International audienceA viscoelastic model for avascular tumor growth

Bresch, Didier

Colin, Thierry

Grenier, Emmanuel

Ribba, Benjamin

Saut, Olivier

Discrete and Continuous Dynamical Systems

In this article, we present a continuous model for tumor growth . This model describes the evolution of three cellular densities: the density of sane tissue, cancer cells and extracellular medium. In order to render correctly the cellular division, this model use a discrete description of the cell cycle (the set of steps a cell has to undergo in order to divide) presented in [7]. To render cellular adhesion and the mechanics which may inﬂuence the growth, we use a viscoelastic approach. This model extends the one presented in [7] with a more realistic description of the movement

Hal - Université Grenoble Alpes

A viscoelastic model for avascular tumor growth

BRESCH, Didier

COLIN, Thierry

GRENIER, Emmanuel

RIBBA, Benjamin

SAUT, Olivier

Oskar Bordeaux

A viscoelastic model for avascular tumor growth.

HAL Université de Savoie

INRIA a CCSD electronic archive server

HAL Id: inria-00267292https://hal.inria.fr/inria-00267292v4Submitted on 10 Jul 2009HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.A viscoelastic model for avascular tumor growthDidier Bresch, Thierry Colin, Emmanuel Grenier, Benjamin Ribba, OlivierSautTo cite this version:Didier Bresch, Thierry Colin, Emmanuel Grenier, Benjamin Ribba, Olivier Saut. A viscoelastic modelfor avascular tumor growth. [Research Report] 2009. ￿inria-00267292v4￿Manuscript submitted to Website: http://AIMsciences.orgAIMS’ JournalsVolume X, Number 0X, XX 200X pp. X–XXA VISCOELASTIC MODEL FOR AVASCULAR TUMORGROWTHDidier BreschLama, Université de Savoie, CNRS UMR 5127,Campus scientifique, 73376 Le Bourget du lac cedex, FranceThierry ColinUniversité Bordeaux 1, Institut de Mathématiques,INRIA Sud-Ouest EPI MC2,351 cours de la libération, 33405 Talence Cedex, FranceEmmanuel GrenierUnité de Mathématiques Pures et Appliquées, CNRS UMR 5669,Ecole Normale Supérieure de Lyon, 69364 Lyon cedex, FranceBenjamin RibbaUniversité de Lyon, Lyon, F-69003, France;Université Lyon 1, Ciblage Thérapeutique en Oncologie (EA3738),Faculté de Médecine Lyon-Sud, Oullins, F-69921, FranceOlivier SautUniversité Bordeaux 1, Institut de Mathématiques,CNRS UMR 5251 and INRIA Sud-Ouest EPI MC2,351 cours de la libération, 33405 Talence Cedex, FranceAbstract. In this article, we present a new continuous model for tumorgrowth. This model describes the evolution of three components: sane tis-sue, cancer cells and extracellular medium. In order to render correctly thecellular division, this model uses a discrete description of the cell cycle (theset of steps a cell has to undergo in order to divide). To account for cellularadhesion and the mechanics which may influence the growth, we assume a vis-coelastic mechanical behavior. This model extends the one presented in [18]with a more realistic description of the forces that drive the movement.1. Introduction. In a sketchy way, a tumor arises after several mutations of cellsthat have made them less sensible to anti-growth factors or lack of nutrients, forinstance. This may lead to uncontrolled division of these cells.In order to divide, a cell needs nutrients (such as oxygen), which is obtainedfrom its close environment in the avascular phase. As the tumor grows, some cellsdo not get any more enough nutrient and turn to a quiescent state where they nolonger divide waiting for the environment to become favorable again. Therefore,for a realistic description of cancer growth, one has to describe the evolution of theconcentration of nutrients. Tumor cells have also the ability to produce their energyfrom glucose, whose production lowers the pH of the medium [12]. This toxicityKey words and phrases. Avascular tumor growth. Multiscale models. Cell cycle modeling.Fluid dynamics.12 D. BRESCH, T. COLIN, E. GRENIER, B. RIBBA AND O. SAUTFigure 1. Scheme of the simplified cell-cycle used in this paperand presented in [18].may harm cancer or healthy cells and favor tumoral invasion. Finally, it is knownthat cellular adhesion has an influence on tumor growth [4, 3].There are many mathematical models describing solid tumor growth. Roughlythey can be divided in two classes. In discrete models, one describes the evolution ofcells individually. In this class, we find for instance cellular automata or agent-basedmodels [1, 17]. These approaches are very efficient to describe the cell-scale aspectsof tumor growth, they are computationally expensive, which makes them difficultto use for a large number of cells. Furthermore, it is also difficult to account for themechanical effects (cell-to-cell mechanical interaction) influencing tumor growth.Yet a precise description at the cellular level is possible [8, 10].Continuous models compute the evolution of cellular densities or of the boundaryof the tumor (when strong geometrical assumptions apply) [9]. They are much lessexpensive to discretize but, as they deal with averages over a large number of cells,they can not always account correctly for the microscopic or genetic aspects oftumor growth. Most of these models are based on partial differential equations,such as reaction-diffusion equations [11] or mass-balance equations [2]. Some ofthese models are accounting for complex phenomena like contact inhibition [20] orcellular adhesion [6]. In this paper, we present a model of this kind, where massflow is obtained through a viscoelastic description of the tumor mass and describingthe mechanical behavior of the healthy tissues, the interstitial liquid and the cancercells in a compact way. The outline of this paper is the following one. In Sec. 2,we present the mathematical model and in Sec. 3, we show a numerical experimentperformed with this model.2. Description of the model. Our aim is to describe the evolution in time ofthe density (number of cells per unit volume) of few cellular species. The densityof healthy tissue at time t in the location x will be denoted by S(t,x), the densityof extra-cellular medium or interstitial liquid by L(t,x). Cancer cells are describedby their densities in each of the two proliferation phases P1(t, a,x), P2(t, a,x) andquiescent phase Q(t,x) respectively. The proliferation phases are age-structured asshown in Fig. 1. We refer the reader to [18] for a detailed presentation.We assume that the total number of species is constant per unit volume, i.e.S + L +∫a(P1 + P2) da + Q = N0. (1)In the sequel, we will take N0 = 1.A VISCOELASTIC MODEL FOR TUMOR GROWTH 3We denote by C(t,x) the concentration of nutrients and H(t,x) the acidity (pro-portional to the concentration of protons). We also consider that the cellular divi-sion generates a movement described by a macroscopic velocity denoted by v(t,x).In this model, we assume that the movement is passive as it is only caused by theincrease of volume due to the mitosis. With this assumption, the various speciesare transported at the same velocity v. However, the species do not have the samemechanical behavior as the rheological properties depend on cellular densities asshown in Eq. (13) and (14).We make the following biological assumptions:• Healthy cells may die if the concentration of nutrient is too low or if the totalcell density is too high or if the acidity is high enough. We neglect division ofthese cells.• Cancer cells may die for the same reasons than healthy cells but they aremore resistant to harsh conditions. (Therefore the corresponding survivalthresholds will be less restrictive.)• Proliferating cells undergo division. If the environment is not favorable enough(in term of hypoxia, overpopulation and acidity) they become quiescent andstop dividing.• Dead cells are metabolized instantly and are accounted for in the extracellularphase.2.1. Equation for the populations of cells. In order to obtain the equationsgiving the evolution of cellular densities, we use the mass-balance principle for everyspecie. The equation for the healthy cells is∂tS + ∇ · (vS) = −αSfASS, (2)where fAS is the function giving the rate of apoptosis for healthy cells. It has theform of a boolean function (with a value of 1 if the environmental conditions arefavorable and 0 otherwise)fAS = 1{C<τ0,S or Σp>KS or H>H0,S}, (3)where Σp is the indicator for overpopulation and is computed asΣp = S +∫ amax,P10P1(a)da +∫ amax,P20P2(a)da + Q.We have denoted by τ0,S , KS and H0,S the thresholds for hypoxia, overpopulationand toxicity respectively.For the tumor cells in phase P1, the equation is:∂tP1 + ∂aP1 + ∇ · (vP1) = −αP fAP P1, (4)with a ranging from 0 to amax,P1 . The function fAP describing apoptosis has thesame expression as Eq. (3) with different (less restrictive) parameters ( τ0,P < τ0,S ,KP > KS and H0,P < H0,S). The equation for cells in phase P2 is similar:∂tP2 + ∂aP2 + ∇ · (vP2) = −αP fAP P2, (5)for a = 0 to amax,P2 . The boundary conditions accounting for the transition betweenphases in Fig. 1 are:P1(a = 0) = 2P2(amax,P2), (6)which describes the mitosis andP2(a = 0) = fQP1(amax,P1) +[ddtfQ]+Q(t−), (7)4 D. BRESCH, T. COLIN, E. GRENIER, B. RIBBA AND O. SAUTwhere fQ is the boolean function checking the environmental factors for a cell to goor come back from the quiescent state:fQ = 1 − 1{C<τQ or Σp>KQ or H<H0,Q}, (8)where again the thresholds are different of those of apoptosis: τ0,P < τ0,S < τQ,KP > KS > KQ and HQ < H0,P < H0,S . That means that if the environmentalconditions are not favorable, then the proliferating cells go into quiescent state andif they become even worse, cells undergo apoptosis.The equation for the quiescent cells is∂tQ + ∇ · (vQ) = (1 − f)P1(a = amax,P1) −[ddtfQ]+Q(t−) − αP fAP Q (9)with the same notation as above.We still have to deal with the extra-cellular phase L:∂tL + ∇ · (vL) = αSfASS +∫ amax,P10αP fAP P1+∫ amax,P20αP fAP P2 + αpfAQQ(10)where we have added every source terms coming from the different equations (2),(4), (5) and (9) in order to ensure conservation (1). This means from the biologicalpoint of view that every waste produced by the death of any kind of cells enter theextracellular medium.Collecting the equations and assumption (1) leads to∇ · v = P2(a = amax,P2). (11)2.2. Equation on the velocity. Eq. (11) is not sufficient to determine the ve-locity v. The medium is described as being a viscoelastic material: the relationbetween the stress, pressure and velocity is:∇ · σ −∇p = −ν∆v, (12)where ν denotes the viscosity of the surrounding liquid like for emulsions. Thephenomenological constitutive law for the stress (see [16] for instance) is∂tσ + v · ∇σ −∇vtσ − σ∇v +1τσ =β(τ)τD(v), (13)where D(v) = ∇v+(∇v)t2 and β(τ) is a function describing the rheological propertiesof the tissue. In this paper, we neglect the nonlinear terms v · ∇σ − (∇v)tσ − σ∇vas the velocities considered are small. The relaxation time τ is given byτ = (1 − L) +1 − L1 − S, (14)and the function β byβ(τ) = β0 + τβ∞. (15)Then, the limit τ → 0 leads to σ = β(0)D(v), which describes the behavior ofa Newtonian liquid (i.e. the extracellular medium L is liquid). When τ → ∞, weobtain ∂tσ = β∞D(v), which is the law for linear elasticity i.e. healthy tissue isconsidered as elastic. Cell-to-cell adhesion is weaker for cancer cells than for healthyones and therefore we consider their behavior as being viscoelastic. In this work, wehave only considered a very limited aspect of cellular adhesion (passive adhesion) asa complete description is out of reach. For a more complete description one can see[19, 14]. In Eq. (14), the expression for τ is rather simple but could be complexifiedA VISCOELASTIC MODEL FOR TUMOR GROWTH 5once more biological insight becomes available on the mechanical behavior of cells.Of course, we have no microscopic justification of such a law. The goal is to have aqualitatively correct behavior of the properties described above.2.3. Nutrients and acidity. The oxygen (or nutrient) is consumed by cancer andhealthy cells. As proliferating cells consume much more oxygen, their consumptionrate is much higher than the one of quiescent or healthy cells. Let us denote by Othe domain corresponding to oxygen sources (for instance blood vessels). In thisdomain the concentration of oxygen is fixed. This leads to the equation−∇ · (D∇C) = −αC(∫ amax,P10P1(a)da +∫ amax,P20P2(a)da)C−αSS − αQQ − ΓCC on Ω/O,C = C0 on ∂Ω \ O,C = Cmax on O,(16)where we have made the stationary assumption because the diffusion time-scale ofoxygen is much lower that the time-scale of cellular division.The acidity is produced by the cancer cells and is transported by the bloodvessels:{−∇ · (D∇H) = αPH(∫ amax,P10 P1(a)da +∫ amax,P20 P2(a)da)+ αQHQ − αBH1OH,H = H0 on ∂Ω.(17)For this matter, we have adapted the model from [11].3. Numerical results. In this section, we present some numerical results of ourmodel.To discretize the mass-balance equations (4), (5), (9), (10), we have used a com-bination of a WENO [15] and an exponential scheme as described in [5]. For Eq.(12), (13) and (11), we adapt the scheme described in [7]. The viscosity ν in Eq.(12) is taken equal to 0.1 [7].Note that it is impossible to have precise values for the functions and parameters.Most of the quantities that we consider are taken in order to obtain behavior thatare concordant to biological and medical values. A way to recover the informationwould be to use image-driven simulation in the spirit of [13].3.1. Heterogeneity of the medium. In this numerical experiment, the focus ison the effect of mechanical stress on the shape of the tumor. The initial tumoris a sphere, the oxygen distribution is isotropic. Hence, with a constant stress inabsence of instabilities, the spheroid should keep its spherical form through thecomputations. In our case, the healthy tissue is softer in a strip around the y axis.This is achieved by changing the density of healthy tissue and liquid according toS = 0.99(1− N) ×{12 if |y −L2 | < 3.5δy1 otherwise,N being the total density of cancer cells and L = 1 − N − S.The oxygen source is represented by the domain O ={|x| > L2 − 4δx}, on thisdomain, we set C = 0.1.For comparison purposes, we have also plotted the contour of the tumoral den-sity 0.3 as obtained from the same experimental conditions with a Newtonian model6 D. BRESCH, T. COLIN, E. GRENIER, B. RIBBA AND O. SAUTbased on [5] where the viscosity ν involved in the Stokes equation is given by the ex-pression ν = 1+S−L (with a simpler expression for the viscosity, the heterogeneityof the medium is not sensible).Figure 2. Evolution of a spheroid when the medium is mechan-ically heterogeneous. The contours corresponding to a density of0.3 are shown in black (Newtonian model) and red (viscoelasticmodel).With the viscoelastic model, a lemon-like shape is observed, whereas with aNewtonian model an ovoid is obtained. A detailed comparison (not presented here)shows that the main difference is indeed observed on the outer rim of the tumor.Even limited, this difference can eventually lead to bigger differences as the outerrim is composed of proliferating cells with exponential growth.With an homogeneous medium (or a simpler model as [18]), the tumor wouldhave stayed disc-shaped. The kind of shape obtained in this run is observed inin-vitro experiments. Detailed comparison will be discussed in forthcoming work.3.2. Influence of the stiffness of the external medium. In this experiment,we consider the influence of the stiffness of the healthy tissue on the tumoral growth.The initial tumor is a spheroid. We plot the final tumoral density for two values(β∞ = 1 and β∞ = 50) of the parameter β∞ in Eq. (15) on Fig. 3.The experimental setup (oxygen sources,. . . ) is the same as in the previousexperiment.For a lower stiffness the tumor is bigger as its growth is less restricted by thesurrounding tissue. Let us also denote the necrotic core in the central part of thetumor, where nutrients are no longer available.4. Conclusion. In this paper, we have presented a mathematical model for avas-cular tumor growth. The movement of the species obeys to a force balance equationof viscoelastic type. The model accounts, the same time, for microscopic (cell-cycle)and macroscopic (limited cellular-adhesion, mechanical effects) aspects of the tu-moral growth.This model was initially described to study avascular growth. Yet, throughthe domain O appearing in Eq. (16) and (17), a coupling could be made with amodel describing the evolution of the density of blood vessels during the process ofangiogenesis.A VISCOELASTIC MODEL FOR TUMOR GROWTH 7Figure 3. Evolution of a spheroid for two different stiffness of thehealthy tissue (through different values of the parameter β∞ in Eq.(15)).Acknowledgements. The authors would like to thank Davide Ambrosi for hishelpful remarks and comments.REFERENCES[1] T. Alarcón, H. M. Byrne, and P. K. Maini. A cellular automaton model for tumour growthin inhomogeneous environment. J. Theoret. Biol., 225(2):257–274, 2003.[2] D. Ambrosi and L. Preziosi. On the closure of mass balance models for tumor growth. Math.Models Methods Appl. Sci., 12(5):737–754, 2002.[3] A.R.A. Anderson. A hybrid mathematical model of solid tumour invasion: the importance ofcell adhesion. Mathematical Medicine and Biology, 22(2):163–186, 2005.[4] R.C. Bast, D.W. Kufe, R.E. Pollock, R.R. Weichselbaum, and J.F. Holland. Cancer Medicine.BC Decker Inc, Canada, 2000.[5] Didier Bresch, Thierry Colin, Emmanuel Grenier, Benjamin Ribba, and Olivier Saut. Com-putational modeling of solid tumor growth: the avascular stage. submitted.[6] H Byrne and M Chaplain. Modelling the role of cell-cell adhesion in the growth and develop-ment of carcinomas. Mathematical and Computer Modelling, Jan 1996.[7] Laurent Chupin. Contribution à l’étude des mélanges de fluides visco-élastiques. PhD thesis,Université Bordeaux 1, 2003.[8] D Drasdo and S Hohme. Individual-based approaches to birth and death in avascu1ar tumors.Mathematical and Computer Modelling, Jan 2003.[9] Avner Friedman. A hierarchy of cancer models and their mathematical challenges. DiscreteContin. Dyn. Syst. Ser. B, 4(1):147–159, 2004. Mathematical models in cancer (Nashville,TN, 2002).[10] J Galle, M Loeffler, and D Drasdo. Modeling the effect of deregulated proliferation andapoptosis on the growth dynamics of epithelial . . . . Biophysical Journal, Jan 2005.[11] R.A. Gatenby. A reaction-diffusion model of cancer invasion. Cancer Research, 56(24):5745–5753, 1996.[12] R.A. Gatenby and R.J. Gillies. Why do cancers have high aerobic glycolysis? Nature ReviewsCancer, 4(11):891–899, 2004.[13] C. Hogea, C. Davatzikos, and G. Biros. An image-driven parameter estimation problem for areaction–diffusion glioma growth model with mass effects. Journal of Mathematical Biology,2007. In press.[14] A Iordan, A Duperray, and C Verdier. A fractal approach to the rheology of concentrated cellsuspensions. Phys. Rev. E, Jan 2008.[15] Guang-Shan Jiang and Danping Peng. Weighted ENO schemes for Hamilton-Jacobi equations.SIAM J. Sci. Comput., 21(6):2126–2143 (electronic), 2000.[16] D.D. Joseph. Fluid dynamics of viscoelastic liquids. Springer, 1990.[17] Y. Mansury, M. Kimura, J. Lobo, and T.S. Deisboeck. Emerging Patterns in Tumor Systems:Simulating the Dynamics of Multicellular Clusters with an Agent-based Spatial AgglomerationModel. Journal of Theoretical Biology, 219(3):343–370, 2002.8 D. BRESCH, T. COLIN, E. GRENIER, B. RIBBA AND O. SAUT[18] B. Ribba, O. Saut, T. Colin, D. Bresch, E. Grenier, and J.P. Boissel. A multiscale mathemat-ical model of avascular tumor growth to investigate the therapeutic benefit of anti-invasiveagents. J. Theoret. Biol., 243(4):532–541, 2006.[19] P Saramito. A new constitutive equation for elastoviscoplastic fluid flows. Journal of Non-Newtonian Fluid Mechanics, 145(1):1–14, 2007.[20] J Sherratt and M Chaplain. A new mathematical model for avascular tumour growth. Journalof Mathematical Biology, Jan 2001.E-mail address: Didier.Bresch@univ-savoie.frE-mail address: Thierry.Colin@math.u-bordeaux1.frE-mail address: Emmanuel.Grenier@umpa.ens-lyon.frE-mail address: Benjamin.Ribba@recherche.univ-lyon1.frE-mail address: Olivier.Saut@math.u-bordeaux1.fr

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A viscoelastic model for avascular tumor growthDidier Bresch, Thierry Colin, Emmanuel Grenier, Benjamin Ribba, OlivierSautTo cite this version:Didier Bresch, Thierry Colin, Emmanuel Grenier, Benjamin Ribba, Olivier Saut. A viscoelasticmodel for avascular tumor growth. [Research Report] 2009. <inria-00267292v4>HAL Id: inria-00267292https://hal.inria.fr/inria-00267292v4Submitted on 10 Jul 2009HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.Manuscript submitted to Website: http://AIMsciences.orgAIMS’ JournalsVolume X, Number 0X, XX 200X pp. X–XXA VISCOELASTIC MODEL FOR AVASCULAR TUMORGROWTHDidier BreschLama, Universite´ de Savoie, CNRS UMR 5127,Campus scientifique, 73376 Le Bourget du lac cedex, FranceThierry ColinUniversite´ Bordeaux 1, Institut de Mathe´matiques,INRIA Sud-Ouest EPI MC2,351 cours de la libe´ration, 33405 Talence Cedex, FranceEmmanuel GrenierUnite´ de Mathe´matiques Pures et Applique´es, CNRS UMR 5669,Ecole Normale Supe´rieure de Lyon, 69364 Lyon cedex, FranceBenjamin RibbaUniversite´ de Lyon, Lyon, F-69003, France;Universite´ Lyon 1, Ciblage The´rapeutique en Oncologie (EA3738),Faculte´ de Me´decine Lyon-Sud, Oullins, F-69921, FranceOlivier SautUniversite´ Bordeaux 1, Institut de Mathe´matiques,CNRS UMR 5251 and INRIA Sud-Ouest EPI MC2,351 cours de la libe´ration, 33405 Talence Cedex, FranceAbstract. In this article, we present a new continuous model for tumorgrowth. This model describes the evolution of three components: sane tis-sue, cancer cells and extracellular medium. In order to render correctly thecellular division, this model uses a discrete description of the cell cycle (theset of steps a cell has to undergo in order to divide). To account for cellularadhesion and the mechanics which may influence the growth, we assume a vis-coelastic mechanical behavior. This model extends the one presented in [18]with a more realistic description of the forces that drive the movement.1. Introduction. In a sketchy way, a tumor arises after several mutations of cellsthat have made them less sensible to anti-growth factors or lack of nutrients, forinstance. This may lead to uncontrolled division of these cells.In order to divide, a cell needs nutrients (such as oxygen), which is obtainedfrom its close environment in the avascular phase. As the tumor grows, some cellsdo not get any more enough nutrient and turn to a quiescent state where they nolonger divide waiting for the environment to become favorable again. Therefore,for a realistic description of cancer growth, one has to describe the evolution of theconcentration of nutrients. Tumor cells have also the ability to produce their energyfrom glucose, whose production lowers the pH of the medium [12]. This toxicityKey words and phrases. Avascular tumor growth. Multiscale models. Cell cycle modeling.Fluid dynamics.12 D. BRESCH, T. COLIN, E. GRENIER, B. RIBBA AND O. SAUTFigure 1. Scheme of the simplified cell-cycle used in this paperand presented in [18].may harm cancer or healthy cells and favor tumoral invasion. Finally, it is knownthat cellular adhesion has an influence on tumor growth [4, 3].There are many mathematical models describing solid tumor growth. Roughlythey can be divided in two classes. In discrete models, one describes the evolution ofcells individually. In this class, we find for instance cellular automata or agent-basedmodels [1, 17]. These approaches are very efficient to describe the cell-scale aspectsof tumor growth, they are computationally expensive, which makes them difficultto use for a large number of cells. Furthermore, it is also difficult to account for themechanical effects (cell-to-cell mechanical interaction) influencing tumor growth.Yet a precise description at the cellular level is possible [8, 10].Continuous models compute the evolution of cellular densities or of the boundaryof the tumor (when strong geometrical assumptions apply) [9]. They are much lessexpensive to discretize but, as they deal with averages over a large number of cells,they can not always account correctly for the microscopic or genetic aspects oftumor growth. Most of these models are based on partial differential equations,such as reaction-diffusion equations [11] or mass-balance equations [2]. Some ofthese models are accounting for complex phenomena like contact inhibition [20] orcellular adhesion [6]. In this paper, we present a model of this kind, where massflow is obtained through a viscoelastic description of the tumor mass and describingthe mechanical behavior of the healthy tissues, the interstitial liquid and the cancercells in a compact way. The outline of this paper is the following one. In Sec. 2,we present the mathematical model and in Sec. 3, we show a numerical experimentperformed with this model.2. Description of the model. Our aim is to describe the evolution in time ofthe density (number of cells per unit volume) of few cellular species. The densityof healthy tissue at time t in the location x will be denoted by S(t,x), the densityof extra-cellular medium or interstitial liquid by L(t,x). Cancer cells are describedby their densities in each of the two proliferation phases P1(t, a,x), P2(t, a,x) andquiescent phase Q(t,x) respectively. The proliferation phases are age-structured asshown in Fig. 1. We refer the reader to [18] for a detailed presentation.We assume that the total number of species is constant per unit volume, i.e.S + L+∫a(P1 + P2) da+Q = N0. (1)In the sequel, we will take N0 = 1.A VISCOELASTIC MODEL FOR TUMOR GROWTH 3We denote by C(t,x) the concentration of nutrients and H(t,x) the acidity (pro-portional to the concentration of protons). We also consider that the cellular divi-sion generates a movement described by a macroscopic velocity denoted by v(t,x).In this model, we assume that the movement is passive as it is only caused by theincrease of volume due to the mitosis. With this assumption, the various speciesare transported at the same velocity v. However, the species do not have the samemechanical behavior as the rheological properties depend on cellular densities asshown in Eq. (13) and (14).We make the following biological assumptions:• Healthy cells may die if the concentration of nutrient is too low or if the totalcell density is too high or if the acidity is high enough. We neglect division ofthese cells.• Cancer cells may die for the same reasons than healthy cells but they aremore resistant to harsh conditions. (Therefore the corresponding survivalthresholds will be less restrictive.)• Proliferating cells undergo division. If the environment is not favorable enough(in term of hypoxia, overpopulation and acidity) they become quiescent andstop dividing.• Dead cells are metabolized instantly and are accounted for in the extracellularphase.2.1. Equation for the populations of cells. In order to obtain the equationsgiving the evolution of cellular densities, we use the mass-balance principle for everyspecie. The equation for the healthy cells is∂tS +∇ · (vS) = −αSfASS, (2)where fAS is the function giving the rate of apoptosis for healthy cells. It has theform of a boolean function (with a value of 1 if the environmental conditions arefavorable and 0 otherwise)fAS = 1{C<τ0,S or Σp>KS or H>H0,S}, (3)where Σp is the indicator for overpopulation and is computed asΣp = S +∫ amax,P10P1(a)da+∫ amax,P20P2(a)da+Q.We have denoted by τ0,S , KS and H0,S the thresholds for hypoxia, overpopulationand toxicity respectively.For the tumor cells in phase P1, the equation is:∂tP1 + ∂aP1 +∇ · (vP1) = −αP fAPP1, (4)with a ranging from 0 to amax,P1 . The function fAP describing apoptosis has thesame expression as Eq. (3) with different (less restrictive) parameters ( τ0,P < τ0,S ,KP > KS and H0,P < H0,S). The equation for cells in phase P2 is similar:∂tP2 + ∂aP2 +∇ · (vP2) = −αP fAPP2, (5)for a = 0 to amax,P2 . The boundary conditions accounting for the transition betweenphases in Fig. 1 are:P1(a = 0) = 2P2(amax,P2), (6)which describes the mitosis andP2(a = 0) = fQP1(amax,P1) +[ddtfQ]+Q(t−), (7)4 D. BRESCH, T. COLIN, E. GRENIER, B. RIBBA AND O. SAUTwhere fQ is the boolean function checking the environmental factors for a cell to goor come back from the quiescent state:fQ = 1− 1{C<τQ or Σp>KQ or H<H0,Q}, (8)where again the thresholds are different of those of apoptosis: τ0,P < τ0,S < τQ,KP > KS > KQ and HQ < H0,P < H0,S . That means that if the environmentalconditions are not favorable, then the proliferating cells go into quiescent state andif they become even worse, cells undergo apoptosis.The equation for the quiescent cells is∂tQ+∇ · (vQ) = (1− f)P1(a = amax,P1)−[ddtfQ]+Q(t−)− αP fAPQ (9)with the same notation as above.We still have to deal with the extra-cellular phase L:∂tL+∇ · (vL) = αSfASS +∫ amax,P10αP fAPP1+∫ amax,P20αP fAPP2 + αpfAQQ(10)where we have added every source terms coming from the different equations (2),(4), (5) and (9) in order to ensure conservation (1). This means from the biologicalpoint of view that every waste produced by the death of any kind of cells enter theextracellular medium.Collecting the equations and assumption (1) leads to∇ · v = P2(a = amax,P2). (11)2.2. Equation on the velocity. Eq. (11) is not sufficient to determine the ve-locity v. The medium is described as being a viscoelastic material: the relationbetween the stress, pressure and velocity is:∇ · σ −∇p = −ν∆v, (12)where ν denotes the viscosity of the surrounding liquid like for emulsions. Thephenomenological constitutive law for the stress (see [16] for instance) is∂tσ + v · ∇σ −∇vtσ − σ∇v +1τσ =β(τ)τD(v), (13)where D(v) = ∇v+(∇v)t2 and β(τ) is a function describing the rheological propertiesof the tissue. In this paper, we neglect the nonlinear terms v · ∇σ− (∇v)tσ− σ∇vas the velocities considered are small. The relaxation time τ is given byτ = (1− L) +1− L1 − S, (14)and the function β byβ(τ) = β0 + τβ∞. (15)Then, the limit τ → 0 leads to σ = β(0)D(v), which describes the behavior ofa Newtonian liquid (i.e. the extracellular medium L is liquid). When τ → ∞, weobtain ∂tσ = β∞D(v), which is the law for linear elasticity i.e. healthy tissue isconsidered as elastic. Cell-to-cell adhesion is weaker for cancer cells than for healthyones and therefore we consider their behavior as being viscoelastic. In this work, wehave only considered a very limited aspect of cellular adhesion (passive adhesion) asa complete description is out of reach. For a more complete description one can see[19, 14]. In Eq. (14), the expression for τ is rather simple but could be complexifiedA VISCOELASTIC MODEL FOR TUMOR GROWTH 5once more biological insight becomes available on the mechanical behavior of cells.Of course, we have no microscopic justification of such a law. The goal is to have aqualitatively correct behavior of the properties described above.2.3. Nutrients and acidity. The oxygen (or nutrient) is consumed by cancer andhealthy cells. As proliferating cells consume much more oxygen, their consumptionrate is much higher than the one of quiescent or healthy cells. Let us denote by Othe domain corresponding to oxygen sources (for instance blood vessels). In thisdomain the concentration of oxygen is fixed. This leads to the equation−∇ · (D∇C) = −αC(∫ amax,P10P1(a)da+∫ amax,P20P2(a)da)C−αSS − αQQ− ΓCC on Ω/O,C = C0 on ∂Ω \O,C = Cmax on O,(16)where we have made the stationary assumption because the diffusion time-scale ofoxygen is much lower that the time-scale of cellular division.The acidity is produced by the cancer cells and is transported by the bloodvessels:{−∇ · (D∇H) = αPH(∫ amax,P10 P1(a)da+∫ amax,P20 P2(a)da)+ αQHQ− αBH1OH,H = H0 on ∂Ω.(17)For this matter, we have adapted the model from [11].3. Numerical results. In this section, we present some numerical results of ourmodel.To discretize the mass-balance equations (4), (5), (9), (10), we have used a com-bination of a WENO [15] and an exponential scheme as described in [5]. For Eq.(12), (13) and (11), we adapt the scheme described in [7]. The viscosity ν in Eq.(12) is taken equal to 0.1 [7].Note that it is impossible to have precise values for the functions and parameters.Most of the quantities that we consider are taken in order to obtain behavior thatare concordant to biological and medical values. A way to recover the informationwould be to use image-driven simulation in the spirit of [13].3.1. Heterogeneity of the medium. In this numerical experiment, the focus ison the effect of mechanical stress on the shape of the tumor. The initial tumoris a sphere, the oxygen distribution is isotropic. Hence, with a constant stress inabsence of instabilities, the spheroid should keep its spherical form through thecomputations. In our case, the healthy tissue is softer in a strip around the y axis.This is achieved by changing the density of healthy tissue and liquid according toS = 0.99(1−N)×{12 if |y −L2 | < 3.5δy1 otherwise,N being the total density of cancer cells and L = 1−N − S.The oxygen source is represented by the domain O ={|x| > L2 − 4δx}, on thisdomain, we set C = 0.1.For comparison purposes, we have also plotted the contour of the tumoral den-sity 0.3 as obtained from the same experimental conditions with a Newtonian model6 D. BRESCH, T. COLIN, E. GRENIER, B. RIBBA AND O. SAUTbased on [5] where the viscosity ν involved in the Stokes equation is given by the ex-pression ν = 1+S−L (with a simpler expression for the viscosity, the heterogeneityof the medium is not sensible).Figure 2. Evolution of a spheroid when the medium is mechan-ically heterogeneous. The contours corresponding to a density of0.3 are shown in black (Newtonian model) and red (viscoelasticmodel).With the viscoelastic model, a lemon-like shape is observed, whereas with aNewtonian model an ovoid is obtained. A detailed comparison (not presented here)shows that the main difference is indeed observed on the outer rim of the tumor.Even limited, this difference can eventually lead to bigger differences as the outerrim is composed of proliferating cells with exponential growth.With an homogeneous medium (or a simpler model as [18]), the tumor wouldhave stayed disc-shaped. The kind of shape obtained in this run is observed inin-vitro experiments. Detailed comparison will be discussed in forthcoming work.3.2. Influence of the stiffness of the external medium. In this experiment,we consider the influence of the stiffness of the healthy tissue on the tumoral growth.The initial tumor is a spheroid. We plot the final tumoral density for two values(β∞ = 1 and β∞ = 50) of the parameter β∞ in Eq. (15) on Fig. 3.The experimental setup (oxygen sources,. . . ) is the same as in the previousexperiment.For a lower stiffness the tumor is bigger as its growth is less restricted by thesurrounding tissue. Let us also denote the necrotic core in the central part of thetumor, where nutrients are no longer available.4. Conclusion. In this paper, we have presented a mathematical model for avas-cular tumor growth. The movement of the species obeys to a force balance equationof viscoelastic type. The model accounts, the same time, for microscopic (cell-cycle)and macroscopic (limited cellular-adhesion, mechanical effects) aspects of the tu-moral growth.This model was initially described to study avascular growth. Yet, throughthe domain O appearing in Eq. (16) and (17), a coupling could be made with amodel describing the evolution of the density of blood vessels during the process ofangiogenesis.A VISCOELASTIC MODEL FOR TUMOR GROWTH 7Figure 3. Evolution of a spheroid for two different stiffness of thehealthy tissue (through different values of the parameter β∞ in Eq.(15)).Acknowledgements. The authors would like to thank Davide Ambrosi for hishelpful remarks and comments.REFERENCES[1] T. Alarco´n, H. M. Byrne, and P. K. Maini. A cellular automaton model for tumour growthin inhomogeneous environment. J. Theoret. Biol., 225(2):257–274, 2003.[2] D. Ambrosi and L. Preziosi. On the closure of mass balance models for tumor growth. Math.Models Methods Appl. Sci., 12(5):737–754, 2002.[3] A.R.A. Anderson. A hybrid mathematical model of solid tumour invasion: the importance ofcell adhesion. Mathematical Medicine and Biology, 22(2):163–186, 2005.[4] R.C. Bast, D.W. Kufe, R.E. Pollock, R.R. Weichselbaum, and J.F. Holland. Cancer Medicine.BC Decker Inc, Canada, 2000.[5] Didier Bresch, Thierry Colin, Emmanuel Grenier, Benjamin Ribba, and Olivier Saut. Com-putational modeling of solid tumor growth: the avascular stage. submitted.[6] H Byrne and M Chaplain. Modelling the role of cell-cell adhesion in the growth and develop-ment of carcinomas. Mathematical and Computer Modelling, Jan 1996.[7] Laurent Chupin. Contribution a` l’e´tude des me´langes de fluides visco-e´lastiques. PhD thesis,Universite´ Bordeaux 1, 2003.[8] D Drasdo and S Hohme. Individual-based approaches to birth and death in avascu1ar tumors.Mathematical and Computer Modelling, Jan 2003.[9] Avner Friedman. A hierarchy of cancer models and their mathematical challenges. DiscreteContin. Dyn. Syst. Ser. B, 4(1):147–159, 2004. Mathematical models in cancer (Nashville,TN, 2002).[10] J Galle, M Loeﬄer, and D Drasdo. Modeling the effect of deregulated proliferation andapoptosis on the growth dynamics of epithelial . . . . 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Emerging Patterns in Tumor Systems:Simulating the Dynamics of Multicellular Clusters with an Agent-based Spatial AgglomerationModel. Journal of Theoretical Biology, 219(3):343–370, 2002.8 D. BRESCH, T. COLIN, E. GRENIER, B. RIBBA AND O. SAUT[18] B. Ribba, O. Saut, T. Colin, D. Bresch, E. Grenier, and J.P. Boissel. A multiscale mathemat-ical model of avascular tumor growth to investigate the therapeutic benefit of anti-invasiveagents. J. Theoret. Biol., 243(4):532–541, 2006.[19] P Saramito. A new constitutive equation for elastoviscoplastic fluid flows. Journal of Non-Newtonian Fluid Mechanics, 145(1):1–14, 2007.[20] J Sherratt and M Chaplain. A new mathematical model for avascular tumour growth. Journalof Mathematical Biology, Jan 2001.E-mail address: Didier.Bresch@univ-savoie.frE-mail address: Thierry.Colin@math.u-bordeaux1.frE-mail address: Emmanuel.Grenier@umpa.ens-lyon.frE-mail address: Benjamin.Ribba@recherche.univ-lyon1.frE-mail address: Olivier.Saut@math.u-bordeaux1.fr

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A viscoelastic model for avascular tumor growth.

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