# Efficient Radial-Shell Model for 3D Tumor Spheroid Dynamics with Radiotherapy

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## Abstract

**:**

## Simple Summary

## Abstract

## 1. Introduction

## 2. Radial-Shell Model (RS Model)

#### 2.1. Dynamics with Radiotherapy

## 3. Parameter Calibration

**Table 1.**The RS model reproduces experimental growth curves with parameters in a realistic biological range. For each experiment, the estimated parameter values and fit result of the calibrated RS Model, literature values from monolayer experiments for the parameters, and fit results from previous models (except for FaDu, for which no model has been proposed) are displayed. The parameters (left to right) are: doubling time $ln\left(2\right)/\gamma $ with the proliferation rate $\gamma $, oxygen consumption rate a, anoxic death rate $\u03f5$, rate $\delta $ at which the volume occupied by membrane-defect cells is reduced, the width of the radial shells $dr=\kappa d{r}^{*}$ expressed in multiples $\kappa $ of the cell diameter $d{r}^{*}$, and the inward transport rate $\lambda $, displayed as the velocity $\lambda \xb7dr$. The goodness of fit is assessed by the coefficient of determination ${R}^{2}$ for the volume ${V}_{\mathrm{spheroid}}$. Note that the two sets of lines on HCT-116 refer to separate experiments which exhibit different spheroid growth curves, probably due to discrepancies in the experimental setup. For the FaDu cell line, we assume a range of $ln\left(2\right)/\gamma \in [20,40]$ h, which is around the reported value of 30 h.

Cell Line | Parameter Estimation | $ln\left(2\right)/\mathit{\gamma}$ [h] | a [mmHg/s] | $\mathit{\u03f5}$ [1/h] | $\mathit{\delta}$ [1/h] | $\mathit{\kappa}$ | $\mathbf{\lambda}\mathit{d}\mathit{r}$ [µm/h] | ${\mathit{R}}^{2}$ |
---|---|---|---|---|---|---|---|---|

HCT-116 Figure 2 | RS-model | $22.8$ | $27.7$ | 2.30 × 10^{−1} | 1.11 × 10^{−2} | $1.12$ | $38.8$ | $0.997$ |

Lit. (Experiment) | ||||||||

Schulte Am Esch et al. [51]; Cowley et al. [52]; Petitprez et al. [53]; Jain et al. [54] | 17.1–36 | - | - | - | - | - | ||

Grimes et al. [23] | $22.1\pm 4.8$ | |||||||

Lit. (3D model) | 28 | $22.1$ | ∼2.78 × 10^{−6} | ∼2.78 × 10^{−7} | ∼2.04 | ∼12.0 | $0.981$ | |

Brüningk et al. [1] | ||||||||

HCT-116 Figure A1 | RS-model | $29.7$ | $33.2$ | 1.5 × 10^{−1} | 6.2 × 10^{−3} | $2.39$ | $46.1$ | $0.997$ |

Lit. (Experiment) | ||||||||

Schulte Am Esch et al. [51]; Cowley et al. [52]; Petitprez et al. [53]; Jain et al. [54] | 17.1–36 | - | - | - | - | - | ||

Grimes et al. [24] | $27.9\pm 6.0$ | |||||||

Lit. (Non-spatial model) | 51.8–88.6 | 21.87–33.97 | $\gtrsim \gamma =0.04$ | - | - | 0.968–0.998 | ||

Grimes et al. [24] | ||||||||

MDA-MB-468 Figure A2 | RS-model | $52.4$ | $15.6$ | 4.7 × 10^{−2} | 3.7 × 10^{−7} | $9.81$ | $4.2$ | $0.994$ |

Lit. (Experiment) | ||||||||

Finlay-Schultz et al. [55]; DSMZ (www.dsmz.de) | <24, 30–80 | - | - | - | - | - | ||

Grimes et al. [24] | $18.1\pm 4.5$ | |||||||

Lit. (Non-spatial model) | 45.8–62.6 | 13.6–22.6 | $\gtrsim \gamma =0.01$ | - | - | 0.893–0.971 | ||

Grimes et al. [24] | ||||||||

LS-174T Figure A3 | RS-model | $34.7$ | $22.5$ | 4.5 × 10^{−2} | 2.0 × 10^{−5} | $6.22$ | $67.8$ | $0.998$ |

Lit. (Experiment) | ||||||||

DSMZ (www.dsmz.de) | 30–40 | - | - | - | - | - | ||

Grimes et al. [24] | $20.6\pm 4.4$ | |||||||

Lit. (Non-spatial model) | 25.68–39.12 | 16.2–25.0 | $\gtrsim \gamma =0.02$ | - | - | 0.956–0.998 | ||

Grimes et al. [24] | ||||||||

SCC-25 Figure A4 | RS-model | $36.9$ | $11.9$ | 1.5 × 10^{−1} | 2.0 × 10^{−2} | $2.85$ | $11.0$ | $0.993$ |

Lit. (Experiment) | ||||||||

Steinbichler et al. [56]; Gavish et al. [57] | 32.8–57.6 | - | - | - | - | |||

Grimes et al. [24] | $11.2\pm 4.6$ | |||||||

Lit. (Non-spatial model) | 88.1–103.0 | 6.6–15.8 | $\gtrsim \gamma =0.02$ | - | - | 0.884–0.968 | ||

Grimes et al. [24] | ||||||||

FaDu Figure A5 | RS-model | $36.0$ | $9.9$ | 3.6 × 10^{−1} | 3.1 × 10^{−2} | $4.15$ | $53.3$ | $0.998$ |

Lit. (Experiment) | ||||||||

DSMZ (www.dsmz.de) | 20–40 | - | - | - | - | - | ||

Leung et al. [58] | $\sim 10.6$ |

**Table 2.**The RS model reproduces the experimentally observed spheroid dynamics after radiation for HCT-116 and FaDu. For HCT-116, the same radiosensitivities as in Brüningk et al. [1] were chosen and the probabilities of mitotic catastrophe ${P}_{\mathrm{mc}}$ were optimized based on the highest dose of 10 Gy while adopting the other parameters for untreated growth (Table 1). Similarly, for FaDu the probabilities of mitotic catastrophe were optimized for 20 Gy, while apt radiosensitivities were chosen from the range observed in previous clonogenic survival assays [17,59] and optimized for 5 Gy. The goodness of fit was assessed using the coefficient of determination ${R}^{2}$ for the volume ${V}_{\mathrm{spheroid}}$. The bold ${R}^{2}$ values indicate the experiments used for validation (not included in the calibration).

${\mathit{P}}_{\mathbf{mc}}^{1}$ | ${\mathit{P}}_{\mathbf{mc}}^{2}$ | ${\mathit{\alpha}}_{\mathbf{RT}}$ | ${\mathit{\beta}}_{\mathbf{RT}}$ | d [Gy] | ${\mathit{R}}^{2}$ | |
---|---|---|---|---|---|---|

HCT-116 RS-model | $0.27$ | $0.67$ | $0.5$ | $0.042$ | 10 (100% control) | $0.9852$ |

5 (100% relapse) | 0.9970 | |||||

2 (100% relapse) | 0.9916 | |||||

HCT-116 Brüningk et al. [1] | $0.44$ | $0.59$ | 10 (100% control) | $0.98$ | ||

5 (100% relapse) | 0.88 | |||||

2 (100% relapse) | 0.98 | |||||

FaDu RS-model | $0.16$ | $0.81$ | $0.35$ | $0.079$ | 20 (100% control) | $0.9867$ |

5 (100% relapse) | $0.9812$ | |||||

$2.5$ (100% relapse) | 0.9873 |

## 4. Results

#### 4.1. Parameter Transfer to a Cellular Automaton

## 5. Discussion

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

RS model | Radial-Shell model |

ODE | ordinary differential equations |

PDE | partial differential equation |

3D | three-dimensional |

2D | two-dimensional |

1D | one-dimensional |

LQ model | linear quadratic model |

OER | oxygen enhancement ratio |

## Appendix A. Additional Comparison of RS Model to Untreated Growth Experiments

**Figure A1.**(

**a**) The RS model reproduces the outer radius (dark green) for untreated growth of the cell line HCT-116, with experimental data from Grimes et al. [24] (black crosses) even better than the model proposed along with the data (light green dashed). Note that the necrotic radius during the experiment can be estimated (gray crosses; see main text Section 3 for details) and that the RS model additionally reproduces this necrotic radius (red). (

**c**) Volumes corresponding to the radii displayed in (

**a**) in a semi-log plot. Multiples of 2 and 5 of the starting volume ${V}_{\mathrm{t}=0}$ are highlighted with horizontal gray dashed and gray dotted lines, respectively. Additionally, the RS model predicts the cell concentration over the radius at any time point. Examples show the cell distribution for (

**b**) the first time point $t=0$ d and (

**d**) the last day $t=17$ d of the experiment. Displayed are the cell concentrations of proliferation-competent cells ${c}_{p}$ (green solid) and membrane-defect cells ${c}_{n}$ (red solid), the total concentration of cells (black dotted), and the oxygen profile (blue solid, right y axis). The marker at the bottom of the panels (

**b**,

**c**) indicates the necrotic and outer radius estimated from the model (red and green crosses) and from the actual experimental data (gray and black arrows), respectively. The used parameters are reported in Table 1.

**Figure A2.**(

**a**) The RS model reproduces the outer radius (dark green) for untreated growth of the cell line MDA-MB-468, with experimental data from Grimes et al. [24] (black crosses) even better than the model proposed along with the data (light green dashed). Note that the necrotic radius during the experiment can be estimated (gray crosses; see main text Section 3 for details) and that the RS model additionally reproduces this necrotic radius (red). (

**c**) Volumes corresponding to the radii displayed in (

**a**) in a semi-log plot. Multiples of 2 and 5 of the starting volume ${V}_{\mathrm{t}=0}$ are highlighted with horizontal gray dashed and gray dotted lines, respectively. Additionally, the RS model predicts the cell concentration over the radius at any time point. Examples show the cell distribution for (

**b**) the first time point $t=0$ d and (

**d**) the last day $t=12$ d of the experiment. Displayed are the cell concentrations of proliferation-competent cells ${c}_{p}$ (green solid) and membrane-defect cells ${c}_{n}$ (red solid), the total concentration of cells (black dotted), and the oxygen profile (blue solid, right y axis). The marker at the bottom of the panels (

**b**,

**c**) indicates the necrotic and outer radius estimated from the model (red and green crosses) and from the actual experimental data (gray and black arrows), respectively. The used parameters are reported in Table 1.

**Figure A3.**The RS model reproduces the outer radius (dark green) for untreated growth of the cell line LS-174T, with experimental data from Grimes et al. [24] (black crosses) even better than the model proposed along with the data (light green dashed). Note that the necrotic radius during the experiment can be estimated (gray crosses; see main text Section 3 for details) and that the RS model additionally reproduces this necrotic radius (red). (

**c**) Volumes corresponding to the radii displayed in (

**a**) in a semi-log plot. Multiples of 2 and 5 of the starting volume ${V}_{\mathrm{t}=0}$ are highlighted with horizontal gray dashed and gray dotted lines, respectively. Additionally, the RS model predicts the cell concentration over the radius at any time point. Examples show the cell distribution for (

**b**) the first time point $t=0$ d and (

**d**) the last day $t=7$ d of the experiment. Displayed are the cell concentrations of proliferation-competent cells ${c}_{p}$ (green solid) ans membrane-defect cells ${c}_{n}$ (red solid), the total concentration of cells (black dotted), and the oxygen profile (blue solid, right y axis). The marker at the bottom of the panels (

**b**,

**c**) indicates the necrotic and outer radius estimated from the model (red and green crosses) and from the actual experimental data (gray and black arrows), respectively. The used parameters are reported in Table 1.

**Figure A4.**The RS model reproduces the outer radius (dark green) for untreated growth of the cell line SCC-25, with experimental data from Grimes et al. [24] (black crosses) even better than the model proposed along with the data (light green dashed). Note that the necrotic radius during the experiment can be estimated (gray crosses; see main text Section 3 for details) and that the RS model additionally reproduces this necrotic radius (red). (

**c**) Volumes corresponding to the radii displayed in (

**a**) in a semi-log plot. Multiples of 2 and 5 of the starting volume ${V}_{\mathrm{t}=0}$ are highlighted with horizontal gray dashed and gray dotted lines, respectively. Additionally, the RS model predicts the cell concentration over the radius at any time point. Examples show the cell distribution for (

**b**) the first time point $t=0$ d and (

**d**) the last day $t=17$ d of the experiment. Displayed are the cell concentrations of proliferation-competent cells ${c}_{p}$ (green solid) and membrane-defect cells ${c}_{n}$ (red solid), the total concentration of cells (black dotted), and the oxygen profile (blue solid, right y axis). The marker at the bottom of the panels (

**b**,

**c**) indicates the necrotic and outer radius estimated from the model (red and green crosses) and from the actual experimental data (gray and black arrows), respectively. The used parameters are reported in Table 1.

**Figure A5.**The RS model reproduces the outer radius (dark green) for untreated growth of the FaDu cell line, with experimental data from Chen et al. [18] (black crosses). Note that the necrotic radius during the experiment can be estimated (gray crosses; see main text Section 3 for details) and that the RS model additionally reproduces this necrotic radius (red). (

**c**) Volumes corresponding to the radii displayed in (

**a**) in a semi-log plot. Multiples of 2 and 5 of the starting volume ${V}_{\mathrm{t}=0}$ are highlighted with horizontal gray dashed and gray dotted lines, respectively. Additionally, the RS model predicts the cell concentration over the radius at any time point. Examples show the cell distribution for (

**b**) the first time point $t=0$ d and (

**d**) the last day $t=14$ d of the experiment. Displayed are the cell concentrations of proliferation-competent cells ${c}_{p}$ (green solid) and membrane-defect cells ${c}_{n}$ (red solid), the total concentration of cells (black dotted), and the oxygen profile (blue solid, right y axis). The marker at the bottom of the panels (

**b**,

**c**) indicates the necrotic and outer radius estimated from the model (red and green crosses) and from the actual experimental data (gray and black arrows), respectively. The used parameters are reported in Table 1.

## Appendix B. Additional Comparison of RS Model to Radiotherapy Experiments

**Figure A6.**(

**a**) The RS model reproduces the outer radius (dark green) for the FaDu cell line with 20 Gy radiation, experimental data from Chen et al. [18] (black crosses) as effectively as the model proposed along with the data (light green dashed). We used the parameters calibrated for untreated growth, only fitted the probabilities of mitotic catastrophe ${P}_{\mathrm{mc}}^{1}$, ${P}_{\mathrm{mc}}^{2}$, and adopted the radiosensitivities ${\alpha}_{\mathrm{RT}}$ and ${\beta}_{\mathrm{RT}}$ from the ranges observed in Chen et al. [17] and Xu et al. [59]. Note that no necrotic core is expected for this regime (experimental estimate shown as gray crosses), which is reproduced by the RS model (red). The used parameters can be found in Table 1 and Table 2. (

**c**) Volumes corresponding to the radii are displayed in (

**a**) in a semi-log plot. Multiples of 2 and 5 of the starting volume ${V}_{\mathrm{t}=0}$ are highlighted with horizontal gray dashed and gray dotted lines, respectively. Additionally, the RS model predicts the cell concentration over the radius at any time point. Examples show the cell distribution for (

**b**) the first time point $t=0$ d and (

**d**) the last day $t=33$ d of the experiment. Displayed are the cell concentrations of proliferation competent cells ${c}_{p}$ (green solid) and membrane-defect cells ${c}_{n}$ (red solid), the total concentration of cells (black dotted), and the oxygen profile (blue solid, right y axis). The marker at the bottom of the panels (

**b**,

**c**) indicates the necrotic and outer radius estimated from the model (red and green crosses) and from the actual experimental data (gray and black arrows), respectively.

**Figure A7.**(

**a**) The RS model reproduces the outer radius (dark green) for the FaDu cell line with 5 Gy radiation, experimental data from Chen et al. [18] (black crosses). We used the parameters calibrated for untreated growth, the values for ${P}_{\mathrm{mc}}^{1}$ and ${P}_{\mathrm{mc}}^{2}$ fitted at 20 Gy, and adopted the radiosensitivities ${\alpha}_{\mathrm{RT}}$ and ${\beta}_{\mathrm{RT}}$ from the ranges observed in Chen et al. [17] and Xu et al. [59]. Note that no necrotic core is expected for this regime (experimental estimate shown as gray crosses), which is reproduced by the RS model (red). The used parameters can be found in Table 1 and Table 2. (

**c**) Volumes corresponding to the radii are displayed in (

**a**) in a semi-log plot. Multiples of 2 and 5 of the starting volume ${V}_{\mathrm{t}=0}$ are highlighted with horizontal gray dashed and gray dotted lines, respectively. Additionally, the RS model predicts the cell concentration over the radius at any time point. Examples show the cell distribution for (

**b**) the first time point $t=0$ d and (

**d**) the last day $t=21$ d of the experiment. Displayed are the cell concentrations of proliferation competent cells ${c}_{p}$ (green solid) and membrane-defect cells ${c}_{n}$ (red solid), the total concentration of cells (black dotted), and the oxygen profile (blue solid, right y axis). The marker at the bottom of the panels (

**b**,

**c**) indicates the necrotic and outer radius estimated from the model (red and green crosses) and from the actual experimental data (gray and black arrows), respectively.

**Figure A8.**(

**a**) The RS model reproduces the outer radius (dark green) for the FaDu cell line, with $2.5$ Gy radiation experimental data from Chen et al. [18] (black crosses). We used the parameters calibrated for untreated growth, the values ${P}_{\mathrm{mc}}^{1}$ and ${P}_{\mathrm{mc}}^{2}$ fitted at 20 Gy, and adopted the radiosensitivities ${\alpha}_{\mathrm{RT}}$ and ${\beta}_{\mathrm{RT}}$ from the ranges observed in Chen et al. [17] and Xu et al. [59]. The necrotic radius of the experiments can be estimated (gray crosses; see Section 3 for details). The RS model can predict the necrotic radius (red) in a good manner. Note that the necrotic radius of the RS model appears jagged due to the spatial discretization of ${r}_{0}$. The used parameters can be found in Table 1 and Table 2. (

**c**) Shows the same data plotted for the volume in a semi-log plot. Multiples of 2 and 5 of the starting volume ${V}_{\mathrm{t}=0}$ are highlighted with horizontal gray dashed and gray dotted lines, respectively. Additionally, the RS model predicts the cell concentration over the radius at any time point. Examples show the cell distribution for (

**b**) the first time point $t=0$ d and (

**d**) the last day $t=14$ d of the experiment. Displayed are the cell concentrations of proliferation competent cells ${c}_{p}$ (green solid) and membrane-defect cells ${c}_{n}$ (red solid), the total concentration of cells (black dotted), and the oxygen profile (blue solid, right y axis). The marker at the bottom of the panels (

**b**,

**c**) indicates the necrotic and outer radius estimated from the model (red and green crosses) and from the actual experimental data (gray and black arrows), respectively.

## Appendix C. Equations for the RS Model

- Net gain by proliferation of proliferation-competent cells, $T=p$, and of damaged cells, $T=d$, with probability $1-{P}_{\mathrm{mc}}$: ${f}_{\gamma}({r}_{i},T,\rho )$
- Net gain due to inward transport for all cell types: ${f}_{\lambda}({r}_{i},T)$
- Anoxic death of proliferation-competent and damaged cells $T\in \{p,d\}$: ${f}_{\u03f5}({r}_{i},T,\rho )$
- Reduction of volume occupied by membrane-defect cells: ${f}_{\delta}\left({r}_{i}\right)$
- Death due to mitotic catastrophe for damaged cells, $T=d$, with probability ${P}_{\mathrm{mc}}$: ${f}_{\mathrm{mc}}({r}_{i},\rho )$.

## Appendix D. Determination of the Radial Oxygen Profile

**Figure A9.**The oxygen profile estimated with the hybrid analytical approach (solid blue) is similar to the previously published method of Grimes et al. [24] (solid black), which assumes hard spheres for the spheroid and its necrotic core. Shown here is a cell concentration for proliferation-competent cells (solid green), membrane-defect cells (solid red), and their sum (dotted black), with the parameter fitted for the FaDu cell line.

**Figure A10.**Using bisection, the anoxic radius ${r}_{\mathrm{an}}$ for the oxygen profile is determined by requiring the minima of the oxygen profile ${\rho}^{*}\left(r\right)$ to be at ${r}_{\mathrm{an}}$, with ${\rho}^{*}\left({r}_{\mathrm{an}}\right)={\rho}_{\mathrm{an}}$. (

**a**) Example cell concentration of oxygen consuming proliferation-competent cells ${c}_{p}$ (solid dark green) with corresponding oxygen profiles for different anoxic radii ${r}_{\mathrm{an}}$ (dashed lines). We used the parameters calibrated for FaDu. (

**b**) Magnification of the black box in (

**a**) around the minima of the oxygen profiles. The solid vertical lines in corresponding colors highlight the anoxic radius ${r}_{\mathrm{an}}$ for each oxygen profile. The correct radius determined by bisection is shown in red.

**Table A1.**The RS model is sufficiently insensitive to the oxygen level ${\rho}_{0}$ assumed at the surface of the spheroid (radius ${r}_{0}$). Here, we show that for the range ${\rho}_{0}\in [80,120]$ mmHg, which can be expected in experiments [45,46], the goodness of fit is only slightly affected. Examples show the cell lines HCT-116 and MDA-MB-468; the first row shows the fit results of Table 1. Below are two rows where we respectively adjusted ${\rho}_{0}$ and kept the previous fitted values, while separated by the dashed line are two more rows in which we respectively adjusted ${\rho}_{0}$ and refitted the parameters.

Cell Line | ${\mathit{\rho}}_{0}$ [mmHg] | $ln\left(2\right)/\mathit{\gamma}$ [h] | a [mmHg/s] | $\mathit{\u03f5}$ [1/h] | $\mathit{\delta}$ [1/h] | $\mathit{\kappa}$ | $\mathit{\lambda}\mathit{d}\mathit{r}$ [µm/h] | ${\mathit{R}}^{2}$ |
---|---|---|---|---|---|---|---|---|

HCT-116 Figure 2 | 100 (main text) | $0.997$ | ||||||

80 | 22.78 | 27.74 | 2.30 × 10^{−1} | 1.11 × 10^{−2} | 1.12 | 38.8 | $0.9838$ | |

120 | $0.9905$ | |||||||

80 | $23.76$ | $29.57$ | 1.41 × 10^{−1} | 9.60 × 10^{−3} | $1.25$ | $42.3$ | $0.9968$ | |

120 | $23.03$ | $25.91$ | 3.38 × 10^{−1} | 1.58 × 10^{−2} | $1.12$ | $38.5$ | $0.9961$ | |

MDA-MB-468 Figure A2 | 100 (main text) | $0.9943$ | ||||||

80 | 52.39 | 15.57 | 4.67 × 10^{−2} | 3.67 × 10^{−7} | 9.81 | 4.2 | $0.9845$ | |

120 | $0.9632$ | |||||||

80 | $49.23$ | $16.3$ | 9.59 × 10^{−2} | 1.18 × 10^{−4} | $9.65$ | $2.5$ | $0.9959$ | |

120 | $53.95$ | $14.51$ | 1.30 × 10^{−1} | 3.53 × 10^{−6} | $8.07$ | $2.1$ | $0.996$ |

**Table A2.**The RS model is sufficiently insensitive to the oxygen level ${\rho}_{0}$ assumed at the surface of the spheroid (radius ${r}_{0}$). Here, we show two that for the range ${\rho}_{0}\in [80,120]$ mmHg, which can be expected in experiments [45,46], the goodness of fit is only slightly affected. Examples show the fit results for radiation doses of 5 Gy and 10 Gy for the cell line HCT-116, where the first row shows the fit results of Table 2. Below are two rows where we respectively adjusted ${\rho}_{0}$ and kept the previous fitted values while separated by the dashed line two more rows where we respectively adjusted ${\rho}_{0}$ and refitted ${P}_{\mathrm{mc}}^{1}$ and ${P}_{\mathrm{mc}}^{2}$. All other parameters are taken from calibration of untreated growth (Table 1).

Cell Line | ${\mathit{\rho}}_{0}$ [mmHg] | ${\mathit{P}}_{\mathbf{mc}}^{1}$ | ${\mathit{P}}_{\mathbf{mc}}^{2}$ | ${\mathit{\alpha}}_{\mathbf{RT}}$ | ${\mathit{\beta}}_{\mathbf{RT}}$ | d [Gy] | ${\mathit{R}}^{2}$ |
---|---|---|---|---|---|---|---|

HCT-116 Figure 3 | 100 (main text) | $0.9852$ | |||||

80 | 0.27 | 0.67 | $0.9852$ | ||||

120 | 0.5 | 0.042 | 10 (full controlled) | $0.9852$ | |||

80 | $0.28$ | $0.66$ | $0.9855$ | ||||

120 | $0.26$ | $0.67$ | $0.9853$ | ||||

HCT-116 Figure 4 | 100 (main text) | $0.9852$ | |||||

80 | 0.27 | 0.67 | $0.9877$ | ||||

120 | 0.5 | 0.042 | 5 (full relapsed) | $0.999$ | |||

80 | $0.28$ | $0.66$ | $0.9982$ | ||||

120 | $0.26$ | $0.67$ | $0.9977$ |

## Appendix E. Radiotherapy

## Appendix F. Analytical Estimates of Spheroid Dynamics

**Figure A11.**The outer radius ${R}_{\mathrm{spheroid}}$ of the RS model agrees with the analytical predictions. We used the parameter fitted for the FaDu cell line and a radiation dose of 20 Gy. Certain parameters were altered to make the characteristics of the spheroid dynamics more pronounced: the initial volume is lower and the probability for a mitotic catastrophe ${P}_{\mathrm{mc}}$ changes after $t=25$ d from ${P}_{\mathrm{mc}}^{1}=0$ to ${P}_{\mathrm{mc}}^{2}=0.87$. From the analytical derivation, a linear scaling (black dashed lines) for larger and an exponential scaling (black dotted lines) for smaller radii is expected; (

**a**,

**b**) display the same results in the linear and semi-log-y scales, respectively. The gray area marks the predicted transition region between the linear and exponential regimes $dr\le {R}_{\mathrm{spheroid}}\le 3\xb7dr$.

**Figure A12.**The dynamics resulting from different choices of $\gamma $ and $dr$ can only be discriminated at small ${R}_{\mathrm{spheroid}}<150$ µm, where the transition between the linear and exponential regimes is observable. As an example, three choices of parameter sets with the same product $\gamma dr$ (same slope in the linear regime) are displayed for the FaDu cell line with 20 Gy irradiation (the solid line corresponds to parameters in Figure A6). The time $ln\left(2\right)/\gamma $ is varied in the fitting range used for the calibration of the RS model (20 h dashed line and 40 h dotted line).

## Appendix G. Computational Speed Comparison

**Figure A13.**The RS model is feasibly fast for parameter fitting. Shown here is a comparison of the simulation time needed to reach a specific outer radius ${R}_{\mathrm{spheroid}}$ for three model types: cell-based (3D cellular automaton), shown by the point-dashed line, the RS model, shown by a solid line, and the non-spatial model from [24], shown by a dashed line. The cell-based model scales exponentially, while the other two models scale linearly. Linear and exponential fits are respectively highlighted by black dotted lines.

## Appendix H. Cellular Automaton

- 1.
- For each possible event for the selected cell type T, find the one where the prerequisites are fulfilled given the oxygen level $\widehat{\rho}(\underline{x},t)$ at the position of the cell and the free space in its neighborhood
- 2.
- If a suitable event is found,
- (a)
- draw a uniform random number ${r}_{\mathrm{andom}}\in [0,1]$.
- (b)
- When the rate ${r}_{\mathrm{ate}}$ of that specific event follows: ${r}_{\mathrm{ate}}\xb7dt>{r}_{\mathrm{andom}}$, perform the event.

- 3.
- Otherwise, the drawn cell does not perform any event.

## Appendix I. Parameter κ for Neighborhood Connection

**Table A3.**The fitted shell size $\kappa $ of the RS model can be mapped to different neighborhood environments of a cell-based model when transferring the parameter to achieve a similar simulation. If two neighborhood condition are shown, e.g., 1-von Neumann/1-Moore, one is chosen randomly for each successive proliferation event. Note that the 3D cellular automaton itself can have variances for the radius ${R}_{\mathrm{spheroid}}$ that influence the $\kappa $ fitting. Therefore, we used the mean of five 3D cellular automaton simulations to reduce this error.

Neighborhood (ca3d) | Shell Size $\mathit{\kappa}$ |
---|---|

1-von Neumann | $0.81$ |

1-von Neumann/1-Moore | $1.11$ |

1-Moore | $1.39$ |

2-von Neumann | $1.48$ |

2-von Neumann/2-Moore | $2.08$ |

3-von Neumann | $2.11$ |

3-von Neumann/2-Moore | $2.42$ |

2-Moore | $2.58$ |

3-von Neumann/3-Moore | $2.94$ |

3-Moore | $3.71$ |

3-Moore/4-Moore | $4.26$ |

4-Moore | $4.76$ |

**Figure A14.**The shell size $\kappa $ of the RS model can be related to the individual neighborhood condition of a 3D cellular automaton. (

**a**) The correlation is independent of the chosen proliferation rate $\gamma $ for the range of experimentally expected values. Note that the variance for each fit of $\kappa $ was smaller than the shown symbols. (

**b**) It is possible to find a linear correlation between the shell size $\kappa $ and the mean distance at which a cell can be placed from its mother cell in a certain neighborhood condition in the 3D cellular automaton. Note that neum1moor1 refers to a random choice between these two neighborhoods for each proliferation event. The mean radius of ${R}_{\mathrm{spheroid}}$ of 5 simulations of the 3D cellular automaton where used to fit the $\kappa $ of the RS model. The fitted parameters of the FaDu cell line were used. Only $\u03f5$ and $\delta $ were set to 0 in order to single out the effect of proliferation. Additionally, the starting volume was multiplied by $2.5$ to increase visibility.

## Appendix J. Initial Starting Conditions of the RS Model

**Figure A15.**The simulation result is independent of the starting condition. We compared two different starting cell concentrations for the radial-shell model. The brighter colors (solid) are from the simulation where the starting condition was set as described in Appendix J and used in all other simulations of the manuscript. The darker colored lines (dashed) are from a simulation in which the initial cell concentration was set with a relatively steep sigmoid function for $20\%$ of the starting volume. The same parameters were used for both simulations.

## Appendix K. Adjustment for Differing Numbers of Experimental Data Points

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**Figure 1.**Illustration of the radial-shell model (RS-model). (

**a**) Sketch of a typical stained histological section of a spheroid with a viable rim (green) surrounding a secondary necrotic core, with cell debris membrane-defect cells (red) suggesting an approximate rotational symmetry of the spheroid. (

**b**) By exploiting the symmetry, the model is able to consider the cell dynamics on radial shells $i\in [0,1,2,...)$ and along the distance r from the center of the spheroid at $i=0$. The shells have equal radial width $dr$, and their maximal volume ${V}_{i}$ is consequently increasing outwards. Each shell i holds a concentration of cells ${c}_{T}\left({r}_{i}\right)$ of type T cells, where $T=p$ denotes proliferation-competent cells (green, proliferate with maximal rate $\gamma $, consume oxygen with rate a) and $T=n$ membrane-defect cells (red, irreversibly lost competence to proliferate, do not consume oxygen). Note that the concentrations ${c}_{T}\in [0,1]$ are normalized according to the available space in each shell. (

**c**) Oxygen consumption of proliferation-competent cells leads to a radial decrease of oxygen pressure from the value ${\rho}_{0}$ at the most outer shell at ${r}_{0}$ down to an anoxic threshold ${\rho}_{\mathrm{an}}$ at ${r}_{\mathrm{an}}$ towards the center of the spheroid. Beyond the threshold ${\rho}_{\mathrm{an}}$, cells are anoxic and become membrane-defect with the anoxic death rate $\u03f5$. The volume occupied by membrane-defect cells is reduced with rate $\delta $ and all cells are transported inward with rate $\lambda $ such that the spheroid remains compact. Then, the outer radius ${R}_{\mathrm{spheroid}}$ and necrotic radius ${R}_{\mathrm{necrotic}}$ are defined as the radius of a sphere with a volume equal to the total cell volume and membrane-defect cell volume, respectively. For reference, the shell width $dr=\kappa d{r}^{*}$ is expressed as multiple $\kappa $ of a single cell diameter $d{r}^{*}$.

**Figure 2.**(

**a**) The RS model reproduces the outer radius (dark green) for untreated growth of the cell line HCT-116, with experimental data from Brüningk et al. [1] (black crosses) even better than the model proposed along with the data (light green dashed). Note that the necrotic radius during the experiment can be estimated (gray crosses; see Section 3 for details) and that the RS model additionally reproduces this necrotic radius (red). (

**c**) Volumes corresponding to the radii displayed in (

**a**) in a semi-log plot. Multiples of 2 and 5 of the starting volume ${V}_{\mathrm{t}=0}$ are highlighted with horizontal gray dashed and gray dotted lines, respectively. The RS model predicts the cell concentration over the radius at any time point. Examples show the cell distributions for (

**b**) the first time point $t=0$ d and (

**d**) the last day $t=21$ d of the experiment. Displayed are the cell concentrations of proliferation-competent cells ${c}_{p}$ (green solid), membrane-defect cells ${c}_{n}$ (red solid), total concentration of cells (black dotted), and oxygen profile (blue solid, right y axis). The marker at the bottom of the panels (

**b**,

**c**) indicates the necrotic and outer radius estimated from the model (red and green crosses) and from the actual experimental data (gray and black arrows), respectively. The used parameters are reported in Table 1.

**Figure 3.**(

**a**) The RS model reproduces the outer radius (dark green) for the cell line HCT-116 with 10 Gy radiation experimental data from Brüningk et al. [1] (black crosses) as effectively as the model proposed along with the data (light green dashed). We used the parameters calibrated for untreated growth, only fitted the probabilities of mitotic catastrophe ${P}_{\mathrm{mc}}^{1}$, ${P}_{\mathrm{mc}}^{2}$, and adopted the radiosensitivities ${\alpha}_{\mathrm{RT}}$ and ${\beta}_{\mathrm{RT}}$ from Brüningk et al. [1]. Note that for this regime no secondary necrotic core is expected (experimental estimate shown as gray crosses), which is reproduced by the RS model (red). The parameters we used can be found in Table 1 and Table 2. (

**c**) Volumes corresponding to the radii displayed in (

**a**) in a semi-log plot. Multiples of 2 and 5 of the starting volume ${V}_{\mathrm{t}=0}$ are highlighted with horizontal gray dashed and gray dotted lines, respectively. The RS model predicts the cell concentration over the radius at any time point. Examples show the cell distribution for (

**b**) the first time point $t=0$ d and (

**d**) the last day $t=21$ d of the experiment. Displayed are the cell concentrations of proliferation-competent cells ${c}_{p}$ (green solid) and membrane-defect cells ${c}_{n}$ (red solid), the total concentration of cells (black dotted), and the oxygen profile (blue solid, right y axis). The marker at the bottom of the panels (

**b**,

**c**) indicates the necrotic and outer radius estimated from the model (red and green crosses) and from the actual experimental data (gray and black arrows), respectively.

**Figure 4.**(

**a**) The RS model reproduces the outer radius (dark green) for the cell line HCT-116, with 5 Gy radiation experimental data from Brüningk et al. [1] (black crosses) even better than the model proposed along with the data (light green dashed). We used the parameters calibrated for untreated growth, the values for ${P}_{\mathrm{mc}}^{1}$ and ${P}_{\mathrm{mc}}^{2}$ fitted at 10 Gy, and the values ${\alpha}_{\mathrm{RT}}$ and ${\beta}_{\mathrm{RT}}$ from Brüningk et al. [1]. Note that the necrotic radius during the experiment can be estimated (gray crosses) (see Section 3 for details) and that the RS model reproduces this necrotic radius (red). Additionally, note that the necrotic radius of the RS model appears jagged due to the spatial discretization of ${r}_{0}$. The used parameters are reported in Table 1 and Table 2. (

**c**) Volumes corresponding to the radii displayed in (

**a**) in a semi-log plot. Multiples of 2 and 5 of the starting volume ${V}_{\mathrm{t}=0}$ are highlighted with horizontal gray dashed and gray dotted lines, respectively. The RS model predicts the cell concentration over the radius at any time point. Examples show the cell distribution for (

**b**) the first time point $t=0$ d and (

**d**) the last day $t=21$ d of the experiment. Displayed are the cell concentrations of proliferation-competent cells ${c}_{p}$ (green solid) and membrane-defect cells ${c}_{n}$ (red solid), the total concentration of cells (black dotted), and the oxygen profile (blue solid, right y axis). The marker at the bottom of the panels (

**b**,

**c**) indicates the necrotic and outer radius estimated from the model (red and green crosses) and from the actual experimental data (gray and black arrows), respectively.

**Figure 5.**(

**a**) The RS model reproduces the outer radius (dark green) for the cell line HCT-116, with 2 Gy radiation experimental data from Brüningk et al. [1] (black crosses) as effectively as the model proposed along with the data (light green dashed). We used the parameters calibrated for untreated growth, the values for ${P}_{\mathrm{mc}}^{1}$ and ${P}_{\mathrm{mc}}^{2}$, and the model proposed along with the data (light green dashed) at 10 Gy, as well as the values ${\alpha}_{\mathrm{RT}}$ and ${\beta}_{\mathrm{RT}}$ from Brüningk et al. [1]. The necrotic radius during the experiment can be estimated (gray crosses) (see Section 3 for details) and the RS model reproduces this necrotic radius (red). Note that the necrotic radius of the RS model appears jagged due to the spatial discretization of ${r}_{0}$. The used parameters can be found in Table 1 and Table 2. (

**c**) Volumes corresponding to the radii displayed in (

**a**) in a semi-log plot. Multiples of 2 and 5 of the starting volume ${V}_{\mathrm{t}=0}$ are highlighted with horizontal gray dashed and gray dotted lines, respectively. Additionally, the RS model predicts the cell concentration over the radius at any time point. Examples show the cell distribution for (

**b**) the first time point $t=0$ d and (

**d**) the last day $t=21$ d of the experiment. Displayed are the cell concentrations of proliferation-competent cells ${c}_{p}$ (green solid) and membrane-defect cells ${c}_{n}$ (red solid), the total concentration of cells (black dotted), and the oxygen profile (blue solid, right y axis). The marker at the bottom of the panels (

**b**,

**c**) indicates the necrotic and outer radius estimated from the model (red and green crosses) and from the actual experimental data (gray and black arrows), respectively. The used parameters are reported in Table 1.

**Figure 6.**The cellular automaton reproduces the experimental growth curve for the FaDu cell line after parameter transfer from the previously fitted RS model. We directly transferred the parameters of the RS model to the cellular automaton using a 50:50 alteration between the 3-Moore and 4-Moore neighborhoods to achieve a value of $\kappa =4.26$. (

**a**) Cell concentration over the radius at $t=14$ d for the 3D cellular automaton; the mean cell concentrations (solid) from ten independent simulation runs are compared with the cell concentrations of the corresponding RS model (dashed). The cell concentrations of proliferation-competent cells ${c}_{p}$ (green solid) and membrane-defect cells ${c}_{n}$ (red solid) are displayed. (

**b**) Growth dynamics over time for the outer radius ${R}_{\mathrm{spheroid}}$ (green) and necrotic radius ${R}_{\mathrm{necrotic}}$ (red) along with the experimental outer radius (black crosses) and corresponding estimated necrotic radius (gray crosses). For the cellular automaton, the maximal and minimal radii from ten independent simulations are shown.

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**MDPI and ACS Style**

Franke, F.; Michlíková, S.; Aland, S.; Kunz-Schughart, L.A.; Voss-Böhme, A.; Lange, S.
Efficient Radial-Shell Model for 3D Tumor Spheroid Dynamics with Radiotherapy. *Cancers* **2023**, *15*, 5645.
https://doi.org/10.3390/cancers15235645

**AMA Style**

Franke F, Michlíková S, Aland S, Kunz-Schughart LA, Voss-Böhme A, Lange S.
Efficient Radial-Shell Model for 3D Tumor Spheroid Dynamics with Radiotherapy. *Cancers*. 2023; 15(23):5645.
https://doi.org/10.3390/cancers15235645

**Chicago/Turabian Style**

Franke, Florian, Soňa Michlíková, Sebastian Aland, Leoni A. Kunz-Schughart, Anja Voss-Böhme, and Steffen Lange.
2023. "Efficient Radial-Shell Model for 3D Tumor Spheroid Dynamics with Radiotherapy" *Cancers* 15, no. 23: 5645.
https://doi.org/10.3390/cancers15235645