# Mathematical Model of Clonal Evolution Proposes a Personalised Multi-Modal Therapy for High-Risk Neuroblastoma

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## Abstract

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## Simple Summary

## Abstract

## 1. Introduction

- 1
- The clonal competition between the treatment-sensitive, treatment-resistant, and non-malignant cell populations weakens the total cell population. For example, an evolution-guided application of paclitaxel was found to keep resistant cancer cells in check, thus prolonging the progression-free survival in preclinical breast cancer models [14].
- 2
- If the oncologist could hypothetically predict which mutations will be selected by the therapeutic agents, an evolutionary trap could theoretically be created (called a sucker’s gambit in the review [12]). In fact, the treatment-sensitive population could theoretically be maintained indefinitely by cycling between two complementary agents (evolutionary herding). It is known from experimental data that targetable mutations and alterations of oncogenic pathways in neuroblastoma are selected by chemotherapeutic agents and enriched at relapse [15]; examples are de novo mutations in ALK [16] and the genes encoding the RAS-MAPK pathway [17]. In fact, drugs targeting specific molecular aberrations in neuroblastoma are under active development and ALK inhibitors are the most notable examples because they are frontline treatment options [18,19]. Although neuroblastoma can develop resistance to ALK inhibitors too, the resistance mechanisms involved create other vulnerabilities, such as hypersensitivity to MEK inhibition [20].
- 3
- Chemotherapy will break the total cell population into smaller, fragmented (spatially distinct) [21], and genetically homogeneous (discussed above) cell populations. They are potentially vulnerable to even tiny stochastic perturbations induced by drugs or hypoxia. For instance, in a tumour, cells must cooperate to generate an angiogenic signal [22], so targeted therapies would kill them most effectively after the tumour breaks into clusters and before they can reconnect to build new blood vessels. To exploit the unique vulnerabilities of small populations, it is necessary to switch therapeutic agents when the vulnerabilities emerge.

## 2. Quick Guide to Methodology

**Mathematical model of clonal evolution and pharmacokinetics:**Our model describes the average dynamics of a structured population of cells, divided into nine different clones (genotypes). A key model assumption is that a cell undergoes only three processes: growth (balance between division and natural death), mutation, and drug-induced death. The rates of these processes depend on the cell’s genotype, phenotype, and environment. It assumes that a cancer cell (neuroblastoma cell) can have three levels of genetically conferred resistance to a chemotherapeutic agent: none, mild, and strong. For the sake of simplicity, the model considers two drugs only, resulting in nine clones whose resistance levels with respect to the drugs are denoted by i and j, respectively; where i and j can take the values of 0, 1, and 2. Although Figure 1 is relevant to the particular case where i denotes the level of resistance to vincristine (VCR) and j denotes the level of resistance to cyclophosphamide (CPM), it also describes the mathematical model’s general population structure. Furthermore, it assumes that regardless of its genotype (the clone it belongs to), the cell can phenotypically adapt (plasticity) to prolonged exposure to the drugs by altering its gene expression pattern, including epigenetic alterations [23]. For example, cancer cells and neuroblastoma cells in particular can upregulate their DNA repair proteins and drug efflux pumps to acquire multi-drug resistance [24]. As these alterations cost energy in the form of ATP, the model assumes phenotypic adaptation is at the expense of biomass production and growth [25]. Although it is known from experiments that chemotherapy enriches targetable mutations and activates oncogenic pathways [15], the model does not include this phenomenon, which we decided to be beyond the scope of this pilot study.

**Summary of model assumptions.**For the sake of clarity and convenience, we summarise the key model assumptions here.

- 1
- A population of neuroblastoma cells (a clone) can undergo three processes only: growth (division minus natural death), mutation, and drug-induced mortality.
- 2
- Each clone follows logistic growth, limited by the other clones and the total carrying capacity (clonal competition).
- 3
- A neuroblastoma cell has three levels of genetic resistance to a drug: none, mild, and strong. It can only mutate and enter a clone whose resistance level is directly above or below its own. Mutation occurs randomly—uniformly in all directions—in the absence of drugs (selective pressures). Therefore, the mutation term is simply the growth term multiplied by the mutation rate.
- 4
- In addition to genetically conferred resistance, a neuroblastoma cell can phenotypically adapt to drugs after prolonged exposure to them. Adaptation costs energy, so both cell death and growth will decrease as a result. The extent of decrease depends linearly on the length of the exposure period.
- 5
- Drug delivery follows first-order pharmacokinetics.
- 6
- Spatial variations and stochastic effects are both assumed to be negligible.

**Aggregation of experimental data for model calibration and validation:**We carried out a review of the chemotherapeutic agents currently used to treat neuroblastoma. Based on our review (see the Supplementary Materials file), vincristine ($d=1$) and cyclophosphamide ($d=2$) affect the M-phase and the entirety of the cell cycle, respectively, so their mechanisms of action are complementary. We also found sufficient data about them for model calibration. Therefore, these drugs were chosen for inclusion in the model. The resistance levels with respect to vincristine and cyclophosphamide are denoted by i and j, respectively. In order to inform the model, experimental data regarding neuroblastoma cells’ responses to vincristine and cyclophosphamide were aggregated from different sources [26,27,28,29,30,31,32,33]. First, armed with in vitro data about the growth kinetics of neuroblastoma cell lines with different levels of drug resistance, collected in the absence of treatment, we calibrated the growth rates (${r}_{i,j}$). Second, we calibrated the drug-induced death rates, specifically the ${m}_{d}^{i,j}\left({c}_{d}\right)$ function for both drugs. This step was informed by in vitro experiments involving differentially resistant cell lines in media with different drug concentrations (deducing the types and magnitude of drug dose-dependences on sensitive neuroblastoma cells, and re-calibrating the magnitude only on resistant neuroblastoma cells), as well as a different set of in vivo experiments performed on mice because compatible in vitro data were not available. Using the data, we deduced how the mortality rates of sensitive neuroblastoma cells and drug doses are related, qualitatively and quantitatively. Then, we adjusted the quantitative aspects of this relation for resistant neuroblastoma cells.. Third, we parameterised the functions modelling phenotypic adaptation—${\varphi}_{1}$ and ${\varphi}_{2}$—based on in vitro data. The remaining parameters—K, $\mu $ and ${z}_{d}$—were taken from the literature directly or indirectly after adjusting them for this model. Finally, after fixing every parametric value, we used the calibrated model to replicate experimentally observed trends with success, thus validating both the model design and calibration. The calibrated parameters indicate that having genetically conferred resistance to a drug comes at the expense of a lower growth rate in the absence of the drug. The Supplementary Materials file includes a more technical discussion of our calibration pipeline, including its limitations. Table 1 shows the calibration results summarising the model parameters, including their symbols, numerical values, units, and physical meanings.

**Optimising drug schedules using a genetic algorithm and gradient descent.**The drug doses and timings of administration, as well as the overall duration of induction chemotherapy, constitute the strategies in the Stackelberg game described in the introduction. Mathematically, this strategy is encoded by the two ${\omega}_{d}$ variables in Equation (2). As indicated in the introduction, it is currently difficult or impossible to infer a tumour’s clonal composition before treatment. Therefore, we decided to evaluate a broad range of resistance levels: zero, five, 10, 15, 20, and 25% of the total cell count before treatment, which was set to be half of the carrying capacity in all cases ($K/2=5\xb7{10}^{9}$ cells, corresponding to 5 cm${}^{3}$). In each case, we found the optimal chemotherapy schedule by minimising the tumour size (sum of the nine ${n}_{i,j}$ values) at the end of the schedule, i.e., after two weeks from the beginning of the last cycle. Although the tumour’s final composition is arguably as important as its final size, we did not include it in our objective function because the induction chemotherapy stage is followed by other modes of treatment, such as surgery, immunotherapy, and radiotherapy, which do not necessarily depend on the tumour’s final composition in terms of the resistance to chemotherapy. The fundamental unit of a two-week cycle was adopted, so only the number of cycles and the two doses in each cycle were varied and optimised. We decided to limit our study to 12 or fewer cycles due to concerns about patient toxicity. In each cycle of a simulation, up to 2 mg m${}^{-2}$ of VCR and 2 g m${}^{-2}$ of CPM—their MTDs—were administered in the first 48 and 56 h, respectively [34,35]. In a clinical setting, VCR is administered intravenously as a solution, while CPM is administered as powder [34,35], hence the different periods of administration in our studies. We found literature values for the clearance rates in the two pharmacokinetic equations [36,37,38,39]. To be consistent with these studies, optimal drug schedules for tumours with different clonal compositions were obtained for a three-year-old child, 80 cm in height and 15 kg in weight, just like the child in whom the chosen clearance rates were measured. The use of a genetic algorithm allowed us to search multiple, disconnected regions of the solution space, while a method based on the gradient descent was used to search the vicinity of the solution proposed by the genetic algorithm more thoroughly. For the technical details pertaining to our optimisation algorithm, please consult the Supplementary Materials file.

## 3. Results

#### 3.1. Mixtures of Fully Sensitive and VCR-Resistant Cells

#### 3.2. Mixtures of Fully Sensitive and CPM-Resistant Cells

**Figure 3.**Population dynamics of the nine clones in a tumour containing fully sensitive and mildly VCR-resistant cells only before treatment. The resistant cells made up 15% of the initial cell population and the population dynamics were driven by strategy A (panel

**a**), strategy B (panels

**b**and

**d**), or strategy C (panel

**c**). S denotes fully sensitive cells; O-mild, mildly VCR-resistant cells; O-strong, strongly VCR-resistant cells; C-mild, mildly CPM-resistant cells; C-strong, strongly CPM-resistant cells; and total, total cell population. MTD represents the total cell population responding to the default schedule: applying VCR and CPM at their MTDs for eight cycles (2688 h). VCR and O both denote vincristine, while CPM and C both denote cyclophosphamide. These trajectories were obtained by solving the model numerically with the ode45 solver in MATLAB.

#### 3.3. Mixtures of Fully Sensitive, VCR-Resistant, and CPM-Resistant Cells

## 4. Discussion

#### 4.1. Therapeutic Strategies Based on General Evolutionary Principles

- 1
- Is the tumour to be treated already resistant to the less cytotoxic drug but not the other drug? If so, the optimal strategy is to apply the more effective drug at its MTD to exploit clonal evolution to kill the resistant clone effectively before adding the other drug to the regimen to shrink the tumour, which is mostly sensitive to the less effective drug at the end of the first stage. Finally, switch to a third drug (or another intervention) to exploit the final state of the tumour. This is strategy A. For instance, strategy A was found to be optimal for mixtures of fully sensitive and mildly VCR-resistant cancer cells (O-mild rows in Figure 2). Figure 3a presents the population dynamics of the nine clones induced by the optimal schedule corresponding to the third O-mild row (15%) in Figure 2.
- 2
- If the tumour is already resistant to the more cytotoxic drug only, is it mildly or strongly resistant to it? If it is strongly resistant, a similar two-stage strategy will work, but only the less effective drug is used in the first stage, while both drugs are used in the second stage. Furthermore, both stages should last longer than in strategy A to prolong clonal competition, thus maintaining a negative selection pressure on the resistant clone. This change is necessary because resistance to the more cytotoxic drug is harder to deal with. This is strategy B. For instance, strategy B was found to be optimal for mixtures of fully sensitive and strongly CPM-resistant cancer cells (C-strong rows in Figure 2). Figure 3b presents the population dynamics of the nine clones induced by the optimal schedule corresponding to the third C-strong row (15%) in Figure 2. During the dynamic simulation’s first stage, the strongly CPM-resistant clone (orange line) shrank even though the whole population (black line) and the fully sensitive clone (grey line) grew. If the tumour is mildly resistant to the more cytotoxic drug only, the optimal strategy is to use both drugs at their MTDs for a short duration to shrink the sensitive clone in the tumour and then switch to a third drug (or another intervention) targeting the presumably reduced and fragmented cell populations. This is strategy C. For instance, strategy C was found to be optimal for mixtures of fully sensitive and mildly CPM-resistant cancer cells (C-mild row in Figure 2). Figure 3c presents the population dynamics of the nine clones induced by the optimal schedule corresponding to the C-mild row in Figure 2; the resistant clone made up 15% of the initial population in this simulation.
- 3
- If the tumour is already resistant to both drugs, is it mildly or strongly resistant to the more cytotoxic drug, or are there both mildly and strongly resistant cells? If the tumour is only mildly resistant, strategy C is recommended.
- 4
- If the tumour is strongly resistant or contains both mildly and strongly resistant cells, what is the total fraction of cells that are resistant? A low fraction favours strategy C, while a high fraction favours strategy B. As a rule of thumb, based on the results presented in Figure 2, a fraction below 15% is considered low in a strongly resistant tumour and a fraction below 25% is considered low if the tumour contains both mildly and strongly resistant cells. For instance, strategy B was found to be optimal for the case where strongly VCR-resistant and strongly CPM-resistant cells constitute 15% of the initial population. Figure 3d shows the population dynamics of the nine clones induced by the optimal schedule corresponding to the second and last both strong rows in Figure 2; the resistant cells made up 15% of the initial population in this simulation. In the first stage, the CPM-resistant clone (orange line) was suppressed by the other clones in the presence of VCR only, but the VCR-resistant clone (blue line) expanded aggressively to dominate the population. In the second stage, the tumour dominated by the VCR-resistant clone responded to CPM effectively.

#### 4.2. Clinical Translation

#### 4.3. Design Choices

#### 4.4. Validity and Future Work

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**The mathematical model’s population structure. Each circle represents a subpopulation (clone) with two drug-specific resistance levels. Each ${n}_{i,j}$ is the size of a clone (cell count). Each arrow indicates a possible mutation step.

**Figure 2.**Optimal chemotherapy schedules for tumours with different initial clonal compositions. The first block is pertinent to mixtures of mildly VCR-resistant and fully sensitive cells; the second block, mixtures of strongly VCR-resistant and fully sensitive cells; the third block, mixtures of mildly CPM-resistant and fully sensitive cells; the fourth block, mixtures of strongly CPM-resistant and fully sensitive cells; the fifth block, mixtures of mildly VCR-resistant, mildly CPM-resistant, and fully sensitive cells; the sixth block, mixtures of strongly VCR-resistant, strongly CPM-resistant, and fully sensitive cells; and the last block, mixtures of VCR-resistant (mildly and strongly), CPM-resistant (mildly and strongly), and fully sensitive cells. VCR and O both denote vincristine, while CPM and C both denote cyclophosphamide. The VCR doses are in mg m${}^{-2}$ and the CPM doses are in g m${}^{-2}$. The percentages quantify the extent to which a virtual tumour was initially occupied by resistant cells. The Supplementary Materials file contains enlarged copies of these two heat maps.

Parameters | Values | Units | Meanings |
---|---|---|---|

${z}_{1}$ | $0.91$ | h ${}^{-1}$ | VCR clearance rate |

${z}_{2}$ | $0.12$ | h${}^{-1}$ | CPM clearance rate |

${r}_{0,0}$ | $8.5\xb7{10}^{-3}$ | h${}^{-1}$ | Sensitive clone’s growth rate |

${r}_{1,0}$ | $7.7\xb7{10}^{-3}$ | h${}^{-1}$ | VCR-10 clone’s growth rate |

${r}_{2,0}$ | $7.5\xb7{10}^{-3}$ | h${}^{-1}$ | VCR-20 clone’s growth rate |

${r}_{0,1}$ | $7.7\xb7{10}^{-3}$ | h${}^{-1}$ | CPM-20 clone’s growth rate |

${r}_{0,2}$ | $7.5\xb7{10}^{-3}$ | h${}^{-1}$ | CPM-32 clone’s growth rate |

${r}_{1,1}$ | $7\xb7{10}^{-3}$ | h${}^{-1}$ | VCR-10-CPM-20 clone’s growth rate |

${r}_{1,2}$ | $6.8\xb7{10}^{-3}$ | h${}^{-1}$ | VCR-10-CPM-32 clone’s growth rate |

${r}_{2,1}$ | $6.8\xb7{10}^{-3}$ | h${}^{-1}$ | VCR-20-CPM-20 clone’s growth rate |

${r}_{2,2}$ | $6.6\xb7{10}^{-3}$ | h${}^{-1}$ | VCR-20-CPM-32 clone’s growth rate |

K | ${10}^{10}$ | cells | Carrying capacity of the tumour |

$\mu $ | ${10}^{-4}$ | Dimensionless | Mutation rate |

${\alpha}_{1}$ | $1.122\xb7{10}^{4}$ | Dimensionless | Shape parameter in mortality function (VCR) |

${\beta}_{1}$ | $0.6704$ | Dimensionless | Shape parameter in mortality function (VCR) |

${m}_{1}^{0,0}$ | $40.4$ | h${}^{-1}$ | Sensitive clone’s maximum mortality rate due to VCR |

${m}_{1}^{0,1}$ | $6.8$ | h${}^{-1}$ | VCR-10 clone’s maximum mortality rate due to VCR |

${m}_{1}^{0,2}$ | 6 | h${}^{-1}$ | VCR-20 clone’s maximum mortality rate due to VCR |

${\alpha}_{2}$ | $2.9507\xb7{10}^{-5}$ | Dimensionless | Shape parameter in mortality function (CPM) |

${\beta}_{2}$ | 1 | Dimensionless | Shape parameter in mortality function (CPM) |

${m}_{2}^{0,0}$ | $3.1474\xb7{10}^{-6}$ | h${}^{-1}$ | Sensitive clone’s maximum mortality rate due to CPM |

${m}_{2}^{0,1}$ | $1.5737\xb7{10}^{-6}$ | h${}^{-1}$ | CPM-20 clone’s maximum mortality rate due to CPM |

${m}_{2}^{0,2}$ | $9.4422\xb7{10}^{-7}$ | h${}^{-1}$ | CPM-32 clone’s maximum mortality rate due to CPM |

nuovo${\mathrm{T}}_{min}$ | 0 | days | nuovoMinimum memory period associated with phenotypic adaptation |

${\mathrm{T}}_{max}$ | 10 | days | nuovoMaximum memory period associated with phenotypic adaptation |

nuovo${\Phi}_{{1}_{min}}$ | 0 | Dimensionless | nuovoMinimum effect of phenotypic adaptation on growth |

${\Phi}_{{1}_{max}}$ | 1 | Dimensionless | Maximum effect of phenotypic adaptation on growth |

nuovo${\Phi}_{{2}_{min}}$ | 0 | Dimensionless | nuovoMinimum effect of phenotypic adaptation on drug-induced mortality |

${\Phi}_{{2}_{max}}$ | 2 | Dimensionless | Maximum effect of phenotypic adaptation on drug-induced mortality |

**Table 2.**Results of applying the optimal chemotherapy schedules for mixtures of fully sensitive and VCR-resistant cells, presented in the top rows (below VCR size); mixtures of fully sensitive and CPM-resistant cells, presented in the middle rows (below CPM size); and mixtures of fully sensitive, VCR-resistant, and CPM-resistant cells, presented in the bottom rows (below Both size). The cell count (N), final size (FS), and gain (G) columns show the final cell count achieved with the optimal schedule for each initial clonal composition, the final size as a percentage of the initial size ($5\xb7{10}^{9}$), and the gain relative to using MTDs for eight cycles (achieving ${N}_{MTD}$), respectively. The gain refers to the percentage difference in tumour size: $\left(\frac{{N}_{MTD}-N}{{N}_{MTD}}\right)\times 100$. Each size (5, 10, 15, 20, and $25\%$) refers to the fraction of the initial population occupied by resistant cells. The strongly VCR-resistant cells were killed more effectively than the mildly VCR-resistant cells in the simulations because the strategy of exploiting clonal competition worked more effectively against the strongly resistant cells. Resistance to VCR comes at the expense of growth in the absence of VCR in our model.

VCR Size | Mild | Strong | |||||||
---|---|---|---|---|---|---|---|---|---|

N ($\xb7{10}^{8}$) | FS (%) | G (%) | N ($\xb7{10}^{8}$) | FS (%) | G (%) | ||||

5% | 1.17 | 2.34 | 16.09 | 0.53 | 1.06 | 50.48 | |||

10% | 1.80 | 3.6 | 21.18 | 0.85 | 1.7 | 50.29 | |||

15% | 2.38 | 4.76 | 21.40 | 1.11 | 2.22 | 51.35 | |||

20% | 2.91 | 5.82 | 21.07 | 1.36 | 2.7 | 51.00 | |||

25% | 3.42 | 6.84 | 19.46 | 1.60 | 3.2 | 50.25 | |||

CPM Size | Mild | Strong | |||||||

N ($\xb7{\mathbf{10}}^{\mathbf{8}}$) | FS (%) | G (%) | N ($\xb7{\mathbf{10}}^{\mathbf{8}}$) | FS (%) | G (%) | ||||

5% | 4.21 | 8.42 | 8.40 | 9.93 | 19.86 | 61.40 | |||

10% | 6.70 | 13.4 | 10.10 | 14.00 | 28 | 55.89 | |||

15% | 8.70 | 17.4 | 8.58 | 16.67 | 33.34 | 51.57 | |||

20% | 10.33 | 20.66 | 6.65 | 19.20 | 38.4 | 46.59 | |||

25% | 11.69 | 23.38 | 4.79 | 21.60 | 43.2 | 41.49 | |||

Both Size | Mild | Strong | Mild and Strong | ||||||

N ($\xb7{\mathbf{10}}^{\mathbf{8}}$) | FS (%) | G (%) | N ($\xb7{\mathbf{10}}^{\mathbf{8}}$) | FS (%) | G (%) | N ($\xb7{\mathbf{10}}^{\mathbf{8}}$) | FS (%) | G (%) | |

5% | 3.04 | 6.08 | 1.31 | 7.67 | 15.30 | 58.60 | 5.42 | 10.84 | 45.06 |

10% | 4.92 | 9.84 | 4.30 | 11.38 | 22.76 | 55.02 | 8.14 | 16.28 | 47.17 |

15% | 6.40 | 12.8 | 4.82 | 12.27 | 24.24 | 57.42 | 10.05 | 20.1 | 46.31 |

20% | 7.72 | 15.44 | 4.01 | 12.53 | 25.06 | 59.58 | 11.83 | 23.66 | 43.90 |

25% | 8.81 | 17.62 | 2.98 | 12.68 | 25.36 | 60.92 | 12.12 | 24.24 | 46.95 |

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**MDPI and ACS Style**

Italia, M.; Wertheim, K.Y.; Taschner-Mandl, S.; Walker, D.; Dercole, F.
Mathematical Model of Clonal Evolution Proposes a Personalised Multi-Modal Therapy for High-Risk Neuroblastoma. *Cancers* **2023**, *15*, 1986.
https://doi.org/10.3390/cancers15071986

**AMA Style**

Italia M, Wertheim KY, Taschner-Mandl S, Walker D, Dercole F.
Mathematical Model of Clonal Evolution Proposes a Personalised Multi-Modal Therapy for High-Risk Neuroblastoma. *Cancers*. 2023; 15(7):1986.
https://doi.org/10.3390/cancers15071986

**Chicago/Turabian Style**

Italia, Matteo, Kenneth Y. Wertheim, Sabine Taschner-Mandl, Dawn Walker, and Fabio Dercole.
2023. "Mathematical Model of Clonal Evolution Proposes a Personalised Multi-Modal Therapy for High-Risk Neuroblastoma" *Cancers* 15, no. 7: 1986.
https://doi.org/10.3390/cancers15071986