.K2.duration = 2.5; parameters.K2.death_rate = 1.05e-3; parameters.K2.initial = 0; parameters.Q.duration = 74.35 ; parameters.Q.death_rate = 1.05e-3; parameters.Q.initial = 1000; parameters.A.duration = 8.6; parameters.A.initial = 0;

Next, we write a function to read in the parameter values, construct the matrix (and all the data structures), find eigenvalues and eigenvectors, and create the theoretical solution. It also finds the positive eigenvalue to determine the long-time values.

function solution = Ki67_exact( parameters ) % allocate memory for the main outputs solution.T = 0:parameters.dt:parameters.t_max; solution.K1 = zeros( 1 , length(solution.T)); solution.K2 = zeros( 1 , length(solution.T)); solution.K = zeros( 1 , length(solution.T)); solution.Q = zeros( 1 , length(solution.T)); solution.A = zeros( 1 , length(solution.T)); solution.Live = zeros( 1 , length(solution.T)); solution.Total = zeros( 1 , length(solution.T)); % allocate memory for cell fractions solution.AI = zeros(1,length(solution.T)); solution.KI1 = zeros(1,length(solution.T)); solution.KI2 = zeros(1,length(solution.T)); solution.KI = zeros(1,length(solution.T)); % get the main parameters T1 = parameters.K1.duration; r1A = parameters.K1.death_rate; T2 = parameters.K2.duration; r2A = parameters.K2.death_rate; TQ = parameters.Q.duration; rQA = parameters.Q.death_rate; TA = parameters.A.duration; % write out the mathematical model: % d[Populations]/dt = Operator*[Populations] Operator = [ -(1/T1 +r1A) , 0 , 1/TQ , 0; ... 2/T1 , -(1/T2 + r2A) ,0 , 0; ... 0 , 1/T2 , -(1/TQ + rQA) , 0; ... r1A , r2A, rQA , -1/TA ]; % eigenvectors and eigenvalues [V,D] = eig(Operator); eigenvalues = diag(D); % save the eigenvectors and eigenvalues in case you want them. solution.V = V; solution.D = D; solution.eigenvalues = eigenvalues; % initial condition VecNow = [ parameters.K1.initial ; parameters.K2.initial ; ... parameters.Q.initial ; parameters.A.initial ] ; solution.K1(1) = VecNow(1); solution.K2(1) = VecNow(2); solution.Q(1) = VecNow(3); solution.A(1) = VecNow(4); solution.K(1) = solution.K1(1) + solution.K2(1); solution.Live(1) = sum( VecNow(1:3) ); solution.Total(1) = sum( VecNow(1:4) ); solution.AI(1) = solution.A(1) / solution.Total(1); solution.KI1(1) = solution.K1(1) / solution.Total(1); solution.KI2(1) = solution.K2(1) / solution.Total(1); solution.KI(1) = solution.KI1(1) + solution.KI2(1); % now, get the coefficients to write the analytic solution % [Populations] = c1*V(:,1)*exp( d(1,1)*t) + c2*V(:,2)*exp( d(2,2)*t ) + % c3*V(:,3)*exp( d(3,3)*t) + c4*V(:,4)*exp( d(4,4)*t ); coeff = linsolve( V , VecNow ); % find the (hopefully one) positive eigenvalue. % eigensolutions with negative eigenvalues decay, % leaving this as the long-time behavior. eigenvalues = diag(D); n = find( real( eigenvalues ) &gt; 0 ) solution.long_time.KI1 = V(1,n) / sum( V(:,n) ); solution.long_time.KI2 = V(2,n) / sum( V(:,n) ); solution.long_time.QI = V(3,n) / sum( V(:,n) ); solution.long_time.AI = V(4,n) / sum( V(:,n) ) ; solution.long_time.KI = solution.long_time.KI1 + solution.long_time.KI2; % now, write out the solution at all the times for i=2:length( solution.T ) % compact way to write the solution VecExact = real( V*( coeff .* exp( eigenvalues*solution.T(i) ) ) ); solution.K1(i) = VecExact(1); solution.K2(i) = VecExact(2); solution.Q(i) = VecExact(3); solution.A(i) = VecExact(4); solution.K(i) = solution.K1(i) + solution.K2(i); solution.Live(i) = sum( VecExact(1:3) ); solution.Total(i) = sum( VecExact(1:4) ); solution.AI(i) = solution.A(i) / solution.Total(i); solution.KI1(i) = solution.K1(i) / solution.Total(i); solution.KI2(i) = solution.K2(i) / solution.Total(i); solution.KI(i) = solution.KI1(i) + solution.KI2(i); end return;

Now, let’s run it and see what this thing looks like:

Next, we plot KI_{1}, KI_{2}, and AI versus time (solid curves), along with the theoretical long-time behavior (dashed curves). Notice how well it matches–it’s neat when theory works! 🙂

Some readers may recognize the long-time fractions: KI_{1} + KI_{2} = KI = 0.1743, and AI = 0.00833, very close to the DCIS patient values from our simulation study in Macklin et al. (2012) and improved calibration work in Hyun and Macklin (2013).

### Comparing simulations and theory

I wrote a small Matlab program to implement the discrete model: start with 1000 cells in the \(Q\) phase, and in each time interval \([t,t+\Delta t]\), each cell “decides” whether to advance to the next phase, stay in the same phase, or apoptose. If we compare a single run against the theoretical curves, we see hints of a match:

If we average 10 simulations and compare, the match is better:

And lastly, if we average 100 simulations and compare, the curves are very difficult to tell apart:

Even in logarithmic space, it’s tough to tell these apart:

### Code

The following matlab files (available here) can be used to reproduce this post:

- Ki67_exact.m
- The function defined above to create the exact solution using the eigenvalue/eignvector approach.
- Ki67_stochastic.m
- Runs a single stochastic simulation, using the supplied parameters.
- script.m
- Runs the theoretical solution first, creates plots, and then runs the stochastic model 100 times for comparison.

To make it all work, simply run “script” at the command prompt. Please note that it will generate some png files in its directory.

### Closing thoughts

In this post, we showed a nice way to check a discrete model against theoretical behavior–both in short-term dynamics and long-time behavior. The same work should apply to validating many discrete models. However, when you add spatial effects (e.g., a cellular automaton model that won’t proliferate without an empty neighbor site), I wouldn’t expect a match. (But simulating cells that initially have a “salt and pepper”, random distribution should match this for early times.)

Moreover, models with deterministic phase durations (e.g., K_{1}, K_{2}, and A have fixed durations) aren’t consistent with the ODE model above, unless the cells they are each initialized with a random amount of “progress” in their initial phases. (Otherwise, the cells in each phase will run synchronized, and there will be fixed delays before cells transition to other phases.) Delay differential equations better describe such models. However, for long simulation times, the slopes of the sub-populations and the cell fractions should start to better and better match the ODE models.

Now that we have verified that the discrete model is performing as expected, we can have greater confidence in its predictions, and start using those predictions to assess the underlying models. In ODE and PDE models, you often validate the code on simpler problems where you have an analytical solution, and then move on to making simulation predictions in cases where you can’t solve analytically. Similarly, we can now move on to variants of the discrete model where we can’t as easily match ODE theory (e.g., time-varying rate parameters, spatial effects), but with the confidence that the phase transitions are working as they should.

## Some quick math to calculate numerical convergence rates

I find myself needing to re-derive this often enough that it’s worth jotting down for current and future students. 🙂

### Introduction

A very common task in our field is to assess the convergence rate of a numerical algorithm: if I shrink \(\Delta t\) (or \(\Delta x\)), how quickly does my error shrink? And in fact, does my error shrink? Assuming you have a method to compute the error for a simulation (say, a simple test case where you know the exact solution), you want a fit an expression like this:

\[ \mathrm{Error}(\Delta t) = C \Delta t^n ,\] where \( C\) is a constant, and \( n \) is the order of convergence. Usually, if \( n \) isn’t at least 1, it’s bad.

So, suppose you are testing an algorithm, and you have the error \( E_1 \) for \( \Delta t_1 \) and \( E_2 \) for \( \Delta t_2 \). Then one way to go about this calculation is to try to cancel out \( C\):

\begin{eqnarray} \frac{ E_1}{E_2} = \frac{C \Delta t_1^n }{C \Delta t_2^n } = \left( \frac{ \Delta t_1 }{\Delta t_2} \right)^n & \Longrightarrow & n = \frac{ \log\left( E_1 / E_2 \right) }{ \log\left( \Delta t_1 / \Delta t_2 \right) } \end{eqnarray}

Another way to look at this problem is to rewrite the error equation in log-log space:

\begin{eqnarray} E = C \Delta t^N & \Longrightarrow & \log{E} = \log{C} + n \log{ \Delta t} \end{eqnarray}

so \(n\) is the slope of the equation in log space. If you only have two points, then,

\[ n = \frac{ \log{E_1} – \log{E_2} }{ \log{\Delta t_1} – \log{\Delta t_1} } = \frac{ \log\left( E_1 / E_2 \right) }{ \log\left( \Delta t_1 / \Delta t_2 \right) }, \] and so we end up with the exact same convergence rate as before.

However, if you have calculated the error \( E_i \) for a whole bunch of values \( \Delta t_i \), then you can extend this idea to get a better sense of the overall convergence rate for all your values of \( \Delta t \), rather than just two values. Just find the linear least squares fit to the points \( \left\{ ( \log\Delta t_i, \log E_i ) \right\} \). If there are just two points, it’s the same as above. If there are many, it’s a better representation of overall convergence.

### Trying it out

Let’s demonstrate this on a simple test problem:

\[ \frac{du}{dt} = -10 u, \hspace{.5in} u(0) = 1.\]

We’ll simulate using (1st-order) forward Euler and (2nd-order) Adams-Bashforth:

dt_vals = [.2 .1 .05 .01 .001 .0001]; min_t = 0; max_t = 1; lambda = 10; initial = 1; errors_forward_euler = []; errors_adams_bashforth = []; for j=1:length( dt_vals ) dt = dt_vals(j); T = min_t:dt:max_t; % allocate memory solution_forward_euler = zeros(1,length(T)); solution_forward_euler(1) = initial; solution_adams_bashforth = solution_forward_euler; % exact solution solution_exact = initial * exp( -lambda * T ); % forward euler for i=2:length(T) solution_forward_euler(i) = solution_forward_euler(i-1)... - dt*lambda*solution_forward_euler(i-1); end % adams-bashforth -- use high-res Euler to jump-start dt_fine = dt * 0.1; t = min_t + dt_fine; temp = initial ; for i=1:10 temp = temp - dt_fine*lambda*temp; end solution_adams_bashforth(2) = temp; for i=3:length(T) solution_adams_bashforth(i) = solution_adams_bashforth(i-1)... - 0.5*dt*lambda*( 3*solution_adams_bashforth(i-1)... - solution_adams_bashforth(i-2 ) ); end % Uncomments if you want to see plots. % figure(1) % clf; % plot( T, solution_exact, 'r*' , T , solution_forward_euler,... % 'b-o', T , solution_adams_bashforth , 'k-s' ); % pause ; errors_forward_euler(j) = ... max(abs( solution_exact - solution_forward_euler ) ); errors_adams_bashforth(j) = ... max(abs( solution_exact - solution_adams_bashforth ) ); end

Here is a plot of the errors:

figure(2) loglog( dt_vals, errors_forward_euler, 'b-s' ,... dt_vals, errors_adams_bashforth, 'r-o' ,... 'linewidth', 2 ); legend( 'Forward Euler', 'Adams-Bashforth' , 4 ); xlabel( '\Delta t (log scale)' , 'fontsize', 14 ); ylabel( 'Absolute errors (log scale)', 'fontsize', 14 ); title( 'Errors vs. \Delta t', 'fontsize' , 16 ); set( gca, 'fontsize' , 14 );

Note that calculating the convergence rate based on the first two errors, and first and last errors, is not terribly

representative, compared with using all the errors:

% Convergence rate based on the first two errors polyfit( log(dt_vals(1:2)) , log(errors_forward_euler(1:2)) , 1 ) polyfit( log(dt_vals(1:2)) , log(errors_adams_bashforth(1:2)) , 1 ) % Convergence rate based on the first and last errors m = length(dt_vals); polyfit( log( [dt_vals(1),dt_vals(m)] ) ,... log( [errors_forward_euler(1),errors_forward_euler(m)]) , 1 ) polyfit( log( [dt_vals(1),dt_vals(m)] ) ,... log( [errors_adams_bashforth(1),errors_adams_bashforth(m)]) , 1 ) % Convergence rate based on all the errors polyfit( log(dt_vals) , log(errors_forward_euler) , 1 ) polyfit( log(dt_vals) , log(errors_adams_bashforth) , 1 ) % Convergence rate based on all the errors but the outlier polyfit( log(dt_vals(2:m)) , log(errors_forward_euler(2:m)) , 1 ) polyfit( log(dt_vals(2:m)) , log(errors_adams_bashforth(2:m)) , 1 )

Using the first two errors gives a convergence rate over 5 for Adams-Bashforth, and around 1.6 for forward Euler. Using first and last is better, but still over-estimates the convergence rates (1.15 and 2.37, FE and AB, respectively). Linear least squares is closer to reality: 1.12 for FE, 2.21 for AB. And lastly, linear least squares but excluding the outliers, we get 1.08 for forward Euler, and 2.03 for Adams-Bashforth. (As expected!)

### Which convergence rates are “right”?

So, which values do you report as your convergence rates? Ideally, use all the errors to avoid bias and/or cherry-picking. It’s the most honest and straightforward way to present the work. However, you may have a good rationale to exclude the clear outliers in this case. But then again, if you have calculated the errors for enough values of \(\Delta t\), there’s no need to do this at all. There’s little value in (or basis for) reporting the convergence rate to three significant digits. I’d instead report these as approximately first-order convergence (forward Euler) and approximately second-order convergence (Adams-Bashforth); we get this result with either linear least squares fit, and using all your data points puts you on more solid ground.

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## Building a Cellular Automaton Model Using BioFVM

**Note:** This is part of a series of “how-to” blog posts to help new users and developers of BioFVM. See below for guides to setting up a C++ compiler in Windows or OSX.

### What you’ll need

- A working C++ development environment with support for OpenMP. See these prior tutorials if you need help.
- A download of BioFVM, available at http://BioFVM.MathCancer.org and http://BioFVM.sf.net. Use Version 1.1.4 or later.
- The source code for this project (see below).

Matlab or Octave for visualization. Matlab might be available for free at your university. Octave is open source and available from a variety of sources.

### Our modeling task

We will implement a basic 3-D cellular automaton model of tumor growth in a well-mixed fluid, containing oxygen pO_{2} (mmHg) and a drug *c* (e.g., doxorubicin, μM), inspired by modeling by Alexander Anderson, Heiko Enderling, Jan Poleszczuk, Gibin Powathil, and others. (I highly suggest seeking out the sophisticated cellular automaton models at Moffitt’s Integrated Mathematical Oncology program!) This example shows you how to extend BioFVM into a new cellular automaton model. I’ll write a similar post on how to add BioFVM into an existing cellular automaton model, which you may already have available.

Tumor growth will be driven by oxygen availability. Tumor cells can be live, apoptotic (going through energy-dependent cell death, or necrotic (undergoing death from energy collapse). Drug exposure can both trigger apoptosis and inhibit cell cycling. We will model this as growth into a well-mixed fluid, with pO_{2} = 38 mmHg (about 5% oxygen: a physioxic value) and *c* = 5 μM.

### Mathematical model

As a cellular automaton model, we will divide 3-D space into a regular lattice of voxels, with length, width, and height of 15 μm. (A typical breast cancer cell has radius around 9-10 μm, giving a typical volume around 3.6×10^{3} μm^{3}. If we make each lattice site have the volume of one cell, this gives an edge length around 15 μm.)

In voxels unoccupied by cells, we approximate a well-mixed fluid with Dirichlet nodes, setting pO_{2} = 38 mmHg, and initially setting *c *= 0. Whenever a cell dies, we replace it with an empty automaton, with no Dirichlet node. Oxygen and drug follow the typical diffusion-reaction equations:

\[ \frac{ \partial \textrm{pO}_2 }{\partial t} = D_\textrm{oxy} \nabla^2 \textrm{pO}_2 – \lambda_\textrm{oxy} \textrm{pO}_2 – \sum_{ \textrm{cells} i} U_{i,\textrm{oxy}} \textrm{pO}_2 \]

\[ \frac{ \partial c}{ \partial t } = D_c \nabla^2 c – \lambda_c c – \sum_{\textrm{cells }i} U_{i,c} c \]

where each uptake rate is applied across the cell’s volume. We start the treatment by setting *c = *5 μM on all Dirichlet nodes at *t *= 504 hours (21 days). For simplicity, we do not model drug degradation (pharmacokinetics), to approximate the *in vitro *conditions.

In any time interval [*t*,*t*+Δ*t*], each live tumor cell *i* has a probability *p _{i,D}* of attempting division, probability

*p*of apoptotic death, and probability

_{i,A}*p*of necrotic death. (For simplicity, we ignore motility in this version.) We relate these to the birth rate

_{i,N}*b*, apoptotic death rate

_{i}*d*, and necrotic death rate

_{i,A}*d*by the linearized equations (from Macklin et al. 2012):

_{i,N}\[ \textrm{Prob} \Bigl( \textrm{cell } i \textrm{ becomes apoptotic in } [t,t+\Delta t] \Bigr) = 1 – \textrm{exp}\Bigl( -d_{i,A}(t) \Delta t\Bigr) \approx d_{i,A}\Delta t \]

\[ \textrm{Prob} \Bigl( \textrm{cell } i \textrm{ attempts division in } [t,t+\Delta t] \Bigr) = 1 – \textrm{exp}\Bigl( -b_i(t) \Delta t\Bigr) \approx b_{i}\Delta t \]

\[ \textrm{Prob} \Bigl( \textrm{cell } i \textrm{ becomes necrotic in } [t,t+\Delta t] \Bigr) = 1 – \textrm{exp}\Bigl( -d_{i,N}(t) \Delta t\Bigr) \approx d_{i,N}\Delta t \]

*T*, which will vary by the type of cell death. Each dead cell automaton has a probability

_{i,D}*p*of lysis (rupture and removal) in any time span [

_{i,L}*t*,

*t+*Δ

*t*]. The duration

*T*is converted to a probability of cell lysis by

_{D}\[ \textrm{Prob} \Bigl( \textrm{dead cell } i \textrm{ lyses in } [t,t+\Delta t] \Bigr) = 1 – \textrm{exp}\Bigl( -\frac{1}{T_{i,D}} \Delta t\Bigr) \approx \frac{ \Delta t}{T_{i,D}} \]

##### (Illustrative) parameter values

We use *D*_{oxy} = 10^{5} μm^{2}/min (Ghaffarizadeh et al. 2016), and we set *U*_{i,oxy} = 20 min^{-1} (to give an oxygen diffusion length scale of about 70 μm, with steeper gradients than our typical 100 μm length scale). We set *λ*_{oxy} = 0.01 min^{-1} for a 1 mm diffusion length scale in fluid.

We set *D _{c}* = 300 μm

^{2}/min, and

*U*

_{c}= 7.2×10

^{-3}min

^{-1}(

*D*from Weinberg et al. (2007), and

_{c}*U*twice as large as the reference value in Weinberg et al. (2007) to get a smaller diffusion length scale of about 204 μm). We set

_{i,c}*λ*= 3.6×10

_{c}^{-5}min

^{-1}to give a drug diffusion length scale of about 2.9 mm in fluid.

We use *T _{D}* = 8.6 hours for apoptotic cells, and

*T*= 60 days for necrotic cells (Macklin et al., 2013). However, note that necrotic and apoptotic cells lose volume quickly, so one may want to revise those time scales to match the point where a cell loses 90% of its volume.

_{D}#### Functional forms for the birth and death rates

We model pharmacodynamics with an area-under-the-curve (AUC) type formulation. If *c*(*t*) is the drug concentration at any cell *i*‘s location at time *t*, then let its integrated exposure *E _{i}*(

*t*) be

\[ E_i(t) = \int_0^t c(s) \: ds \]

and we model its response with a Hill function

\[ R_i(t) = \frac{ E_i^h(t) }{ \alpha_i^h + E_i^h(t) }, \]

where *h *is the drug’s Hill exponent for the cell line, and *α *is the exposure for a half-maximum effect.

We model the microenvironment-dependent birth rate by:

\[ b_i(t) = \left\{ \begin{array}{lr} b_{i,P} \left( 1 – \eta_i R_i(t) \right) & \textrm{ if } \textrm{pO}_{2,P} < \textrm{pO}_2 \\ \\ b_{i,P} \left( \frac{\textrm{pO}_{2}-\textrm{pO}_{2,N}}{\textrm{pO}_{2,P}-\textrm{pO}_{2,N}}\right) \Bigl( 1 – \eta_i R_i(t) \Bigr) & \textrm{ if } \textrm{pO}_{2,N} < \textrm{pO}_2 \le \textrm{pO}_{2,P} \\ \\ 0 & \textrm{ if } \textrm{pO}_2 \le \textrm{pO}_{2,N}\end{array} \right. \]

where pO_{2,P} is the physioxic oxygen value (38 mmHg), and pO_{2,N} is a necrotic threshold (we use 5 mmHg), and 0 < *η* < 1 the drug’s birth inhibition. (A fully cytostatic drug has *η* = 1.)

We model the microenvironment-dependent apoptosis rate by:

\[ d_{i,A}(t) = d_{i,A}^* + \Bigl( d_{i,A}^\textrm{max} – d_{i,A}^* \Bigr) R_i(t) \]

*d*

_{i,A}^{max}is the maximum apoptotic death rate. We model the microenvironment-dependent necrosis rate by:

\[ d_{i,N}(t) = \left\{ \begin{array}{lr} 0 & \textrm{ if } \textrm{pO}_{2,N} < \textrm{pO}_{2} \\ \\ d_{i,N}^* & \textrm{ if } \textrm{pO}_{2} \le \textrm{pO}_{2,N} \end{array}\right. \]

*d*

_{i,N}^{*}.

##### (Illustrative) parameter values

We use *b _{i,P}* = 0.05 hour

^{-1}(for a 20 hour cell cycle in physioxic conditions),

*d*

_{i,A}^{*}= 0.01

*b*, and

_{i,P}*d*

_{i,N}^{*}= 0.04 hour

^{-1}(so necrotic cells survive around 25 hours in low oxygen conditions).

We set *α* = 30 μM*hour (so that cells reach half max response after 6 hours’ exposure at a maximum concentration *c* = 5 μM), *h* = 2 (for a smooth effect), *η* = 0.25 (so that the drug is partly cytostatic), and *d _{i,A}*

^{max}= 0.1 hour^-1 (so that cells survive about 10 hours after reaching maximum response).

### Building the Cellular Automaton Model in BioFVM

BioFVM already includes Basic_Agents for cell-based substrate sources and sinks. We can extend these basic agents into full-fledged automata, and then arrange them in a lattice to create a full cellular automata model. Let’s sketch that out now.

#### Extending Basic_Agents to Automata

The main idea here is to define an Automaton class which extends (and therefore includes) the Basic_Agent class. This will give each Automaton full access to the microenvironment defined in BioFVM, including the ability to secrete and uptake substrates. We also make sure each Automaton “knows” which microenvironment it lives in (contains a pointer pMicroenvironment), and “knows” where it lives in the cellular automaton lattice. (More on that in the following paragraphs.)

So, as a schematic (just sketching out the most important members of the class):

class Standard_Data; // define per-cell biological data, such as phenotype, // cell cycle status, etc.. class Custom_Data; // user-defined custom data, specific to a model. class Automaton : public Basic_Agent { private: Microenvironment* pMicroenvironment; CA_Mesh* pCA_mesh; int voxel_index; protected: public: // neighbor connectivity information std::vector<Automaton*> neighbors; std::vector<double> neighbor_weights; Standard_Data standard_data; void (*current_state_rule)( Automaton& A , double ); Automaton(); void copy_parameters( Standard_Data& SD ); void overwrite_from_automaton( Automaton& A ); void set_cellular_automaton_mesh( CA_Mesh* pMesh ); CA_Mesh* get_cellular_automaton_mesh( void ) const; void set_voxel_index( int ); int get_voxel_index( void ) const; void set_microenvironment( Microenvironment* pME ); Microenvironment* get_microenvironment( void ); // standard state changes bool attempt_division( void ); void become_apoptotic( void ); void become_necrotic( void ); void perform_lysis( void ); // things the user needs to define Custom_Data custom_data; // use this rule to add custom logic void (*custom_rule)( Automaton& A , double); };

So, the Automaton class includes everything in the Basic_Agent class, some Standard_Data (things like the cell state and phenotype, and per-cell settings), (user-defined) Custom_Data, basic cell behaviors like attempting division into an empty neighbor lattice site, and user-defined custom logic that can be applied to any automaton. To avoid lots of switch/case and if/then logic, each Automaton has a function pointer for its current activity (current_state_rule), which can be overwritten any time.

Each Automaton also has a list of neighbor Automata (their memory addresses), and weights for each of these neighbors. Thus, you can distance-weight the neighbors (so that corner elements are farther away), and very generalized neighbor models are possible (e.g., all lattice sites within a certain distance). When updating a cellular automaton model, such as to kill a cell, divide it, or move it, you leave the neighbor information alone, and copy/edit the information (standard_data, custom_data, current_state_rule, custom_rule). In many ways, an Automaton is just a bucket with a cell’s information in it.

Note that each Automaton also “knows” where it lives (pMicroenvironment and voxel_index), and knows what CA_Mesh it is attached to (more below).

#### Connecting Automata into a Lattice

An automaton by itself is lost in the world–it needs to link up into a lattice organization. Here’s where we define a CA_Mesh class, to hold the entire collection of Automata, setup functions (to match to the microenvironment), and two fundamental operations at the mesh level: copying automata (for cell division), and swapping them (for motility). We have provided two functions to accomplish these tasks, while automatically keeping the indexing and BioFVM functions correctly in sync. Here’s what it looks like:

class CA_Mesh{ private: Microenvironment* pMicroenvironment; Cartesian_Mesh* pMesh; std::vector<Automaton> automata; std::vector<int> iteration_order; protected: public: CA_Mesh(); // setup to match a supplied microenvironment void setup( Microenvironment& M ); // setup to match the default microenvironment void setup( void ); int number_of_automata( void ) const; void randomize_iteration_order( void ); void swap_automata( int i, int j ); void overwrite_automaton( int source_i, int destination_i ); // return the automaton sitting in the ith lattice site Automaton& operator[]( int i ); // go through all nodes according to random shuffled order void update_automata( double dt ); };

So, the CA_Mesh has a vector of Automata (which are never themselves moved), pointers to the microenvironment and its mesh, and a vector of automata indices that gives the iteration order (so that we can sample the automata in a random order). You can easily access an automaton with operator[], and copy the data from one Automaton to another with overwrite_automaton() (e.g, for cell division), and swap two Automata’s data (e.g., for cell migration) with swap_automata(). Finally, calling update_automata(dt) iterates through all the automata according to iteration_order, calls their current_state_rules and custom_rules, and advances the automata by dt.

#### Interfacing Automata with the BioFVM Microenvironment

The setup function ensures that the CA_Mesh is the same size as the Microenvironment.mesh, with same indexing, and that all automata have the correct volume, and dimension of uptake/secretion rates and parameters. If you declare and set up the Microenvironment first, all this is take care of just by declaring a CA_Mesh, as it seeks out the default microenvironment and sizes itself accordingly:

// declare a microenvironment Microenvironment M; // do things to set it up -- see prior tutorials // declare a Cellular_Automaton_Mesh CA_Mesh CA_model; // it's already good to go, initialized to empty automata: CA_model.display();

If you for some reason declare the CA_Mesh fist, you can set it up against the microenvironment:

// declare a CA_Mesh CA_Mesh CA_model; // declare a microenvironment Microenvironment M; // do things to set it up -- see prior tutorials // initialize the CA_Mesh to match the microenvironment CA_model.setup( M ); // it's already good to go, initialized to empty automata: CA_model.display();

Because each Automaton is in the microenvironment and inherits functions from Basic_Agent, it can secrete or uptake. For example, we can use functions like this one:

void set_uptake( Automaton& A, std::vector<double>& uptake_rates ) { extern double BioFVM_CA_diffusion_dt; // update the uptake_rates in the standard_data A.standard_data.uptake_rates = uptake_rates; // now, transfer them to the underlying Basic_Agent *(A.uptake_rates) = A.standard_data.uptake_rates; // and make sure the internal constants are self-consistent A.set_internal_uptake_constants( BioFVM_CA_diffusion_dt ); }

A function acting on an automaton can sample the microenvironment to change parameters and state. For example:

void do_nothing( Automaton& A, double dt ) { return; } void microenvironment_based_rule( Automaton& A, double dt ) { // sample the microenvironment std::vector<double> MS = (*A.get_microenvironment())( A.get_voxel_index() ); // if pO2 < 5 mmHg, set the cell to a necrotic state if( MS[0] < 5.0 ) { A.become_necrotic(); } // if drug > 5 uM, set the birth rate to zero if( MS[1] > 5 ) { A.standard_data.birth_rate = 0.0; } // set the custom rule to something else A.custom_rule = do_nothing; return; }

#### Implementing the mathematical model in this framework

We give each tumor cell a tumor_cell_rule (using this for custom_rule):

void viable_tumor_rule( Automaton& A, double dt ) { // If there's no cell here, don't bother. if( A.standard_data.state_code == BioFVM_CA_empty ) { return; } // sample the microenvironment std::vector<double> MS = (*A.get_microenvironment())( A.get_voxel_index() ); // integrate drug exposure A.standard_data.integrated_drug_exposure += ( MS[1]*dt ); A.standard_data.drug_response_function_value = pow( A.standard_data.integrated_drug_exposure, A.standard_data.drug_hill_exponent ); double temp = pow( A.standard_data.drug_half_max_drug_exposure, A.standard_data.drug_hill_exponent ); temp += A.standard_data.drug_response_function_value; A.standard_data.drug_response_function_value /= temp; // update birth rates (which themselves update probabilities) update_birth_rate( A, MS, dt ); update_apoptotic_death_rate( A, MS, dt ); update_necrotic_death_rate( A, MS, dt ); return; }

The functional tumor birth and death rates are implemented as:

void update_birth_rate( Automaton& A, std::vector<double>& MS, double dt ) { static double O2_denominator = BioFVM_CA_physioxic_O2 - BioFVM_CA_necrotic_O2; A.standard_data.birth_rate = A.standard_data.drug_response_function_value; // response A.standard_data.birth_rate *= A.standard_data.drug_max_birth_inhibition; // inhibition*response; A.standard_data.birth_rate *= -1.0; // - inhibition*response A.standard_data.birth_rate += 1.0; // 1 - inhibition*response A.standard_data.birth_rate *= viable_tumor_cell.birth_rate; // birth_rate0*(1 - inhibition*response) double temp1 = MS[0] ; // O2 temp1 -= BioFVM_CA_necrotic_O2; temp1 /= O2_denominator; A.standard_data.birth_rate *= temp1; if( A.standard_data.birth_rate < 0 ) { A.standard_data.birth_rate = 0.0; } A.standard_data.probability_of_division = A.standard_data.birth_rate; A.standard_data.probability_of_division *= dt; // dt*birth_rate*(1 - inhibition*repsonse) // linearized probability return; } void update_apoptotic_death_rate( Automaton& A, std::vector<double>& MS, double dt ) { A.standard_data.apoptotic_death_rate = A.standard_data.drug_max_death_rate; // max_rate A.standard_data.apoptotic_death_rate -= viable_tumor_cell.apoptotic_death_rate; // max_rate - background_rate A.standard_data.apoptotic_death_rate *= A.standard_data.drug_response_function_value; // (max_rate-background_rate)*response A.standard_data.apoptotic_death_rate += viable_tumor_cell.apoptotic_death_rate; // background_rate + (max_rate-background_rate)*response A.standard_data.probability_of_apoptotic_death = A.standard_data.apoptotic_death_rate; A.standard_data.probability_of_apoptotic_death *= dt; // dt*( background_rate + (max_rate-background_rate)*response ) // linearized probability return; } void update_necrotic_death_rate( Automaton& A, std::vector<double>& MS, double dt ) { A.standard_data.necrotic_death_rate = 0.0; A.standard_data.probability_of_necrotic_death = 0.0; if( MS[0] > BioFVM_CA_necrotic_O2 ) { return; } A.standard_data.necrotic_death_rate = perinecrotic_tumor_cell.necrotic_death_rate; A.standard_data.probability_of_necrotic_death = A.standard_data.necrotic_death_rate; A.standard_data.probability_of_necrotic_death *= dt; // dt*necrotic_death_rate return; }

And each fluid voxel (Dirichlet nodes) is implemented as the following (to turn on therapy at 21 days):

void fluid_rule( Automaton& A, double dt ) { static double activation_time = 504; static double activation_dose = 5.0; static std::vector<double> activated_dirichlet( 2 , BioFVM_CA_physioxic_O2 ); static bool initialized = false; if( !initialized ) { activated_dirichlet[1] = activation_dose; initialized = true; } if( fabs( BioFVM_CA_elapsed_time - activation_time ) < 0.01 ) { int ind = A.get_voxel_index(); if( A.get_microenvironment()->mesh.voxels[ind].is_Dirichlet ) { A.get_microenvironment()->update_dirichlet_node( ind, activated_dirichlet ); } } }

At the start of the simulation, each non-cell automaton has its custom_rule set to fluid_rule, and each tumor cell Automaton has its custom_rule set to viable_tumor_rule. Here’s how:

void setup_cellular_automata_model( Microenvironment& M, CA_Mesh& CAM ) { // Fill in this environment double tumor_radius = 150; double tumor_radius_squared = tumor_radius * tumor_radius; std::vector<double> tumor_center( 3, 0.0 ); std::vector<double> dirichlet_value( 2 , 1.0 ); dirichlet_value[0] = 38; //physioxia dirichlet_value[1] = 0; // drug for( int i=0 ; i < M.number_of_voxels() ;i++ ) { std::vector<double> displacement( 3, 0.0 ); displacement = M.mesh.voxels[i].center; displacement -= tumor_center; double r2 = norm_squared( displacement ); if( r2 > tumor_radius_squared ) // well_mixed_fluid { M.add_dirichlet_node( i, dirichlet_value ); CAM[i].copy_parameters( well_mixed_fluid ); CAM[i].custom_rule = fluid_rule; CAM[i].current_state_rule = do_nothing; } else // tumor { CAM[i].copy_parameters( viable_tumor_cell ); CAM[i].custom_rule = viable_tumor_rule; CAM[i].current_state_rule = advance_live_state; } } }

#### Overall program loop

There are two inherent time scales in this problem: cell processes like division and death (happen on the scale of hours), and transport (happens on the order of minutes). We take advantage of this by defining two step sizes:

double BioFVM_CA_dt = 3; std::string BioFVM_CA_time_units = "hr"; double BioFVM_CA_save_interval = 12; double BioFVM_CA_max_time = 24*28; double BioFVM_CA_elapsed_time = 0.0; double BioFVM_CA_diffusion_dt = 0.05; std::string BioFVM_CA_transport_time_units = "min"; double BioFVM_CA_diffusion_max_time = 5.0;

Every time the simulation advances by BioFVM_CA_dt (on the order of hours), we run diffusion to quasi-steady state (for BioFVM_CA_diffusion_max_time, on the order of minutes), using time steps of size BioFVM_CA_diffusion time. We performed numerical stability and convergence analyses to determine 0.05 min works pretty well for regular lattice arrangements of cells, but you should always perform your own testing!

Here’s how it all looks, in a main program loop:

BioFVM_CA_elapsed_time = 0.0; double next_output_time = BioFVM_CA_elapsed_time; // next time you save data while( BioFVM_CA_elapsed_time < BioFVM_CA_max_time + 1e-10 ) { // if it's time, save the simulation if( fabs( BioFVM_CA_elapsed_time - next_output_time ) < BioFVM_CA_dt/2.0 ) { std::cout << "simulation time: " << BioFVM_CA_elapsed_time << " " << BioFVM_CA_time_units << " (" << BioFVM_CA_max_time << " " << BioFVM_CA_time_units << " max)" << std::endl; char* filename; filename = new char [1024]; sprintf( filename, "output_%6f" , next_output_time ); save_BioFVM_cellular_automata_to_MultiCellDS_xml_pugi( filename , M , CA_model , BioFVM_CA_elapsed_time ); cell_counts( CA_model ); delete [] filename; next_output_time += BioFVM_CA_save_interval; } // do the cellular automaton step CA_model.update_automata( BioFVM_CA_dt ); BioFVM_CA_elapsed_time += BioFVM_CA_dt; // simulate biotransport to quasi-steady state double t_diffusion = 0.0; while( t_diffusion < BioFVM_CA_diffusion_max_time + 1e-10 ) { M.simulate_diffusion_decay( BioFVM_CA_diffusion_dt ); M.simulate_cell_sources_and_sinks( BioFVM_CA_diffusion_dt ); t_diffusion += BioFVM_CA_diffusion_dt; } }

### Getting and Running the Code

**Start a project:**Create a new directory for your project (I’d recommend “BioFVM_CA_tumor”), and enter the directory. Place a copy of BioFVM (the zip file) into your directory. Unzip BioFVM, and copy BioFVM*.h, BioFVM*.cpp, and pugixml* files into that directory.**Download the demo source code:**Download the source code for this tutorial: BioFVM_CA_Example_1, version 1.0.0 or later. Unzip its contents into your project directory.**Go ahead and overwrite the Makefile**.**Edit the makefile (if needed):**Note that if you are using OSX, you’ll probably need to change from “g++” to your installed compiler. See these tutorials.**Test the code:**Go to a command line (see previous tutorials), and test:make ./BioFVM_CA_Example_1

(If you’re on windows, run BioFVM_CA_Example_1.exe.)

### Simulation Result

If you run the code to completion, you will simulate 3 weeks of *in vitro * growth, followed by a bolus “injection” of drug. The code will simulate one one additional week under the drug. (This should take 5-10 minutes, including full simulation saves every 12 hours.)

In matlab, you can load a saved dataset and check the minimum oxygenation value like this:

MCDS = read_MultiCellDS_xml( 'output_504.000000.xml' ); min(min(min( MCDS.continuum_variables(1).data )))

And then you can start visualizing like this:

contourf( MCDS.mesh.X_coordinates , MCDS.mesh.Y_coordinates , ... MCDS.continuum_variables(1).data(:,:,33)' ) ; axis image; colorbar xlabel('x (\mum)' , 'fontsize' , 12 ); ylabel( 'y (\mum)' , 'fontsize', 12 ); set(gca, 'fontsize', 12 ); title('Oxygenation (mmHg) at z = 0 \mum', 'fontsize', 14 ); print('-dpng', 'Tumor_o2_3_weeks.png' ); plot_cellular_automata( MCDS , 'Tumor spheroid at 3 weeks');

#### Simulation plots

Here are some plots, showing (left from right) pO_{2} concentration, a cross-section of the tumor (red = live cells, green = apoptotic, and blue = necrotic), and the drug concentration (after start of therapy):

##### 1 week:

Oxygen- and space-limited growth are restricted to the outer boundary of the tumor spheroid.

##### 2 weeks:

Oxygenation is dipped below 5 mmHg in the center, leading to necrosis.

##### 3 weeks:

As the tumor grows, the hypoxic gradient increases, and the necrotic core grows. The code turns on a constant 5 micromolar dose of doxorubicin at this point

##### Treatment + 12 hours:

The drug has started to penetrate the tumor, triggering apoptotic death towards the outer periphery where exposure has been greatest.

##### Treatment + 24 hours:

The drug profile hasn’t changed much, but the interior cells have now had greater exposure to drug, and hence greater response. Now apoptosis is observed throughout the non-necrotic tumor. The tumor has decreased in volume somewhat.

##### Treatment + 36 hours:

The non-necrotic tumor is now substantially apoptotic. We would require some pharamcokinetic effects (e.g., drug clearance, inactivation, or removal) to avoid the inevitable, presences of a pre-existing resistant strain, or emergence of resistance.

##### Treatment + 48 hours:

By now, almost all cells are apoptotic.

##### Treatment + 60 hours:

The non-necrotic tumor is nearly completed eliminated, leaving a leftover core of previously-necrotic cells (which did not change state in response to the drug–they were already dead!)

### Source files

You can download completed source for this example here: https://sourceforge.net/projects/biofvm/files/Tutorials/Cellular_Automaton_1/

This file will include the following:

- BioFVM_cellular_automata.h
- BioFVM_cellular_automata.cpp
- BioFVM_CA_example_1.cpp
- read_MultiCellDS_xml.m (updated)
- plot_cellular_automata.m
- Makefile

### What’s next

I plan to update this source code with extra cell motility, and potentially more realistic parameter values. Also, I plan to more formally separate out the example from the generic cell capabilities, so that this source code can work as a *bona fide *cellular automaton framework.

More immediately, my next tutorial will use the reverse strategy: start with an existing cellular automaton model, and integrate BioFVM capabilities.

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## Moving the blog to MathCancer.org

Hi, everyone!

Blogspot has been a great platform for me, but in the end, editing posts with source code and mathematics has been too much of a headache in the neglected blogspot and google UIs.

Elsewhere in the universe, WordPress has developed and encouraged a great ecosystem of plugins that let you do LaTeX and code syntax highlighting directly in your posts with ease. I can’t spend hours and hours on fixing mangled posts. It’s time to move on.

So as of today, I am moving to a self-hosted blog at http://MathCancer.org/blog/

I will leave old posts at http://MathCancer.blogspot.com and gradually migrate them here to MathCancer.org/blog. Thanks for following me over the last few years.

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## Paul Macklin featured in Biotechniques article

Paul Macklin’s work on mathematical modeling of breast cancer and BioFVM was recently featured in Kelly Rae Chi’s article in *Biotechniques *on virtual cell cultures. It also included great work by James Glazier (CompuCell3D) and Kristin Swanson (glioblastoma modeling).

**Read the article:** http://www.biotechniques.com/news/Mighty-Modelers-The-Art-of-Virtual-Cell-Culture/biotechniques-364893.html (July 20, 2016)

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## Saving MultiCellDS data from BioFVM

**Note:** This is part of a series of “how-to” blog posts to help new users and developers of BioFVM.

### Introduction

A major initiative for my lab has been MultiCellDS: a standard for multicellular data. The project aims to create model-neutral representations of simulation data (for both discrete and continuum models), which can also work for segmented experimental and clinical data. A single-time output is called a *digital snapshot*. An interdisciplinary, multi-institutional review panel has been hard at work to nail down the draft standard.

A BioFVM MultiCellDS digital snapshot includes program and user metadata (more information to be included in a forthcoming publication), an output of the microenvironment, and any cells that are secreting or uptaking substrates.

As of Version 1.1.0, BioFVM supports output saved to MultiCellDS XML files. Each download also includes a matlab function for importing MultiCellDS snapshots saved by BioFVM programs. This tutorial will get you going.

BioFVM (finite volume method for biological problems) is an open source code for solving 3-D diffusion of 1 or more substrates. It was recently published as open access in *Bioinformatics* here:

http://dx.doi.org/10.1093/bioinformatics/btv730

The project website is at http://BioFVM.MathCancer.org, and downloads are at http://BioFVM.sf.net.

### Working with MultiCellDS in BioFVM programs

We include a MultiCellDS_test.cpp file in the examples directory of every BioFVM download (Version 1.1.0 or later). Create a new project directory, copy the following files to it:

- BioFVM*.cpp and BioFVM*.h (from the main BioFVM directory)
- pugixml.* (from the main BioFVM directory)
- Makefile and MultiCellDS_test.cpp (from the examples directory)

Open the MultiCellDS_test.cpp file to see the syntax as you read the rest of this post.

See earlier tutorials (below) if you have troubles with this.

#### Setting metadata values

There are few key bits of metadata. First, the program used for the simulation (all these fields are optional):

// the program name, version, and project website: BioFVM_metadata.program.program_name = "BioFVM MultiCellDS Test"; BioFVM_metadata.program.program_version = "1.0"; BioFVM_metadata.program.program_URL = "http://BioFVM.MathCancer.org"; // who created the program (if known) BioFVM_metadata.program.creator.surname = "Macklin"; BioFVM_metadata.program.creator.given_names = "Paul"; BioFVM_metadata.program.creator.email = "Paul.Macklin@usc.edu"; BioFVM_metadata.program.creator.URL = "http://BioFVM.MathCancer.org"; BioFVM_metadata.program.creator.organization = "University of Southern California"; BioFVM_metadata.program.creator.department = "Center for Applied Molecular Medicine"; BioFVM_metadata.program.creator.ORCID = "0000-0002-9925-0151"; // (generally peer-reviewed) citation information for the program BioFVM_metadata.program.citation.DOI = "10.1093/bioinformatics/btv730"; BioFVM_metadata.program.citation.PMID = "26656933"; BioFVM_metadata.program.citation.PMCID = "PMC1234567"; BioFVM_metadata.program.citation.text = "A. Ghaffarizadeh, S.H. Friedman, and P. Macklin, BioFVM: an efficient parallelized diffusive transport solver for 3-D biological simulations, Bioinformatics, 2015. DOI: 10.1093/bioinformatics/btv730."; BioFVM_metadata.program.citation.notes = "notes here"; BioFVM_metadata.program.citation.URL = "http://dx.doi.org/10.1093/bioinformatics/btv730"; // user information: who ran the program BioFVM_metadata.program.user.surname = "Kirk"; BioFVM_metadata.program.user.given_names = "James T."; BioFVM_metadata.program.user.email = "Jimmy.Kirk@starfleet.mil"; BioFVM_metadata.program.user.organization = "Starfleet"; BioFVM_metadata.program.user.department = "U.S.S. Enterprise (NCC 1701)"; BioFVM_metadata.program.user.ORCID = "0000-0000-0000-0000"; // And finally, data citation information (the publication where this simulation snapshot appeared) BioFVM_metadata.data_citation.DOI = "10.1093/bioinformatics/btv730"; BioFVM_metadata.data_citation.PMID = "12345678"; BioFVM_metadata.data_citation.PMCID = "PMC1234567"; BioFVM_metadata.data_citation.text = "A. Ghaffarizadeh, S.H. Friedman, and P. Macklin, BioFVM: an efficient parallelized diffusive transport solver for 3-D biological simulations, Bioinformatics, 2015. DOI: 10.1093/bioinformatics/btv730."; BioFVM_metadata.data_citation.notes = "notes here"; BioFVM_metadata.data_citation.URL = "http://dx.doi.org/10.1093/bioinformatics/btv730";

You can sync the metadata current time, program runtime (wall time), and dimensional units using the following command. (This command is automatically run whenever you use the save command below.)

BioFVM_metadata.sync_to_microenvironment( M );

You can display a basic summary of the metadata via:

BioFVM_metadata.display_information( std::cout );

#### Setting options

By default (to save time and disk space), BioFVM saves the mesh as a Level 3 matlab file, whose location is embedded into the MultiCellDS XML file. You can disable this feature and revert to full XML (e.g., for human-readable cross-model reporting) via:

set_save_biofvm_mesh_as_matlab( false );

Similarly, BioFVM defaults to saving the values of the substrates in a compact Level 3 matlab file. You can override this with:

set_save_biofvm_data_as_matlab( false );

BioFVM by default saves the cell-centered sources and sinks. These take a lot of time to parse because they require very hierarchical data structures. You can disable saving the cells (basic_agents) via:

set_save_biofvm_cell_data( false );

Lastly, when you do save the cells, we default to a customized, minimal matlab format. You can revert to a more standard (but much larger) XML format with:

set_save_biofvm_cell_data_as_custom_matlab( false )

#### Saving a file

Saving the data is *very *straightforward:

save_BioFVM_to_MultiCellDS_xml_pugi( "sample" , M , current_simulation_time );

Your data will be saved in sample.xml. (Depending upon your options, it may generate several .mat files beginning with “sample”.)

If you’d like the filename to depend upon the simulation time, use something more like this:

double current_simulation_time = 10.347; char filename_base [1024]; sprintf( &filename_base , "sample_%f", current_simulation_time ); save_BioFVM_to_MultiCellDS_xml_pugi( filename_base , M, current_simulation_time );

Your data will be saved in sample_10.347000.xml. (Depending upon your options, it may generate several .mat files beginning with “sample_10.347000”.)

#### Compiling and running the program:

Edit the Makefile as below:

PROGRAM_NAME := MCDS_test all: $(BioFVM_OBJECTS) $(pugixml_OBJECTS) MultiCellDS_test.cpp $(COMPILE_COMMAND) -o $(PROGRAM_NAME) $(BioFVM_OBJECTS) $(pugixml_OBJECTS) MultiCellDS_test.cpp

If you’re running OSX, you’ll probably need to update the compiler from “g++”. See these tutorials.

Then, at the command prompt:

make ./MCDS_test

On Windows, you’ll need to run without the ./:

make MCDS_test

### Working with MultiCellDS data in Matlab

#### Reading data in Matlab

Copy the read_MultiCellDS_xml.m file from the matlab directory (included in every MultiCellDS download). To read the data, just do this:

MCDS = read_MultiCellDS_xml( 'sample.xml' );

This should take around 30 seconds for larger data files (500,000 to 1,000,000 voxels with a few substrates, and around 250,000 cells). The long execution time is primarily because Matlab is ghastly inefficient at loops over hierarchical data structures. Increasing to 1,000,000 cells requires around 80-90 seconds to parse in matlab.

#### Plotting data in Matlab

##### Plotting the 3-D substrate data

First, let’s do some basic contour and surface plotting:

mid_index = round( length(MCDS.mesh.Z_coordinates)/2 ); contourf( MCDS.mesh.X(:,:,mid_index), ... MCDS.mesh.Y(:,:,mid_index), ... MCDS.continuum_variables(2).data(:,:,mid_index) , 20 ) ; axis image colorbar xlabel( sprintf( 'x (%s)' , MCDS.metadata.spatial_units) ); ylabel( sprintf( 'y (%s)' , MCDS.metadata.spatial_units) ); title( sprintf('%s (%s) at t = %f %s, z = %f %s', MCDS.continuum_variables(2).name , ... MCDS.continuum_variables(2).units , ... MCDS.metadata.current_time , ... MCDS.metadata.time_units, ... MCDS.mesh.Z_coordinates(mid_index), ... MCDS.metadata.spatial_units ) );

OR

contourf( MCDS.mesh.X_coordinates , MCDS.mesh.Y_coordinates, ... MCDS.continuum_variables(2).data(:,:,mid_index) , 20 ) ; axis image colorbar xlabel( sprintf( 'x (%s)' , MCDS.metadata.spatial_units) ); ylabel( sprintf( 'y (%s)' , MCDS.metadata.spatial_units) ); title( sprintf('%s (%s) at t = %f %s, z = %f %s', ... MCDS.continuum_variables(2).name , ... MCDS.continuum_variables(2).units , ... MCDS.metadata.current_time , ... MCDS.metadata.time_units, ... MCDS.mesh.Z_coordinates(mid_index), ... MCDS.metadata.spatial_units ) );

Here’s a surface plot:

surf( MCDS.mesh.X_coordinates , MCDS.mesh.Y_coordinates, ... MCDS.continuum_variables(1).data(:,:,mid_index) ) ; colorbar axis tight xlabel( sprintf( 'x (%s)' , MCDS.metadata.spatial_units) ); ylabel( sprintf( 'y (%s)' , MCDS.metadata.spatial_units) ); zlabel( sprintf( '%s (%s)', MCDS.continuum_variables(1).name, ... MCDS.continuum_variables(1).units ) ); title( sprintf('%s (%s) at t = %f %s, z = %f %s', MCDS.continuum_variables(1).name , ... MCDS.continuum_variables(1).units , ... MCDS.metadata.current_time , ... MCDS.metadata.time_units, ... MCDS.mesh.Z_coordinates(mid_index), ... MCDS.metadata.spatial_units ) );

Finally, here are some more advanced plots. The first is an “exploded” stack of contour plots:

clf contourslice( MCDS.mesh.X , MCDS.mesh.Y, MCDS.mesh.Z , ... MCDS.continuum_variables(2).data , [],[], ... MCDS.mesh.Z_coordinates(1:15:length(MCDS.mesh.Z_coordinates)),20); view([-45 10]); axis tight; xlabel( sprintf( 'x (%s)' , MCDS.metadata.spatial_units) ); ylabel( sprintf( 'y (%s)' , MCDS.metadata.spatial_units) ); zlabel( sprintf( 'z (%s)' , MCDS.metadata.spatial_units) ); title( sprintf('%s (%s) at t = %f %s', ... MCDS.continuum_variables(2).name , ... MCDS.continuum_variables(2).units , ... MCDS.metadata.current_time, ... MCDS.metadata.time_units ) );

Next, we show how to use isosurfaces with transparency

clf patch( isosurface( MCDS.mesh.X , MCDS.mesh.Y, MCDS.mesh.Z, ... MCDS.continuum_variables(1).data, 1000 ), 'edgecolor', ... 'none', 'facecolor', 'r' , 'facealpha' , 1 ); hold on patch( isosurface( MCDS.mesh.X , MCDS.mesh.Y, MCDS.mesh.Z, ... MCDS.continuum_variables(1).data, 5000 ), 'edgecolor', ... 'none', 'facecolor', 'b' , 'facealpha' , 0.7 ); patch( isosurface( MCDS.mesh.X , MCDS.mesh.Y, MCDS.mesh.Z, ... MCDS.continuum_variables(1).data, 10000 ), 'edgecolor', ... 'none', 'facecolor', 'g' , 'facealpha' , 0.5 ); hold off % shading interp camlight view(3) axis image axis tightcamlight lighting gouraud xlabel( sprintf( 'x (%s)' , MCDS.metadata.spatial_units) ); ylabel( sprintf( 'y (%s)' , MCDS.metadata.spatial_units) ); zlabel( sprintf( 'z (%s)' , MCDS.metadata.spatial_units) ); title( sprintf('%s (%s) at t = %f %s', ... MCDS.continuum_variables(1).name , ... MCDS.continuum_variables(1).units , ... MCDS.metadata.current_time, ... MCDS.metadata.time_units ) );

You can get more 3-D volumetric visualization ideas at Matlab’s website. This visualization post at MIT also has some great tips.

##### Plotting the cells

Here is a basic 3-D plot for the cells:

plot3( MCDS.discrete_cells.state.position(:,1) , ... MCDS.discrete_cells.state.position(:,2) , ... MCDS.discrete_cells.state.position(:,3) , 'bo' ); view(3) axis tight xlabel( sprintf( 'x (%s)' , MCDS.metadata.spatial_units) ); ylabel( sprintf( 'y (%s)' , MCDS.metadata.spatial_units) ); zlabel( sprintf( 'z (%s)' , MCDS.metadata.spatial_units) ); title( sprintf('Cells at t = %f %s', MCDS.metadata.current_time, ... MCDS.metadata.time_units ) );

plot3 is more efficient than scatter3, but scatter3 will give more coloring options. Here is the syntax:

scatter3( MCDS.discrete_cells.state.position(:,1), ... MCDS.discrete_cells.state.position(:,2), ... MCDS.discrete_cells.state.position(:,3) , 'bo' ); view(3) axis tight xlabel( sprintf( 'x (%s)' , MCDS.metadata.spatial_units) ); ylabel( sprintf( 'y (%s)' , MCDS.metadata.spatial_units) ); zlabel( sprintf( 'z (%s)' , MCDS.metadata.spatial_units) ); title( sprintf('Cells at t = %f %s', MCDS.metadata.current_time, ... MCDS.metadata.time_units ) );

Jan Poleszczuk gives some great insights on plotting many cells in 3D at his blog. I’d recommend checking out his post on visualizing a cellular automaton model. At some point, I’ll update this post with prettier plotting based on his methods.

### What’s next

Future releases of BioFVM will support reading MultiCellDS snapshots (for model initialization).

Matlab is pretty slow at parsing and visualizing large amounts of data. We also plan to include resources for accessing MultiCellDS data in VTK / Paraview and Python.

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## BioFVM warmup: 2D continuum simulation of tumor growth

**Note:** This is part of a series of “how-to” blog posts to help new users and developers of BioFVM. See also the guides to setting up a C++ compiler in Windows or OSX.

### What you’ll need

- A working C++ development environment with support for OpenMP. See these prior tutorials if you need help.
- A download of BioFVM, available at http://BioFVM.MathCancer.org and http://BioFVM.sf.net. Use Version 1.0.3 or later.
- Matlab or Octave for visualization. Matlab might be available for free at your university. Octave is open source and available from a variety of sources.

### Our modeling task

We will implement a basic 2-D model of tumor growth in a heterogeneous microenvironment, with inspiration by glioblastoma models by Kristin Swanson, Russell Rockne and others (e.g., this work), and continuum tumor growth models by Hermann Frieboes, John Lowengrub, and our own lab (e.g., this paper and this paper).

We will model tumor growth driven by a growth substrate, where cells die when the growth substrate is insufficient. The tumor cells will have motility. A continuum blood vasculature will supply the growth substrate, but tumor cells can degrade this existing vasculature. We will revisit and extend this model from time to time in future tutorials.

### Mathematical model

Taking inspiration from the groups mentioned above, we’ll model a live cell density *ρ* of a relatively low-adhesion tumor cell species (e.g., glioblastoma multiforme). We’ll assume that tumor cells move randomly towards regions of low cell density (modeled as diffusion with motility *μ*). We’ll assume that that the net birth rate *r _{B}* is proportional to the concentration of growth substrate

*σ*, which is released by blood vasculature with density

*b*. Tumor cells can degrade the tissue and hence this existing vasculature. Tumor cells die at rate

*r*when the growth substrate level is too low. We assume that the tumor cell density cannot exceed a max level

_{D}*ρ*

_{max}. A model that includes these effects is:

\[ \frac{ \partial \rho}{\partial t} = \mu \nabla^2 \rho + r_B(\sigma)\rho \left( 1 – \frac{ \rho}{\rho_\textrm{max}} \right) – r_D(\sigma) \rho \]

\[ \frac{ \partial b}{\partial t} = – r_\textrm{degrade} \rho b \]

\[ \frac{\partial \sigma}{ \partial t} = D\nabla^2 \sigma – \lambda_a \sigma – \lambda_2 \rho \sigma + r_\textrm{deliv}b \left( \sigma_\textrm{max} – \sigma \right) \]

where for the birth and death rates, we’ll use the constitutive relations:

\[ r_B(\sigma) = r_B \textrm{ max} \left( \frac{\sigma – \sigma_\textrm{min}}{ \sigma_\textrm{ max} – \sigma_\textrm{min} } , 0 \right)\]

\[r_D(\sigma) = r_D \textrm{ max} \left( \frac{ \sigma_\textrm{min} – \sigma}{\sigma_\textrm{min}} , 0 \right) \]

### Mapping the model onto BioFVM

BioFVM solves on a vector **u** of substrates. We’ll set **u** = [*ρ* , *b*, *σ* ]. The code expects PDEs of the general form:

\[ \frac{\partial q}{\partial t} = D\nabla^2 q – \lambda q + S\left( q^* – q \right) – Uq\]

So, we determine the decay rate (*λ*), source function (*S*), and uptake function (*U*) for the cell density *ρ* and the growth substrate *σ*.

#### Cell density

We first slightly rewrite the PDE:

\[ \frac{ \partial \rho}{\partial t} = \mu \nabla^2 \rho + r_B(\sigma) \frac{ \rho}{\rho_\textrm{max}} \left( \rho_\textrm{max} – \rho \right) – r_D(\sigma)\rho \]

and then try to match to the general form term-by-term. While BioFVM wasn’t intended for solving nonlinear PDEs of this form, we can make it work by quasi-linearizing, with the following functions:

\[ S = r_B(\sigma) \frac{ \rho }{\rho_\textrm{max}} \hspace{1in} U = r_D(\sigma). \]

When implementing this, we’ll evaluate *σ* and *ρ* at the previous time step. The diffusion coefficient is *μ*, and the decay rate is zero. The target or saturation density is *ρ*_{max}.

#### Growth substrate

Similarly, by matching the PDE for *σ* term-by-term with the general form, we use:

\[ S = r_\textrm{deliv}b, \hspace{1in} U = \lambda_2 \rho. \]

The diffusion coefficient is *D*, the decay rate is *λ*_{1}, and the saturation density is *σ*_{max}.

#### Blood vessels

Lastly, a term-by-term matching of the blood vessel equation gives the following functions:

\[ S=0 \hspace{1in} U = r_\textrm{degrade}\rho. \]

The diffusion coefficient, decay rate, and saturation density are all zero.

### Implementation in BioFVM

**Start a project:**Create a new directory for your project (I’d recommend “BioFVM_2D_tumor”), and enter the directory. Place a copy of BioFVM (the zip file) into your directory. Unzip BioFVM, and copy BioFVM*.h, BioFVM*.cpp, and pugixml* files into that directory.**Copy the matlab visualization files:**To help read and plot BioFVM data, we have provided matlab files. Copy all the *.m files from the matlab subdirectory to your project.**Copy the empty project:**BioFVM Version 1.0.3 or later includes a template project and Makefile to make it easier to get started. Copy the Makefile and template_project.cpp file to your project. Rename template_project.cpp to something useful, like 2D_tumor_example.cpp.**Edit the makefile:**Open a terminal window and browse to your project. Tailor the makefile to your new project:notepad++ Makefile

Change the PROGRAM_NAME to 2Dtumor.

Also, rename main to 2D_tumor_example throughout the Makefile.

Lastly, note that if you are using OSX, you’ll probably need to change from “g++” to your installed compiler. See these tutorials.

**Start adapting 2D_tumor_example.cpp:**First, open 2D_tumor_example.cpp:notepad++ 2D_tumor_example.cpp

Just after the “using namespace BioFVM” section of the code, define useful globals. Here and throughout, new and/or modified code is in blue:

using namespace BioFVM: // helpful -- have indices for each "species" int live_cells = 0; int blood_vessels = 1; int oxygen = 2; // some globals double prolif_rate = 1.0 /24.0; double death_rate = 1.0 / 6; // double cell_motility = 50.0 / 365.25 / 24.0 ; // 50 mm^2 / year --> mm^2 / hour double o2_uptake_rate = 3.673 * 60.0; // 165 micron length scale double vessel_degradation_rate = 1.0 / 2.0 / 24.0 ; // 2 days to disrupt tissue double max_cell_density = 1.0; double o2_supply_rate = 10.0; double o2_normoxic = 1.0; double o2_hypoxic = 0.2;

**Set up the microenvironment:**Within main(), make sure we have the right number of substrates, and set them up:// create a microenvironment, and set units Microenvironment M; M.name = "Tumor microenvironment"; M.time_units = "hr"; M.spatial_units = "mm"; M.mesh.units = M.spatial_units; // set up and add all the densities you plan M.set_density( 0 , "live cells" , "cells" ); M.add_density( "blood vessels" , "vessels/mm^2" ); M.add_density( "oxygen" , "cells" ); // set the properties of the diffusing substrates M.diffusion_coefficients[live_cells] = cell_motility; M.diffusion_coefficients[blood_vessels] = 0; M.diffusion_coefficients[oxygen] = 6.0; // 1e5 microns^2/min in units mm^2 / hr M.decay_rates[live_cells] = 0; M.decay_rates[blood_vessels] = 0; M.decay_rates[oxygen] = 0.01 * o2_uptake_rate; // 1650 micron length scale

Notice how our earlier global definitions of “live_cells”, “blood_vessels”, and “oxygen” makes it easier to make sure we’re referencing the correct substrates in lines like these.

**Resize the domain and test:**For this example (and so the code runs very quickly), we’ll work in 2D in a 2 cm × 2 cm domain:// set the mesh size double dx = 0.05; // 50 microns M.resize_space( 0.0 , 20.0 , 0, 20.0 , -dx/2.0, dx/2.0 , dx, dx, dx );

Notice that we use a tissue thickness of dx/2 to use the 3D code for a 2D simulation. Now, let’s test:

make 2Dtumor

Go ahead and cancel the simulation [Control]+C after a few seconds. You should see something like this:

Starting program ... Microenvironment summary: Tumor microenvironment: Mesh information: type: uniform Cartesian Domain: [0,20] mm x [0,20] mm x [-0.025,0.025] mm resolution: dx = 0.05 mm voxels: 160000 voxel faces: 0 volume: 20 cubic mm Densities: (3 total) live cells: units: cells diffusion coefficient: 0.00570386 mm^2 / hr decay rate: 0 hr^-1 diffusion length scale: 75523.9 mm blood vessels: units: vessels/mm^2 diffusion coefficient: 0 mm^2 / hr decay rate: 0 hr^-1 diffusion length scale: 0 mm oxygen: units: cells diffusion coefficient: 6 mm^2 / hr decay rate: 2.2038 hr^-1 diffusion length scale: 1.65002 mm simulation time: 0 hr (100 hr max) Using method diffusion_decay_solver__constant_coefficients_LOD_3D (implicit 3-D LOD with Thomas Algorithm) ... simulation time: 10 hr (100 hr max) simulation time: 20 hr (100 hr max)

**Set up initial conditions:**We’re going to make a small central focus of tumor cells, and a “bumpy” field of blood vessels.// set initial conditions // use this syntax to create a zero vector of length 3 // std::vector<double> zero(3,0.0); std::vector<double> center(3); center[0] = M.mesh.x_coordinates[M.mesh.x_coordinates.size()-1] /2.0; center[1] = M.mesh.y_coordinates[M.mesh.y_coordinates.size()-1] /2.0; center[2] = 0; double radius = 1.0; std::vector<double> one( M.density_vector(0).size() , 1.0 ); double pi = 2.0 * asin( 1.0 ); // use this syntax for a parallelized loop over all the // voxels in your mesh: #pragma omp parallel for for( int i=0; i < M.number_of_voxels() ; i++ ) { std::vector<double> displacement = M.voxels(i).center – center; double distance = norm( displacement ); if( distance < radius ) { M.density_vector(i)[live_cells] = 0.1; } M.density_vector(i)[blood_vessels]= 0.5 + 0.5*cos(0.4* pi * M.voxels(i).center[0])*cos(0.3*pi *M.voxels(i).center[1]); M.density_vector(i)[oxygen] = o2_normoxic; }

**Change to a 2D diffusion solver:**// set up the diffusion solver, sources and sinks M.diffusion_decay_solver = diffusion_decay_solver__constant_coefficients_LOD_2D;

**Set the simulation times:**We’ll simulate 10 days, with output every 12 hours.double t = 0.0; double t_max = 10.0 * 24.0; // 10 days double dt = 0.1; double output_interval = 12.0; // how often you save data double next_output_time = t; // next time you save data

**Set up the source function:**void supply_function( Microenvironment* microenvironment, int voxel_index, std::vector<double>* write_here ) { // use this syntax to access the jth substrate write_here // (*write_here)[j] // use this syntax to access the jth substrate in voxel voxel_index of microenvironment: // microenvironment->density_vector(voxel_index)[j] static double temp1 = prolif_rate / ( o2_normoxic – o2_hypoxic ); (*write_here)[live_cells] = microenvironment->density_vector(voxel_index)[oxygen]; (*write_here)[live_cells] -= o2_hypoxic; if( (*write_here)[live_cells] < 0.0 ) { (*write_here)[live_cells] = 0.0; } else { (*write_here)[live_cells] = temp1; (*write_here)[live_cells] *= microenvironment->density_vector(voxel_index)[live_cells]; } (*write_here)[blood_vessels] = 0.0; (*write_here)[oxygen] = o2_supply_rate; (*write_here)[oxygen] *= microenvironment->density_vector(voxel_index)[blood_vessels]; return; }

Notice the use of the static internal variable temp1: the first time this function is called, it declares this helper variable (to save some multiplication operations down the road). The static variable is available to all subsequent calls of this function.

**Set up the target function (substrate saturation densities):**void supply_target_function( Microenvironment* microenvironment, int voxel_index, std::vector<double>* write_here ) { // use this syntax to access the jth substrate write_here // (*write_here)[j] // use this syntax to access the jth substrate in voxel voxel_index of microenvironment: // microenvironment->density_vector(voxel_index)[j] (*write_here)[live_cells] = max_cell_density; (*write_here)[blood_vessels] = 1.0; (*write_here)[oxygen] = o2_normoxic; return; }

**Set up the uptake function:**void uptake_function( Microenvironment* microenvironment, int voxel_index, std::vector<double>* write_here ) { // use this syntax to access the jth substrate write_here // (*write_here)[j] // use this syntax to access the jth substrate in voxel voxel_index of microenvironment: // microenvironment->density_vector(voxel_index)[j] (*write_here)[live_cells] = o2_hypoxic; (*write_here)[live_cells] -= microenvironment->density_vector(voxel_index)[oxygen]; if( (*write_here)[live_cells] < 0.0 ) { (*write_here)[live_cells] = 0.0; } else { (*write_here)[live_cells] *= death_rate; } (*write_here)[oxygen] = o2_uptake_rate ; (*write_here)[oxygen] *= microenvironment->density_vector(voxel_index)[live_cells]; (*write_here)[blood_vessels] = vessel_degradation_rate ; (*write_here)[blood_vessels] *= microenvironment->density_vector(voxel_index)[live_cells]; return; }

And that’s it. The source should be ready to go!

### Source files

You can download completed source for this example here:

### Using the code

#### Running the code

First, compile and run the code:

make 2Dtumor

The output should look like this.

Starting program … Microenvironment summary: Tumor microenvironment: Mesh information: type: uniform Cartesian Domain: [0,20] mm x [0,20] mm x [-0.025,0.025] mm resolution: dx = 0.05 mm voxels: 160000 voxel faces: 0 volume: 20 cubic mm Densities: (3 total) live cells: units: cells diffusion coefficient: 0.00570386 mm^2 / hr decay rate: 0 hr^-1 diffusion length scale: 75523.9 mm blood vessels: units: vessels/mm^2 diffusion coefficient: 0 mm^2 / hr decay rate: 0 hr^-1 diffusion length scale: 0 mm oxygen: units: cells diffusion coefficient: 6 mm^2 / hr decay rate: 2.2038 hr^-1 diffusion length scale: 1.65002 mm simulation time: 0 hr (240 hr max) Using method diffusion_decay_solver__constant_coefficients_LOD_2D (2D LOD with Thomas Algorithm) … simulation time: 12 hr (240 hr max) simulation time: 24 hr (240 hr max) simulation time: 36 hr (240 hr max) simulation time: 48 hr (240 hr max) simulation time: 60 hr (240 hr max) simulation time: 72 hr (240 hr max) simulation time: 84 hr (240 hr max) simulation time: 96 hr (240 hr max) simulation time: 108 hr (240 hr max) simulation time: 120 hr (240 hr max) simulation time: 132 hr (240 hr max) simulation time: 144 hr (240 hr max) simulation time: 156 hr (240 hr max) simulation time: 168 hr (240 hr max) simulation time: 180 hr (240 hr max) simulation time: 192 hr (240 hr max) simulation time: 204 hr (240 hr max) simulation time: 216 hr (240 hr max) simulation time: 228 hr (240 hr max) simulation time: 240 hr (240 hr max) Done!

#### Looking at the data

Now, let’s pop it open in matlab (or octave):

matlab

To load and plot a single time (e.g., the last tim)

!ls *.mat M = read_microenvironment( 'output_240.000000.mat' ); plot_microenvironment( M );

To add some labels:

labels{1} = 'tumor cells'; labels{2} = 'blood vessel density'; labels{3} = 'growth substrate'; plot_microenvironment( M ,labels );

Your output should look a bit like this:

Lastly, you might want to script the code to create and save plots of all the times.

labels{1} = 'tumor cells'; labels{2} = 'blood vessel density'; labels{3} = 'growth substrate'; for i=0:20 t = i*12; input_file = sprintf( 'output_%3.6f.mat', t ); output_file = sprintf( 'output_%3.6f.png', t ); M = read_microenvironment( input_file ); plot_microenvironment( M , labels ); print( gcf , '-dpng' , output_file ); end

### What’s next

We’ll continue posting new tutorials on adapting BioFVM to existing and new simulators, as well as guides to new features as we roll them out.

Stay tuned and watch this blog!

## Setting up gcc / OpenMP on OSX (Homebrew edition)

**Note:** This is part of a series of “how-to” blog posts to help new users and developers of BioFVM and PhysiCell. This guide is for OSX users. Windows users should use this guide instead. A Linux guide is expected soon.

These instructions should get you up and running with a minimal environment for compiling 64-bit C++ projects with OpenMP (e.g., BioFVM and PhysiCell) using gcc. These instructions were tested with OSX 10.11 (El Capitan), but they should work on any reasonably recent version of OSX.

In the end result, you’ll have a compiler and key makefile capabilities. The entire toolchain is free and open source.

Of course, you can use other compilers and more sophisticated integrated desktop environments, but these instructions will get you a good baseline system with support for 64-bit binaries and OpenMP parallelization.

** Note 1:** OSX / Xcode

*appears*to have gcc out of the box (you can type “gcc” in a Terminal window), but this really just maps back onto Apple’s build of clang. Alas, this will not support OpenMP for parallelization.

** Note 2:** Yesterday in this post, we showed how to set up gcc using the popular MacPorts package manager. Because MacPorts builds gcc (and all its dependencies!) from source, it takes a very, very long time. On my 2012 Macbook Air, this step took

*16 hours*. This tutorial uses Homebrew to dramatically speed up the process!

* Note 3:* This is an update over the previous version. It incorporates new information that Xcode command line tools can be installed without the full 4.41 GB download / installation of Xcode. Many thanks to Walter de Back and Tim at the Homebrew project for their help!

### What you’ll need:

**XCode Command Line Tools:**These command line tools are needed for Homebrew and related package managers. Installation instructions are now very simple and included below. As of January 18, 2016, this will install Version 2343.**Homebrew**: This is a package manager for OSX, which will let you easily download and install many linux utilities*without building them from source*. You’ll particularly need it for getting gcc. Installation is a simple command-line script, as detailed below. As of January 17, 2016, this will download Version 0.9.5.**gcc5**(from Homebrew)**:**This will be an up-to-date 64-bit version of gcc, with support for OpenMP. As of January 17, 2016, this will download Version 5.2.0.

### Main steps:

#### 1) Install the XCode Command Line Tools

Open a terminal window (Open Launchpad, then “Other”, then “Terminal”), and run:

> xcode-select --install

A window should pop up asking you to either get Xcode or install. Choose the “install” option to avoid the huge 4+ GB Xcode download. It should only take a few minutes to complete.

#### 2) Install Homebrew

Open a terminal window (Open Launchpad, then “Other”, then “Terminal”), and run:

> ruby -e "$(curl -fsSL https://raw.githubusercontent.com/Homebrew/install/master/install)"

Let the script run, and answer “y” whenever asked. This will not take very long.

#### 3) Get, install, and prepare gcc

Open a terminal window (see above), and search for gcc, version 5.x or above

> brew search gcc5

You should see a list of packages, including gcc5. Take a note of what is found. (In my case, it found homebrew/versions/gcc5.)

Then, download and install gcc5:

> brew install homebrew/versions/gcc5

This will download whatever dependencies are needed, generally already pre-compiled. The whole process should only take five or ten minutes.

Lastly, you need to get the exact name of your compiler. In your terminal window, type g++, and then hit tab twice to see a list. On my system, I see this:

Pauls-MBA:~ pmacklin$ g++ g++ g++-5 g++-mp-5

Look for the version of g++ without an “mp” (from MacPorts) in its name. In my case, it’s g++-5. Double-check that you have the right one by checking its version. It should look something like this:

Pauls-MBA:~ pmacklin$ g++-5 --version g++-5 (Homebrew gcc5 5.2.0) 5.2.0 Copyright (C) 2015 Free Software Foundation, Inc. This is free software; see the source for copying conditions. There is NO warranty; not even for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

Notice that Homebrew shows up in the information. The correct compiler is g++-5.

#### 5) Test the compiler

__Write a basic parallelized program:__

__Write a basic parallelized program:__

Open Terminal (see above). You should be in your user profile’s root directory. Make a new subdirectory, enter it, and create a new file:

> mkdir omp_test > cd omp_test > nano omp_test.cpp

Then, write your basic OpenMP test:

#include <iostream> #include <cmath> #include <vector> #include <omp.h> int main( int argc, char* argv[] ) { omp_set_num_threads( 8 ); double pi = acos( -1.0 ); std::cout << "Allocating memory ..." << std::endl; std::vector<double> my_vector( 128000000, 0.0 ); std::cout << "Done!" << std::endl << std::endl; std::cout << "Entering main loop ... " << std::endl; #pragma omp parallel for for( int i=0; i < my_vector.size(); i++ ) { my_vector[i] = exp( -sin( i*i + pi*log(i+1) ) ); } std::cout << "Done!" << std::endl; return 0; }

Save the file (as omp_test.cpp). (In nano, use [Control]+X, Y, and then confirm the choice of filename.)

In the omp_set_num_threads() line above, replace 8 with the maximum number of virtual processors on your CPU. (For a quad-core CPU with hyperthreading, this number is 8. On a hex-core CPU without hyperthreading, this number is 6.) If in doubt, leave it alone for now.

__Write a makefile:__

Next, create a Makefile to start editing:

> nano Makefile

Add the following contents:

CC := g++-5 # replace this with your correct compiler as identified above ARCH := core2 # Replace this with your CPU architecture. # core2 is pretty safe for most modern machines. CFLAGS := -march=$(ARCH) -O3 -fopenmp -m64 -std=c++11 COMPILE_COMMAND := $(CC) $(CFLAGS) OUTPUT := my_test all: omp_test.cpp $(COMPILE_COMMAND) -o $(OUTPUT) omp_test.cpp clean: rm -f *.o $(OUTPUT).*

Go ahead and save this (as Makefile). ([Control]+X, Y, confirm the filename.)

__Compile and run the test:__

Go back to your (still open) command prompt. Compile and run the program:

> make > ./my_test

The output should look something like this:

Allocating memory ... Done! Entering main loop ... Done!

**Note 1:** If the make command gives errors like “**** missing separator”, then you need to replace the white space (e.g., one or more spaces) at the start of the “$(COMPILE_COMMAND)” and “rm -f” lines with a single tab character.

**Note 2:** If the compiler gives an error like “fatal error: ‘omp.h’ not found”, you probably used Apple’s build of clang, which does not include OpenMP support. You’ll need to make sure that you specify your compiler on the CC line of your makefile.

Now, let’s verify that the code is using OpenMP.

Open another Terminal window. While the code is running, run top. Take a look at the performance, particularly CPU usage. While your program is running, you should see CPU usage fairly close to ‘100% user’. (This is a good indication that your code is running the OpenMP parallelization as expected.)

### What’s next?

Download a copy of BioFVM and try out the included examples! Visit BioFVM at MathCancer.org.

- BioFVM announcement on this blog: [click here]
- BioFVM on MathCancer.org: http://BioFVM.MathCancer.org
- BioFVM on SourceForge: http://BioFVM.sf.net
- BioFVM Method Paper in BioInformatics: http://dx.doi.org/10.1093/bioinformatics/btv730
- BioFVM tutorials: [click here]

*
*

## Setting up gcc / OpenMP on OSX (Homebrew edition) (outdated)

**Note 1:** This is the part of a series of “how-to” blog posts to help new users and developers of BioFVM and PhysiCell. This guide is for OSX users. Windows users should use this guide instead. A Linux guide is expected soon.

**Note 2:** This tutorial is outdated. Please see this updated version.

These instructions should get you up and running with a minimal environment for compiling 64-bit C++ projects with OpenMP (e.g., BioFVM and PhysiCell) using gcc. These instructions were tested with OSX 10.11 (El Capitan), but they should work on any reasonably recent version of OSX.

In the end result, you’ll have a compiler and key makefile capabilities. The entire toolchain is free and open source.

Of course, you can use other compilers and more sophisticated integrated desktop environments, but these instructions will get you a good baseline system with support for 64-bit binaries and OpenMP parallelization.

** Note 3:** OSX / Xcode

*appears*to have gcc out of the box (you can type “gcc” in a Terminal window), but this really just maps back onto Apple’s build of clang. Alas, this will not support OpenMP for parallelization.

** Note 4:** Yesterday in this post, we showed how to set up gcc using the popular MacPorts package manager. Because MacPorts builds gcc (and all its dependencies!) from source, it takes a very, very long time. On my 2012 Macbook Air, this step took

*16 hours*. This tutorial uses Homebrew to dramatically speed up the process!

### What you’ll need:

**XCode:**This includes command line development tools. Evidently, it is required for both Homebrew and its competitors (e.g., MacPorts). Download the latest version in the App Store. (Search for xcode.) As of January 15, 2016, the App Store will install Version 7.2.**Please note that this is a 4.41 GB download!****Homebrew**: This is a package manager for OSX, which will let you easily download and install many linux utilities*without building them from source*. You’ll particularly need it for getting gcc. Installation is a simple command-line script, as detailed below. As of January 17, 2016, this will download Version 0.9.5.**gcc5**(from Homebrew)**:**This will be an up-to-date 64-bit version of gcc, with support for OpenMP. As of January 17, 2016, this will download Version 5.2.0.

### Main steps:

#### 1) Download, install, and prepare XCode

As mentioned above, open the App Store, search for Xcode, and start the download / install. Go ahead and grab a coffee while it’s downloading and installing 4+ GB. Once it has installed, open Xcode, agree to the license, and let it install whatever components it needs.

Now, you need to get the command line tools. Go to the Xcode menu, select “Open Developer Tool”, and choose “More Developer Tools …”. This will open up a site in Safari and prompt you to log in.

Sign on with your AppleID, agree to yet more licensing terms, and then search for “command line tools” for your version of Xcode and OSX. (In my case, this is OSX 10.11 with Xcode 7.2) Click the + next to the correct version, and then the link for the dmg file. (Command_Line_Tools_OS_X_10.11_for_Xcode_7.2.dmg).

Double-click the dmg file. Double-click pkg file it contains. Click “continue”, “OK”, and “agree” as much as it takes to install. Once done, go ahead and exit the installer and close the dmg file.

#### 2) Install Homebrew

Open a terminal window (Open Launchpad, then “Other”, then “Terminal”), and run:

> ruby -e "$(curl -fsSL https://raw.githubusercontent.com/Homebrew/install/master/install)"

Let the script run, and answer “y” whenever asked. This will not take very long.

#### 3) Get, install, and prepare gcc

Open a terminal window (see above), and search for gcc, version 5.x or above

> brew search gcc5

You should see a list of packages, including gcc5. Take a note of what is found. (In my case, it found homebrew/versions/gcc5.)

Then, download and install gcc5:

> brew install homebrew/versions/gcc5

This will download whatever dependencies are needed, generally already pre-compiled. The whole process should only take five or ten minutes.

Lastly, you need to get the exact name of your compiler. In your terminal window, type g++, and then hit tab twice to see a list. On my system, I see this:

Pauls-MBA:~ pmacklin$ g++ g++ g++-5 g++-mp-5

Look for the version of g++ without an “mp” (for MacPorts) in its name. In my case, it’s g++-5. Double-check that you have the right one by checking its version. It should look something like this:

Pauls-MBA:~ pmacklin$ g++-5 --version g++-5 (Homebrew gcc5 5.2.0) 5.2.0 Copyright (C) 2015 Free Software Foundation, Inc. This is free software; see the source for copying conditions. There is NO warranty; not even for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

Notice that Homebrew shows up in the information. The correct compiler is g++-5.

#### 5) Test the compiler

##### Write a basic parallelized program:

Open Terminal (see above). You should be in your user profile’s root directory. Make a new subdirectory, enter it, and create a new file:

> mkdir omp_test > cd omp_test > nano omp_test.cpp

Then, write your basic OpenMP test:

#include <iostream> #include <cmath> #include <vector> #include <omp.h> int main( int argc, char* argv[] ) { omp_set_num_threads( 8 ); double pi = acos( -1.0 ); std::cout << "Allocating memory ..." << std::endl; std::vector<double> my_vector( 128000000, 0.0 ); std::cout << "Done!" << std::endl << std::endl; std::cout << "Entering main loop ... " << std::endl; #pragma omp parallel for for( int i=0; i < my_vector.size(); i++ ) { my_vector[i] = exp( -sin( i*i + pi*log(i+1) ) ); } std::cout << "Done!" << std::endl; return 0; }

Save the file (as omp_test.cpp). (In nano, use [Control]+X, Y, and then confirm the choice of filename.)

In the omp_set_num_threads() line above, replace 8 with the maximum number of virtual processors on your CPU. (For a quad-core CPU with hyperthreading, this number is 8. On a hex-core CPU without hyperthreading, this number is 6.) If in doubt, leave it alone for now.

##### Write a makefile:

Next, create a Makefile to start editing:

> nano Makefile

Add the following contents:

CC := g++-5 # replace this with your correct compiler as identified above ARCH := core2 # Replace this with your CPU architecture. # core2 is pretty safe for most modern machines. CFLAGS := -march=$(ARCH) -O3 -fopenmp -m64 -std=c++11 COMPILE_COMMAND := $(CC) $(CFLAGS) OUTPUT := my_test all: omp_test.cpp $(COMPILE_COMMAND) -o $(OUTPUT) omp_test.cpp clean: rm -f *.o $(OUTPUT).*

Go ahead and save this (as Makefile). ([Control]-X, Y, confirm the filename.)

##### Compile and run the test:

Go back to your (still open) command prompt. Compile and run the program:

> make > ./my_test

The output should look something like this:

Allocating memory ... Done! Entering main loop ... Done!

**Note 1: **If the make command gives errors like “**** missing separator”, then you need to replace the white space (e.g., one or more spaces) at the start of the “$(COMPILE_COMMAND)” and “rm -f” lines with a single tab character.

**Note 2: **If the compiler gives an error like “fatal error: ‘omp.h’ not found”, you probably used Apple’s build of clang, which does not include OpenMP support. You’ll need to make sure that you specify your compiler on the CC line of your makefile.

Now, let’s verify that the code is using OpenMP.

Open another Terminal window. While the code is running, run top. Take a look at the performance, particularly CPU usage. While your program is running, you should see CPU usage fairly close to ‘100% user’. (This is a good indication that your code is running the OpenMP parallelization as expected.)

### What’s next?

Download a copy of BioFVM and try out the included examples! Visit BioFVM at MathCancer.org.

- BioFVM announcement on this blog: [click here]
- BioFVM on MathCancer.org: http://BioFVM.MathCancer.org
- BioFVM on SourceForge: http://BioFVM.sf.net
- BioFVM Method Paper in BioInformatics: http://dx.doi.org/10.1093/bioinformatics/btv730
- BioFVM tutorials: [click here]