Running Tumour Spheroid Simulations

This tutorial is automatically generated from TestRunningTumourSpheroidSimulationsTutorial.hpp at revision 1684df04656d. Note that the code is given in full at the bottom of the page.

An example showing how to run tumour spheroid simulations

Introduction

In this tutorial we show how Chaste can be used to simulate a growing cell monolayer culture or multicellular tumour spheroid. Like the crypt simulations, tumour spheroid simulations include cell-cycle models and force laws to determine how cells divide and move. In tumour spheroid simulations, however, these are also coupled to a system of partial differential equations (PDEs) that determine the concentration of specified nutrients (e.g. oxygen) throughout the cell population. Also, unlike in a crypt simulation (for example), the cell population may grow substantially as the simulation progresses.

In summary, the main difference between this tutorial and the other cell-based simulation tutorials is that a PDE is defined, which is used in the simulation.

The test

As in the other cell-based simulation tutorials, we begin by including the necessary header files. We have encountered some of these files already. Recall that often CheckpointArchiveTypes.hpp or CellBasedSimulationArchiver.hpp must be included as the first Chaste header.

#include <cxxtest/TestSuite.h>
#include "CheckpointArchiveTypes.hpp"
#include "AbstractCellBasedTestSuite.hpp"
#include "HoneycombMeshGenerator.hpp"
#include "GeneralisedLinearSpringForce.hpp"
#include "RandomNumberGenerator.hpp"
#include "SmartPointers.hpp"

The SimpleOxygenBasedCellCycleModel header file defines a cell-cycle model in which a cell’s rate of progress through G1 phase changes over time in a simple manner, according to the local oxygen concentration. We also include the WildTypeCellMutationState header file, which defines a wild type cell mutation state that we will use to construct cells. A cell mutation state is always required when constructing a cell, however in earlier simulation tutorials we used helper classes (CellsGenerator and CryptCellsGenerator) that allowed us to avoid having to construct cells directly.

#include "SimpleOxygenBasedCellCycleModel.hpp"
#include "WildTypeCellMutationState.hpp"
#include "StemCellProliferativeType.hpp"

The next three header files define: a PDE that describes how oxygen is transported through the domain via diffusion and is consumed by live cells; a constant-valued boundary condition to associate with the PDE; and a PDE modifier class, which is passed to the simulation object and handles the numerical solution of any PDEs.

#include "CellwiseSourceEllipticPde.hpp"
#include "ConstBoundaryCondition.hpp"
#include "EllipticGrowingDomainPdeModifier.hpp"

We use an OffLatticeSimulation.

#include "OffLatticeSimulation.hpp"

The header file PetscSetupAndFinalize.hpp must be included in all tests which use Petsc. This is a suite of data structures and routines that are used in the finite element PDE solvers, which is how we solve the oxygen transport PDE.

#include "PetscSetupAndFinalize.hpp"

Having included all the necessary header files, we proceed by defining the test class.

class TestRunningTumourSpheroidSimulationsTutorial : public AbstractCellBasedTestSuite
{
public:
    void TestSpheroidTutorial()
    {

This first line can be ignored: it is a macro which just says don’t run this test if in parallel.

        EXIT_IF_PARALLEL;

First we want to create a non-periodic ‘honeycomb’ mesh. We use the honeycomb mesh generator, as before, saying 10 cells wide and 10 cells high. Note that the thickness of the ghost nodes layer is 0, i.e. there are no ghost nodes. In contrast to the crypt simulation tutorial, here we call GetMesh() on the HoneycombMeshGenerator object to return the mesh, which is of type MutableMesh.

        HoneycombMeshGenerator generator(10, 10, 0);
        boost::shared_ptr<MutableMesh<2,2> > p_mesh = generator.GetMesh();

Next, we need to create some cells. Unlike in the the crypt simulation tutorial, we don’t just use a CellsGenerator class, but do it manually, in a loop. First, we define a std::vector of cell pointers.

        std::vector<CellPtr> cells;

This line defines a mutation state to be used for all cells, of type WildTypeCellMutationState (i.e. ‘healthy’):

        MAKE_PTR(WildTypeCellMutationState, p_state);
        MAKE_PTR(StemCellProliferativeType, p_stem_type);

Now we loop over the nodes…

        for (unsigned i=0; i<p_mesh->GetNumNodes(); i++)
        {

…then create a cell, giving it a SimpleOxygenBasedCellCycleModel. The spatial dimension (1, 2 or 3) needs to be set on the cell-cycle model before it is passed to the cell.

            SimpleOxygenBasedCellCycleModel* p_model = new SimpleOxygenBasedCellCycleModel;
            p_model->SetDimension(2);
            CellPtr p_cell(new Cell(p_state, p_model));
            p_cell->SetCellProliferativeType(p_stem_type);

We also alter the default cell-cycle times.

            p_model->SetStemCellG1Duration(8.0);
            p_model->SetTransitCellG1Duration(8.0);

We now define a random birth time, chosen from $[-T,0]$, where $T = t_1 + t_2$, where $t_1$ is a parameter representing the $G_1$ duration of a ‘stem’ cell, and $t_2$ is the basic $S+G_2+M$ phases duration…

            double birth_time = - RandomNumberGenerator::Instance()->ranf() *
                                 (  p_model->GetStemCellG1Duration()
                                  + p_model->GetSG2MDuration() );

…then we set the birth time and push the cell back into the vector of cells.

            p_cell->SetBirthTime(birth_time);
            cells.push_back(p_cell);
        }

Now that we have defined the cells, we can define the CellPopulation. We use a MeshBasedCellPopulation, rather than a MeshBasedCellPopulationWithGhostNodes since this particular cell population does not include any ghost nodes. The constructor takes in the mesh and the cells vector.

        MeshBasedCellPopulation<2> cell_population(*p_mesh, cells);

Next we instantiate an instance of the PDE class which we defined above. This will be passed into the OffLatticeSimulationWithPdes object. The CellwiseSourceEllipticPde is a PDE class which inherits from AbstractLinearEllipticPde and represents the PDE $u_{xx} + u_{yy} = k(x,y)u$, where $u(x,y)$ denotes the oxygen concentration at position $(x,y)$ and the function $k(x,y)$ specifies the rate of consumption by live cells there. Here $k(x,y)$ takes the value $-0.03$ (the coefficient below) if the cell located at $(x,y)$ is a live cell, and zero if the cell has died due to oxygen deprivation.

        MAKE_PTR_ARGS(CellwiseSourceEllipticPde<2>, p_pde, (cell_population, -0.03));

We also create a constant-valued boundary condition to associate with the PDE. This boundary condition object takes in a single argument in its constructor, the value at the boundary. We also introduce a boolean to specify whether this value is the flux at the boundary (a Neumann boundary condition) or the value of the state variable at the boundary (a Dirichlet boundary condition) below.

        MAKE_PTR_ARGS(ConstBoundaryCondition<2>, p_bc, (1.0));
        bool is_neumann_bc = false;

To pass the PDE to our simulator, it must first be encapsulated in a cell-based PDE modifier object, together with the boundary condition for the PDE. The latter is specified by the second and third arguments of the constructor below: the second argument defines the value of the boundary condition and the third argument defines whether it is of Neumann type (true) or Dirichlet type (false). Thus, in our case, we are a specifying a constant-value boundary condition. Note that we currently cannot impose more than one boundary condition for each PDE (so that e.g. we cannot impose a zero-flux boundary condition on some part of the boundary and a fixed-value boundary condition on the rest), although the boundary condition itself can be made spatially varying or time-dependent.

The PDE is tagged to show that the quantity to be solved for (the quantity of interest in the cells’ data is "oxygen").

The CellData class is used to store the value of the current nutrient concentration for each cell.

        MAKE_PTR_ARGS(EllipticGrowingDomainPdeModifier<2>, p_pde_modifier, (p_pde, p_bc, is_neumann_bc));
        p_pde_modifier->SetDependentVariableName("oxygen");

We are now in a position to construct an OffLatticeSimulationWithPdes object, using the cell population. We then pass the PDE modifier object to the simulation.

        OffLatticeSimulation<2> simulator(cell_population);
        simulator.AddSimulationModifier(p_pde_modifier);

We next set the output directory and end time.

        simulator.SetOutputDirectory("SpheroidTutorial");
        simulator.SetEndTime(1.0);

We must now create one or more force laws, which determine the mechanics of the cell population. As in the crypt simulation tutorial, we assume that a cell experiences a force from each neighbour that can be represented as a linear overdamped spring, so we use a GeneralisedLinearSpringForce object. Note that we have called the method SetCutOffLength on the GeneralisedLinearSpringForce before passing it to the simulator: this call modifies the force law so that two neighbouring cells do not impose a force on each other if they are located more than 3 units (=3 cell widths) away from each other. This modification is necessary when no ghost nodes are used, for example to avoid artificially large forces between cells that lie close together on the spheroid boundary.

        MAKE_PTR(GeneralisedLinearSpringForce<2>, p_linear_force);
        p_linear_force->SetCutOffLength(3);
        simulator.AddForce(p_linear_force);

We call Solve() on the simulator to run the simulation.

        simulator.Solve();
    }
};

To visualize the results, use Paraview.

Open SpheroidTutorial/results_from_time_0/pde_results_oxygen_..vtu.

Click the green Apply button.
Select oxygen from the Active Variable Controls (default is: Solid Color)
Select Surface With Edges from the Representation Toolbar (default is: Surface)
Press Play to visualize the simulation

Full code

#include <cxxtest/TestSuite.h>
#include "CheckpointArchiveTypes.hpp"
#include "AbstractCellBasedTestSuite.hpp"
#include "HoneycombMeshGenerator.hpp"
#include "GeneralisedLinearSpringForce.hpp"
#include "RandomNumberGenerator.hpp"
#include "SmartPointers.hpp"
#include "SimpleOxygenBasedCellCycleModel.hpp"
#include "WildTypeCellMutationState.hpp"
#include "StemCellProliferativeType.hpp"
#include "CellwiseSourceEllipticPde.hpp"
#include "ConstBoundaryCondition.hpp"
#include "EllipticGrowingDomainPdeModifier.hpp"

#include "OffLatticeSimulation.hpp"
#include "PetscSetupAndFinalize.hpp"

class TestRunningTumourSpheroidSimulationsTutorial : public AbstractCellBasedTestSuite
{
public:
    void TestSpheroidTutorial()
    {
        EXIT_IF_PARALLEL;

        HoneycombMeshGenerator generator(10, 10, 0);
        boost::shared_ptr<MutableMesh<2,2> > p_mesh = generator.GetMesh();

        std::vector<CellPtr> cells;

        MAKE_PTR(WildTypeCellMutationState, p_state);
        MAKE_PTR(StemCellProliferativeType, p_stem_type);

        for (unsigned i=0; i<p_mesh->GetNumNodes(); i++)
        {
            SimpleOxygenBasedCellCycleModel* p_model = new SimpleOxygenBasedCellCycleModel;
            p_model->SetDimension(2);
            CellPtr p_cell(new Cell(p_state, p_model));
            p_cell->SetCellProliferativeType(p_stem_type);

            p_model->SetStemCellG1Duration(8.0);
            p_model->SetTransitCellG1Duration(8.0);

            double birth_time = - RandomNumberGenerator::Instance()->ranf() *
                                 (  p_model->GetStemCellG1Duration()
                                  + p_model->GetSG2MDuration() );
            p_cell->SetBirthTime(birth_time);
            cells.push_back(p_cell);
        }

        MeshBasedCellPopulation<2> cell_population(*p_mesh, cells);

        MAKE_PTR_ARGS(CellwiseSourceEllipticPde<2>, p_pde, (cell_population, -0.03));

        MAKE_PTR_ARGS(ConstBoundaryCondition<2>, p_bc, (1.0));
        bool is_neumann_bc = false;

        MAKE_PTR_ARGS(EllipticGrowingDomainPdeModifier<2>, p_pde_modifier, (p_pde, p_bc, is_neumann_bc));
        p_pde_modifier->SetDependentVariableName("oxygen");

        OffLatticeSimulation<2> simulator(cell_population);
        simulator.AddSimulationModifier(p_pde_modifier);

        simulator.SetOutputDirectory("SpheroidTutorial");
        simulator.SetEndTime(1.0);

        MAKE_PTR(GeneralisedLinearSpringForce<2>, p_linear_force);
        p_linear_force->SetCutOffLength(3);
        simulator.AddForce(p_linear_force);

        simulator.Solve();
    }
};

Running Potts Based Simulations

Running Vertex Based Crypt Simulations

Running Tumour Spheroid Simulations

An example showing how to run tumour spheroid simulations#

Introduction#

The test#

Full code#

An example showing how to run tumour spheroid simulations

Introduction

The test

Full code