A mathematical model and numerical method for studying platelet adhesion and aggregation during blood clotting

Abstract

The repair of small blood vessels and the pathological growth of internal blood clots involve the formation of platelet aggregates adhering to portions of the vessel wall. Our microscopic model represents blood by a suspension of discrete massless platelets in a viscous incompressible fluid. Platelets are initially noncohesive; however, if stimulated by an above-threshold concentration of the chemical ADP or by contact with the adhesive injured region of the vessel wall, they become cohesive and secrete more ADP into the fluid. Cohesion between platelets and adhesion of a platelet to the injured wall are modeled by creating elastic links. Repulsive forces prevent a platelet from coming too close to another platelet or to the wall. The forces affect the fluid motion in the neighborhood of an aggregate. The platelets and secreted ADP both move by fluid advection and diffusion. The equations of the model are studied numerically in two dimensions. The platelet forces are calculated implicitly by minimizing a nonlinear energy function. Our minimization scheme merges Gill and Murray's ( Math. Programming 7 (1974) , 311) modified Newton's method with elements of the Yale sparse matrix package. The stream-function' formulation' of the Stokes' equations for the fluid motion under the influence of platelet forces is solved using Bjorstad's biharmonic solver (“Numerical Solution of the Biharmonic Equation,” Ph. D. Thesis, Stanford University, 1980). The ADP transport equation is solved with an alternating-direction implicit scheme. A linked-list data structure is introduced to keep track of changing platelet states and changing configurations of interplatelet links. Results of calculations with healthy platelets and with diseased platelets are presented.