computation

Article Deformable Cell Model of Tissue Growth

Nikolai Bessonov 1 and Vitaly Volpert 2,* 1 Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, 199178 Saint Petersburg, Russia; [email protected] 2 Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, 69622 Villeurbanne, France * Correspondence: [email protected]; Tel.: +3-347-243-2765 Received: 30 September 2017; Accepted: 24 October 2017; Published: 30 October 2017

Abstract: This paper is devoted to modelling tissue growth with a deformable cell model. Each cell represents a polygon with particles located at its vertices. Stretching, bending and pressure forces act on particles and determine their displacement. Pressure-dependent cell proliferation is considered. Various patterns of growing tissue are observed. An application of the model to tissue regeneration is illustrated. Approximate analytical models of tissue growth are developed.

Keywords: deformable cells; tissue growth; pattern formation; analytical approximation

1. Introduction Mathematical and computer modelling of tissue growth is used in various biological problems such as wound healing and regeneration, morphogenesis, tumor growth, etc. Tissue growth can be described with continuous models: partial differentiation equations for cell concentrations (reaction–diffusion equations), Navier–Stokes or Darcy equations for the velocity of the medium, and elasticity equations for the distribution of mechanical stresses. There is a vast literature devoted to these models (see, e.g., [1–5] and references therein). Another approach deals with individual-based models where cells are considered as individual objects. It allows a more detailed description at the level of individual cells though analytical investigation of such models becomes impossible and their numerical simulations are often more involved than for the continuous models. There are various lattice and off-lattice cell-based models which can be regrouped as spherical particle models, cellular automata and deformable cell models. Spherical particle models originate from molecular dynamics. Each cell is considered as a sphere which can interact with the surrounding cells. The force acting between them depends on the distance between their centers. Additional random and dissipative forces can be introduced, as is the case in dissipative particle dynamics [6]. The motion of particles is determined by Newton’s second law. Moreover, cells can divide, die and change their type. Different cell types can have different behavior from the point of view of their motion, division, and death. Already in the 1980s, a similar model, with the equation of motion replaced by some algorithm of cell displacement, was used to study embryogenesis [7]. The spherical particle method to model morphogenesis was developed in a recent work [8]. In [9], one-cell layered tissues were studied under the assumption that there were additional adhesion forces between cells. Hematopoiesis and leukemia development were investigated in [10–13]. Consecutive cell proliferation and differentiation allowed the description of normal and leukemic hematopoiesis in the bone marrow. Dynamics of cell populations and the conditions of leukemia development were studied. In more complex models, cells can be represented as two connected spherical particles [14]. In the cell-center models, cells are considered as polygons with their centers connected by springs, and cell shape determined by Voronoi tessellation. This approach was used for crypt modelling in [15,16] and brain cancer in [17,18]. Particle models can be combined with continuous models,

Computation 2017, 5, 45; doi:10.3390/computation5040045 www.mdpi.com/journal/computation Computation 2017, 5, 45 2 of 18 ordinary and partial differential equations for various intracellular and extracellular concentrations. This hybrid method was used for modelling of cancer [19,20], hematopoiesis and blood diseases [21–24] and blood clotting in flow [25,26]. In some cases, it is possible to justify transition from particle models to continuous models [27–29]. Another approach to tissue growth is based on cellular automata (see [5,30,31] and references therein). This method can be easier to implement and it is better adapted for the transition from discrete to continuous models. On the other hand, it imposes a square lattice geometry which can influence the results. This method is used to study tumor growth [32,33] and cell invasion [34,35], angiogenesis [36,37], tissue engineering [38], and development of avian gut tissue [39]. This approach was further developed in the cellular Potts model [30,40] (see also [41,42] and references therein). Deformable cell models take into account cell geometry and possible deformation due to mechanical forces. The cell membrane can be considered as an ensemble of particles with various forces acting on them (Section2). This approach is developed for modeling blood flows where blood cells are considered as individual objects (see [43] and references therein). It is close to the spherical particle models though the particles here do not correspond to individual cells. This method gives more detailed information about cell geometry, adhesion and deformation but it is more difficult to realize, especially in the case of cell division (not considered in the works on blood flows). The questions about the direction of cell division and the mechanical properties of the cell wall become of primary importance here. If they are not sufficiently well described, then the model will fail because of geometrical constraints or because of an accumulation of mechanical stresses. There are various realizations of the deformable cell model, including the immersed boundary method [44], the subcellular element method [45], and the level set method [46] (see [30] and references therein). The deformable cell model coupled with Navier–Stokes equations was used to study the development of epithelial acini [47,48] and ductal tumor [49]. The interaction of cell geometry and migration for the development of cancer was studied in [50]. A phase field model for collective cell migration was developed in [51]. Deformable cell models for plant growth were developed in [52,53]. Growing shoots and roots have a particular cell organization where dividing cells are located in outer layers of the meristem. Differentiated cells do not divide. The direction of cell division depends on cell location. A precise description of the location of dividing cells and of the direction of their division allowed us to reproduce a self-similar growing structure of the plant root [52]. Directions of cell division and visco-elasto-plastic properties of cell walls are important in modelling root growth with a deformable cell model. In this work, we continue to develop the deformable cell model of tissue growth. The main assumptions of the model are similar to those in [52] except for the location of dividing cells and the direction of their division. Instead of a particular cell structure and division specific for root and shoot meristem, we consider uniform tissue growth where dividing cells can be located everywhere. The direction of their division is determined by a special algorithm described in the next section. We will describe the deformable cell model in Section2. Some properties of growing tissues and applications are discussed in Section3. Section4 is devoted to an approximate analytical model of tissue growth.

2. Deformable Cell Model

2.1. Forces We begin the description of the model with mechanical forces acting in cellular structures. An individual cell at its mechanical equilibrium (without deformation) represents a regular polygon with n vertices. Initially, it has an area S0, the length of the sides l0 and the angle between any neighboring sides α0. These parameters can be different for different cells. We place a particle of mass m at each vertex of the polygon. Computation 2017, 5, 45 3 of 18

If the polygon changes its shape, then there are three forces which tend to return it to its original configuration. They depend on the change of side length, angles and volume. Similar to other deformable cell models, we define them as follows. If the length l of a side changes, then there is a force acting on each of the two vertices located at this side:

  l  dl  −→f 1 = k1 1 + µ1 −→τ , −→f 2 = −→f 1 (1) l0 − dt −

(Figure1b). Here, −→τ is the unit vector along the side. If the angle α between two sides changes (Figure1c), then there is momentum

 α  dα M = k2 1 µ2 (2) − α0 − dt which creates a pair of forces applied to the vertices and acting on each side in the direction perpendicular to it, α0 = 180◦. Finally, if the cell area S changes, then there is a force (pressure)

 S  dS P = k3 1 µ3 (3) − S0 − dt applied to each side and acting in the direction perpendicular to it (Figure1d). Let us note that the first terms in (1)–(3) describe elastic force and the second term viscous dissipation. They correspond to visco-elastic properties of biological cells. The problem is considered in dimensionless variables. Coefficients ki are material parameters which determine the resistance to stretching, bending and compression. Parameters µi represent artificial viscosity introduced in order to avoid numerical instability.

Figure 1. Schematic representation of the model: a structure which consists of four deformable cells (a); stretching force acts between two neighboring vertices along the line connecting them (b); bending force acts on neighboring sides as the angle between them differs from the angle α0 (c); pressure acts on all sides in the direction perpendicular to them (d). Computation 2017, 5, 45 4 of 18

The total force −→F i acting on the i-th vertex is a sum of these three forces. The position −→r i of this vertex is determined by the equations

d r d v −→i = v , m −→i = µ v + −→F , (4) dt −→i i dt − −→i i where −→v i is the vertex velocity, µ is the coefficient which determines viscous friction from the surrounding medium. For further description of the model, we denote by l0 the side length at mechanical equilibrium and by l its length in the process of cell growth and deformation. Due to plastic deformations, l0 can i change. So, we will distinguish the initial value l0 which corresponds to the moment of cell appearance, i and the current value l0, which can be different from l0. We introduce two additional features in the model:

i 1. If the length of a side becomes twice as large as its initial length l0, then an additional vertex is introduced in the middle of the side. At the moment of side division, the stretching force is preserved. Hence the number of vertices depends on cell deformation. This allows us to better describe cell shape in the case of large deformations. 2. If the length l of a side becomes sufficiently large, that is li /l γ, with some γ, 0 < γ < 1, 0 ≤ then the initial length l0 increases irreversibly in such a way that it satisfies the equality l0 = γl. In this case, if the stretching force is removed, then the spring will not return to its initial length but to some greater length. This corresponds to the irreversible deformation of the cell wall when it grows. Here, γ is a material parameter characterizing plastic deformation observed for biological tissues. In the simulations presented below, we set γ = 0.8.

2.2. Cell Growth and Division

2.2.1. Cell Growth

The cell area S0 (without deformation) of growing cells changes in time with the rate:

dS 0 = σ(t), dt where σ(t) is some given function. We set σ = 0, if a cell does not grow and σ(t) = const in the case of linear growth. Other growth rates can also be considered [52]. When S0 increases two-fold, the cell divides. The area of the daughter cell (without deformation) returns to the two-fold lower value. When studying root growth, we also introduced cell growth without division [52].

2.2.2. Cell Division One of the important characteristics of growing tissue is the directions of cell division. If these directions are not coordinated, then cells can become strongly deformed. This contradicts biological observations and can result in failure of the numerical method. We will choose the direction of cell division according to the algorithm explained below. This algorithm is specific for unpolarized cells [47]. Let us recall that cells are represented as polygons. They grow, increasing their volume, and divide into two cells with approximately equal areas. Cell division occurs along a section connected to vertices (diagonal) of the polygon. Only such sections are considered. For each cell, the algorithm of choice of the section consists of two steps. In the first step, we choose all sections which divide the polygon into two parts with approximately equal area. Since these two areas are determined by the vertices of the polygon, in general they are not exactly equal to each other. Therefore, we introduce a maximal possible difference e. Hence we choose all sections such that the areas S1 and S2 of two resulting post-division polygons satisfy the condition S S < e. Here, S + S = S . In the second step of | 1 − 2| 1 2 0 Computation 2016, 5, x 5 of 17

the algorithm, we choose the shortest section among all of those chosen in the first step. Let us note that this algorithm depends on the location of cell nodes. ComputationThus,2017, we5, 45 can summarize this algorithm as a choice of the shortest section by5 of dividing 18 the polygon into two subpolygons with approximately equal areas. Due to this algorithm of cell division, thecells algorithm, cannot be we strongly choose the deformed shortest section (elongated). amongWe all of will those see chosen that it in provides the first step. stable Let tissue us note growth in a thatwide this range algorithm of parameters. depends on Other the location approaches of cell tonodes. the direction of cell division were used in [52] in order Thus, we can summarize this algorithm as a choice of the shortest section by dividing the to model plantComputation root2016 growth, 5, x and other polarized cells in [47]. 5 of 17 polygon into two subpolygons with approximately equal areas. Due to this algorithm of cell division, cells3. Numerical cannot be strongly Modelling deformed of Tissue (elongated). Growth We will see that it provides stable tissue growth in a wide range ofthe parameters. algorithm, we Other choose approaches the shortest to section the direction among ofallcell of those division chosen were in the used first in step. [52] Let in us order note that this algorithm depends on the location of cell nodes. to3.1. model All Cellsplant Grow root growth and Divide and other polarized cells in [47]. Thus, we can summarize this algorithm as a choice of the shortest section by dividing the polygon into two subpolygons with approximately equal areas. Due to this algorithm of cell division, 3. NumericalModelling Modelling of tissue of Tissue growth Growth with a deformable cell model implies certain restrictions on the location ofcells dividing cannot be cells strongly and deformed on the direction (elongated). of We their will seedivision. that it provides If these stable condition tissue growth are not in a satisfied, wide range of parameters. Other approaches to the direction of cell division were used in [52] in order 3.1. All Cells Grow and Divide then it willto lead model to plant a rapid root accumulation growth and other of polarized mechanical cells in stresses [47]. and to failure of the cellular structure. OneModelling possible cell of tissue organization growth with was a suggested deformable in cell [52] model for the implies description certain of restrictions root growth. onAccording the to 3. Numerical Modelling of Tissue Growth locationthe biological of dividing observations, cells and on dividing the direction cells of are their located division. inthis If these case condition close to are the not outer satisfied, surface of the thengrowing it will root.lead3.1. to All Directions a Cells rapid Grow accumulation and of cellDivide division of mechanical are also stresses precisely and determined. to failure of the In cellular this work, structure. we consider a Oneuniform possible tissue cellModelling organization growth where of tissue was all suggested growth cells of with the in a [52 deformabletissue] for thecan description cell divide. model implies of root certain growth. restrictionsAccording on tothe the biological observations, dividing cells are located in this case close to the outer surface of the Let uslocation recall of that dividing each cells cell and is on characterized the direction of their by itsdivision. equilibrium If these condition area S are0 and not satisfied, current area S. growingThe equilibrium root.then Directions it willarea lead grows of to cell a rapid division here accumulation as are a linear also of precisely mechanical function determined. stresses of time. and An to In failure this example work, of the cellularwe of aconsider growing structure. a tissue is uniform tissueOne growth possible where cell organization all cells of was the suggestedtissue can in divide. [52] for the description of root growth. According to shown in Figure2. Each cell grows and divides. The direction of division is determined by the Let us recallthe biological that each observations, cell is characterized dividing cells are by located its equilibrium in this case area closeS to theand outer current surface area of theS. shortest diagonal algorithm described above. Initially, there are 20 nodes0 at the surface of the cell in The equilibriumgrowingarea root. grows Directions here as of a cell linear division function are also of precisely time. An determined. example In of this a growing work, we considertissue is a shownthis simulation. in Figureuniform2. Other Each tissue cell growth nodes grows where can and be all cellsadded divides. of the if Thetissue the distance direction can divide.between of division two is determined neighboring by nodes the exceeds Let us recall that each cell is characterized by its equilibrium area S and current area S. shortestsome given diagonal value. algorithm We will described use the above. values Initially, of parameters there are 20k1 nodes= 40, atk2 the= surface0.1,0 k3 of= the100 cell, dS in0/dt = 0.1, The equilibrium area grows here as a linear function of time. An example of a growing tissue is µ = 30, = 0.04 (time step), (0) = 20, µ = 1, µ = 0.01, µ = 1, γ = 0.8, unless other values this simulation.dt shown Other in Figure nodes2. can Each beS cell0 added grows if and the divides.distance1 The between2 direction two of3 neighboring division is determined nodes exceeds by the someare specified. given value.shortest We diagonal will use algorithm the values described of parameters above. Initially,k1 = there40, arek2 20= nodes0.1, k3 at= the100 surface, dS0/ ofdt the= cell0.1 in, µ = 30, dt =this0.04 simulation.(time step), OtherS0 nodes(0) = can20, beµ1 added= 1, ifµ2 the= distance0.01, µ3 between= 1, γ two= 0.8 neighboring, unless other nodes values exceeds are specified.some given value. We will use the values of parameters k1 = 40, k2 = 0.1, k3 = 100, dS0/dt = 0.1, µ = 30, dt = 0.04 (time step), S0(0) = 20, µ1 = 1, µ2 = 0.01, µ3 = 1, γ = 0.8, unless other values are specified.

(a)(b) Figure 2. Example of simulations where all cells grow and divide; S0 is a linear function of time; Figure 2. Exampledirection of of cell simulations division is chosen where along all the cells shortest grow diagonal. and divide; First divisionsS0 is are a shownlinear on function the left, of time; Figure 2. Example of simulations where all cells grow and divide; S0 is a linear function of time; and a snapshot of the growing cell structure is shown on the right. directiondirection of cellof cell division division is chosen is chosen along along the shortest the shortest diagonal. diagonal. First divisions First divisionsare shown arein (a shown); and a on the left, snapshotand a snapshot of the growing of the cell growing structure cell is structure shown in (isb). shown on the right. ComputationComputation20162016, 5, x5, x 6 of6 of 17 17

TheThe first first division division occurs occurs along along a diametera diameter of of the the circular circular cell. cell. The The second second division division takes takes place place alongalong the the perpendicular perpendicular diameter. diameter. These These first first two two divisions divisions form form the the axis axis of of the the growing growing cell cell structure. structure. TheyThey determine determine the the next next several several divisions. divisions. However, However, during during further further growth growth of of the the tissue, tissue, the the axes axes areare not not preserved preserved and and radial radial structure structure begins begins to to emerge. emerge. The The outer outer cells cells divide divide either either in in the the direction direction parallelparallel to to the the outer outer surface surface or or perpendicular perpendicular to to it. it. This This cell cell organization organization becomes becomes more more obvious obvious during during

Computationfurtherfurther2017 growth, 5, growth 45 (Figure (Figure4 4Left).Left). Let Let us us also also note note that that the the shapes shapes of of the6 the of 18 cells cells represent represent mostly mostly polygons polygons withwith four four or or five five vertices. vertices. The The cell cell structure structure symmetry symmetry is is lost lost after after several several divisions divisions because because cells cells haveThehave first approximately approximately division occurs along equal equal a diameter area area but of but the not circularnot exactly exactly cell. The equal. equal. second Therefore, divisionTherefore, takes even place even after after the the first first division, division, the the two two along the perpendicular diameter. These first two divisions form the axis of the growing cell structure. Theydaughterdaughter determine cells the cells next are areseveral slightly slightly divisions. different. different. However, during further growth of the tissue, the axes are not preserved and radial structure begins to emerge. The outer cells divide either in the direction parallel3.2.3.2. Pressure-Dependentto Pressure-Dependent the outer surface or perpendicular Proliferation Proliferation to it. This cell organization becomes more obvious during further growth (Figures3 and4a). Let us also note that the shapes of the cells represent mostly polygonsCell withCell four division division or five vertices. leads leads to The to cells’ cell cells’ structure compression compression symmetry and is lostand to after to internal severalinternal divisions pressure pressure because growth. growth. Initially, Initially, internal internal pressure pressure cellsin havein a separate a approximately separate cell cell equal equals equals area zero but zero not because exactly because equal. the the Therefore, cell cell has has even its its equilibrium after equilibrium the first division, shape. shape. the When When it itgrows, grows,itsits equilibrium equilibrium two daughter cells are slightly different. areaareaS0Sincreases0 increases linearly linearly in in time. time. Its Its actual actual area area also also grows grows but but slower slower because because the the cell cell membrane membrane 3.2.resists Pressure-Dependentresists elongation. elongation. Proliferation This This difference difference creates creates internal internal cell cell pressure. pressure. When When the the cell cell divides, divides,itsits equilibrium equilibrium Cell division leads to cells’ compression and to internal pressure growth. Initially, internal pressure areaareaS0Sisis divided divided by by2 2andand its its actual actual area area is is divided divided approximately approximately by by2.2.Therefore,Therefore, the the value valueofof in a separate cell0 equals zero because the cell has its equilibrium shape. When it grows, its equilibrium areapressurepressureS0 increases remains remains linearly in approximately time. approximately Its actual area the also the same. grows same. butMoreover, Moreover, slower because since since the cells cell cells membrane are are attached attached to to each each other, other, they they cannot cannot resistsreturnreturn elongation. to to the the This circular circulardifference shape creates shape forinternal for which which cell pressure. the the area Whenarea is theis maximal. maximal. cell divides, Cellits Cell equilibrium deformation deformation increases increases the the pressure. pressure. area S0 is divided by 2 and its actual area is divided approximately by 2. Therefore, the value of pressure remainsMaximalMaximal approximately pressure pressure the (among same. (among Moreover, all all cells) since cells) cells is is reached are reached attached at to at the each the center other, center they of ofcannot the the domain domain filled filled by by cells. cells. It It grows grows returnexponentiallyexponentially to the circular shape in in time for time which (Figure (Figure the area5 5 isLeft) maximal.Left) because because Cell deformation cells cells at at increases the the center center the pressure. need need to to push push surrounding surrounding cells cells more more andMaximaland more more pressure in in order order (among to to allexpand. expand. cells) is reached This This is at is therelated related center toof tothe the the domain exponential exponential filled by cells. growth growth It grows of of cell cell number number (Section (Section4).4). For For a a exponentially in time (Figure5a) because cells at the center need to push surrounding cells more and morefixedfixed in order time, time, to expand. the the maximal maximal This is related pressure pressure to the exponential depends depends growth on on parameters. of parameters. cell number It(Section It grows grows4). For approximately approximately a as as a lineara linear function function fixedofof time, viscosity viscosity the maximal (Figure (Figure pressure5 5Right). dependsRight). on It parameters. It can can be be compared It compared grows approximately with with fluid fluid as a flow linear flow functionin in a porousa porous medium medium where where pressure pressure of viscosity (Figure5b). It can be compared with fluid flow in a porous medium where pressure differencedifferencedifference is proportional is is proportional proportional to the viscosity to to(Section the the viscosity4 viscosity). (Section (Section4).4).

(a)(b) FigureFigure 3. 3.SnapSnapshotsshots of of growing growing tissue tissue with with the the maximal maximal division division pressure pressurepmaxpmax==1515inin the the consecutive consecutive Figure 3. Cont. momentsmoments of of time. time. Non-dividing Non-dividing cells cells with with pressure pressure greater greater than thanpmaxpmaxareare shown shown in in dark dark grey. grey.

BiologicalBiological cells cells cannot cannot divide divide when when they they are are too too compressed compressed (see (see [54 [54] and] and references references therein). therein). InIn order ordertoto describe describe this this effect, effect, we we introduce introduce the the maximal maximal pressure pressure beyond beyond which which cells cells do do not not divide. divide. LetLet us us recall recall that that pressure pressureP Pinsideinside cells cells is is determined determined by by Equation Equation (3 ().3). We We impose impose an an additional additional conditioncondition that that if ifP Pisis greater greater than than some some critical critical value valuepmaxpmax, then, then the the cell cell does does not not divide. divide.FigureFigure4 Left4 Left showsshows a a snapshot snapshot of of the the growing growing cells’ cells’ structure structure for for the the maximal maximal division division pressure pressurepmaxpmax==1010. . WeWe obtain obtainaa regular regular structure structure close close to to a a circle circle with with radial radial cell cell organization organization near near its its outer outer border. border. CellsCells there there divide divide either either in in the the direction direction parallel parallel to to the the boundary boundary or or perpendicular perpendicular to to it. it. The The latter latter increasesincreases the the number number of of cells cells in in the the outer outer layer layer and and provides provides growth growth of of the the perimeter. perimeter. Similar Similar cell cell structuresstructures can can be be observed observed in in the the cross-section cross-section of of growing growing plant plant branches. branches. Computation 2016, 5, x 6 of 17

The first division occurs along a diameter of the circular cell. The second division takes place along the perpendicular diameter. These first two divisions form the axis of the growing cell structure. They determine the next several divisions. However, during further growth of the tissue, the axes are not preserved and radial structure begins to emerge. The outer cells divide either in the direction parallel to the outer surface or perpendicular to it. This cell organization becomes more obvious during further growth (Figure4 Left). Let us also note that the shapes of the cells represent mostly polygons with four or five vertices. The cell structure symmetry is lost after several divisions because cells have approximately equal area but not exactly equal. Therefore, even after the first division, the two daughter cells are slightly different.

3.2. Pressure-Dependent Proliferation Cell division leads to cells’ compression and to internal pressure growth. Initially, internal pressure in a separate cell equals zero because the cell has its equilibrium shape. When it grows, its equilibrium area S0 increases linearly in time. Its actual area also grows but slower because the cell membrane resists elongation. This difference creates internal cell pressure. When the cell divides, its equilibrium area S0 is divided by 2 and its actual area is divided approximately by 2. Therefore, the value of pressure remains approximately the same. Moreover, since cells are attached to each other, they cannot return to the circular shape for which the area is maximal. Cell deformation increases the pressure. Maximal pressure (among all cells) is reached at the center of the domain filled by cells. It grows exponentially in time (Figure5 Left) because cells at the center need to push surrounding cells more and more in order to expand. This is related to the exponential growth of cell number (Section4). For a fixed time, the maximal pressure depends on parameters. It grows approximately as a linear function of viscosity (Figure5 Right). It can be compared with fluid flow in a porous medium where pressure difference is proportional to the viscosity (SectionComputation4). 2017, 5, 45 7 of 18

(c)

Figure 3. Snapshots of growing tissue with the maximalFigure division 3. Snapshots pressure of growingpmax tissue= with15 thein maximal the consecutive division pressure pmax = 15 in the consecutive moments of time. Non-dividing cells with pressure greater than p are shown in dark grey. moments of time. Non-dividing cells with pressure greater than pmax are shown in dark grey. max

Biological cells cannot divide when they are too compressed (see [54] and references therein). Biological cells cannot divide when they areIn order tooto compressed describe this effect, (see we [ introduce54] and the references maximal pressure therein). beyond which cells do not divide. In order to describe this effect, we introduce theLet maximal us recall that pressure pressure beyondP inside cells which is determined cells do by not Equation divide. (3). We impose an additional condition that if P is greater than some critical value pmax, then the cell does not divide. Figure4a shows Let us recall that pressure P inside cells is determineda snapshot of by the Equation growing cells’ (3 structure). We for impose the maximal an additional division pressure pmax = 10. We obtain a regular structure close to a circle with radial cell organization near its outer border. Cells there divide condition that if P is greater than some critical value pmax, then the cell does not divide. Figure4 Left either inComputation the direction2016 parallel, 5, x to the boundary or perpendicular to it. The latter increases the number 7 of 17 shows a snapshot of the growing cells’ structureof cells forComputation in the the outer maximal2016 layer, 5 and, x provides division growth pressure of the perimeter.pmax Similar= 10 cell. structures can be observed 7 of 17 We obtain a regular structure close to a circlein with the cross-section radial cell of growing organization plant branches. near its outer border. Cells there divide either in the direction parallel to the boundary or perpendicular to it. The latter increases the number of cells in the outer layer and provides growth of the perimeter. Similar cell structures can be observed in the cross-section of growing plant branches.

(a)(b)

Figure 4. Cont. FigureFigure 4. 4.AA snapshot snapshot of of the the growing growing cell cell structure structure with with the the maximal maximal division division pressure pressurepmaxpmax= 10=(Left10 (),Left), ppmaxmax==44 (Middle (Middle)) and and with with the the disappearance disappearance of ofnon-dividing non-dividing cells cells (Right (Right). ).

Figure 5. Maximal pressure in the growing cell tissue. (Left) maximal pressure as a function of time; Figure 5. Maximal pressure in the growing cell tissue. (Left) maximal pressure as a function of time; (Right) maximal pressure at time t = 700 for different values of viscosity coefficient µ. Dimensionless (Right) maximal pressure at time t = 700 for different values of viscosity coefficient µ. Dimensionless time and pressure units are used. time and pressure units are used.

The value of the maximal division pressure can influence the properties of the growing cell structure.The Anothervalue of example the maximal is shown division in Figure pressure4 Middle. can Here, influence the maximal the properties division pressure of the growing is set at cell structure. Another example is shown in Figure4 Middle. Here, the maximal division pressure is set at pmax = 4 after a short initial period when pmax = 5 (otherwise, the first divisions do not occur). In this case,pmax there= 4 after are cells a short which initial do not period divide when evenpmax at the= outer5 (otherwise, surface. theThis first results divisions in the appearance do not occur). of the In this specificcase, there asymmetric are cells which structure do withnot divide a curved even surface at the and outer “leaves” surface. separated This results by an in theinterface. appearance Cells at of the thespecific interface asymmetric do not divide structure since with the pressure a curved there surface is greater and “leaves” than the separated critical value byp anmax interface.. Cells at the interfaceThe appearance do not divide of curved since structures the pressure can there be considered is greater than as instability the critical of value symmetricpmax. circular structures.The appearance This instability of curved results structures from cell competition can be considered for space. as More instability compressed of symmetric cells do circular not grow,structures. and less This compressed instability cells results grow from and take cell even competition more space. for As space. happens More in compressedmany other examples, cells do not thegrow, instability and less occurs compressed near the cells extinction grow and limit take where even the more cell space. structure As happens stops growing in many because other ofexamples, the excessivelythe instability high occurs pressure. near the extinction limit where the cell structure stops growing because of the excessivelyWe consider high pressure.one more example where cells die by apoptosis if they do not divide during some time.We This consider is related one to the more check example points where during cells the celldie cycle.by apoptosis In this case, if they they do are not removed divide during from the some computationaltime. This is related domain. to Inthe this check case, points there during is a curved the cellfront cycle. of cells In slowly this case, propagating they are removed outwards from and the numerouscomputational cell clusters domain. inside In this (Figure case,4 there Right). is a This curved is afront non-stationary of cells slowly structure propagating where cells outwards divide and andnumerous expand cell if they clusters have enoughinside (Figure space; they4 Right). stopdividing This is a and non-stationary disappear if structure there is not where enough cells space. divide and expand if they have enough space; they stop dividing and disappear if there is not enough space. Computation 2016, 5, x 7 of 17

Computation 2017, 5, 45 8 of 18

Computation 2016, 5, x 7 of 17

(c)

Figure 4. A snapshot of the growing cell structure with the maximal division pressure pmax = 10 (a); Figure 4. A snapshot of the growing cell structure with the maximal division pressure pmax = 10 (Left), pmax = 4 (b) and with the disappearance of non-dividing cells (c). pmax = 4 (Middle) and with the disappearance of non-dividing cells (Right). The value of the maximal division pressure can influence the properties of the growing cell structure. Another example is shown in Figure4b. Here, the maximal division pressure is set at pmax = 4 after a short initial period when pmax = 5 (otherwise, the first divisions do not occur). In this case, there are cells which do not divide even at the outer surface. This results in the appearance of the specific asymmetric structure with a curved surface and “leaves” separated by an interface. Cells at the interface do not divide since the pressure there is greater than the critical value pmax. The appearance of curved structures can be considered as instability of symmetric circular structures. This instability results from cell competition for space. More compressed cells do not Figure 4. A snapshot of the growing cell structure with the maximal division pressure pmax = 10 (Left), grow, and less compressed cells grow and take even more space. As happens in many other examples, the instability occurs nearpmax the= extinction4 (Middle limit) where and with the cell the structure disappearance stops growing of non-dividing because of the cells (Right). excessively high pressure.

Figure 5. Maximal pressure in the growing cell tissue. (Left) maximal pressure as a function of time; (Right) maximal pressure at time t = 700 for different values of viscosity coefficient µ. Dimensionless time and pressure units are used.

The value of the maximal division pressure can influence the properties of the growing cell structure. Another example is shown in Figure4 Middle. Here, the maximal division pressure is set at pmax = 4 after a short initial period when pmax = 5 (otherwise, the first divisions do not occur). In this case, there are cells which do not divide even at the outer surface. This results in the appearance(a) of the specific asymmetric structure with a curved surface and “leaves” separated by an interface. Cells at Figure 5. MaximalFigure pressure 5. Cont. in the growing cell tissue. (Left) maximal pressure as a function of time; the interface do not divide since the pressure there is greater than(Right the critical) maximal value pressurepmax. at time t = 700 for different values of viscosity coefficient µ. Dimensionless The appearance of curved structures can be considered astime instability and pressure of symmetric units are used. circular structures. This instability results from cell competition for space. More compressed cells do not grow, and less compressed cells grow and take even more space. As happens in many other examples, The value of the maximal division pressure can influence the properties of the growing cell the instability occurs near the extinction limit where the cell structure stops growing because of the structure. Another example is shown in Figure4 Middle. Here, the maximal division pressure is set at excessively high pressure. p = 4 after a short initial period when p = 5 (otherwise, the first divisions do not occur). In this We consider one more example where cells die by apoptosismax if they do not divide during some max case, there are cells which do not divide even at the outer surface. This results in the appearance of the time. This is related to the check points during the cell cycle. In this case, they are removed from the computational domain. In this case, there is a curved front ofspecific cells slowly asymmetric propagating structure outwards with a and curved surface and “leaves” separated by an interface. Cells at numerous cell clusters inside (Figure4 Right). This is a non-stationarythe interface structure do not divide where since cells dividethe pressure there is greater than the critical value pmax. and expand if they have enough space; they stop dividing and disappearThe appearance if there is of not curved enough structures space. can be considered as instability of symmetric circular structures. This instability results from cell competition for space. More compressed cells do not grow, and less compressed cells grow and take even more space. As happens in many other examples, the instability occurs near the extinction limit where the cell structure stops growing because of the excessively high pressure. We consider one more example where cells die by apoptosis if they do not divide during some time. This is related to the check points during the cell cycle. In this case, they are removed from the computational domain. In this case, there is a curved front of cells slowly propagating outwards and numerous cell clusters inside (Figure4 Right). This is a non-stationary structure where cells divide and expand if they have enough space; they stop dividing and disappear if there is not enough space. Computation 2016, 5, x 7 of 17

Figure 4. A snapshot of the growing cell structure with the maximal division pressure pmax = 10 (Left), pmax = 4 (Middle) and with the disappearance of non-dividing cells (Right). Computation 2017, 5, 45 9 of 18

(b)

Figure 5. Maximal pressure in the growingFigure cell 5. Maximal tissue. pressure (Left) inmaximal the growing pressure cell tissue. as (a a) maximal function pressure of time; as a function of time; (Right) maximal pressure at time t = 700(b)for maximal differentpressure values at time oft = viscosity700 for different coefficient values of viscosityµ. Dimensionless coefficient µ. Dimensionless time and pressure units are used. time and pressure units are used.

We consider one more example where cells die by apoptosis if they do not divide during some The value of the maximal divisiontime. This pressure is related can to the influence check points the during properties the cell cycle. of In the this case,growing they are cell removed from the computational domain. In this case, there is a curved front of cells slowly propagating outwards and structure. Another example is shownnumerous in Figure cell4 Middle. clusters inside Here, (Figure the4c). maximal This is a non-stationary division pressure structure is where set at cells divide and pmax = 4 after a short initial period whenexpandpmax if they= have5 (otherwise, enough space; the they first stop dividing divisions and do disappear not occur). if there is In not this enough space. case, there are cells which do not divide3.3. even External at Supply the outer of Nutrients surface. This results in the appearance of the specific asymmetric structure with a curved surface and “leaves” separated by an interface. Cells at If nutrients come from outside of the growing tissue, as is the case for tumor growth, then exterior the interface do not divide since the pressurecells have more there nutrients is greater than the than cells the inside critical the tissue. value Theyp willmax grow. and divide faster than cells The appearance of curved structuresinside thecan tissue. be They considered will also compete as for instability nutrients. This of competition symmetric can lead circular to nonuniform tissue growth and to the emergence of asymmetric structures. This effect was also observed in hybrid models structures. This instability results fromof tumor cell growth competition [55]. for space. More compressed cells do not grow, and less compressed cells grow andIn take this section, even we more illustrate space. it with As the happens deformable in cell many model. other We assume examples, that the growth rate of the instability occurs near the extinctionexterior limit cells iswhere proportional the cell to the structure length of their stops exterior growing boundary. because This assumption of the implies that the concentration of nutrients in the medium surrounding the tissue is constant and their consumption by excessively high pressure. cells is proportional to the length of the cell boundary. We consider one more example whereIf we cells assume die that by nutrients apoptosis can be transported if they do inside not the divide tissue, during then interior some cells will also have time. This is related to the check pointssome during nutrients. the In order cell cycle.to simplify In the this model, case, we they will suppose are removed that the concentration from the of nutrients in the extracellular matrix at some point inside the tissue is the same as in the cell located at this point. computational domain. In this case, there is a curved front of cells slowly propagating outwards and The flux of nutrients q between cells is proportional to the difference of concentrations c1 and c2 in numerous cell clusters inside (Figurethe4 Right). neighboring This cells is and a to non-stationary the length L of theboundary structure between where them, cellsq = divideλ(c1 c2)L. Here, λ is a − and expand if they have enough space;parameter they stop which dividing determines and the flux disappear intensity. if there is not enough space. Figure6 shows examples of numerical simulations of tissue growth with an external supply of nutrients only to the outer cells and to the interior cells in the case of nutrient diffusion in the tissue. We observe asymmetric growth which results from cell competition for nutrients. A cell which has a longer exterior boundary will grow and divide faster and will compress surrounding cells, reducing their exterior boundary and growth rate. Moreover, the pressure-dependent proliferation increases this competition because compressed cells can stop their growth completely. Emerging structures are more pronounced here than in the model without nutrients (Section 3.2). Computation 2016, 5, x 8 of 17 Computation 2016, 5, x 8 of 17

3.3. External Supply of Nutrients 3.3. External Supply of Nutrients If nutrients come from outside of the growing tissue, as is the case for tumor growth, then exterior cells haveIf nutrients more nutrients come from than outside the cells of the inside growing the tissue. tissue, They as is thewill case grow for and tumor divide growth, faster then than exterior cells insidecells the have tissue. more They nutrients will also than compete the cells for inside nutrients. the tissue. This They competition will grow can and lead divide to nonuniform faster than tissue cells growthinside and the tissue.to the emergence They will also of asymmetric compete for structures. nutrients. This This effect competition was also can observed lead to nonuniform in hybrid models tissue ofgrowth tumor growthand to the [55 emergence]. of asymmetric structures. This effect was also observed in hybrid models of tumorIn this growth section, [55 we]. illustrate it with the deformable cell model. We assume that the growth rate of exteriorIn cells this is section, proportional we illustrate to the it length with the of their deformable exterior cell boundary. model. We This assume assumption that the implies growth that rate the of exterior cells is proportional to the length of their exterior boundary. This assumption implies that the concentration of nutrients in the medium surrounding the tissue is constant and their consumption by concentration of nutrients in the medium surrounding the tissue is constant and their consumption by cells is proportional to the length of the cell boundary. cells is proportional to the length of the cell boundary. If we assume that nutrients can be transported inside the tissue, then interior cells will also have If we assume that nutrients can be transported inside the tissue, then interior cells will also have some nutrients. In order to simplify the model, we will suppose that the concentration of nutrients in some nutrients. In order to simplify the model, we will suppose that the concentration of nutrients in the extracellular matrix at some point inside the tissue is the same as in the cell located at this point. the extracellular matrix at some point inside the tissue is the same as in the cell located at this point. The flux of nutrients q between cells is proportional to the difference of concentrations c1 and c2 in The flux of nutrients q between cells is proportional to the difference of concentrations c1 and c2 in the neighboring cells and to the length L of the boundary between them, q = λ(c1 c2)L. Here, λ is a the neighboring cells and to the length L of the boundary between them, q = λ(c − c )L. Here, λ is a parameter which determines the flux intensity. 1 − 2 parameter which determines the flux intensity. Figure6 shows examples of numerical simulations of tissue growth with an external supply of Figure6 shows examples of numerical simulations of tissue growth with an external supply of nutrients only to the outer cells and to the interior cells in the case of nutrient diffusion in the tissue. nutrients only to the outer cells and to the interior cells in the case of nutrient diffusion in the tissue. We observe asymmetric growth which results from cell competition for nutrients. A cell which has a We observe asymmetric growth which results from cell competition for nutrients. A cell which has a longer exterior boundary will grow and divide faster and will compress surrounding cells, reducing longer exterior boundary will grow and divide faster and will compress surrounding cells, reducing their exterior boundary and growth rate. Moreover, the pressure-dependent proliferation increases their exterior boundary and growth rate. Moreover, the pressure-dependent proliferation increases this competition because compressed cells can stop their growth completely. Emerging structures are Computationthis competition2017, 5, 45 because compressed cells can stop their growth completely. Emerging structures10 ofare 18 more pronounced here than in the model without nutrients (Section 3.2). more pronounced here than in the model without nutrients (Section 3.2).

(a)(b) FigureFigure 6. 6.AA snapshot snapshot of of growing growing tissue tissue with with an an external external supply supply of of nutrients nutrients only only to to the the outer outer cells cells Figure(Left(Left) 6.and) andA with snapshot with nutrient nutrient of growing diffusing diffusing tissue inside inside with the the tissue an tissue external (Right (Right supply).). of nutrients only to the outer cells (a) and with nutrient diffusing inside the tissue (b). 3.4.3.4. Wound Wound Healing Healing 3.4. WoundTogetherTogether Healing with with tumor tumor growth, growth, wound wound healing healing is is one one of of the the main main applications applications of of tissue tissue growth growth models.models.Together An An with extensive extensive tumor literature growth,literature woundreview review onhealing on this this subject is subject one of can can the be bemain found found applications in in [56 [56].]. A of A conventional tissue conventional growth models.approachapproach An to to extensive study study wound wound literature healing healing review consists consists on in this in the the subject analysis analysis can of of be travelling travelling found in wave wave [56]. solutions solutions A conventional of of the the reaction–diffusion system of equations for the cell density and for the concentration of some approachreaction–diffusion to study wound system healing of equations consists for in the the cell analysis density of and travelling for the concentration wave solutions ofof some the substances which influence cell proliferation (e.g., oxygen). The wave of cell proliferation fills the reaction–diffusionsubstances which system influence of cell equations proliferation for the (e.g., cell oxygen). density The and wave for theof cell concentration proliferation of fills some the substances which influence cell proliferation (e.g., oxygen). The wave of cell proliferation fills the spatial domain, which corresponds to the wound and which is surrounded by the remaining tissue. In this section, we illustrate modelling of wound healing with the deformable cell model. We will see that pressure-dependent proliferation is an appropriate mechanism to control wound closure. Figure7 shows an example of numerical simulations of tissue regeneration. The original tissue (left image) is obtained from a single cell as described above. The cells of this tissue are fixed—they do not grow or divide any more. After that, we remove the internal part of the tissue. The cells at the boundary of the removed tissue can grow and divide. They fill the available place. They stop growing and divide when the tissue is regenerated due to pressure-dependent proliferation. Cells push each other, internal pressure grows and they no longer grow and divide. The new tissue fits the form of the wound well. However, some damage remains at the place where two parts of the tissue meet each other. Its size depends on the mechanical properties of cells. It decreases if cells are more deformable and better fit each other. Another example of tissue regeneration is shown in Figure8. The main difference here in comparison with the previous example is that the region where a part of the tissue is removed is not bounded by the remaining tissue (Figure8b). If we use the same approach as in the previous case, we obtain a different form of the regenerated tissue compared with the initial tissue (Figure8c). In order to reproduce the form of the original tissue, we need to introduce different growth rates for different cells. Consider cells located at the wound surface in Figure8b. We impose a particular growth rate for each of these cells. Their descendants will preserve these growth rates. The cells located closer to the outer surface grow and divide slowly. The growth rate and division increase as much as the cells go deeper into the damaged tissue. The maximal growth rate is reached at the center. Such distribution of the growth rate allows us to approximate the initial tissue form by the regenerated tissue (Figure8d). It should also be noted that this tissue continues to grow even when the initial form is regenerated. Therefore, there are additional mechanisms which control the size of the tissue [57–59]. ComputationComputationComputation20162016, 52016,, x5,, x5, x 9 of9 17 of9 17 of 17

spatialspatialspatial domain, domain, domain, which which which corresponds corresponds corresponds to theto theto wound the wound wound and and which and which which is surrounded is surrounded is surrounded by by the by the remaining the remaining remaining tissue. tissue. tissue. InIn thisIn this section this section section, we, we illustrate, we illustrate illustrate modelling modelling modelling of woundof woundof wound healing healing healing with with thewith the deformable the deformable deformable cell cell model. cell model. model. We We will We will see will see see thatthat pressure-dependentthat pressure-dependent pressure-dependent proliferation proliferation proliferation is an is an appropriateis an appropriate appropriate mechanism mechanism mechanism to controlto controlto control wound wound wound closure. closure. closure. FigureFigureFigure7 shows7 shows7 shows an an example an example example of numerical of numerical of numerical simulations simulations simulations of tissue of tissue of tissue regeneration. regeneration. regeneration. The The original The original original tissue tissue tissue (left(left image)(left image) image) is obtained is obtained is obtained from from froma single a single a single cell cell as cell asdescribed describedas described above. above. above. The The cells The cells of cells thisof thisof tissue this tissue tissue are are fixed—they are fixed—they fixed—they do do do notnot grownot grow grow or ordivide ordivide divide any any more. any more. more. After After After that, that, wethat, we remove we remove remove the the internal the internal internal part part of part ofthe ofthe tissue. the tissue. tissue. The The cells The cells atcells theat theat the boundaryboundaryboundary of theof theof removed the removed removed tissue tissue tissue can can grow can grow grow and and divide. and divide. divide. They They They fill fill the fill the available the available available place. place. place. They They They stop stop growing stop growing growing andand divideand divide divide when when when the the tissue the tissue tissue is regenerated is regenerated is regenerated due due to due pressure-dependentto pressure-dependentto pressure-dependent proliferation. proliferation. proliferation. Cells Cells Cells push push push each each each other,other,other, internal internal internal pressure pressure pressure grows grows grows and and they and they no they no longer no longer longer grow grow grow and and divide. and divide. divide. The The new The new tissue new tissue tissue fits fits the fits the form the form formof theof theof the woundwoundwound well. well. well. However, However, However, some some some damage damage damage remains remains remains at theat theat place the place place where where where two two parts two parts parts of ofthe ofthe tissue the tissue tissue meet meet meet each each each Computationother.other.other. Its Its size2017 Its size depends, size5, depends 45 depends on on the on the mechanical the mechanical mechanical properties properties properties of cells.of cells.of cells. It decreases It decreases It decreases if cells if cells if are cells are more are more more deformable deformable deformable11 of 18 andand betterand better better fit fiteach fiteach other.each other. other.

(a)(b)(c) FigureFigureFigure 7. Numerical 7. Numerical 7. Numerical simulations simulations simulations of wound of wound of wound healing. healing. healing. Original Original Original tissue tissue tissue (Left (Left), (Left damaged), damaged), damaged tissue tissue tissue (Middle (Middle (Middle), ), ), tissueFiguretissuetissue regeneration 7. regeneration regenerationNumerical (Right (Rightsimulations ().Right Grey). Grey). cells Grey cellsof are cells are wound fixed, are fixed, fixed, while healing. while while white white white cells Original cells divide cells divide tissue divide and and fill( anda fill); the fill damagedthe missing the missing missing part tissue part of part of (b of); thetissuethe tissue.the tissue.regeneration tissue. (c). Grey cells are fixed, while white cells divide and fill the missing part of the tissue. AnotherAnotherAnother example example example of of tissue of tissue tissue regeneration regeneration regeneration is shownis shownis shown in in Figure in Figure Figure8.8 The.8 The. main The main main difference difference difference here here herein in in comparisoncomparisoncomparison with with thewith the previous the previous previous example example example is that is that is the that the region the region region where where where a part a part a of part theof theof tissue the tissue tissue is removed is removed is removed is not is not is not Let us note that if the critical value pmax is high enough, it will not influence cell division. On the bounded by the remaining tissue (Figure8b). If we use the same approach as in the previous case, we otherboundedbounded hand, by there by the the is remaining a remaining biochemical tissue tissue regulation(Figure (Figure8b).8b). ofIf we theIf we use cell use the division the same same approach rate, approach implying as asin thein that the previous some previous cells case, case, divide we we obtainobtain a different a different form form of theof the regenerated regenerated tissue tissue compared compared with with the the initial initial tissue tissue (Figure (Figure8c).8c). In Inorder order fasterobtain than others. a different This form regulation of the regenerated is not explicitly tissue introduced compared in with the modelthe initial but tissue it acts (Figure implicitly8c). Inthrough order to reproduceto reproduceto reproduce the the form the form form of theof theof original the original original tissue, tissue, tissue, we we need we need need to introduceto introduceto introduce different different different growth growth growth rates rates ratesfor for different for different different the cell location. cells.cells.cells. Consider Consider Consider cells cells located cells located located at theat the at wound the wound wound surface surface surface in Figurein Figurein Figure8b.8 Web.8 Web. impose We impose impose a particular a particular a particular growth growth growth rate rate for rate for for eacheach ofeach theseof theseof these cells. cells. cells. Their Their Their descendants descendants descendants will will preserve will preserve preserve these these these growth growth growth rates. rates. rates. The The cells The cells located cells located located closer closer closer to theto theto the outerouterouter surface surface surface grow grow grow and and divide and divide divide slowly. slowly. slowly. The The growth The growth growth rate rate and rate and division and division division increase increase increase as asmuch asmuch much as asthe asthe cells the cells go cells go go deeperdeeperdeeper into into the into the damaged the damaged damaged tissue. tissue. tissue. The The maximal The maximal maximal growth growth growth rate rate is rate reached is reached is reached at theat the at center. the center. center. Such Such Suchdistribution distribution distribution of of of thethe growththe growth growth rate rate allows rate allows allows us us to us approximateto approximateto approximate the the initial the initial initial tissue tissue tissue form form form by by the by the regenerated the regenerated regenerated tissue tissue tissue (Figure (Figure (Figure8d).8d).8d). It shouldIt shouldIt should also also be also be noted be noted noted that that this that this tissue this tissue tissue continues continues continues to growto growto grow even even even when when when the the initial the initial initial form form form is regenerated. is regenerated. is regenerated. Therefore,Therefore,Therefore, there there there are are additional are additional additional mechanisms mechanisms mechanisms which which which control control control the the size the size of size theof theof tissue the tissue tissue [57 [–5759 [–57].59–].59]. LetLet usLet us note usnote that note that if that the if the if critical the critical critical value value valuepmaxpmaxispmax highis highis enough, high enough, enough, it will it will it not will not influence not influence influence cell cell division. cell division. division.OnOn theOn the the otherotherother hand, hand, hand, there there there is a is biochemical a is biochemical a biochemical regulation regulation regulation of theof theof cell the cell division cell division division rate, rate, implying rate, implying implying that that some that some some cells cells divide cells divide divide fasterfasterfaster than than others. than others. others. This This regulation This regulation regulation is not is not is explicitly not explicitly explicitly introduced introduced introduced in the in the in model the model model but but it but acts it acts it implicitly acts implicitly implicitly through through through thethe cellthe cell location. cell location. location.

Figure 8. Numerical simulations of wound healing. Original tissue (a); damaged tissue (b); tissue regeneration with equal rates of cell growth (c); cell growth rate depends on the initial position of the first dividing cell in its lineage (d). Computation 2017, 5, 45 12 of 18

4. Analytical Approximation of the Growth Rate

4.1. Approximate Model Passing from the equation of motion of individual particles to an approximate averaged equation of motion of the medium, we get ∂v ∂p + µv + k = 0, (5) ∂t ∂x where v is the velocity of the medium and p is the pressure (see [27] for more detail). We consider this equation in the one-dimensional case in order to obtain an approximate analytical solution. It corresponds to the two-dimensional case with plane geometry or to the circular tissue with a sufficiently large radius. Equation (5) is a macroscopic analogue of Equation (4) with an average force proportional to the pressure gradient. It should be noted, however, that force in (4) also contains other components. So, the analytical continuous model is an approximation. Parameter µ in both equations corresponds to viscous friction. We will consider this equation in the quasi-stationary approximation,

∂p µv = k , (6) − ∂x which takes the form of the Darcy equation for the porous medium. Next, we will assume that the medium is incompressible. This assumption is justified for sufficiently large values of the pressure force coefficient k3 in the cell model. In this case, cells preserve their volume and, in the limit of large coefficients, the medium becomes incompressible. In this case, the mass conservation (or continuity) equation can be written as follows: div v = W, (7) where W is the rate of production of mass. In the one-dimensional case, divergence is the space derivative ∂v/∂x. Production of mass is determined by the rate of cell division W = f (p). We will suppose that it depends on the pressure. When biological cells are strongly compressed, they do not divide. This assumption is consistent with pressure-dependent proliferation in the cell model (Section3), which can be considered as discretization of the continuous model. From Equations (6) and (7), we get

kp00 + µ f (p) = 0, (8) where prime denotes the space derivative.

4.2. Constant Proliferation If the proliferation rate does not depend on pressure, then we set f (p) = b, where b is some positive constant. We solve Equation (8) in the interval ξ < x < ξ with the boundary condition − p( ξ) = p0. We obtain ± µb p(x) = (x2 ξ2) + p . − 2k − 0 Since dξ/dt = v(ξ), then taking into account (6), we have

dξ = bξ. dt Hence the length of the interval grows exponentially. If we consider a circular tissue with the radius ξ, then neglecting the influence of the curvature on the growth rate, we get the formula for the tissue area A = πξ2:

2bt A(t) = e A0, Computation 2017, 5, 45 13 of 18

where A0 is the initial area. Figure9a shows an example of numerical simulations of tissue growth with a constant proliferation rate. The linear dependence of the logarithm of tissue area on time, log A(t), shows that the growth rate is exponential. Three different proliferation rates are considered, dS0(t)/dt = 0.1, 0.05, 0.025. The slopes of the straight lines are in the same proportions. For large amounts of time, the assumptions of the model may not be satisfied, and the growth rate slows down in comparison with the exponential one.

(a)

(b)

Figure 9. Comparison of numerical simulations (continuous curves) with the analytical model (dots) for different values of parameters. (a) tissue growth with a constant proliferation rate in logarithmic coordinates. Linear dependence corresponds to the exponential growth. 1. dS0(t)/dt = 0.1, 2. = 0.05, 3. = 0.025;(b) quadratic growth for pressure-dependent proliferation, 1. pm = 6, 2. pm = 5. Dimensionless time and area units are used. Computation 2017, 5, 45 14 of 18

4.3. Pressure-Dependent Proliferation We will consider a piece-wise constant function f (p): ( b , 0 < p < p f (p) = m . 0 , p p ≥ m Let us suppose that the medium fills the half-axis x ξ. We should complete the formulation of ≤ the problem by the boundary condition for the pressure

p(ξ) = p0, (9) where p0 is some non-negative constant, p0 < pm. We can now solve problem (8) and (9). Suppose that the value p = pm is reached at x = 0. Taking into account the continuity of pressure and of its first derivative at x = 0, we obtain ( p , x 0 p(x) = 0 ≤ , p αx2/2 , 0 < x ξ m − ≤ where α = bµ/(2k). From the boundary condition (9), we find the value of ξ: r 2(p p ) ξ = m − 0 . α Hence

r s k k 2(pm p0) 2bk(pm p0) v(ξ) = p0(ξ) = α − = − . − µ µ α µ In the case of circular tissue growth, neglecting curvature, we can approximate the growth rate of irs radius R by the formula

dR = v. dt Hence we obtain the formula for the tissue area A = πR2:

2bk(p p ) A(t) = π m − 0 t2. (10) µ

Figure9b shows the results of numerical simulations of tissue growth and comparison with the analytical approximation. At the beginning of growth, when the pressure-dependent proliferation is not yet established, all cells divide, and the growth rate is exponential. Therefore, instead of Formula (10), we approximate it by the expression

A(t) = at2 + b, (11) where the coefficients a and b are chosen to fit the simulations presented in Section 3.1 with two different values of maximal pressure: pm = 5 and pm = 6. The curves are well approximated by Formula (11) with a = 0.0033, b = 1142 and a = 0.0045, b = 2227. − − In order to compare more precisely the values of a obtained in numerical simulations with the coefficient 2bk(p p ) a = π m − 0 (12) µ in (10), we need to specify the value of pressure p0 at the boundary. This is the pressure in the cells at the outer layer. It varies during cell growth. Its average value depends on the values of parameters. Computation 2017, 5, 45 15 of 18

For the basic set of parameters, p 4. Hence the right-hand side in (12) increases twice when we 0 ≈ compare pm = 5 and pm = 6 while the coefficient a on the left-hand side obtained from numerical simulations in 0.0045/0.0033 1.5 times. This difference is related to the fact that intracellular pressure ≈ in the numerical model (Section3) is only a part of the total pressure in the analytical model. The latter also takes into account the contribution from elastic forces. If we decrease the value of the stretching coefficient, k1 = 30 (instead of k1 = 40 above), then pressure in cells at the outer layer decreases. Therefore, growth accelerates and we get a = 0.004. This example shows how mechanical properties of cells influence the growth rate of the tissue and how they influence macroscopic phenomenological parameters.

5. Discussion In this work, we developed a deformable cell model of growing tissue where cells are attached to each other. Two neighboring cells have the same boundary and cannot be separated. Another possible approach is to consider separated cells which can adhere due to additional forces acting between them [47,48]. The direction of cell division is chosen along the shortest diagonal which divides the cell into two cells with approximately equal areas. This algorithm appears to be very robust and allows us to simulate tissue growth with more than 104 cells. Pressure-dependent proliferation, which corresponds to biological observations, can result in the appearance of asymmetric tissue structures. The spring model of cells (Section2) can be considered as an approximation of an elastic medium. Together with viscosity in the equation for particle displacement, we obtain a viscoelastic medium. Using the porous medium equation to describe it, we simplify the model but we obtain a good approximation of the tissue growth rate. Modelling of tissue growth with the deformable cell model can be applied to study tumor growth. In the case of pressure-dependent proliferation, it describes the linear growth rate of the tissue radius specific for tumors after a short initial stage of exponential growth [54]. The growing tumor can lose its radial symmetry, resulting in the appearance of spatial structures. This instability can be related to cell competition for nutrients, and it can be modeled with partial differential equations [60–64] or with cell-based models [32,55] (see also references in [54]). However, as is indicated in [54], in vitro experiments show a smoother shape of the growing tumor. It is suggested that another possible mechanism of pattern formation can be related to mechanical stresses: cells diffuse along the surface searching for a free space. Results presented in this work show that indeed, competition for nutrients is not the only possible mechanism of instability. Mechanical constraints lead to the emergence of spatial structures. The introduction of nutrients increases their roughness. Hence pressure-dependent proliferation, a linear growth rate (exponential rate at the beginning) and various mechanisms of pattern formation of growing tissues observed in numerical simulations are in agreement with the experiments in vitro. Another classic example of tissue growth is wound healing. It is conventionally modelled with reaction–diffusion equations for the concentrations of cells and some substances which determine their proliferation. A wave of cell proliferation propagates in the wound region and fills it. The deformable cell model of tissue growth is an appropriate tool to study wound healing. Even without additional concentrations, pressure-dependent proliferation allows us to control wound closure. When two borders of the wound meet and cells touch each other, their proliferation stops. However, if we consider the wound cross-section perpendicular to the epidermis, the wound area is not bounded by the remaining tissue from one side. Therefore, the problem here is different: how to control the form and the size of the growing tissue in an open domain. In this case, we need to introduce the rate of cell proliferation which depends on its position. This question requires further investigation. The model presented in this work has certain limitations. First of all, we consider a two-dimensional tissue (cell sheet) while most biological tissues are essentially three-dimensional. Numerical simulations in this case become more involved but we expect that the main properties discussed above remain the same. Cell division and death are determined by the intracellular Computation 2017, 5, 45 16 of 18 and extracellular regulations which can be described by ordinary or partial differential equations. Such hybrid discrete-continuous models [65] are used in the case of a simplified cell representation (e.g., soft sphere model) but not for the deformable cell models. Let us also note that mechanical properties of the tissue can influence molecular transport [66] and, consequently, cell fate. Mechanical forces can influence cell division and migration [67]. These and other questions require further investigation and the development of more specific models.

Author Contributions: N.B. developed computational software and V.V. performed the simulations. N.B. and V.V. wrote the paper. Conflicts of Interest: The authors declare no conflicts of interest.

References

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