Network structures, such as polycrystals, foams, and biological tissues consist of space-filling polyhedral grains, bubbles, or cells, respectively. The irregularity of such physical polyhedra derives from the fact that their faces consist of mixed shapes, their edges are of varying length, and the positions of their vertices are arranged non-symmetrically. The geometric complexity of irregular polyhedra forming 3-d networks makes their analysis difficult. To help circumvent this difficulty, average N-hedra (ANHs) are proposed as a set of regular polyhedra consisting of N identical faces. ANHsonly a few of which are constructibleact as abstract proxies for each corresponding topological class of irregular network polyhedra with the same number of faces. This study provides a comparison of the intrinsic areas and volumes of ANHs with data estimated numerically using Brakkes Surface Evolver simulations for a range of constructible polyhedral cells. Evolver data show that for every topological class, ANHs always provide a sharp upper bound for the isoperimetric quotient, which is a measure of the inverse energy cost of the cell. Of special interest also is demonstrating that the critical ANH, which has both zero mean and Gaussian curvature, satisfies Kusners bound for the average number of faces in a minimally partitioned network. In Euclidean 3-space the requisite number of faces is satisfied exactly by the critical ANH. The critical ANH, therefore, statistically represents the abstract unit cell that solves the so-called Kelvin Problem for a space-filling 3-d network exhibiting the minimum partition energy or surface area. This limit remains of particular interest in the case of annealed polycrystals and dry foams, as it establishes the lower bound of the energy cost of cellular random structures with a given metric gauge.

Based on Griffith's criterion for crack growth,
a method was recently proposed for determining
crack paths by taking continuous-time limits
of discrete-time variational problems. This has
been successfully carried out, but an important
difference remains between these solutions and
Griffith's model. I will explain the method and
the main issues in its implementation, and
then describe the remaining gap and some
efforts to remove it.

We will compute homogenized limits for the diffusion
of second messengers in the Ros Outer Segments (ROS)
of vertebrates, in visual transduction, when spike--like
incisures are presents in the receptors discs. The layered
structure of the cytoplasm and the presence of incisures,
present topological limitations to compactness. We will
present an homogenization limiting procedure based on
extension of functions with concave modulus of continuity.

We discuss also the physiological significance of the incisures
and present a spectrum of numerical simulations.

Normal intravascular blood clotting (hemostasis) and its pathological
counterpart, thrombosis, occur under flow and this can profoundly
influence the progress of clot formation. This talk will focus on two
different aspects of our efforts to model and probe the interactions
of flow and clotting. One involves the biochemistry of the
coagulation enzyme network and how the behavior of this system is
affected by flow-mediated platelet deposition on an injury and by
flow-mediated transport of the enzymes and their precursors. The
other involves a continuum model that describes platelet thrombosis
initiated by a ruptured atherosclerotic plaque in a
coronary-artery-sized vessel. This model includes full treatment of
the fluid dynamics, and the aggregation of platelets in response to
the plaque rupture and further chemical signals. Among the behaviors
seen with this model are the growth of wall-adherent platelet thrombi
to occlude the vessel and stop the flow, and the transient growth and
subsequent embolization of thrombi leaving behind a passivated injured
surface.

Growth is involved in many fundamental biological processes such as
morphogenesis, physiological regulation, or pathological disorders.
It is, in general, a process of enormous complexity involving
genetic, biochemical, and physical components at many different
scales and with complex interactions. In this talk, I will consider
the modeling of elastic growth in elastic materials and investigate
its mechanical consequences. First, starting with simple system in
one two and three dimensions, I will show how to generalize the
classical theory of exact elasticity to include growth. Second, we
will see that growth affects the geometry of a body by changing
typical length scales but also its mechanics by inducing residual
stresses. The competition between these two effects can be used to
regulate the physical properties of a material during regular
physiological conditions. It can also lead to interesting spontaneous
instabilities in growing materials as observed in simple physical
systems.