Deriving Tumor Control Thresholds from T-Cell Expansion and Killing

Why a Control Threshold Is a Balance, Not a Label

An increase in tumor-reactive T cells is not, by itself, evidence that tumor burden will fall. The relevant question is whether the T cells’ killing effect is large enough, at that time and tumor burden, to offset tumor-cell growth. A minimal growth-and-killing model makes that balance explicit and gives a threshold for the effector abundance required for control.

The threshold is conditional on the model. It depends on tumor growth, killing potency, and, when growth is density dependent, current tumor burden. It is an instantaneous condition on the rate of change, not a guarantee of long-term tumor elimination. The derivation below uses a simple model to make those distinctions precise.

A Minimal Model and Its Assumptions

Assume tumor burden follows logistic growth in the absence of immune killing, and that each unit of effector abundance adds a constant per-capita tumor killing rate. The model is

Here \(r\) has units of inverse time, \(B\) and \(K\) have the same burden units, and \(E\) has a specified abundance unit. Accordingly, \(k\) has units of inverse time per abundance unit, so \(kE\) is a per-capita killing rate. For positive \(B\), the sign of \(dB/dt\) is the sign of the bracketed net per-capita rate.

The assumptions are deliberately restrictive: tumor growth is logistic with fixed \(r\) and \(K\); killing is proportional to both \(B\) and \(E\); effector function is homogeneous across the tumor; and the parameters do not change over the interval considered. For the threshold calculation at a particular time, \(E\) can be treated as its current value even if it changes over time. This model does not represent spatial access, changing antigen, or a detailed immune-state network.

Deriving the Instantaneous Threshold

For a positive tumor burden, divide the equation by \(B\). Tumor burden is stationary at that instant when the per-capita growth rate equals the per-capita killing rate:

Solving for effector abundance gives the burden-dependent threshold:

Thus, for \(0<B<K\), the model predicts instantaneous decline when \(E>E_{\mathrm{crit}}(B)\), no instantaneous change when \(E=E_{\mathrm{crit}}(B)\), and growth when \(E<E_{\mathrm{crit}}(B)\). Equivalently, define the control ratio \(C=kE/\{r(1-B/K)\}\) when the tumor’s logistic per-capita growth rate is positive. Then \(C>1\) indicates killing exceeds growth, \(C=1\) is balance, and \(C<1\) indicates growth exceeds killing.

At low burden, \(B/K\) is near zero and the threshold approaches \(r/k\), the abundance needed to offset the maximum logistic per-capita growth rate. At higher burden, logistic growth per tumor cell is slower, so the instantaneous threshold is lower. This does not mean a large tumor is necessarily easier to treat: burden, model validity, and the possibility of regrowth as burden falls all matter.

What Expansion Adds to the Threshold

To ask whether expanding T cells can reach the threshold, suppose effector abundance changes according to

where \(g\) is the net per-capita effector expansion rate, equal in a simple proliferation-loss description to proliferation rate minus loss rate. This exponential approximation assumes a constant \(g\) and no upper limit on expansion. It is useful over a defined interval, not as a claim that T cells expand exponentially without bound.

In the low-burden approximation \(B\ll K\), tumor per-capita growth is approximately \(r\). The tumor’s net per-capita rate changes sign when \(kE(t)=r\). If \(g>0\) and \(kE_0<r\), the crossing time is

Before that time, the tumor grows under this approximation; after it, the instantaneous rate is negative, provided the model and expansion persist. If \(kE_0\geq r\), control is already present at \(t=0\). If \(g\leq 0\) and \(kE_0<r\), the effector abundance does not rise to the threshold in this model.

Crossing the Threshold Does Not Erase Earlier Growth

The tumor burden at threshold crossing reflects the entire preceding history of growth and killing. For the low-burden model with \(E(t)=E_0e^{gt}\), \(g>0\), and constant \(r\), dividing by \(B\) and integrating yields

The first equation follows from \(d\ln B/dt=(1/B)dB/dt\); integrating \(r\) gives \(rt\), and integrating \(kE_0e^{gt}\) gives \((kE_0/g)(e^{gt}-1)\). If the rate changes from positive to negative at \(t_{\mathrm{crit}}\), the burden reaches its maximum at that time in this simplified model. The maximum may be larger than the initial burden because tumor growth occurred before crossing.

Checks, Alternatives, and Limits

A threshold calculation is most useful when its assumptions and units are checked alongside the arithmetic. In particular, \(r\) and \(kE\) must both be rates with units of inverse time. A calculated threshold should be nonnegative for \(0\leq B\leq K\). If a model or parameter set implies negative effector abundance, or a tumor burden outside the intended range, the issue is not repaired by interpreting the threshold literally.

• Check the direction of the inequality. In the stated model, more effectors increase killing, so \(E>E_{\mathrm{crit}}\) must correspond to a negative net rate.
• Check whether expansion is net expansion. A positive proliferation rate does not ensure \(g>0\) when effector loss is also present.
• Check the time horizon. Exponential effector expansion and constant killing potency may be reasonable approximations over a short interval but need not persist.
• Check what the model calls control. The condition \(dB/dt\leq0\) describes an instantaneous direction of change; it does not alone establish durable elimination or a clinical response.

A common mistake is to conclude that any T-cell expansion guarantees tumor control. The relevant comparison is not whether \(E\) increases, but whether \(kE\) reaches the tumor’s per-capita growth rate. For example, if \(kE_0<r\) and \(g\leq0\), this model predicts no threshold crossing from expansion. Another mistake is to infer immediate shrinkage from crossing: a tumor can grow before the crossing and reach a larger peak, as the integrated example shows.

This threshold model is simpler than alternatives that include saturable killing, explicit effector-to-target dependence, or time-varying functional states. With saturable killing, the killing term no longer rises linearly with \(E\), so the threshold must be derived from that model’s specific rate law. A purely exponential tumor-growth model omits the burden-dependent reduction in growth near \(K\); it gives the low-burden threshold \(r/k\) directly. More detailed models can represent additional biology, but each requires parameters and data that support its extra structure.

Finally, the threshold identifies a balance of model rates, not a uniquely measured biological constant. Different combinations of effector abundance and killing potency can produce the same product \(kE\). Estimates of that product may therefore be more directly constrained by tumor trajectories than separate estimates of \(k\) and \(E\). Predictions should be revisited if measured tumor trajectories or effector dynamics systematically depart from the assumed rates.