MTSs are considered to be a convenient simulation benchmark, as they are easily accessible to observation and represent a good in vitro model of the first stage ...
Mesoscopic Modeling of Multicellular Tumor Spheroids: Validation through a Quantitative Comparison with Experimental Data M. Griffa♣,†,‡, S. Delsanto♦,♠,‡, L. Morra♦,♠,‡, P.P. Delsanto♣,†,‡ Dept. of Physics♣ and Dept. of Computer Engineering♦ Politecnico of Torino Corso Duca degli Abruzzi 24, 10129, Torino (Italy) INFM, Research Unit of Torino-Politecnico† and of Torino-University♠ ‡ LIMA Bioinformatics, Bioindustry Park of Canavese, Via Ribes 5, 10100, Colleretto Giacosa (TO) (Italy) Abstract A recently proposed mathematical model for the growth of Multicellular Tumor Spheroids [1] is here improved in its formulation and validated against in vitro experimental data, in order to analyze the role of its parameters and estimate their values. Large scale numerical simulations have been conducted for the best fitting of a set of in vitro data concerning a single neoplastic cell line (EMT6/Ro), in order to obtain a quantitative parameter assessment. Key-Words: tumor growth; multicellular tumor spheroids; mathematical modeling; numerical simulations; tumor biology.
1 Introduction The modeling of tumor growth mechanisms, as a fundamental issue in oncological research, has recently assumed an increasingly significant role. In fact, the exploitation of in silico simulations leads the way to a truly quantitative comprehension of tumor biology [2,3], as a prerequisite for the optimalization of tumor therapy protocols [4,5]. Ideally, the goal is to devise models, which not only should allow a realistic prevision of tumor behavior (e.g. remission or latency vs. indefinite growth), but which should be capable of relating mechanisms such as cellular reproduction, or cell migration, with clinical observations. This would not only allow a more effective therapy, but could also lend insight on the relative role of the different mechanisms affecting tumor growth behavior. In this scenario, “mesoscopic” or “intermediate-scale” models emerge as a particularly suitable simulation tools. They may be based on a Local Interaction Simulation Approach (LISA) [6], which implements a spatio-temporal discretization of the system and thereby considers the interactions and mean behaviors of groups of cells “contained” in each simulation gridcell [7,8]. Typical orders of magnitude for the number of cancerous cells in each grid cell are around 500 to 1500 cells for capturing the interrelated processes at a “cellular mesoscopic scale”, i.e. processes involving small amounts of cells. Mesoscopic models are intrinsically equipped to relate individual cell mechanisms with the overall state of the tumor, thus enhancing our knowledge of the relations between cellular level conditions and prognosis. They also yield informations about system-level features, such as the volume, surface morphology or mass density of the whole tumor mass,
which are usually the only data whose temporal changes can be detected with current available imaging techniques. Qualitative results in this sense have in fact already been achieved [9]. However, in order for these relations to be quantitatively, it is imperative to have a realistic evaluation of the parameter values. It should be noted that the correspondence between the theoretical mesoscopic parameters and the biological ones is not always too clear. Thus it is often difficult to infer from observational data not only their values, but even their order of magnitude [1,10]. We have recently proposed [1] a preliminary mesoscopic model, which aims at minimizing the uncertainty due to the large number of unknown parameters by considering only the most relevant mechanisms in tumor growth, such as cell proliferation, death and migration rates, as related to nutrients availability. In this paper, we draw the first step towards a quantitative model by deriving the relations which connect the model parameters to experimentally obtained values. Also, the model of Ref. [1] is modified in order to account for density-driven passive migration (mass transport) and inhibition of migration due to extreme nutrient unavailability. In Section 2, the biophysical properties of the system are illustrated together with their implementations in the model. Then, the relations between the model parameter values and the experimentally obtained data are explicited. In Section 3 the simulation setup and results are illustrated and discussed, while in Section 4 some conclusions and suggestions for further improvements are drawn.
2 The Model In order to facilitate a comparison with experimental data, we have chosen to model tumor growth in the
avascular phase. We have also chosen to model Multicellular Tumor Spheroids (MTS), which are composed by aggregates of malignant cells, cultured in vitro after implantation in a gel matrix. MTSs are considered to be a convenient simulation benchmark, as they are easily accessible to observation and represent a good in vitro model of the first stage of avascular tumor growth. It is possible to obtain MTS cultures in gel matrices with different agar concentrations [11], thus mimicking different micro-environmental stress conditions. Our model is based on a spatio-temporal discretization of the tumor growth system. Exploiting the MTS spherical symmetry, the space is divided into N+1 concentrical and isovolumetric gridcells [1], henceforth called shells, each having an external boundary surface at radial distance rn (n = 0,1, …, N) from the center of the lattice. Time is also discretized with step τ. In the basic model only two kinds of populations, i.e. viable and dead cells, are considered. For each generic shell n (n = 0, …, N) the following quantities are defined: cn ≡ number of cancerous cells, dn ≡ number of dead cells and νn ≡ number of nutrient units. An initial cancerous seed is assumed present in the innermost shell at the beginning of the simulation (t = 0). In order to limit as much as possible the number of unknown parameters, while retaining a good description of the system behavior, we have restricted the number of interaction mechanisms to be included to only four: reproduction, migration, death and feeding [10,12-13]. We assume that nutrients’ consumption for each shell is proportional (with coefficient γ) to the number of cancerous cells in it. The other three mechanisms are enacted by the cells in each shell depending on the levels of locally available nutrient, which lends the energy necessary to execute the processes. Specifically, if the nutrient is above a given threshold QR the cancerous cells proliferate with probability per unit time ρ. As the nutrient level falls, the cells are initially quiescent (i.e. not capable of reproducing, yet alive). If the nutrients in the shell further decrease below a threshold QM, cells migrate with probability per unit time per unit surface µ towards the adjacent shells in search of “greener pastures” (chemotaxis). Finally, nutrients below death threshold QD (QD< QM< QR) cause the interruption of the migration process (cells no longer have the energy necessary to migrate) and cell death occurs with probability per unit time δ. All these mechanisms are implemented by assigning to the n-th shell (n = 0, …, N) the probabilities, which are functions of the nutrients level. In particular, the reproduction (ρn) and the death (δn) probabilities per unit time are modeled with sigmoidal functions, and tend asymptotically to their maximum values ρ and δ. The sigmoidal functions are very steep, (almost a step function), and were introduced in order to ensure a greater numerical stability of the code. Likewise, the chemotactical migration
probability per unit time per unit surface, µn, is obtained by a smoothed rectangular pulse (also modeled by two sigmoids, see Fig.1-a). Nutrients diffuse through the medium from one shell to the adjacent ones with probability per unit time per unit surface α. Since we are considering MTSs cultured in free suspensions, we do not introduce mechanisms accounting for the influence of external pressure on the growing spheroid. However, the reciprocal pressure exerted by adjacent cells is modeled by assuming that above a given cellular density threshold migration by mass transport occurs, with the same probability µ. The mass transport migration probability is again a sigmoidal function, here depending on the density, i.e. on the number of cells (see Fig. 1-b). Thus, in case of high density, migration may occur even in abundance of nutrients. The resulting migration probability is then given by the higher between the chemotactical and stress-driven ones. The equations describing the system are then:
c = cn (1 − τδn + τρn )+ (r0 ) (n) 2
x n
+ (r0 ) (n +1) 2
d n =d n x
2 3
(τµn−1cn−1 − τµn cn ) +
(1)
(τµn+1cn+1 − τµn cn ) n
cn
(2)
νnx = νn − τγcn + τα(r0 ) (n) 2
+ τα(r0 ) (n +1) 2
2 3
2 3
2 3
(νn−1 − νn ) +
(3)
(νn+1 − νn )
Fig. 1 – Migration probabilities per unit time per unit surface, due respectively to chemotaxis (a) and stressdriven mass transport (b).
where all quantities are defined at time t, except when an asterisk denotes time t+1. The two diffusion terms refer to the migration exchange with the inner and outer adjacent
shells respectively, and are proportional to the interface surfaces, hence the (n)2/3 and (n+1)2/3 coefficients.
3 Numerical simulations for the validation of the model In order to validate our model and arrive to a quantitative evaluation of the parameters, we have chosen a specific set of experiments performed by J.P. Freyer, R.M. Sutherland and co-workers [14-19]. Our choice was motivated by the following reasons: • the experiments were performed with the use of MTSs derived from only one cellular line, the EMT6/Ro mouse mammary carcinoma line; • similar experimental techniques were adopted in all the quoted experiments; • the experiments were devised in order to evaluate several relevant variables of the MTS growth dynamics. Among them: the temporal evolution of the volume and the total number of cells [14], the thickness of the viable rim at different times [14], the temporal evolution of the fraction of cells in different cycle phases [15], the diffusivity of metabolites (specifically glucose) in the inner regions of the MTS [19]. Concerning the in silico experimental setup, a grid lattice has been adapted with a radius rN = 600 µm for the outer shell (index n = N). Such a value is larger than all the MTS radii experimentally considered up to 500 µm [14]. Thus no problems due to boundary conditions could arise. The mean mesoscopic number of cell, cm, is taken at 500 in order to consider cellular interactions among sufficiently large homogeneous amounts of cells with different properties. Following Ref. [14] and [18], we assume a spheroid “density” of mV ≈ 2.125 * 108 cells/cm3. We then select the value of the innermost shell radius ro, assuming that under normal equilibrium conditions it contains cm cells. This results in an isovolumetric spherical grid lattice with each grid cell having a volume V0 ≈ 2.346 * 10-6 cm3 and N = 384. The nutrients are assumed to be initially uniformly distributed through the entire grid in normalized units equal to ν0 = 1 per shell. The initial time t = 0 corresponds to a previous time of MTS culture of about 100 h. The diffusion probability α has been arbitrarily assumed to be given by D α= (4) Vo ⋅ ∆ ro where ∆ro is the grid spacing around the innermost shell and D is the Fick-like diffusion coefficient for nutrient molecules. The latter has been fixed at D = 104 h-1 µm2 in
agreement with the order degrees reported in [19] (of course the grid spacing varies when one moves to the outer shells). Correspondingly one finds α ≈ 2.7 * 10-3 h-1µm-2. Another important parameter to be fixed in order to perform a comparison with experimental data is the number Nmax of cells per shell triggering the stress-driven migration process. In particular, we suppose that the migration process, which we assume to be enacted in shells totally occupied by cancerous cells. We model the viable and dead cells as rigid spherical bodies with radii rc and K*rc (rc ≈ 7 µm [14]) respectively, where K is a factor less or equal to 1, since dead cells may not occupy a larger volume than live cells. For simplicity we assume here K equal to 1.Then the stress driven migration process occurs when:
r c n + K d n ≥ o rc so r N max = o rc
3
(5)
3
≈ 1633
(6)
The time step τ is chosen in order to adequately model the nutrient diffusion process, and corresponds to the condition that the maximum number of nutrients units which may diffuse from one shell to its neighbors at each time step is less than the total number of nutrient units in it. As the outermost shells have greater interface surfaces, and consequently higher diffusion terms (the diffusion term is proportional to the surface), the upper bound for τ was obtained by considering the N-th shell. We obtain:
τ=
1 4απ ro N
2 3
≈ 0.0027 h
(7)
for a total time of growth simulation equal to ∆t = T*τ ≈ 315 h. This value for τ is in agreement with the purpose of performing mesoscopic simulations, because it’s too high for following single events at the molecular and sub-cellular levels, too low for macroscopic growth events, but just adequate for tracing inter-cellular interactions and their dynamics. To conclude, we can classify the whole amount of parameters in three different categories: • parameters directly obtained by experimental data: rN, rc, c0, D, mV, d0, ν0, T; • lattice discretization parameters, inferred from a comparison with the experimental set-up and justified by numerical stability requirements: r0, α, N, NMAX, τ, cm;
•
parameters related to measurable variables with biological/biochemical meaning: δ, µ, ρ, γ, K, QD, QM, QR (See Table 1).
Parameter -1
δ = 10 h
-1
µ = 3 * 10-3 h-1 µm-2 ρ = 2.5 * 10-2 h-1
QD = 0.5 QM = 0.6 QR = 0.8 NMAX ≈ 1633
Figure 2: time series for the volume of the spheroid (semi-log scale). Square points: experimental data. Round points with continuous line: best-fitting numerical simulation.
Meaning Probability per unit of time for cell death Probability per unit of time for cell migration Probability per unit of time for cell reproduction Cell death threshold on nutrient concentration Cell chemotactic migration threshold on nutrient concentration Cell reproduction threshold on nutrient concentration Cell stress-driven migration threshold on cell concentration
Table I: cellular features and processes-related parameters
3.2 Results As previously discussed, we have considered the experimental data about temporal evolution of volume and number of viable cells of EMT6/Ro MTSs as the main test set for the model. We have consequently searched the best parameter setup to fit the two time series reported [14]. Fig. 2 shows the comparison between the experimental time series for the volume and the in silico corresponding results. The dynamics can be characterized as an exponential growth with a growth coefficient decaying also exponentially (the so called Gompertzian [20]), a typical of neoplastic spheroids [21].
Figure 3: time series for the total cell number of the spheroid (semi-log scale). Square points: experimental data. Round points with continous line: best-fitting numerical simulation. The experimental data taken from [14] don’t report the error bars.
The discrepancy at the end of the curve (t ≥ 250 h) is due to that for the moment only basic mechanisms are taken into account (e.g. we do not consider a consumption of nutrient depending on local nutrient availability). Fig.3 illustrates the evolution of the number of viable cells vs. time. The discrepancy between experimental and numerical results here is larger, probably due to the fact that in our model the impact of density is only partially taken into account (work to include explicitly the stress dependence of the density is in progress).
Figure 4: radial profile for viable and dead cells (top) and for nutrients (bottom). Snapshopt at time t = 100 h.
Figure 6: radial profile for viable and dead cells (top) and for nutrients (bottom). Snapshopt at time t = 312 h. The onset of the recurrent necrotic death leads to the subdivision of the MTS in a central dead core surrounded by viable cells.
At later stages (Fig. 6), the sudden onset and expansion of the necrotic core (in correspondence with central depletion of nutrients) is associated with a thinning of the external layer of viable cells and a slowing down of their outward expansion, as observed in vitro [14,23]. The model thus confirms experimental observations about the interrelation between the onset of necrosis and of growth saturation [14,24].
4 Conclusion
Figure 5: radial profile for viable and dead cells (top) and for nutrients (bottom). Snapshopt at time t = 200 h.
Fig. 4, 5 and 6 illustrate the spatial distribution of viable cells, dead cells and nutrients at three successive time steps. It may be observed that for a long time the virtual MTS grows without a compact dead core (Fig. 4 and 5), with a kind of ‘wavefront’ of viable (proliferant) cells moving outwards, as confirmed experimentally [14,22].
The main goal of this work has been the validation of the proposed basic model for MTS growth. This has been achieved by reducing the space of unknown parameters, both through inference and comparison with experimental observations and by a careful selection of the basic mechanisms regulating tumor growth. This work thus represents a first step in the direction of designing models directly related to the experimental reality through a quantitative determination of all parameters. Future work will be in the direction of progressively introducing additional mechanisms regulating tumor growth, such as the influence of local stress and/or local heterogeneity in nutrients consumptions. In this work a single cellular line has been analyzed. It is conceivable, however, that different cellular lines might require other values of the parameters. Evolutions of the in silico model might also lend insight into the different behavior of different cellular lines. Finally, in the analysis of the interdependence among different mechanisms regulating growth, our model emerges as an interesting tool, both due to its closeness with the experimental reality and to the significant computational efficiency tied to its one dimensional representation of the three dimensional space. Work in progress concerns the application of genetic algorithms to the exploration of the parameter space, an approach which would hardly be feasible with a classical three dimensional representation.
Acknowledgements We wish to thank Dr. M. Scalerandi, for fruitful discussions and suggestions. We are also grateful to Dr. GP. Perego, Aethia s.r.l., Bioindustry Park of Canavese. for technical assistance in the use of the Beowulf PC cluster. References: [1]: Delsanto, P.P., Morra, L., Delsanto, S., Griffa, M., Guiot, C., Towards a Model of Local and
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