Physical modeling of clot formation

Jeremy Gunawardena pointed me to a pair of papers documenting an impressive effort in multiscale modeling, aimed at connecting biochemical events with events that happen on the cellular and super-cellular scales (Xu et al. 2010. doi:10.1016/j.bpj.2009.12.4331; Xu et al. 2008  doi: 10.1098/​rsif.2007.1202; full references below).  These papers are fascinating for many reasons: first, they describe a model for the formation of a blood clot that does quite a good job of recreating the dynamics of clot formation and the complex, inhomogeneous structure of the clot itself.  Second, the model is a great demonstration of how to merge models of different types of biological events that happen at different length scales (tens of nanometers to hundreds of micrometers) into one coherent whole. This is the kind of modeling we’ll need to get good at if we want to understand how molecular interactions influence the physiology of tissues and whole organisms, and the authors offer a very thoughtful discussion of why they selected the types of models they used at each scale and how they merged them.  And third, the authors are able to test model predictions against (their own) in vivo data; and they are finding interesting ways in which the model is wrong, so we’re learning something.

Before we dive into the details, let’s look at a snippet of the main result.  This is a movie showing the formation of a clot.  It’s a 2-D simulation, so you’re effectively looking at a cross-section of a blood vessel that has been damaged.  Red blobs, “quiescent” platelets; green blobs, activated platelets; blue blobs, platelets that are covered in fibrin.  Black blobs, other blood cells (red cells, white cells, it doesn’t matter).  The flow of the blood is from left to right, and the injury is the dark bar on the bottom edge.

[Incidentally, much of the action is happening right at the bottom edge, so please remember to click on any other portion of the page after you start the movie.  Otherwise the progress bar will cover the most interesting bits.]

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See how the black blobs — the non-platelet blood cells — get involved in the forming clot and participate in the breaking-off of aggregates of activated platelets?  This is something that hasn’t been captured by other efforts to model clot formation, and it’s important: broken-off bits of clots can have very nasty consequences, including causing pulmonary embolisms.  So the ability to model the structure of a clot in detail, and get a sense of how likely it is to break apart under different conditions, could be important.  If you model only what the platelets are doing, you can’t get at that question.  Similarly, the simulation allows you to get an understanding of the way the blood flows over the clot, and the turbulence at the back of the clot as the clot grows.  You can see the turbulence in the swirling behavior of cells behind the forming clot, especially as the clot gets large.

If you have any interest in multiscale modeling at all, I urge you to read these papers — at least Xu et al. 2008.  You don’t have to know or care about blood clotting; the paper provides enough background to understand how the biology played into the modeling, at least at a superficial level.  The authors walk you step by step through the problem — clot formation involves events at many different scales, from biochemical events (such as the production of thrombin and the conversion of fibrinogen into fibrin fibrils; all happening on a scale <0.1µm), to events that deal with the properties and behavior of cells (such as activation, adhesion, aggregation, movement, and interactions with the shear force of the flowing blood; all at a length scale of ~1µm), to the mechanical forces of the blood flow itself (~100µm).  The approach they use is to combine four submodels into one uber-model: an ODE-based biochemical model describing the behavior of the clotting factors, a stochastic cellular Potts model to deal with the cell state transitions the platelets undergo, as well as the interactions among cells and between the cells and the blood vessel wall; a model of blood flow based on the Navier-Stokes equations; and a model that deals with the interface between the surface of the developing thrombus (the outer edges of the cells on the exterior of the thrombus) and the blood flowing over it.  The thought process that goes into all this is clearly, though briefly, explained, and the compromises that are made to reduce the computational time required to a manageable level are also discussed.  An education.

In Xu et al. 2010, the authors extend this 4-part model with a much more detailed model of the coagulation pathway, including both solution-phase reactions (for example the activation of factors VII and VIII) and reactions that take place on the cell membrane.  This submodel consists of 22 PDEs and 23 ODEs, using parameter values mainly from the literature.  Using this model, they aim to understand the difference in clotting behavior between wild-type and Factor VII (FVII)-deficient mice.  First, they examine clotting behavior experimentally. They find that in the FVII-deficient mice, clots start off well but are unstable; chunks of platelets fall off instead of sticking, and the clot never reaches the size or fibrin density of the wild-type clot.  When they run the model with a very low FVII concentration, the simulation shows that thrombin generation is both delayed, and reduced to about 1/3 of the level seen with wild type; this therefore reduces fibrin accumulation.  This is consistent with the experimentally-observed reduction in size and stability of the clot. The authors also show that these effects of FVII deficiency kick in only when the FVII level is below 1% of wild type; this is consistent with the ability of certain partially FVII-deficient mice (with FVII levels ~1% of wild type) to survive much longer than mice that lack FVII entirely.

One thing the model doesn’t explain is that in the FVII-deficient mice the number of platelets attaching to the site of damage is reduced early on.  This implies that thrombin is needed even to get the very first platelets to stick — we didn’t previously suspect thrombin might be important at this early stage.

I challenge anyone — whether you’re interested in blood clotting or not — to read these papers and come away thinking that modeling is unnecessary in dealing with a system of this complexity.  There are so many interacting parts, some minute, some tiny, and some merely small, all with multiple potential states; there is absolutely no way to keep it all straight in your head.  Sure, once you know the answer you can wave your hands around and explain why the result is obvious in retrospect.  That’s not science.  If you’re dealing with a system this complicated, you can only propose rigorous, testable hypotheses if you have a way of pulling together all the information you already know — everything from the viscosity of plasma to the rate of dissociation of thrombin from a platelet surface — in a useful form.  The next trick is getting the coarse-graining right so that you can actually interpret what the model is telling you.  I’m not saying it’s easy… just that it’s becoming essential.

Xu Z, Lioi J, Mu J, Kamocka MM, Liu X, Chen DZ, Rosen ED, & Alber M (2010). A multiscale model of venous thrombus formation with surface-mediated control of blood coagulation cascade. Biophysical journal, 98 (9), 1723-32 PMID: 20441735
Xu Z, Chen N, Kamocka MM, Rosen ED, & Alber M (2008). A multiscale model of thrombus development. Journal of the Royal Society, Interface / the Royal Society, 5 (24), 705-22 PMID: 17925274