Gompertzian Growth
Definition
Gompertzian growth is a mathematical model of population growth in which the relative growth rate declines exponentially as the population size increases, producing an asymmetric S-shaped (sigmoid) curve that plateaus at a carrying capacity. The model is given by the differential equation:
[ \frac{dN}{dt} = r N \ln\left(\frac{K}{N}\right) ]
where N is the number of tumor cells, r is the intrinsic growth rate, and K is the carrying capacity (asymptotic maximum tumor size). The specific growth rate (dN/dt divided by N) is r ln(K/N), which decreases logarithmically as N approaches K (Sarapata, 2013).
Originally formulated by Benjamin Gompertz (1825) for actuarial survival tables, the model was applied to tumor growth by Laird (1964, 1965) and has since become the dominant growth model in mathematical oncology.
Key Properties
Asymmetric sigmoid. Unlike the logistic model (dN/dt = rN(1 − N/K)), which is symmetric about its inflection point at K/2, the Gompertz curve has its inflection point at K/e ≈ 0.37K. This means growth begins to decelerate earlier — when the tumor is at only 37% of its final size — which better matches empirical tumor growth data where deceleration due to nutrient limitation, spatial constraint, and angiogenic bottlenecks occurs well before the carrying capacity is approached.
Decelerating relative growth rate. The key distinction from exponential growth is that in Gompertzian growth, the relative growth rate — the rate of growth divided by the current size — falls exponentially with time. Small tumors grow near-exponentially; large tumors grow slowly or imperceptibly. This matches clinical observations: many tumors exhibit rapid early growth followed by prolonged indolent phases (Traina et al., 2010).
Size, not time, governs growth rate. Two tumors of different sizes have different growth rates even if they are the same chronological age. This has therapeutic implications: the optimal timing of chemotherapy depends on tumor size, not calendar time (Traina et al., 2010; Castorina et al., 2009).
Fit to Empirical Data
Across systematic comparisons, Gompertz is among the best-fitting models but not universally dominant. Sarapata (2013) found that the power law model ranked best for 6 of 10 tumor types, logistic for 3, and Gompertz for 1 (head and neck squamous cell carcinoma); the von Bertalanffy model never ranked best and placed last for 7 of 10 types. Hassan & Al-Saedi (2024) found that the Bertalanffy model achieved the best fit to their single dataset (SSR = 367,276 after optimization), with Gompertz second (SSR = 893,826 after optimization, a 97.5% reduction from its pre-optimization SSR). The Gompertz parameters are highly sensitive to the fitting method — small errors in r or K can produce large errors in predicted growth trajectories (Hassan & Al-Saedi, 2024).
The finding that no single model fits all tumor types (Sarapata, 2013; Hassan & Al-Saedi, 2024) means that assuming Gompertzian growth when the true dynamics are Bertalanffy or power law can introduce systematic bias in evolutionary inference.
Relevance to Clonal Evolution
Growth model determines the null expectation for neutrality
The neutral-evolution null model — the 1/f distribution of mutation frequencies against which selection is tested — is derived under the assumption of exponential growth (Graham & Sottoriva, 2017). Under Gompertzian growth, the population expansion rate decelerates over time, which alters the expected VAF distribution: in a decelerating population, fewer new mutations are generated late in growth (because fewer cell divisions occur), so the low-frequency tail of the distribution is depleted relative to the exponential expectation.
This means that a Gompertzian-growing tumor with no selection could produce a VAF distribution that deviates from the 1/f expectation — potentially being misclassified as showing selection. Conversely, a tumor with weak selection under Gompertzian growth could produce a distribution indistinguishable from the 1/f neutral null.
Clone detectability depends on when in the growth curve a clone arises
A driver-mutation arising early in Gompertzian growth — when the population is small and expanding rapidly — has both time and population growth to reach detectable frequency. The same driver arising late — when the population is near carrying capacity and growth is near zero — may never reach detectable size regardless of its selective advantage. This is the temporal constraint described by the Big Bang model (Sottoriva et al., 2015): early-arising private mutations become pervasive because they had more doublings to expand; late-arising mutations remain localized not because they lack fitness but because they lack time.
Therapeutic scheduling
The Norton-Simon hypothesis — that the rate of tumor regression under chemotherapy is proportional to the Gompertzian-predicted unperturbed growth rate — forms the basis of dose-dense chemotherapy scheduling (Traina et al., 2010). Because Gompertzian growth rate depends on tumor size, the optimal drug dose depends on when in the growth curve treatment begins. Castorina et al. (2009) showed that when intratumor-heterogeneity is present — multiple subclones with different Gompertzian growth parameters — the single-population Norton-Simon schedule becomes suboptimal, requiring modification based on the balance of subclone growth rates.
Open Question
Whether the 1/f neutral null model — derived under the assumption of exponential growth (Graham & Sottoriva, 2017) — needs Gompertzian correction is unresolved. If the null model assumes exponential growth, then non-exponential growth would predictably alter the expected VAF distribution, but the magnitude of this effect across tumor types remains unquantified (Hassan & Al-Saedi, 2024; Sarapata, 2013).