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Multi-Agent Systems and Blood Cell Formation
Bessonov Nikolai 1 , Demin Ivan 2 , Kurbatova Polina^2 ,
Pujo-Menjouet Laurent 2 and Volpert Vitaly 2
1 Institute of Mechanical Engineering Problems, 199178 Saint Petersburg
2 Université de Lyon, Université Lyon1, CNRS UMR 5208 Institut Camille Jordan
F - 69200 Villeurbanne Cedex,
1 Russia
2 France
1. Introduction
The objective of this chapter is to give an insight of the mathematical modelling of hematopoiesis using multi-agent systems. Several questions may arise then: what is hematopoiesis and why is it interesting to study this problem from a mathematical point of view? Has the multi-agent system approach been the only attempt done until now? What does it bring more than other techniques? What were the results obtained? What is there left to do? We hope that the following will give the reader all the answers to these questions. And even more, we would be delighted if after reading it, you would like to know more on this subject and try to work on it to contribute to the understanding of this complex field. Let us start with the biological background in order to get a clear idea of the problem behind the model.
1.1 Hematopoiesis: what is it? Hematopoiesis (from the ancient Greek meaning to make (ǑǐNJdžÂǎ) blood (ǂÂǍǂ)) is the scientific name used to describe the blood cell formation.
1.2 Where does it occur? It appears in the yolk sac or blood islands during early embryogenesis. Then, with the development of the individual, it reaches the spleen, liver and lymph nodes to eventually settle down in the medulla, also known as bone marrow once this latter has been completely formed. This process takes place in the femur, tibia or any other long bones for children to finally moves to the pelvis, cranium, vertebrae and sternum in the adult bodies.
1.3 How does it work? There are two main branches in hematopoiesis: myeloid and lymphoid (see Fig. 1). These two branches originate from the same cell type: the hematopoietic stem cells (HSC). The lymphoid branch gives birth to the T and B cells, antibodies and memory cells. Maturation, activation and some of proliferation of these latter are developed mostly into secondary
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lymphoid organs such as the spleen, thymus and lymph nodes. This is the reason why we shall not be focused on this branch. We might mention it from time to time though during this chapter when we would like to describe hematopoiesis in its whole. Consequently, our main attention will be given to the myeloid branch. Three blood cell types arise from this branch through three cell lineages: red blood cells (erythrocytes), white blood cells and platelets (megakaryocytes) (see Fig. 1). Their daily production is fairly high: each second for instance, the body produces 2 millions of erythrocytes, also 2 millions of platelets and 700,000 granulocytes. Their lifetime differs from one type to another (120 days for erythrocytes, about 7 to 10 for the thrombocytes, and 6 to 14 hours only for the granulocytes (the shortest lifetime of these cell types).
Fig. 1. Illustration of hematopoiesis: all blood cells originate from the stem cell compartment on the left and are released in the blood stream on the right. The lymphoid branch, on top, releases T and B-lymphocytes. The myeloid branch consists of the red lineage (bottom), white lineage in blue and platelets in green.
1.4 The myeloid branch: an insight Let us have a closer look at the myeloid branch. But before doing this, it seems important to remind that all cells in each lineage originate from the HSC. These particular cells are able to self-renew. Their lifetime and number are still unknown, even if some attempts were done to predict their number in the body (Dingli et al., 2007a, b). Besides self-renewing, each stem cell can differentiate to a more mature cell, also called progenitor cells or it can die by apoptosis (natural cell death). HSC when differentiating give birth to early progenitor cells,
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Fig. 2. Illustration of the differentiation pathways in the myeloid giving a representation of control of the platelet (in green), red blood cell (in red) and granulocyte or monocyte (G/M) (including neutrophils, basophils and eosinophils) production (in blue). The control loops, respectively mediated by thrombopoietin (TPO), erythropoietin (EPO) and the granulocyte colony stimulating factors (G-CSF) are indicated.
2. Objective of the chapter: what is the problem about?
Even if the regulating hormones presented in the previous paragraph are widely investigated in the biologist community, several questions and important issues remain open: how, for instance does thrombopoietin exactly act on the megakaryocytic lineage? Does it act on the apoptosis rate also, like EPO on erythrocytes or does it act only on the differentiation rate? What about G-CSF? Moreover, is spatial distribution of the cell in the bone marrow important or not in homeostasis? Do cells communicate between each other? If yes, how do they proceed? How do some diseases spread in the blood system, while some others do not? Is it possible for anyone to develop leukemia without knowing it, and to recover without any cure? How do stem cells and progenitor cells choose their lineage? Is this due to the environment of the cell, that is some external information or does it come from some stochasticity, some random noise inside the cell itself that leads its decision to prefer one lineage rather than another one? Some of these questions have been tackled for almost 50 years now from a mathematical point of view, with different models and techniques. Some researchers used non linear
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partial differential equations with delay or not, some studied several systems of ordinary differential equations, linear or not, stochastic differential equations, etc. However, among all the existing models, and to the best of our knowledge, very few attempts were made to take the bone marrow structure and cell interaction into account. The objective of this chapter is to try to answer some of the previous questions by the use of multi-agent systems and taking the bone marrow structure and space competition between cells into consideration. Before this, it seems necessary to justify the reason why the approach of the problem with the use of multi-agent system would be a good technique in comparison with other models. This is the reason why, in the next section we set up the state of the art, reminding some of the previous models and results obtained in the past decades. Then, we introduce different multi-agent approaches used to model hematopoiesis. In section 4, we give some of our contributions to this field and eventually conclude with what we believe was successful, what needs to be improved and the work planned for future investigations.
3. State of the art: different approach to model hematopoiesis
3.1 Deterministic models: 3.1.1 The first models Deterministic models are considered to be the first models describing cell cycle. In 1959, Lajtah et al. were the first to introduce a cell cycle model with a resting phase. Then, in the early 1970's, Burns and Tannock (1970) as well as Smith and Martin (1973) carried on Lajtah's work using a two phase model: the proliferating phase and the resting phase. This study was then generalized by Mackey in the late 1970's and applied to the study of hematopoietic stem cell. All these models consist in systems of ordinary differential equations, linear or not. In 1976, Wazewska-Czyzewska and Lasota (1976) proposed a similar model but they introduced a discrete model and applied it to erythrocyte production.
3.1.2 Development of the models: going to more realistic and complex models Each model afterward was more or less built from the first ones, with significant improvement, adding nonlinearity, delay. Several systems of partial differential equations arose then from that time; some were structured in age, size or maturity, sometimes two of them at the same time, sometimes with discrete delays, other times with distributed delays. They have been used to describe different parts of hematopoiesis: it could have been the stem cell compartment only, the red blood cell lineage, the myeloid lineage, with feedback or not regulating the production. The objective of each model was to understand the possible dysfunction of the system leading to diseases like anemia, leukemia, neutropenia or thrombocytopenia. Some of these diseases are chronic (this will be developed below) and oscillations of the size of cell populations could occur: this is the case for chronic myelogenous leukemia, cyclic neutropenia or cyclic thrombocytopenia. Incorporating one or several feedback loops on one or more lineages in the model was then necessary to simulate a possible regulation of the blood cell in the bone marrow.
3.1.3 Some success stories All these deterministic models combined with the study of the influence of different parameters allowed the authors of these researches to obtain accurate results and even predictions in rather good agreement with the experiments.
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The last question would be answered yes. Under the assumption that a study is mainly focused on very few niches at the same time in a tiny part of the medulla; a discrete and stochastic approach would make sense. This is what is developed in the next paragraph.
3.2 A different approach: the multi-agent models Before the introduction of the multi-agent models, it appears necessary to have an insight of what has been done in term of stochastic models. They have been used to describe the mechanism of cell proliferation determined with a certain probability, and not by previous deterministic mechanisms. It is also important to remind one of the rare existing models with reaction diffusion taking the spatial structure of the bone marrow into account. This will help to justify the use of our software based on the multi-agent systems taking the medulla structure into account.
3.2.1 Stochastic models In every deterministic model, any cell fate such as differentiation, apoptosis or self-renewal is predicted by specific processes well defined, like a good engine that self-regulates. In case of deregulation, the whole mechanism reacts and tries to reach back its non pathological equilibrium, also called homeostasis. Sometimes things do not occur in this way, and other equilibria can be reached, changing the population fate and subsequently the whole system. However, in vivo, the cell decisions may not originate from well determined laws, and the parameters involved in the problem can exhibit great sensitivities to tiny changes. These small variations could appear in the inside of the cell (intrinsic) as well as its external environment (extrinsic). This problem has been investigated in the study of stem cells in the late 1990's and year 2000's with the work of Abkowitz (1996, 2000), Dingli, (2006, 2007a, b), Newton (1995), Roeder and Loeffler (2002, 2006b). The authors used discrete models where decisions ruling the cell future could be made following stochastic processes. Some studies have shown the high sensitivity of stability of the HSC system to perturbation and death rates but not to proliferation rates (Lei and Mackey, 2007). The influence of extrinsic fluctuation has been modelled by Gillespie (1992) and Shahrezael (2008). Concerning the intrinsic perturbation, the influence of intern information and variation inside the cell nucleus leading to a drastic change of its fate is still in discussion amongst biologists. It is currently being investigated by mathematicians who would like to understand the influence of these changes to the lineage choice of a progenitor cells due to the different changes occurring randomly in the nucleus (variation during transcription of a gene, translation or mRNA, etc.). Sensitivity to such modifications would decrease as cells increase their maturity. In other words, it would be more difficult for a precursor cell to change its lineage, while, an immature cell, let say a MEP progenitor could be easily influenced to become either a megakaryocyte or an erythrocyte. This decision could occur as explained above at the molecular level, when stochasticity would have a greater influence. Thus multi-scale models would be necessary. This has been already proposed by Crauste et al. in 2010. All these works are of great interest, but still, one important thing is missing in the models: space. Consequently, cell competition for space, their communication depending on their position, and of course the bone marrow structure should be taken into account. However, one deterministic approach exists and is briefly explained in the next paragraph.
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3.2.2 Spatial models In 2005, Bessonov et al. proposed a spatial model in order to describe the influence of the medullar stroma. Indeed, in these compact areas, spatial position and competition are important. This is even more a crucial matter in case for instance of acute leukemia. When this pathology develops in the bone marrow, immature cells, mostly white cells, overtake the space dedicated to more mature cells. These latter are pushed out from the marrow directly to the blood stream without completing their maturation process. The whole system is rapidly invaded by immature cells unable to satisfy the function requested. Furthermore, they stop the development of cells from other lineages which causes anemia due to a lack of erythrocytes and hemorrhages because of a lack of platelets. The approach introduced by Bessonov et al. in 2005 consists in a simplified continuous model describing cell movement in the stroma. It is built with reaction-diffusion systems of partial differential equations with convection. The role of cell diffusion was used to illustrate a random motion in the stroma, mechanical pressure between cells was set up explain the space competition in the marrow and Darcy law in porous medium allowed the authors to simulate the medullar stroma. Existence of a diffusion threshold for leukemic cells was proven, below which the healthy state loses its stability and let the leukemic cells overtake the system. Their simulations showed also the action of chemotherapy on the proliferation velocity of the cells.
3.2.3 Multi-agent models: a compromise There exist two ways to combine spatial models with stochasticity. One way could be to include some stochasticity in the continuous reaction-diffusion system of partial differential equations. The second way would be to consider discrete multi-agent models. To the best of our knowledge, the first way has not been tackled yet. This is the reason why we focus our attention here on the second approach: the multi-agent models. The main objective of the use of the multi-agent systems applied to hematopoiesis is to simulate cells as individual capable of self-renewal, apoptosis or differentiation in a closed space representing a part of the bone marrow. The first models appeared in 2006 with Pimentel 2006 who introduced a simple interface based on the early 1978's Mackey model on hematopoiesis. In 2008, D'Inverno et al. worked on a multi-agent model simulating stem cells but the problem was more adapted for the intestinal crypt cells. Ramas at the same period developed a software package named Netlogo (http://ccl.northwestern.edu/netlogo/), a "programmable modeling environment for simulating natural and social phenomena", with one application to the blood cell formation. However, Netlogo's aim is not to model hematopoiesis only. Thus, many specificities related to the bone marrow do not appear. This is the reason why in 2006, Bessonov et al. created a new multi-agent based software dedicated only to the cell interaction in the bone marrow. This work integrated complex processes that have not been taken into account by the previous studies, such as cell communication, size difference, cell differentiation, space competition, pathological and non pathological cell mutations, spread of diseases like leukemia and the bone marrow niche. All the details of this new interface are developed in the next section.
4. A specific multi-agent model adapted to hematopoiesis
How is it possible to model hematopoiesis in the bone marrow in a realistic way using at the same time the space structure and the cell population dynamics? The aim of this section is to
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lineage, green for platelets and blue for the white lineage. This depends totally on the user's decision. Each lineage can be simulated at the same time or not. One can focus only on one branch, each one having a specific property or not: the size of the white cells or erythrocytes decreases as their maturity increases but this is exactly the opposite for megakaryocytes for instance. The size of each cell type can be determined by the user just by changing the radius of the disks illustrating the cells.
4.1.1.2 Stochasticity in proliferation
A cell can die by apoptosis at any time. The proliferation duration can be constant and defined by the user for each cell type, but it can be determined and occur within a time distribution: a constant time plus or minus a range whose value is defined by the user. This is one of the various specificities of this software. It is also possible to decide what could be the probability for a stem cell for instance to give birth to a red white or platelet progenitor. This ability of the software seems quite important for hematologists in the sense that almost 45 % of the blood consists of red blood cells that is about 99% of the hematocrit (portion of cells in the blood, the other portion consist of plasma, that is the remaining 55% of blood). The rest of the hematocrit is composed of white cells for 0.2%, and megakaryocytes between 0.6 to 1%. Thus, it seems realistic to assume that a stem cell has more chances to give birth to a red blood cell rather than a cell from another lineage. The software offers this possibility by choosing different probability for a cell to give birth to a certain cell type. This possibility includes also the probability to die. Thus, even apoptosis can be given a random rate that can be determined by the user (see Fig. 4). This is also relevant in the sense that apoptosis is rather important in the erythrocyte lineage, and this rate can be reduced under the influence of EPO. The influence of EPO and other simulating factors will be described later on.
4.1.2 The bone marrow structure The bone marrow is set up as a simple rectangle in the software. Any time that a cell is pushed away from the rectangle border, it is assumed to reach the blood stream. The size of the rectangle can be chosen easily by the users, and modified anytime. Moreover, in order to be more realistic, it is possible to introduce fixed segments of different size anywhere in the rectangle to simulate the porosity of the bone marrow. These segments cannot be crossed by the cells and they are considered as walls necessary to bypass for the cells. The user can place the segments in different ways: they can be put like a bottle neck to force the cells to use only one way out to the blood stream, they can be in 3 of the 4 borders of the rectangle to give only one possible side for cells to reach the stream, they can also represent different niches where stem cells could develop colonies forming grapes of new born cells. Viscosity of the blood cells in the bone marrow can also be decided by changing a parameter value in the run window of the software. When dividing, each cell giving birth to two or more daughter cells pushes away the other individuals. Space competition is then described. Consequently, if one cell type divides more rapidly than others, the bone marrow would swiftly be full of this type of cells and offspring, the other cells would be pushed away out in the bone marrow, or would have no room to develop their lineage. This phenomenon can occur for instance in case of acute myeloid leukemia described in the next section. It seems more realistic for stem cells to be fixed in a niche and the more a cell is mature, the more its ability to get detached from the colony would be important. This has not yet been taken into account in the software, but it is under current investigation. For the moment, all the cells have the same ability to move out from the bone marrow walls.
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Fig. 4. Presentation of the different windows given by the software. The top left window represent the running application with different parameters such as viscosity. The left window gives the illustration of the tree consisting of all the cells and offspring, each differentiation probability are set up in this tree, each colour represents a cell type, the red frame corresponds to some of cell properties such as the life time, size (radius), etc. The green frame deals with the space property (size of the domain, and addition of segments). The small disks in the grids represent the different cells dividing, differentiating and dying with a focus on one part of the simulation at the bottom left of the figure.
4.2 Normal and pathological hematopoiesis After setting up all the adjustments for the specific problem chosen by the user: number of cell lineages, cell fate, different probabilities (differentiation rates, apoptosis rates, etc.), size, bone marrow structure... It is possible to simulate normal and pathological hematopoiesis.
4.2.1 Normal hematopoiesis To get an accurate model of normal hematopoiesis it is important to collect as many information about the different parameters as possible. This is the reason why it is necessary to exchange many discussions with hematologists. One attempt has been made, taking each size of cell type into account, with different proliferating times, apoptosis probabilities and different niches. The result has been plotted. However, many parameters need to be set up properly in good agreement with the experimental observations. This is under current investigation (see Fig. 5).
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Fig. 6. Illustration of the spread of a disease. A malignant cell is placed in the bone marrow during normal hematopoiesis (top left) and develops to eventually invade the whole domain. This could correspond to a case of acute leukemia (illustration taken from Bessonov et al. 2006).
4.2.2.1 Acute leukemia
In this example, we consider a malignant cell that mutates. The cell chosen is not a stem cell. This choice was made in order to show that the disease can spread and settle down the marrow even if it is not a stem cell. It all depends on the properties given on the mutant cell. In this case, the pathological cells proliferate more rapidly than the average cells. At the early stage of the disease, it seems that leukemic cells will not stay in the marrow. But rapidly, they start to wash out the non pathological cells from the bone marrow and spread all around the place to eventually occupy almost all the medullar medium (see Fig. 6). It is important at this point to note that production of normal cells does not decrease in presence of the pathology. The only event occurring here is the strong local pressure that pushes other cells out to the blood systems. As a consequence, a large number of immature cells invade the blood stream which gives the onset of the symptoms. The proliferation time is of great importance in the spread velocity of the disease. It has been illustrated in Fig. 7. For a long proliferation time, malignant cells remain localized in a small region and do not seem to spread in the whole system. On the contrary, for a short proliferation time the disease would easily spread out and invade the bone marrow. A combination of different
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parameters would then allow the system to reach different equilibria depending on the threshold reached. A mathematical analysis corresponding to the simulation has not yet been done but should be investigated in the future showing a great range of dynamics. It is interesting for instance to note that if the density of stem cells is increased, and the same values for parameters are kept, then leukemia in these specific simulations has less chances to develop. Furthermore, if a mutation is given to a mature cell, this cell will have more difficulties to multiply and let the disease spread, and vice versa. Each choice could be driven by a specific type of leukemia one wants to model.
Fig. 7. Comparison between three simulations with different proliferation times for leukemic cells. The rst one is modelled with a proliferation time of 100 time unit, the second 50, and the third 20 (illustration taken from Bessonov et al. 2006).
4.2.2.2 Chronic myelogenous leukemia (CML)
Chronic myelogenous leukemia occurs when concentration of cells of all types oscillate periodically. This kind of behavior has been observed in the 1970's and studied in the late 1990's and beginning of the 2000's (see section 3.1.3). It is possible, using a specific choice of parameters to obtain such oscillations with the Bessonov's software. Moreover, this application is able to get an output of the cell population leaving the bone marrow. Each cell type can be counted and put in a specific file. This file can be analyzed by any mathematical software able to analyze sets of data. The result we obtain with the example shown here seems qualitatively equivalent to the clinical data obtained in the 1970's (see Fig. 8). Getting quantitative results would be quite interesting for the biologist community in the sense that it would be then possible to replicate quite accurately the experimental or clinical data. This is still under investigation.
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rapid increase of the pathology in the bone marrow. But after a certain period of time, the malignant cell population is washed out from the bone marrow and is replaced by normal cells (see Fig. 9). A question may arise then from this point: how is it possible for a population of mutant cells able to develop rapidly to disappear from the space representing the bone marrow here? The answer could enlightened by a simple focus of the individual level. In the simulation proposed here, all the stem cells are attached to the left wall of the domain, which is not the case for all the other cells. Thus, each cell except the stem cells is either condemned to die by apoptosis, differentiate, self-renew or leave the blood stream after a certain period of time. This is also the case for the cells defined as mutant cells in the example here. They are also part of the dynamics rules of the software. They can self-renew or differentiate but do not increase their number. It may thus be impossible for these cells to overcome their loss and eventually, they are pushed out of the bone marrow by the younger generations of healthy cells. In some cases however, self-renewing mutant cells can divide with a rate large enough to spread out and settle down in the medulla.
Fig. 9. Evolution of the disease spread: malignant (black) cell population grows from a malignant focus but not for a long time but eventually get washed out by the healthy cells. The disease free steady state is then stable in this example. This could correspond to what happens frequently in everyone's body (illustration taken from Bessonov et al. 2006).
In the next section, we go further into the cell environment, by taking the cell communication into account. Thus, not only space competition in the bone marrow plays a role for the development of different cells, but also the influence of the environment and the communication between cells in one neighborhood via some exchange of molecules.
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4.3 Cell communication Cell communication corresponds to another application of the Bessonov's software. In this section, we give an overview of the possibilities given by this application and the influence of different parameters on the cell population dynamics.
4.3.1 How does cell communication is taken into account in the software?
4.3.1.1 A simple example
It is known now that cell differentiation, self-renewal and apoptosis properties can be ruled by complex dynamics of molecules produced inside and outside each cells. For instance, it has been discovered that the stimulation hormones like EPO (see 1.5) would decrease the apoptosis rate in the red blood cell lineage. On the other hand, cell differentiation and the choice of one of the lineages can be regulated by a system of transcription factors. Some mathematical models have been attempted to tackle this problem (Roeder 2006, Crauste 2010). In Huang et al. (2007), the authors developed a model of binary cell fate decisions combining stochastic and deterministic instructions. In our work, we decided to give the possibility for the user to add this ability of cell differentiation through communication with the environment to the existing other applications provided by our software and described above. As mentioned by Roeder et al. (2006a) lineage specification is "a competition process between different interacting lineages propensities". Even if our software allows the user to simulate several cell lineage specifications and communication with as many stimulating factors as wanted, we believe that a description of the application use with the simplest model of only two subpopulations would be more understandable.
4.3.1.2 An exchange of information
Let us then start with one population of undifferentiated cells denoted by A. These cells can divide, giving two daughter cells. One of the daughter cell would be exactly of the same type of its mother (self-renewing), and the other would be of either type G or type F. Two possible lineages are then given to the undifferentiated cells. The color given to the A-type cell would be white, if would be blue for the G-type and red for the F-type (see Fig. 10). Each cell denoted i- not even a cell type but really each individual - is characterized by two functions: fi and gi depending on time the time t, which could correspond for instance to a certain amount of two types of molecules. We assume in this simulation that every new born cell is undifferentiated, that is white. In other word, every time that a stem cell divides, it gives rise to two white cells. These cells are prescribed by the same initial amount of molecules, i.e. f0, g0. This content changes with time with a rate defined by two differential equation describing the evolution of fi and gi with respect to time:
dfi dgi =a(Fi-fi), and =a(Gi-gi), dt dt
where a is a constant, and the Fi and Gi can be chosen to satisfy the so-called "average rule", that is
j i j i
Fi= fj/N, and Gi= gj/N, z z
¦ ¦ (2)
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dgi =Q(fi,gi), fi=fi =constant * dt
for blue cells, where fi* and gi* stand for the concentration of molecules of F and G-type at the moment when differentiation has been decided. In other words, if a cell is of type F, the amount of "F-molecules" would change depending on the defined function P but the "G- molecule" content will remain constant and vice versa with function Q. This property gives the specific shape of the figures representing the simulations on the cell type evolution with a blank squared shape on the upper right part of the plot (see Fig. 11). The function P and Q are defined to be quadratic functions as follows
2 P(f,g)=a +a f+a f +a f 1 2 3 4 g+a 5 g, (6)
and
2 Q(f,g)=b +b 1 2 g+b 3 g +b f 4 g+a f, 5 (7)
where ai and bi, i=1,...,5 are some constants defined by the user. The quadratic form of P and Q was chosen for simplicity, but can be modified anytime by the user.
Fig. 11. Cell position in the f–g plane. A white cell remains undifferentiated as long as fi (t)+gi (t)^2 2 d ǔ(inside the disk (figure in the left)), where ǔ is a given parameter. If the
concentrations of f and g become sufficiently high (greater than ǔ), the cell chooses its type. Once the differentiation occurs and the cell chooses its type, further evolution of f and g becomes different. For red cells df i/dt =P(fi,gi), gi = gi (it remains constant), for blue cells dgi/dt = Q(fi,gi), fi=fi (it remains constant). In other words, after differentiation, the value of f in red cells increases and the value of g remains constant; for blue cells g increases and f remains constant which gives the specific shape of the f ï g graph (right) (illustration taken from Bessonov et al. 2009).
In each figure it is then possible to represent the cell of all types with undifferentiated cells on the quarter of disk on the bottom left angle with radius equal to ǔ (corresponding to the distance between the origin of the axis and the bottom left part of the blank squared shape.
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Everything above this disk, between the y-axis and the blank squared shape would correspond of cells of blue G-type and the higher the cell is located the more mature it is. On the other hand, all cells on the right side after the disk and below the blank squared shape are the red F-type cells. And thus, the choice of the lineage can be well defined depending on the zone a cell can be plot. Consequently, if all cells remain within the disk, this means that all cells remain undifferentiated; this could correspond to a case of acute leukemia. They can also be located in one or two lineages, or the three depending on the sets of parameters chosen. This will be developed in the next paragraph. Furthermore, cell generations can be observed by the "circular stripes" appearing in the simulations. This is correlated to the ratio of the proliferation time between the stem cells and the first daughters. Let us see some examples in the next paragraph.
4.3.2 Examples of cell communication and differentiation We remind the reader here the starting bases of the cell communication application. At the beginning of the simulation, only undifferentiated cells are produced by the stem cells. Once the whole domain has been filled up with all the white cells, the process stops and each cell is prescribed randomly one of the two (red or blue) types with some value fi and gi. The application starts again and then, all new born undifferentiated cells are obliged to choose one of the types depending on their environment and the parameters set up as explained in the previous paragraph. Some structures can appear. This process starts with a random distribution of the cells, but specific structures can appear depending on the different sets of parameters. The application can give different outputs: the number of cells of each type can appear in a specific file as explain in a previous section. In other words, after a certain time Ǖ (^) i, which represents the moment when the ith cell leaves the bone marrow. The cell is then registered into the file with its fi and gi content, which determines its type depending on the level of red or blue molecule inside. To be more convenient, the software plots directly all these cells on the (f,g)-plane. In other words, it (f(Ǖ (^) i ),g(Ǖ (^) i)) plots corresponding to each cell leaving the bone marrow. The graph obtains represent then the population of blood cells released in the system. Let us give three examples corresponding to the influence of the main parameters: starting with the proliferation time, then the cell size, and finally a combination of the cell communication parameters with the self-renewal processes.
4.3.2.1 Cell communication and the proliferation time parameter
In this example, let us assume that cell of the second generation cannot differentiate anymore. This may represent a simplified case of normal hematopoiesis. Almost no undifferentiated cells are found in the bone marrow, and a great proportion of cells clearly belong to one of the two types and increasing the proliferation time would improve the cell differentiation process. It is possible to give a biological explanation behind these simulations. It is indeed, easy to understand that in the case of non pathological hematopoiesis, a cell being given more time to mature will leave the bone marrow with a more complete functional material than cells having little time to fulfill the maturation process. It has been shown for instance that in case of stress erythropoiesis (like anemia or blood loss), cell proliferation is accelerated due to the combination of self-renewal process of progenitors, but also the apoptosis rate decreases in the bone marrow, due to the effect of EPO, but once in the blood stream, cell death is greater than normal due to the fact that they
