Thrombosis Research 133 (2014) S12–S14

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An overview of mathematical modeling of thrombus formation under flow☆ Karin Leiderman a,⁎, Aaron Fogelson b a b

Applied Mathematics Unit, School of Natural Sciences, University of California Merced, Merced, CA, USA Department of Mathematics, University of Utah, Salt Lake City, UT, USA

a r t i c l e

i n f o

Keywords: Coagulation Thrombus formation Thrombus permeability Mathematical model Multiscale model

a b s t r a c t In the last decade numerous mathematical models have been formulated to investigate specific components of the clotting system such as the tissue factor pathway of coagulation. Sophisticated, multiscale models were developed to better understand the interplay of flow-mediated transport, platelet deposition, and coagulation kinetics, and their overall effect on thrombus formation. Promotion of thrombus growth is partially due to the wellknown pro-coagulant roles of platelets like the surface-dependent coagulation reactions. Iterations of theoretical model predictions and experiment have helped to elucidate anticoagulant roles of platelets as well. These roles include paving over the subendothelium and hindering transport of substrates to and from enzyme complexes which can strongly affect thrombus formation. In this review, we give a brief overview of theoretical models of thrombus formation under flow and some of the experiments that motivated them and were motivated by them. © 2014 Elsevier Ltd. All rights reserved.

Introduction Extensive studies, both experimental and theoretical, have been performed to determine the complex processes by which thrombi form. Substantial progress has been made in identifying the players (proteins, cells, etc.) and many of the biochemical and cell-biological interactions in which they participate. Much less progress has been made in understanding how physical processes such as flow-mediated transport and platelet deposition might affect thrombus formation. It is also unclear how these physical processes might modify the view of thrombus formation that has been developed from the biochemical studies. The aim of this review is to highlight the progress made toward these goals through the use of theoretical modeling. Theoretical models of clotting are typically based on systems of equations that describe specific dynamic processes within the clotting system; these processes could be biochemical, biophysical, biomechanical, or a combination. The equations keep track of changes in certain desired quantities (e.g., thrombin concentration, platelet concentration, velocity) from their prescribed initial values. Ordinary differential equation (ODE) models work under the assumption that quantities are well-mixed, i.e., changes in these quantities are tracked only in time and there are no spatial variations. Partial differential equation

☆ Review Article for the Conference Proceedings for the 7th Symposium on Hemostasis: Old System, New Players, New Directions. ⁎ Corresponding author at: 5200 North Lake Road, School of Natural Sciences, University of California Merced, Merced, CA 95343, USA. E-mail address: [email protected] (K. Leiderman). 0049-3848/© 2014 Elsevier Ltd. All rights reserved.

(PDE) models have the ability to track quantities in both space and time so that complex spatial information can be extracted from them, e.g., thrombus density and permeability. Modeling efforts can be further characterized by the specific components of the clotting system on which they focus. These include coagulation kinetics without flow using ODEs [1–4], with simplified treatment of flow using ODEs [5–8], and more recently, multiscale PDE models that aim to integrate coagulation kinetics, platelet deposition, as well as flow [6,9–13]. One of these models also incorporated trained neural networks for patientspecific platelet responses [13].

Coagulation Kinetics ODE models of the tissue factor (TF) pathway of coagulation in a static system have been instrumental in delineating TF regulatory mechanisms and sensitivity of this pathway to initial TF concentration [1,3,4]. These models, and some ODE and PDE models based on them [11,14,15], simulate a biochemical environment in which there is an unlimited supply of phospholipid. This means that enzyme-complex formation is limited only by enzyme and cofactor concentrations when, in vivo, complex formation takes place on cell surfaces, where binding sites are limited and there is competition for these sites. Hoffman and Monroe were the first to emphasize that coagulation is regulated by properties of cell surfaces [16]. They developed a cell-based experimental model of coagulation which led them to propose that coagulation occurs not as a “cascade”, but in three overlapping stages: initiation, propagation, and amplification, all stages occurring on a

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cellular surfaces [17]. This new notion of coagulation has been widely accepted into the biological literature. The Kuharsky-Fogelson model captures the interplay between the biochemical kinetics, platelet deposition, and flow, and includes surface-dependent reactions [5]. The model treats platelet adhesion to a small injury; platelet activation in response to chemical agonists and incorporates a full description of TF-pathway biochemistry (right side of Fig. 1) with simplified mass transport of platelets and proteins into or out of the injury zone by flow and diffusion. This model predicted thrombin production to show threshold dependence on the extent of TF exposure, which was later confirmed experimentally [18]. The model also provided kinetic explanations for reduced thrombin production with hemophilia A and B and proposed that platelets adhering to the subendothelium cover enzyme complexes there and so serve to physically inhibit their activity. This mechanism, also confirmed experimentally [19] implies that platelets have an important anticoagulant role in addition to their well-recognized pro-coagulant ones. The balance between these roles underlies many of the models’ behaviors including the TF threshold. That a TF threshold was seen for experiments done with platelets and flow [18], but not without them [14] indicates the importance of these biophysical factors. The original model was extended to include APC production on endothelial cells [6], and FXI reactions [7]. The latter study predicts that low platelet counts attenuate the reduction in thrombin generation in hemophilia C (FXI deficiency), a prediction that underscores the intimate relationship between coagulation, platelet function and transport. Multiscale Models of Thrombus Formation Theoretical studies are often motivated by experimental studies and experimental investigation of the interplay between the biochemical and biophysical processes has recently been given more attention. Specifically, there has been interest in thrombus permeability and structural heterogeneity during formation under flow. To investigate this, researchers have utilized intravital microscopy in animal models [20,21] and in vitro flow chambers [18,20,22–25]. Furthermore, measuring and visualizing thrombin concentrations produced during thrombus formation under flow has only recently become possible in vitro [26]. Xu et al. developed a model in which they tracked individual discrete platelets and red blood cells in flow [11] and include coagulation biochemistry from [1]. Results predicted that red blood cell entrapment within a growing thrombus led to thrombus heterogeneity and structural instability. The model was later extended [12] to include updated coagulation biochemistry from [3], some of the platelet-surface reactions from [5] and a porous media representation for regions with


aggregated platelets. The new model predicted that thrombin production was not significantly impacted by thrombus permeability, a conclusion at odds with some other recent studies described below. Flamm et al. developed a multiscale model in which they trained neural networks of platelet signaling to predict patient-specific platelet responses [13]. This was embedded into a model that tracked discrete platelets and concentrations of two chemical agonists in a dynamic flow field. The models included highly robust treatment of intracellular platelet signaling and activation but did not account for coagulation biochemistry and thrombin production. Leiderman and Fogelson [9] developed a model of thrombus formation under flow that accounted for coagulation biochemistry, including surface-dependent reactions, as well as platelet activation and deposition. The model extends [5] to treat spatial variations in concentrations, heterogeneities within the thrombus, and flow disturbances caused by the growing thrombus. Platelet concentrations rather than discrete platelets were tracked. The thrombus was treated as a porous material which allowed for analysis of intra-thrombus fluid and solute transport, and revealed that diffusive solute transport aided in upstream thrombus growth. Results also showed that thrombus growth was strongly influenced by wall shear rate and the near-wall excess of platelets and that thrombus permeability strongly impacted growth and structure. This model was also able to predict coagulation timecourses in agreement with in vitro measurements for hemophiliac blood [27]. Motivated by recent experiments [20] that suggest an important role for intra-thrombus transport, the Leiderman-Fogelson model was extended [10] to further explore the effect of hindered transport of proteins within the thrombus. Results showed that early growth was promoted because platelets sheltered enzymes from the flow, and that later growth was greatly slowed because enzymes were confined to the interior core of the thrombus and the supply of substrates to them was greatly reduced (Fig. 1). With the transport hindrance, the final thrombus that emerged from the model was smaller, had a dense core and a less dense shell similar to thrombi described in [20,25]. Recently, two other studies have also focused on the implications of intra-clot transport, in the context of a core and shell structure, and that integrate modeling and experiment. Kim et al. investigated the role of a fibrin shell on thrombus growth [28]. Their experimentally measured permeability of the fibrin shell did not limit protein diffusion in simulations but did impede platelets from entering the thrombus, which could reduce platelet activation and thrombus stability (see also [29]). Using a thrombus geometry reconstructed from their experiments, Voronov et al. found similar results [30]; they measured thrombus permeability via simulation and report significant variance between

Fig. 1. (Left) Leiderman-Fogelson Model [10] A. Velocity vectors (arrows) and bound platelet density are colored from 0 (blue) to maximum packing (red). B. During thrombus growth, thrombin production becomes weaker and restricted to the progressively thinner regions within the thrombus because transport limitations prevent prothrombin from reaching the platelet-bound prothrombinase complex. (Right) Schematic of coagulation reactions in [5–7,9,10].


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shell and core. In both of these studies, the shell and core structure was prescribed rather than an emergent property of thrombus growth. Conclusions Ultimately, the goals of theoretical modeling are to gain biological insight and correctly predict overall clotting system responses. Doing so is not possible unless the most influential interactions of the system are accounted for quantitatively. As highlighted in this review, theoretical models that account for biochemical, biophysical, and biomechanical factors give a more comprehensive view of thrombus formation under flow that i) is not possible without this combination of factors, and ii) is motivated by and complementary to in vitro and in vivo experimentation. Iteration of testable theoretical predictions followed by experimentation is an approach that will lead to a better understanding of connections within the clotting system and thus elucidation of the mechanisms behind them. With these rapid advances in model sophistication as well as computer technology and experimental techniques with flow, we expect that continuing this approach will lead to development of novel approaches to reduce cardiovascular disease. Conflict of Interest Statement The authors wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome. Acknowledgements This work was supported, in part, by NIH grant 1R01GM090203-01 and NSF grant DMS-1160432. References [1] Jones KC, Mann KG. A model for the tissue factor pathway to thrombin. II. A mathematical simulation. J Biol Chem 1994;269:23367–73. [2] Chatterjee MS, Purvis JE, Brass LF, Diamond SL. Pairwise agonist scanning predicts cellular signaling responses to combinatorial stimuli. Nat Biotechnol 2010;28:727–32. [3] Hockin MF, Jones KC, Everse SJ, Mann KG. A model for the stoichiometric regulation of blood coagulation. J Biol Chem 2002;277:18322–33. [4] Butenas S, Orfeo T, Gissel MT, Brummel KE, Mann KG. The significance of circulating factor IXa in blood. J Biol Chem 2004;279:22875–82. [5] Kuharsky AL, Fogelson AL. Surface-mediated control of blood coagulation: the role of binding site densities and platelet deposition. Biophys J 2001;80:1050–74. [6] Fogelson AL, Tania N. Coagulation under flow: the influence of flow-mediated transport on the initiation and inhibition of coagulation. Pathophysiol Haemost Thromb 2005;34:91–108. [7] Fogelson AL, Hussain YH, Leiderman K. Blood clot formation under flow: the importance of factor XI depends strongly on platelet count. Biophys J 2012;102:10–8.

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An overview of mathematical modeling of thrombus formation under flow.

In the last decade numerous mathematical models have been formulated to investigate specific components of the clotting system such as the tissue fact...
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