Noname manuscript No. (will be inserted by the editor) Hybrid model of erythropoiesis P. Kurbatova · N. Eymard · V. Volpert Received: date / Accepted: date Abstract A hybrid model of cell dynamics is presented. It is illustrated by model examples and applied to study erythropoiesis (red blood cell produc- tion). In this approach, cells are considered as discrete objects while intra- cellular proteins and extra-cellular biochemical substances are described with continuous models. Spatial organization of erythropoiesis occurring in specific structures of the bone marrow, called erythroblastic island, is investigated. Keywords hybrid model · erythropoiesis · erythroblastic island · macrophage · regulatory mechanisms Mathematics Subject Classification (2000) 92C37 · 68U20 · 35Q70 · 35Q92 1 Hybrid models of cell dynamics Hybrid discrete-continuous models are widely used in the investigation of dy- namics of cell populations in biological tissues and organisms that involve processes at different scales. In this approach biological cells are considered as discrete objects described either by cellular automata ([9], [13], [14], [21], [26], [27]) or by various on-lattice or off-lattice models ([8], [15], [22]) while intracellular and extracellular concentrations are described with continuous models, ordinary or partial differential equations. In cellular automaton model each individual cell can be represented as a single site of lattice, as several connected lattice sites or the lattice site can be larger than an individual cell. A generalized cellular automaton approach N. Eymard Institut Camille Jordan, University Lyon 1, UMR 5208 CNRS 69622 Villeurbanne, France E-mail: [email protected]P. Kurbatova E-mail: [email protected]V. Volpert E-mail: [email protected]
Transcript
Noname manuscript No.(will be inserted by the editor)
Hybrid model of erythropoiesis
P. Kurbatova · N. Eymard · V. Volpert
Received: date / Accepted: date
Abstract A hybrid model of cell dynamics is presented. It is illustrated bymodel examples and applied to study erythropoiesis (red blood cell produc-tion). In this approach, cells are considered as discrete objects while intra-cellular proteins and extra-cellular biochemical substances are described withcontinuous models. Spatial organization of erythropoiesis occurring in specificstructures of the bone marrow, called erythroblastic island, is investigated.
Keywords hybrid model · erythropoiesis · erythroblastic island ·
Hybrid discrete-continuous models are widely used in the investigation of dy-namics of cell populations in biological tissues and organisms that involveprocesses at different scales. In this approach biological cells are consideredas discrete objects described either by cellular automata ([9], [13], [14], [21],[26], [27]) or by various on-lattice or off-lattice models ([8], [15], [22]) whileintracellular and extracellular concentrations are described with continuousmodels, ordinary or partial differential equations.
In cellular automaton model each individual cell can be represented as asingle site of lattice, as several connected lattice sites or the lattice site canbe larger than an individual cell. A generalized cellular automaton approach
N. EymardInstitut Camille Jordan, University Lyon 1, UMR 5208 CNRS 69622 Villeurbanne, FranceE-mail: [email protected]
is presented by the cellular Potts models (CPM). The CPM is a more sophis-ticated cellular automaton that describes individual cells, occupying multiplelattice sites, as extended objects of variable shapes. These models take intoaccount surface energy of cell membrane. The CPM effective energy can con-trol cell behaviors including cell adhesion, signalling, volume and surface areaor even chemotaxis, elongation and haptotaxis [13], [26]. In each particularCA model, the rules which determine cell motion should be specified. It canbe influenced by the interaction of cells with the elements of their immediatesurrounding and by processes that involve cellular response to external signalslike chemotaxis. The numerous models with gradient fields of chemical concen-trations that govern motility of cells have been suggested. Cellular automatonhave been used extensively to model a wide range of problems. Different stagesof tumor development from initial avascular phase([9], [14]) to invasion ([2])and angiogenesis ([21], [27]) are studied.
Off-lattice models are important to those biological situations in which theshape of individual cells can influence the dynamics or geometry of the wholepopulation of cells. In off-lattice models, shape of cells can be explicitly mod-elled and response to local mechanical forces, interaction with neighboring cellsand environment can be investigated. Hybrid off-lattice models, not limited inpossible directions of cell motion, are widely applied to the modelling of tumorgrowth and invasion where cell migration should be taken into account ([15],[22]). Another type of off-lattice models, called fluid-based elastic cell model,approach that takes into account cell elasticity, is also applied in tumor growthmodelling ([2], [8]).
In this work we use a hybrid modelling approach with off-lattice cell dy-namics to describe erythropoiesis. Cells are considered as discrete objects andthey are represented as soft spheres. Intracellular and extracellular concentra-tions are described with continuous models.
In Section 2, we introduce a hybrid modeling approach with off-lattice celldynamics. In Section 3 this approach is applied to the modelling of erythro-poiesis.
2 1D model example
We restrict ourself here to the case where cells are considered as soft spheres.They can interact with each other and with the surrounding medium mechani-cally and biochemically, they can divide, differentiate and die due to apoptosis.Cell behavior is determined by intracellular regulatory networks described byordinary differential equations and by extracellular bio-chemical substancesdescribed by partial differential equations.
We describe the motion of each cell by the displacement of its center ac-cording to Newton’s second law. More detailed description can be found in[4], [5], [10], [19]. We begin with the 1D model example where cells are lo-cated along a straight line. Consider two elastic balls with the centers at thepoints x1 and x2 and with the radii, respectively, r1 and r2. If the distance
Hybrid model of erythropoiesis 3
d12 between the centers is less then the sum of the radii, r1 + r2, then thereis a repulsive force f12 between them which depends on the distance d12. Ifa particle with the center at xi is surrounded by several other particles withthe centers at the points xj , j = 1, ..., k, then we consider the pairwise forcesfij assuming that they are independent of each other. This assumption corre-sponds to small deformation of the particles. Hence, we find the total force Fi
acting on the i-th particle from all other particles, Fi =∑
j 6=i fij . The motionof the particles can now be described as the motion of their centers:
mxi + µmxi − Fi = 0, (1)
where m is the mass of the particle, xi is the center of the ith particle, thesecond term in the left-hand side describes the friction by the surroundingmedium with the coefficient of friction µ, the third term is the potential forcebetween cells. Dissipative forces can also be written in a different form. Thisis related to dissipative particle dynamics [16]. We consider the potential forcebetween particles in the following form
fij =
{
Kh0−hij
hij−(h0−h1), h0 − h1 < hij < h0
0 , hij ≥ h0
(2)
where hij is the distance between the particles i and j, h0 is the sum of cellradii, K is a positive parameter, and h1 accounts for the incompressible partof each cell. This means that the internal part of the cell is incompressible.It allows us to control compressibility of the meduim. The force between theparticles tends to infinity when hij decrease to h0 − h1.
For each particular application, intracellular and extracellular regulationshould be specified for each cell type as well as how cell fate (proliferation,differentiation and apoptosis) depends on the regulatory networks. In thissection we consider a model example where cells can only divide or die byapoptosis. After division a cell gives two cells identical to itself. Cell fate isdetermined by two intracellular proteins ui and vi. We suppose that if theconcentration ui attains some critical value uc, then the cell divides. If vi
becomes equal vc, the cell dies. Intracellular concentrations are described byordinary differential equations:
dui
dt= k
(1)1 u(xi, t) − k
(1)2 ui(t) + H1,
dvi
dt= k
(2)1 v(xi, t) − k
(2)2 vi(t) + H2.
(3)
Here and in what follows we write equations for intra-cellular concentrationsneglecting the change of the cell volume. This approximation is justified sincethe volume changes only twice before cell division and this change is relativelyslow. The first term in the right-hand side of the first equation shows that theintra-cellular concentration ui grows proportionally to the value of the extra-cellular concentration u(x, t) at the space point xi where the cell is located.
4 P. Kurbatova et al.
It is similar for the second equation. These equations contain the degradationterms and constant production terms, H1 and H2. When a new cell appears,we put initially the concentrations ui and vi equal zero. It is also possible torelate these concentrations to the corresponding concentrations in the mothercell (Section 3).
Consider the case where k(1)1 is different from zero. If it is positive, then
cells stimulate proliferation of the surrounding cells, if it is negative, they sup-press it. Both cases can be observed experimentally. We restrict ourselves here
by the example of negative k(1)1 . All other coefficients are taken equal to zero.
Therefore, cells have a fixed life time τv. If they do not divide during this time,
they die. We notice that as k(2)1 equals zero, the extracellular concentration v
does not influence the evolution of vi. We achieve similar qualitative behaviorof cell population, if instead of the variable u, which decelerate cell prolifer-ation, we consider the variable v assuming that it accelerates cell apoptosis.Example where k2
1 6= 0 is studied in [3]. We carry out the 1D simulation wherecells can move along the straight line. Initially, there are two cells in the middleof the interval. Figure 1 shows the evolution of this population in time. Foreach moment of time (vertical axis) we have the positions of cells (horizontalaxis) indicated with blue points.
The distribution of u in the extracellular matrix is shown in Figure 1e.Similar to the cell population (Figure 1d) it spreads in space as a generalizedtravelling wave. The evolution of the cell population in Figure 1 (upper, left)can be characterized by two main properties. First of all, it expands to theleft and to the right with approximately constant speed. Second, the totalpopulation consists of relatively small sub-populations. Each of them startsfrom a small number of cells. Usually, these are two cells at the right and atthe left of the previous sub-population. During some time, the sub-populationgrows, reaches certain size and disappears giving birth to new sub-populations.
This behavior can be explained as follows. The characteristic time of celldivision is less than that of cell death. When the cell sub-population is small,the quantity of u is also small, and its influence on cell division is not sig-nificant. When the sub-population becomes larger, it slows down cell divisionbecause of growth of u. As a result, the sub-population disappears. The outercells can survive because the level of u there is less.
The geometrical pattern of cell distribution for these values of parametersreminds Serpinsky carpet (Figure 1, left), an example of fractal sets. Thepattern of cell distribution depends on the parameters. Three other examplesare shown in Figure 1.
3 Modelling erythropoiesis
3.1 Organization and modelling of erythropoiesis
Erythropoiesis represents a continuous process maintaining an optimal num-ber of circulating red blood cells and tissue oxygen tension. It occurs mainly
Hybrid model of erythropoiesis 5
3 3.5 4 4.5 5 5.5 6 6.5 70
1000
2000
3000
4000
5000
6000
7000
8000
(a) vc = 2, k(1)1 = −5 · 10−3 ,
d1 = 0.0001, b1 = 0.2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
1000
2000
3000
4000
5000
6000
7000
8000
(b) vc = 2, k(1)1 = −5.5 ·10−3,
d1 = 0.0001, b1 = 0.2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
1000
2000
3000
4000
5000
6000
(c) vc = 3, k(1)1 = −5 · 10−3,
d1 = 0.0001, b1 = 0.2
0 0.5 1 1.5 2 2.5 30
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
(d) vc = 1.01, k(1)1 = 10−4, d1 =
0.0001, b1 = 0.2
0 0.5 1 1.5 2 2.5 3 3.50
1
2
3
4
5
6
7
8
9
10t=1500t=3000U
x
(e) Distribution of u at differentmoments of time for the valuesof parameters vc = 1.01, k
(1)1 =
10−4, d1 = 0.0001, b1 = 0.2
Fig. 1 Dynamics of cell population in the case where cells either self-renew or die byapoptosis. Cells are shown with blue dots. Horizontal axis shows cell position, vertical axis -
in the bone marrow where erythroid progenitors, immature blood cells, whichcan proliferate and differentiate, undergo a series of transformations to becomeerythroblasts (mature progenitors) and then reticulocytes which subsequentlyenter the bloodstream and mature into erythrocytes. At every step of thisdifferentiation process, erythroid cells can die by apoptosis (programmed celldeath) or self-renew [11], [12]. Numerous external regulations control cell fateby modifying the activity of intracellular proteins. Erythropoietin (Epo) is ahormone synthesized in the kidney in response of decrease in tissue oxygenlevel. Epo promotes survival of early erythroblast subsets by negative regula-tion of their apoptosis through the action on the death receptor Fas [18].
Glucocorticoids [11], [12] and some intracellular autocrine loops [12], [25]induce self-renewal. Previously considered by the authors [7], [10], Erk (fromthe MAPK family) promoting cell self-renewal inhibits Fas (a TNF familymember) [24]. The cell fate depends on the level of these proteins. In additionto global feedbacks, there is a local feedback control through cell-cell interac-tion, during which Fas-ligand produced by mature cells binds to the membraneprotein Fas inducing both differentiation and death by apoptosis [20].
6 P. Kurbatova et al.
The process of erythroid maturation occurs in erythroblastic islands, thespecialized niches of bone marrow, in which erythroblasts surround a centralmacrophage which influences their proliferation and differentiation [6], [28].However, erythropoiesis has been mainly studied under the influence of Epo,which can induce differentiation and proliferation in vitro without the presenceof the macrophage. Hence, the roles of the macrophage and the erythroblasticisland have been more or less neglected.
In the next section we will use a hybrid discrete-continuous model devel-oped in [4], [5] in order to bring together intracellular and extracellular levelsof erythropoiesis as well as to study the importance of spatial structure of ery-throblastic islands in the regulation of erythropoiesis. We focus in particularon the role of the macrophage in erythroid cell proliferation and differentia-tion and its role in the erythroblastic island robustness. To our knowledge,this is the first attempt to model erythropoiesis by taking these aspects intoaccount. In the previous models [10], [19] it was supposed that Fas-ligand wasproduced by mature erythroid cells [20]. This assumption corresponds to hu-man erythropoiesis. In this work we study the production of red blood cell inmice taking into account the co-expression of Fas and Fas-ligand by immatureerythroid progenitors, particularly in spleen [17].
3.2 Erythroblastic islands
Production of red blood cells in the bone marrow occurs in small units callederythroblastic islands. They consist from a central macrophage surroundedby erythroid cells with a various level of maturity. Their number can varyfrom several cells up to about 30 cells. Some of the erythroid cells produceFas-Ligand which influences the surrounding cells by increasing intracellularFas activity. These are immature cells in murine erythropoiesis [17] and moremature cells in human erythropoiesis [20]. On the other hand, macrophagesproduce a growth factor (like SCF, Ephrin-2 or BMP-4 [23]) which stimulatesproliferation. In addition, immature cells are subject to a feedback controlmediated by mature red blood cells circulating in the bloodstream, represent-ing the action of Epo. Concentration of Epo in the computational domain issupposed to be uniform, so all cells are similarly influenced by Epo (Figure 2).
The concentrations of Fas-ligand FL and of the growth factor G, producedby macrophages, in the extracellular matrix are described by the reaction-diffusion equations:
∂FL
∂t= D1∆F + W1 − σ1FL, (4)
∂G
∂t= D2∆G + W2 − σ2G. (5)
where W1 and W2 are the constant source terms, the last terms in the right-hand sides of these equations describe their degradation, D1 and D2 are dif-fusion coefficients.
Hybrid model of erythropoiesis 7
apoptosis
Kidney Epo
production
Macrophage
Erythroid progenitor
(Fas)
Fig. 2 Intracellular and extracellular regulation that determine erythroid progenitor fate.
Complete intracellular regulatory mechanisms involved in erythroid pro-genitor cell fate are very complex and not yet elucidated. Based on the cur-rent knowledge (Section 3.1), we consider a simplified regulatory network withthree groups of proteins and hormones. The first group, denoted by u, consistsof proteins and hormones, such as protein Erk, BMP4 and glucocorticoids,involved into self-renewal. The second group, denoted by v is represented byGATA-1 [1], promoting erythroid progenitor differentiation. The last group,denoted by w, is represented by protein Fas inducing apoptosis. Moreover, self-renewal proteins inhibits apoptotis and differentiation. Intracellular regulationin erythroid progenitors is described by the ordinary differential equations:
du
dt= γ1 (6)
dv
dt= γ2(1 − β1uv) (7)
dw
dt= γ3(1 − β2uw), (8)
At the end of cell cycle, if u < v, progenitors differentiate, if u > v, thenprogenitors self-renew. If w > wcr, then the cell dies by apoptosis at any mo-ment of time during cell cycle. The quantity w being different in different cells,some of them die and some other cells survive. The intracellular regulation isinfluenced by the extracellular variables through the coefficients γ1 and γ3:
γ1 = γ01 + γ1
1G, γ2 = γ02 + γ1
2Epo, γ3 = γ03 + γ1
3FL.
The concentration of Epo is constant in these simulations. Therefore the valueof the coefficient γ2 is the same for all cells, while the values of the coefficientsγ1 and γ3 depend on the cell position with respect to the macrophage and toother erythroid cells. Indeed, γ1 depends on the quantity of G produced by themacrophage. This quantity is calculated for each cell at each moment of time.
8 P. Kurbatova et al.
Similarly, the quantity of FL influences γ3. Let us note, that the thresholdwcr also depends on the concentration of Epo. Since Epo downregulates cellapoptosis, wcr increases with the increase of Epo.
After each cell division, initial concentrations u, v, w in the daughter cellsare set equal to the half of the concentration in the mother cell. Cell cycleis taken 24 hours plus/minus random value between 0 and 12 hours. Thefirst generation of progenitors has the initial concentrations of u, v, w given asa random variable uniformly distributed in the intervals [0, u0], [0, v0] and[0, w0]. Random perturbations in initial conditions are important to describeexperiments in cell culture without macrophages (not presented here). In thiscase, if the initial protein concentrations are the same for all cells, they willhave the same fate. However the experiments show this fate can be differentfor different cells.
A typical structure of erythroblastic islands is shown in Figure 3. It consistsof a macrophage and two other types of cells, immature progenitors (yellow)and reticulocytes (blue). Macrophage expresses a growth factor (green) drivingnearby erythroblasts toward self-renewal. In mice Fas-ligand (red) is producedby undifferentiated cells (Figure 3, left), while in human by differentiated cells(Figure 3, right). We will consider below only erythropoiesis in mice. Theresults of modelling will be compared with experiments in the subsequentwork. Model of human erythropoiesis can be found in [10].
Mice Human
Fig. 3 The structure of erythroblastic islands. A large cell represent the macrophage pro-ducing growth factor (green). Immature cells are yellow, mature cells are blue. In humanerythropoiesis (right) Fas-ligand (red) is produced by differentiated cells. In mice (left), itis produced by progenitors. Black circle inside cells shows their incompressible part. Cellsare born with radius r0. During the cell cycle, the radius increases until it reaches 2r0. Atthis moment, the cell divides.
With a suitable choice of extracellular and intracellular parameters, func-tioning of the island is stable (Figure 4, left). It contains in average 25 erythroidprogenitors and 5 reticulocytes. The influence of the most important param-eters on the erythroblastic island is estimated and shown in Figure 4 and 5.Periodic oscillations in the number of cells are related to the cell cycle.
An important characteristic of erythroblastic islands is the rate of produc-tion of progenitors and reticulocytes, which determine the number of erythro-
Hybrid model of erythropoiesis 9
cytes in blood. We study how this production depends on the parameters wcr,γ1 and γ0
2 . First, we will focus on the global feedback control mediated byEpo. When wcr increases, the size of the island also increases due to decreasedapoptosis of immature progenitors. Consequently, there are more cells at theperimeter of the island. They have a tendency to differentiate since they arefar from the macrophage which promotes self-renewal. Therefore, increasingof the Epo level increases the production of reticulocytes (Figure 4, right).
Increase of γ1 promotes self-renewal and augments the number of immatureprogenitors. Consequently, the number of reticulocytes also increases (Figure5, left). When we increase the value of γ0
2 , which influences the differentiationrate, the number of reticulocytes decreases. For sufficiently large values ofthis parameter, the island disappears after several cell cycles (Figure 5, right).Let us note that we split here the action of Epo on cell differentiation fromits action to downregulate apoptosis. In vivo, Epo acting on both of themstimulates production of erythrocytes.
Fig. 4 The number of progenitors (red curve) and reticulocytes (blue curve) in time (left).Average in time number of progenitors and reticulocytes for different values of the parameterwcr (right). Each curve is obtained as a mean value for 10 simulations. Cells cycle lengthis 24h. Cells cycle variation is 12h. Values of intracellular parameters: γ1 = 0.00075; γ0
We proposed a new model of erythropoiesis taking into account intracellularand extracellular regulations and cell-cell interaction. Competition betweenthree groups of proteins that determine the cell fate has been considered withcontinuous models (ordinary differential equations) whereas cells have beenstudied as discrete objects. Extracellular regulatory network are describedby partial differential equations. The model suggests an important role ofmacrophages in functioning of erythroblastic islands and production of ma-ture red blood cells. Without macrophages, erythroblastic islands quickly lose
10 P. Kurbatova et al.
Fig. 5 Average in time number of progenitors and reticulocytes for different values of γ1
(left) and γ02 (right) (mean values for 10 simulations). Other parameter values are the same
as in Figure 4.
their stability and either die out or abnormally proliferate [10]. Macrophagescontrol the size of erythroblastic islands which show their capacity to rapidlyincrease production of mature cells in response to stress (anemia, hypoxia).This response is based on the increased production of erythropoietin and glu-cocorticoids. Analysis of feedback by Epo and of the role of central macrophagein erythroblastic islands give a new insight into the mechanisms of control oferythroid cell proliferation and differentiation. More detailed description ofthe intracellular regulation and comparison with experimental data will bepresented in the subsequent papers.
Acknowledgments
Authors thank Prof. Mark Koury and all members of the INRIA Team DRAC-ULA for fruitful discussions.
References
1. G.R. Aispuru, M.V. Aguirre, J.A. Aquino-Esperanza, C.N. Lettieri, J.A. Juaristi, N.C.Brandan, Erythroid expansion and survival in response to acute anemia stress: the role ofEPO receptor, GATA-1, Bcl-xL and caspase-3, Cell Biol. Int. 32 (8), (2008), pp. 966–78.
2. A.R.A. Anderson, K.A. Rejniaka, P. Gerleea, V. Quaranta, Modelling of Cancer Growth,Evolution and Invasion: Bridging Scales and Models, Math. Model. Nat. Phenom. 2 (3),pp. 1–29, (2007).
3. N. Bessonov, P. Kurbatova, V. Volpert, (2013), Pattern Formation in Hybrid Mod-els of Cell Populations. V. Capasso et al. (eds.), Pattern Formation in Morphogenesis,Springer Proceedings in Mathematics 15, Springer-Verlag Berlin Heidelberg 2013, pp.107-119 (2012).
4. N. Bessonov, F. Crauste, S. Fischer, P. Kurbatova, and V. Volpert, Application of Hy-brid Models to Blood Cell Production in the Bone Marrow., Math. Model. Nat. Phenom,Volume 6, Number 7, pp. 2–12, (2011).
5. N. Bessonov, P. Kurbatova, V. Volpert,Particle dynamics modelling of cell populations.Math. Model. Nat. Phenom., JANO9 The 9th International Conference on NumericalAnalysis and Optimization, Volume 5, Number 7, pp. 42 – 47, (2010).
Hybrid model of erythropoiesis 11
6. J.A. Chasis and N. Mohandas, Erythroblastic islands: niches for erythropoiesis, Blood112, (2008).
7. F. Crauste, I. Demin, O. Gandrillon, V. Volpert, Mathematical study of feedback controlroles and relevance in stress erythropoiesis J. Theor. Biology 263, pp. 303–316, (2010).
8. R. Dillon, M. Owen, and K. Painter, A single-cell-based model of multicellular growthusing the immersed boundary method, AMS Contemp. Math. 466, pp. 1–15, (2008).
9. S. Dormann and A. Deutsch, Modeling of self-organized avascular tumor growth with ahybrid cellular automaton, In Silico Biol., pp. 393–406, (2002).
10. S. Fischer, P. Kurbatova, N. Bessonov, O. Gandrillon, V. Volpert et F. Crauste.Modelling erythroblastic islands: using a hybrid model to assess the function of centralmacrophage. J. Theo. Biol., 298, 92-106 (2012).
11. O. Gandrillon, The v-erbA oncogene. Assessing its differentiation-blocking ability usingnormal chicken erythrocytic progenitor cells, Methods Mol. Biol. 202, pp. 91–107, (2002).
12. O. Gandrillon , U. Schmidt, H. Beug, J. Samarut, TGF-beta cooperates with TGF-alphato induce the self-renewal of normal erythrocytic progenitors: evidence for an autocrinemechanism, Embo J. 18, pp. 2764–2781, (1999).
13. C. Giverso, M. Scianna, L. Preziosi, N. Lo Buono and A. Funaro, Individual Cell-BasedModel for In-Vitro Mesothelial Invasion of Ovarian Cancer, Math. Model. Nat. Phenom.5 (1), pp. 203–223, (2010).
14. Y. Jiang, J. Pjesivac-Grbovic, C. Cantrell, and J. P. Freyer, A Multiscale Model forAvascular Tumor Growth, Biophys J. 289 (6), pp. 3884–3894, (2005).
15. J. Jeon, V. Quaranta, and P. T. Cummings, An Off-Lattice Hybrid Discrete-ContinuumModel of Tumor Growth and Invasion, Biophys. J. 98 (1), pp. 37–47, (2010).
16. M. Karttunen, I. Vattulainen, A.Lukkarinen. A novel methods in soft matter simula-tions. Springer, Berlin, (2004).
17. M. Koulnis, Y. Liu, K. Hallstrom, M. Socolovsky, Negative autoregulation by Fas sta-bilizes adult erythropoiesis and accelerates its stress response, PLoS One, (2011).
18. M. J. Koury and M.C. Bondurant, Erythropoietin retards DNA breakdown and preventsprogrammed death in erythroid progenitor cells, Science 248, pp. 378–381, (1990).
19. P. Kurbatova, S. Bernard, N. Bessonov, F. Crauste, I. Demin, C. Dumontet, S. Fis-cher et V. Volpert. Hybrid model of erythropoiesis and leukemia treatment with cytosinearabinoside. SIAM J. App. Math., 71 (6), 2246-2268 (2011).
20. R. De Maria, U. Testa, L. Luchetti, A. Zeuner, G. Stassi, E. Pelosi, R. Riccioni, N. Felli,P. Samoggia and C. Peschle, Apoptotic Role of Fas/Fas Ligand System in the Regulationof Erythropoiesis, Blood 93, pp. 796–803, (1999).
21. S.R. McDougall, A.R.A. Anderson, M.A.J. Chaplain and J.A. Sherratt, Mathematicalmodelling of flow through vascular networks: Implications for tumour-induced angiogen-esis and chemotherapy strategies, Bulletin of Mathematical Biology 64 (4), pp. 673–702,(2002).
22. I. Ramis-Conde, D. Drasdo, Alexander R. A. Anderson, and Mark A. J. Chaplain,Modeling the Influence of the E-Cadherin--Catenin Pathway in Cancer Cell Invasion: AMultiscale Approach, Biophys. J., 95 (1), pp. 155–165, (2008).
23. M.M. Rhodes, P. Kopsombut, M.C. Bondurant, J.O. Price, M.J. Koury. Adherence tomacrophages in erythroblastic islands enhances erythroblast proliferation and increaseserythrocyte production by a different mechanism than erythropoietin, Blood, 111, (2008).
24. C. Rubiolo, D. Piazzolla, K. Meissl, H. Beug, J.C. Huber, A. Kolbus and M. Baccarini,A balance between Raf-1 and Fas expression sets the pace of erythroid differentiation,Blood 108, pp. 152–159, (2006).
25. S.T. Sawyer, S.M. Jacobs-Helber, Unraveling distinct intracellular signals that promotesurvival and proliferation: study of erythropoietin, stem cell factor, and constitutive sig-naling in leukemic cells, J. Hematother. Stem Cell Res. 9, pp. 21–29, (2000).
26. M. Scianna, R. M. H. Merks, L. Preziosi and E. Medico, Individual cell-based models ofcell scatter of ARO and MLP-29 cells in response to hepatocyte growth factor, J. Theor.Biol. 260 (1), pp. 151–160, (2009).
27. A. Stephanou, S.R. McDougall, A.R.A. Anderson and M.A.J. Chaplain, Mathematicalmodelling of flow in 2D and 3D vascular networks: Applications to anti-angiogenic andchemotherapeutic drug strategies, Mathematical and Computer Modelling 41 (10), pp.1137-1156, (2005).
28. A.S. Tsiftsoglou, I.S. Vizirianakis, J. Strouboulis, Erythropoiesis: model systems, molec-ular regulators, and developmental programs, IUBMB Life., 61(8), pp. 800–830, (2009).
