The construction, maintenance and disruption of tissues emerge from the interactions of cells with each other, the extracellular microenvironment that the cells create and their external boundary conditions. Our ability to make biomedically meaningful predictions at the organ or organism level is limited because of the difficulty of predicting the emergent properties of large ensembles of cells. A middle-out approach to model building starting from cell behaviors and combining subcellular molecular reaction kinetics models, the physical and mechanical behaviors of cells and the longer range effects of the extracellular environment, allows us to address such emergence. I will discuss CompuCell3D as a multi-scale, multi-cell modeling platform to study such emergent phenomena and to connect them to their physiological outcomes. I will illustrate two projects using CompuCell3D, the development and of blood vessels and its effect on the growth of a generic model solid tumor and Choroidal Neovascularization (CNV) in Age-Related Macular Degeneration (the most common cause of blindness among the elderly). Time permitting, I will also briefly discuss our proof-of-concept simulations of somatic evolution in solid tumors.
 

In 1982 Calabi proposed studying gradient flow of the L^2 norm
of the scalar curvature (now called Calabi flow) as a tool for finding
canonical metrics within a given Kahler class. The main motivating
conjecture behind this flow (due to Calabi-Chen) asserts the smooth long
time existence of this flow with arbitrary initial data. By exploiting
aspects of the Mabuchi-Semmes-Donaldson metric on the space of Kahler
metrics I will construct a kind of weak solution to this flow, known as a
minimizing movement, which exists for all time.

Finite-time Lyapunov exponents and vectors are used to define and diagnose boundary-layer type, two-timescale behavior and to determine the associated manifold structure in the flow. Two-timescale behavior is characterized by a slow-fast splitting of the tangent bundle for a state space region. The slow-fast splitting, defined by finite-time Lyapunov exponents and vectors, is interpreted in relation to the asymptotic theory of partially hyperbolic sets. The finite-time Lyapunov approach relies more heavily on the Lyapunov vectors due to their relatively fast convergence compared to that of the corresponding exponents. Examples of determining slow manifolds and solving Hamiltonian boundary-value problems associated with optimal control are described.