Gompertzian Tumor Growth and Size-Dependent Growth Rates

Why Tumor Growth Can Slow as Size Increases

A tumor may become larger while the fraction of its volume added per day becomes smaller. This distinction between absolute growth and per-capita growth matters when interpreting longitudinal tumor measurements: a declining fraction does not necessarily mean that fewer volume units are being added each day. The Gompertz model represents this size dependence with a per-capita rate that declines with the logarithm of tumor size.

Here, $V(t)$ denotes tumor volume at time $t$. As in other tumor-growth models, volume is a measured proxy for tumor burden; it need not equal the number of viable tumor cells. The model describes untreated net growth over a specified range. It does not, by itself, explain the biological mechanism causing growth to slow or describe a drug effect.

Definition: The Gompertz tumor-growth model is $$ \frac{dV}{dt}=\alpha V\ln\!\left(\frac{K}{V}\right), $$ where $V>0$ is tumor volume, $K>0$ is the model's asymptotic size parameter, and $\alpha>0$ has units of inverse time. The per-capita growth rate is the rate of change divided by current volume: $$ \frac{1}{V}\frac{dV}{dt}=\alpha\ln\!\left(\frac{K}{V}\right). $$

If volume is measured in $\text{mm}^3$ and time in days, then $K$ is in $\text{mm}^3$, while $\alpha$ is in $\text{day}^{-1}$. The logarithm's argument, $K/V$, is dimensionless because its numerator and denominator have the same units. The absolute growth rate $dV/dt$ is measured in $\text{mm}^3/\text{day}$.

The usual interpretation for an untreated growing tumor is $0<V<K$. In this range, $\ln(K/V)>0$, so $dV/dt>0$. At $V=K$, the model's instantaneous growth rate is zero. If the equation is applied mathematically for $V>K$, it predicts negative net growth; that prediction should not automatically be taken as a biological explanation for tumor regression.

Two Different Rates, Two Different Patterns

The per-capita rate answers, “What fraction of the current volume is added per unit time?” The absolute rate answers, “How many volume units are added per unit time?” From the model, $r(V)=\alpha\ln(K/V)$ is the per-capita rate, whereas $g(V)=\alpha V\ln(K/V)$ is the absolute rate. These quantities have different units and need not move in the same direction as tumor size changes.

For fixed $K$ and $\alpha$, the per-capita rate decreases as $V$ increases. Its derivative with respect to volume is $-\alpha/V$, which is negative for every $V>0$. Absolute growth, however, first rises and then falls over $0<V<K$. Its maximum occurs at $V=K/e$, as shown below.

Result: In the Gompertz model, the per-capita growth rate declines monotonically with size. The absolute growth rate is maximal at $V=K/e$, where it equals $\alpha K/e$. Thus, the tumor can keep growing even after its absolute growth rate has begun to decline.

Differentiate the absolute rate.
For fixed parameters, $g(V)=\alpha V\ln(K/V)$. Its derivative is $\alpha[\ln(K/V)-1]$, since the derivative of $V\ln(K/V)$ is $\ln(K/V)-1$.

Find the stationary size.
Set the derivative to zero: $\ln(K/V)-1=0$. Therefore $\ln(K/V)=1$, so $K/V=e$ and $V=K/e$.

Check that it is a maximum.
The second derivative is $-\alpha/V$, which is negative for all $V>0$. The stationary point is therefore a maximum. Equivalently, $g'(V)>0$ below $K/e$ and $g'(V)<0$ above it.

This calculation establishes the location of peak absolute growth without solving for the tumor's full trajectory over time. The trajectory and its derivation are a separate question. For interpretation at a particular time, the key is to evaluate the rate at the current $V$, then distinguish its per-capita and absolute forms.

Worked Example: Comparing Per-Capita and Absolute Growth

Suppose a hypothetical untreated tumor is described by $K=1{,}000\ \text{mm}^3$ and $\alpha=0.12\ \text{day}^{-1}$. Compare the model's instantaneous rates at $V=200\ \text{mm}^3$ and $V=400\ \text{mm}^3$.

At $200\ \text{mm}^3$, the per-capita rate is $$ r(200)=0.12\ln(1{,}000/200)=0.12\ln(5) =0.1931\ \text{day}^{-1}. $$ The absolute rate is $$ g(200)=200(0.12\ln 5)=38.63\ \text{mm}^3/\text{day}. $$

At $400\ \text{mm}^3$, the per-capita rate is $$ r(400)=0.12\ln(1{,}000/400)=0.12\ln(2.5) =0.10995\ \text{day}^{-1}. $$ The absolute rate is $$ g(400)=400(0.12\ln 2.5)=43.98\ \text{mm}^3/\text{day}. $$

The larger tumor has a lower per-capita rate but a higher absolute rate in this comparison. The peak absolute-growth size is $K/e=1{,}000/e\approx367.9\ \text{mm}^3$: $200\ \text{mm}^3$ is below the peak and $400\ \text{mm}^3$ is above it. Absolute growth rises up to $K/e$ and falls thereafter, yet the rate at $400\ \text{mm}^3$ remains greater than at $200\ \text{mm}^3$ (about 43.98 versus 38.63 $\text{mm}^3/\text{day}$). Being above the peak does not mean that the rate must already be lower than it was at every smaller size.

Estimating and Interpreting Model Parameters

The parameter $\alpha$ controls the time scale of change: for a given $V/K$, a larger $\alpha$ gives a proportionally larger instantaneous rate. The parameter $K$ sets the size at which the model's instantaneous growth rate reaches zero and, under the usual positive-growth interpretation, the asymptotic size. Neither parameter should be treated as a direct measurement of a unique biological mechanism.

If an instantaneous absolute rate and volume are available, the model can be rearranged to calculate $\alpha$, provided $K$ is specified. Alternatively, per-capita rates at two sizes can be used to calculate both parameters under the model. These calculations assume that the rates are compatible with the same untreated model and that the sizes and rates are measured on a common scale.

Worked Example: Calculating the Time-Scale Parameter

Suppose a hypothetical tumor has volume $200\ \text{mm}^3$, the model uses $K=800\ \text{mm}^3$, and its estimated instantaneous absolute growth rate is $30\ \text{mm}^3/\text{day}$. Calculate $\alpha$.

Rearranging $dV/dt=\alpha V\ln(K/V)$ gives $$ \alpha=\frac{dV/dt}{V\ln(K/V)} =\frac{30}{200\ln(800/200)} =\frac{30}{200\ln(4)}. $$

Since $\ln(4)=1.38629$, the denominator is $200(1.38629)=277.258\ \text{mm}^3$. Thus $$ \alpha=\frac{30}{277.258}=0.1082\ \text{day}^{-1}. $$

The units reduce to inverse days because the observed rate has units $\text{mm}^3/\text{day}$ and the denominator has units $\text{mm}^3$. With this parameter and $K$, the model's per-capita rate at $200\ \text{mm}^3$ is $\alpha\ln 4\approx0.1500\ \text{day}^{-1}$. Multiplying by $200\ \text{mm}^3$ recovers approximately $30.00\ \text{mm}^3/\text{day}$, checking the calculation.

Worked Example: Inferring Parameters From Two Per-Capita Rates

Suppose hypothetical instantaneous per-capita growth rates are $0.300\ \text{day}^{-1}$ at $100\ \text{mm}^3$, and $0.100\ \text{day}^{-1}$ at $400\ \text{mm}^3$. Assume both measurements follow one Gompertz model. Estimate $\alpha$ and $K$, then check the implied rate at $200\ \text{mm}^3$.

Write the model's per-capita rate at each size: $$ 0.300=\alpha\ln(K/100),\qquad 0.100=\alpha\ln(K/400). $$ Subtract the second equation from the first. The difference between the logarithms is $\ln(400/100)=\ln 4$, so $$ 0.200=\alpha\ln 4,\qquad \alpha=\frac{0.200}{1.38629}=0.1443\ \text{day}^{-1}. $$

Substitute into the first equation: $$ \ln(K/100)=\frac{0.300}{0.1443}\approx2.0790, \qquad K\approx100e^{2.0790}=800\ \text{mm}^3. $$

At $200\ \text{mm}^3$, the implied per-capita rate is $$ 0.1443\ln(800/200)=0.1443\ln 4\approx0.2000\ \text{day}^{-1}. $$ This intermediate value is consistent with a rate that declines as size increases from 100 to 400 $\text{mm}^3$. The inference is conditional on the assumed model and the rates being instantaneous per-capita rates, not simply interval averages.

Assumptions, Diagnostics, and Alternatives

The equation is a deterministic, single-burden growth law with fixed $K$ and $\alpha$ over the modeled interval. It assumes that current volume is sufficient to determine the instantaneous rate: two tumors with the same $V$ and parameters have the same predicted rate, regardless of their history. It also assumes a smooth trajectory and does not explicitly represent measurement error, changing treatment, changing tumor composition, or spatial variation within the tumor.

A basic model check is whether estimated per-capita rates decrease with size and whether the decrease is compatible with the model's specific dependence on $\ln(K/V)$. For example, with fixed parameters, the absolute rate must peak at $K/e$; observations strongly inconsistent with that pattern may signal model mismatch, noisy rates, or changing conditions. In practice, derivatives estimated from sparse, noisy volume measurements can be unstable, so apparent short-term increases or decreases should not be treated as decisive evidence on their own.

Check scale and units. $K$ and $V$ must use the same volume units; $\alpha$ must use the reciprocal of the time unit.
Check the domain. The conventional growing-tumor interpretation requires $0<V<K$. A fitted $K$ below observed volumes needs careful investigation rather than a routine positive-growth interpretation.
Check residual patterns and parameter plausibility. Systematic disagreement across size ranges can indicate that one fixed Gompertz law is inadequate. A good fit over a narrow range does not establish that its asymptote or rate pattern will remain valid outside that range.
Check what the measurement represents. A measured volume may include nonviable material or reflect measurement conventions. A model for measured volume is not automatically a model for viable-cell number.

The exponential model uses a constant per-capita growth rate, so it cannot represent a rate that declines with size without additional structure. The logistic model introduced in the earlier material on logistic tumor growth also has a size-dependent per-capita rate, but its dependence on size differs from the Gompertz model: logistic per-capita growth declines linearly with $V$, whereas Gompertz per-capita growth declines with $\ln V$, for fixed parameters. These alternatives are not interchangeable at every size range; choosing among them requires considering the data, the scientific question, and how extrapolation will be used.

Common Pitfalls and Practical Meaning

A frequent error is to read “growth slows” as meaning that absolute growth must fall whenever the tumor gets larger. In this model, it is the per-capita rate that always declines with size. Absolute growth rises below $K/e$, reaches its maximum there, and falls above it. Another error is to call $K$ the tumor's observed maximum size: it is a model parameter that the trajectory approaches under the usual model, not necessarily a measured boundary or a mechanistically established capacity.

The distinction matters in pharmacology because a baseline growth law supplies the untreated reference against which treatment-related changes may later be assessed. It does not identify treatment effects by itself. A lower observed growth rate could reflect the natural size-dependent pattern, a treatment effect, changing measurements, or other processes; attribution requires a model and study design that distinguish these possibilities.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

For $K=600\ \text{mm}^3$, at what volume is absolute growth maximal? Give the approximate value and units.
If a tumor's volume increases while its Gompertz per-capita growth rate declines, must its absolute growth rate also decline? Explain.
At $V=K/2$, express the per-capita growth rate in terms of $\alpha$, and state its units.
Why is it inappropriate to interpret a fitted $K$ automatically as a directly measured biological carrying capacity?

Solutions. (1) The maximum occurs at $K/e=600/e\approx220.7\ \text{mm}^3$. (2) No. Absolute growth increases with size below $K/e$, peaks at $K/e$, and decreases above it; a declining per-capita rate alone does not determine the direction of absolute growth. (3) $r(K/2)=\alpha\ln 2$, with units of inverse time. (4) $K$ is the asymptotic size parameter in a specified mathematical model. Its interpretation depends on the model assumptions and data; it is not necessarily an observed maximum or a uniquely established biological mechanism.