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The model was developed to further investigate spatial effects on the threshold behavior of thrombin, to explore how high
near-wall platelet concentrations and shear rate affect thrombus growth, and to understand how solute transport within a
developing clot affects its growth in space and time. The model tracks concentrations of coagulation proteins, inhibitors,
and platelets throughout a two-dimensional domain that mimics a portion of an in vitro microfluidic channel or blood
vessel with an exposed patch of TF. Fluid motion is governed by the incompressible NaviereStokes equation with an
additional porous media term to add frictional resistance to the fluid due to the presence of bound platelets; the permeability
of the bound platelets varies in space and time as the clotting process progresses. PDEs are used to simulate the transport of
platelets and proteins by advection and diffusion through the fluid and through the porous, heterogeneous clot. As in the
KF model, platelets that adhere to the TF patch physically inhibit the activity there, but as an extension of the KF model,
there is explicit treatment of the release of ADP, which can activate platelets. ADP that is released into the fluid by
activated platelets over a short time period after activation can activate more platelets or get carried away by flow. A
near-wall excess of platelet s is also included to mimic the effects of platelet margination [96e98].
Model results showed that thrombin production demonstrates a threshold behavior with respect to TF. As with the ODE
models described earlier, generation of a thrombin burst is dependent on there being enough IXa and VIIIa produced by
TF:VIIa before the subendothelium is covered with platelets. The spatial distributions reveal something new about the
threshold dependence. The thrombin concentrations plateau because there is little to no spatial overlap between tenase
complexes and available binding sites to bring X to the tenase to become activated; Xa production is limited and so
prothombinase levels, and therefore thrombin levels, plateau. The model showed that shear rate, thrombus permeability,
and a near-wall excess of platelets each could significantly affect the thrombus structure and heterogeneity, as well as the
growth rate. This mathematical model has a physical analog [99,100] and correctly predicted experimental observations of
platelet-dependent thrombin and fibrin generation in individuals with mild hemophilia A, including their response to
treatment with rVIIa [100]; see the top of Fig. 5.8.
Leiderman and Fogelson extended this model to include a more complete hindered motion as a result of the porous
thrombus [101]. The previous model hindered transport for platelet species in and around the forming thrombus because
platelets were modeled as a continuum and the hindrance, in part, accounted for their physical size. In the extend ed version,
advection and diffusion of solutes were hindered within a clot and resulted in slower growing, denser clots. The limited
solute transport resulted in restriction of tenase and prothrombinase to areas deep within the interior of the clot and in
restriction of their corresponding substrates from outside the clot from meeting them. This inside-out, outside-in hindrance
slowed the production of thrombin and thus activation of platelets, and hence, clot growth slowed. This notion that clots
have a dense core and a less dense periphery is in accordance with observations described in animal models [75,83], and
clot profiles from this model are in qualitative agreement with platelet profiles extracted from flow assay data [80]; see
Fig. 5.9. The model is, as of this writing, being updated to simulate extravascular clot formation in a microfluidic device;
the mathematical model and microfluidic device are being developed simultaneously. This development is in the ea rly
stages, but the fluid dynamics simulated in a complex geometry show excellent agreement with the microfluidic model (see
the bottom of Fig. 5.8), which has demonstrated a hemostatic prototype sensitive to both coagulation and platelet function
[102]. We are developing these models in tandem to study coagulopathies and platelet dysfunction that result in excessive
blood loss; we refer the reader to other studies using microfluidic devices to study bleeding disorders for more information
[103,104].
Govindarajan et al. extended the first LF model to include fi brin formation by a simple assumption that fibrin forms at a
rate directly proportional to the thrombin concentration [143]. In addition, they modified platelet diffusivity and allowed
for shear-rate-dependent adhesion. Fibrin was assumed to contribute to the mechanical properties of the clot, and the
resistance imparted by fibrin was found to be about 30 times higher than that imparted by platelets. Fibrin formation did not
extend past the platelets in the clot, similar to other experimental studies by Colace et al. [80] and computational studies by
Tosenberger et al. [105,106], but not in support of the hypothesis of a fibrin cap [107,108].
The models of clot formation under flow by Tosenberger and colleagues are based on the dissipative particle dynamics
(DPD) method. With this method, the platelets and fluid are modeled as mesoscale particles subjected to repulsive,
dissipative, and random forces. Results of these types of models are sensitive to the choice of parameters describing such
forces. The authors allowed platelets to aggregate prior to activation. Fibrin formed primarily in the core of the clots, aiding
in rigidity and stability [105,106,109]. Karniadakis and colleagues have helped develop the field of DPD in the context of
platelet margination and aggregation in complex geometries [57,110,111].
Other studies have focused on intraclot transport in terms of the “core” and “shell” structure as proposed in the
experimental study of Stalker and colleagues [83]. The role of a confining fibrin shell on clot growth was investigated by
Kim and colleagues [107]. They demonstrated that the fibrin shell, prescribed with their own experimenta lly measured
permeability, allowed thrombin to leak out of the clot and be washed away, but also limited exposure of platelets on the
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FIGURE 5.8 Mathematical modeling of bypass therapy in hemophilia and hemostasis. A patient with severe factor VIII deficiency that had developed
inhibitors was treated with 90 mg/kg recombinant factor VIIa (rFVIIa). Top left: Overlay of platelet (blue) and fibrin (green) accumulation pre- and
posttreatment on a collageneTF surface following perfusion for 5 min at a wall shear rate of 150/s. Top center: Fibrin (Fbn) deposition pre- and
posttreatment in comparison with a healthy control in a microfluidic assay (MFA). Top right: Predicted average thrombin concentration in the growing
thrombus as a function of rFVIIa and FVIII levels using the spatial-temporal (ST) LeidermaneFogelson model [7]. (A and B) Passive tracer particles
moving through the computational analog of a microfluidic model of hemostasis shown in (C) using bright-field imaging [102].
FIGURE 5.9 Limiting clot growth with hindered solute transport. Simulations of clot growth in the model of platelet deposition and blood coagulation
without (A and C) and with (B and D) hindered solute transport [7,45]. (A and B) The spatial distribution of prothrombin bound to the platelet-surface
complex prothrombinase. These distributions show the regions where thrombin is produced; with hindered solute transport there is only a small regionof
overlap between the substrate and its enzyme, and thus thrombin production and platelet activation slow the growth the clot. (C and D) The corresponding
spatial distribution of bound platelets.
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clot surface, ultimately limiting clot growth. Using a clot geometry reconstructed from their own experiments, Voronov
et al. measured thrombus permeability via simulation and reported a significant difference between core and shell [112].In
both studies, the shell and core structure was prescribed rather than an emergent property of thrombus growth. Stalker and
colleagues looked further into the implications of a coreeshell structure and combined experiments and simulations in a
series of three papers [78,79,113]. In these studies, they found that initial platelet accumulation has narrow gaps that
produce an altered microenvironment that affects agonist distribution and subsequent thrombus growth. Through
computational studies, they found that hindered solute diffusion becomes the dominant mode of transport within a
thrombus, but that in the core, compared with the shell, molecules are completely restricted. Further, they demonstrated
that intraclot transport regulates thrombin activity and thus platelet activation in vivo.
MODELS OF FIBRIN POLYMERIZATION
Only a few models of fibrin production, polymerization, and deposition have been developed as of this writing. Naski and
Shafer developed a kinetic ODE model of fibrin production in the presence of AT [144]. Using their own previously
determined kinetic parameters for release of fibrinopeptides [114,115], they proposed a kinetic scheme and corresponding
system of ODEs to track the concentration of fibrinogen, fibrin, thrombin, and AT. Their model quantitatively matches the
experimentally measured release of fibrinopeptides A and B and provides an estimate for the thrombin concentration
necessary to produce a clot of known composition. This model was included in the ODE model of Chatterjee et al.,
and Shafer model monitors concentrations but does not track the polymerization/aggregation process. Weisel and
Nagaswami formulated an ODE model of polymerization kinetics including the cleavage of fibrinopeptides, formation of
protofibrils, and lateral aggregation into fibers [116]. Under certain length constraints of protofibrils prior to aggregation,
the model results are in good agreement with lag periods observed in turbidit y profiles. By varying kinetic rates, the model
can capture a wide variety of fibrin polymerization dynamics.
Fibrin gel structure is highly correlated with the thrombin concentration that induced its formation; as thrombin
concentration increases, fibers tend to be thinner with a higher degree of branching and form a gel that is less susceptible to
fibrinolysis [8]. To better understand the branching process, Fogelson and Keener generalized a kinetic gelation model of
linear aggregation from Ziff and Snell [117], to include clusters and branches [118]. Their proposed mechanism of fibrin
branching led to different structures that depended on the supply rate of monomers, in qualitative accord with the
experimental observations.
Since fibrin gels form under flow in vivo, the deposition of fibrin under flow must also be considered. The structure and
density of a fibrin gel that forms under flow are very different from those of one that forms under static conditions; fibrin
gels that form under flow are dense and have much lower water content than static gels [119]. Further, the rate of fibrin
deposition is highly dependent on shear rate, as shown in in vitro flow assays [120]. Using a numerical simulation in
conjunction with microfluidic assays, Neeves et al. found that low shear rates (w10/s) allow for fibrin deposition to occur,
but the deposition is limited to a thin boundary layer near the wall [121]. Guy and Fogelson used a kinetic gelation model
to investigate fibrin gel formation under the in
combines the kinetic gelation model with a one-dimensional (1D) PDE to allow for transport of prothrombin, thrombin,
fibrinogen, and fibrin. Thrombin produced at the boundary (wall) converts fibrinogen into fibrin, and the fibrin polymerizes
and feeds back onto the flow with a porous mediaelike term. At low shear rates, the gel growth is limited by the
availability of thrombin. At high shear rates, fibrin is removed by flow before gelation can occur, similar to results
described earlier. Interestingly, the transition between these two regimes is dictated by the permeability of the gel.
fluence of a shear flow [122]. To predict the height of fibrin gels, their model
MODELS OF FIBRINOLYSIS
Understanding the interactions between all proteins involved in the fibrinolytic cascade is critical for effective
thrombolysis, the clinical initiation of fibrinolysis. Current thrombolytic stroke treatment involves an infusion of
recombinant tPA administered within 4.5 h of stroke onset. However, bleeding complications often occur posttreatment
[123], so safer, more effective thrombolytic therapies are desirable. Mathematical models of fibrinolysis, developed since
the late 1980s, are being used to understand observed phenomena like the accumulation of enzymes at the lysis front and
the effects of blood flow, diffusion, and clot structure on lysis, and to propose changes to thrombolytic strategy.
Many models of fibrinolysis are 1D reactioneadvectionediffusion equations. Zidansek and Blinc studied the spatial
distribution of plasmin within the clot and the progression of the bloodeclot boundary [124]. uPA is introduced into the
homogeneous fibrin clot at a uniform flow velocity, and no distinction is made between fibrin-bound and plasma-phase
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proteins. Model results confirm experimental results: clots degrade about 2 orders of magnitude faster when exposed to
pressure-induced uPA permeation compared with diffusion alone. A more detai led model by Diamond and Anand accounts
for uPA and both fibrin-bound and fluid-phase tPA, plasminogen, and plasmin [125]. The clot is a uniform porous gel
composed of fibrin fibers whose diameters shrink uniformly as degradation proceeds. As fiber diameter shrinks, the local
specific permeability changes, which in turn affects the overall permeation velocity across the clot. Model results suggest
that thrombolytic therapy is effective over realistic time scales only if tPA is pressure driven; diffusion alone will not
dissolve the clot in a reasonable time. Modifications to the Diamond and Anand model include fibrinolytic inhibitors PAI-1
and a
-AP and solubilization of enzymes back into solution [126], adsorption and reaction of fibrinolytic enzymes from the
2
systemic circulation (to model intravenous thrombolytic therapy) [127], and the effect of outer convection on tPA-mediated
lysis of fibrin clots under various flow conditions [128]. These models reasonably predict experimental lysis front velocities and show a high accumulation of tPA and plasmin at the lysis front [126], accurately predict measured lysis front
positions and systemic tPA concentrations and show that the rate of lysis increases with increasing permeation [127],
display an initial delay in tPA-mediated lysis followed by the formation of a lysis front that moves at a constant rate, and
predict that outer convection modulates lysis at shear stresses below 10 dyn/cm
18 dyn/cm
2
[128].
2
, but has no effect at shear stresses above
Some models aim to study coagulation and fibrinolysis simultaneously. Anand and colleagues built a 1D
reactioneadvectionediffusion model that includes the extrinsi c coagulation pathway and fibrinolysis in a viscoelastic fluid
[129]. The fibrinolysis portion of the model is fairly simple: tPA is considered to be in steady state while plasminogen is
depleted (contrary to current knowledge); plasmin is created by tPA and is inhibited by a
-AP; fibrin (created through
2
the interaction of thrombin and fibrinogen) is degraded through an interaction with plasmin. A more biochemically
comprehensive 1D reactionediffusion model, which includes both the extrinsic and the intrinsic coagulation
pathway, treats fibrinolysis in a more realistic manner [130]. Although tPA is still assumed to be in steady state, the
MichaeliseMenten activation of plasminogen to plasmin by tPA is an improvement from the previous model. Using the
Anand et al. models as a starting point, Sequeira and colleagues developed a 3D model of coagulation and fibrinolysis in
flowing blood [131]. The model is simplified to neglect blood flow, however, and the stability of the continuum of
equilibria is determined. It is hard to compare these types of models with experimental data, since the models that account
for coagulation and fibrinolysis simultaneously are oversimplified.
A clot is not a homogeneous concentration of fibrin, as many early models assume, but rather a tangled mesh of fibrin
fibers. Fibrin concentration is high in regions of the clot containing fiber and nonexistent in regions of the clot between
fibers. To account for this fibrin heterogeneity, Bannish and colleagues developed a 1D reactionediffusion model in which
the fibrin concentration varies in space [132]. The model includes the major biochemical kinetics considered in earlier
models (tPA converting plasminogen to plasmin, plasmin degrading fibrin, tPA and plasminogen binding to, and
unbinding from, fibrin) and also considers plasmin-mediated exposure of lysine binding sites. While the model predicts
patterns and rates of lysis similar to earlier models and experiments, it is unable to describe the observ ed differences in lysis
of fine and coarse clots. Bannish and colleagues conclude that higher d imensional models are necessary to understand
how clot structure affect s the patterns and rates of lysis.
To better study the geometry of clot dissolution, higher-dimensional models were created. Anand and colleagues
modified the Diamond and Anand model [125] to account for pressure-driven permeation of plasmin (rather than tPA) into
a2Dfibrin clot [133]. Results show dissolution “fingers” in the clot, consistent with experimental observations. A 2D
random walk model by Zidansek and colleagues predicts similar lysing patterns [134]. This model uses the same kinetic
equations for fibrinolytic enzymes (uPA, plasminogen, plasmin, a
-AP) as the Zidansek and Blinc model [124], but the
2
reactions are characterized phenomenologically by a single reaction time representing the time lag between arrival of uPA
and clot degradation. For sufficiently large values of this reaction time, finger-like patterns of clot dissolution are obtained.
While Zidansek et al. modeled how a channel forms in the clot, Pleydell and colleagues created a 2D model of fibrinolysis
of a recanalized blood vessel in low-velocity flow to study what happens to the remaining clot after the channel forms
[135]. The clot is assumed to be an annulus and a steady-state model tracks free plasminogen, tPA, plasmin, a
-AP,
2
fibrinogen, and fibrin degradation products, as well as bound tPA, plasminogen, and plasmin. Model results show that the
rate of clot degradation will change as the clot dissolves because the kinetic reactions depend on flow parameters, which
change as the channel widens. Sersa and colleagues extended the ideas of Pleydell et al. to study thrombolysis of
nonocclusive clots in high-velocity axially directed blood flow that is either strictly laminar [136] or allowed to be turbulent
[137]. As in Pleydell et al., there is a channel through the clot, and lysis is modeled as the radial expansion of the channel.
Also, biochemical reactions are not explicitly modeled, but rather summarized phenomenologically in two parameters
representing time lags in the fibrinolytic system. By comparing model results with experimental data, the authors show that
fast flow increases the clot dissolution rate due to the work of mechanical forces of streaming blood, not just due to better
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permeation of fibrinolytic enzymes into the clot. As previous models treated clots homogeneously, Bajd and Sersa
developed a 3D model of clot dissolution that accounts for complex microscopic structure by defining the clot to be a
collection of randomly distributed spherical blood cells connected by elastic fibrin bonds [138]. These bonds can be
corroded via biochemical degradation or eroded via mechanical forces of streaming blood. As in Sersa et al. [136], the
authors find that clot dissolution is enhanced in high-velocity flow, and also that the size of the removed clot fragments
depends on flow velocity. Piebalgs and Xu developed a more complete 2D model that describes thrombol ysis of the initial
occlusive clot through the recanalized mural clot [139]. The model is a higher-dimensional extension/modification of the
Diamond and Anand and Wootton et al. models [125,128], which explicitly includes equations for basic fibrinolytic
reactions and includes both inner and outer convection. Results of this model show lysis patterns similar to those in Anand
et al. and Zidansek et al. [133,134], and suggest that both the lysis pattern and the lysis time depend on the pressure drop
across the clot.
The most biochemically and structurally detailed model as of this writing is the 3D stochastic multiscale model from
Bannish and colleagues [140,141]. A microscale model of a fiber cross section is used to model degradation of a single
fiber. Binding sites are distributed throughout the fiber cross section in accordance with known protofibril spacing. tPA can
convert plasminogen to plasmin, which can degrade binding sites or expose new sites. Each “protofibril” cross section has
six total binding sites (the exposed site plus five initial ly cryptic sites that can be exposed by plasmin), representing the six
chains of two overlapping fibrin monomers. Plasmin can crawl to neighboring binding sites, and the fiber is considered
degraded when 2/3 of the binding sites within the cross section have been degraded by plasmin. In this way, single-fiber
degradation is accurately modeled as “transverse cutting.” In the macroscale model, a fibrin clot is modeled as a 3D square
lattice, with each lattice edge representing a fibrin fiber. The diameter of the fibers and the pore size between them are
adjustable. A bolus of tPA molecules is introduced into a fibrin-free region abutting the clot, and the model tracks each
molecule as it binds to and unbinds from fibrin and diffuses through the clot. When a tPA molecule binds, a “tPA leaving
time” and fiber degradation time are calculated from distributions obtained from the microscale model. With this stochastic
multiscale model, it is possible to study the lysis of clots of different structure. In fact, Bannish et al. were able to reconcile
the conflict in the literature about whether coarse clots degrade faster or slower than fine clots [140,141]. They show that
the number of tPA molecules in the system determines what type of clot degrades faster (Fig. 5.10). Coarse clots have an
advantage when there are few er tPA molecules, because coarse clots have fewer fibers. Even though those fibers take
longer to degrade, degradation can start on all fibers at the clot front. When there is a lot of tPA present, fine clots degrade
faster because there is enough tPA to start degradation of all fibers at the clot front, and the individual fibers composing the
fine clot degrade more quickly than the fibers in the coarse clot.
FIGURE 5.10 Lysis front velocity, in mm/min, of a fine clot (red triangles) and a coarse clot (black circles) as a function of the ratio of the number of
tissue-type plasminogen activator (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/mm
(B) Laboratory experiments were run with five different tPA-to-surface-area ratios varying from 600 to 12,000 molecules/mm
mean of three independent trials. Clots were formed from the pooled plasma of six donors. Reproduced from Bannish BE, Chernysh IN, Keener JP,
Fogelson AL, Weisel JW. Molecular and physical mechanisms of fibrinolysis and thrombolysis from mathematical modeling and experiments. Sci Rep
2017;7:6914.
2
. Each symbol is the mean of 10 independent simulations.
2
. Each symbol indicates the
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While the Bannish et al. model accounts for detailed biochemistry and fibrin structure, it does not include blood flow or
the mechanical properties of fibrin fibers. Models that include flow and/or mechanics do not treat the kinetics and structure
in a realistic way (for example, Diamond and Anand [125] and Piebalgs and Xu [139] treat degradation as the uniform
shrinking of a fiber diameter, while Zidansek et al. [134] and Sersa et al. [136,137] treat the biochemical reactions
phenomenologically). There is a need for a comprehensive, mechanistic model that includes detailed information about the
biochemistry of lysis and the structure of fibrin fibers and fibrin clots under physiological blood flow. This type of model
would be an invaluable tool for innovating thrombolytic therapy.
SUMMARY
Mathematical models of blood clotting have made important contributions to the field of hemostasis by revealing new
mechanisms that regulate this complex, nonlinear process. We chose to highlight examples of such models where the real
power of mathemat ical models has been exploited. Because mathematical simulations give the concentrations of every
possible protein and enzyme complex at any point in space and time, one can and should use this information to
systematically and seamlessly probe the system to uncover the underlying mechanisms. Too often, models are used to
qualitatively reproduce experimental results without offering any mechanistic explanations of the results. We believe this is
where the field of mathematical modeling could be significantly improved. In addition, descriptions of mathematical
models can be highly technical and accessible only to the expert. It is the job of the mathematicians to analyze model
results and put them into an appropriate scientific context. Better yet, mathematicians should work across disciplines to
acquire other expert input on their modeling results. Multidisciplinary teams that include mathematicians, engineers,
physical scientists, biologists, and clinicians are likely to generate the most successful models that will yield outputs to
guide the development of therapeutic interventions and cater these interventions to individual patients.
ACKNOWLEDGMENTS
This work was supported in part by the National Institutes of Health (R01HL120728).
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