A Comparison and Catalog of Intrinsic Tumor Growth Models

Bibliographic Reference

Sarapata, E. A. (2013). A comparison and catalog of intrinsic tumor growth models [Senior thesis, Harvey Mudd College]. Scholarship @ Claremont. https://scholarship.claremont.edu/hmc_theses/52

Core Argument

The choice of tumor growth model (exponential, power law, logistic, Gompertz, or von Bertalanffy) substantially affects estimated growth parameters and predicted dynamics, yet no single model is universally best — the optimal model depends on tumor type. Systematic fitting across 10 tumor types and 5 models provides a catalog of recommended parameters and a basis for selecting growth models appropriate to specific tumor types. The author explicitly discourages the power law model despite it achieving the lowest least-squares residuals for 6 of 10 cancer types, citing extreme parameter sensitivity that makes its fitted parameters biologically unjustifiable to vary. The author also questions the biological veracity of the von Bertalanffy best-fit parameters because their intrinsic growth rates are consistently two to three orders of magnitude smaller than those of the other models.

Methods

Five ordinary differential equation (ODE) growth models were fit to published experimental tumor growth data spanning 10 tumor types (bladder, breast, colon, head and neck squamous cell carcinoma, hepatocellular carcinoma, lung, melanoma, ovarian, pancreatic, renal cell carcinoma), with at least 7 datasets per type drawn from a minimum of 5 different papers each. In vitro data (lacking immune interference) were used to determine intrinsic growth rates; in vivo data were used to determine carrying capacities.

Models compared:

• Exponential: dP/dt = CP (1 parameter)
• Power law: dP/dt = CP^a (2 parameters)
• Logistic: dP/dt = rP(1 − P/K) (2 parameters)
• Gompertz: dP/dt = r log(K/P) P (2 parameters)
• Von Bertalanffy: dP/dt = r(K − P) (2 parameters, linear in P)

A hybrid local-global least-squares minimization algorithm was implemented in MATLAB: Nelder-Mead simplex direct search (fminsearch) initialized parameters, which were passed to a Markov chain Monte Carlo fitting with simulated annealing (200 iterations, 10 annealing steps) to escape local minima, followed by a second round of fminsearch to converge precisely to the deepest local minimum. ODEs were solved numerically using MATLAB’s ode45 (4th/5th order Runge-Kutta).

Fitting evaluation used both least-squares residuals and the Bayesian Information Criterion (BIC), with BIC defined as BIC = n * ln(d(p)/(n−1)) + k * ln(n), where k is the number of model parameters. The BIC penalizes models for additional parameters to guard against overfitting. Rankings were determined primarily by least-squares residuals summed across all individual and combined trials for each cancer type, with BIC used as a secondary metric.

Two types of parameter sensitivity analysis were performed: (1) local parameter sensitivity analysis (univariate, varying each parameter by ±10%) and (2) the Partial Rank Correlation Coefficient (PRCC) test using Latin Hypercube Sampling (LHS) with 1000 randomized parameter vectors, which measures the statistical influence of parameters accounting for interactions between them.

Key Findings

• No universal best model. Model fit quality varied by tumor type. According to Table 3.3 rankings by least-squares residuals: the power law model ranked best for 6 of 10 cancer types (bladder, colon, melanoma, ovarian, pancreatic, renal cell carcinoma), logistic for 3 (breast, liver, lung), and Gompertz for 1 (head and neck squamous cell carcinoma). The von Bertalanffy model never ranked best.

• Von Bertalanffy consistently performed worst. The von Bertalanffy model ranked last (5th) for 7 of 10 tumor types; exponential ranked last for the remaining 3 (colon, liver, lung). Von Bertalanffy’s poor performance is compounded by the fact that its best-fit intrinsic growth rates are consistently two to three orders of magnitude smaller than those of other models, casting doubt on the biological veracity of its fitted parameters.

• Power law model is discouraged despite strong fitting performance. Although the power law model achieved the lowest least-squares residuals for 6 of 10 cancer types, the author explicitly discourages its use. The power law exponent (a) is extremely sensitive: increasing it by only 10% caused the predicted tumor size to grow by nearly 35000% in 10 days. The PRCC analysis indicates a highly nonlinear relationship between the power law parameters r and a, meaning researchers cannot justifiably vary one parameter while adjusting the other to preserve curve behavior. This rigidity and extreme sensitivity make the power law “a less than ideal choice.”

• Logistic model sometimes degenerates to exponential. In multiple cases (in vivo breast trial 3, in vivo HNSCC trial 4, several lung and ovarian trials), the logistic fitting returned a carrying capacity orders of magnitude higher than comparable trials with an intrinsic growth rate identical to the exponential fit. This indicates the logistic model effectively degenerated to exponential growth, because the carrying capacity was too high to constrain growth within the observed time window.

• BIC and least-squares residuals agree only 65% of the time. The trial with the lowest least-squares residuals also had the lowest BIC in 65% of cases; the two metrics disagreed in 35% of cases. The thesis ranked models primarily by least-squares residuals because all models had either 1 or 2 parameters, making BIC’s penalty for additional parameters redundant in this context. The author notes that choosing a different evaluation metric would produce a different ranked ordering.

• Data limitations constrain carrying capacity estimation. Very few datasets allowed tumors to grow large enough for proper estimation of carrying capacity. The thesis specifically sought out datasets including large-population measurements to enable fair comparison between models with and without carrying capacities.

Concepts Introduced or Used

• Exponential model: dP/dt = CP; unlimited growth, 1 parameter
• Power law model: dP/dt = CP^a; sub-exponential growth without carrying capacity, 2 parameters
• Logistic model: dP/dt = rP(1 − P/K); symmetric sigmoid, carrying capacity K, 2 parameters
• Gompertz model: dP/dt = r log(K/P) P; asymmetric sigmoid, carrying capacity K, 2 parameters
• Von Bertalanffy model: dP/dt = r(K − P); the simplest growth model incorporating a carrying capacity, linear in P, 2 parameters
• Carrying capacity (K): maximum sustainable tumor size; estimated from in vivo trials only
• BIC (Bayesian Information Criterion): model selection metric penalizing additional parameters
• PRCC (Partial Rank Correlation Coefficient): multivariate sensitivity analysis using Latin Hypercube Sampling
• clonal-expansion — growth model choice constrains interpretation of clonal expansion dynamics

Entities Referenced

• Tumor types fitted: bladder, breast, colon, head and neck squamous cell carcinoma (HNSCC), hepatocellular carcinoma (liver), lung, melanoma, ovarian, pancreatic, renal cell carcinoma (RCC) — 10 types total
• Software: MATLAB (custom hybrid Nelder-Mead + MCMC implementation, ode45, fminsearch, lhsdesign, corrcoef)
• Prior models referenced: de Pillis and Radunskaya (2006) on melanoma growth model comparison; Hart et al. (1998) on breast cancer power law growth; Aroesty et al. (1973) on Gompertz vs logistic; Lopez et al. (2004) on bacterial growth model comparison; Norton (1988) on Gompertzian breast cancer growth
• Data sources: 70+ individual datasets from at least 5 peer-reviewed publications per tumor type; includes in vitro and in vivo (SCID mice, nude mice, normal mice, hamsters, human) data

Limitations (as stated by author)

• Power law parameter instability. The power law model’s parameters are highly sensitive and unstable; intrinsic growth rates rise unpredictably to accommodate lower exponents and vice versa, preventing researchers from justifiably varying parameters within a fitted range. The author explicitly discourages its use.
• Von Bertalanffy biological veracity. Best-fit von Bertalanffy intrinsic growth rates are consistently two to three orders of magnitude smaller than those of other models, raising doubts about their biological accuracy.
• Fitting algorithm dependence on initial conditions. While the hybrid algorithm improves on Nelder-Mead alone, its outcomes still depend on initial parameter guesses, and the stochastic MCMC component does not always converge to the deepest global minimum.
• Evaluation metric dependence. BIC and least-squares residuals disagree on the best model in 35% of cases. Choosing a different evaluation metric would produce a different ranking of models.
• Unit conversion assumptions. Converting volume, area, and linear measurements to cell counts required assuming cubic or spherical tumor geometry and estimated cell volumes, introducing systematic error.
• Limited large-tumor data. Very few datasets allowed tumors to grow large enough for reliable carrying capacity estimation, limiting the ability to distinguish sigmoidal models from unconstrained growth models.
• Single-cancer sensitivity analysis. The PRCC and local sensitivity analyses were performed using parameters from in vitro colon cancer trials only; sensitivity behavior may differ for other tumor types and parameter regimes.
• CD47 model incomplete. The proposed CD47 treatment model could not be parameterized because the required macrophage and dendritic cell experimental data were either difficult to find or nonexistent.

Relevance to Clonal Evolution

This work is a practical reference for anyone interpreting clonal expansion dynamics from sequencing data. The choice of growth model directly affects estimates of when clones arose, how fast they expanded, and whether their observed frequencies suggest selection. If a Gompertz or logistic model is assumed but the true growth follows a different functional form, inferences about selective advantage from cancer-cell-fraction distributions may be systematically biased. The finding that no single model fits all tumor types reinforces the need for tumor-type-specific growth assumptions in evolutionary inference. However, the author’s explicit discouragement of the power law model — despite its top ranking in 6/10 cancers — and the questionable biological veracity of the von Bertalanffy parameters serve as important cautions: goodness-of-fit alone does not justify a model’s biological plausibility.