# Hybrid Cellular Automata Modeling Reveals the Effects of Glucose Gradients on Tumour Spheroid Growth

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## Abstract

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## Simple Summary

## Abstract

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Model Development

#### 2.1.1. Domain Building

#### 2.1.2. Glucose Layer

**x**= (x,y) and time t. The dynamics of the glucose concentration field C(

**x**,t) in time and space is determined by solving the classical Fickian reaction–diffusion equation:

**x**,t) is the consumption rate of glucose by the cells. The computational cost to obtain the numerical solution of Equation (1) can thus be high, given the considerable disparity between the time scales of cell division (hours to days) and glucose diffusion (seconds); cellular proliferation was treated as an adiabatic perturbation in the chemical field [29]. Thus, using the adiabatic perturbation approximation, Equation (1) is approximated as a pseudo-stationary problem [39]:

_{x,y}as:

_{act}, y

_{act}) and (x

_{star}, y

_{starv}) are the coordinates of the LPs where active and starving cells are located. The use of the Dirac delta function allows for the representation of a point-like object, in this case, the presence of a cell in a specific LP.

#### 2.1.3. Cellular Layer and Cell Phenotypes

_{starv}> C

_{nec}). A cell was active if the local glucose concentration C

_{x,y}(i.e., concentration in the LP occupied) was higher than both these predefined thresholds, C

_{x,y}> C

_{starv}> C

_{nec}. An active cell consumed glucose according to Equation (6) and its metabolism included the possibility to migrate and eventually duplicate upon the completion of its cell cycle.

_{nec}< C

_{x,y}< C

_{starv}, the cells did not have enough nutrients to duplicate, and thus starved. The cells still consumed residual glucose and could migrate, looking for more nutrient-rich LPs. As time progressed, if C

_{x,y}increased back to levels higher than C

_{starv}, a starved cell could become active again, unless it remained in the starved state for too long a period, going in apoptosis (see subsequent section).

_{x,y}< C

_{nec}, the cell underwent necrosis. This process is irreversible, independent of any future change in the glucose concentration, and from that time on, the necrotic cell cannot proliferate, migrate, or consume glucose.

#### 2.1.4. Rules Governing Cellular Dynamics

_{m}and duplication time T

_{d}.

- (a)
- Cell migration

_{m}, a cell can migrate. The probability of the cell to migrate is affected by cell–cell interactions if the cell under evaluation is attached to a cell cluster and not isolated (detached). If a migration event occurs, the cell occupies one of its eight neighboring LPs, provided the target location is empty (Figure 1b); alternatively, if migration does not occur, the cell remains in the same position until the subsequent time interval, T

_{m}.

_{adh}, defined as:

_{m}. If $\sigma >{P}_{\mathrm{a}\mathrm{d}\mathrm{h}}$, migration is allowed and a change in the cell adhesion state can be induced; in our model, for simplicity, we assume that cell detachment is irreversible, i.e., once detached, a cell is not allowed to re-attach. If the cell of interest is already detached $\left({P}_{\mathrm{a}\mathrm{d}\mathrm{h}}=0\right)$, it shows the typical behavior of isolated cells and is expected to migrate every T

_{m}.

_{5}), is associated with a numeric interval ${R}_{i}=\left[{a}_{i-1},\text{}{a}_{i}\right[$. The 9 intervals have different sizes $(\left|{R}_{i}\right|={a}_{i}-{a}_{i-1})$, are contiguous and non overlapping, and span the entire range 0–1 ($0<\left|{R}_{i}\right|<1,\text{}\sum _{i=1}^{9}\left|{R}_{i}\right|=1)$ (Figure 1c). A further random number $0\le \lambda \le 1$ is generated and compared to the ${R}_{i}$ intervals; if $\lambda \in {R}_{i}$, the cell migrates towards direction ${P}_{i}$, if the corresponding LP is empty. In particular, if $\lambda \in {R}_{5}$, i.e., the LP already occupied by the cell, the cell does not migrate. If the cell, upon arrival at its new location, is not contiguous with any attached cells, it undergoes a transition into the detached state, provided it is not already in that state.

- (b)
- Cell proliferation

_{d}, active cells proliferate, generating a new daughter cell, identical to its progenitor. The new cell is randomly located in one of the free positions among the eight LPs surrounding the progenitor. If no empty spot is available, the new cell’s location is chosen by identifying the direction where the minimum number of cells separate the progenitor from the edge of the cluster. All the cells along the selected direction shift one position away from the progenitor cell, and the new-born cell occupies the vacancy. It is worth mentioning our model considered the memory of the history of each cell to consider the possible cell-dependent variables, such as the random mutations of cell parameters, even if, in this work, this feature was not used. In each time step, all the cells were considered the same, with the only differences being among the active, starved, and necrotic phenotypes, while cell dynamic evolution was dependent on the nutrient availability only. The code implemented in our model also allowed for more complex interactions. Another simplification of our model was that only active cells attached to the spheroid could duplicate, while detached cells were expected to enter irreversibly in a migration state, unless nutrient availability induced their deaths.

- (c)
- Cell death

_{apop}), it could spontaneously die. This biological event, known as apoptosis [41], is a mechanism of defense of the cells to prevent the propagation of lesions to the future generation. In our model, apoptosis corresponded to the degradation of the cell, which was dissolved in the ECM, and left the previously occupied LP empty. It is worth mentioning that, in the case where the glucose concentration reduces further to values below C

_{nec}, while the cell is already in a starved state, the cell does not enter a state of apoptosis, but evolves into a necrotic state, where it remains indefinitely. Necrotic cells, according to our model, do not proliferate nor migrate, but continue indefinitely to occupy the same LP. The flowchart of the whole cell dynamic algorithm is presented in Figure 2. The model implementation of cell dynamics is also summarized as a sequence of operations in the figure caption.

#### 2.1.5. Analysis

#### 2.2. Statistical Analysis

_{d}, T

_{m}, K, and α.

#### 2.2.1. Average and Variance Convergence

_{d}, T

_{m}, K, and α) were not enough to guarantee the statistical significance of the results. Therefore, for each set of input parameters, the simulation was replicated $n$ times, defining the actual outcomes as the arithmetic average of the n iteration. To identify a value of $n$ reliable from the statistical point of view, a convergence analysis was performed to study the effect of the number of simulation replicates on the model outputs (see Supplementary Materials, Figure S2). Convergence was verified on 4 different sets of model parameters, for the gradient experimental condition, running up to 100 simulations for each set. For n ≅ 10, all the model outcomes became constant and independent on n (data reported in Figure S2) for each of the 4 sets of parameters investigated. Therefore, in the results presented below, each simulation was reiterated and mediated for n = 40.

#### 2.2.2. Global and Sensitivity Analyses

_{d}∈ [10, 60 h] [45,46], and T

_{m}∈ [10, 120 min] [47]).

_{d}, T

_{m}, K, and α), and nine outputs, ${Y}_{i}$, which were ${N}_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}^{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$, ${N}_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}^{\mathrm{m}\mathrm{i}\mathrm{g}}$, ${N}_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}^{E}$, ${N}_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}^{S}$, ${N}_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}^{W}$ ${N}_{\%}^{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$, ${N}_{\%}^{E}$, ${N}_{\%}^{S}$, and ${N}_{\%}^{W}$. This subset of outputs, ${Y}_{i}$, was chosen to be linearly independent, as required by the MLRA. The entire procedure (LHS and MLRA) was repeated 10 times, on 10 different sets of 500 combinations of the input parameter ${X}_{j}$ value combinations. As result, four regression coefficient distributions were obtained. Each distribution (one for each input parameter, ${X}_{j}$) represented the measure of the sensitivity index (SI) of the respective parameter.

#### 2.2.3. Local Sensitivity Analysis

_{d}[10, 60 h] (basal value T

_{d}= 18 h), and migration time T

_{m}[10, 120 min] (basal value T

_{m}= 30 min). For α, the range was set between −30 and 30 to account for a chemo-repellent effect. The baseline value of 6 was arbitrarily chosen based on trial and error in order to represent a moderate chemotactic force. For K, the range was still between 0 and 4, with a baseline value of 0.2. Furthermore, the baseline value was selected through trial and error to ensure a balanced level of adhesion strength. For each parameter, 150 values were sampled uniformly and randomly within the specified range. Also, in this case, the LSA was conducted for both the isotropic and gradient cases.

## 3. Results

#### 3.1. Baseline Case

_{d}= 18 h, and T

_{m}= 30 min) both under isotropic and gradient conditions.

#### 3.2. Global and Sensitivity Analysis (GSA)

_{d}and K (Figure 5c–e), with the former being the most relevant, while ${N}_{\%}^{E}$, ${N}_{\%}^{S}$, and ${N}_{\%}^{W}$ were only affected by ${T}_{m}$ (Figure 5g–i). From these observations, we can see that $\alpha $ only influences the outputs related to the directionality of cell migration, solely in the presence of a gradient, while it has no effect in isotropic conditions; parameters K and T

_{d}affect both the size of the spheroid and the number of cells detaching from it, in both isotropic and gradient conditions; and Tm has a reduced effect on spheroid growth compared to K and T

_{d}, but it significantly influences the directionality of cell migration, only in the presence of a gradient.

#### 3.3. Local Sensitivity Analysis (LSA)

#### 3.3.1. Chemotaxis Sensitivity Index α

#### 3.3.2. Cell–Cell Adhesion Parameter K

#### 3.3.3. Doubling Time T_{d}

_{d}mostly impacts the number of adherents ${(N}_{cell}^{core})$ and migrating cells ${(N}_{cell}^{mig})$. Figure 6e,f display the trends of ${N}_{cell}^{core}$ and ${N}_{cell}^{mig}$, evaluated at four different time points (12, 24, 36, and 48 h) as T

_{d}variates. Again, we present the results only for the gradient case. Upon increasing ${T}_{d}$, the ${N}_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}^{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$ count (Figure 6e) rapidly decreases, and a similar, but less marked, trend is followed by ${N}_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}^{\mathrm{m}\mathrm{i}\mathrm{g}}$ (Figure 6f), reconfirming the results of the GSA.

#### 3.3.4. Migration Time T_{m}

_{m}. In the gradient case, the effect of ${\text{}T}_{m}$ is observed by an increase in the cell population percentage migrating West (a higher glucose concentration) when ${\text{}T}_{m}$ decreases (Figure 6h). However, when ${\text{}T}_{m}$ becomes sufficiently large, the four percentages (${N}_{\%}^{N},{N}_{\%}^{S},{N}_{\%}^{E},{N}_{\%}^{W}$) seem to converge to a ~25% value, which can be attributed to the reduced frequency of all cells to migrate.

## 4. Discussion

_{m}had a similar but less effective influence on tumor growth and invasiveness.

_{d}led to a drastic decrease in the numbers of core and invading cells, while perturbing T

_{m}had the opposite effect, accelerating tumor growth.

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Sung, H.; Ferlay, J.; Siegel, R.L.; Laversanne, M.; Soerjomataram, I.; Jemal, A.; Bray, F. Global Cancer Statistics 2020: GLOBOCAN Estimates of Incidence and Mortality Worldwide for 36 Cancers in 185 Countries. CA Cancer J. Clin.
**2021**, 71, 209–249. [Google Scholar] [CrossRef] - World Health Organization. Global Health Estimates 2020: Deaths by Cause, Age, Sex, by Country and by Region, 2000–2019; World Health Organization: Geneva, Switzerland, 2020. [Google Scholar]
- Siegel, R.L.; Miller, K.D.; Fuchs, H.E.; Jemal, A. Cancer statistics, 2022. CA Cancer J. Clin.
**2022**, 72, 7–33. [Google Scholar] [CrossRef] - Dillekås, H.; Rogers, M.S.; Straume, O. Are 90% of deaths from cancer caused by metastases? Cancer Med.
**2019**, 8, 5574–5576. [Google Scholar] [CrossRef] - Fares, J.; Fares, M.Y.; Khachfe, H.H.; Salhab, H.A.; Fares, Y. Molecular principles of metastasis: A hallmark of cancer revisited. Signal Transduct. Target. Ther.
**2020**, 5, 28. [Google Scholar] [CrossRef] - Roussos, E.T.; Condeelis, J.S.; Patsialou, A. Chemotaxis in cancer. Nat. Rev. Cancer
**2011**, 11, 573–587. [Google Scholar] [CrossRef] - Vasaturo, A.; Caserta, S.; Russo, I.; Preziosi, V.; Ciacci, C.; Guido, S. A novel chemotaxis assay in 3-D collagen gels by time-lapse microscopy. PLoS ONE
**2012**, 7, e52251. [Google Scholar] [CrossRef] - Caserta, S.; Campello, S.; Tomaiuolo, G.; Sabetta, L.; Guido, S. A methodology to study chemotaxis in 3-D collagen gels. AIChE J.
**2013**, 59, 4025–4035. [Google Scholar] [CrossRef] - Cristini, V.; Frieboes, H.B.; Gatenby, R.; Caserta, S.; Ferrari, M.; Sinek, J. Morphologic instability and cancer invasion. Clin. Cancer Res.
**2005**, 11, 6772–6779. [Google Scholar] [CrossRef] - Bearer, E.L.; Lowengrub, J.S.; Frieboes, H.B.; Chuang, Y.-L.; Jin, F.; Wise, S.M.; Ferrari, M.; Agus, D.B.; Cristini, V. Multiparameter computational modeling of tumor invasion. Cancer Res.
**2009**, 69, 4493–4501. [Google Scholar] [CrossRef] - Cristini, V.; Lowengrub, J. Multiscale Modeling of Cancer: An Integrated Experimental and Mathematical Modeling Approach; Cambridge University Press: Cambridge, UK, 2010. [Google Scholar]
- Sixt, M.; Lämmermann, T. In vitro analysis of chemotactic leukocyte migration in 3D environments. In Cell Migration; Springer: Berlin/Heidelberg, Germany, 2011; pp. 149–165. [Google Scholar]
- Lämmermann, T.; Bader, B.L.; Monkley, S.J.; Worbs, T.; Wedlich-Söldner, R.; Hirsch, K.; Keller, M.; Förster, R.; Critchley, D.R.; Fässler, R. Rapid leukocyte migration by integrin-independent flowing and squeezing. Nature
**2008**, 453, 51–55. [Google Scholar] [CrossRef] - Friedl, P.; Bröcker, E.B. The biology of cell locomotion within three-dimensional extracellular matrix. Cell. Mol. Life Sci. CMLS
**2000**, 57, 41–64. [Google Scholar] [CrossRef] [PubMed] - Sant, S.; Johnston, P.A. The production of 3D tumor spheroids for cancer drug discovery. Drug Discov. Today Technol.
**2017**, 23, 27–36. [Google Scholar] [CrossRef] [PubMed] - Ng, K.W.; Leong, D.T.W.; Hutmacher, D.W. The challenge to measure cell proliferation in two and three dimensions. Tissue Eng.
**2005**, 11, 182–191. [Google Scholar] [CrossRef] - Rhodes, N.P.; Srivastava, J.K.; Smith, R.F.; Longinotti, C. Metabolic and histological analysis of mesenchymal stem cells grown in 3-D hyaluronan-based scaffolds. J. Mater. Sci. Mater. Med.
**2004**, 15, 391–395. [Google Scholar] [CrossRef] - Nyga, A.; Cheema, U.; Loizidou, M. 3D tumour models: Novel in vitro approaches to cancer studies. J. Cell Commun. Signal.
**2011**, 5, 239–248. [Google Scholar] [CrossRef] [PubMed] - Yamada, K.M.; Cukierman, E. Modeling tissue morphogenesis and cancer in 3D. Cell
**2007**, 130, 601–610. [Google Scholar] [CrossRef] [PubMed] - Hoarau-Véchot, J.; Rafii, A.; Touboul, C.; Pasquier, J. Halfway between 2D and animal models: Are 3D cultures the ideal tool to study cancer-microenvironment interactions? Int. J. Mol. Sci.
**2018**, 19, 181. [Google Scholar] [CrossRef] - Li, C.K.N. The glucose distribution in 9L rat brain multicell tumor spheroids and its effect on cell necrosis. Cancer
**1982**, 50, 2066–2073. [Google Scholar] [CrossRef] - Kam, Y.; Rejniak, K.A.; Anderson, A.R.A. Cellular modeling of cancer invasion: Integration of in silico and in vitro approaches. J. Cell. Physiol.
**2012**, 227, 431–438. [Google Scholar] [CrossRef] - Stillman, N.R.; Kovacevic, M.; Balaz, I.; Hauert, S. In silico modelling of cancer nanomedicine, across scales and transport barriers. NPJ Comput. Mater.
**2020**, 6, 92. [Google Scholar] [CrossRef] - Karolak, A.; Rejniak, K.A. Micropharmacology: An in silico approach for assessing drug efficacy within a tumor tissue. Bull. Math. Biol.
**2019**, 81, 3623–3641. [Google Scholar] [CrossRef] - Dogra, P.; Butner, J.D.; Chuang, Y.-l.; Caserta, S.; Goel, S.; Brinker, C.J.; Cristini, V.; Wang, Z. Mathematical modeling in cancer nanomedicine: A review. Biomed. Microdevices
**2019**, 21, 40. [Google Scholar] [CrossRef] - Burdett, E.; Kasper, F.K.; Mikos, A.G.; Ludwig, J.A. Engineering tumors: A tissue engineering perspective in cancer biology. Tissue Eng. Part B Rev.
**2010**, 16, 351–359. [Google Scholar] [CrossRef] [PubMed] - Norton, K.-A.; Gong, C.; Jamalian, S.; Popel, A.S. Multiscale agent-based and hybrid modeling of the tumor immune microenvironment. Processes
**2019**, 7, 37. [Google Scholar] [CrossRef] [PubMed] - Kapałczyńska, M.; Kolenda, T.; Przybyła, W.; Zajączkowska, M.; Teresiak, A.; Filas, V.; Ibbs, M.; Bliźniak, R.; Łuczewski, Ł.; Lamperska, K. 2D and 3D cell cultures–a comparison of different types of cancer cell cultures. Arch. Med. Sci. AMS
**2018**, 14, 910. [Google Scholar] [CrossRef] - Patel, A.A.; Gawlinski, E.T.; Lemieux, S.K.; Gatenby, R.A. A cellular automaton model of early tumor growth and invasion: The effects of native tissue vascularity and increased anaerobic tumor metabolism. J. Theor. Biol.
**2001**, 213, 315–331. [Google Scholar] [CrossRef] [PubMed] - Wang, Z.; Zhang, L.; Sagotsky, J.; Deisboeck, T.S. Simulating non-small cell lung cancer with a multiscale agent-based model. Theor. Biol. Med. Model.
**2007**, 4, 50. [Google Scholar] [CrossRef] - Costa, E.C.; Moreira, A.F.; de Melo-Diogo, D.; Gaspar, V.M.; Carvalho, M.P.; Correia, I.J. 3D tumor spheroids: An overview on the tools and techniques used for their analysis. Biotechnol. Adv.
**2016**, 34, 1427–1441. [Google Scholar] [CrossRef] - Chang, S.L.; Cavnar, S.P.; Takayama, S.; Luker, G.D.; Linderman, J.J. Cell, isoform, and environment factors shape gradients and modulate chemotaxis. PLoS ONE
**2015**, 10, e0123450. [Google Scholar] [CrossRef] - Gatenby, R.A.; Gawlinski, E.T. A reaction-diffusion model of cancer invasion. Cancer Res.
**1996**, 56, 5745–5753. [Google Scholar] - Masur, K.; Vetter, C.; Hinz, A.; Tomas, N.; Henrich, H.; Niggemann, B.; Zänker, K.S. Diabetogenic glucose and insulin concentrations modulate transcriptom and protein levels involved in tumour cell migration, adhesion and proliferation. Br. J. Cancer
**2011**, 104, 345–352. [Google Scholar] [CrossRef] [PubMed] - Lamers, M.L.; Almeida, M.E.S.; Vicente-Manzanares, M.; Horwitz, A.F.; Santos, M.F. High glucose-mediated oxidative stress impairs cell migration. PLoS ONE
**2011**, 6, e22865. [Google Scholar] [CrossRef] [PubMed] - Palorini, R.; Votta, G.; Pirola, Y.; De Vitto, H.; De Palma, S.; Airoldi, C.; Vasso, M.; Ricciardiello, F.; Lombardi, P.P.; Cirulli, C. Protein kinase A activation promotes cancer cell resistance to glucose starvation and anoikis. PLoS Genet.
**2016**, 12, e1005931. [Google Scholar] [CrossRef] [PubMed] - Leithner, K.; Hrzenjak, A.; Trötzmüller, M.; Moustafa, T.; Köfeler, H.C.; Wohlkoenig, C.; Stacher, E.; Lindenmann, J.; Harris, A.L.; Olschewski, A. PCK2 activation mediates an adaptive response to glucose depletion in lung cancer. Oncogene
**2015**, 34, 1044–1050. [Google Scholar] [CrossRef] [PubMed] - Ferraro, R.; Ascione, F.; Dogra, P.; Cristini, V.; Guido, S.; Caserta, S. Diffusion-induced anisotropic cancer invasion: A novel experimental method based on tumor spheroids. AIChE J.
**2022**, 68, e17678. [Google Scholar] [CrossRef] - Levenspiel, O. Chemical Reaction Engineering; John Wiley & Sons: Hoboken, NJ, USA, 1998. [Google Scholar]
- Schornack, P.A.; Gillies, R.J. Contributions of cell metabolism and H+ diffusion to the acidic pH of tumors. Neoplasia
**2003**, 5, 135–145. [Google Scholar] [CrossRef] - Morana, O.; Wood, W.; Gregory, C.D. The apoptosis paradox in cancer. Int. J. Mol. Sci.
**2022**, 23, 1328. [Google Scholar] [CrossRef] - Dogra, P.; Butner, J.D.; Ramírez, J.R.; Chuang, Y.-l.; Noureddine, A.; Brinker, C.J.; Cristini, V.; Wang, Z. A mathematical model to predict nanomedicine pharmacokinetics and tumor delivery. Comput. Struct. Biotechnol. J.
**2020**, 18, 518–531. [Google Scholar] [CrossRef] - Saltelli, A. Global sensitivity analysis: An introduction. In Proceedings of the 4th International Conference on Sensitivity Analysis of Model Output (SAMO 2004), Santa Fe, NM, USA, 8–11 March 2004. [Google Scholar]
- Wang, Z.; Deisboeck, T.S.; Cristini, V. Development of a sampling-based global sensitivity analysis workflow for multiscale computational cancer models. IET Syst. Biol.
**2014**, 8, 191–197. [Google Scholar] [CrossRef] - Pollock, K.; Stroemer, P.; Patel, S.; Stevanato, L.; Hope, A.; Miljan, E.; Dong, Z.; Hodges, H.; Price, J.; Sinden, J.D. A conditionally immortal clonal stem cell line from human cortical neuroepithelium for the treatment of ischemic stroke. Exp. Neurol.
**2006**, 199, 143–155. [Google Scholar] [CrossRef] - Herlyn, M.; Steplewski, Z.; Herlyn, D.; Koprowski, H. Colorectal carcinoma-specific antigen: Detection by means of monoclonal antibodies. Proc. Natl. Acad. Sci. USA
**1979**, 76, 1438–1442. [Google Scholar] [CrossRef] [PubMed] - Milo, R.; Phillips, R. Cell Biology by the Numbers; Garland Science: New York, NY, USA, 2015. [Google Scholar]

**Figure 1.**Model schematic showing cellular and glucose field layers. (

**a**) Typical initial configuration of the cellular layer (upper grid) with cell-free LPs (dark blue) surrounding cell-occupied LPs (orange). Not in scale. In the glucose layer (bottom grid) glucose concentration is calculated in a pseudo-stationary condition in each lattice point (LP), according to Equation (6). In the scheme, a typical concentration profile in a cell spheroid is reported using a color scale. (

**b**) Eight possible migration directions for a representative cell (located in P5); (

**c**) schematic representation of the migration direction probabilities (R

_{i}values).

**Figure 2.**Cell dynamic simulation cycle. The CA model iteratively scans the cellular layer, where each LP represents a cell. If the LP is occupied by a cell, the glucose concentration C

_{x,y}is sampled from the glucose layer; if C

_{x,y}< C

_{nec}, the cell is classified as necrotic; if C

_{nec}< C

_{x,y}< C

_{starv}, the cell is classified as starved; if C

_{x,y}> C

_{starv}, the cell is classified as active. Starved cells after T

_{apop}undergo apoptosis and release the occupied LP. Active cells after T

_{d}proliferate. Active cells after T

_{m}can migrate, according to the migration rules.

**Figure 3.**Baseline numerical solution. Snapshots of cellular layer (

**a**,

**c**) and glucose layer (

**b**,

**d**) at 0, 12, 24, 36, and 48 h following seeding of cancerous cells in the center of the simulation domain at 0 h, under isotropic (

**a**,

**b**) and gradient (

**c**,

**d**) conditions. Cellular layers display cells attached to the core (orange) and cells detached from the core (light blue), which migrate to the collagen matrix (dark blue).

**Figure 4.**Quantification of baseline solution. Results of the simulation assuming basal conditions with input parameters α = 6; K = 0.2, T

_{d}= 18 h, T

_{m}= 30 min in isotropic (

**a**–

**d**) and gradient cases (

**e**–

**h**); (

**a**,

**e**) core (orange) and migrated cell numbers over time (light blue); (

**b**,

**f**) core (orange) and migrated cell percentages (light blue); (

**c**,

**g**) migrated cell number ratio ϕ; (

**d**,

**h**) percentage of cells migrated in the four directions: North (light blue), East (yellow), South (orange), and West (purple) over time. The solid lines represent the means (n = 40) and the ribbons represents the standard errors.

**Figure 5.**MLRA regression coefficients (SI) in isotropic (

**a**–

**i**) and gradient (

**j**–

**r**) cases. The columns refer to the key parameters $\alpha $, $K$, ${T}_{d}$, and ${T}_{m}$. The rows refer to the outputs of interest: (

**a**,

**j**) core cell number; (

**b**,

**k**) migrated cell number; (

**c**,

**l**) number of cells migrating East; (

**d**,

**m**) number of cells migrating South; (

**e**,

**n**) number of cells migrating West; (

**f**,

**o**) core cell percentage; (

**g**,

**p**) percentage of cells migrating East; (

**h**,

**q**) percentage of cells migrating South; (

**i,r**) percentage of cells migrating West. The asterisk indicates the significance parameters (p < 0.05) for the related outputs.

**Figure 6.**Most relevant LSA results according to the GSA. Results of LSA perturbing input parameter α: (

**a**) percentages of cells migrating North (light blue), East (yellow), South (orange), and West (purple) at 48 h in the isotropic case; (

**b**) percentages of cells migrating North (light blue), East (yellow), South (orange), and West (purple) at 48 h in the gradient case. Results of LSA perturbing input parameter K: (

**c**) core cell numbers at 12 (green), 24 (blue), 36 (yellow), and 48 h (red) in gradient case; (

**d**) migrating cell numbers at 12 (green), 24 (blue), 36 (yellow), and 48 h (red) in gradient case. Results of LSA perturbing input parameter T

_{d}: (

**e**) core cell numbers at 12 (green), 24 (blue), 36 (yellow), and 48 h (red); (

**f**) migrating cell numbers at 12 (green), 24 (blue), 36 (yellow), and 48 h (red). Results of LSA perturbing input parameter T

_{m}: (

**g**) core percentages at 12 (green), 24 (blue), 36 (yellow), and 48 h (red); (

**h**) percentages of cells migrating North (light blue), East (yellow), South (orange), and West (purple) at 48 h. The solid lines represent the means (n = 40) and the ribbons represent the standard deviations.

Output Variable | Formula | Description |
---|---|---|

${N}_{cell}^{core}$ | Number of cells adhered to the spheroid | |

${N}_{cell}^{mig}$ | Number of cells that migrated away from the spheroid | |

${N}_{cell}^{TOT}$ | ${N}_{cell}^{core}+{N}_{cell}^{mig}$ | Total number of cells |

${N}_{\%}^{core}$ | $({N}_{cell}^{core}/{N}_{cell}^{TOT})\bullet 100$ | Percentage of cells adhered to the spheroid |

${N}_{\%}^{mig}$ | $({N}_{cell}^{mig}/{N}_{cell}^{TOT})\bullet 100$ | Percentage of cells that migrated away from the spheroid |

$\varphi $ | ${N}_{cell}^{core}/{N}_{cell}^{mig}$ | Ratio of adhered cells to migrated cells |

${N}_{cell}^{N}$ | Number of migrated cells located in the North quadrant | |

${N}_{cell}^{S}$ | Number of migrated cells located in the South quadrant | |

${N}_{cell}^{E}$ | Number of migrated cells located in the East quadrant | |

${N}_{cell}^{W}$ | Number of migrated cells located in the West quadrant | |

${N}_{\%}^{N}$ | $({N}_{cell}^{N}/{N}_{cell}^{mig})\bullet 100$ | Percentage of migrated cells located in the North quadrant |

${N}_{\%}^{S}$ | $({N}_{cell}^{S}/{N}_{cell}^{mig})\bullet 100$ | Percentage of migrated cells located in the South quadrant |

${N}_{\%}^{E}$ | $({N}_{cell}^{E}/{N}_{cell}^{mig})\bullet 100$ | Percentage of migrated cells located in the East quadrant |

${N}_{\%}^{W}$ | $({N}_{cell}^{W}/{N}_{cell}^{mig})\bullet 100$ | Percentage of migrated cells located in the West quadrant |

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**MDPI and ACS Style**

Messina, L.; Ferraro, R.; Peláez, M.J.; Wang, Z.; Cristini, V.; Dogra, P.; Caserta, S.
Hybrid Cellular Automata Modeling Reveals the Effects of Glucose Gradients on Tumour Spheroid Growth. *Cancers* **2023**, *15*, 5660.
https://doi.org/10.3390/cancers15235660

**AMA Style**

Messina L, Ferraro R, Peláez MJ, Wang Z, Cristini V, Dogra P, Caserta S.
Hybrid Cellular Automata Modeling Reveals the Effects of Glucose Gradients on Tumour Spheroid Growth. *Cancers*. 2023; 15(23):5660.
https://doi.org/10.3390/cancers15235660

**Chicago/Turabian Style**

Messina, Luca, Rosalia Ferraro, Maria J. Peláez, Zhihui Wang, Vittorio Cristini, Prashant Dogra, and Sergio Caserta.
2023. "Hybrid Cellular Automata Modeling Reveals the Effects of Glucose Gradients on Tumour Spheroid Growth" *Cancers* 15, no. 23: 5660.
https://doi.org/10.3390/cancers15235660