Heterogeneous Sensitivity Across Tumor Cell Subpopulations

Why One Tumor Can Show More Than One Response

A measured tumor burden can combine cells that do not respond to a drug equally. Some subpopulations may be killed quickly, while others are killed more slowly or continue to grow under the same exposure. The observed tumor trajectory is therefore not necessarily the trajectory of any one cell type: it is the sum of the contributions from all modeled subpopulations.

This distinction matters even when treatment exposure is constant. Initially, the most abundant subpopulation may dominate the measured response. Later, its contribution may have fallen, so a less sensitive subgroup can make up a larger share of the remaining tumor. The total burden can then shrink more slowly than it did at first, despite no change in the drug or in any subgroup’s sensitivity.

Definition: A sensitivity-defined subpopulation is a modeled group of tumor cells assigned a common set of growth and drug-response parameters. If $N_i(t)$ is the burden of subpopulation $i$, the total modeled burden is $N(t)=\sum_{i=1}^{m}N_i(t)$. The subgroup proportion, or composition weight, is $w_i(t)=N_i(t)/N(t)$, so $\sum_i w_i(t)=1$.

Subpopulation Dynamics and Total Burden

A simple model treats each subpopulation as growing and being killed independently, with its own net rate. For constant drug concentration $C$, let $g_i$ be the untreated per-capita growth rate of subgroup $i$, and let $k_i(C)$ be its drug-induced per-capita kill rate. Then its net rate is $a_i(C)=g_i-k_i(C)$. Under a constant exposure and constant rates, the model is

\frac{dN_i}{dt}=a_i(C)N_i,\qquad N_i(t)=N_i(0)e^{a_i(C)t},\qquad N(t)=\sum_{i=1}^{m}N_i(0)e^{a_i(C)t}.

The exponential solution assumes each subgroup has a fixed net rate over the time interval considered. The sum is appropriate when the measured burden is proportional to the total number of cells (or to a comparable additive quantity) and subgroup contributions are not double-counted. The model also assumes no switching between modeled groups, no migration into or out of the tumor, and no interactions that make one subgroup’s rate depend on another subgroup’s abundance. These are modeling assumptions, not automatic properties of tumor biology.

Drug sensitivity can be represented through the kill function. For example, a subgroup-specific Emax form is $k_i(C)=k_{\max,i}C/(EC_{50,i}+C)$. Under the same maximum kill rate, a lower $EC_{50,i}$ means greater sensitivity at a given positive concentration. If maximum kill rates also differ, sensitivity rankings can depend on concentration; a single label such as “more sensitive” may not describe every exposure.

Key result: The instantaneous proportional change in total burden is the composition-weighted mean of subgroup net rates: $$ \frac{d\log N}{dt}=\sum_{i=1}^{m}w_i(t)a_i. $$ Thus the initial aggregate rate depends on the initial composition, and the aggregate rate can change as the composition changes, even when every $a_i$ remains constant.

Why the Aggregate Rate Changes

The weighted-rate result follows by differentiating the total. Since $dN/dt=\sum_i a_iN_i$, division by $N$ gives $d\log N/dt=(1/N)\sum_i a_iN_i=\sum_iw_i a_i$. Each subgroup’s weight changes according to its rate relative to the current aggregate rate:

\frac{dw_i}{dt} =\frac{a_iN_iN-N_i(dN/dt)}{N^2} =w_i\left(a_i-\sum_jw_ja_j\right).

For constant rates, differentiating the aggregate log-rate once more gives

\frac{d^2\log N}{dt^2} =\sum_i a_i\frac{dw_i}{dt} =\sum_iw_i a_i^2-\left(\sum_iw_i a_i\right)^2 =\sum_iw_i(a_i-\bar a_w)^2\geq 0, \qquad \bar a_w=\sum_iw_i a_i.

The right-hand side is the composition-weighted variance of the net rates. It is zero when all represented subgroups have the same rate, and positive when rates differ and more than one subgroup is present. Consequently, with constant subgroup rates, the total log-burden slope becomes less negative over time if all subgroups are shrinking but at different speeds. This curvature does not mean any subgroup has changed its drug response: it can arise solely because the weights have changed. If some subgroups grow while others shrink, the total burden may still decline initially, but the growing subgroup can eventually dominate.

Worked Example: Three Subpopulations Under Constant Exposure

Consider a hypothetical tumor with initial burden 100 arbitrary burden units: 70 units in a sensitive subgroup, 20 in an intermediate subgroup, and 10 in a less sensitive subgroup. At the stated constant exposure, suppose their net rates are $-0.0500$, $-0.0200$, and $0.0100$ day$^{-1}$, respectively. The groups are assumed not to switch or interact.

At day 0, the composition weights are $0.70$, $0.20$, and $0.10$. The initial total log-burden slope is their weighted mean:

0.70(-0.0500)+0.20(-0.0200)+0.10(0.0100) =-0.0350-0.0040+0.0010 =-0.0380\ \text{day}^{-1}.

The negative value means total burden is initially declining. To calculate the burden at day 10, apply each subgroup’s own exponential rate, then add the subgroup burdens:

N(10)=70e^{-0.0500(10)}+20e^{-0.0200(10)}+10e^{0.0100(10)} \approx42.46+16.37+11.05 =69.88.

The average log-burden slope over these 10 days, calculated from the displayed rounded total, is

\frac{\log(69.88/100)}{10} \approx -0.03584\ \text{day}^{-1}.

This is less negative than the initial slope of $-0.0380$ day$^{-1}$. At day 10, the less sensitive, growing subgroup contributes about $11.05/69.88=0.1581$, or 15.81%, of the total, compared with 10% initially. The total has declined even though that subgroup has grown: its burden is about 11.05 units, while the two shrinking groups together have fallen from 90 to about 58.83 units.

Exposure, Sensitivity, and Starting Composition

A difference in sensitivity affects the total only in combination with exposure and subgroup abundance. To see this, consider a shared growth rate but subgroup-specific Emax kill functions. The calculation below assumes the concentration remains fixed, so each resulting net rate is constant during the interval. It illustrates how subgroup-specific response can be translated into a total trajectory; it does not establish that a particular Emax model is appropriate for real data.

Worked Example: Emax Kill Rates Produce Different Net Rates

Suppose, hypothetically, that both subgroups grow at $0.0400$ day$^{-1}$, and both have $k_{\max}=0.0800$ day$^{-1}$. Their $EC_{50}$ values are $0.500$ and $4.00$ concentration units. At a constant concentration of $2.00$ units, the kill rates are

k_1=\frac{0.0800(2.00)}{0.500+2.00}=0.0640\ \text{day}^{-1}, \qquad k_2=\frac{0.0800(2.00)}{4.00+2.00}=0.02667\ \text{day}^{-1}.

The corresponding net rates are $a_1=0.0400-0.0640=-0.0240$ day$^{-1}$ and $a_2=0.0400-0.02667=0.01333$ day$^{-1}$. Thus the first subgroup shrinks and the second grows at this exposure. If initial burdens are 80 and 20 units, respectively, the initial total log-rate is

0.80(-0.0240)+0.20(0.01333) =-0.01920+0.002667 =-0.01653\ \text{day}^{-1}.

After 10 days, the subgroup burdens are $80e^{-0.240}\approx62.93$ units and $20e^{0.1333}\approx22.85$ units. The total is about $85.78$ units, of which the second subgroup contributes $22.85/85.78\approx26.64\%$. It began at 20% of the tumor. The increase in its proportion follows from the difference in net rates; it does not require a change in the subgroup’s $EC_{50}$.

An observed total response therefore depends on both the response parameters and the initial mixture. Two tumors with the same subgroup-specific rates can show different early total slopes if their starting proportions differ. Conversely, the same initial total slope can arise from different combinations of proportions and rates. A single aggregate measurement, or even an initial aggregate slope, generally cannot identify all subgroup-specific parameters and proportions without additional information or constraints.

Worked Example: The Same Initial Rate Does Not Persist

Suppose a hypothetical tumor has two subpopulations with constant net rates $-0.0400$ and $0.0200$ day$^{-1}$, and initial burdens of 75 and 25 units. Its initial aggregate log-rate is $0.75(-0.0400)+0.25(0.0200)=-0.0250$ day$^{-1}$. A homogeneous model that holds this initial rate constant would predict a day-20 burden of

100e^{-0.0250(20)}=100e^{-0.500}\approx60.65\ \text{units}.

The heterogeneous model instead gives

N(20)=75e^{-0.0400(20)}+25e^{0.0200(20)} \approx33.70+37.30=71.00\ \text{units}.

The average log-burden slope over 20 days is $\log(71.00/100)/20\approx-0.0171$ day$^{-1}$, less negative than the initial value. By day 20, the second subgroup contributes about $37.30/71.00=52.5\%$ of total burden. The homogeneous calculation is a local approximation based on the initial slope; it does not reproduce the later trajectory in this example.

Diagnostics, Alternatives, and Limits

A heterogeneous model should be assessed against the structure it assumes, not merely against whether it can produce a curved total trajectory. Useful checks include the following:

Check the total against subgroup sums. At every time point, verify $N(t)=\sum_iN_i(t)$, and ensure the measured burden is plausibly additive across the modeled groups.
Check the implied composition. Under constant rates, a subgroup with a larger net rate must gain proportion relative to one with a smaller net rate. A fitted model implying the opposite signals a calculation or model mismatch.
Check the exposure assumption. If concentration changes materially, a constant-rate exponential over the whole interval may be inappropriate. For a known time-varying concentration and no switching, the subgroup solution is $N_i(t)=N_i(0)\exp\{\int_0^t[g_i-k_i(C(u))]\,du\}$.
Check whether curvature has other explanations. Changing exposure, changing growth conditions, measurement error, or unmodeled biology can also alter the observed slope. Curvature alone does not prove sensitivity heterogeneity.
Check identifiability. If only total burden is observed, several parameter combinations may produce similar trajectories. More subgroups and more free parameters do not automatically yield more reliable biological conclusions.

A homogeneous effective-rate model is simpler and may be adequate over a short interval or when subgroup rates are nearly equal. Its limitation is that it does not represent composition change: under a constant effective rate, the total is a single exponential and its log-slope is fixed. A discrete subpopulation model represents a small number of interpretable groups, but requires estimates of their rates and starting proportions. A continuous distribution of sensitivities can represent gradual variation rather than a few groups, but adds assumptions about that distribution and can be difficult to identify from aggregate data. The appropriate choice depends on the scientific question and the information available, not solely on which model fits most flexibly.

Common pitfall: “The aggregate tumor’s response slowed, so the drug became less effective over time.” That conclusion does not follow from the total trajectory alone. If faster-shrinking cells become a smaller fraction of the tumor, the aggregate slope can become less negative while every subgroup’s drug effect remains unchanged. Check the subgroup model and exposure assumptions before attributing the change to altered drug action.

The main limitation is that a total-burden curve usually does not reveal subgroup composition by itself. The model can explain how a specified mixture behaves, but its explanation is not uniquely established simply because the summed curve matches observed burden. Independent information about subgroup abundance or response can strengthen interpretation. Likewise, if cells switch between states, interact, or experience different exposures, the independent fixed-rate equations above need to be extended rather than interpreted literally.

Check Your Understanding

Work through the questions, then compare your reasoning with the solutions below.

Two subpopulations have current weights 0.60 and 0.40, with net rates $-0.0300$ and $0.0100$ day$^{-1}$. What is the instantaneous log-rate of total burden?
Under constant subgroup net rates, what happens to the total log-rate if all represented subpopulations have exactly the same net rate?
A growing subgroup has a lower initial weight than a shrinking subgroup. Under the no-switching, constant-rate model, how does its weight change relative to the shrinking subgroup?
Why does a total trajectory that becomes less steep during treatment not by itself show that drug sensitivity changed?

Solutions. 1. The weighted rate is $0.60(-0.0300)+0.40(0.0100)=-0.0180+0.00400=-0.0140$ day$^{-1}$, so total burden is initially declining. 2. If every rate equals $a$, then $d\log N/dt=\sum_iw_i a=a$, because the weights sum to 1. The aggregate log-rate is constant, and subgroup proportions remain unchanged. 3. Its weight increases relative to the shrinking subgroup because the growing subgroup has the larger net rate; the exact change in its share also depends on all subgroup rates and current weights. 4. Different fixed subgroup rates change the composition weights over time. Since the total log-rate is their weighted mean, that mean can become less negative as the faster-shrinking groups contribute less, without any change in drug sensitivity or exposure.