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Chapter 5
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Mathematical Models of Thrombus
Formation and Fibrinolysis
Karin Leiderman1, Brittany E. Bannish2, Michael A. Kelley1and Ada M. Palmisano
1
Colorado School of Mines, Golden, CO, United States;2University of Central Oklahoma, Edmond, OK, United States
1
INTRODUCTION
Hemostasis is the normal process by which the body stops bleeding in the event of an injury. Primary hemostasis is the
initial formation of a platelet plug to slow plasma leakage from the vessel. The plug forms via platelet deposition and
adhesion to exposed collagen at the injury site and via platelet aggregation. Secondary hemostasis includes the formation of
a fibrin mesh to stabilize the platelet plug so that the injury can heal; blood coagulation is the major process of secondary
hemostasis, in which dozens of proteins act collectively to produce the enzyme thrombin, which cleaves fibrinogen
into fibrin. Although the descriptions of primary and secondary hemostasis might suggest that these processes work
independently, they are probably intimately intertwine d, coupled through positive feedback loops and agonist activation
of platelets. Once the injured tissue is healed, the clot is dissolved via fibrinolysis. These steps involve complex
biochemical processes that occur on multiple spatial and temporal scales and are strongly influenced by biophysical and
biomechanical effects. Due to the large number of proteins, chemicals, and cells, in addition to the combination of
biochemical, biophysical, and biomechanical effects on the system, it is difficult to intuit how the system will respond
without quantitative methods. In the past few decades, multiple mathematical models have been developed to address this
challenge. In this chapter, we give a comprehensive survey of such models, as well as some of the experiments that have
motivated them and been motivated by them. We focus specifically on models that consider biochemical and biophysical
effects; we reference and give brief summaries of other types of models. The aim of this chapter is to provide an overview
of the sophisticated models and methods that have been developed and to highlight some of the insight gained about an
amazingly intricate, complex system using the power of mathematics.
OVERVIEW OF DIFFERENTIAL EQU ATIONS
Most mathematical models of thrombosis, hemostasis, and fibrinolysis are built from differential equations. A differential
equation relates an unknown quantity to its derivative. For example, if function F(t) describes a quantity that de pends only
on time (not space), then the differential equation
says that F(t) is a function whose derivative, or instantaneous rate of change
function. A solution to the differential equation is a function F(t) satisfying the equation. In this case, the solution is
FðtÞ¼Fð0Þe
an ordinary differential equati on (ODE) because the function F depends on only one independent variable, t.
The complex reactions and motion of clotting factors involved in thrombosis often change in space, as well as time. For
example, functions F(x,t) and G(x,y,z,t) represent quantities that depend on time and one spatial dimension (x), or time and
Cardiovascular Thrombus. https://doi.org/10.1016/B978-0-12-812615-8.00005-3
Copyright © 2018 Elsevier Inc. All rights reserved.
dF
¼ aF
dt
at
, where F(0) is the initial condition, or the value of the quantity at time 0. The equation above is called
dF
; is equal to a constant a times the
dt
67
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three spatial dimensions (x, y, and z), respectively. A partial differential equation (PDE) is used to relate an unknown
quantity of several independent variables to its partial derivatives. For example,
vF
D
kF
vx
vF
and by diffusion
vx
v
vF
D
vx
vx
represents a function whose change in time
vF
vt
vF
vt
vF
vx
v
þ
vx
¼u
depends on transport by advectionu
and on decay ð kFÞ. This is known as a reactioneadvectionediffusion equation; the quantity described by F is carried
with velocity u by advection, is spread with coefficient D by diffusion , and reacts (in this case with itself) by decaying at
rate k. Solutions to these types of equations can be found analytically or numerically. To find a solution, it is necessary to
have an initial condition (F(x,0)) and boundary conditions,(F(0,t) and F(1,t), for example, if the spatial domain is defined
by 0 x 1).
The higher dimensional analog of this type of reactioneadvectionediffusion PDE is
vG
¼u$VG þ V$ðDVGÞkG
vt
where u is the velocity vector, V is the gradient (the multivariable generalization of the derivative), and V$ is the
divergence (a measure of the extent to which a vector field behaves like a source at an infinitesimal region of space).
Most of the models reviewed in this chapter are coupled differential equations: equations that involve multiple
unknown quantities. For example, if the aforementioned quantity G can irreversibly react at rate r with quantity H to
produce the complex G:H, then a system of coupled PDEs could be
vG
¼u$VG þ V$ðDVGÞkG rGH
vt
vH
¼u$VH þ V$ðD V H ÞkH rGH
vt
,
vðG: HÞ
¼u$VðG: HÞþV$ðDVðG : HÞÞ kðG: HÞþrGH
vt
Note that the new interaction term affects G and H negatively (rGH), because G and H are lost in the reaction, while
G:H increases (þrGH) because the interaction of G and H creates a new complex.
BRIEF BIOLOGICAL BACKGROUND
Coagulation and Platelet Aggregation
A break in a blood vessel triggers coagulation and platelet aggregation, two intertwined processes that are d esigned to seal
the injury. Platelet aggregation begins when flowing platelets form bonds between their receptors and molecules embedded
in the subendothelium that has become exposed at a site of injury (see Fig. 5.1). These bonds trigger the platelets to become
activated; activated platelets secrete chemical agonists that activate other platelets, resulting in a change in platelet
membrane properties to produce a surface that can support enzymatic reactions. In addition, activated platelets can form
molecular bonds and cohere to one another. Initially the platelet aggregates are relatively unstable and require products
from coagulation to remain intact. Coagulation is a complex biochemical process that culminates in the generation of the
enzyme thrombin, and progresses by means of enzymatic reactions on subendothelial, endothelial, and activated platelet
surfaces (see Fig. 5.2). Thrombin plays multiple critical roles in the overall process of clot formation. First, it is a powerful
platelet agonist, activating platelets at concentrations near 1 nM. Next, it acts on the soluble plasma protein fibrinogen to
produce insoluble fibrin that polymerizes and forms a stabilizing gel to hold the platelet aggregate together. Finally, it plays
a role in both positive and negative feedback of its own production in the coagulation network (see Fig. 5.2).
Here we briefly summarize the major features of coagulation and platelet aggregation. We use the following notation
and abbreviations: roman numerals represent the coagulation factors; tissue factor (TF), activated protein C (APC), tissue
factor pathway inhibitor (TFPI), antithrombin III (AT), thrombomodulin (TM).
There are dozens of proteins involved in thrombin generation and many have an inactive and an active form. The
inactive enzyme precursors are factors VII, IX, X, and II (prothrombin), each of which has a corresponding active enzyme
factor, VIIa, IXa, Xa, and IIa (thrombin), respective ly. The inactive/active cofactor pairs are V/Va and VIII/VIIIa. Note that
although the cofactors are in an active form, they do not function as enzymes, i.e., they do not activate other proteins. The
α
α
α2β
α
α
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GPlb
Platelet
llbβ3
llbβ3
Fibrin(ogen)
vWF
GPlb
llbβ3
Platelet
1
EC EC
GPVI
llbβ3
vWF
GPlb
Collagen
Subendothelium
FIGURE 5.1 Schematic of platelet aggregation. Depiction of platelet adhesion and aggregation receptors and their ligands. Each platelet surface has
approximately 25,000 glycoprotein (GP) Ib receptors that bind to subendothelium-bound von Willebrand factor (vWF), approximately 50,000 integrin
receptors that bind to the plasma protein fibrinogen and to vWF, approximately 4000 GP VI receptors, and 1000e4000 integrin a2b1receptors
a
IIbb3
that bind to various types of collagen. The integrin receptors must be activated to form strong long-lived bonds. Collagen is a key component of the
subendothelial matrix; fibrinogen is a soluble plasma protein; and vWF is adsorbed to collagen, circulates in plasma, and is secreted by endothelial cells
(EC). Adapted from Fogelson AL, Neeves KB. Fluid mechanics of blood clot formation. Annu Rev Fluid Mech January 2015;47(1):377e403.
FIGURE 5.2 Schematic of blood coagulation reactions; see detailed description in the text. Dashed magenta arrows show cellular or chemical activation
processes. Blue arrows show chemical transport in the fluid or on a surface. Green segments with two arrowheads depict binding and unbinding from a
surface. Rectangular boxes denote surface-bound species. Solid black lines with open arrows show enzyme action in a forward direction, while dashed
black lines with open arrows show feedback action of enzymes. Red disks show chemical inhibitors. Roman numerals represent the coagulation factors.
APC, activated protein C; AT, antithrombin III; EC, endothelial cell; PC, protein C; TF, tissue factor; TM, thrombomodulin.
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cofactors TF, VIIIa, and Va significantly increase the catalytic efficiency (up to 5 or 6 orders of magnitude) of their
corresponding enzyme (VIIa, IXa, and Xa, respectively) compared with that of the enzyme without its cofactor.
Many of the reactions occur only on a cellular surface, and there are three cell-bound enzymeecofactor
complexesdTF:VIIa on the subendothelium and VIIIa:IXa (“tenase”) and Va:Xa (“prothrombinase” ) on activated platelet
surfacesdthat play major roles in thrombin generation. Many of their substrates (i.e., the proteins that the enzyme
complexes activate) must also be bound to the cellular surface to become activated. Hoffman and Monroe were the first to
emphasize that coagulation is regulated by properties of cell surfaces [1]. They developed a cell-based experimental model
of coagulation, which led them to propose that coagulation occurs not as a “cascade” as previously thought, but in three
overlapping stages, initiation, amplification, and propagation, each occurring on a cell ular surface [1e3]. Here, we
summarize these stages (and the action of inhibitors that funct ion throughout each stage).
1. Initiation: Small amounts of Xa and thrombin are generated on a TF-bearing surface.
The TF complex TF:VIIa produces IXa and Xa on the subendothelial surface. The Xa may activate its cofactor V on the
subendothelium, form Va:Xa, and generate small amounts of thrombin. Platelets that adhere and become activated may
also provide V or Va for this initial step [4].
2. Amplification: Thrombin and Xa activate cofactors on lipid or activated platelet surfaces.
The small amounts of thrombin are enough to activate platelets. In addition, this thrombin activates platelet-bound
coagulation cofactors V and VIII (also activated by small amounts of Xa) and XI. Thus, Va and VIIIa become present
on the activated platelet surfaces and VIIIa:IXa complexes begin to form.
3. Propagation: A burst of thrombin is produced on lipid or activated platelet surfaces.
The VIIIa:IXa complexes rapidly produce Xa, more Va:Xa is formed, and an explosive burst of thrombin results.
Factor XI is activated by thrombin primarily on platelet surfaces and boosts the thrombin generation even more via IXa.
4. Inhibition: Coagulation enzymes/cofactors are inhibited in solution and on lipid and activated platelet surfaces.
APC is formed by a complex of TM and thrombin on the surfaces of endothelial cells and may inactivate Va and VIIIa.
AT inhibits thrombin, Xa, IXa, and XIa, and TFPI inhibits Xa and TF:VIIa.
There are specific binding sites expressed on the activated platelet surfaces; there are binding sites for each of the
zymogen/enzyme pairs IX/IXa and X/Xa, for each of the inactive/active cofactor pairs V/Va and VIII/VIIIa, and for
prothrombin. Thrombin and XI/XIa bind nonspecifically to the surfaces, but become associated nonetheless. The quantity
of available binding sites controls the formation and/or activity of the platelet-bound enzyme complexes [5e7].
Fibrin Polymerization
Thrombin converts the plasma protein fibrinogen into fibrin, which forms a gel that stabilizes the platelet aggregates. This
process occurs in several steps as sketched in Fig. 5.3. Fibrinogen is a molecule with trinodular structure; it has a central
region (E domain) that contains two pairs of fibrinopeptides, A and B, and two distal end nodules (D domains). Thrombin
cleaves the peptides in the E domain of the fibrinogen molecule, converting it to a fibrin monomer. This process exposes
binding sites, termed “knobs,” which fit into complementary “holes” on the distal end of separate fibrin monomers. The
monomers assemble in a half-staggered configurat ion and yield protofibrils, which then laterally aggregate to form fibers.
During lateral aggregation, branch points form and aggregated fibers cross-link with one another, leading to the formation
of a fibrin gel. Details of the fibrinogen a nd fibrin processes have been reviewed elsewhere [8,9].
Fibrinolysis
Fibrinolysis, the enzymatic degradation of the mesh of fibrin fibers, is initiated by plasminogen activators such as urokinase
(uPA) and tissue-type plasminogen activator (tPA). These plasminogen activators convert plasminogen to the enzyme
plasmin, which degrades fibrin. Unlike uPA, tPA is a fibrin-specific plasminogen activator; plasminogen bound to fibrin
near tPA can be activated to plasmin. Plasmin degrades fibers by cutting across them laterally (rather than by uniformly
shrinking the fiber diameter) [10,11], and exposes new lysine residue binding sites in the process (see Fig. 5.4).
Degradation of fibrin clots proceeds as a front, with a high accumulation of both tPA and plasminogen at the front [12].Itis
generally accepted that fine clots composed of thin, densely packed fibers degrade more slowly than coarse clots composed
of thicker, less densely packed fibers [13,14], though some experiments have shown the opposite result [15,16]. Three
main inhibitors regulate the fibrinolytic cascade: a
activated thrombin-activatable fibrinolysis inhibitor (TAFIa). PAI-1 and a
and plasmin, respectively. TAFIa is an indirect inhibitor that cleaves the binding sites exposed by plasmin, effectively
preventing the positive feedback otherwise present in lysis (see Fig. 5.4).
-antiplasmin (a2-AP), plasminogen activator inhibitor-1 (PAI-1), and
2
-AP are direct inhibitors, inhibiting tPA
2
α
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FIGURE 5.3 Schematic of the knobehole interaction in fibrin polymerization. Fibrinopeptides in the central E domain cover knobs that are
complementary to holes that are always exposed at the ends of the protein in the D domain. After the fibrinopeptides are removed upon cleavage by
thrombin, knobehole interactions occur and give rise to the two-stranded protofibril made up of half-staggered molecules. Adapted from Weisel JW.
Fibrinogen and fibrin. In: Advances in protein chemistry [internet]. Fibrous proteins: coiled-coils, collagen and elastomers, vol. 70. Academic Press;
2005. p. 247e99. Available from: http://www.sciencedirect.com/science/article/pii/S0065323305700085; Wolberg AS. Thrombin generation and fibrin
clot structure. Blood Rev May 2007;21(3):131e42.
tPA
PAI -1
PLi
-AP
2
PLG
FIGURE 5.4 Schematic of fibrinolysis. Plasminogen (PLG) and tissue-type plasminogen activator (tPA) diffuse in the plasma and can bind to fibrin. If
tPA and PLG bind in close proximity to one another, the tPA can convert the PLG to plasmin (PLi). PLi then degrades the fibrin by cutting across the fiber.
New binding sites for tPA and PLG are exposed as PLi cleaves fibrin. Activated thrombin-activatable fibrinolysis inhibitor (TAFIa) cleaves these newly
exposed binding sites. Plasminogen activator inhibitor-1 (PAI-1) and a
tPA
tPA tPA
PLG
PLG
Fibrin fiber
-antiplasmin (a2-AP) inhibit unbound tPA and PLi, respectively.
2
Plasma
PLG
TAF la
PLi
PLG
binding
sites
PLi
MODELS OF THROMBIN GENERAT ION AND THROMBUS FORMATION
Zero-Dimensional Models
Multiple ODE models of thrombin generation have been developed since the early 1990s. A primary purpose for the
development of such models has been to simulate thrombin generation that occurs in a test tube, in the presence of
phospholipid vesicles [17e21] or in the presence of platelets [22]. ODE models in this context work under the assumption
that constituents in the test tube are well or instantaneously mixed. The HockineMann (HM) model, developed in the
Mann laboratory, is the most widely used [18]. The HM model has been instrumental to understanding the sensitivity of
this complex system to initial TF concentrations and inhibitors, and the activation threshold behavior of coagulation in
general. In addition, the model has been used as is, or slightly extended, to investigate sensitivity in normal variations of
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clotting factors in healthy individuals [23], to assess risk of disease [24], and to investigate mechanisms underlying
complications of trauma and coagulopathies [25e27].
A major assumption in all but one of these models is that the environment has an unlimited supply of cell (lipid)
surface, and thus the binding to and from lipid surfaces is not accounted for. Under this assumption, enzyme-complex
formation is limited only by enzyme and cofactor concentrations, but because complex formation takes place on cell
surfaces, there may be competition for limited binding sites that is unaccounted for. Negl ecting this competition could
affect the ability and/or timing of complex formation, which could change the model predictions of timing, rate, and
maximum of a thrombin burst. Bungay and Gentry [21] took a slightly different approach from the HM model and
considered competition of a finite concentration of lipids and included the APC inhibition pathway. They reported that
thrombin concentration had a threshold-like dependence on lipid concentration and that the three inhibitors, AT, APC, and
TFPI, had maximum effects at different concentrations of lipids. Chatterjee, Diamond, and colleagues developed a static
model that accounted for platelets and platelet activation to investigate the behavior of blood treated with corn trypsin
inhibitor (CTI; an inhibitor thought to inhibit the intrinsic pathway via XIa) in the absence of TF [22]. They did not
consider association and dissociation from the platelet surface, but instead assumed that the reaction rates increased as a
function of the number of activated platelets. In that study, with experi mental confirmation, they predicted that XIIa
initiates coagulation via the intrinsic pathway, even in the presence of CTI.
Mathematical models can also aid in the design and test the efficiency of anticoagulants. Mathematical models are used
to simulate clinical assays such as the test of prothrombin time and the activated partial thromboplastin time [28,29],to
estimate the plasma concentrations of coagulation factors in vivo using a pharmacokineticepharmacodynamic approach
[30], and to predict drug distribution at a tissue/ organ level using physiologically based pharmacokinetic models [31].
Direct Xa inhibitors have been the focus of multiple applications of these models. The HM model was used to compare the
effects of a direct, reversible Xa inhibitor, rivaroxaban, with those from a cofactor for AT, fondaparinux, on a thrombin
generation assay. They concluded that rivaroxaban was more effective than fondaparinux in suppressing an ongoing
coagulation process, due to its direct inhibition of the prothrombinase complex. They also found that fondaparinux had a
significant effect on IXa, in addition to Xa. A different study investigating the efficacy and safety of rivaroxaban modeled
plasma concentrations in vivo by including a flow component [32]. Model predictions suggested a broad therapeutic
window in accordance with the final dose recommendation based on clinical studies. The model was later extended to
inform pediatric dosing of rivaroxaban [33] and to include warfarin to provide a mechanistic rationale for dosing schedules
that switch between the drugs [34].
In summary, there are a variety of mathematical models of thrombin generation that are appropriate for simulating
specific clinical and research assays. However, this chapter will focus on the spatialetemporal model s that incorporate both
biochemical and biophysical effects on the blood clotting system, so we refer the reader to other excellent reviews on ODE
models of coagulation [35e39].
Coagulation Under Flow: Zero-Dimensional Models
Kuharsky and Fogelson developed an ODE model (KF model) to simulate thrombin generation at the site of a small
vascular injury under flow [6]. This model was an application and extension of previous work in which they showed that
regulating the density of membrane binding sites can act as a threshold switch for surface-mediated enzyme activity [5].
The flow model was proposed as a means to extend that conclusion to a more physiologically relevant system of
coagulation. The KF model addressed questions about whether a threshold behavior of thrombin generation woul d exist
due to TF density at the site of an injury, whether the major inhibitors of the coagulation reactions were chemical or
physical, and whether the model could kinetically explain the bleeding disorders caused by hemophilias A and B.
The KF model accounts for platelets, platelet deposition, and surface-mediated coagulation reactions, all under flow.
Each of the proteins and platelet species are transported to and from a reaction zone by flow and diffusion via mass transfer
coefficients. The reaction zone is a region located just above a small patch of exposed subendothelium, defined by the
height a particle can be above the injury and still diffuse to the injury before being carried away by the flow. The reaction
zone increases in thickness as platelets adhere to the subendothelium or deposit on already-bound platelets. The authors
proposed an inhibitory, anticoagulant role for the adherent platelets, namely that they physically inhibit the enzyme
complexes embedded in the injury patch. This notion of “paving over” the injury was later confirmed experimentally [40].
Three species of platelets are considered: mobile unactivated, subendothelium-bound activated, and bound (not to
subendothelium) activated. Platelets become activated by adhering directly to the subendothelium or by exposure to a
sufficient level of thrombin or ADP. ADP release is not explicitly modeled, so exposure to activated platelets acts as a
surrogate for chemical agonists like ADP released from the platelets.
y
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(A) (B)
100
100/s
500/s
1000/s
Shear Rate
10
Thrombin concentration
Control
TF Tirtation
TF densit
FIGURE 5.5 Importance of biophysical mechanism. For various shear rates, thrombin production shows sharp threshold behavior (A) in the
mathematical model of coagulation under flow [6] in an experiment with platelets and flow [41], (B) but not without flow [22]. TF, tissue factor.
TF concentration
A systematic study varying the amount of exposed TF showed that thrombin generation behaves in a threshold-like
manner: below some TF density there is little to no thrombin generated; above that density, there is a strong thrombin
response with a final concentration that saturates as TF density increases. This threshold behavior prediction was
later confirmed experimentally with platelets under flow [41], but not without [22], which highlights the importance of
biophysical factors (see Fig. 5.5). It was also shown that the model could account for the bleeding in hemophilias A and B
only when TF:VIIa activation is significantly blocked, as is the case when platelets pave over the subendothelium. Another
interesting prediction from the model was that dilution of protein species was a more potent inhibitor than chemical
inhibitors such as TFPI, AT, and APC.
To further investigate this notion, the model was extended [42] by Fogelson and Tania to incorporate APC and a
modified mechanism of action for TFPI [43], but even in those cases, flow was still found to be the primary inhibitory
mechanism. Subsequent modeling studies from a different group proposed an additional kinetic scheme for TFPI [44],but
these schemes have yet to be tested in the KF model. An additional prediction from the extended version of the KF model
was that the appearance of a thrombin burst depends on the outcome of a race between platelet coverage of TF activity in
the subendothelium and establishment of a sufficiently high density of the tenase complex on the adherent platelets within
the reaction zone (see Fig. 5.6). Even though flow is determined to be the dominant inhibitor at the site of injury, the
authors suggest that APC and TFPI near the site of an injury, along with AT in the plasma, could play a key role in limiting
the spread of coagulation processes away from the injury site.
The KF model was extended yet again by Fogelson, Hussain, and Leiderman to include the activation of XI into XIa by
thrombin and the subsequent activation of IX into IXa by XIa in the plasma and on the surface of activated platelets [45].
With a typical physiological platelet count (250,000 platelets/mL) at a relatively low shear rate (<200/s), XIa activated by
thrombin was shown to play an important role in overall thrombin production only at lower TF levels, by effectively
reducing the TF-density threshold. The findings show, however, that if the near-wall concentration of platelets is high (due
to a higher systemic platelet count or an increased shear-rate-dependent platelet margination), XIa plays a much more
crucial role in thrombin production for high levels of TF. This behavior is due to the increased rate of platelet coverage of
TF activity, which results in attenuated production of IXa on that surface. To overcome that limitation, XIa provides
another mechanism by which IXa can be produced both in the plasma and on cell surfaces.
In conjunction with in vitro thrombin generation assays, Leiderman and colleagues used the extended KF model to
explore possible synergistic effects between exogenous XIa (E-XIa) and TF on thrombin generation [46]. Motivation for
this study was the mounting evidence that either XI or XIa is involved in the development of thrombotic events such as
immunoglobulin intravenous treatments [47e52]. In the thrombin generation assays, for increasing concentrations of
E-XIa with low (but not with high) TF concentrations, the peak thrombin significan tly increases while lag time and time to
peak significantly decrease. There is a similar dependence on XIa concentration of lag time and thrombin generation rates
in the mathematical model results (see Fig. 5.7).
Two specific cases were further investigated with the mathematical model: high TF density with no E-XIa and low TF
density with low E-XIa. In both cases a thrombin burst was reported, although in the latter case the burst occurred much
later. The model revealed that the major difference in these thrombin generation time courses is in the pathway to forming
the tenase complex (IXa:VIIIa). Recall that this mathematical model had previously predicted that the appearance of a
thrombin burst depends on the outcome of a race between platelet coverage of TF activity and establishment of a
sufficiently high density of the tenase. For the high-TF-density case, tenase is formed on the platelet in the following way:
(1) sufficient IXa and Xa are produced by TF, (2) Xa binds to the platelet and activates VIII to VIIIa, and (3) TF-activated
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FIGURE 5.6 Thrombin burst depends on tenase formation and platelet coverage. Scaled concentrations showing fluid-phase factor IXa (FIXa), available
platelet-bound FVIIIa, and platelet coverage of the subendothelium. Left: TF density above threshold level; there is a time at which FIXa and FVIIIa
overlap, showing the potential to form tenase complexes and produce local FXa before the subendothelium is paved over by platelets; a significant
thrombin burst (not shown) occurs in this case. Right: below-threshold TF density; there is no time at which FIXa and FVIIIa overlap; no thrombin burst
occurs in this case. SE, subendothelium; TF, tissue factor. Adapted from Fogelson. Fogelson AL, Tania, N Coagulation under flow: the influence of
flow-mediated transport on the initiation and inhibition of coagulation. Pathophysiol Haemost Thromb 2005;34(2e3):91e108
IXa binds to the platelet surface, predominantly to the binding sites exclusive to IXa. For the low-TF-density, low-E-XIa
case, tenase is formed by: (1) a positive-feedback loop in which very small amounts of TF-activated FXa bind to the
platelet surface and use bootstrapping to create more platelet-bound FXa, (2) IX binding to shared IX/IXa binding sites on
the platelet surface, and (3) platelet-bound IX being activated by platelet-bound XIa.
Thus, the model predicts that blocking IX binding to shared IX/IXa binding sites, in the presence of E-XIa and small
amounts of TF, would reduce thrombin generation without changing the system response at higher densities of TF. This
observation may provide a possible avenue for treating thrombosis without significantly affecting hemostasis.
Coagulation and Biophysical Factors: Two- and Three-Dimensional Models
The past few decades have seen a burst of experimental and theoretical studies aimed at understanding the complex process
of blood clot formation and degradation. Significant advances have been made to determine the precise biochemical
schemes and players involved in coagulation, but only recently has more attention been focused on biophysical and
biomechanical processes such as flow, platelet deposition, and aggregation. Moreover, it is still not completely understood
how coagulation is affected by, and coupled to, these physical processes. There has been interest in understanding blood as
an excitable medium, in size thresholds for initiating clotting, and in the structural heterogeneity and permeability of clots
as they form under flow. Theoretical studies are often motivated by experimental studies, and here we give a summary of
mathematical modeling efforts that include a significant biochemical and biophysical component. We note that an
enormous amount of progress has been made in computational modeling of blood cells in flow, platelet adhesion,
activation, aggregation, and thromboembolism, in terms of cell-membrane mechanics, physics, hydrodynamics, the
multiphase nature of blood, and many-particle suspensions in complex geometries. Many of these models focus heavily on
the details of platelets and red blood cells with extremely simplified biochemistry (e.g., only prothrombin, thrombin, AT,
and fibrin). There are relatively few models that couple the physical processes to the biochemistry, even though
coagulation biochemistry is inherent to secondary hemostasis and necessary to understanding the intricate control and
regulatory mechanisms that comprise the hemostatic response. In this chapter, we concentrate on the modeling efforts that
have significant focus on both biochemistry and biophysical processes, and we refer the reader to reviews on models of
platelet aggregation and margination, multiphase-fluid models, and mesoscale particle methods [53e60].
Blood as an Excitable Medium
There is experimental evidence that under conditions in which the diffusive contribution to mass transfer is much higher
than the advective contribution (i.e., static or very low flow conditions), blood acts as an active medium and coagulation
propagates by a self-sustaining traveling wave of thrombin [61]. Interestingly, it was shown that TF controls the initiation,
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(A) (B)
1200
1000
800
600
400
Lag Time (sec)
200
10
–2
10
–1
10
0
10
1
TF (fmol/cm2)
(C)
120
FXIa (pM)
(D)
5.4
100
1.8
0.6
80
0.2
0.067
60
0.020
0.007
40
20
Thrombin Lag Time (min)
0
0.000
0.01 0.03
0
0.1 0.3 0.000 0.01 0.03 0.1 0.3
TF (pM) TF (pM)
100
80
60
40
[Thrombin] (nM)
20
0
10
–2
10
–1
TF (fmol/cm2)
1.2
1.0
0.8
0.6
0.4
0.2
0.0
Thrombin Peak Height (normalized)
10
0
10
1
Subendothelium
(E) (F)
Xa
VIII
VIIIa
Xa:VIII
Xa
X
TF:VIIa
Xa
X:VIIIa:IXa
IX IXa
X
XIa
IX
IXa
Xa
VIIIa:IXa VIIIa:IXa
AP
IXa
IXa
XIa:IX
IXa
IXa
IXa
Xa:VIII
Xa
Xa:VIII
Xa
Xa
Xa
Xa:VIII
VIIIa
VIIIa
Xa
VIIIa
TF:VIIa
AP
IXa
IXX
IXa
IXa
IXa
IXa
IXa
IXa
IXa
IXa
IXa
IXa
IXa
FIGURE 5.7 Model and experiment showing synergy between exogenous XIa and TF levels. (A and B) Thrombin generation assays (TGAs) showing
the effects of varying exogenously added TF and XIa. (C and D) Mathematical model results showing the effects of varying TF density and exogenously
added XIa. The colors of the lines indicate the concentration of initial exogenous XIa; the colors in (A) and (B) correspond to the colors in (C) and (D). In
the TGAs, thrombin lag times and peak height are sensitive to changes in TF density for low values of exogenous XIa. In the math model, thrombin
plateau values are sensitive to changes in TF density for low values of exogenous XIa. For the higher doses of XIa used, the peak heights were like those
seen for higher TF but the lag times were progressively longer with smaller TF. For lower levels of XIa that still produced a burst, the resulting lag times
were longer, with corresponding peak heights reduced. However, small doses of TF and exogenous XIa, each far from sufficient to elicit a burst in
thrombin production alone, act synergistically to cause substantial thrombin production. The mathematical model was used to uncover two different routes
to tenase formation. Schematics illustrating these routes for (E) high TF with no exogenous XIa and for (F) low TF together with a low concentration of
exogenous XIa are shown. (E) Sufficient Xa binds to the activated platelet (AP) surface to activate platelet-bound VIII and thus allows for tenase
formation. (F) The feedback loop in which small amounts of plasma Xa bind to the platelet surface and use bootstrapping to create more platelet-bound
FXa is shown. Note that all reactions shown take place on the platelet surface only. TF, tissue factor.
76 Cardiovascular Thrombus
https://t.me/med1917
but the tenase complex controls the speed of the wave [62]. In subsequent experimental studies, it was found that
coagulation initially propagates at a constant speed, then abruptly stops and becomes surrounded by an “inhibition zone” in
which coagul ation is strongly suppressed [63]. This view of coagulation lends itself well to mathematical analysis of
reactionediffusion equations. A series of PDE models (all comprising systems of reactionediffusion equations) have been
developed in this context to better understand the mechanisms underlying these observations [64e69]. The models predict
a mechanistic switch in the system via APC. The models also show a secondary inhibitory wave (moving faster than
the thrombin wave) that stops clot growth and identifies roles of specific coagulation pathways during the initiation,
amplification, and propagation stages of clot formation.
Size Thresholds
In the aforementioned experiments and simulations, the patch of TF-bearing material was always large enough to initiate
coagulation. One natural question then arises: is there a minimum size patch to trigger a strong coagul ation response? Size
thresholds in the context of coagulation could limit the clotting response to sufficiently serious injuries. Beltrami and Jesty
developed a mathematical model that accounted for binding sites and surface diff usion within a TF-bearing patch. Their
model predicted a minimum patch size of 5 mm to initiate coagulation under static conditions, but reported that larger
patches are necessary under flow [70]. Ismagilov and colleagues conducted laboratory experiments and numerical
simulations to investigate regulation of coagulation by TF patch size under both static and flow conditions [71e74].In
their studies, reactions produce enzymes at the patch, and diffusion removes enzymes from the patch. For specified TF
density in the patches, a threshold patch size greater than 90 mm is necessary to initiate clotting. In later studies that
included flow, they found that the threshold size increases with increasing flow shear rate. They suggest that blood could be
exposed to a significant amount of TF without initiating clotting. Jordan and Chaikof developed a mathematical model of
thrombin generation under flow in which multiple TF patches are separated by patches of TM that could generate the
inhibitor APC [142]. Interestingly, they found that multiple TF sites, each that individually failed to activate the coagulation pathway, interact in an additive manner to yield a prothrombotic system.
AdvectioneReactioneDiffusion Models and Spatial Heterogeneity
Experimental efforts are aimed at characterizing permeability and spatial heterogeneity of blood clots in animal models
[75e79] and in vitro flow chambers [41,75,80e83]. Since measuring and visualizing thrombin concentrations produced
during thrombus formation under flow have only recently become possible in vitro [84], a major challenge for theoretical
studies is going beyond simple qualitative agreement with experiments.
Diamond and colleagues have taken a data-driven approach to modeling clot formation under flow [85]. Using a
neural-net trained model of platelet activation [86], the platelet motion is simulated by their lattice kinetic Monte Carlo
method (i.e., computational algorithms that rely on repeated random sampling to obtain numerical results; in
physics-related problems, Monte Carlo methods are useful for simulating systems with many coupled degrees of freedom,
such as fluids, disordered materials, strongly coupled solids, and cellular structures) [87,88], but only a few coagulation
proteins are considered. They use donor platelets and predict individual responses to various agonists and deposition rates.
They extended the model to include the biochemistry from Chatterjee et al. [22] and their simulations make accurate
predictions of occlusion times observed with whole blood flowed over collagen and TF in microfluidic devices [80].
Xu et al. developed a model in which they track individual discrete platelets and red blood cells in flow [89], coupled to
a reduced set of coagulation reactions [90]. Fibrin is modeled as a background concentration that affects platelet
aggregation. Platelets and blood cells are represented using a cellular Potts model [91,92] with specified thresholds for
aggregation and the release of clotting factors. As platelets become activated, they add to the mass of the growing clot and,
in turn, affect where the fluid can flow. The major prediction from this model is that red blood cell entrapment within a
growing clot leads to clot heterogeneity and structural instability.
This model was extended to investigate the effects of pulsatile flow and non-Newtonian viscosity (blood viscosity
that varies with local shear rate); biochemical reactions were updated to mimic those from the HM model, some of the
surface-mediated reactions from the KF model were included, and a porous media representation of the clot was used to
account for flow [93e95]. The model predicts that low levels of VII in blood cause a delay in throm bin production early in
the development of venous clots. In addition, the model predicts that thrombin production is not significantly influenced by
thrombus permeability, a conclusion contrary to other studies described next.
Leiderman and Fogelson (LF) developed a comprehensive spatiale temporal model of blood coagulation under flow [7]
that accounts for detailed biochemistry from the KF model, in addition to platelet deposition and aggregation under flow.
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