Aaron L. Fogelson

Surface-mediated control of blood coagulation: the role of binding site densities and platelet deposition.

A mathematical model of the extrinsic or tissue factor (TF) pathway of blood coagulation is formulated and results from a computational study of its behavior are presented. The model takes into account plasma-phase and surface-bound enzymes and zymogens, coagulation inhibitors, and activated and unactivated platelets. It includes both plasma-phase and membrane-phase reactions, and accounts for chemical and cellular transport by flow and diffusion, albeit in a simplified manner by assuming the existence of a thin, well-mixed fluid layer, near the surface, whose thickness depends on flow. There are three main conclusions from these studies. (i) The model system responds in a threshold manner to changes in the availability of particular surface binding sites; an increase in TF binding sites, as would occur with vascular injury, changes the systems production of thrombin dramatically. (ii) The model suggests that platelets adhering to and covering the subendothelium, rather than chemical inhibitors, may play the dominant role in blocking the activity of the TF:VIIa enzyme complex. This, in turn, suggests that a role of the IXa-tenase pathway for activating factor X to Xa is to continue factor Xa production after platelets have covered the TF:VIIa complexes on the subendothelium. (iii) The model gives a kinetic explanation of the reduced thrombin production in hemophilias A and B.

Presynaptic calcium diffusion from various arrays of single channels. Implications for transmitter release and synaptic facilitation

A one-dimensional model of presynaptic calcium diffusion away from the membrane, with cytoplasmic binding, extrusion by a surface pump, and influx during action potentials, can account for the rapid decay of phasic transmitter release and the slower decay of synaptic facilitation following one spike, as well as the very slow decline in total free calcium observed experimentally. However, simulations using this model, and alternative versions in which calcium uptake into organelles and saturable binding are included, fail to preserve phasic transmitter release to spikes in a long tetanus. A three-dimensional diffusion model was developed, in which calcium enters through discrete membrane channels and acts to release transmitter within 50 nm of entry points. Analytic solutions of the equations of this model, in which calcium channels were distributed in active zone patches based on ultrastructural observations, were successful in predicting synaptic facilitation, phasic release to tetanic spikes, and the accumulation of total free calcium. The effects of varying calcium buffering, pump rate, and channel number and distribution were explored. Versions appropriate to squid giant synapses and frog neuromuscular junctions were simulated. Limitations of key assumptions, particularly rapid nonsaturable binding, are discussed.

Grow with the flow: A spatial-temporal model of platelet deposition and blood coagulation under flow

The bodys response to vascular injury involves two intertwined processes: platelet aggregation and coagulation. Platelet aggregation is a predominantly physical process, whereby platelets clump together, and coagulation is a cascade of biochemical enzyme reactions. Thrombin, the major product of coagulation, directly couples the biochemical system to platelet aggregation by activating platelets and by cleaving fibrinogen into fibrin monomers that polymerize to form a mesh that stabilizes platelet aggregates. Together, the fibrin mesh and the platelet aggregates comprise a thrombus that can grow to occlusive diameters. Transport of coagulation proteins and platelets to and from an injury is controlled largely by the dynamics of the blood flow. To explore how blood flow affects the growth of thrombi and how the growing masses, in turn, feed back and affect the flow, we have developed the first spatial-temporal mathematical model of platelet aggregation and blood coagulation under flow that includes detailed descriptions of coagulation biochemistry, chemical activation and deposition of blood platelets, as well as the two-way interaction between the fluid dynamics and the growing platelet mass. We present this model and use it to explain what underlies the threshold behaviour of the coagulation systems production of thrombin and to show how wall shear rate and near-wall enhanced platelet concentrations affect the development of growing thrombi. By accounting for the porous nature of the thrombus, we also demonstrate how advective and diffusive transport to and within the thrombus affects its growth at different stages and spatial locations.

Continuum models of platelet aggregation: formulation and mechanical properties

Platelet aggregation is an important component of the blood’s clotting response, and is associated as well with many forms of cardiovascular disease. A new class of continuum models of platelet aggregation is presented. The models describe interactions among a viscous, incompressible fluid and populations of nonactivated (nonsticky) and activated (sticky) platelets suspended in this fluid. Cohesion between activated platelets can profoundly influence the motion of the suspending fluid. A platelet-activating chemical triggers a platelet’s transition from nonactivated to activated status, and induces the secretion of more of the same chemical. Investigation of the mechanical properties of the fluid-platelet system in the absence of new activation shows that platelet cohesion can generate extra pressure and extra viscous and elastic stresses. The last is sufficient to maintain the integrity of an aggregate that is subject to substantial external stress.

TNPACK—A truncated Newton minimization package for large-scale problems: I. Algorithm and usage

We present a FORTRAN package of subprograms for minimizing multivariate functions without constraints by a truncated Newton algorithm. The algorithm is especially suited for problems involving a large number of variables. Truncated Newton methods allow approximate, rather than exact, solutions to the Newton equations. Truncation is accomplished in the present version by using the preconditioned Conjugate Gradient algorithm (PCG) to solve approximately the Newton equations. The preconditioner M is factored in PCG using a sparse modified Cholesky factorization based on the Yale Sparse Matrix Package. In this paper we briefly describe the method and provide details for program usage

Unconditionally stable discretizations of the immersed boundary equations

The immersed boundary (IB) method is known to require small timesteps to maintain stability when solved with an explicit or approximately implicit method. Many implicit methods have been proposed to try to mitigate this timestep restriction, but none are known to be unconditionally stable, and the observed instability of even some of the fully implicit methods is not well understood. In this paper, we prove that particular backward Euler and Crank-Nicolson-like discretizations of the nonlinear immersed boundary terms of the IB equations in conjunction with unsteady Stokes Flow can yield unconditionally stable methods. We also show that the position at which the spreading and interpolation operators are evaluated is not relevant to stability so as long as both operators are evaluated at the same location in time and space. We further demonstrate through computational tests that approximate projection methods (which do not provide a discretely divergence-free velocity field) appear to have a stabilizing influence for these problems; and that the implicit methods of this paper, when used with the full Navier-Stokes equations, are no longer subject to such a strict timestep restriction and can be run up to the CFL constraint of the advection terms.

Coagulation under Flow: The Influence of Flow-Mediated Transport on the Initiation and Inhibition of Coagulation

A mathematical model of intravascular coagulation is presented; it encompasses the biochemistry of the tissue factor pathway, platelet activation and deposition on the subendothelium, and flow- and diffusion-mediated transport of coagulation proteins and platelets. Simulation experiments carried out with the model indicate the predominant role played by the physical processes of platelet deposition and flow-mediated removal of enzymes in inhibiting coagulation in the vicinity of vascular injury. Sufficiently rapid production of factors IXa and Xa by the TF:VIIa complex can overcome this inhibition and lead to formation of significant amounts of the tenase complex on the surface of activated platelets and, as a consequence, to substantial thrombin production. Chemical inhibitors are seen to play almost no (TFPI) or little (AT-III and APC) role in determining whether substantial thrombin production will occur. The role of APC is limited by the necessity for diffusion of thrombin from the site of injury to nearby endothelial cells to form the thrombomodulin-thrombin complex and for diffusion in the reverse direction of the APC made by this complex. TFPI plays an insignificant part in inhibiting the TF:VIIa complex under the conditions studied whether its action involves sequential binding of TFPI to Xa and then TFPI:Xa to TF:VIIa, or direct binding of TFPI to Xa already bound to the TF:VIIa complex.

Immersed Interface Methods for Neumann and Related Problems in Two and Three Dimensions

We develop and apply a finite-difference method to discretize the Laplacian operator with Neumann boundary conditions on an irregular domain using a regular Cartesian grid. The method is an extension of the immersed interface method developed by LeVeque and Li [SIAM J. Numer. Anal., 31 (1994) 1019--1044]. With careful selection of stencils, the method is second order accurate and produces a matrix that is stable (diagonally semidominant). The method is illustrated on several two-dimensional problems and one three-dimensional problem.

TNPACK—a truncated Newton minimization package for large-scale problems: II. Implementation examples

We have recently presented a FORTRAN package for solving unconstrained optimization problems by a truncated Newton algorithm. TNPACK is intended to solve problems for which some separability and sparsity-structure information of the Hessian is available. The Newton equations are solved approximately at each step by a Preconditioned Conjugate Gradient method, with adaptations to indefinite systems; the linear system involving the preconditioned is solved by a sparse modified Cholesky factorization based on the Yale Sparse Matrix Package. In this paper we describe implemental ion examples on two standard optimization problams and two real-life applications. Our intent is to aid users in their own applications, to highlight key options and parameters that may require tailoring to the problem and to describe application areas for which TNPACK is most !suited. These examples will illustrate various strategies for formulating preconditioners, applying reordering to them in order to minimize fill-in, enforcing truncation, and dealing with indefinite regions.

The Influence of Hindered Transport on the Development of Platelet Thrombi Under Flow

Vascular injury triggers two intertwined processes, platelet deposition and coagulation, and can lead to the formation of an intravascular clot (thrombus) that may grow to occlude the vessel. Formation of the thrombus involves complex biochemical, biophysical, and biomechanical interactions that are also dynamic and spatially-distributed, and occur on multiple spatial and temporal scales. We previously developed a spatial-temporal mathematical model of these interactions and looked at the interplay between physical factors (flow, transport to the clot, platelet distribution within the blood) and biochemical ones in determining the growth of the clot. Here, we extend this model to include reduction of the advection and diffusion of the coagulation proteins in regions of the clot with high platelet number density. The effect of this reduction, in conjunction with limitations on fluid and platelet transport through dense regions of the clot can be profound. We found that hindered transport leads to the formation of smaller and denser clots compared to the case with no protein hindrance. The limitation on protein transport confines the important activating complexes to small regions in the interior of the thrombus and greatly reduces the supply of substrates to these complexes. Ultimately, this decreases the rate and amount of thrombin production and leads to greatly slowed growth and smaller thrombus size. Our results suggest a possible physical mechanism for limiting thrombus growth.