Evolutionary antifragile therapy

Abstract

A population of cells within a tumor can be described as antifragile (the opposite of fragile) if they derive a benefit from fluctuations or perturbations in environmental conditions induced by treatment. We hypothesized that treatment fluctuations may either promote (antifragile tumor) or inhibit (fragile tumor) the evolution of treatment resistance to targeted therapies. Analysis of the convexity of dose response curves provides a direct prediction of response to prescribed fluctuations in treatment. Theory predicts that continuous treatment protocols (i.e. zero prescribed fluctuations in treatment) will outperform uneven protocols when dose response is convex. We apply this theory to predict in vivo response to targeted therapy and validate these predictions by experimentally testing high/low intermittent dosing (uneven dosing) and continuous dosing (even dosing) schedules. Convexity (and its inverse, concavity) explains two phenomena: dose response is a convex (fragile) function but resistance onset rate is a concave (antifragile) function. Thus, we design, propose, and validate alternative treatment schedules which maximize response while maintaining prolonged sensitivity to treatment. These analyses provide supporting evidence that fluctuations alter the evolutionary trajectory of tumors in response to targeted therapy, even without altering the cumulative dose. This insight is used to design alternative, switching treatment protocols to limit the evolution of resistance, a process which we call evolutionary antifragile therapy.

Figure 1.. Schematic of convex and concave curvature

(A) A schematic of sample dose schedules with equivalent mean dose, $\bar{x}=\frac{1}{T}{\int }_0^{T}x\left(t\right)dt$. Shown for a range of dose variance, $\sigma$, from even (purple) to uneven (green).(B,C) Example dose response curves: convex (fragile) shown in B, and concave (antifragile) shown in C. By Jensen’s inequality, the optimal kill is achieved by even treatment if convex (B), or uneven treatment if concave (C). (D) The curvature at a given dose $\bar{x}$, may be cell line dependent (left). Curvature predicts optimal schedule (right). (E) Classification of tumors: antifragile is a situation where dose variance causes benefit to patient, and harm to tumor. (F) Schematic of fragility, F, as a function of dose variance. (G) Example dose response (eqn. 4), delineated with antifragile (left) and fragile (right) regions. (H) Using the dose response in G, model predicted fragility as a function of variance (see eqn. 3) (I) Using the dose response in G, model predicted fragility as a function of dose mean, $\bar{x}$ (see eqn. 3)

Figure 2.. Model parameterization of high and low dose treatment with even and uneven schedules

(A) Two cell state model consisting of drug-naive cells, $C_1$, which transition into drugged cells, $C_2$ as a function of the drug concentration (see Methods eqn. 6 and 7). (B) In vivo dose response curve determines dosing model parameters: ${\hat{b}}_1,{\hat{b}}_2,k_{on},k_{off}$ (see Methods eqn. 8). (C) Due to the convex shape of the dose response function in B, the model-predicted fragility, $F$, is fragile ($F>0$) for the full range of dose variance ($\sigma >0$) and dose mean ($\bar{x}>0$). (D) Four cohorts of 6–8 mice each (see figure 3): high dose (even, uneven dosing) and low dose (even, uneven dosing) (E) Static dose response model (see Methods eqns. 6, 7 where $k_{on}=\text{const}.$) initially fits the data well in the first week, but fits poorly after onset of resistance. (F) Dynamic dose response model (see Methods eqns. 6, 7, with 9) improves the model fit by including ${\alpha }_i$, the rate of resistance onset, where the subscript $i$ indicates the dose delivered each week: $i\in \left\{\mathrm{0,12.5,25,50}\right\}$.

Figure 3.. Mouse-specific parameterization of high and low dose treatment with even and uneven schedules

(A) Uneven high dose (50 mpk for one week followed by 0 mpk for one week). (B) Even, high dose (25 mpk). (C) Uneven low dose (25 mpk for one week followed by 0 mpk for one week). (D) Even, low dose (12.5 mpk). Tumor volume measurements are normalized by initial measurement, $V_0$, and a best fit parameterization to the two cell-state model (see Methods eqns. 6, 7). Each single mouse shares dose response parameters across all cohorts (${\hat{b}}_1,{\hat{b}}_2,k_{on}{,k}_{off}$; see figure 2B black dashed curve) but retains a mouse-specific value(s) for ${\alpha }_i$, the rate of resistance onset, where the subscript $i$ indicates the dose delivered each week: $i\in \left\{\mathrm{0,12.5,25,50}\right\}$.

Figure 4.. Rate of resistance is a concave function of dose

(A) Parameterization of the resistance rates $\alpha$ (see Methods eqns. 9) from figure 3 as a function of dose, $x$. The data are fit to a concave best fit (dashed line). (B) Model-predicted EC₅₀ values are much lower for uneven schedules (green) than even schedules (purple) due to the concave shaped $\alpha \left(x\right)$. (C) Model-predicted dose response for the high dose uneven schedules (green) and even schedules (purple). (D) (C) Model-predicted dose response for the low dose uneven schedules (green) and even schedules (purple).

Figure 5.. Modeling predicts that alternative dosing schedules can preserve sensitivity while maximizing tumor kill

(A,B) Based on parameterization in figure 3, two alternative schedules are simulated: Uneven dosing (6 weeks) followed by even dosing (8 weeks) shown in blue. Even dosing (6 weeks) followed by uneven dosing (8 weeks) shown in red. (C) Modeling-predicted final tumor volume. The even-uneven or uneven-even switching schedules are non-inferior to even dosing, despite maintaining a superior EC₅₀ value in (D). (D) Modeling-predicted final EC₅₀ value. Uneven schedules minimize EC₅₀, followed by even-uneven, then uneven-even switching schedules.

Figure 6.. Validation experiments for proposed alternative treatment schedules

(A) Even, high dose (25 mpk) (B) Even-uneven switching: 25 mpk for six weeks followed by 50/0 mpk uneven weekly (C) Uneven-even switching: 50/0 mpk uneven for six weeks followed by 25 mpk weekly. (D) Uneven 50/0 schedule. Note: weeks 12–13 and 18–19 were administered the same {0,50} mpk and {0,25} mpk across all cohorts to evaluate sensitivity to same drug dose for each cohort (see figure 7).

Figure 7.. Evaluation of treatment sensitivity across treatment schedules

Top: sensitivity to treatment (proliferation rate, DIP) at evaluation time point 1 (A) Response to zero dose in week 13 (B) response to high dose (50mpk) in week 14 (C) net response to both doses across weeks 13–14. Bottom: Model-predicted response and sensitivity for theoretical dosing compared to true dosing. (D) Final normalized tumor volume, shown for the true dosing schema (including evaluation time points). (E) Model-predicted final normalized volume with evaluation time points removed and simulated for the theoretical proposed alternative schedules in 5C. (E) Corresponding model-predicted EC₅₀ values for the same theoretical proposed alternative schedules.