Molecular and Physical Mechanisms of Fibrinolysis and Thrombolysis from Mathematical Modeling and Experiments

Abstract

Despite the common use of thrombolytic drugs, especially in stroke treatment, there are many conflicting studies on factors affecting fibrinolysis. Because of the complexity of the fibrinolytic system, mathematical models closely tied with experiments can be used to understand relationships within the system. When tPA is introduced at the clot or thrombus edge, lysis proceeds as a front. We developed a multiscale model of fibrinolysis that includes the main chemical reactions: the microscale model represents a single fiber cross-section; the macroscale model represents a three-dimensional fibrin clot. The model successfully simulates the spatial and temporal locations of all components and elucidates how lysis rates are determined by the interplay between the number of tPA molecules in the system and clot structure. We used the model to identify kinetic conditions necessary for fibrinolysis to proceed as a front. We found that plasmin regulates the local concentration of tPA through forced unbinding via degradation of fibrin and tPA release. The mechanism of action of tPA is affected by the number of molecules present with respect to fibrin fibers. The physical mechanism of plasmin action (crawling) and avoidance of inhibition is defined. Many of these new findings have significant implications for thrombolytic treatment.

Figure 1

Schematic diagram of multiscale model. (A) The microscale model is a cross section (the square grid of gray squares) of the fibrin fiber. For simplicity, we approximate the circular cross section of the cylindrical fiber by a square of equal area. The microscale model is initialized with a single tPA molecule (black dot) randomly placed on the outer edge of the fiber. (B) On the left is the fiber cross section. Each gray square in the cross section represents 6 binding doublets (as seen on the right). A binding doublet is a pair of binding sites to which tPA, plasminogen, and plasmin can bind. Initially, there is one exposed binding doublet at each location, and 5 cryptic doublets that can be exposed by plasmin during degradation. A tPA molecule on a binding doublet may convert plasminogen on any doublet at the same binding location to plasmin. (C) The macroscale model represents a 3-D fibrin clot formed in a small chamber. The black and gray lines comprising the lattice represent the fibrin fibers. The size of the chamber is 100 × 100 × (100 + H) μm³. For simplicity, we only consider one periodic slab of clot, one fiber thick (black). “p.s.” means “pore size”, the distance between fibers. (D) The macroscale model is initialized by randomly placing a fixed number of tPA molecules at the top of the chamber. The tPA molecules (black and white dots) can diffuse and bind to fibrin fibers (black). Unbound tPA is shown in black, bound tPA is shown in white. (E) When tPA binds to a fiber, it initiates the lytic process, so that some time after tPA has bound to a fiber, the fiber degrades.

Figure 2

Lysis front velocity as a function of clot structure and tPA concentration. Lysis front velocity, in μm/min, of a fine clot (red triangles) and a coarse clot (black circles) as a function of the ratio of the number of tPA molecules to the surface area of clot exposed to the fibrin-free region. (A) Mathematical model simulations were run with 11 different tPA-to-surface-area ratios varying from 8 to 1600 molecules/μm². Each symbol indicates the mean of ten independent simulations. For a tPA-to-surface-area ratio of 150, a pair of distinct experiments were run for each clot type, one with an 85 nM tPA solution and the other with a 5 nM tPA solution, and the results for each pair were almost the same (as seen by the double symbols). (B) Laboratory experiments were run with 5 different tPA-to-surface-area ratios varying from 600 to 12000 molecules/μm². Each symbol indicates the mean of three independent trials. Clots were formed from the pooled plasma of 6 donors.

Figure 3

Confocal micrographs of a fibrin clot during fibrinolysis initiated by tPA. Fluorescently labeled fibrin is green, tPA is red, and the overlap of fibrin and tPA thus makes yellow. Micrographs are (A) initial image, (B) 147 sec later, (C) 196 sec later, (D) 245 sec later. Magnification bar = 5 μm.

Figure 4

Diagram of protofibril spacing in a fiber cross section. The circular fiber cross section of a 97.5-nm diameter fiber is approximated by a square of equal area. Black circles are protofibril cross sections, each with diameter 4.8 nm. The distance from the edge of one protofibril to the edge of a neighboring protofibril is also 4.8 nm.

Figure 5

Basic model for plasmin crawling. (A) The 4 possible states a plasmin molecule with 2 limbs can be in. A filled circle represents fibrin with a plasmin limb bound and an open circle represents fibrin with a plasmin limb unbound. (B) Reaction diagram for a plasmin molecule transitioning between states. We redefine variables such that S ₀₀ = $\hat{\mathrm{S}}$ ₀₀ (which represents all states with two unbound limbs), S ₁₁ = $\hat{\mathrm{S}}$ ₁₁ (which represents all states with 2 bound limbs), and S ₁₀ = $\hat{\mathrm{S}}$ ₁₀ + $\hat{\mathrm{S}}$ ₀₁ (which represents all states with 1 bound and 1 unbound limb). The possibility of an unbound plasmin (S ₀₀) binding is neglected since we are interested in what happens prior to the plasmin molecule unbinding. Since it is unlikely for two limbs to unbind at the exact same time, we assume that S ₁₁ cannot transition directly to S ₀₀, but must first transition to S ₁₀.

Figure 6

Schematic of the interplay between clot structure and tPA. Each subfigure depicts a coarse clot with thick fibers (left) and a fine clot with thin fibers (right) contained in the same volume. Round symbols represent bound tPA molecules. (A,B) When there is not much tPA present, the coarse clot degrades faster because it has fewer fibers; the small number of tPA molecules is able to start degradation on all fibers at the coarse clot front, but is unable to do so at the fine clot front. (C,D) When there is a lot of tPA present, degradation begins on all fibers at the front of both clot types. In this case, the fine clot degrades faster since the individual thin fibers in the fine clot are lysed faster than the individual thick fibers in the coarse clot.