Category: tutorial

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:

  1. The experimental model may be a bad choice for the disease or process (not our fault).
  2. Second, the mathematical model of the experimental system may have flawed assumptions (something we have to evaluate).
  3. 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, G2, M, and part of G1 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 ) > 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:

ode_solutions

 

Next, we plot KI1, KI2, 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! ūüôā

ode_fractions

Some readers may recognize the long-time fractions: KI1 + KI2 = 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:

ode_1runsfractions_1runs

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

ode_10runsfractions_10runs

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

ode_100runsfractions_100runs

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

ode_100runs_log

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., K1, K2, 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.

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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

  1. A working C++ development environment with support for OpenMP. See these prior tutorials if you need help.
  2. A download of BioFVM, available at http://BioFVM.MathCancer.org and http://BioFVM.sf.net. Use Version 1.0.3 or later.
  3. 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 rB 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 rD when the growth substrate level is too low. We assume that the tumor cell density cannot exceed a max level ρmax. A model that includes these effects is:

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

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

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

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 λ1, and the saturation density is σmax.

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

  1. 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.
  2. 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.
  3. 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.
  4. 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.

  5. 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; 
    
  6. 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.

  7. Resize the domain and test: For this example (and so the code runs very quickly), we’ll work in 2D in a 2 cm √ó 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)
    
  8. 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 ‚Äď 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;
    }
    
  9. 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;
    
  10. 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
    
  11. 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 ‚Äď 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.

  12. 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; 
    }
    
  13. 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:

  1. 2D_tumor_example.cpp
  2. Makefile

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:


BioFVM 2D warmup -- final output

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!


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