A SINGLE CELL APPROACH IN MODELING THE DYNAMICS OF TUMOR MICROREGIONS

Interactions between tumor cells and their environment lead to the formation of microregions containing nonhomogeneous subpopulations of cells and steep gradients in oxygen, glucose, and other metabolites. To address the formation of tumor microregions on the level of single cells, I propose a new two-dimensional time-dependent mathematical model taking explicitly into account the individually regulated biomechanical processes of tumor cells and the effect of oxygen consumption on their metabolism. Numerical simulations of the self-organized formation of tumor microregions are presented and the dynamics of such a process is discussed.

1. Introduction.Most tumors in vivo arise from a single cell that has escaped the growth controlling mechanism.However, even at very early stages of their growth, tumors become highly non-homogeneous.Fast growing tumor cells can significantly change their environment leading to formation of gradients of different metabolites, such as oxygen, glucose, growth factors and other nutrients.Changes in metabolites concentration cause in turn the development of microregions occupied by tumor cells of different phenotypes, such as proliferating, quiescent or necrotic cells.Those subpopulations of cells are characterized by different growth and functional properties as well as by diverse responses to therapeutic factors.The biology of tumor microregions has been investigated experimentally using the multicell spheroid model (for review see [12], [13], [18]).In experimental setting, the spheroids are usually initiated from aggregates consisting of several cells, but as their size increases, their growth kinetics become similar to those of tumor in vivo, such as micrometastases or pre-vascular primary tumors.The multicell spheroids develop layered structure with a central necrotic core surrounded by quiescent cells and a tin rim of proliferating cells.Steep gradients in oxygen, glucose and other metabolites are also observed in such spheroids.Because of similarity to tumor in vivo, multicell spheroids are appropriate models to study novel antitumors therapies and they become mandatory test systems in therapeutic screening programs (for review see [5], [14]).
In order to address the formation of tumor microregions, we present a twodimensional time-dependent mathematical model in which every tumor cell is treated as an individual entity characterized by its own geometry and individually controlled cell processes.The formation of tumor clusters and tumor microregions in this model depends entirely on the local dynamics of separate cells and on the cellcell interactions.This model allows one to follow fate of each individual cell and to investigate how changes occurring in individual cells can influence behavior of the whole tumor tissue.We have applied this model previously to study formation of tumor clusters with random distribution of proliferating cells, not dependent on any external factors [17].In this paper we extend our model by relating the progression of cell processes to the external sources of energy, such as oxygen, glucose or other nutrients.For simplicity, we introduce in our model only one external metabolic factor and take explicitly into account only the effect of oxygen consumption on cell growth and metabolism.We present two-dimensional simulations showing the selforganized formation of layered structure of tumor microregions occupied by cells with different phenotypes.We describe the dynamics of microregions development and present comparison with experimental data.
The presented single cell approach in modeling the avascular phase of tumor growth is significantly different from previous mathematical models (for review see [1] and [9]), such as the continuum and multiphase models which both view tumors as cell densities and the discrete-continuous hybrid models or cellular automata models which both view tumor cells as single points.Our model allows for a more realistic representation of a single cell and individually regulated cell processes.It also allows for a more realistic representation of the tumor tissue as a conglomerate of cells non-homogeneous in shape and function.This model is suitable to investigate those biological phenomena that occur on the scale of from one to several hundreds of cells.Therefore, it can be used to bridge research conducted on different scales.Results of intracellular research can serve as source of parameters employed in our model and similarly results obtained form our model can be used in defining attributes of the large-scale approaches, such as the hybrid-type or continuum models.
2. Mathematical formulation.Our mathematical model is based on the immersed boundary method [15] and captures interactions between immersed elastic tumor cells and a viscous incompressible fluid, representing the cytoplasm inside the cells and the extracellular matrix outside the tumor tissue.The fluid flow is influenced by sources of fluid located in the nuclei of growing cells, as well as forces generated by the immersed, elastic boundaries, while at the same time the elastic structures move at the local fluid velocity.The cell cycle and cell processes are related to the concentration of external factors, such as oxygen, sensed by the cell from its local environment.
Our model is defined on the square two-dimensional domain Ω.The elastic membranes of the tumor cells form a collection Γ of closed curves defined in the curvilinear coordinates X(l, t), where l is a position along the cell boundary.We model the fluid inside and outside the cells as a homogeneous continuum with the same constant density ρ and viscosity µ.The cell nuclei form a finite set Ξ of material points Y k in which sources of fluid S(Y k , t) are located if the host cells are growing.Oxygen concentration γ(x, t) is defined in a domain which coincides with the fluid domain.The rate of change of oxygen concentration depends on oxygen consumption by the tumor cells and on its diffusion.The complete system of equations describing the interaction between tumor cells, the extracellular matrix and concentration of oxygen is given in Equations 1-9.
In this system, Equations 1-2 are Navier-Stokes equations of a viscous incompressible fluid.Equation 1 is the Newton's second law, where mass density times the material acceleration is equal to the sum of various force densities arising from the fluid pressure p, the viscosity µ, the local fluid expansion s, and the external force f .Equation 2 is the law of mass conservation, where the source distribution s is identically equal to zero on the whole fluid domain except at the isolated points sources which are used to model the growth of cells.It is required that the conservation of mass is preserved globally in the fluid domain Ω at each time t, i.e. ρ Ω (∇ • u) dx = 0.
Equations 3-6 represent interactions between the fluid and the immersed points of cell boundaries and cell nuclei.In Equations 3-4, the force density defined on cell boundaries and sources defined at the cell nuclei are applied to the fluid, while in Equations 5-6 all material points are carried along with the fluid.The Dirac delta function in these equations is two-dimensional, thus δ(x) = δ(x 1 ) • δ(x 2 ).Equation 7states that the boundary force F(l, t) is determined by the boundary configuration at time t, where the function F satisfies a generalized Hooke's law.The boundary forces include the adjacent, adhesion and contractile forces and will be described bellow.Equation 8 states that the source distribution S(Y k , t) at time t is determined by the locations of the nuclei of all growing cells, where the function S is a step function that takes non-zero values only during the time of cell growth.In order to satisfy conservation of mass globally, the balancing sinks of fluid are located at the fixed positions on the domain boundary.
Equation 9 is a diffusion-reaction equation defining the rate of change of oxygen concentration γ.We assume that oxygen is not carried with the moving cytoplasm and its dynamics comes only from diffusion and consumption near the boundaries Γ of the tumor cells, i.e. within the cell microenvironment Θ Γ .Here χ is a set characteristic function.
2.1.Cell structure.A eukaryotic cell in our model consists of the elastic membrane enclosing the fluid cytoplasm and the point nucleus.The elasticity of the plasma membrane comes from the fact that each point X(l, t) on the cell boundary is connected to its two adjacent neighboring points X(l + 1, t) and X(l − 1, t) by a linear spring F adj satisfying Hooke's law with the constant resting length L adj and constant spring stiffness F adj : 2.2.Cell-cell adhesion.Separate cells can adhere to their neighbors by using the transmembrane adhesion proteins, which are represented in our model as short linear springs attached at the boundary points X(l, t) and X(k, t) of two distinct cells if they are within the distance L max adh .The adhesive force F adh satisfies Hooke's law with the constant resting length L adh and constant spring stiffness F adh : provided X(l, t) − X(k, t) ≤ L max adh and is zero otherwise.Figure 1 shows a small cluster of eukaryotic cells with boundary points (stars), cell nuclei (circles), linear springs between adjacent boundary points (thin lines) and adhesive connections between distinct cells (thick lines).2.4.Cell life cycle.All cells are consider viable and ready for proliferation unless the concentration of oxygen in the cell microenvironment falls below a certain level γ min when the cell become necrotic.All viable and proliferating cells consume oxygen dissolved in their microenvironments on the rate which is relatively independent of oxygen concentration until the very low oxygen levels are reached.At such concentrations the rate of oxygen consumption decreases with decreasing oxygen concentration [3].We model this process in Equation 9 by using the Michaelis-Menten formulation.A process of cell proliferation is initiated periodically in a viable cell with the highest level of oxygen in its vicinity.This process is completed by producing two viable daughter cells.All necrotic cells remain inside the tumor cluster, but stop participating in oxygen consumption.
2.5.Cell proliferation.A process of cell proliferation is modeled by introducing two point sources inside the cell, that correspond to two sets of sister chromatids (Figure 2a).The fluid created by both sources causes the cell growth by pushing cell boundaries and by increasing cell area (Figure 2b).The sources are inactivated when the cell area is doubled.At this time, the contractile ring is created by attaching the contractile forces on the opposite sites of the cell boundary.This results in a formation of the contractile furrow (Figure 2c) and causes division of the cell into two daughter cells of approximately equal areas and with their own nuclei (Figure 2d).The strength of each fluid source has a non-zero value S 0 over the time of cell growth, but once the cell area is doubled, the fluid sources are inactivated.The value of S 0 is determined computationally or experimentally, to match dynamics of the growing living cell.
The contractile forces F div are defined between two opposite boundary points X(l, t) and X(k, t) of the same cell.They act on the cell boundaries until the opposite points reach a distance L min div at which point the dividing cell is split into two daughter cells.Each contractile force satisfies Hooke's law with the constant resting length L div and constant spring stiffness F div : provided X(l, t) − X(k, t) ≥ L min div and is zero otherwise.The value of the spring stiffness constant F div is determined computationally in a way to resemble the dynamics of cytokinesis in the living cell.
2.6.Necrotic cells.A change in cell phenotype from viable to necrotic depends entirely on the level of oxygen concentration in the cell vicinity.Once the oxygen concentration in the cell microenvironment falls below a certain level γ min , this cell looses its ability to proliferate and becomes necrotic.
3. Numerical Implementation.We implement our model using the finite difference schemes for discretizing the Navier-Stokes equations and the reaction-diffusion equation.We use the same regularly spaced fluid and oxygen grid.Interactions between those grids and the material points on cell boundaries and at cell nuclei are implemented using the discrete approximation δ h (x) to the Dirac delta function, where δ h (x) = δ h (x 1 ) • δ h (x 2 ) and 3.1.Algorithm.The numerical algorithm for the full coupled system can be outlined as follows.At the end of time step n we are given values of the fluid velocity field u n , the concentration of oxygen γ n , the configuration of the immersed boundary points X n on the cell membranes and positions of cell nuclei Y n .These values are updated at the next time step in the following way: 1. Inspect the concentration of oxygen γ n in the microenvironments of all quiescent cells to determine which cells become necrotic; update the levels of oxygen concentration γ n+1 by computing rates of its consumption in the vicinity of all quiescent and proliferating cells.2. Determine the source distribution S n at the nuclei of growing cells and the balancing sinks at the boundary of fluid domain; spread these values to the neighboring grid points to find the local growth rate s n of the fluid.3. Calculate the total force density F n at the cell boundaries, including the membrane adjacent forces, cell-cell adhesion forces and contractile forces in the dividing cells; spread these values to the neighboring grid points to determine the forces f n on the fluid.4. Solve the Navier-Stokes equations for the fluid velocity field u n+1 by using the fast Fourier algorithm.5. Interpolate the fluid velocity field u n+1 to each immersed boundary point on cell membranes and to each cell nuclei; compute new positions of material points X n+1 and Y n+1 by moving them at the local fluid velocity using a forward Euler step.More details about an numerical implementation of this algorithm and the numerical solution of the Navier-Stokes equations, can be found in [16] and [17].

3.2.
Choice of computational parameters.Our model of growing tumors is defined on a two-dimensional square domain corresponding to the tissue of size 0.7mm × 0.7mm.This domain is discretized using the 512 × 512 grid, which gives the mesh width h = 1.37µm.Resting lengths of all adjacent L adj , adherent L adh and contractile springs L div are equal to the half of mesh width h/2, whereas radii of all adhesion L max adh and contractile L min div microenvironments are equal to the mesh width h.The time step used in all simulations is equal to ∆t = 0.001 seconds.
Fluid viscosity and density have been chosen to reflect the rheological properties of the network part of the cytoplasm as reported in [10] and [4].Stiffness of the adjacent springs in the cell membrane corresponds to rigidity of the cortical cytoskeletal network associated with the plasma membrane [10].Stiffness of the adherent connections in normal cells has been reported in [2], however, we choose a smaller value for our computations, since the tumor cells are known to have weaker adhesion connections than those in normal cells.Strength of sources in the growing cells and stiffness of the contractile forces have been determined computationally in our previous work [17] to match the dynamics of growing and dividing cells, respectively.The values of oxygen concentration and consumption rates have been reported in the series of experimental papers on the in vitro growth of the multicell spheroids ([6], [7], [8], [3]).The value of oxygen diffusion in the living tissues has been reported in [19].The values of all physical parameters used in our model are summarized in Table 1.

Physical quantity
Computational Table 1.Numerical values of computational parameters.
4. Computational results.We start our numerical simulations by placing a small cluster of seven viable tumor cells at the center of the computational domain.
Initially the whole domain is filled uniformly with oxygen of optimal concentration γ 0 .Periodically, the process of proliferation is started by a viable cell with the highest concentration of oxygen in its vicinity.The tumor cell grows until its area is doubled, then it is divided into two daughter cells.All viable and proliferating cells consume oxygen dissolved in their environment.If oxygen concentration falls below the γ min level in the vicinity of a viable cell, this cell becomes necrotic.Therefore, the faith of each tumor cell depends entirely on the evolution of oxygen concentration in the cell microenvironment.All computer simulations have been carried over the period corresponding to 14 days of tumor development.The results discussed below show formation of tumor microregions occupied separately by cells with different phenotypes and characterized by different concentrations of oxygen.time, the percentage of quiescent cells decreases to about 40% on the day 14th.A subpopulation of necrotic cells arises first around the fifth day of tumor growth and is characterized by a fast exponential expansion.The percentage of proliferating cells starts to decrease with the increasing subpopulations of quiescent and necrotic cells.It reaches less than 10% of all tumor cells at the end of the 14th day.This general pattern in formation of subpopulations of tumor cells, predicted by our model, is consistent with results of biological experiments reported in ([6], [7], [8]).In particular, Figure 3(a) corresponds to pre-vascular tumor growth shown in Figure 2 of [12].4.2.Spatial distribution of tumor cell subpopulations.Figure 3(b) presents the temporal evolution of tumor microregions, while he spatial distribution of emerging tumor cell subpopulations over the same time of 14 days is shown in Figure 4. Our model predicts that initially the distribution of proliferating cells is uniform in the whole tumor cluster.After about four days, however, the populations of quiescent and proliferating cells become spatially separated.The outer rim of actively mitotic cells is formed, while the quiescent cells occupy the center of the cluster.The first necrotic cells arise in the center of the tumor cluster after five days of its growth.Further development leads to expansion of the necrotic core, while the surrounding layer of quiescent cells has relatively stable radius-around 48 − 50 µm, Figures 4(f)-(h).The outer rim of proliferating cells has a width of a few tumor cells.These predictions of our model are consistent with biological experiments ([6], [7], [8]).In particular, the three-layered structure of the multicell spheroid is shown in Figure 2 of [11].4.3.Development of oxygen gradients.The spatial distribution of tumor cell subpopulations depends on oxygen concentration sensed by the cells from their microenvironment.Initially this concentration is uniformly equal to 0.28 mM , however, extensive oxygen consumption by the quickly growing tumor cells creates a steep gradient of oxygen in the tumor tissue and its surroundings.Figure 5 shows spatial distribution of oxygen concentration after 14 days of tumor development that corresponds to the shape of tumor cluster presented in Figure 4(i).Dark blue region in the center of the domain corresponds to the area of high oxygen reduction (less than 0.01 mM ) where the necrotic tumor cells are located.Dark red region far from the tumor cluster contains oxygen of high concentration (more than 0.25 mM ).The well pronounce gradient of oxygen is visible around the rim of proliferating cells (the green-yellow region).
The representative profiles of oxygen concentration taken at four different time points during the tumor development are shown in Figure 6.These profiles are captured along a horizontal line crossing the center of the domain.All profiles are characterized by decrease in oxygen concentration in the region directly surrounding the tumor cluster, thus lowering the concentration of oxygen at the tumor surface (indicated by vertical arrows).In small tumor clusters, the shapes of oxygen profiles are parabolic (graphs a and b).In larger clusters, gradients are steeper and show central plateau with near-zero concentration of oxygen (graphs c and d).The profiles of oxygen concentration presented in Figure 6 are analogous to the profiles of partial oxygen pressure in multicell spheroids discussed in [11] and [12].Since those two quantities are proportional by Henry's law, our predictions are consistent with biological experiments.

5.
Discussion.We have presented a new, single cell approach in modeling the formation and development of tumor microregions.The novelty of our model reveals in treating every cell as a separate entity with its own geometry and individually regulated cell processes.Development of the whole tumor cluster is a result of local dynamics of separate growing cells that need to compete for space and resources with their neighbors.This model takes explicitly into account the effect of oxygen consumption on cell metabolism.It also includes the effects of cell-cell interactions on the geometry of separate cells and the whole tumor tissue.
We have applied this model in numerical simulations of the oxygen-dependent growth of tumor clusters that can be considered as models for micrometastatic or avascular primary tumors.We have shown that predictions of our model regarding the dynamics of tumor growth, the emergence of tumor cell subpopulations, the temporal and spatial evolutions of tumor microregions and the profiles of oxygen concentration, all are biologically adequate.This approach allows one to follow the faith of a single tumor cell in the cluster or the whole clone of cells arising from a common predecessor.As an illustrative example we present evolution of seven color-coded clones emerging from single cells shown in Figure 7(a).The final configuration of separate clones, containing about 900 cells in total, is presented in Figure 7 7(b) is in extinction, because it has emerged from a centrally located predecessor cell and due to this location all red cells needed to compete for space and oxygen with all other clones.Six other clones have arisen from symmetrically located predecessor cells, however, they did not develop uniformly.Some of them have been dominated by their neighbors during the tumor growth, such as the light-blue clone which probably will not survive during the further development of the tumor cluster.This asymmetry in clonal development follows from the fact that cells with higher concentration of oxygen in their vicinity proliferate more often, allowing tumor cells to spread widely in search for well-oxygenated environments.This may be an indication of tumor aggressiveness.
The possibility to observe a lifespan of each separate cell and a faith of all its daughter cells in many succeeding generations gives us opportunity to model biomechanical consequences of cell genetic mutations.By altering properties of a single cell or of a small group of cells (for instance, the cell doubling time can be shorten or elongated, cell resistance to hypoxia can be altered or cell-cell adherent connections can be loosen or strengthen) and assuming that all daughter cells will carry on those genetic mutation, we can observe changes in the tumor growth and its responses to the external factors.
We have presented here a very simple situation with only one source of energy regulating life of each tumor cell, whereas biological processes in living cells involve not only oxygen concentration, but also other metabolites, such as glucose, lactate or pH.In our further research, we are planning to investigate growth of tumors taking into account cell behavior in more complex environments and cell responsiveness to more than one metabolic factor.
Our model allows also for investigation of other biological phenomena occurring on the scale of from one to several hundreds of cells.It can be used as a intermediate model between the intracellular and large-scale models.Results of intracellular research can serve as source of parameters employed in our model and similarly results obtained form our model can be used in defining attributes of the largescale approaches, such as the hybrid-type or continuum models.Therefore, the computation approach presented in this paper can bridge the micro-and macroscale models.

Figure 1 .
Figure 1.A cluster of eukaryotic cells connected by adhesive links.Each cell consists of the nucleus (circle) enclosed by the plasma membrane (stars) connected by linear adjacent springs (thin lines).

2. 3 .
Cell microenvironment.All boundary points located on the cell membrane play an additional role of cell membrane receptors.They are used by the cell to sense concentration of oxygen from the cell microenvironment Θ Γ = X∈Γ {x : ||x − X|| < ε} defined as a set of points lying within a distance ε from the cell boundary Γ.

Figure 2 .
Figure 2. Main phases of the cell cycle.(a) nucleus division, (b) cell growth and elongation, (c) formation of the contractile ring, (d) cellular division and formation of two daughter cells.

4. 1 .Figure 3 .
Figure 3. Dynamics of tumor growth.(a) Three-phased evolution of the tumor area-fast expansion, growth reduction and growth saturation.(b) Development of tumor cells subpopulations-percentage of necrotic (N), quiescent (Q) and proliferating (P) cells in the whole tumor cluster.

Figure 4 .
Figure 4. Spatial development of three tumor microregions containing proliferating cells (red), necrotic cells (blue), quiescent cells (gray).Time of growth and tumor diameters are shown.

Figure 5 .
Figure 5. Spatial distribution of oxygen concentration after 14 days of tumor development corresponding to the shape of tumor cluster shown in Figure 4(i).Oxygen concentration measured in mM , distance in cm.

Figure 7 .
Figure 7. Seven color-coded predecessor cells (a) give rise to asymmetrically developed clones shown after 14 days of tumor growth (b).