Comparative Study of Tumor Growth Based on Single Species Models

Bibliographic Reference

Hassan, S. S., & Al-Saedi, H. M. (2024). Comparative study of tumor growth based on single species models. BIO Web of Conferences, 97, 00118. https://doi.org/10.1051/bioconf/20249700118

Core Argument

No consensus exists on which ODE growth model best describes a given cancer type, yet model choice substantially affects predictions of tumor growth trajectory. By fitting five single-species models (exponential, logistic, Allee effect, Gompertz, Bertalanffy) to one experimental dataset and comparing their sum of squared residuals, the authors show that SSR values differ by orders of magnitude across models and conclude that the Bertalanffy model provides the best fit after parameter optimization. The abstract mentions deriving equations for the minimum chemotherapy dose needed for tumor suppression, but no treatment simulations or dose calculations appear in the results — the paper’s empirical contribution is limited to model fitting and SSR comparison against untreated tumor growth data.

Methods

Five ordinary differential equation models of tumor growth were fit to experimental tumor volume data (0–120 days, 15 time points) extracted via WebPlotDigitizer from Worschech et al. (2009). The exponential model’s growth rate r was estimated analytically from the data. Parameters for logistic, Gompertz, and Bertalanffy models were optimized using least-squares minimization in MATLAB R2023a; Wolfram Mathematica 13.2 was used for numerical computations and visualization. Model selection was by minimum sum of squared residuals (SSR). The Allee effect model was excluded from optimization because its pre-optimization SSR was already low (528,352.56). The carrying capacity M was initially set to 4000 (the maximum observed tumor volume) for all models.

Key Findings

• Exponential model performed worst (SSR = 133,754,783.3), failing at later growth stages because it cannot account for growth deceleration from nutrient limitation, angiogenesis dependence, or spatial constraints. Adequate only for early-phase growth.

• Bertalanffy model achieved the best fit after optimization (SSR = 367,275.99; r = 0.299; M = 4,214), suggesting that surface-to-volume ratio constraints dominated growth deceleration in this dataset. The authors conclude that “the extrapolation of the Bertalanffy is the most accurate which would be useful to use in predicting how the tumour would develop in the future.”

• The Allee effect model performed best before optimization (SSR = 528,352.56), incorporating a threshold m below which the tumor cannot sustain itself — directly relevant to whether small tumor populations (micrometastases, resistant subclones) establish or go extinct.

• The Bertalanffy model showed the largest improvement from parameter optimization (SSR dropping from 37,198,029 to 367,276 — a ~99.0% reduction), surpassing the Gompertz model’s improvement (35,811,734.1 to 893,826.1 — a ~97.5% reduction). The logistic model’s SSR paradoxically worsened after optimization (7,509,636.24 to 29,081,933.73), suggesting the optimization procedure may not be reliable across all model forms.

Concepts Introduced or Used

• Gompertz model: dw/dt = r w ln(M/w); asymmetric sigmoid growth with decelerating relative growth rate (Gompertz, 1825)
• Logistic model: dw/dt = r w (1 − w/M); symmetric sigmoid with linear density dependence (Verhulst, 1838)
• Bertalanffy model: dw/dt = α w^(2/3) − β w; growth proportional to surface area, death proportional to volume (Bertalanffy, 1949)
• Allee effect model: dw/dt = r w (1 − w/M)(1 − m/w); includes a minimum viable population threshold m below which net growth is negative
• Exponential model: dw/dt = r w; unbounded growth (Malthus, 1798)
• Carrying capacity (M): asymptotic maximum tumor size or volume
• Sum of squared residuals (SSR): model selection criterion; lower = better fit
• clonal-expansion — growth model choice constrains the dynamics of clonal expansions
• therapy-resistance — model misspecification affects predicted drug doses needed for suppression

Entities Referenced

• Data source: Worschech et al. (2009), experimental tumor volume measurements in mice
• Growth models: Exponential (Malthus 1798), Logistic (Verhulst 1838), Gompertz (1825), Bertalanffy (1949), Allee effect (Allee 1931)
• Software: MATLAB R2023a, Wolfram Mathematica 13.2, WebPlotDigitizer
• Authors: Sokaina Sabah Hassan and Hayder M. Al-Saedi, Department of Mathematics, College of Science for Women, University of Baghdad

Limitations

The paper contains no explicit limitations section, and the authors present an optimistic view of their findings — particularly regarding the Bertalanffy model’s extrapolation reliability. The following limitations are identifiable from the study design but are not acknowledged by the authors:

• Single dataset: Only one experimental dataset (Worschech et al., 2009) was used. The introduction itself notes that model selection is tumor-type-dependent (citing Benzekry et al., 2014; Sarapata & de Pillis, 2014), yet the paper generalizes Bertalanffy as the best model without testing on multiple tumor types or datasets.
• Single-species framework: The models do not account for clonal heterogeneity, immune interactions, stromal coupling, or spatial structure — factors known to affect tumor growth dynamics in vivo.
• No formal model selection criteria: Model comparison used raw SSR only, without information criteria (AIC, BIC) that penalize parameter count, and without any form of cross-validation or out-of-sample testing.
• No treatment simulations performed: Despite the abstract’s mention of determining “the least level of chemotherapy needed for suppressing the tumour,” the results contain zero chemotherapy dose calculations, treatment scenarios, or dose-response modeling. The connection between growth model selection and treatment optimization is asserted but never demonstrated.
• Unreliable parameter optimization: The logistic model’s SSR increased more than threefold after optimization (7.5M to 29.1M), indicating either a flawed optimization routine, convergence to a poor local minimum, or an error in the reported values.
• Conference proceedings paper: Published in BIO Web of Conferences, which has a lighter review process than a full journal article. Findings should be treated as preliminary.

Relevance to Clonal Evolution

This paper underscores a practical problem for clonal evolution inference: the growth model assumed when interpreting cancer-cell-fraction distributions or reconstructing subclonal-architecture is rarely validated against the individual tumor’s actual growth dynamics. If the assumed growth model is wrong (exponential vs. Gompertz vs. Bertalanffy), estimates of when clones arose, how fast they expanded, and whether their frequencies imply selection are all systematically biased. The finding that SSR values span three orders of magnitude across models (from ~367k to ~134M) highlights the scale of potential error from model misspecification in evolutionary inference pipelines. The fact that this paper — which explicitly set out to compare model predictions for treatment — never actually performed any treatment simulations illustrates a broader gap: the link between growth model selection and therapeutic decision-making is more often asserted than empirically demonstrated.