A group of biophysicists, including representatives from MIPT, has developed a mathematical model of arterial thrombus formation, which is the main cause of heart attacks and strokes. The scientists described the process of platelet aggregation as being similar to the popular video game Tetris and derived equations that allowed them to reproduce the wave process of platelet aggregate formation in a blood vessel.

Activated thrombocyte on a slide with immobilized fibrinogen, one of the proteins found in the blood. Scanning electron microscopy. Photograph courtesy of Sergey Obydenny / Wikimedia

Researchers from the Dmitry Rogachev Federal Research and Clinical Center of Pediatric Hematology, Oncology and Immunology (FRCC PHOI), the Center for Theoretical Problems of Physico-Chemical Pharmacology, MIPT, Moscow State University, the Institute for Problems of Mechanical Engineering (St Petersburg), and the Free University of Berlin described the new model in the journal PLOS ONE. When discussing the development of the model and the members of the research team, Mikhail Panteleev, one of the authors of the publication, said:

“Fazly Ataullakhanov and I formulated the problem and derived the equation, and Evgenia Babushkina (now a graduate student in Berlin), under the guidance of her mentor Nikolay Bessonov, developed solution methods for the two-dimensional case taking into account the changing hydrodynamics of the flow in which the thrombus is formed. She also performed all the simulations and analyses. Fazly and I are professors of the Department of Translational and Regenerative Medicine of MIPT’s Faculty of Biological and Medical Physics, which is based at FRCC PHOI”.

Develpoment of a continuous adsorption equation under assumption that thrombus formation occurs in the same way as the tiles become stacked up in the classic video game Tetris is a key aspect of the new model. In Tetris, the tiles either drop down onto a flat surface, or become attached to parts sticking out from the rest of the block. The only difference between thrombus formation and the game is that when a layer is complete, it does not disappear, therefore as time passes a thrombus is capable of obstructing the space it is in. In addition, the falling shapes are always the same: the model describes the aggregation of platelets, specialized blood cells.

Having described the mathematical process of how vacant areas on the surface of a growing thrombus are filled, the scientists were able to build first a one-dimensional model (as in Tetris), and then a two-dimensional model (platelets are deposited in a dimensional plane). And at one point, the scientists began to consider certain platelets as being infinitely small, and the thrombus itself as being continuous; in other words, the scientists went from a discrete model to a continuous model.

Discrete model: the system under study consists of individual particles, and the behaviour of each particle can be tracked individually. This makes it possible to simulate, for example, gas molecules in the problem of Brownian motion – representing each molecule as a particle colliding with a larger particle.

Continuous model: the system under study consists of solid objects that can freely change their size or any other characteristic. This can be used to model temperature increase in a functioning boiler for example – the output will be the temperature field in the volume under study.

The sequential solution of the equations obtained enabled the researchers to reproduce the dynamics of the growth of a real thrombus and study its behaviour under various conditions – in the case of damage to the vascular wall for example.

Active media and autowaves

In their paper, the researchers emphasize that the process of thrombus formation resembles a travelling wave, and this similarity is by no means accidental. It was previously demonstrated that the thrombus formation process is like an autowave – the blood, which carries blood platelets and a number of special proteins for blood coagulation, is an active medium. At that time the conclusion drawn by the researchers was concerned with blood coagulation as a result of a cascade of biochemical reactions involving proteins, but it is also possible to talk of an active medium in the case of adhered platelets.

The term “active medium” plays a key role in non-linear dynamics – the science of mathematical modelling of a whole range of systems, from mixtures of interacting chemicals and lasers to forest fires and even social networks. The easiest way to describe an active medium is to use the example of a forest fire: every dry tree is not simply a passive object such as a molecule, but a potential source of thermal energy. If there is a fire near a dry tree, it too will start to burn and provide more heat, which can then ignite other trees. The ability of elements in the system to release energy is a key feature of an active medium.

What is there in common between a forest fire:

...and a laser?

Molecules of carbon dioxide emitting photons of infrared radiation in a fluorescent glass tube of a gas laser, and trees that ignite as a fire approaches are both part of active media. They all have a certain amount of energy that can be released during a particular process. Blood is also an active medium and can also be described using the methods of non-linear dynamics.

In active media, a local event (lightning striking a tree for example) can initiate a transition process in a system from one state to another (in this case a dry tree becomes a burning tree). This process spreads like a wave in space and the specific physical nature of the system is not so important; the same equation can be used to describe entirely different cases. The term “autowave” means that wave propagation process is not passive, as in the case of seismic waves travelling from an earthquake’s epicentre, but active – at each point the wave receives more energy. In the case of thrombus formation, instead of dry and burning trees, we need to think of platelets flowing in blood plasma. These platelets can go from a free-flowing state to a deposited state.

Thrombi: essential, but also dangerous

Blood platelets play an important role in forming thrombi, which block the blood vessels of clots. Under normal circumstances, they flow freely in the bloodstream, but if the vascular wall becomes damaged, they start to adhere to one another and to the vascular wall; in addition to this, the blood also contains many proteins required for thrombus formation. Even if there are no platelets, reactions with these proteins are able to help form a clot to block a damaged vessel and these reactions also occur in the form of autowaves. Normally, thrombi prevent blood loss in the human body when a blood vessel has become damaged. Sometimes, however, thrombus formation occurs not as a result of an injury with damage to a blood vessel, but as a result of a reaction to a pathological process, such as the build-up of fatty plaque within an artery in cases of atherosclerosis. This type of thrombus formation can block a vessel completely and cut off the blood supply to tissues and organs: this in turn can lead to myocardial infarction (blockage of the arteries to the heart), stroke (blockage of the arteries supplying blood to the brain), or gangrene of the extremities. The new model correctly describes arterial thrombus formation – these particular thrombi consist mainly of platelets; blood proteins play a relatively small role in the process.

Thrombus formation in various circumstances: on the left, the area of the vascular wall that is damaged is ten times smaller than on the right. However, over time, the thrombus growth in both cases is approximately the same size. Image: 10.1371/journal.pone.0141068

Prospects

Mikhail Panteleev also told MIPT’s press service about the prospects of studying thrombi and why the researchers chose to look at arteries, rather than venous thrombi or the blood clotting process in the capillaries:

We are working on various issues in the field of hemostasis – the “physiological” process of wound clotting, as well as venous and arterial blood clots. Arterial thrombi in particular are very interesting, due to their social importance (heart attacks and ischemic strokes) and complex mechanisms (blood proteins also play a key role here that is not fully understood). However, we have always had difficulty working with arterial thrombosis in terms of developing and implementing computer models, because the subject involves a very complicated combination of mechanics (cell attachment), hydrodynamics with variable geometry, and incredible biochemistry. There is no standard software that can adequately address these problems, and only collaboration with Nikolai and Evguenia, specialist in computational methods from the Institute of Problems of Mechanical Engineering, made this study possible.

In our paper in PLOS ONE, we tried to use the most primitive description of a thrombus as a continuous medium, rather than discrete particles. This approximation is rough in many respects and it limits the scope of the research, but it is able to give us some common patterns. On the one hand, we plan to continue to apply it to specific tasks, as far as it is applicable, and on the other hand, we are developing more sophisticated and advanced models with three-dimensional blood cells, the full mechanics of their interaction, and the proper biochemistry. This will be done only on supercomputers of course. Another aspect is the “inside” of a platelet, modelling intracellular signalling, calcium oscillations, and the collapse of mitochondria in thrombocytes. This topic has been addressed in a number of papers that have already been published, including a very recent paper in Molecular BioSystems. Over time, I hope that all these projects will merge together into full multi-scale model of thrombus formation.

In order to understand the social importance of the problem described by Mikhail Panteleev, it should be noted that every day around 1000 people in Russia suffer a stroke and of those who survive, 2 in 3 are left with a disability. Contrary to popular belief, strokes not only affect the elderly, but also people under the age of 45 – in approximately 1 in 8 cases. Signs of a stroke include partial paralysis and a sudden onset of weakness. In a number of cases, the symptoms, which include slurred speech and a staggering gait, may make it appear as if the sufferer is under the influence of alcohol: this error of judgement could cost the lives of victims of an acute cerebrovascular accident.

Source: https://mipt.ru/