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

*(English)*Zbl 0558.92009Author’s 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 neighbourhood 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 P. E. Gill and W. Murray’s [Math. Programming 7, 311-350 (1974; Zbl 0297.90082)] 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 P. Bjørstad’s biharmonic solver [Efficient solution of the biharmonic equation. Ph. D. Thesis, Stanford University (1980), see also Elliptic problem solvers, Proc. Conf., Santa Fee/N.M. 1980, 203-217 (1981; Zbl 0467.65010)]. 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.

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 neighbourhood 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 P. E. Gill and W. Murray’s [Math. Programming 7, 311-350 (1974; Zbl 0297.90082)] 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 P. Bjørstad’s biharmonic solver [Efficient solution of the biharmonic equation. Ph. D. Thesis, Stanford University (1980), see also Elliptic problem solvers, Proc. Conf., Santa Fee/N.M. 1980, 203-217 (1981; Zbl 0467.65010)]. 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.

Reviewer: J.Cronin

##### MSC:

92Cxx | Physiological, cellular and medical topics |

92-08 | Computational methods for problems pertaining to biology |

65K05 | Numerical mathematical programming methods |

65-04 | Software, source code, etc. for problems pertaining to numerical analysis |

65K10 | Numerical optimization and variational techniques |

76Z05 | Physiological flows |

##### Keywords:

blood clotting; repair of small blood vessels; platelet aggregates; suspension of discrete massless platelets; viscous incompressible fluid; Cohesion; adhesion; Repulsive forces; advection; diffusion; minimization scheme; modified Newton’s method; stream-function formulation of the Stokes’ equations; biharmonic solver; ADP transport equation; alternating-direction implicit scheme##### Software:

YSMP
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DOI

**OpenURL**

##### References:

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