BB16: A framework for hybrid discrete/continuous multiscale modelling of biological tissues

Project completed November 9, 2012

Background

Mechanisms involved in biological tissue function operate over a large spectrum of spatial and temporal scales. Although there have been significant advances in computational capabilities, simulating organs on the molecular scale over a period of several years is currently unfeasible.

In order to address this issue, a variety of multiscale approaches have been developed to model biological tissues. This led to the emergence of two paradigms: discrete (cellular) and continuum (tissular) representations. These two classes of models have their respective strengths and weaknesses. Cellular-scale descriptions can incorporate a substantial amount of information, which may be particularly relevant to medical applications. However, they are computationally intensive. Continuum-based approaches, on the other hand, are less demanding computationally, but they describe only an average behaviour of the tissue and have domains of validity that can be difficult to determine.

Researchers at the Oxford Centre for Collaborative Applied Mathematics (OCCAM) have proposed hybrid frameworks that combine these two representations, with a particular focus on tissue mechanics. Such frameworks rely on the idea that, in many cases, a cellular-scale description is needed only in a limited region of the tissue where processes vary over short spatial or temporal scales. In the rest of the tissue, where characteristic times and lengths are significantly larger, continuum descriptions may be used to reduce computation time.

Techniques and Challenges

The challenge lies in establishing relationships between the discrete and continuum frameworks. Tissular models are often expressed as systems of partial differential equations (PDEs), while cellular models are formulated in a discrete way as, for example, cellular automata, Potts or particle-based models.

In the field of crystal mechanics, the discrete-to-continuum connection is often made by assuming that local atom displacements can be mapped homogeneously from the deformation gradient. This assumption is known as the Cauchy–Born rule (CBR - see Figure 1). Although the CBR does not hold exactly for disordered systems, it may still be used as a leading order approximation for analytic calculations of strain energies.

The researchers numerically investigated the applicability of the CBR to 2D cellular-scale models by assessing the mechanical behaviour of model biological tissues, including crystalline (honeycomb, such as in Figure 2) and non-crystalline reference states, using the cancer, heart and soft tissue environment (Chaste, an open-source library of C++ code developed at the Department of Computer Science, University of Oxford). The numerical procedure applies an affine deformation to the boundary cells and computes the quasi-static position of internal cells. The position of internal cells was compared with the prediction of the CBR and an average deviation was calculated in the strain domain.

Results

For one class of cellular model, termed centre-based (see Figure 2), the researchers showed that the CBR holds exactly when the deformation gradient is relatively small and the reference stress-free configuration is defined by a honeycomb lattice. They further demonstrated that the CBR may be used approximately when the reference state is perturbed from the honeycomb configuration.

By contrast, for another class of cellular model, termed vertex-based (see Figure 2), a similar analysis revealed that the CBR does not provide a good representation of the tissue mechanics, even when the reference configuration is defined by a honeycomb lattice. These results have important implications for concurrent discrete/continuous modelling, adaptation of atom-to-continuum (AtC) techniques to biological tissues.

The Future

Future work will focus on comparison with experimental data on tissue deformation, notably developing theories that apply to non-affine situations, exploring the influence of the reference state upon the macroscale mechanical properties of the tissue in more detail (in particular the impact of pre-stress), adapting the CBR to account for homogeneous cellular growth, implementing hybrid concurrent models of biological tissues in Chaste, and adapting other AtC strategies to the biological context.

Related Publications

[12/52] Davit Y., Osborne J.M., Byrne H.M., Gavaghan D., Pitt-Francis J.: Validity of the Cauchy-Born rule applied to discrete cellular-scale models of biological tissues

[12/05] Davit Y., Byrne H.M., Osborne J.M., Pitt-Francis J., Gavaghan D., Quintard M.: Solute transport within porous biofilms: diffusion or dispersion?

[11/68] Davit Y., Quintard M.: Comment on “Frequency-dependent dispersion in porous media", Physical Review E, Vol 86 No. 1, 2012