Mathematical Modeling of Blood Clotting

The body’s 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. 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 a ffects the clot growth and how the growing masses, in turn, feed back and aff ect the fluid motion, we developed a spatial-temporal model of platelet aggregation and blood coagulation under flow. This is the fi rst spatial-temporal model 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.

HOW DOES BLOOD CLOT?

thrombosis_diagram_clotting_stages1bAn intact, healthy vessel with a continuous lining of endothelial cells carries blood comprised of blood cells and plasma factors.

thrombosis_diagram_clotting_stages3Upon injury, tissue factor and collagen in the subendothelium become exposed: coagulation begins, platelets adhere and become activated. Activated platelets become sticky and begin to support chemical reactions on their surfaces.

thrombosis_diagram_clotting_stages5
Thrombin cleaves fibrinogen into fibrin which polymerizes into a mesh and holds platelet aggregates together: stable blood clot.
SCHEMATIC OF COAGULATION REACTIONS:
CoagSchematicNoFibrinADPw2

 

RELATED PUBLICATIONS:

  1. Leiderman, K. and A.L. Fogelson. An Overview of Mathematical Modeling of Thrombus Formation Under Flow. Thromb. Res. 2014; 133(S1):S12-S14.
  2. Onasoga A.A., Leiderman, K., Fogelson, A.L., Wang, M., Manco-Johnson, M.J., Di Paola, J.A., and K.B. Neeves. The effect of FVIII deficiencies and replacement and bypass therapies on thrombus formation under venous flow conditions in microfluidic and computational models. PLoS ONE. 2013; 8(11): e78732.
  3.  Leiderman, K. and A.L. Fogelson. The Influence of Hindered Transport on the Development of Platelet Thrombi Under Flow. Bull. Math. Biol. Oct, 2012:1-29
  4. Fogelson, A.L., Hussein, Y., and K. Leiderman. The Influence of Thrombin-activated FXIa on Thrombin Production is Predicted to Depend Strongly on Platelet Count. Biophys. J. 2012; 102 (1): 10-18 Link
  5. Wolberg, A.S., Aleman, M.M., Leiderman, K, and K.R. Machlus. Procoagulant Activity in Hemostasis and Thrombosis: Virchow’s Triad Revisited. Anesth. Analg. 2010; 114 (2):275:285 Link
  6. Leiderman, K. and A.L. Fogelson. Grow with the Flow: A Spatial-Temporal Model of Platelet Deposition and Blood Coagulation Under Flow. Math. Med. Biol. 2011 Mar;28(1):47-84. (Winner of SIAM Student Paper Prize in 2010) Link

Computational Modeling of Flagellar Motion in a Brinkman Fluid

The scienti c theme of this work is to study the interaction of cilia and flagella with their complex fluid environments. For example, our group would like to know how the fluid environment influences the emergent beat patterns of cilia and waveforms of flagella. Answering these questions has important implications, some of which include improvement of aerosol drug delivery and a better understanding of the fluid mechanics of reproduction. Our work has mostly focused on the development of numerical methods to solve the underlying PDEs of the fluid-structure interaction problems.

Presentation1b

Figure showing that added fluid resistance prevents two finite-length swimmers from attracting. Compare top row (attraction after t=90) to bottom row where resistance has been added to the Stokes equations via the Brinkman term.

RELATED PUBLICATIONS:

  1. K. Leiderman and S.D. Olson. Swimming in a 2D Brinkman fluid: Computational modeling and regularized solutions. Phys. Fluids (In revision)
  2. H.N. Nguyen, K. Leiderman and S.D. Olson. A fast method to compute triply-periodic Brinkman flows. Computer and Fluids (Submitted)
  3. S.D. Olson and K. Leiderman. Effect of Fluid Resistance on Symmetric and Asymmetric Flagellar Waveforms. J. Aero Aqua Bio-mech., 2015; 4:12-17.
  4. H.N. Nguyen and K. Leiderman. Computation of the singular and regularized image systems for doubly-periodic Stokes flow in the presence of a wall. J. Comp. Phys. 2015; 297:442-461
  5. Buchmann, A.L. Fauci, L.J., Leiderman, K., Strawbridge, E.M, and Zhao, L. Flow induced by bacterial carpets and transport of microscale loads. IMA Proceedings, 2014;
  6. Leiderman, K., Bouzarth, E.L., H.N. Nguyen. A Regularization Method for the Numerical Solution of Doubly-Periodic Stokes Flow. Contemporary Mathematics Series of AMS, 2014.
  7. Leiderman, K.,  Bouzarth, E.L., Cortez, R. and  A.T. Layton. A Regularization Method for the Numerical Solution of Periodic Stokes Flow. J. Comp. Phys. 2013; 236:187-202.
  8. Cortez, R., Cummins, B., Leiderman, K. and D. Varela. Computation of Brinkman Flows Using Regularized Methods. J. Comp. Phys. 229 (2010) 76097624. Link