Resistant Subclones Present Before Treatment

Why Baseline Subclones Matter

A tumor can contain cells that respond differently to the same treatment before therapy begins. If a small population is already less sensitive to a drug, the overall tumor may initially shrink while that population contributes to the remaining burden. A model that treats every cell as having the same response can therefore miss an important source of variation in predicted response.

This tutorial represents that baseline heterogeneity with two subclones: drug-sensitive cells and drug-resistant cells. “Resistant” is used in a relative, model-specific sense: at the specified treatment exposure, resistant cells have a lower drug-induced kill rate than sensitive cells. Resistance does not necessarily mean that treatment has no effect on those cells.

Definition: Let \(N_S(t)\) and \(N_R(t)\) denote the numbers of sensitive and resistant tumor cells at time \(t\). Their baseline counts are \(N_{S,0}\) and \(N_{R,0}\), and the total baseline burden is \(N_0=N_{S,0}+N_{R,0}\). The baseline resistant fraction is \(f_{R,0}=N_{R,0}/N_0\), and the sensitive fraction is \(f_{S,0}=N_{S,0}/N_0=1-f_{R,0}\).

A fraction is dimensionless and lies between 0 and 1. For example, \(f_{R,0}=0.01\) means that one percent of the modeled baseline cells are resistant. A fraction estimated from a sample describes that sample and may not perfectly represent the whole tumor; the distinction matters when interpreting predictions.

Two Clone-Specific Rates

For clone \(i\), where \(i\) is \(S\) or \(R\), let \(r_i\) be its untreated per-capita growth rate and \(k_i\) its per-capita drug-induced kill rate during treatment. Both rates have units of inverse time, such as day\(^{-1}\). In this formulation, a positive \(k_i\) removes cells at a rate proportional to the number of cells in that clone. The net rate during treatment is \(\lambda_i=r_i-k_i\): a negative value means that clone’s count declines, zero means it is constant, and a positive value means it grows.

The model assumes constant rates over the time interval being considered. In particular, it treats the drug exposure and its resulting kill rates as constant; this can be a useful approximation over a specified period, but not for every dosing schedule. Under treatment, the two clone-specific equations are:

\frac{dN_S}{dt}=(r_S-k_S)N_S=\lambda_S N_S, \qquad \frac{dN_R}{dt}=(r_R-k_R)N_R=\lambda_R N_R.

The total modeled burden is \(N(t)=N_S(t)+N_R(t)\). Since each clone follows constant-rate exponential dynamics, its count at time \(t\) is its baseline count multiplied by its own exponential growth factor. Consequently, total burden is a sum of two contributions, not generally a single exponential with one fixed rate:

N(t)=N_{S,0}e^{\lambda_S t}+N_{R,0}e^{\lambda_R t} =N_0\left[(1-f_{R,0})e^{\lambda_S t}+f_{R,0}e^{\lambda_R t}\right].

This expression makes the role of baseline composition explicit: each clone’s contribution to the total is weighted by its starting fraction. A negative sensitive net rate can produce substantial early shrinkage, while a positive resistant net rate adds an increasing contribution. The total can still be declining over an interval even if one clone is growing.

Justifying the Constant-Rate Solution

The solution follows directly from the clone-specific differential equation. For either clone, write \(dN_i/dt=\lambda_iN_i\) with constant \(\lambda_i\). When \(N_i>0\), divide by \(N_i\) and integrate from baseline time 0 to time \(t\):

\frac{dN_i}{N_i}=\lambda_i\,dt, \qquad \int_{N_{i,0}}^{N_i(t)}\frac{dN_i}{N_i} =\int_0^t\lambda_i\,dt, \qquad \ln\!\left(\frac{N_i(t)}{N_{i,0}}\right)=\lambda_i t.

Exponentiating gives \(N_i(t)=N_{i,0}e^{\lambda_i t}\). Adding the sensitive and resistant solutions gives the total-burden formula above. The calculation assumes that the two clone counts change only through their own growth and drug-kill terms. It does not include conversion between clone types, density-dependent interactions, or other sources and losses.

Worked Example: A Small Resistant Fraction During Treatment

Consider a hypothetical tumor with \(N_0=1.00\times10^9\) cells and baseline resistant fraction \(f_{R,0}=0.00100\). Suppose both clones have untreated growth rate \(r_S=r_R=0.0300\) day\(^{-1}\). During a constant treatment, let \(k_S=0.0700\) day\(^{-1}\) and \(k_R=0.0200\) day\(^{-1}\). Calculate the modeled total burden after 30 days.

The fractions are \(f_{R,0}=0.00100\) and \(f_{S,0}=1-0.00100=0.99900\). The net rates are \(\lambda_S=0.0300-0.0700=-0.0400\) day\(^{-1}\) and \(\lambda_R=0.0300-0.0200=0.0100\) day\(^{-1}\). Thus the sensitive clone declines under treatment, while the resistant clone grows under the model assumptions.

At \(t=30\) days, the clone multipliers are \(e^{\lambda_S t}=e^{-0.0400(30)}=e^{-1.20}\approx0.3012\) and \(e^{\lambda_R t}=e^{0.0100(30)}=e^{0.300}\approx1.3499\). Retaining the unrounded exponential values in the weighted calculation:

\begin{aligned} \frac{N(30)}{N_0} &=0.99900e^{-1.20}+0.00100e^{0.300}\\ &\approx0.99900(0.3011942)+0.00100(1.3498588)\\ &\approx0.3022429,\\ N(30)&=(1.00\times10^9)(0.3022429) \approx3.022\times10^8\ \text{cells}. \end{aligned}

The modeled burden is about 30.2% of its baseline value, a decrease of about \(100\%(1-0.3022429)=69.8\%\). The resistant fraction was only 0.100% at baseline, yet its positive net rate makes its contribution important to the total predicted burden. This is a model-based result for the specified rates and duration, not a claim that a measured tumor will follow those exact dynamics.

Worked Example: How Baseline Fraction Changes the Predicted Response

Use the same hypothetical rates and 30-day treatment as above, but compare two tumors that each begin with \(1.00\times10^9\) cells. Tumor A has a resistant fraction of 0.00100; Tumor B has a resistant fraction of 0.0500. Calculate the 30-day burden ratio for each.

For Tumor A, the weighted ratio is \(0.99900e^{-1.20}+0.00100e^{0.300}\), which is approximately 0.3022429. Its modeled burden is \(1.00\times10^9(0.3022429)\approx3.022\times10^8\) cells.

For Tumor B, the sensitive fraction is \(1-0.0500=0.9500\). Using the same unrounded multipliers:

\begin{aligned} \frac{N_B(30)}{N_{B,0}} &=0.9500e^{-1.20}+0.0500e^{0.300}\\ &\approx0.9500(0.3011942)+0.0500(1.3498588)\\ &\approx0.2861345+0.06749294\\ &\approx0.3536274,\\ N_B(30)&=(1.00\times10^9)(0.3536274) \approx3.536\times10^8\ \text{cells}. \end{aligned}

Tumor B’s modeled burden is about 35.4% of its baseline value, a decrease of about \(100\%(1-0.3536274)=64.6\%\). Under these assumptions, the higher baseline resistant fraction corresponds to less total shrinkage at day 30. This comparison isolates the effect of the chosen starting fractions: it holds the rates, baseline total burden, and treatment duration fixed.

Using a Response Target to Assess Baseline Composition

The same weighted-burden equation can be rearranged to ask which baseline resistant fractions are compatible with a specified response target. This is a model-based threshold calculation, not a procedure for establishing a patient’s clone fraction. It assumes the clone-specific rates and treatment duration are already specified and remain constant.

Worked Example: Maximum Baseline Resistant Fraction for a Burden Target

Use the rates in the previous examples and ask what baseline resistant fractions would yield a total burden no greater than 50% of baseline after 30 days. The 30-day multipliers are \(a=e^{-1.20}\approx0.3011942\) for sensitive cells and \(b=e^{0.300}\approx1.3498588\) for resistant cells. For resistant fraction \(f_{R,0}\), the total burden ratio is \((1-f_{R,0})a+f_{R,0}b\).

Set this ratio no greater than 0.500. Since \(b>a\), rearranging the inequality gives:

(1-f_{R,0})a+f_{R,0}b\leq0.500 \quad\Longrightarrow\quad f_{R,0}\leq\frac{0.500-a}{b-a} =\frac{0.500-0.3011942}{1.3498588-0.3011942} \approx0.1896.

Thus, for these hypothetical rates and a 30-day interval, the model predicts a burden at or below half the baseline burden when the baseline resistant fraction is about 19.0% or less. At the boundary \(f_{R,0}\approx0.1896\), the modeled ratio is approximately 0.500. This threshold changes if the rates, exposure, or duration change.

Assumptions, Diagnostics, and Alternatives

The two-clone model is interpretable because it makes a small number of explicit assumptions. Before using its predictions, check whether these are reasonable for the question and observation period:

The tumor is adequately represented by two groups with distinct, stable treatment responses during the modeled interval.
Each clone has a constant per-capita untreated growth rate and a constant per-capita kill rate under the specified treatment.
There is no switching between sensitive and resistant states, and no additional clone source or loss term.
The baseline fractions and clone-specific rates are specified or estimated with enough information to support the intended prediction.
The modeled cell burden is the quantity being compared. If the data are tumor volume, relating volume to viable-cell burden requires an appropriate measurement model.

Diagnostics should focus on mismatches between model implications and observations. For example, a single constant-rate model predicts exponential change for each clone under fixed treatment. Systematic changes in the observed rate of burden change, or inconsistent responses across treatment periods, may indicate time-varying exposure, changing growth conditions, an inadequate two-clone representation, or measurement limitations. Aggregate tumor measurements alone generally cannot identify both baseline fractions and clone-specific rates uniquely: different combinations can yield similar total burdens over a limited time range.

A single-population model is simpler and may be adequate when differential response is not material to the question or when only a short interval is being summarized. The two-clone model is preferable when there is evidence or a clear scientific reason to represent baseline response heterogeneity and when the available information can constrain its extra parameters. More elaborate models may represent changing exposure or more complex tumor biology, but additional detail does not guarantee more reliable inference; it also requires data that can inform the added structure.

A common mistake is to treat “resistant” as synonymous with “unaffected by treatment” and set \(k_R=0\) automatically. Resistance here is relative: it means a lower kill rate than \(k_S\) at the specified exposure, and \(k_R\) can still be positive. Another mistake is to assume that a very small resistant fraction is irrelevant because it contributes little at baseline. Its effect on a prediction depends jointly on its starting fraction, its net rate, and the duration considered; calculate its weighted contribution rather than dismissing it by inspection.

Key takeaway: Under constant clone-specific rates, total tumor burden is the sum of sensitive and resistant contributions, each weighted by its baseline fraction. Baseline composition can therefore change predicted shrinkage even when total starting burden and treatment are the same.

Check Your Understanding

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

A tumor has \(N_0=2.00\times10^8\) cells and baseline resistant fraction \(f_{R,0}=0.00400\). What are the baseline resistant and sensitive cell counts?
For a clone, \(r=0.0250\) day\(^{-1}\) and \(k=0.0250\) day\(^{-1}\). What is its net rate under treatment, and what does that imply for its modeled count while the rates remain constant?
Why can a tumor with a small baseline resistant fraction still have less predicted shrinkage than an otherwise identical tumor with no resistant cells?
For the 30-day rates used in the worked examples, would a baseline resistant fraction of 0.200 produce a burden ratio below or above 0.500? Show the calculation.

Solutions. (1) \(N_{R,0}=f_{R,0}N_0=0.00400(2.00\times10^8)=8.00\times10^5\) cells. The sensitive count is \(N_{S,0}=2.00\times10^8-8.00\times10^5=1.992\times10^8\) cells. (2) \(\lambda=r-k=0.0250-0.0250=0\) day\(^{-1}\), so \(N(t)=N_0e^{0t}=N_0\); the modeled clone count is constant during that interval. (3) Even if its initial contribution is small, the resistant clone has a larger treatment-period multiplier than the sensitive clone in the examples. Its weighted contribution raises the total relative to an otherwise identical tumor without that clone. (4) With \(f_{R,0}=0.200\), the ratio is \(0.800(0.3011942)+0.200(1.3498588)=0.2409554+0.2699718=0.5109272\), approximately 0.511. It is above 0.500, so the model predicts a burden greater than half of baseline at day 30.