## MathCancer C++ Style and Practices Guide

As PhysiCell, BioFVM, and other open source projects start to gain new users and contributors, it’s time to lay out a coding style. We have three goals here:

1. Consistency: It’s easier to understand and contribute to the code if it’s written in a consistent way.
2. Readability: We want the code to be as readable as possible.
3. Reducing errors: We want to avoid coding styles that are more prone to errors. (e.g., code that can be broken by introducing whitespace).

So, here is the guide (revised June 2017). I expect to revise this guide from time to time.

### Place braces on separate lines in functions and classes.

I find it much easier to read a class if the braces are on separate lines, with good use of whitespace. Remember: whitespace costs almost nothing, but reading and understanding (and time!) are expensive.

#### DON’T

class Cell{
public:
double some_variable;
bool some_extra_variable;

Cell(); };

class Phenotype{
public:
double some_variable;
bool some_extra_variable;

Phenotype();
};


#### DO:

class Cell
{
public:
double some_variable;
bool some_extra_variable;

Cell();
};

class Phenotype
{
public:
double some_variable;
bool some_extra_variable;

Phenotype();
};


### Enclose all logic in braces, even when optional.

In C/C++, you can omit the curly braces in some cases. For example, this is legal

if( distance > 1.5*cell_radius )
interaction = false;
force = 0.0; // is this part of the logic, or a separate statement?
error = false;


However, this code is ambiguous to interpret. Moreover, small changes to whitespace–or small additions to the logic–could mess things up here. Use braces to make the logic crystal clear:

#### DON’T

if( distance > 1.5*cell_radius )
interaction = false;
force = 0.0; // is this part of the logic, or a separate statement?
error = false;

if( condition1 == true )
do_something1 = true;
elseif( condition2 == true )
do_something2 = true;
else
do_something3 = true;


#### DO

if( distance > 1.5*cell_radius )
{
interaction = false;
force = 0.0;
}
error = false;

if( condition1 == true )
{ do_something1 = true; }
elseif( condition2 == true )
{ do_something2 = true; }
else
{ do_something3 = true; }


### Put braces on separate lines in logic, except for single-line logic.

This style rule relates to the previous point, to improve readability.

#### DON’T

if( distance > 1.5*cell_radius ){
interaction = false;
force = 0.0; }

if( condition1 == true ){ do_something1 = true; }
elseif( condition2 == true ){
do_something2 = true; }
else
{ do_something3 = true; error = true; }


#### DO

if( distance > 1.5*cell_radius )
{
interaction = false;
force = 0.0;
}

if( condition1 == true )
{ do_something1 = true; } // this is fine
elseif( condition2 == true )
{
do_something2 = true; // this is better
}
else
{
do_something3 = true;
error = true;
}


See how much easier that code is to read? The logical structure is crystal clear, and adding more to the logic is simple.

### End all functions with a return, even if void.

For clarity, definitively state that a function is done by using return.

#### DON’T

void my_function( Cell& cell )
{
cell.phenotype.volume.total *= 2.0;
cell.phenotype.death.rates[0] = 0.02;
// Are we done, or did we forget something?
// is somebody still working here?
}


#### DO

void my_function( Cell& cell )
{
cell.phenotype.volume.total *= 2.0;
cell.phenotype.death.rates[0] = 0.02;
return;
}


### Use tabs to indent the contents of a class or function.

This is to make the code easier to read. (Unfortunately PHP/HTML makes me use five spaces here instead of tabs.)

#### DON’T

class Secretion
{
public:
std::vector<double> secretion_rates;
std::vector<double> uptake_rates;
std::vector<double> saturation_densities;
};

void my_function( Cell& cell )
{
cell.phenotype.volume.total *= 2.0;
cell.phenotype.death.rates[0] = 0.02;
return;
}


#### DO

class Secretion
{
public:
std::vector<double> secretion_rates;
std::vector<double> uptake_rates;
std::vector<double> saturation_densities;
};

void my_function( Cell& cell )
{
cell.phenotype.volume.total *= 2.0;
cell.phenotype.death.rates[0] = 0.02;
return;
}


### Use a single space to indent public and other keywords in a class.

This gets us some nice formatting in classes, without needing two tabs everywhere.

#### DON’T

class Secretion
{
public:
std::vector<double> secretion_rates;
std::vector<double> uptake_rates;
std::vector<double> saturation_densities;
}; // not enough whitespace

class Errors
{
private:
public:
std::string error_message;
int error_code;
}; // too much whitespace!


#### DO

class Secretion
{
private:
public:
std::vector<double> secretion_rates;
std::vector<double> uptake_rates;
std::vector<double> saturation_densities;
};

class Errors
{
private:
public:
std::string error_message;
int error_code;
};


### Avoid arcane operators, when clear logic statements will do.

It can be difficult to decipher code with statements like this:

phenotype.volume.fluid=phenotype.volume.fluid<0?0:phenotype.volume.fluid;


Moreover, C and C++ can treat precedence of ternary operators very differently, so subtle bugs can creep in when using the “fancier” compact operators. Variations in how these operators work across languages are an additional source of error for programmers switching between languages in their daily scientific workflows. Wherever possible (and unless there is a significant performance reason to do so), use clear logical structures that are easy to read even if you only dabble in C/C++. Compiler-time optimizations will most likely eliminate any performance gains from these goofy operators.

#### DON’T

// if the fluid volume is negative, set it to zero
phenotype.volume.fluid=phenotype.volume.fluid<0.0?0.0:pCell->phenotype.volume.fluid;


#### DO

if( phenotype.volume.fluid < 0.0 )
{
phenotype.volume.fluid = 0.0;
}


Here’s the funny thing: the second logic is much clearer, and it took fewer characters, even with extra whitespace for readability!

### Pass by reference where possible.

Passing by reference is a great way to boost performance: we can avoid (1) allocating new temporary memory, (2) copying data into the temporary memory, (3) passing the temporary data to the function, and (4) deallocating the temporary memory once finished.

#### DON’T

double some_function( Cell cell )
{
return = cell.phenotype.volume.total + 3.0;
}
// This copies cell and all its member data!


#### DO

double some_function( Cell& cell )
{
return = cell.phenotype.volume.total + 3.0;
}
// This just accesses the original cell data without recopying it.


### Where possible, pass by reference instead of by pointer.

There is no performance advantage to passing by pointers over passing by reference, but the code is simpler / clearer when you can pass by reference. It makes code easier to write and understand if you can do so. (If nothing else, you save yourself character of typing each time you can replace “->” by “.”!)

#### DON’T

double some_function( Cell* pCell )
{
return = pCell->phenotype.volume.total + 3.0;
}
// Writing and debugging this code can be error-prone.


#### DO

double some_function( Cell& cell )
{
return = cell.phenotype.volume.total + 3.0;
}
// This is much easier to write.


### Be careful with static variables. Be thread safe!

PhysiCell relies heavily on parallelization by OpenMP, and so you should write functions under the assumption that they may be executed many times simultaneously. One easy source of errors is in static variables:

#### DON’T

double some_function( Cell& cell )
{
static double four_pi = 12.566370614359172;
static double output;
output *= output;
output *= four_pi;
return output;
}
// If two instances of some_function are running, they will both modify
// the *same copy* of output


#### DO

double some_function( Cell& cell )
{
static double four_pi = 12.566370614359172;
double output;
output *= output;
output *= four_pi;
return output;
}
// If two instances of some_function are running, they will both modify
// the their own copy of output, but still use the more efficient, once-
// allocated copy of four_pi. This one is safe for OpenMP.


### Use std:: instead of “using namespace std”

PhysiCell uses the BioFVM and PhysiCell namespaces to avoid potential collision with other codes. Other codes using PhysiCell may use functions that collide with the standard namespace. So, we formally use std:: whenever using functions in the standard namespace.

#### DON’T

using namespace std;

cout << "Hi, Mom, I learned to code today!" << endl;
string my_string = "Cheetos are good, but Doritos are better.";
cout << my_string << endl;

vector<double> my_vector;
vector.resize( 3, 0.0 );


#### DO

std::cout << "Hi, Mom, I learned to code today!" << std::endl;
std::string my_string = "Cheetos are good, but Doritos are better.";
std::cout << my_string << std::endl;

std::vector<double> my_vector;
my_vector.resize( 3, 0.0 );


### Camelcase is ugly. Use underscores.

This is purely an aesthetic distinction, but CamelCaseCodeIsUglyAndDoYouUseDNAorDna?

#### DON’T

double MyVariable1;
bool ProteinsInExosomes;
int RNAtranscriptionCount;

void MyFunctionDoesSomething( Cell& ImmuneCell );


#### DO

double my_variable1;
bool proteins_in_exosomes;
int RNA_transcription_count;

void my_function_does_something( Cell& immune_cell );


### Use capital letters to declare a class. Use lowercase for instances.

To help in readability and consistency, declare classes with capital letters (but no camelcase), and use lowercase for instances of those classes.

#### DON’T

class phenotype;

class cell
{
public:
std::vector<double> position;
phenotype Phenotype;
};

class ImmuneCell : public cell
{
public:
std::vector<double> surface_receptors;
};

void do_something( cell& MyCell , ImmuneCell& immuneCell );

cell Cell;
ImmuneCell MyImmune_cell;

do_something( Cell, MyImmune_cell );


#### DO

class Phenotype;

class Cell
{
public:
std::vector<double> position;
Phenotype phenotype;
};

class Immune_Cell : public Cell
{
public:
std::vector<double> surface_receptors;
};

void do_something( Cell& my_cell , Immune_Cell& immune_cell );

Cell cell;
Immune_Cell my_immune_cell;

do_something( cell, my_immune_cell );


## 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

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.

We will implement a basic 3-D cellular automaton model of tumor growth in a well-mixed fluid, containing oxygen pO2 (mmHg) and a drug c (e.g., doxorubicin, μM), inspired by modeling by Alexander Anderson, Heiko Enderling, Jan PoleszczukGibin 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 pO2 = 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 3.6×103 μm3. 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 pO2 = 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,tt], each live tumor cell i has a probability pi,D of attempting division, probability pi,A of apoptotic death, and probability pi,N of necrotic death. (For simplicity, we ignore motility in this version.) We relate these to the birth rate bi, apoptotic death rate di,A, and necrotic death rate di,N by the linearized equations (from Macklin et al. 2012):

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

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

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

Each dead cell has a mean duration Ti,D, which will vary by the type of cell death. Each dead cell automaton has a probability pi,L of lysis (rupture and removal) in any time span [t,t+Δt]. The duration TD is converted to a probability of cell lysis by

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

##### (Illustrative) parameter values

We use Doxy = 105 μm2/min (Ghaffarizadeh et al. 2016), and we set Ui,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 λoxy = 0.01 min-1 for a 1 mm diffusion length scale in fluid.

We set Dc = 300 μm2/min, and Uc = 7.2×10-3 min-1 (Dc from Weinberg et al. (2007), and Ui,c 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 λc = 3.6×10-5 min-1 to give a drug diffusion length scale of about 2.9 mm in fluid.

We use TD = 8.6 hours for apoptotic cells, and TD = 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.

#### 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 Ei(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 pO2,P is the physioxic oxygen value (38 mmHg), and pO2,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:

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

where di,Amax is the maximum apoptotic death rate. We model the microenvironment-dependent necrosis rate by:

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

for a constant value di,N*.
##### (Illustrative) parameter values

We use bi,P = 0.05 hour-1 (for a 20 hour cell cycle in physioxic conditions), di,A* = 0.01 bi,P, and di,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 di,Amax = 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&amp; 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

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
{
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;
}

}
}


#### 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

1. 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.
3. 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.
4. 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) pO2 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

This file will include the following:

1. BioFVM_cellular_automata.h
2. BioFVM_cellular_automata.cpp
3. BioFVM_CA_example_1.cpp
5. plot_cellular_automata.m
6. 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.

## 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)

## 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

### 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:

1. BioFVM*.cpp and BioFVM*.h (from the main BioFVM directory)
2. pugixml.* (from the main BioFVM directory)
3. 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.

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:

// who created the program (if known)
BioFVM_metadata.program.creator.organization = "University of Southern California";
BioFVM_metadata.program.creator.department = "Center for Applied Molecular Medicine";

// (generally peer-reviewed) citation information for the program
BioFVM: an efficient parallelized diffusive transport solver for 3-D biological
simulations, Bioinformatics, 2015. DOI: 10.1093/bioinformatics/btv730.";

// user information: who ran the program
BioFVM_metadata.program.user.department = "U.S.S. Enterprise (NCC 1701)";

// And finally, data citation information (the publication where this simulation snapshot appeared)
an efficient parallelized diffusive transport solver for 3-D biological simulations, Bioinformatics,
2015. DOI: 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

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.mesh.Z_coordinates(mid_index), ...


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.mesh.Z_coordinates(mid_index), ...


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.mesh.Z_coordinates(mid_index), ...


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 , ...


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
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 , ...


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, ...


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, ...


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.

## 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.

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:

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

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

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

// 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 × 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;

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
#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 );

{
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 – 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] *=
microenvironment->density_vector(voxel_index)[live_cells];

return;
}


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

### 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
plot_microenvironment( M );


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 );
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$g++-5 --version g++-5 (Homebrew gcc5 5.2.0) 5.2.0 Copyright (C) 2015 Free Software Foundation, Inc. This is free software; see the source for copying conditions. There is NO warranty; not even for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  Notice that Homebrew shows up in the information. The correct compiler is g++-5. #### 5) Test the compiler ##### Write a basic parallelized program: Open Terminal (see above). You should be in your user profile’s root directory. Make a new subdirectory, enter it, and create a new file: > mkdir omp_test > cd omp_test > nano omp_test.cpp  Then, write your basic OpenMP test: #include <iostream> #include <cmath> #include <vector> #include <omp.h> int main( int argc, char* argv[] ) { omp_set_num_threads( 8 ); double pi = acos( -1.0 ); std::cout << "Allocating memory ..." << std::endl; std::vector<double> my_vector( 128000000, 0.0 ); std::cout << "Done!" << std::endl << std::endl; std::cout << "Entering main loop ... " << std::endl; #pragma omp parallel for for( int i=0; i < my_vector.size(); i++ ) { my_vector[i] = exp( -sin( i*i + pi*log(i+1) ) ); } std::cout << "Done!" << std::endl; return 0; }  Save the file (as omp_test.cpp). (In nano, use [Control]+X, Y, and then confirm the choice of filename.) In the omp_set_num_threads() line above, replace 8 with the maximum number of virtual processors on your CPU. (For a quad-core CPU with hyperthreading, this number is 8. On a hex-core CPU without hyperthreading, this number is 6.) If in doubt, leave it alone for now. ##### Write a makefile: Next, create a Makefile to start editing: > nano Makefile  Add the following contents: CC := g++-5 # replace this with your correct compiler as identified above ARCH := core2 # Replace this with your CPU architecture. # core2 is pretty safe for most modern machines. CFLAGS := -march=$(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.

2. BioFVM on MathCancer.org: http://BioFVM.MathCancer.org
3. BioFVM on SourceForge: http://BioFVM.sf.net
4. BioFVM Method Paper in BioInformatics: http://dx.doi.org/10.1093/bioinformatics/btv730

## 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:

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.

2. BioFVM on MathCancer.org: http://BioFVM.MathCancer.org
3. BioFVM on SourceForge: http://BioFVM.sf.net
4. BioFVM Method Paper in BioInformatics: http://dx.doi.org/10.1093/bioinformatics/btv730

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

Note: 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.

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: This process is somewhat painful because MacPorts compiles everything from source, rather than using pre-compiled binaries. This tutorial uses Homebrew: a newer package manager that uses pre-compiled binaries to dramatically speed up the process. I highly recommend using the Homebrew version of this tutorial.

### What you’ll need:

1. XCode: This includes command line development tools. Evidently, it is required for both Macports and its competitors (e.g., Homebrew). 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!
2. MacPorts: This is a package manager for OSX, which will let you easily download, build and install many linux utilities. You’ll particularly need it for getting gcc. Download the latest installer (MacPorts-2.3.4-10.11-ElCapitan.pkg) here. As of January 15, 2016, this will download Version 2.3.4.
3. gcc5 (from MacPorts): This will be an up-to-date 64-bit version of gcc, with support for OpenMP. As of January 15, 2016, this will download Version 5.3.0.

### Main steps:

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 Macports

Double-click the MacPorts pkg file you downloaded above. OSX may complain with a message like this:

“MacPorts-2.3.4-10.11-ElCapitan.pkg” can’t be opened because it is from an unidentified developer.

If so, follow the directions here.

Leave all the default choices as they are in the installer. Click OK a bunch of times. The package scripts might take awhile.

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

sudo port -v selfupdate


to make sure that everything is up-to-date.

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

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

port search gcc5


You should see a list of packages, including gcc5.

sudo port install gcc5


You should see a list of packages, including gcc5.

This will download, build, and install any dependencies necessary for gcc5, including llvm and many, many other things. This takes even longer than the 4.4 GB download of Xcode. Go get dinner and a coffee. You may well need to let this run overnight. (On my 2012 Macbook Air, it required 16 hours to fully build gcc5 and its dependencies. We’ll discuss this point further below.)

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$g++ g++ g++-mp-5  Look for the version of g++ with an “mp” in its name. In my case, it’s g++-mp-5. Double-check that you have the right one by checking its version. It should look something like this: Pauls-MBA:~ pmacklin$ g++-mp-5 --version

g++-mp-5 (MacPorts gcc5 5.3.0_0) 5.3.0
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 MacPorts shows up in the information. The correct compiler is g++-mp-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[] )
{

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, or set it at a relatively safe value of 4.

##### Write a makefile:

Next,  create a Makefile to start editing:

nano Makefile


CC := g++-mp-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.) ### MacPorts and Pain MacPorts builds all the tools from source. While this ensures that you get very up-to-date binaries, it is very, very slow! However, all hope is not lost. It turns out that Homebrew will install pre-compiled binaries, so the 16-hour process of installing gcc is reduced to about 5-10 minutes. Check back tomorrow for a follow-up tutorial on how to use Homebrew to set up gcc. ### What’s next? Download a copy of BioFVM and try out the included examples! Visit BioFVM at MathCancer.org. 1. BioFVM announcement on this blog: [click here] 2. BioFVM on MathCancer.org: http://BioFVM.MathCancer.org 3. BioFVM on SourceForge: http://BioFVM.sf.net 4. BioFVM Method Paper in BioInformatics: http://dx.doi.org/10.1093/bioinformatics/btv730 5. BioFVM tutorials: [click here] Share this: ## Setting up a 64-bit gcc/OpenMP environment on Windows Note: 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 Windows users. OSX users should use this guide for Homebrew (preferred method) or this guide for MacPorts (much slower but reliable). 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 a 64-bit Windows port of gcc. These instructions should work for any modern Windows installation, say Windows 7 or above. This tutorial assumes you have a 64-bit CPU running on a 64-bit operating system. In the end result, you’ll have a compiler, key makefile capabilities, and a decent text editor. 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. ### What you’ll need: 1. MinGW-w64 compiler: This is a native port of the venerable gcc compiler for windows, with support for 64-bit executables. Download the latest installer (mingw-w64-install.exe) here. As of January 8, 2016, this installer will download gcc 5.3.0. 2. MSYS tools: This gets you some of the common command-line utilities from Linux, Unix, and BSD systems (make, touch, etc.). Download the latest installer (mingw-get-setup.exe) here. 3. Notepad++ text editor: This is a full-featured text editor for Windows, including syntax highlighting, easy commenting, tabbed editing, etc. Download the latest version here. As of January 8, 2016, this will download Version 6.8.8. ### Main steps: #### 1) Install the compiler Run the mingw-w64-install.exe. When asked, select: Version: 5.3.0 (or later) Architecture: x86_64 Threads: win32 (While I have tested posix, the native threading should be faster.) Exception: seh (While sjlj works and should be more compatible with various GNU tools, the native SEH should be faster.) Build version: 0 (or the default) Leave the destination folder wherever the installer wants to put it. In my case, it is: c:\Program Files\mingw-w64\x86_64-5.3.0-win32-seh-rt_v4_rev0  Let MinGW-w64 download and install whatever it needs. #### 2) Install the MSYS tools Run mingw-get-setup.exe. Leave the default installation directory and any other defaults on the initial screen. Click “continue” and let it download what it needs to get started. (a bunch of XML files, for the most part.) Click “Continue” when it’s done. This will open up a new package manager. Our goal here is just to grab MSYS, rather than the full (but merely 32-bit) compiler. Scroll through and select (“mark for installation”) the following: • mingw-developer-toolkit. (Note: This should automatically select msys-base.) Next, click “Apply Changes” in the “Installation” menu. When prompted, click “Apply.” Let the package manager download and install what it needs (on the order of 95 packages). Go ahead and close things once the installation is done, including the package manager. #### 3) Install the text editor Run the Notepad++ installer. You can stick with the defaults. #### 4) Add these tools to your system path Adding the compiler, text editor, and MSYS tools to your path helps you to run make files from the compiler. First, get the path of your compiler: 1. Open Windows Explorer ( [Windows]+E ) 2. Browse through C:\, then Program Files, mingw-w64, then a messy path name corresponding to our installation choices (in my case, x86_64-5.3.0-win32-seh_rt_v4-rev0), then mingw64, and finally bin. 3. Open notepad ([Windows]+R , notepad), 4. Go to the address bar of Explorer and copy ([Control]+C) the full path of the directory you just browsed to. 5. Paste it into notepad ([Control]+V), preceded and followed by a semicolon, and a final slash if it isn’t there. In my case: ;c:\Program Files\mingw-w64\x86_64-5.3.0-win32-seh-rt_v4_rev0\mingw64\bin\;  Then, get the path to Notepad++. 1. Go back to Explorer, and choose “This PC” or “My Computer” from the left column. 2. Browse through C:\, then Program Files (x86), then Notepad++. 3. Copy the path from the Explorer address bar. 4. Paste this path just after the final semicolon in notepad, followed by a backslash and a semicolon. In my case: ;c:\Program Files\mingw-w64\x86_64-5.3.0-win32-seh-rt_v4_rev0\mingw64\bin\;c:\Program Files (x86)\Notepad++\;  Then, get the path for MSYS: 1. Go back to Explorer, and choose “This PC” or “My Computer” from the left column. 2. Browse through C:\, then MinGW, then msys, then 1.0, and finally bin. 3. Copy the path from the Explorer address bar. 4. Paste this path just after the final semicolon in notepad, followed by a backslash and a semicolon. In my case: ;c:\Program Files\mingw-w64\x86_64-5.3.0-win32-seh-rt_v4_rev0\mingw64\bin\;c:\Program Files (x86)\Notepad++\;C:\MinGW\msys\1.0\bin\;  Lastly, add these paths to the system path: 1. Go the Start Menu, the right-click “This PC” or “My Computer”, and chose “Properties.” 2. Click on “Advanced system settings” 3. Click on “Environment Variables…” in the “Advanced” tab 4. Scroll through the “System Variables” below until you find Path. 5. Select “Path”, then click “Edit…” 6. At the very end of “Variable Value”, paste what you made in Notepad in the prior steps. Make sure to paste at the end of the existing value, rather than overwriting it! 7. Hit OK, OK, and OK to completely exit the “Advanced system settings.” #### 5) Test the compiler ##### Write a basic parallelized program: Enter a command prompt ( [windows]+R, then cmd ). 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 notepad++ 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 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, open up a new tab in Notepad++, and save it as “Makefile”. Add the following contents: CC := g++ ARCH := core2 # Replace this 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:

### What’s next?

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

2. BioFVM on MathCancer.org: http://BioFVM.MathCancer.org
3. BioFVM on SourceForge: http://BioFVM.sf.net
4. BioFVM Method Paper in BioInformatics: http://dx.doi.org/10.1093/bioinformatics/btv730

## BioFVM: an efficient, parallelized diffusive transport solver for 3-D biological simulations

I’m very excited to announce that our 3-D diffusion solver has been accepted for publication and is now online at Bioinformatics. Click here to check out the open access preprint!

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 (free; open access)

BioFVM (stands for “Finite Volume Method for biological problems) is an open source package to solve for 3-D diffusion of several substrates with desktop workstations, single supercomputer nodes, or even laptops (for smaller problems). We built it from the ground up for biological problems, with optimizations in C++ and OpenMP to take advantage of all those cores on your CPU. The code is available at SourceForge and BioFVM.MathCancer.org.

The main idea here is to make it easier to simulate big, cool problems in 3-D multicellular biology. We’ll take care of secretion, diffusion, and uptake of things like oxygen, glucose, metabolic waste products, signaling factors, and drugs, so you can focus on the rest of your model.

### Design philosophy and main capabilities

Solving diffusion equations efficiently and accurately is hard, especially in 3D. Almost all biological simulations deal with this, many by using explicit finite differences (easy to code and accurate, but very slow!) or implicit methods like ADI (accurate and relatively fast, but difficult to code with complex linking to libraries). While real biological systems often depend upon many diffusing things (lots of signaling factors for cell-cell communication, growth substrates, drugs, etc.), most solvers only scale well to simulating two or three. We solve a system of PDEs of the following form:

$\frac{\partial \vec{\rho}}{\partial t} = \overbrace{ \vec{D} \nabla^2 \vec{\rho} }^\textrm{diffusion} – \overbrace{ \vec{\lambda} \vec{\rho} }^\textrm{decay} + \overbrace{ \vec{S} \left( \vec{\rho}^* – \vec{\rho} \right) }^{\textrm{bulk source}} – \overbrace{ \vec{U} \vec{\rho} }^{\textrm{bulk uptake}} + \overbrace{\sum_{\textrm{cells } k} 1_k(\vec{x}) \left[ \vec{S}_k \left( \vec{\rho}^*_k – \vec{\rho} \right) – \vec{U}_k \vec{\rho} \right] }^\textrm{sources and sinks by cells}$
Above, all vector-vector products are term-by-term.

#### Solving for many diffusing substrates

We set out to write a package that could simulate many diffusing substrates using algorithms that were fast but simple enough to optimize. To do this, we wrote the entire solver to work on vectors of substrates, rather than on individual PDEs. In performance testing, we found that simulating 10 diffusing things only takes about 2.6 times longer than simulating one. (In traditional codes, simulating ten things takes ten times as long as simulating one.) We tried our hardest to break the code in our testing, but we failed. We simulated all the way from 1 diffusing substrate up to 128 without any problems. Adding new substrates increases the computational cost linearly.

#### Combining simple but tailored solvers

We used an approach called operator splitting: breaking a complicated PDE into a series of simpler PDEs and ODEs, which can be solved one at a time with implicit methods.  This allowed us to write a very fast diffusion/decay solver, a bulk supply/uptake solver, and a cell-based secretion/uptake solver. Each of these individual solvers was individually optimized. Theory tells us that if each individual solver is first-order accurate in time and stable, then the overall approach is first-order accurate in time and stable.

The beauty of the approach is that each solver can individually be improved over time. For example, in BioFVM 1.0.2, we doubled the performance of the cell-based secretion/uptake solver. The operator splitting approach also lets us add new terms to the “main” PDE by writing new solvers, rather than rewriting a large, monolithic solver. We will take advantage of this to add advective terms (critical for interstitial flow) in future releases.

#### Optimizing the diffusion solver for large 3-D domains

For the first main release of BioFVM, we restricted ourselves to Cartesian meshes, which allowed us to write very tailored mesh data structures and diffusion solvers. (Note: the finite volume method reduces to finite differences on Cartesian meshes with trivial Neumann boundary conditions.) We intend to work on more general Voronoi meshes in a future release. (This will be particularly helpful for sources/sinks along blood vessels.)

By using constant diffusion and decay coefficients, we were able to write very fast solvers for Cartesian meshes. We use the locally one-dimensional (LOD) method–a specialized form of operator splitting–to break the 3-D diffusion problem into a series of 1-D diffusion problems. For each (y,z) in our mesh, we have a 1-D diffusion problem along x. This yields a tridiagonal linear system which we can solve efficiently with the Thomas algorithm. Moreover, because the forward-sweep steps only depend upon the coefficient matrix (which is unchanging over time), we can pre-compute and store the results in memory for all the x-diffusion problems. In fact, the structure of the matrix allows us to pre-compute part of the back-substitution steps as well. Same for y- and z-diffusion. This gives a big speedup.

Next, we can use all those CPU cores to speed up our work. While the back-substitution steps of the Thomas algorithm can’t be easily parallelized (it’s a serial operation), we can solve many x-diffusion problems at the same time, using independent copies (instances) of the Thomas solver. So, we break up all the x-diffusion problems up across a big OpenMP loop, and repeat for y– and z-diffusion.

Lastly, we used overloaded +=, axpy and similar operations on the vector of substrates, to avoid unnecessary (and very expensive) memory allocation and copy operations wherever we could. This was a really fun code to write!

The work seems to have payed off: we have found that solving on 1 million voxel meshes (about 8 mm3 at 20 μm resolution) is easy even for laptops.

#### Simulating many cells

We tailored the solver to allow both lattice- and off-lattice cell sources and sinks. Desktop workstations should have no trouble with 1,000,000 cells secreting and uptaking a few substrates.

#### Simplifying the non-science

We worked to minimize external dependencies, because few things are more frustrating than tracking down a bunch of libraries that may not work together on your platform. The first release BioFVM only has one external dependency: pugixml (an XML parser). We didn’t link an entire linear algebra library just to get axpy and a Thomas solver–it wouldn’t have been optimized for our system anyway. We implemented what we needed of the freely available .mat file specification, rather than requiring a separate library for that. (We have used these matlab read/write routines in house for several years.)

Similarly, we stuck to a very simple mesh data structure so we wouldn’t have to maintain compatibility with general mesh libraries (which can tend to favor feature sets and generality over performance and simplicity).  Rather than use general-purpose ODE solvers (with yet more library dependencies, and more work for maintaining compatibility), we wrote simple solvers tailored specifically to our equations.

The upshot of this is that you don’t have to do anything fancy to replicate results with BioFVM. Just grab a copy of the source, drop it into your project directory, include it in your project (e.g., your makefile), and you’re good to go.

### All the juicy details

The Bioinformatics paper is just 2 pages long, using the standard “Applications Note” format. It’s a fantastic format for announcing and disseminating a piece of code, and we’re grateful to be published there. But you should pop open the supplementary materials, because all the fun mathematics are there:

• The full details of the numerical algorithm, including information on our optimizations.
• Convergence tests: For several examples, we showed:
• First-order convergence in time (with respect to Δt), and stability
• Second-order convergence in space (with respect to Δx)
• Accuracy tests: For each convergence test, we looked at how small Δt has to be to ensure 5% relative accuracy at Δx = 20 μm resolution. For oxygen-like problems with cell-based sources and sinks, Δt = 0.01 min will do the trick. This is about 15 times larger than the stability-restricted time step for explicit methods.
• Performance tests:
• Computational cost (wall time to simulate a fixed problem on a fixed domain size with fixed time/spatial resolution) increases linearly with the number of substrates. 5-10 substrates are very feasible on desktop workstations.
• Computational cost increases linearly with the number of voxels
• Computational cost increases linearly in the number of cell-based source/sinks

And of course because this code is open sourced, you can dig through the implementation details all you like! (And improvements are welcome!)

### What’s next?

• As MultiCellDS (multicellular data standard) matures, we will implement read/write support for  <microenvironment> data in digital snapshots.
• We have a few ideas to improve the speed of the cell-based sources and sinks. In particular, switching to a higher-order accurate solver may allow larger time step sizes, so long as the method is still stable. For the specific form of the sources/sinks, the trapezoid rule could work well here.
• I’d like to allow a spatially-varying diffusion coefficient. We could probably do this (at very great memory cost) by writing separate Thomas solvers for each strip in x, y, and z, or by giving up the pre-computation part of the optimization. I’m still mulling this one over.
• I’d also like to implement non-Cartesian meshes. The data structure isn’t a big deal, but we lose the LOD optimization and Thomas solvers. In this case, we’d either use explicit methods (very slow!), use an iterative matrix solver (trickier to parallelize nicely, except in matrix-vector multiplication operations), or start with quasi-steady problems that let us use Gauss-Seidel iterative type methods, like this old paper.
• Since advective flow (particularly interstitial flow) is so important for many problems, I’d like to add an advective solver. This will require some sort of upwinding to maintain stability.
• At some point, we’d like to port this to GPUs. However, I don’t currently have time / resources to maintain a separate CUDA or OpenCL branch. (Perhaps this will be an excuse to learn Julia on GPUs.)

Well, we hope you find BioFVM useful. If you give it a shot, I’d love to hear back from you!

Very best — Paul