An Agent Based Simulation of the Blood Coagulation Process in the Human Body Iman Yazdanbod1 Bachelor of Health Sciences Honours, Bioinformatics Major, University of Calgary, 2500 University Drive N.W., Calgary, Alberta, Canada T2N 1N4
[email protected]
Abstract. This study is to describe the integration of an agent-based model of the blood coagulation process into the LINDSAY Virtual Human project, which aims to simulate and visualize physiological processes inside the human body. A generic modelling language is used to design the interaction behaviours of the involved bio-agents (cellular and chemical structures found in the blood). Physical interactions among the agents, such as collisions and binding, are computed by an embedded physics engine. In order to effectively retrace and to accurately model coagulation, comparisons with the results of established differential equation-based models is drawn. The blood coagulation simulation accounts for the formation, expression, and propagation of the blood clots within the injured area of a blood vessel. The agent-based model as well as the simulation framework that is used allow one to change the configuration of the simulation interactively and to have it rendered in three dimensions. These features turn this simulation into an invaluable tool for research and education.
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Introduction
The human circulatory system is responsible for transporting materials (such as nutrients, hormones, and immune cells) around the entire body [1]. One of the important aspects of this system is its coordination with the haemostatic system to limit the loss of blood following an injury. The natural process of stopping the flow or the loss of blood, occurs in three stages: (a) vascular spasm, or intense contraction of blood vessels in the area of the injury, (b) formation of a platelet plug, and (c) blood clotting. Once blood loss has stopped, the tissue healing process can begin [2]. Understanding the biological and physiological process of blood coagulation has important biomedical values. There are many disorders such as improper regulation of thrombus growth, thrombotic complication with cancer, vessel blocking affects of clots, etc, that can be resolved if the blood coagulation process is well understood [3]. Computational and mathematical models can be used as one of the available tools for understanding biological processes. There have been numerous studies to simulate the blood coagulation process computationally. However, this process is an extremely complicated haemostatic mechanism, wherein various biochemical and mechanical factors are involved.
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Therefore, most of these attempts are focused on a few aspects of the entire system. For instance, there are a few models that couple the formation of the platelet plug with the formation of the clot. Also, almost all of these studies have attempted to model this process using nonlinear differential equations (ODE) or partial differential equations (PDE) [3]. These studies generally lack stochasticity, which is greatly important for biological systems, and are not able to propose an easy way to demonstrate accurate 3D visualization by using differential equations. Blood coagulation, as well as other biological processes can be computationally modeled using the agent-based modelling (ABM) paradigm. ABM is a computational modelling approach that models the interactions between autonomous decision-making entities called agents within a simulation. As a whole, the local interactions between these agents may result in a system-wide change. In fact, systems that are modelled using this paradigm can be highly sensitive to small details, such as specific local agent interactions, the order of execution of the agents, etc. Therefore, it is critical to develop such models using a unified modelling language, wherein all the small details are taken into consideration. Swarm Script provides a unified algorithmic language for developing agent-based models. This modelling language is able to address the complexity of biological systems across different scales. The behaviour of an Swarm agent is described by a set of rules. Each rule consists of a set of predicates that are graphically shown in a bin known as a selection bin and a set of actions which are graphically presented in a bin known as an action bin. Swarm script will be discussed in detail in section 3. The blood coagulation simulation is developed within a computational framework, called the LINDSAY Composer (LC). This framework utilizes efficient embedded computational engines with visualization technologies and is considered an ideal simulation environment for developing visual and complex biological and physiological models. Based on the materials discussed above, the goals of this project are: (1) Developing an accurate agent-based simulation to study the blood coagulation process using Swarm Script (2) integration of the simulation into the context of the Lindsay virtual human, where the location of the wound could be selected by the user and (3) providing a semi-realistic 3D visualization and an easy way for computing and demonstrating different model scenarios. The remainder of this proposal is organized as follows. In section 2, the related work in the respective area is presented. Section 3 is dedicated to the details of the project from the biological aspects as well as the modeling paradigm used. Section 4 introduces the computational framework of the project. Lastly, in section ??, the timeline of the project is expressed.
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Related Work
Due to the complex nature of coagulation and its serious clinical consequences, there have been several studies focusing on this topic. Each of these studies has explored the process from a different perspective. Therefore, over the past
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30 years, a reasonably comprehensive understanding of this processes has been developed. Here, a quick overview of the most significant research conducted in this area is presented. The kinetics of the extrinsic and intrinsic pathways and their feedback loops have been modeled using a PDE system in a study conducted in 1989 [4]. Later on in 1993 and 1994, this study has been enriched with more details to have closer correlation with experimental results [5, ?]. In 2001, it was shown that the activation threshold for the coagulation chemical cascade is affected by the flow rate, size of the wound site, and the initial concentration of the chemicals [6]. In 2004, a PDE model was introduced to describe the interactions between platelets and sub-endothelial layers and cohesion of activated platelets [7]. A relatively comprehensive model of coagulation was proposed in 2005, in which rheological properties of the clot, Newtonian fluid model, biochemical interactions, and fibrogenesis were incorporated. This study employed convection-reaction-diffusion equations to model platelet activation, whereas the formation of the clot is modeled as a sheer thinning viscoelastic region, which first forms within time and then dissolves afterwards [8]. In this section two recent studies are presented in some more detail, as they proved seminal in informing our work. The first study is a PDE model of the coagulation, which encompasses all the chemicals that have been known to be involved in the process [9]. The second one is a hybrid model of PDEs and Cellular Potts Model (CPM) and is considered as one of the most comprehensive models of the coagulation up to the present [3]. These two studies are not only used to compare some of the obtained results but also to show how the approach used in this project can be an improvement for the study of the complex biological systems such as coagulation. 2.1
A Model for Stoichiometric Regulation of Blood Coagulation
A PDE system is used to describe the vitamin K-dependent coagulation process [9]. The coagulation kinetics in this model are described by 27 equations and 42 chemical species. The final result of this system describes how the concentration of thrombin changes over the course of simulation at the injury site. Based on the initial concentration of tissue factors released by the wound site, the concentration of thrombin varies. The initial concentration of tissue factors is a function of the size and characteristics of the wound site. Even though there is a relatively close correlation between the results of this model and in vivo experiments, this model did not consider the role of platelets in the formation of the clot. In addition, the physical and hydrodynamical factors were not reflected in this study. 2.2
A multiscale model of thrombus development
Xu et al. present a two dimensional multi-scale model of the blood coagulation process [3]. The biochemical structures used in this model are blood cells, blood plasma, activated and inactivated platelets, fibrinogen, activating chemicals, and
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vessel walls. The multi-scale hierarchy of this model is divided into three levels; 100 micrometer, 1 micrometer for cellular structures, and 0.1 micrometer for sub cellular and chemical structures. The modelling approach used is a hybrid of three main sub-models: cell sub-model, biochemical reaction sub-model, and flow sub-model. Cell sub-model is a discrete stochastic Cellular Potts Model (CPM) that represents different types of the cells such as platelets and blood cells [10]. CPM is a lattice based computational model, wherein the simulation environment is divided into many lattice sites represented as pixels. Each pixel is associated with an index and an energy state. Pixels that share the same index belong to the same cell. The behaviours of the cells and their interactions are defined in terms of effective energies. At each CPM time step, a pixel is randomly chosen and based on its energy state the change of its index is attempted. When a pixel changes its index, it means that the pixel is shfted from one type of the cells to another type. Over the course of the simulation, the CPM evolves to minimize the energy of the total energy of the simulation [10]. The biochemical reaction sub-model is basically a PDE system that is used to describe the biochemical nature of the system. Navier-Stokes equations are used to describe the hydrodynamics of blood flow. The intention for hybridizing these modelling paradigms is to capture the discrete aspects of the coagulation process at the micro-scale level, such as enzyme threshold effects in cells as well as the continuous aspects of it at the macro-scale level, such as continuous extension of the clot at the wound site. At each iteration of this simulation, both equations systems are solved and their results are used for updating the CPM. The resulting simulation successfully demonstrates the formation of the clot at the wound site. This model is fairly comprehensive. However, it does not fully describe some aspects of coagulation. For instance, the biophysical complexity of the system is not well captured, as the physical interactions are limited to the 2D environment of the CPM. Therefore, it is suggested that there are other modelling approaches that are easier to employ and still more successful to simulate complex biological systems [11].
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Project Details
In order to describe the details of the project, we will first explain the biological mechanisms and the bio-agents that are involved in the coagulation process and then we introduce the modeling method that is used to capture this system in a virtual environment. 3.1
Biological Aspects & Agent-based Model
The coagulation process is initiated after any vascular damage is sustained, which results in the contact between blood cells and thrombogenic factors and sub-endothelial proteins such as collagen. According to the classic view, there are two ways to initiate the blood coagulation process: via the intrinsic and extrinsic pathways. The intrinsic pathway is initiated when blood cells touch
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the chemicals released from the damaged membranes of the endothelial and subendothelial cells at the injury site. These activating chemicals are able to activate the inactivated platelets within the blood flow. The extrinsic pathway takes place when the chemicals released from activated platelets meet those that have not been activated yet [12]. After these two pathways, the coagulation process enters the common pathway, where the complex structure of fibril networks is formed. These pathways are complex and involve many chemical activities (Fig. 1). The outcome of this process is the formation of a clot, which is basically a multi-layer mesh of platelet plugs covered with fibril networks. In order to model the blood coagulation process the body should be considered as a set of multiple swarms. In fact, the human body is essentially a swarm of swarms, in which multiple types of swarms interact with each other [13–15]. In this study, it is assumed that all the agents involved essentially behave like swarms. For instance, platelets would be swarm agents that interact and aggregate together based on a set of interaction rules. By studying the process of blood coagulation from a swarm perspective, the interaction between the agents can be formulated as a set of rules. All the interactions are local and occur via collisions. There is no central control in the system directing the agents to form a clot, or even to direct them toward the wound site. This means that the formation of the clot is the result of the independent interactions of individual agents. The strength of this approach is that the agents interact based on simple rules, which eventually results in the complex structure of the clot.
Fig. 1. The interactions between the agents involved in the blood coagulation process
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Our simulation contains 12 types of agents (Fig. 1). The main interactions in the simulation are adhesion, activation, and secretion. For example, in a simple scenario, collagen proteins at the wound site activate inactivated platelets within the blood flow. Activated platelets then secrete thrombin, ADP, and serotonin. The activating chemicals activate other inactivated platelets in the blood flow. Activated platelets adhere to each other to form the platelet plug. Any fibrinogen that comes into contact with thrombin molecules becomes activated. Activated fibrinogens are called fibrins, which adhere to each other to form a fibril network. The combination of the platelet plug and fibril networks form the clot at the wound site. These interactions are specified in the form of simple rules using Swarm Script . 3.2
Swarm Script
Swarm Script is a graphical, interaction-based modeling language that can be used to describe the behaviours of swarm agents. The graphical representation of swarm script contains three elements: entities, bins, and ropes. There are three different types of entities. The first type are generator entities, which select a set of agents among all the agents that are in the simulation based on an interaction pattern. For instance, a generator entity can select a set of agents which collide with the wound site in each time step. The second type of the entities are called filters. These entities take the output set of a preceding entry and apply a filter to it. For instance, filter entities can filter a set of agents by their name, type, size, state, etc. Predicates are the third type of the entities, and are a boolean statement which is evaluated at each time step. If the result is true then they pass the set that they received from the preceding entities, and if the result is false, they return nil. The entities can be combined in any order and capsulated in a bin called selection bin. A selection bin performs a selection by iterating top-down through its contained entries. Selection bins are connected to action bins by ropes. An action bin contains a set of action entities. When the action bin receives a set of agents from a selection bin that is connected to it, it executes the set of actions on the agents. The action bin can impose additional conditions to the set of agents, which are independent of any action entries. Figure 2 shows two different types of swarm script behaviours of fibrinogen molecules. The difference between these two are in the way that selection and action bins are interconnected. In type A, a set of agents are selected by the selection bin on the left hand to check whether certain conditions are met in the simulation environment. When the conditions are met, then the actions of the action bin will be performed on a different set of agents, which are selected by the selection bin on the right hand. In other words, the selection bin on the left hand is the source of the actions and the selection bin on the right hand is the target of the actions. In this specific example, a fibrinogen molecules searches through all the agents that it has collided with in the past simulation frame. If it finds a collision with a thrombin molecule, then it acknowledges the action bin called ”ActivateFibrinogen”. As a result, the action bin performs its action on its target, selected by the selection bin on the right hand. This selection bin tests
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whether the fibrinogen molecule is not activated yet. It returns the fibrinogen (itself), if it is not activated, or returns null if fibrinogen is already activated. In type B, the selection bin is evaluated and the resulted agents are passed to the action bin to be performed by the actions. In this particular case, the selection bin selects its own agent and the action is to randomly move the agent in the simulation environment.
Fig. 2. Different types of Swarm Scripts Behaviours. These behaviours are parts of implemented behaviours for fibrinogen molecules of the simulation.
In this simulation, all the agents interactions are implemented using swarm script. The interface provided for this modeling language, as shown in figure 2, is self-explanatory and therefore swarm script can be particularly useful for developing biological simulations, wherein computer scientists and biologists work along each other.
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Software Tools
LINDSAY Composer (LC) is a 3-dimensional component-based modelling framework, which utilizes various computational engines and visualization technologies. The objective of this framework is to integrate the computational models of physiological functions across the human body. Each object in the LC is represented as a hierarchy of components. Each of these components are handled by one of the engines of the LC [16]. For instance, if an object has a rigid body component, its physical interactions with other physical objects in the simulation context are handled by the LC physics engine. Dependencies among components, such as binding the 3D representation of an object to its physical state, are described through hierarchal relationships between components. In addition there
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is a LC Swarm Script Engine to simulate agent interactions. For instance, when two or more agents collide, a message is sent to this engine, which then executes any possible rules between those agents. The component assembly in the composer enables the developers to place all the required components of a simulation in a hierarchal arrangement of a single object. Therefore, the simulation can be placed anywhere within the simulation environment. The method used to build the blood coagulation simulation is described in the section 5, ”Achievements”. One of the benefits of the framework is that it provides various graphical user interfaces (GUI) that could be used to adjust simulation parameters dynamically. For instance, the GUIs enable users of the system to add or remove agents, or apply new behaviours to agents within the simulation. These features make the LINDSAY Composer an ideal framework for developing a blood coagulation simulation within the human body.
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Achievements
Table 1 summarizes the goals claimed in the proposal of the project. In this section, we describe how each one of these goals are achieved. Table 1. Project’s Goals Goals Convert the graphics of the simulation to OGRE Implementation of the agents’ behaviours by Swarm Script language Integration of the simulation into Lindsay Virtual Human & Simulation parametrization
5.1
Graphics of the simulation
In the previous version of the simulation, the graphics were handled by an embedded graphics engine called ”LCGraphics”. One of the goals for this project was to convert the graphics of the simulation to Object Oriented Graphics Rendering Engine (OGRE). This engine is written in C++ and is designed to accelerate the development of applications with 3D graphics. Lindsay composer is utilized by OGRE. The challenge for the developer of this project was to make graphics files for individual objects and agents of the simulation, which could be handled by OGRE. Figure 3 demonstrates how OGRE has led to a visual improvement for the simulation. 5.2
Agents’ Interactions
In the previous version of the simulation, we have implemented the behaviours of the agents using a modeling language called Swarm Graph Grammars(SGG).
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(Simulation in OGRE)
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(Simulation in LCGraphics)
Fig. 3. The comparison between the previous version of the simulation, where the graphics engine used was LCGraphics with the new version, where the graphics engine is OGRE.
SGG is a unified modelling language for developing complex agent-based systems, where the behaviour of an agent is described by a set of rules. Each rule is composed of a set of if conditions known as predicates and a set of associated actions. The actions are executed if and only if the set of predicates is true. The application of each rule can also be associated with a probability [17]. Even though SGG served its purpose to model the coagulation process, but it had its own complications. For instance, SGG does not provide any interface, where the behaviours of the agents can be observed by users. Thus, biological validation of the behaviours is very complex for non-computer scientists. Also, the execution of rules was not well-optimized and therefore the simulation turned very slow within the course of the experiments. In addition, ideally we would like to have building blocks for behaviours that can be assembled by biologists to constitute complex biological behaviours of different agents. Using SGG this matter could not be achieved, as every rule needed to be hard-coded. These complications motivated us to implement the new version of the simulation using Swarm Script. Swarm Script can be considered as the continuation and evolution of SGG, which not only embraces the strength of SGG, also provides other advantages such as a user-friendly interface, well-optimized execution, and encapsulation of selection criteria as well as associated actions. Therefore, having the behaviours implemented in swarm script can be viewed as an important step, where not only the execution of the simulation is more efficient, but also it can better be used as an educational tool by non-computer scientist users. 5.3
Integration of the Simulation
The first step to integrate the simulation in the context of Lindsay Virtual Human is to interconnect various components of the simulation with each other, so
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that they can be placed anywhere within the simulation environment. The next step is to perform graphical works over the mesh file of the circulatory system of the virtual human, so that when the simulation is placed, the overlapping mesh gets deleted from the circulatory system. Each of these steps are discussed separately. Interconnection of Components: As mentioned, each object in the composer contains a transform component, which determines the spatial location of the object in the simulation environment. For a simulation like blood coagulation, every agent/object should be positioned in the simulation in respect to other agents and objects. In order to be able to place the simulation in various locations in the simulation environment, the respective positions of agents/objects to each other should be maintained. This goal is achieved by placing all the agents/objects in a hierarchal arrangement, where all the transform components are interconnected. Figure 4 demonstrates the hierarchal arrangement of the simulation. All the required components are placed in a component called ”Coagulation”. Coagulation components are listed in table 2. Table 2. Blood Coagulation Components Component Name Function wound The wound site is the place where the clot forms Vertical Destroyer Located at the end of the vessel Horizontal Destroyer Located underneath of the wound site Shear Force Downward force field right before the wound site Blood Pressure The l force field, which directs the agents along the vessel Blood Vessel The main component, contains all the agents of the simulation
All the agents that constitute the blood flow are placed inside the blood vessel. These agents are in the form of inactive and in each simulation step, the blood vessel activates certain number of them and release them inside itself. The number of these agents in the blood flow is determined by users. A.1 arrow directs us to the inner arrangement of the blood vessel. It contains platelet, red blood cell, white blood cell, and fibrinogen as well as its own required components such as rigid body and transform components. Where these agents are released is determined by the values of the transform component of the vessel. A.2 arrow indicates the inner arrangement of a platelet. When activated, platelets secrete activating chemicals such as serotonin and ADP. The activating chemicals are placed inside the hierarchal structure of the platelet in their inactive form. When the platelet gets activated, secretes these chemicals based on the rate determined by users. When released, the transform component of the activating chemicals are set based on the transform component of the secreting platelet. A.3 arrow is connected to the inner structure of serotonin. Serotonin is a simple agent, which does not secrete any other agents. Therefore, it only
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contains the general components such as transform and graphics as well as the component which indicates its behaviour, SWAgent. A.4, A.5, and A.6 are connected to values of transform component, graphics component, and rigid body component respectively. These values can be viewed by user to change them as desired. B.1 directs us to the inner structure of one of the objects of the simulation, blood pressure. Blood pressure in fact is a force field located inside the vessel, which applies force to all the agents along the vessel. Each agent has a random walk and this force direct them toward the right direction. Blood pressure direction, magnitude, and density can be manipulated by reaching to the LCVelocityVector component, shown by B.2. As shown, all the components of the simulation are interconnected. This enables us to place the simulation as we want in the simulation environment. Also, this arrangement enables users to manipulate all the characteristics of the agents on the fly using the provided interfaces. Parameterization of the simulation will be discussed in section 5.4. Graphical Work: It is relatively very difficult to manipulate a massive mesh such as circulatory system mesh on the fly to integrate the simulation. However, for demo purposes, we have selected a single vessel called right anterior ulnar recurrent artery, and prepared a wound site in its structure. Therefore, the simulation can be run on this location. 5.4
Parameterization
The parameters involved in the simulation can be categorized into two different types based on their nature. The first type of parameters attribute the characteristics of the agent and objects of the simulation. Each agent has various parameters such as size, spatial position and orientation, friction, and mass. These parameters can be manipulated through the introspector window of the composer, where all the agents and objects are listed. The desired agent can be selected and any of its parameters can be manually changed. For instance, in order to change the mass of platelet agents, the user should open up the introspector window, find platelet in the inner structure of the blood vessel, go to rigid body component and set the new mass for it. Another example, which is particularly important for this simulation is to change the blood pressure. In order to do so, ”Blood Pressure” component should be opened and its ”LCVelocityVector” component should be selected. At this point, the direction, magnitude, and density of the blood pressure can be changed as desired. Another type of parameters are ones concerned with the agent-based simulation directly. In order to simulate different scenarios, the users should be able to change and control the composition of the blood flow (the number of each type of agent entering the vessel) as well as the activity of the agents. This matter is achieved by providing a unique interface for this simulation, shown in figure 5. Using this interface, the users are able to manipulate the number of platelets, red blood cells, white blood cells, and fibrinogen in the blood flow. Also, they are able to control the secretion rate of the activating chemicals from activated
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Fig. 4. The hierarchal arrangement of the simulation components.
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Fig. 5. Coagulation simulation interface, required to manipulate the number and activity of involved agents.
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platelets. In fact, we can claim that the simulation is fully parameterized, based on the following reasons: (1) The simulation can be placed anywhere within the simulation environment. (2) The hierarchal arrangement of the components enables us to not only manipulate the characteristics of individual agents but also all the agents of the same type. For instance, we can change the mass of a platelet by finding it in the introspector window, or we can change the mass of all the platelets by finding its component in the hierarchal structure of the blood vessel. (3) The blood pressure and the shear force at the wound site can be changed through the introspector window of the simulation. (4) The simulation parameters can be controlled using the coagulation interface provided.
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Results
For such a simulation, it is very important to be able to compare and combine the results of multiple trials. We have developed an agent called ”Statistics”, which creates text files and records the number of agents at each type step. The text files are saved in a specified directory. The behaviour of Statistics is shown in figure 6. The agent that we are interested to trace its trends through the simulation can be filtered in the selection bin indicated by the red box. When placed in the simulation with its current configuration, Statistics records the number of all the agents in the text file. However by adding filters, we can specify what agents should be traced. We can have multiple of Statistics to record the trends of multiple agents in one simulation.
Fig. 6. The agent used to create text files for the results of each trial of the simulation.
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6.1
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Biological Validation
We used previous experimental and modelling studies to calibrate our own model, with parameters adopted from [3]. Figure 7 shows different steps of a typical run of our simulation. At the beginning of the simulation, red blood cells and other blood components are flowing out of the ruptured vessel (t1 ). Some platelets are activated and adhere to the wound site as a result of their interactions with collagen proteins. These activated platelets secrete activating chemicals such as thrombins (shown as blue spheres), which in turn activate fibrinogen molecules and other inactivated platelets (t2 ). The simulation proceeds with the formation of the platelet plug as well as some fibril networks (shown in white, t3 ). Finally, the clot is completely formed and prevents the blood components to drip out of the vessel (t4 ). Activated platelets also reached their deactivated state, with no further thrombogenesis. Continued generation of thrombin could lead to the condition known as stasis, where the normal flow of blood stops within the vessel as the clot blocks its way, which we discuss in more detail in subsection 6.1. Normal Blood Coagulation: Figure 8 shows the graphs plotted for a typical blood coagulation experiment. Graph 1 compares the number of activated (cyan) and inactivated (green) platelets. At the beginning of the simulation, the number of activated platelets increases rapidly. Over time, the rate of platelet activation decreases until there is no more platelet activation by the end of the simulation. This activation trend closely correlates with the results presented in [?]. Graph 2 shows the number of fibrins (dark blue) and number of fibrinogens (purple). Graph 3 plots changes in the number of thrombin molecules (green) and the number of all the activating chemicals together (blue) . As greater numbers of platelets get activated, the number of activating chemicals also increases. However, as the simulation proceeds more activated platelets reach their deactivated state. Consequently, the number of activating chemicals drops to zero. The trend observed for changes in the concentrations of activating chemicals is similar to the trends presented in [1]. Graph 4 compares the number of platelets that are activated by collagen proteins at the wound site (red) with the number of platelets that are activated by activating chemicals (blue). At the beginning of the simulation, clot formation is initiated by collagen proteins that are exposed to the bloodstream after the injury. However, as the wound is filled by platelet plugs and as there is no more room left for further interactions between flowing platelets and collagen proteins, the activating chemicals play a major role for the activation of other platelets. Thrombosis: The presented model can also be used to study pathologic cases such as thrombosis, which results in the pathological development of a clot. There are many causes for this disorder, with one being overreaction of coagulation response. The response may be observed as an increase in the number of platelets in the bloodstream. Figure 9 shows a simulation in which the number of platelets
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(t1 - Inside)
(t1 - Outside)
(t2 - Inside)
(t2 - Outside)
(t3 - Inside)
(t3 - Outside)
(t4 - Inside)
(t4 - Outside)
Fig. 7. The blood coagulation simulation at different time steps (t1 ¡ t2 ¡ t3 ¡ t4 ). The process is observed from two different perspectives: inside and outside of the vessel.
in the bloodstream is 1.5 times more than normal, with their secretion rate increased by 25 percent. The simulation time is exactly the same as in the normal case. The expansion of the clot (thrombogenesis) is not regulated properly and the clot occupies the cross section of the vessel and blocks the blood flow. Figure 8 shows the graphs plotted for this experiment. The number of activated platelets constantly increases (graph 1, cyan line), which leads to the oversecretion of activating chemicals (graph 4, blue line and both lines in graph 3). The oversecretion of activating chemicals leads to the formation of increased number of fibrin molecules as well (graph 2, blue line). As a result, thrombogenesis is not stopped appropriately and blocks the blood components (such as red and white blood cells) to flow through the vessel (Figure. 10).
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(a)
(c)
(b)
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Fig. 8. Agent counts in the case of normal coagulation. (a) activated platelets (cyan) and inactivated platelets (lime green), (b) fibrinogens (purple) and fibrins (dark blue), (c) all the activating chemicals (azure blue) and changes in the number of thrombins (green), (d) collagen proteins in activation of platelets (red) and activating chemicals in activating platelets (blue).
(a)
(c)
(b)
(d)
Fig. 9. Agent counts in the case of abnormal coagulation (thrombosis). (a) activated platelets (cyan) and inactivated platelets (lime green), (b) fibrinogens (purple) and fibrins (dark blue), (c) all the activating chemicals (azure blue) and changes in the number of thrombins (green), (d) collagen proteins in activation of platelets (red) and activating chemicals in activating platelets (blue).
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Fig. 10. Clot structure in an abnormal case (thrombosis). The clot blocks the blood flow, which means the vessel is no longer functional for the transportation of blood.
7
Conclusion and Future Work
We have presented a 3-dimensional, interactive, agent-based model of the blood coagulation process. Simulation accounts for biologically validated agent-based interactions, as well as a semi-realistic visualization of the process. Users are able to manipulate the parameters and observe their effects on the simulation in real time. This simulation is rendered in an interactive 3D environment and can be observed from different camera perspectives. The result of this study has the potential to be used as a tool for both educational exploration and research. Blood coagulation is a complex process, in which at least 42 bio-chemicals are known to be involved in. The current simulation considers 12 agents, which have been identified as the most important ones in the formation of the clot. However, this set of agents is not sufficient to capture the complexity of the actual process. It is also not efficient to have all the biochemicals involved as agents in the simulation, as they will computationally exhaust the physics engine. In order to capture the biochemical effects without an overwhelming number of physical interactions, a multi-scale approach is proposed, wherein each time step in the simulation is divided into 2 phases: a differential equations phase, and an agent based phase. As mentioned earlier, the differential equations (DE) system discussed solves the changes in the concentrations of the chemicals involved in two different pathways of the clot formation (extrinsic and intrinsic pathways). These chemicals are considered to be at the scale of less than 0.1 um. In the first phase of each time step, the DEs could be solved to obtain the concentration of chemicals. The final result of the DE system is the concentration of thrombin [9]. Currently, the blood clotting DE system is implemented by us in Mathematica using the equations provided in [9]. After obtaining the concentration of thrombin, this can be used to release appropriate number of thrombin agents in the simulation. Cellular structures of the agent based simulation are considered to be in the scale of 1 um or more. We believe that this hybrid approach has an
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advantage over both the DE system solely or the agent based simulation. The DE system does not take into account the macromolecules and cellular structures and their interactions. Therefore, this system is not able to describe the biophysical complexity of the formation of a clot. On the other hand, the agentbased simulation does not consider many of the bio-chemical species involved and therefore can not capture the biochemical complexity of the system. These two approaches could be integrated to yield a multi-scale simulation, where the micro-interactions of the bio-chemicals are captured by the DE system, and the macro-interactions of the cellular components and macromolecules are captured using agent based approach. As mentioned, this approach has been used before, in a way that DE system was coupled with CPM model [3].
References 1. Hockin, F., Jones, C., Everes, S.: A model for the stoichiometric regulation of blood coagulation. The American Society for Biochemistry and Molecular Biology (2002) 2. Johnson, M.: 13. In: Human Biology; Concepts and Current Issues. 3rd edn. Pearson (2006) 612–618 3. Xu, Z., Chen, N., Kamocka, M., Rosen, M., Alber, M.: A multiscale model of thrombus development. The Journal of The Royal Society, Interface (2007) 4. Khanin, M., Semenov, V.: A mathematical model of kinetics of blood coagulation. In: Theory of Biology. Volume 136. (1989) 127 – 134 5. Jesty, J., Beltrami, E., Willems, G.: Mathematical analysis of a proteolytic positivefeedback loop: dependence of lag time and enzyme yeilds on the initial conditions and kinetic parameters. In: Biochemistry. Volume 32. (1993) 6266 – 6274 6. Beltrami, E., Jesty, J.: The role of membrance patch size and flow in regulation a proteilytic feedback threshold on a membrane: Possible application in blood coagulation. In: Mathematical Bioscience. Volume 172. (2001) 1 – 13 7. Fogelson, L., Guy, R.: Platelet-wall interaction in continuum models of platelet thrombosis: formulation and numerical solution. In: Mathematical Medical Biology. Volume 21. (2004) 293 – 334 8. Ananed, M., Rajagopal, K.: A model for formation and lysis of blood clots. In: Pathophysiolgy. Volume 34. (2005) 109 – 120 9. Hochim, M., Jones, K., Everse, S., Mann, K.: A model for stoichimetric regulation of blood coagulation. The American Society for Biochemistry and Molecular Biology (2002) 10. Chen, N., Glazier, J., Alber, M.: A parallel implementation of the cellular potts model for simulation of cell-based morphogenesis. In: National Institute of Health. Volume 176. (2007) 670 – 680 11. von Mammen, S., Davison, T., Sarraf, A., Sarpe, V., Yazdanbod, I., Jacob, C.: Swarm intelligence for everyone, accessible modelling and simulation of complex biological systems. In: Swarm Intelligence. (2010) 12. Boron, W., Boulpaep, E.: Medical Physiology. Updated edn. Elsevier Saunders Inc (2005) 13. Hoffmeyer, J.: The swarming body. In: The Fifth Congress of the International Association for Semiotic Studies. (1997) 937–940 14. Jacob, C., Burleigh, I.: Biomolecular swarms: An agent-based model of the lactose operon. Natural Computing 3(4) (2004) 361–376
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15. Jacob, C., Steil, S., Bergmann, K.: The swarming body: Simulating the decentralized defenses of immunity. In: Artificial Immune Systems, ICARIS 2006, 5th International Conference, Oeiras, Portugal, Springer (2006) 16. von Mammen, S., Davison, T., Baghi, H., Jacob, C.: Component-based networking for simulations in medical education. In: IEEE Symposium on Computers and Communications (ISCC), IEEE Press (2010) 975–979 17. Yazdanbod, I., Marcus, S., Jacob, C.: An agent based simulation of the blood coagulation in human body. Journal of Undergraduate Research in Alberta 1 (2011)
