Biophysics Tools and Techniques for the Physics of Life (Mark C. Leake) (1)

Chapter 8 – Theoretical Biophysics 381

as elements in a FEA. Typically, the mechanical properties of each cell are modeled as single

scaler parameters such as stiffness and damping factor, assumed to act homogeneously across

the extent of the cell. Cell–cell boundaries are allowed to be dynamic in the model, within

certain size and shape constraints.

An example of this is the cancer, heart, and soft tissue environment (Chaste) simulation

toolkit developed primarily by research teams in the University of Oxford. Chaste is aimed

at multiple length-scale tissue simulations, optimized to cover a range of real biomechan

ical tissue-level processes as well as implementing effects of electrophysiology feedback in,

for example, regulating the behavior of heart muscle contraction. The simulation packages

tissue- and cell-level electrophysiology, discrete tissue modeling, and soft tissue modeling.

A specific emphasis of Chaste tools includes continuum modeling of cardiac electrophysi

ology and of cell populations, in addition to tissue homeostasis and cancer formation.

The electrical properties of the heart muscle can be modeled by a set of coupled PDEs

that represents the tissue as two distinct continua either inside the cells or outside, which

are interfaced via the cell membrane, which can be represented through a set of complex,

nonlinear ODEs that describe transmembrane ionic current (see Chapter 6). Under cer

tain conditions, the effect of the extracellular environment can be neglected and the model

reduced to a single PDE. Simulation time scales of 10–100 ms relevant to electrical excitation

of heart muscle takes typically a few tens of minutes of a computation time on a typical 32

CPU core supercomputer. Cell-population-based simulations have focused to date on the

time-resolved mechanisms of bowel cancers. These include realistic features to the geometry

of the surface inside the bowel, for example, specialized crypt regions.

Chaste and other biomechanical simulation models also operate at smaller length scale

effects of single cells and subcellular structures and include both deterministic and sto

chastic models of varying degrees of complexity, for example, incorporating the transport of

nutrient and signaling molecules. Each cell can be coarse-grained into a reduced number of

components, which appear to have largely independent mechanical properties. For example,

the nucleus can be modeled as having a distinct stiffness compared to the cytoplasm.

Similarly, the cell membrane and coupled cytoskeleton. Refinements to these models include

tensor approximations to stiffness values (i.e., implementing a directional dependence on the

stiffness parameters).

An interesting analytical approach for modeling the dynamic localization of cells in

developing tissues is to apply arguments from foam physics. This has been applied using

numerical simulations to the data from complex experimental tissue systems, such as growth

regulation of the wings of the fruit fly and fin regeneration in zebrafish. These systems are

good models since they have been characterized extensively through genetics and biochem

ical techniques, and thus many of the key molecular components are well known. However,

standard biochemical and genetics signal regulation do not explain the observed level of

biological regulation that results in the ultimate patterning of cells in these tissues. What

does show promise, however, is to use mechanical modeling, which in essence treats the

juxtaposed cells as bubbles in a tightly packed foam.

The key point here is that the geometry of bubbles in foam has many similarities with those

of cells in developing tissues, with examples including both ordered and disordered states,

but also in the similarities between the topology, size, and shape of foam bubbles compared

to tissue cells. A valuable model for the nearest-neighbor interactions between foam bubbles

is the 2D granocentric model (see Miklius and Hilgenfelt, 2012); this is a packing model based

purely on trigonometric arguments in which the mean free space around any particle (the

problem applied to the packing of grain, hence the name) in the system is minimized. Here,

the optimum packing density, even with a distribution of different cell sizes, results in each

cell in a 2D array having a mean of six neighbors (see Figure 8.11a). The contact angle Φ

between any two tightly packed cells can be calculated from trigonometry (in reference to

Figure 8.11a) as

(8.127)

Φ =

(

)

(

)

−

sin