Blood coagulation: A puzzle for biologists, a maze for mathematicians

Pore-Scale Modeling of Non-Newtonian Shear-Thinning Fluids in Blood Oxygenator Design

This paper reviews and further develops pore-scale computational flow modeling techniques used for creeping flow through orthotropic fiber bundles used in blood oxygenators. Porous model significantly reduces geometrical complexity by taking a homogenization approach to model the fiber bundles. This significantly simplifies meshing and can avoid large time-consuming simulations. Analytical relationships between permeability and porosity exist for Newtonian flow through regular arrangements of fibers and are commonly used in macroscale porous models by introducing a Darcy viscous term in the flow momentum equations. To this extent, verification of analytical Newtonian permeability–porosity relationships has been conducted for parallel and transverse flow through square and staggered arrangements of fibers. Similar procedures are then used to determine the permeability–porosity relationship for non-Newtonian blood. The results demonstrate that modeling non-Newtonian shear-thinning fluids in porous media can be performed via a generalized Darcy equation with a porous medium viscosity decomposed into a constant term and a directional expression through least squares fitting. This concept is then investigated for various non-Newtonian blood viscosity models. The proposed methodology is conducted with two different porous model approaches, homogeneous and heterogeneous, and validated against a high-fidelity model. The results of the heterogeneous porous model approach yield improved pressure and velocity distribution which highlights the importance of wall effects.

Early stages of bone fracture healing: formation of a fibrin-collagen scaffold in the fracture hematoma

This work is concerned with the sequence of events taking place during the first stages of bone fracture healing, from bone breakup until the formation of early fibrous callus (EFC). The latter provides a scaffold over which subsequent remodeling processes will eventually result in successful bone repair. Specifically, some mathematical models are proposed to estimate the time required for (1) the formation immediately after fracture of a fibrin clot, described in terms of a phase transition in a polymerization process, and (2) the onset of EFC which is produced when fibroblasts arising from differentiation of chemotactically recruited mesenchymal stem cells remodel a previous fibrin clot by releasing a collagen matrix over it. An attempt has been made to keep models as simple as possible, so that a explicit dependence of the estimates obtained on relevant biochemical parameters involved is obtained.

Spatial aspects of blood coagulation: two decades of research on the self-sustained traveling wave of thrombin

In a number of experimental studies, it has been demonstrated that the forefront of blood coagulation can propagate in the manner of a signal relay. These data strongly support the concept that the formation of a blood clot is governed by a self-sustained traveling wave of thrombin. The present review critically appraises the experimental data obtained in recent decades concerning the self-sustained spatial propagation of thrombin. Open questions regarding the experimental detection of the self-sustained propagation of thrombin are discussed.