In order to capture the behaviour of cells in a simulation, the mechanisms of the cell cycle as described in the previous sections were implemented in a cell-class within this thesis. The basis for this is the agent-based representation of cells (see section 2.2) within an object-oriented programming language such as C++. This section will introduce the chosen implementation.
Insertion of cells at simulation start
Basic information about each cell is stored within its state vector. This vector of variables contains information such as the current cycle phase and the remaining phase length (see also section 8.1 on page 73). Member functions of the cell handle the regulation of the cell cycle progress. phase length At the initialisation of a simulation run, a predetermined number of cells is inserted. Inserted cells can be marked as malign or benign and inser-tion is done using a Monte Carlo algorithm to obtain a chosen configurainser-tion (spherical e.g. malign cells surrounded by benign ones, mixed or pointwise). Upon instantiation cell agents are assigned a random phase (G1,S orG2), a random phaselength is drawn from a given distribution (e.g. Gaussian around a typical phaselength with given width) and an according size is calculated (initial size plus already performed growth). Simulations starting with few malignant cells will show an oscillating behaviour with respect to the overall cell cycle phases of cells since progression will be synchronised to large extents depending on the chosen parameters for the phaselength distribution. However, this oscil-lation is damped due to the partial randomisation of phaselenghts and other effects such as cell cycle arrest inG0. Synchronisation is completely lost after a few cell cycles (as can be seen in figure 9.9).
Important cell cycle control functions The cycle of each cell agent is controlled by the
void check_phase ( double& dtass, Triangulator<double,Cell>& tri )
function. It handles all of the cell’s internal event triggering, growth and shrinking. Fur-thermore, it checks if the available nutrient concentration has fallen below the critical limit for cell necrosis. The remaining phaselength of a cell agent is decreased according to the
5 Cell cycle and metabolism
N G0 M
anytime
G2
S G1
Cycle checkpoints
Cycle direction
Figure 5.7: Sketch of the cell cycle as implemented in the simulation.
passed timestepdtass. If the remaining phaselength reaches zero,check_phaseinitiates a phase transition handled by thevoid update_phase-function.
update_phase includes all the cell cycle checkpoints described in the previous sec-tion. At the G1/S-transition the nutrient availability is checked on the diffusion grids Soluble<double> DIFF_GLUCOSE and Soluble<double> DIFF_OXYGEN. The cell can be sent into quiescence if a minimum nutrient threshold is not reached. Apart from the nu-trient check, at the restriction point update_phase performs a check of the pressure on the cellular agent mediated by its neighbours, via the check_mechanical_inhibition function. If this pressure exceeds the threshold, cell cycle progression is delayed and the cell is sent into quiescence. All checkpoints can be disabled if i.e. entry into quiescence should only be possible by high pressure on a cell and not mediated by nutrient conditions.
If all checks done byupdate_phaseare positive, the cell is advanced into the next cycle phase, changing its internal variables and drawing a new phaselength from the distribution via the set_phaselength function. So far no external growth signals are processed and the molecular network around the CDK4/6 inhibition is not taken into account directly for reasons of simplicity. However modelling the cyclin interaction is possible once a molecular network is coupled to theupdate_phasefunction.
Sphase is characterised by the replication of the DNA. This is mainly an internal process of the cell without impact on the tissue. DNA replication can be marked as incorrect with a certain probability, forcing the cell to prolongS phase or go into quiescence until repairs were successful. As DNA replication is a highly accurate process this mechanism is not used by default.
In G2 the cell grows to reach its final size for mitosis. TheG2/M-transition is accom-panied by a DNA damage checkpoint. If the cell is marked as damaged (e.g. by the irradiate function) repairs are initiated that succeed with a defined probability. Upon
5 Cell cycle and metabolism
success of the DNA repair the cell enters mitosis. If repairs are unsuccessful, the cell stays in G2 phase attempting further repairs. Upon failure of a determined number of repair attempts, the cell is sent into senescence or apoptosis if possible. If the cells DNA was not damaged at the G2/M checkpoint the cell enters mitosis immediately. Until now Cdc25 dynamics and the Mitosis Promoting Factor, MPF are not taken into account directly.
This is feasible in the same way as for the G1/S checkpoint, through the coupling of a molecular network to the cell. The MPF activates the CDK in response to environmental conditions being right for the cell, and allows the cell to begin DNA replication.
Upon initiation of mitosis the update_phase function calls the proliferate function to handle cytokinesis. Karyokinesis is taken to be an fully internal process of the cell.
However, an anaphase checkpoint can be inserted, halting the progression through mitosis in the beginning due to an attachment error of the chromosomes with a defined probability.
Quiescence of a cell inG0 is associated with a predetermined phaselength decreased by the check_phase function. At zero phaselength the update_phase functions checks for nutrient availability and cell pressure. If both checks are positive the cell is re-entered into the cycle exactly where it was sent intoG0, but if one of the checks is negative quiescence is prolonged. Cells which have been flagged as senescent due to irreparable DNA-damage or terminal differentiation stay inG0 for unlimited time.
Necrosis is determined by a given phaselength during which the cell will shrink con-stantly (handled by thecheck_phasefunction) and eventually be deleted from the simu-lation. Toxic or hazardous products of necrosis can be modelled on the
Soluble<double> DIFF_NECROSIS diffusion grid such as lysic enzymes. If their concen-tration is above a certain concenconcen-tration threshold, neighbouring cells can be affected.
Sample cycle
always nutrients checked and necrosis initiated if below necrosis threshold G1 phase growth with constant rate
G1/S checkpoint nutrient check for G0 threshold; pressure check G2 phase growth with constant rate
G2/M checkpoint DNA damage check; stall and repair or send into senescence if all repair tries failed
Necrosis and apoptosis
Cells undergoing necrosis within the tumour volume are removed from the cell cycle.
They are shrinked to a predefined end radius over the duration of the necrosis phaselength τN. After the expiration of their phaselenght they are removed from the simulation. It is assumed, that the necrotic debris of cells is removed through the decomposition by immuno-agents and diffusion. Effects of toxic substances or other chemicals produced
5 Cell cycle and metabolism
by cell necrosis, which might affect other cells, can be examined through modelling of their diffusing according to section 5.4. Due to the fact that the apoptotic pathway is commonly disabled in cancerous cells, apoptosis is generally not used within the simulation.
Implementation is done similar to necrosis with the difference of a faster cell removal (through the activity of phagocytes) and the apoptotic cell acting as a minor nutrient source for the time of its degradation.