Multiscale Modeling in Biology

Dr. Mark Alber, Department of Mathematics, Center for the Study of Biocomplexity, University of Notre Dame

Multiscale modeling approach typical of systems biology tends to mix continuous, discrete, deterministic, and probabilistic submodels.
To prevent the loss of blood following a break in blood vessels, components in blood and the vessel wall interact rapidly to form a clot to limit hemorrhage. In this talk we will describe a multiscale hybrid model of thrombus formation consisting of components for modeling viscous, incompressible blood plasma; coagulation pathway; quiescent and activated platelets; blood cells; activating chemicals; fibrinogen; the vessel walls and their interactions. At macro scale blood flow is described by the incompressible Navier-Stokes equations and is numerically solved using the projection method. At micro scale, cell movement, cell-cell adhesion, cell-flow and cell-vessel wall interactions are described through an extended stochastic discrete Cellular Potts Model (CPM). Simulation results show development of an inhomogeneous internal structure of the clot confirmed by the preliminary experimental data. It is also demonstrated that dependence of the clot size on the blood flow rate in simulations is close to the one observed experimentally.
In the second half of the talk a continuous limit will be discussed of a two-dimensional stochastic CPM describing cells moving in a medium and reacting to each other through direct contact, cell-cell adhesion, and long range chemotaxis. Contrary to classical Keller-Segel model, solutions of the obtained equation do not collapse in finite time and can be used even when relative volume occupied by cells is quite large. A very good agreement was demonstrated between CPM Monte Carlo simulations and numerical solutions of the obtained macroscopic nonlinear diffusion equation. Combination of microscopic and macroscopic models was used to simulate growth of structures similar to early vascular networks.

Xu, Z., Chen, N., Kamocka, M.M., Rosen, E.D., and M.S. Alber [2008], Multiscale Model of Thrombus Development, Journal of the Royal Society Interface 5 705-722.

Xu, Z., Chen, N., Shadden, S., Marsden, J.E., Kamocka, M.M., Rosen, E.D., and M.S. Alber, Study of Blood Flow Impact on Growth of Thrombi Using a Multiscale Model,
Soft Matter DOI:10.1039/b300001a (to appear).

Alber, M., Chen, N., Lushnikov, P., and S. Newman [2007], Continuous macroscopic limit of a discrete stochastic model for interaction of living cells, Physical Review Letters 99 168102.

Lushnikov, P.P., Chen, N., and M.S. Alber, Macroscopic dynamics of biological cells interacting via chemotaxis and direct contact, Physical Review E (to appear).