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Computational Modelling of Cancer Development and Growth: Modelling at Multiple Scales and Multiscale Modelling

  • Special Issue : Mathematical Oncology
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Abstract

In this paper, we present two mathematical models related to different aspects and scales of cancer growth. The first model is a stochastic spatiotemporal model of both a synthetic gene regulatory network (the example of a three-gene repressilator is given) and an actual gene regulatory network, the NF-\(\upkappa \)B pathway. The second model is a force-based individual-based model of the development of a solid avascular tumour with specific application to tumour cords, i.e. a mass of cancer cells growing around a central blood vessel. In each case, we compare our computational simulation results with experimental data. In the final discussion section, we outline how to take the work forward through the development of a multiscale model focussed at the cell level. This would incorporate key intracellular signalling pathways associated with cancer within each cell (e.g. p53–Mdm2, NF-\(\upkappa \)B) and through the use of high-performance computing be capable of simulating up to \(10^9\) cells, i.e. the tissue scale. In this way, mathematical models at multiple scales would be combined to formulate a multiscale computational model.

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Fig. 1

Reprinted from Hanahan and Weinberg (2011), Copyright (2011), with permission from Elsevier

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From Nelson et al. (2004). Reprinted with permission from AAAS

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From Hlatky et al. (2002), by permission of Oxford University Press

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Acknowledgements

ZS acknowledges the support of the National Science Centre Poland Grant 2011/01/D/ST1/04133, National Science Centre Poland Grant 2014/15/B/ST6/05082 and The National Centre for Research and Development Grant STRATEGMED1/233224/10/NCBR/2014. MAJC and CKM gratefully acknowledge support of EPSRC Grant No. EP/N014642/1 (EPSRC Centre for Multiscale Soft Tissue Mechanics—With Application to Heart & Cancer). EM was supported by an EASTBIO Ph.D. Fellowship. The authors thank Bartosz Borucki from ICM for his help in with the VisNow medical imaging software.

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Appendices

Appendix 1: The Reaction–Diffusion Master Equation

Here we describe the formulation of the spatial stochastic model for intracellular GRN dynamics. We describe the computational domain and how reaction and diffusion events are incorporated into the reaction-diffusion master equation (RDME). We also provide some notes on how simulations are produced. For a more detailed description, see the supplementary material of Sturrock et al. (2013).

1.1 The Computational Domain

The computational domain (see Fig. 2, for example) is set up using COMSOL and a mesh is imposed. In general, the domain \(\varOmega \) is meshed into V tetrahedra-shaped subvolumes, voxels, \(\varOmega _k, k \in \{1,\ldots ,V\}\) such that,

$$\begin{aligned} \varOmega = \bigcup ^V_{k=1}{\varOmega _k}, \quad \text {and} \quad \varOmega _i \cap \varOmega _j = \emptyset , \forall i \ne j,\quad i,j \in \{1,\ldots ,V\}. \end{aligned}$$

At any given time, the state of the system is described by the number of each chemical species within the domain. Changes to the state will either be by the chemical reactions at the voxel level or the movement (diffusion jumps) of a molecule between neighbouring voxels.

1.2 Chemical Reactions

We consider reactions that occur due to molecular contact. We assume that the species of our system, within each subvolume, are uniformly distributed and in thermal equilibrium, such that the motion of each molecule is random. We consider the probability of a collision occurring between two reactant molecules. The likelihood of a reaction occurring, changing the state of the system from x to \(x+N_r\), is determined by its reaction rate, described by the reaction propensity function \(\omega _r(x)\). As such, reactions can be described by

$$\begin{aligned} {x} \xrightarrow {\omega _r(x)} {x} + {N}_{{r}}, \end{aligned}$$

where \(N_r \in \mathbb {Z}^S\) is the transition step, defined by the rth column of the stoichiometric matrix M and \(\omega _r(x)\) is the probability that the reaction occurs during a infinitesimal time interval, i.e.

$$\begin{aligned} \omega _r(x) = \lim _{\mathrm{d}t \rightarrow 0} \dfrac{P(x+N_r, t +\mathrm{d}t)-P(x, t)}{\mathrm{d}t} \end{aligned}$$

1.3 Molecular Diffusion

The movement of a chemical species \(S_l\) from a voxel \(\psi _i\) to a randomly selected adjacent voxel \(\psi _j\) describes the molecular diffusion and is modelled as a first-order event. As such, we treat the diffusive process in a similar way to a reactive process and consider the probability of a transition taking place, i.e. the probability for one of the lth species to make a jump from the ith subvolume to an adjacent jth subvolume. Hence, we consider the following,

$$\begin{aligned} {S}_{{li}} \xrightarrow {{q}_{{lij}} {x}_{{li}}} {S}_{{lj}}, \end{aligned}$$

where \(x_{{li}}\) is number of species l located in voxel i and \(q_{lij}\) is the diffusion rate constant that depends on the macroscopic diffusion coefficient of species l (\(D_l\)) and the mesh of the domain, specifically the shape and size of voxels \(\psi _i\) and \(\psi _j\). Note each \(q_{lij}\) is only non-zero for connected mesh elements and of the types of species we model only mRNAs and proteins diffuse, the free and occupied promoters remain permanently within the voxel assigned as the promoter site.

1.4 Solving the System

The temporal evolution of the probability distribution of each state in the state space is governed by the RDME. We complete the model set-up with zero-flux boundary conditions at the cell membrane, while we impose continuity of flux on the nuclear membrane. For initialisation, we suppose that there is only a single free promoter within each promoter site. All simulations found in this paper are produced using the URDME (Unstructured Reaction-Diffusion Master Equation) software framework (Drawert et al. 2012), which implements the next subvolume method (NSM) (Gibson and Bruck 2000); the NSM being far more computationally efficient than the classical SSA for a 3D domain such as ours. URDME uses unstructured tetrahedral and triangular meshes (such as shown in Fig. 2) for which diffusion rate constants \(q_{lij}\) are automatically computed (Engblom et al. 2009; Drawert et al. 2012).

Appendix 2: NF-\(\upkappa \)B Reactions

Here we give the reactions for the NF-\(\upkappa \)B pathway (Tables 6, 7, 8).

Table 6 Cytoplasmic reactions
Table 7 Reactions at gene sites
Table 8 Global reactions

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Szymańska, Z., Cytowski, M., Mitchell, E. et al. Computational Modelling of Cancer Development and Growth: Modelling at Multiple Scales and Multiscale Modelling. Bull Math Biol 80, 1366–1403 (2018). https://doi.org/10.1007/s11538-017-0292-3

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