## Category: matlab

## 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.

## 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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