Earlier Modeling of Blood Clotting

Earlier Clotting Work

Platelet Deposition and Coagulation Under Flow

We introduced the first models coupling platelet deposition and coagulation with flow-mediated transport. A series of papers describes our work in this area including extensions to the model and applications to clinical issues.

Andrew L. Kuharsky and Aaron L. Fogelson, Surface-mediated Control of Blood Coagulation: The Role of Binding Site Densities and Platelet Deposition, Biophysical Journal, 80, (2001), 1050-1074.

This paper introduced our model which looked at platelet deposition and coagulation in a thin reaction zone above a small vascular injury in which all species were regarded as well-mixed. The model tracks concentrations of three populations of platelets and of all of the main coagulation proteins, in both their inactive and active forms, and complexes of these proteins. The model distinguishes between fluid phase proteins and those bound either to the subendothelial surface or to the surfaces of activated platelets, and the availability of binding sites on activated platelets for the coagulation proteins is a regulator of the system’s response. Critical enzyme complexes are formed on the subendothelial surface and on the surfaces of activated platelets and transport through the fluid of enzymes produced on the subendothelium to the activated platelets is necessary for the reactions to proceed. Inactive proteins are brought by flow and diffusion to the reaction zone and active proteins produced in the reaction zone and not bound to surfaces are carried away by flow. In this context, flow-mediated washout of active proteins is the dominant inhibitor of the coagulation reactions. The model predicted a threshold response in how much thrombin was produced to the amount of tissue factor exposure in the injury; a switch mechanism that allows the system to respond quickly and powerfully but only in response to a sufficiently large stimulus. The attempt to capture the effect of Hemophila A with the model led to a novel hypothesis that platelets depositing on the subendothelium are physical inhibitors of enzyme activity on that surface. The model gave kinetic explanations of Hemophilias A and B and of the effects of moderate and severe reduction in platelet count. Model predictions about platelets surface as inhibitors and about a threshold response were subsequently confirmed experimentally.

Aaron L. Fogelson and Nessy Tania, Coagulation under flow: The influence of flow-mediated transport on the initiation and inhibition of coagulation, Pathophysiology of Haemostasis and Thrombosis, 34, (2005), 91-108.

Here we extended our model of platelet deposition and coagulation under flow to include the important activated protein C (APC) inhibitory pathway. We showed that in the context of small arterial injuries in which thrombomodulin, a key molecule required to form APC, was located outside the injury area, APC had little effect in inhibiting coagulation reactions in the reaction zone. In this paper, we also elucidated the mechanisms underlying the tissue factor threshold. It is essentially a race to establish platelet-bound FVIIIa:FIXa on activated platelet surfaces before adherent platelets shut down the TF:VIIa complex on the subendothelium.

Aaron L. Fogelson, Yasmeen H. Hussain, and Karin M. Leiderman, Blood Clot Formation Under Flow: The Importance of Factor XI on Thrombin Production Depends Strongly on Platelet Count, Biophysical Journal, 102, (2012) 10-18.

In this paper, we added the reaction loop in which thrombin activates FXI to FXIa and FXIa activates FIX to FIXa, thus providing a new positive feedback loop to the reaction network. We investigated the circumstances under which a Factor XI deficiency, also called Hemophilia C, had a significant effect on thrombin production. We found the effect was minimal except when platelet counts were high and the subendothelium was therefore covered more rapidly by adherent platelets than for a normal platelet count. We suggested that this need for an additional modification in platelet count, might explain the weak but variable bleeding behavior seen in persons with FXI deficiency.

Karin Leiderman, William Chang, Mikhail Ovanesov, Aaron L. Fogelson, Synergy Between Tissue Factor and Factor XIa in Initiating Coagulation, Arteriosclerosis, Thrombosis, and Vascular Biology, 2016, 36, 2334-2345.

We showed that very low levels of TF combined in combination with very low levels of circulating FXIa can initiate robust thrombin production. The TF levels are orders of magnitude less than those required for TF on its own to initiate a powerful thrombin response. The mechanisms underlying thrombin production and its timing differ when initiated by the combination of low TF and low FXIa than when triggered by TF alone. Our results may be relevant to the occurrence of thrombosis in persons administered IGG solutions contaminated by low levels of FXIa.

Priscilla Elizondo, Aaron L. Fogelson, A model of venous thrombosis initiation, Biophysical Journal, 2016, 111, 2722-2734.

In this paper we presented a model of the onset of venous thrombosis triggered by reactions on activated endothelial cells (AECs). Neither collagen nor TF is exposed on these cells, but blood-borne tissue factor on microvesicles can deposit on the vessel wall and initiate coagulation. Because collagen is not exposed there is not a rapid build up of wall-adherent platelets. Instead, partially-activating circulating blood cells, e.g., platelets and monocytes, slowly accumulate on the AECs as the slow moving fluid brings them near the wall. A coagulation burst typically occurs hours rather than minutes (as in arterial thrombosis) after the initiation events occur. A much lower TF level is needed to eventually produce a robust thrombin response; whether a thrombin burst occurs depends in a threshold way on TF, the level of thrombomodulin on the AECs, and on the degree of acceleration of antithrombin activity by heparin.

Kathryn G Link, Michael T Stobb, Jorge A Di Poala, Keith B Neeves, Aaron L Fogelson Suzanne S Sindi, Karin Leiderman, A local and global sensitivity analysis of a mathematical model of coagulation and platelet deposition under flow, PLoS One, 2018, https://doi.org/10.1371/journal.pone.0200917.

We developed a suite of local and global sensitivity measures and applied them to our platelet deposition and coagulation under flow ODE-based model. The effects of varying plasma levels of coagulation proteins, varying kinetic rate constants, and varying platelet characteristics on three thrombin metrics were determined for blood from a healthy individual. The metrics are the lag time until a 1 nM thrombin concentration was achieved, the maximum relative rate of thrombin generation, and the final concentration of thrombin after 20 minutes. Our local analysis shows that varying parameters within 50-150% of

baseline values, in a one-at-a-time fashion, always leads to significant thrombin generation in 20 minutes. Our global analysis gave a different and novel result highlighting groups of parameters, still varying within the normal 50-150%, that produced little or no thrombin in 20 minutes.

Karin M. Leiderman and Aaron L. Fogelson, Grow with the Flow: A spatial-temporal model of coagulation and platelet deposition under flow, Mathematical Medicine and Biology, 28, (2011) 47-84.

Karin M. Leiderman and Aaron L. Fogelson, The Influence of Hindered Transport on the Development of Platelet Thrombi Under Flow, Bulletin of Mathematical Biology, 75, (2013) 1255-1283.

In these papers, we presented the first spatial-temporal model of platelet deposition and 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. The growing thrombus is represented as a porous medium whose porosity decreases as the platelet number density in the thrombus increases. We showed how transport of proteins and platelets to/from the thrombus and transport of proteins within the thrombus affect the structure of the final thrombus that results.

Microscale models of Platelet Aggregation

Aaron L. Fogelson, A Mathematical Model and Numerical Method for Studying Platelet Adhesion and Aggregation Adhesion and Aggregation During Blood Clotting, Journal of Computational Physics, 56, (1984), 111-134.

Lisa J. Fauci and Aaron L. Fogelson, Truncated Newton Methods and the Modeling of Complex Immersed Elastic Structures, Communications on Pure and Applied Mathematics, 46, (1993), 787-818.

Haoyu Yu , Three-dimensional Computational Modeling and Simulation of Platelet Aggregation on Parallel Computers. 1999, PhD thesis, University of Utah.

Aaron L. Fogelson}, Haoyu Yu, and Andrew L. Kuharsky, `Computational Modeling of Blood Clotting: Coagulation and Three-dimensional Platelet Aggregation', in Polymer and Cell Dynamics: Multicsale Modeling and Numerical Simulations, Alt et al (Editors), Birkhaeuser-Verlag, Basel, 2003.

Aaron L. Fogelson and Robert D. Guy, Immersed-Boundary-Type Models of Intravascular Platelet Aggregation, Computer

Methods in Applied Mechanics and Engineering, 197, (2008), 2087-2104.

Tyler Skorczewski, Boyce Griffith Aaron L. Fogelson, Multi-bond Models for Platelet Adhesion and Cohesion, Contemporary Mathematics,

2014, 628, 149-173.

In these models, platelets are represented as elastic deformable objects immersed in a Newtonian fluid. The model makes use of the Immersed Boundary Method framework. Force-generating bonds can be created between points on distinct platelets or between a platelet and the injured vessel wall. The bonds may break if subject to sufficiently high forces. The representation of each platelet evolved from a single point in my PhD thesis, to rings of points in 2d simulations and surfaces of points in 3d simulations. The treatment of interplatelet bonds also evolved to account more faithfully for the mechanics and kinetics of actual bonds and for the multiple types of bonds that can form between platelets.

Continuum Models of Platelet Aggregation

Aaron L. Fogelson, Continuum Models of Platelet Aggregation: Formulation and Mechanical Properties, SIAM Journal of Applied Mathematics, 52, (1992), 1089-1110.

Aaron L. Fogelson, Continuum Models of Platelet Aggregation: Mechanical Properties and Chemically-induced Phase Transitions, Fluid Dynamics in Biology, (A.Y. Cheer and C.P. van Dam, Eds.), Contemporary Mathematics Series, American Mathematical Society, Providence, RI, 1993.

Nien-Tzu Wang and Aaron L. Fogelson, Computational Methods for Continuum Models of Platelet Aggregation, Journal on Computational Physics, 151, (1999), 649-675.

Aaron L. Fogelson and Robert D. Guy, Platelet-Wall Interactions in Continuum Models of Platelet Aggregation: Formulation and Numerical Solution, Mathematical Medicine and Biology, 21, (2004), 293-334.

Aaron L. Fogelson and Robert D. Guy, Immersed-Boundary-Type Models of Intravascular Platelet Aggregation, Computer Methods in Applied Mechanics and Engineering, 197, (2008), 2087-2104.

In these models, platelets and the bonds between them are represented by concentration functions which evolve due to transport and reactions including platelet activation and bond formation and breaking. The bonds generate platelet-platelet stresses and the divergence of the platelet stress tensor contributes to determining the fluid motion. The models are macroscale models aimed at thrombus formation in vessels the size of coronary arteries (1 - 2 mm diameter) and are derived from an underlying two-scale model in which processes on the vessel scale and on the platelet scale (2 microns) are both represented. An Oldroyd-B type PDE for the stress-tensor can be derived without approximations if the bonds act like linear springs with zero rest length and if they break at a constant rate. The latter produced some non-physical behaviors and motivated our development of an approximate closure form of the PDE for the stress tensor that allows bond breaking to depend on the local average bond length. The models successfully capture the buildup of a wall-bound thrombus in response to vessel injury; the thrombus can develop until the vessel is occluded, or it can breakup (embolize) due to strong fluid forces. These models are single-phase models meaning that the fluid and platelets move in the same velocity field. The lack of relative motion is a significant weakness of the models as it requires us to greatly accelerate the speed with which platelet activation occurs, and it limits the platelet number density within the thrombus to being at most slightly more than the number density of platelets circulating with the blood. To remove these limitations, we derived a two-phase model in which fluid and individual platelets move at the fluid velocity and bound platelets move at a different velocity. The two-phase model captures thrombus growth on a realistic timescale and can produce thrombi with platelet number densities 100-fold or more that in the bulk blood as occurs in real thrombi. The two-phase models are described in the papers by Du and Fogelson.

Fibrin Polymerization and Fibrinolysis

Robert D. Guy, Aaron L. Fogelson, and James P. Keener, Fibrin gel formation in a shear flow, Mathematical Medicine and Biology, 24, (2007), 111-130.

Aaron L. Fogelson and James P. Keener, Toward an understanding of fibrin branching structure, Physical Review E, 81, 051922 (2010).

Aaron L. Fogelson, James P. Keener, A Framework for Exploring the Post-gelation Behavior of Ziff and Stells Polymerization Models, SIAM Journal on Applied Mathematics, 2015, 75, 1346-68.

Cheryl Zapata PhD thesis. Mathematical Modeling of Fibrin Gelation Dynamics and Structure Formation Under Flow, 2016, University of Utah.

Anna C. Nelson, James P. Keener, Aaron L. Fogelson, Kinetic model of two monomer polymerization, Physical Review E, 2020, 101(2-1):022501.

After fibrin monomers are produced from the plasma protein fibrinogen through cleavage by the enzyme thrombin, fibrin monomers polymerize to form a three-dimensional mesh between and around platelets in a thrombus. This mesh helps to mechanically stabilize the thrombus. Fibrin and platelets comprise different fractions of the thrombus depending on the flow conditions under which it forms. Under very slow flow fibrin dominates and platelets play a small, but essential, role. This is the situation during the formation of venous thrombi in the pockets behind venous valves. Under fast flow, thrombi are mostly made up of platelets. This is typical of thrombi formed in arteries and arterioles. In a continuing series of papers we looked at the effect of flow shear rate on the formation of a fibrin gel layer and we developed a model of fibrin polymerization that includes a plausible mechanism of branch formation. In that model, the structure of the fibrin mesh is sensitive to the concentration of thrombin present during its formation, as for real fibrin clots. We developed a framework for continuing the study of our kinetic gelation model past the time a gel occurs and we incorporated spatial heterogeneity and transport by advection and diffusion into our fibrin polymerization models. We lay the groundwork for including fibrinogen binding to a growing fibrin mesh by looking at two monomer polymerization. Our fibrin polymerization models continue to evolve as described above.

Brittney E. Bannish, James P. Keener, and Aaron L. Fogelson, `Modeling Fibrinolysis: A 3-Dimensional Stochastic Multiscale Model, Mathematical Medicine and Biology, 31, (2014) 17-44.

Brittney E. Bannish, James P. Keener, Michael Woodbury, John W. Weisel, and Aaron L. Fogelson, Modeling Fibrinolysis: 1-Dimensional Continuum Models, Mathematical Medicine and Biology, 31, (2014) 45-64.

Brittany E. Bannish, Irina N. Chernysh, James P. Keener, Aaron L. Fogelson, John W. Weisel Molecular and Physical Mechanisms of Fibrinolysis and Thrombolysis from Mathematical Modeling and Experiments, Scientific Reports, 2017, 7(1):6914. doi:10.1038/s41598-017-06383-w.

Fibrinolysis is the process by which a fibrin clot is enzymatically degraded. When a fibrin mesh is present, plasminogen and tissue plasminogen activator (tPA) can bind to fibrin strands in close proximity and, there, tPA can activated plasminogen into the enzyme plasmin. Plasmin can degrade fibrin by cutting across fibrin fibers. Most models of fibrinolysis are continuum models but these are limited in their ability to explain important experimental observations. To overcome these limitations, we developed a three-dimensional stochastic multi-scale model of fibrinolysis. The model is stochastic both at the scale of plasmin activation and cleavage of a given fiber cross section and at the scale of a meshwork were movement of tPA through the mesh is important in determining the speed of lysis. Probability distributions that account for the biochemistry of the lysis process are determined by random simulations at the fiber level. This information is used for stochastic simulations at the level of a fibrin mesh. The multi-scale model can explain the seemingly contradictory observations about whether fine or coarse clots lyse more quickly, and gives insights into other features of the fibrinolytic process.