A mathematical model of evolving mechanical properties of intraluminal thrombus

Abstract

Quantifying mechanical properties of blood clots is fundamental to understanding many aspects of cardiovascular disease and its treatment. Nevertheless, there has been little attention to quantifying the evolving composition, structure and properties when a clot transforms from an initial fibrin-based mesh to a predominantly collagenous mass. Although more data are needed to formulate a complete mathematical model of the evolution of clot properties, we propose a general constrained mixture model based on diverse data available from in vitro tests on fibrinogenesis, the stiffness of fibrin gels, and fibrinolysis as well as histological and mechanical data from clots retrieved from patients at surgery or autopsy. In particular, albeit resulting from complex kinetics involving many clotting factors, we show that the rapid (minutes) in vitro production of fibrin from fibrinogen can be modeled well by an Avrami-type relation and similarly that the fast (tens of minutes) in vitro degradation of fibrin in response to different concentrations of plasmin can be captured via a single "master function" parameterized by appropriate half-times that can be inferred from laboratory or clinical data. Accounting simultaneously for the production and removal of fibrin as well as chemo-mechano-stimulated production of fibrillar collagens yields predictions of changing mass fractions and bulk mechanical properties that correspond well to experimentally available data. Constrained mixture models thus hold considerable promise for modeling the biomechanics of clot evolution and can guide the design and interpretation of needed experiments and stress analyses.

Figure 1

Uniaxial/Biaxial Cauchy stress - stretch behaviors for fibrin [34] and mature thrombus [43]. These results serve as bounds for the expected predictions from the constrained mixture model.

Figure 2

Fibrinogenesis according to [35].

Figure 3

(a) Kinetics of the degradation of fibrin at different concentrations of plasmin (nM). Note that these data reflect families of sigmoidal responses and that half-times ${\tau }_{1/2}^{f-d}$ (at which the amount of fibrin is 50% of the initial value) can be extracted directly from the data for three of the plasmin concentrations (unfortunately, results for the remaining two plasmin concentrations were not collected over long enough periods). (b) Data from panel a plotted versus nondimensional time and fit with a single Avrami function, with time scaled by concentration dependent half-times. The existence of a “master curve” under laboratory conditions suggests its applicability over longer periods relevant to in vivo settings.

Figure 4

Predicted evolution of the composition (mass fraction ϕ^k) of the structurally significant constituents comprising a clot. Note that the production of collagen III precedes that of collagen I, but eventually there is a greater production of collagen I. Note, too, that the steady state, long-term response begins around 1 month.

Figure 5

Predicted evolution of bulk uniaxial/biaxial mechanical properties (i.e., Cauchy stress - stretch responses) of the clot. Note the similarities with Figure 1 at the early and late times.

Figure 6

Predicted time course of the overall stress in a clot, at a fixed value of extension of 15 %, during development.

Figure 7

Predicted radial gradients of pO₂ (solid curve) through the thickness of a hypothetical thick clot. Shown two are normalized measurements of the distribution of macrophages and voids (canniculi) from [1]. The near direct correlation between oxygen and macrophages suggests the importance of long-term hypoxia or anoxia in dictating the ability of cells to reorganize the clot locally. The inverse correlation between oxygen and voids similarly suggests an oxygen-dependent reorganization.

uniaxial, according to Wang - luminal up to 2.5 and medial 2

biaxial (according to Vande Geest et al 2006, in biaxial peak stretch is 1.2 with stress of 200 kPa