Second-Order Effects of Chemotherapy Pharmacodynamics and Pharmacokinetics on Tumor Regression and Cachexia

Abstract

Drug dose response curves are ubiquitous in cancer biology, but these curves are often used to measure differential response in first-order effects: the effectiveness of increasing the cumulative dose delivered. In contrast, second-order effects (the variance of drug dose) are often ignored. Knowledge of second-order effects may improve the design of chemotherapy scheduling protocols, leading to improvements in tumor response without changing the total dose delivered. By considering treatment schedules with identical cumulative dose delivered, we characterize differential treatment outcomes resulting from high variance schedules (e.g. high dose, low dose) and low variance schedules (constant dose). We extend a previous framework used to quantify second-order effects, known as antifragility theory, to investigate the role of drug pharmacokinetics. Using a simple one-compartment model, we find that high variance schedules are effective for a wide range of cumulative dose values. Next, using a mouse-parameterized two-compartment model of 5-fluorouracil, we show that schedule viability depends on initial tumor volume. Finally, we illustrate the trade-off between tumor response and lean mass preservation. Mathematical modeling indicates that high variance dose schedules provide a potential path forward in mitigating the risk of chemotherapy-associated cachexia by preserving lean mass without sacrificing tumor response.

1. Introduction

First-order effects relate the system’s response to altering the input mean while second-order effects relate the system’s response to modifying the input variance. A mathematical framework used to quantify these so-called second-order effects is known as antifragile theory (Taleb 2012; Axenie et al. 2023). The term antifragile was coined to describe systems that benefit from volatility in the input distribution, serving as a contrast to fragile systems that are harmed by input volatility. Thus far, the concept has been heavily applied to financial markets, where an individual’s investment portfolio may be susceptible to market volatility (fragile), leading them to financial ruin (Taleb and Douady 2013). In contrast, it’s possible to design an investment strategy based on the premise of deriving financial benefit from volatile fluctuations in market prices.

These concepts, rooted in probability theory, can be readily translated into oncology (Taleb and West 2023; Bayer and West 2023). For example, cancer chemotherapy is intended to induce dramatic perturbations to a tumor, leading to increased cell death. The timing of these perturbations is determined by a treatment scheduling protocol. By definition, each chemotherapy treatment scheduling protocol has an associated mean and variance. Thus, we hypothesize that treatment design should account for both first-order and second-order effects. Historically, the design of chemotherapeutic schedules was intended to maximize the first-order effect, typically through a “maximum-tolerable dose” (MTD) paradigm. Improving a first-order effect is achieved either through increasing the cumulative dose delivered (Piccart et al. 2000) or shortening the time between subsequent doses (Citron 2008). In order to improve the design of chemotherapy scheduling, we investigate how second-order effects influence treatment outcomes.

1.1. Antifragility in Cancer

Mathematical modeling is highly suitable to quantify second-order effects and to predict the effectiveness of drug scheduling protocols (Taleb and West 2023; Altrock et al. 2015; Anderson and Quaranta 2008). Previously, we introduced a framework for “antifragile therapy” wherein the curvature of the dose response curve determines preferable treatment schedule (West et al. 2021). In this initial work, we considered two treatment protocols: (1) even dosing: continual administration of the same dose at regular intervals, and (2) uneven dosing: a high dose followed by a low dose, in alternating fashion. Examples of this bolus dosing are shown in Fig. 1A. Assuming no drug decay (Fig. 1B), the treatment outcomes can be determined by observing the shape (curvature) of the dose response function (Fig. 1C).

Fig. 1.

Drug-induced proliferation convexity predicts tumor evolution. A, B Schematics of different dose schedules from even to uneven treatmentwith same cumulative dose delivered.C Examples of drug-induced proliferation (DIP) curves with differing convexity. Green dot indicates difference between the average DIP for two doses versus the DIP at the average dose delivered. D Predicted tumor evolution based on the convexity of the DIP curve. When the DIP curve is concave, more tumor is eliminated by uneven treatment cycles, so the tumor is antifragile. Inversely, if the DIP curve is convex, then even treatment cycles are preferable and the tumor is fragile (Color figure online)

In the event that a dose response curve is convex, even dosing is preferable (Fig. 1D; see also Supplemental Section S1). Conversely, if the underlying dose response is concave, uneven dosing is preferable: a high dose followed by a low dose. Others have followed similar approaches in applying antifragile theory to cancer applications including immune-tumor interactions (Axenie et al. 2022), biological boolean networks (Pineda et al. 2019), molecular biology and evolution (Danchin et al. 2011).

A major limitation of previous work applying antifragility to treatment scheduling has been the lack of pharmacokinetic considerations. Our previous work (West et al. 2021; Taleb and West 2023; Bayer and West 2023) has considered drug pharmacodynamics in isolation, where the curvature of dose response curves may provide predictive value in a limited preclinical (e.g. in vitro) or theoretical context. Figure 1 illustrates the predictive power of this graphical approach that links dose response convexity and concavity to outcomes, but this graphical approach is confounded by the drug’s pharmacokinetic decay and absorption before the drug reaches the target tumor. To rectify this limitation, we introduce a mathematical model of tumor growth, response, and chemotherapy-induced cachexia as a test-bed for quantifying the second-order effects of treatment. Here, we go beyond previous results to include both drug pharmacodynamics (PD) and pharmacokinetics (PK) to quantify the second-order effects of chemotherapy on tumor regression.

To do so, we begin with a simple, single-compartment PK/PD model to illustrate the influence of PK decay rates on emergent second-order treatment effects. We introduce a metric that describes tumor fragility (or antifragility) that predicts whether even or uneven dosing achieves greater tumor regression. In the subsequent sections, we apply this metric to a previously published mouse-parameterized two-compartment PK/PD model of 5-fluorouracil (Farhang-Sardroodi and Wilkie 2020; Farhang-Sardroodi et al. 2022). This model allows us to explore the effect of an additional pharmacokinetic compartment on tumor fragility. Importantly, the model also considers a potential side effect of chemotherapy through cachexia.

1.2. Chemotherapy-Induced Cachexia

Cancer-associated cachexia, defined as the progressive wasting and weakness of skeletal muscle with or without fat loss (Fearon 2011) is a debilitating complication of cancer that reduces quality of life and response to anticancer therapies compromising survival. However, both preclinical (Barreto et al. 2016; Pin et al. 2019) and clinical studies (Naito 2017; Poterucha et al. 2012; Pin et al. 2018) have shown that anticancer therapies themselves can promote or exacerbate cachexia. Often, there is a trade-off between tumor regression and chemotherapy-induced side effects, where dose escalation may result in increased tumor regression, but at the risk of increased patient toxicity or other side effects including skeletal muscle atrophy and weakness; a specific off-target systemic effect that manifests clinically as cachexia. Hence chemo-induced cachexia has recently emerged as a critical clinical problem.

In the second half of this manuscript, we quantify the second-order treatment effects on both tumor regression and lean mass. Second-order effects have not, to our knowledge, been quantified in cachexia and remains an unexplored potential avenue for reducing the rates of cachexia in patients receiving chemotherapy. For example, the first-order effect of increasing mean dose has an adverse effect on increasing the likelihood of cachexia. In this manuscript, we advocate for the quantification of second-order effects so that both the mean dose and variance can be exploited for reducing tumor burden and maintaining lean mass to avoid cachexia. Incorporating second-order effects may provide a potential path forward in mitigating the risk of cancer cachexia by preserving lean mass without sacrificing tumor response.

1.3. The Mathematics of Antifragility

We consider a suite of dosing schedules like those in panel A in Fig. 1, where a pair of doses is administered over a time interval T . This is repeated for n cycles (here n = 6), where each cycle has the same cumulative dose. Therefore, the total drug delivered to the patient is the same for schedules A, B, and C. Treatment schedules A, B, and C are characterized by a dosing vector $\overrightarrow{X}$ with 2-dose cycles governed by a prescribed dose optimize $\sigma$ and denoted as $\overrightarrow{X}=[\overline{x}+\sigma ,\overline{x}-\sigma ,\dots ]$ for n cycles. As shown, schedules A, B, and C transition from purple to green indicating an increase in $\sigma$.

The drug-induced proliferation (DIP, or symbolically $\gamma$) rate determines the pharmacodynamics of tumor killing. DIP represents the steady state proliferation rate of cells in the presence of a drug (Harris 2016; Meyer 2019; Wooten et al. 2021), and thus tumors are decaying if DIP is less than zero. The best treatment protocols would minimize DIP, hence maximizing tumor regression. As shown in Fig. 1C, its shape can determine whether uneven or even treatment gives a higher effective $\gamma$. In particular, the concave $\gamma$ in Fig. 1C predicts an uneven schedule is more beneficial (the green dot representing the average of high and low dose is below the purple dot representing the instantaneous dose). The convex $\gamma$ indicates even treatment would be preferable. Finally, a linear DIP curve leads to identical outcomes for even and uneven treatment (no advantage).

Panel D illustrates examples for all three scenarios under different treatment schedules described earlier. Under a convex DIP, we have uneven treatment being preferable for tumor suppression, so we say the system is antifragile (benefiting from dose volatility) while the concave DIP predicts even treatment is more beneficial, so the system is fragile (benefiting from dose consistency). This motivates the definition of fragility in cancer:

$$\begin{gathered} Fragility,\mathbf{F} \\ =\frac{(\mathbf{uneven}\text{schedule tumor final volume})-(\mathbf{even}\text{schedule tumor final volume})}{\text{initial tumor volume}} \end{gathered}$$ (1)

Fragility provides a quantifiable metric delineating when an uneven schedule or even schedule is preferable. The normalization assists comparison of different tumor size regimes where treatment preference is expected to differ. If fragility is negative (F < 0), an uneven schedule causes greater tumor regression. We term this situation antifragile (Fig. 1D, top). If fragility is positive (F > 0), an even schedule is preferable to an uneven schedule. We term this situation fragile (Fig. 1D, bottom). If fragility is zero, then neither schedule outperforms the other (Fig. 1D, middle). In the subsequent sections below, we focus on defining the boundary between fragile and antifragile regions in parameter space.

2. Methods (Model Development)

2.1. Fragility for Daily Dosing

As stated above, we employ antifragile theory to compare the outcomes between even and uneven dosing schedules. A treatment cycle is defined as two sequential doses, $\overrightarrow{X}=\{x_1,x_2\}$. Here, we restrict ourselves to treatment cycles with identical mean dose, $\bar{x}$, such that $\overline{x}=\frac{1}{2}(x_1+x_2)$. We can define all such schedules with two control parameters, $\bar{x}$ and $\sigma$:

$$\overrightarrow{X}(\overline{x},\sigma )=\{\overline{x}+\sigma ,\overline{x}-\sigma \},$$ (2)

where $\bar{x}$ is the mean dose and $\sigma$ is the dose variation. Note that $\sigma =0$ corresponds to an even treatment schedule. The corresponding fragility of a two-day treatment protocol is characterized by Eq. (1); letting V denote the tumor volume following a treatment cycle, the fragility, F is expressed as

$$F(\overline{x},\sigma )=\frac{V(\overline{x},\sigma )-V(\overline{x},\sigma =0)}{V_0},$$ (3)

where V₀ is the tumor volume immediately prior to treatment.

2.2. Fragility for Weekly Dosing

Next, we expand the range of possible treatment schedules by considering N timepoints which can characterize moments of treatment (e.g. daily administration for a week, N = 7). Within these N timepoints, we administer a fixed number of doses, x, followed by h treatment holidays:

$$\overrightarrow{X}=\{\underset{N-h}{\underbrace{x,\dots ,x}},\underset{h}{\underbrace{0,\dots ,0}}\}.$$ (4)

In this way, we can compare weekly dosing schedules that range from even administration (h = 0) with a daily dose of $\overline{x}$, compared to uneven administration of N – h doses, x′, followed by H number of zero dose holidays. The set of dosing schedules is restricted by the cumulative dose delivered among the N timepoints ($\sum _i^N{x}_i=\bar{x}N$). This implies that the treatment dosage for uneven administration depends on the amount of treatment holidays through

$$x^{\prime }=\frac{N\bar{x}}{N-h}.$$ (5)

The cumulative dose restriction encompasses commonly tested chemotherapy regimens. For example, below we repurpose data from Farhang-Sardroodi et al. (2022) (see Sect. 3.3) that compares daily administration of 5-fluorouracil (h = 0) to a (5,2) schedule of five daily doses followed by two holidays (h = 2). We thus compare final tumor volumes between an even and uneven treatment through the following expression for fragility:

$$F(\bar{x},h)=\frac{V(\bar{x},h)-V(\bar{x},h=0)}{V_0}.$$ (6)

2.3. 1-Compartment Model

Next, we apply this definition of fragility to a 1-compartment model pharmacokinetic model of drug concentration, x(t), coupled to our DIP pharmacodynamic model of tumor growth and decay, $\alpha (x)$ (see Fig. 2A for a schamatic of the model). The tumor is exponentially growing (or decaying) according to the DIP rate, $\gamma$, as a function of the dose, x. Let $V(t)$ be a tumor volume at time t (treatment ends at t = T) and $\gamma (x)$ be the DIP growth rate of the tumor. We then have the following expression for tumor volume:

$$\frac{dV}{dt}=V\gamma (x(t)),$$ (7)

where x(t) is the instantaneous drug concentration near the tumor at time t. In practice, $\gamma (x)$ can be expressed as the Hill function, H(x), a commonly used empirical pharmacodynamic equation (Vakil and Trappe 2021; Köhn-Luque et al. 2023; Harris 2016):

$$H(x)=E_0+\frac{x^n\left(E_1-E_0\right)}{x^n+C^n}.$$ (8)

Fig. 2.

Pharmacokinetic-induced fragility in a 1-Compartment model. A, F 1-compartment model setup for drug delivery and tumor elimination without pharmacokinetics and with fast pharmacokinetic decay. B, G Schematic dose schedules for even and uneven treatments without and with fast pharmacokinetics for ($\overline{x}=25$, $\sigma =25$) and ($\overline{x}=25$, $\sigma =25$) treatment cycles. C, H Example tumor trajectories resulting from the dosing schedules in B, G; while tumor size is smaller following even treatment in C, there is a smaller tumor size following uneven treatment in H. D, I A DIP curve for a tumor and its dependence on treatment decay. For α = [0 .3 .5], increasing pharmacokinetics correspond to an increased region of concavity. E Fragility values across a range of average doses, $\overline{x}$ and variation σ = [1357]; J Fragility values across a range of average doses $\overline{x}$ and pharmacokinetics α = [0 3 5]. Move from antifragile to fragile treatment as $\overline{x}$ increases in both E and J, but as $\alpha$ increases, the range of doses which give an antifragile tumor increases. While this panel considers a “slow” $\alpha =0$, this parameter produces pharmacodynamic-only fragility, and thus implies that residual effects become important when accounting for pharamacokinetics in general (see Sect. 3.2.2). For all figures, the following Hill function parameters were used: E₀ = 1, E₁ = −1, C = 20, n = 10 (Color figure online)

Given the treatment dynamics, x(t), and initial conditions, we can solve for the tumor volume over time. This tumor volume can be used to evaluate the fragility of our tumor in response to different treatment schedules.

2.4. 2-Compartment PK Model of Tumor Response and Cachexia

It is common practice to extend pharmacokinetics to consider two compartments: drug concentration in the plasma, x₁(t), and in the tissue, x₂(t). In this manuscript, we explore a previously published two-compartment PK/PD model by Farhang-Sardroodi et al parameterized by an in vivo study to determine the utility of fragility (Farhang-Sardroodi et al. 2022). Here we focus on the time-evolution of the tumor volume, but additional compartments which characterize lean mass volume also exist and will be used (see Sect. 3.3 and Supplementary). Of note is the two-compartment model for drug transfer between plasma (x₁) and tissue (x₂), representing model 7 of Win-NonLin (Phoenix WinNonlin User’s Guide 2020). The tumor (V) undergoes exponential-linear growth with a decreasing factor based on tissue concentration:

$$\frac{d{x}_1}{dt}=-\alpha x_1(t)+\frac{1}{V_1}\sum _i{d}_i\delta (t-t_i)+k_{21}{x}_2(t)\frac{V_2}{V_1}-k_{12}{x}_1(t),$$ (9)

$$\frac{d{x}_2}{dt}=k_{12}{x}_1(t)\frac{V_1}{V_2}-k_{21}{x}_2(t),$$ (10)

$$\frac{dV}{dt}={\mu }_{0}V{\left(1+{\left(\frac{{\mu }_0}{{\mu }_1}V\right)}^{\eta }\right)}^{-\frac{1}{\eta }}-\kappa x_{2}V.$$ (11)

Instantaneous drug transfer is expressed by a sequence of Dirac delta functions at times t_i and concentrations d_i, with a drug clearance rate determined by $\alpha$. The values V₁ and V₂ characterize the volumes of the tissue and lean mass, while k₁₂ and k₂₁ represent the drug transfer rates between these compartments. The tumor death rate per unit tissue-muscle volume is given by $\kappa$. Finally, μ₀, μ₁ are the exponential and linear growth rates characterizing tumor evolution, with $\eta$ as a transition factor between growth regimes. Figure 2F illustrates the model’s basic structure.

3. Results

3.1. 1-Compartment Model Without Pharmacokinetics

Using the 1-compartment model in Sect. 2.3, we consider the simple case of a single treatment cycle: a pair of doses administered in the time interval T. This is the smallest cycle that will allow us to evaluate the influence of second-order effects. We first look at pharmacokynamic effects only, as first considered by West et al. (2021). To do this, we simply impose the dose concentration experienced by the tumor at all times, ignoring biological mechanisms that may change change this concentration profile. The resulting x(t) without pharmacokinetics is given by:

$$x(t)=\{\begin{array}{ll}\overline{x}+\sigma ,\hfill & 0<t<\frac{T}{2}\hfill \\ \overline{x}-\sigma ,\hfill & \frac{T}{2}<t<T\hfill \end{array}$$ (12)

where $\sigma$ is a parameter determining the unevenness of the dosing schedule and $\overline{x}$ is the mean dose experienced by the tumor ($\overline{x}\ge \sigma$).

Inserting x(t) into the tumor volume from Eq. (7), we can derive an expression for V(t) which only depends on our DIP curve, $\gamma$:

$$V(\overline{x},\sigma )=V_0\text{exp}(\frac{T}{2}((\gamma (\overline{x}+\sigma )+\gamma (\overline{x}-\sigma ))).$$ (13)

Given this expression, we can describe a corresponding fragility defined by Eq. (1) as the normalized difference in final tumor size for uneven and even treatment:

$$F(\overline{x},\sigma )=\text{exp}\left((\gamma (\overline{x}+\sigma )+\gamma (\overline{x}-\sigma ))\frac{T}{2}\right)-\text{exp}(\gamma (\overline{x})T).$$ (14)

It is instructive to consider the conditions when a tumor is antifragile in the case without pharmacodynamics. If uneven treatment is preferable, then F < 0 and the tumor is antifragile. Some manipulation of Eq. (14) reveals that the tumor is antifragile when

$$\frac{\gamma (\overline{x}+\sigma )+\gamma (\overline{x}-\sigma )}{2}<\gamma (\overline{x}).$$ (15)

This implies some structure on the DIP curve. In particular, if the DIP curve is concave in the region $(\overline{x}-\sigma ,\overline{x}+\sigma )$, the inequality is satisfied. If the DIP function is parameterized by the Hill function in Eq. (8), then there exist regions of concavity and convexity which predict antifragile and fragile tumors, respectively. The inflection point ${\overline{x}}_B$ acts as a good approximation between the regions of convexity that characterizes how a tumor responds to uneven treatment, and was found to be given by

$${\overline{x}}_B=C{\left(\frac{n-1}{n+1}\right)}^{1/n}.$$ (16)

The tumor dynamics and fragility resulting from this model configuration are presented in Panels A–E in Fig. 2. Panel A shows the model’s compartments with the treatment dose directly influencing tumor volume. Panel B illustrates the even ($\sigma =0$) schedule in purple and uneven ($\overline{x}=\sigma$) schedule in green, both without PK clearance. The schedules in Panel B create tumor trajectories according to Panel C, where the even treatment produces a smaller tumor than the uneven treatment. Panel D shows an example hypothetical DIP curve determining tumor growth that has a characteristic sigmoidal pattern parameterized from the Hill function. The interaction between the DIP curve and treatment schedule is shown in Panel E, where the fragility is given over a range of average doses $\overline{x}$ and schedules. Here, the concave region of the DIP curve in D corresponds to an antifragile tumor and the convex region of the DIP curve corresponds to a fragile tumor. Thus, uneven is preferable for low doses (F < 0) while even is preferable for high doses (F > 0).

3.2. 1-Compartment Model with Pharmacokinetics

3.2.1. Fast Pharamacokinetics

To determine the role of pharmacokinetics in the 1-compartment model, we now consider a similar single-cycle schedule where the initial drug concentration undergoes exponential decay. This model follows the structure of model 1 from Win-NonLin (Phoenix WinNonlin User’s Guide 2020) with one-compartment, bolus input, and first-order output (Fig. 2F). Before observing the fragility for a general treatment cycle and drug decay rate parameters, we analyze the limiting case of fast pharamcokinetics which has value in its similarity to the pharmacodynamic model considered in Sect. 3.1.

A drug concentration that undergoes “fast pharmacokinetics” has a negligible drug concentration once a second dose in a cycle is delivered. In this case, the drug con centration profile x(t) experienced by the tumor is given by

$$x(t)=\{\begin{array}{ll}(\overline{x}+\sigma )e^{-\alpha t},\hfill & 0<t<\frac{T}{2}\hfill \\ (\overline{x}-\sigma )e^{-\alpha \left(t-\frac{T}{2}\right)},\hfill & \frac{T}{2}<t<T\hfill \end{array}$$ (17)

This expression is accurate when either PK clearance is very rapid (large $\alpha$) or the time between doses delivered is large (large T). Note that it is proportional to Eq. (12) with an additional exponential factor. Using Eq. (7), and assuming the DIP curve can be represented by the Hill Function from Eq. (8), the tumor volume after a treatment cycle can be written as

$$V(t=T)=V_0{e}^{K_{\text{fast}}(\overline{x},\sigma ,\alpha ,T)},$$ (18)

where

$$\begin{gathered} K_{\text{fast}}(\overline{x},\sigma ,\alpha ,T)=E_{0}T-\frac{E_1-E_0}{\alpha n} \\ \left[\text{ln}\left(\frac{{(\overline{x}+\sigma )}^n{e}^{-\alpha n\frac{T}{2}}+C^n}{{(\overline{x}+\sigma )}^n+C^n}\right)+\text{ln}\left(\frac{{(\overline{x}-\sigma )}^n{e}^{-\alpha n\frac{T}{2}}+C^n}{{(\overline{x}-\sigma )}^n+C^n}\right)\right]. \end{gathered}$$ (19)

The antifragile-fragile boundary ${\overline{x}}_B$ for fast pharmacokinetics is then given by

$$\begin{gathered} \text{ln}\left(\frac{{({\overline{x}}_B+\sigma )}^n{e}^{-\alpha n\frac{T}{2}}+C^n}{{({\overline{x}}_B+\sigma )}^n+C^n}\right) \\ +\text{ln}\left(\frac{{({\overline{x}}_B-\sigma )}^n{e}^{-\alpha n\frac{T}{2}}+C^n}{{({\overline{x}}_B-\sigma )}^n+C^n}\right)-2\text{ln}\left(\frac{{\overline{x}}_B^n{e}^{-\alpha n\frac{T}{2}}+C^n}{{\overline{x}}_B^n+C^n}\right)=0. \end{gathered}$$ (20)

An important special case considers infinitely fast pharmacokinetics, α → ∞. Taking this limit for the above expression implies

$$\text{ln}\left(\frac{C^n}{{({\overline{x}}_B+\sigma )}^n+C^n}\right)+\text{ln}\left(\frac{C^n}{{({\overline{x}}_B-\sigma )}^n+C^n}\right)=2\text{ln}\left(\frac{C^n}{{\overline{x}}_B^n+C^n}\right).$$ (21)

Applying logarithm rules and completing some algebra rearrangment gives an implicit function for the antifragile-fragile boundary with fast pharmacokinetics:

$${\overline{x}}_B^{2n}+2C^n{\overline{x}}_B^n-C^n\left[{({\overline{x}}_B-\sigma )}^n+{({\overline{x}}_B+\sigma )}^n\right]-{({\overline{x}}_B-\sigma )}^n{({\overline{x}}_B+\sigma )}^n=0.$$ (22)

Panels F–J in Fig. 2 illustrate the one-compartment model with fast decay and how it influences treatment outcomes. Panel F and Panel G show the introduction of decay and the resulting exponential decay in drug dose experienced by the tumor. Panel H shows the resulting tumor trajectories, with the uneven schedule eliminating more tumor than the even schedule. Panel I shows that as the $\alpha$ decay parameter increases, the DIP curve shifts to the right, increasing the region of concavity. This corresponds to the shifting fragility in Panel J, where increasing PK clearance rate, $\alpha$, corresponds to an extended antifragile treatment regime (black to yellow lines).

These observations result from the decreased amount of drug available to eliminate the cancer. Higher doses are necessary to compensate for fast drug clearance and the time-averaged drug exposure to the tumor decreases for high $\alpha$. Importantly, they illustrate that when residual concentration is negligible, the regions of fragility and antifragility are modified due to a change in the geometry of the drug-induced proliferation.

3.2.2. General Pharmacokinetics

We now consider a general drug decay rate $\alpha$ that is shared for a two-dose cycle of treatment. The concentration profile resulting from the schedule $\{\overline{x}+\sigma ,\overline{x}-\sigma \}$ is given by

$$x(t)=\{\begin{array}{ll}(\overline{x}+\sigma )e^{-\alpha t},\hfill & 0<t<\frac{T}{2}\hfill \\ (\overline{x}+\sigma )e^{-\alpha t}+(\overline{x}-\sigma )e^{-\alpha \left(t-\frac{T}{2}\right)},\hfill & \frac{T}{2}<t<T\hfill \end{array}$$ (23)

The tumor volume following treatment is then given by

$$V(t=T)=V_0{e}^{K(\overline{x},\sigma ,\alpha ,T)},$$ (24)

where integration reveals

$$\begin{gathered} K(\overline{x},\sigma ,\alpha ,T)=E_{0}T-\frac{E_1-E_0}{\alpha n}\{\text{ln}\left(\frac{{(\overline{x}+\sigma )}^n{e}^{-n\alpha \frac{T}{2}}+C^n}{{(\overline{x}+\sigma )}^n+C^n}\right) \\ +\text{ln}\left(\frac{{\left[(\overline{x}+\sigma )+(\overline{x}-\sigma )e^{\alpha \frac{T}{2}}\right]}^n{e}^{-n\alpha T}+C^n}{{\left[(\overline{x}+\sigma )+(\overline{x}-\sigma )e^{\alpha \frac{T}{2}}\right]}^n{e}^{-n\alpha \frac{T}{2}}+C^n}\right)\}. \end{gathered}$$ (25)

The fragility is finally expressed as

$$F(\overline{x},\sigma ,\alpha ,T)=e^{K(\overline{x},\sigma ,\alpha ,T)}-e^{K(\overline{x},0,\alpha ,T)}.$$ (26)

Setting Eq. (26) to zero and completing algebraic simplifications shows that the antifragile-fragile boundary ${\overline{x}}_B$ satisfies:

$$\begin{gathered} \text{ln}\left(\frac{{({\overline{x}}_B+\sigma )}^n{e}^{-n\alpha \frac{T}{2}}+C^n}{{({\overline{x}}_B+\sigma )}^n+C^n}\right)+\text{ln}\left(\frac{{\left[({\overline{x}}_B+\sigma )+({\overline{x}}_B-\sigma )e^{\alpha \frac{T}{2}}\right]}^n{e}^{-n\alpha T}+C^n}{{\left[({\overline{x}}_B+\sigma )+({\overline{x}}_B-\sigma )e^{\alpha \frac{T}{2}}\right]}^n{e}^{-n\alpha \frac{T}{2}}+C^n}\right) \\ =\text{ln}\left(\frac{{\overline{x}}_B^n{e}^{-n\alpha \frac{T}{2}}+C^n}{{\overline{x}}_B^n+C^n}\right)+\text{ln}\left(\frac{{\overline{x}}_B^n{\left[1+e^{\alpha \frac{T}{2}}\right]}^n{e}^{-n\alpha T}+C^n}{{\overline{x}}_B^n{\left[1+e^{\alpha \frac{T}{2}}\right]}^n{e}^{-n\alpha \frac{T}{2}}+C^n}\right). \end{gathered}$$ (27)

Note that a boundary ${\overline{x}}_B>0$ does not necessarily exist for all $\alpha >0$. In fact, as $\alpha \to 0$, there is no antifragile-fragile boundary, as shown in Sect. 3.2.3.

3.2.3. Zero-Rate Pharmacokinetics

We now look at the special limiting case of zero-rate pharamcokinetics, or α → 0. From Eq. (23), the drug dynamics are given by

$$x(t)=\{\begin{array}{ll}\overline{x}+\sigma ,\hfill & 0<t<\frac{T}{2}\hfill \\ 2\overline{x},\hfill & \frac{T}{2}<t<T\hfill \end{array}$$ (28)

It follows that the fragility from Eq. (26) becomes

$$\begin{gathered} F(\overline{x},\sigma ,0,T)=\text{exp}\left(E_{0}T+(E_1-E_0)\frac{T}{2}\frac{{(2\overline{x})}^n}{[{(2\overline{x})}^n+C^n]}\right) \\ \times \left[\text{exp}\left((E_1-E_0)\frac{T}{2}\frac{{(\overline{x}+\sigma )}^n}{[{(\overline{x}+\sigma )}^n+C^n]}\right)-\text{exp}\left((E_1-E_0)\frac{T}{2}\frac{{\overline{x}}^n}{[{\overline{x}}^n+C^n]}\right)\right]. \end{gathered}$$ (29)

For $\sigma >0$, it follows from the above expression that F < 0 for all positive $\overline{x}$ and $\sigma$. So for zero-rate pharamacokinetics, it is always beneficial to apply treatment unevenly for the 1-compartment model. This is not unique to the Hill function representation of the DIP curve, as rewriting Eq. (29) for a general $\gamma (x)$ gives

$$F(\overline{x},\sigma ,0,T)=\text{exp}(\gamma (2\overline{x}))\left[\text{exp}\left(\frac{T}{2}(\gamma (\overline{x}+\sigma ))\right)-\text{exp}\left(\frac{T}{2}\gamma (\overline{x})\right)\right].$$ (30)

So as long as the $\gamma (x)$ is a monotonically decreasing function (which is biologically reasonable), then a tumor is always antifragile.

It is worthwhile to note the difference between Eqs. (12) and (28). The first expression imposes a hypothetical drug concentration over time to characterize tumor response, while the latter expression is a necessary concentration profile when assuming bolus dosing and zero-rate pharmacokinetics. This difference critically illustrates how increasing biological relevance in modeling can have significant effects on modeling interpretation. In our case, it is the difference between realizing that fragile treatment is only preferable for small reference dose $\overline{x}$ and fragile treatment is preferable for all reference doses $\overline{x}$ and $\sigma >0$.

3.2.4. High or Low Dose First

Thus far, we have primarily assumed a high does is followed by a low dose during a treatment cycle. But in general the order of dosing delivery affects the average drug concentration of a tumor. The time-averaged drug concentration 〈x〉 can be determined with

$$\langle x\rangle =\frac{1}{T}\underset{0}{\overset{T}{\int }}x(t)dt.$$ (31)

We first look at the average drug concentration with an intial high dose: $\{\overline{x}+\sigma ,\overline{x}-\sigma \}$. Substituting x(t) from Eq. (23) into Eq. (31) gives this value:

$${\langle x\rangle }_{\{\overline{x}+\sigma ,\overline{x}-\sigma \}}=\frac{1}{T}({\int }_0^T(\overline{x}+\sigma )e^{-\alpha t}\mathit{dt}+{\int }_{T/2}^T(\overline{x}-\sigma )e^{-\alpha (t-\frac{T}{2})}\mathit{dt}),$$ (32)

$$\Rightarrow {\langle x\rangle }_{\{\overline{x}+\sigma ,\overline{x}-\sigma \}}=\frac{1}{T\alpha }\left(2\overline{x}-\left((\overline{x}+\sigma )e^{-\alpha \frac{T}{2}}+(\overline{x}-\sigma )\right)e^{-\alpha \frac{T}{2}}\right).$$ (33)

A similar calculation can determine the average concentration for a schedule with a low first dose: $\{\overline{x}-\sigma ,\overline{x}+\sigma \}$. Subtracting these concentrations yields a difference in first order effects given by

$$\text{Δ}=\frac{\sigma e^{-\alpha \frac{T}{2}}}{\alpha \frac{T}{2}}\left(1-e^{-\alpha \frac{T}{2}}\right)>0,$$ (34)

where Δ is the difference in average drug concentration delivered between high dose first and low dose first schedules. The difference Δ is a positive function for $\alpha ,T,\sigma >0$, so the time-averaged dose concentration is always larger with an higher first dose. Note that this difference decreases as a function of $\alpha$ and T, so when there are fast pharmacokinetics, there is no difference in first-order effects. In general, this difference does not necessarily imply higher treatment efficacy for a higher initial dose due to second-order effects. Figure 3 demonstrates treatment outcomes for high and low-dose first across a range of average initial dose concentrations $\overline{x}$ and clearance rate $\alpha$. We also note that the pharmacokinetic rate $\alpha$ is coupled to the time period, T, whereby this difference, Δ, can be decreased with increasing time period or faster decay.

Fig. 3.

The effect of pharmacokinetic clearance rate on fragility Fragility shown for a range of mean dose, $\overline{x}$, and clearance rate, $\alpha$. A High dose followed by low dose. B low dose followed by high dose. When clearance is sufficient fast (large $\alpha$), higher doses tend to be fragile (blue) while lower doses tend to be antifragile (red). At low clearance rates, residual dose concentration after the second dose plays a larger role, and thus the order of doses matter. Figure generated from dose profiles using Eq. (23) with dose response parameters $\sigma =2$, E₀ = 1, E₁ = −1, C = 20, n = 10 (Color figure online)

3.3. 2-Compartment Pharmacokinetic Model Results

Next, we investigate the influence of pharmacokinetics on second-order effects by examining a two-compartment model of dose concentration in plasma and tissue. A schematic of the model is shown in Fig. 4A where the drug is intravenously delivered to a plasma compartment. The model accounts for plasma clearance at a rate α and transfer rates, r_ij, between plasma and tissue compartments. The model tracks both the plasma concentration, x₁(t), and tissue concentration, x₂(t), over time. Tissue concentration has a direct effect on tumor kill [see Eqs. (9)–(11) in Methods]. Additionally, this system includes a lean mass compartment inhibited by the drug concentration in the tissue compartment. Therefore fragility can be quantified for both tumor response and lean mass. Panel B demonstrates the exponential-linear growth rate assumed for the tumor without treatment (see Methods). This mathematical model of chemotherapy (5-fluorouracil) was previously parameterized to mouse experiments (Farhang-Sardroodi and Wilkie 2020; Farhang-Sardroodi et al. 2022) by deriving a set of parameters to fit tumor volume data and lean mass data over time. We use this model and its associated parameterization as a point of departure for our purpose here to define the effect of pharmacokinetics on second-order effects in a biologically realistic mathematical model.

Fig. 4.

Antifragility in a 2-Compartment model with pharmacokinetics. A A schematic for the 2-compartment model, where drug is intravenously delivered to the blood plasma where decay occurs and is sent to the tissue compartment. Once in the tissue compartment, the drug affects the growth of the tumor and lean mass compartments. B Shows the untreated dynamics for the tumor compartment of the system, undergoing exponential-linear growth. C Shows experimental data and model predictions for lean mass volume and treatment schedules from Farhang-Sardroodi et al. (2022). The two purple schedules indicate daily doses of 35mg and 17mg over 28 days, while the two green schedules indicate doses of 50mg and 24mg on a 5 day-on, 2 day-off schedule. Model parameterization completed by minimizing the sum of squared errors. D A schematic for even and uneven bolus treatment injections across one week treatment and a cumulative drug total of 80 mg. E Tumor trajectories with treatment schedules in D and initial tumor sizes of 10, 100, and 1000mm³. As tumor size increases, even treatment becomes preferable. F, G Fragility of the tumor and lean mass compartments across a range of cumulative dose values and for repeated treatment schedules of 1, 2, and 4 weeks. For high tumor sizes, the tumor responds better to even treatment and is fragile, while uneven treatment is preferable for eliminating smaller tumors. For high tumor sizes, schedules eliminating tumor volume correspond with lean mass preservation (even treatment). Simulations demonstrated in D–G, used the global parameterization reported by Farhang-Sardroodi et al. (2022) (Color figure online)

Data of the lean mass, repurposed from Farhang-Sardroodi et al. (2022), are shown in Fig. 4C with the model fitted to uneven (green) and even dosing schedules (purple). Lean mass is measured over time in response to an uneven schedule (5 days on treatment, 2 days on holiday, shown in green) and an even schedule (7 days on treatment, shown in purple). The purpose of this previous work was to design chemotherapy schedules that preserve lean mass across high dose (Fig. 4C, top) and low dose (Fig. 4C, bottom) schedules. However, both the cumulative dose delivered and the dosing variation (the number of treatment holidays) were altered in tandem, obfuscating firstand second-order effects. In contrast, we consider second-order effects in isolation and optimize the number of treatment holidays across identical cumulative dose by quantifying fragility in the model.

For example, Panel D in Fig. 4 shows a range of even (purple) to uneven (green) schedules in the plasma compartment over time. We now examine the effect on tumor response (E, F) and lean mass (G) separately. Panel E shows these schedules’ effect on tumor volume as a function of the initial tumor sizes. For a mean dose of $\overline{x}=80$, the even (no holiday) treatment schedule maximally reduces the tumor when the initial size, V₀, is large.

Large tumors (V₀ = 1000) tend to be very fragile (F > 0), while small tumors (V₀ = 10) are slightly antifragile (F < 0). Panel F shows simulation results for tumor fragility across a range of mean dose values, with the dashed line corresponding to Panel E. Larger tumors are reduced more under schedules without holidays, while smaller tumors respond to treatments with holidays. As seen in panel F, large tumors are fragile (F > 0) across a large range of mean dose values, indicating the effectiveness of even treatment at reducing tumor burden. This suggests that large tumors should initially be treated with an even schedule and switched to an uneven treatment schedule at a later point in time when the tumor is sufficiently small so as to be fragile. The superiority of switching based on tumor size is shown in Fig. S1, illustrating the promise of fragility-guided treatment scheduling.

This model assumes no feedback of tumor size on lean mass, and thus lean mass treatment dynamics are independent of tumor size. However, lean mass has a strong dependence on time. Lean mass reduction is a function of the average exposure of the tissue to drug concentration over the last $\tau$ days, which is estimated as $\tau =8$ days (Farhang-Sardroodi et al. 2022). This time-exposure effect on lean mass leads to a shift in lean mass fragility after the first week. As seen in panel G, lean mass is initially fragile (1 week), but shifts to antifragile after 2 and 4 weeks. In the long term (>1 week), preservation of lean mass is best achieved using treatment schedules with dose holidays.

4. Discussion

Dose response curvature (or synonymously, antifragile theory) has been previously proposed as a graphical approach to predict outcomes based on dose response convexity and concavity (Taleb and West 2023; West et al. 2021). In previous literature, this approach ignored pharmacokinetic considerations. We have constructed an extension of previous frameworks to include pharmacokinetic clearance of drug (a 1-compartment PK model). This framework allowed us to find analytical expressions for abritray values of pharmacokinetic decay rates, with simplified expressions for certain special cases including fast pharmacokinetic decay ($\alpha \to \infty$), no decay ($\alpha =0$), as well as quantifying the effectiveness of dose ordering (high or low dose first).

Our results indicate that in both simple PK/PD models (Figs. 2, 3) as well as in a more detailed PK/PD mouse-parameterized model (Fig. 4), there exist tumor and muscle lean mass regimes which may benefit from dosing schedules with higher or lower variance. Thus, we confirm our hypothesis that second-order effects are relevant in designing treatment schedules and cannot be ignored.

These theoretical motivate an investigation of second-order effects to help explain chemotherapy response in a mouse-parameterized 2-compartment model. Interestingly, the dominant factor in determining fragility was the tumor size, in addition to the dose value. Large tumors are typically fragile (even is preferable) while small tumors are marginally antifragile. This implies a switching point: large tumors should be treated with even therapy, until such a tumor size that is antifragile, at which point the treatment should be switched to uneven.

The modeling here also investigates the effectiveness of treatment holidays in preserving lean mass. On short time scales, the high doses of uneven treatment results in a decline in lean mass (compared to even treatment), but over longer time scales, treatment holidays prove to be a viable strategy for “resetting the clock” of the time exposure detrimental effect of chemotherapy concentration on lean mass.

Across the single and two-compartment models, some comparisons may be made. First, despite differences between the two models, second-order effects are non-zero in both, and thus are important considerations in designing and implementing treatment schedules. Secondly, tumors tend to be antifragile at low doses with two exceptions: slow pharmacokinetics (Fig. 3) or large tumor sizes (Fig. 4). It’s difficult to draw further generalized observations and therefore we recommend careful parameterization of the model and subsequent simulation and quantification of second-order effects to apply these findings in other contexts.

While here we restrict ourselves to quantifying second-order effects in treatment schedules with identical cumulative dose, optimal control theory can aid in the search of optimal treatment schedules without this restriction (Jarrett 2020; Sbeity and Younes 2015; Schättler and Ledzewicz 2015). Gains resulting from switching an antifragile tumor to an uneven schedule can also be used as a dose de-escalation strategy by implementing less overall number of cycles while achieving the same burden reduction than an even counterpart.

Second-order effects may be driving experimental results but remain hidden due to different mean dose values, making it difficult to quantify their impact. Here we have illustrated the importance of comparing protocols with identical first-order metrics, since only then can second-order effects be properly understood. Our historic focus on mean dose and its response in cancer treatment has meant that variation in drug dose has been severely understudied. However, given the importance of dose modulation (including treatment holidays) as a means of regulating cachexia, it is advisable to begin leveraging second-order effects in treatments.

Supplementary Material

The online version contains supplementary material available at https://doi.org/10.1007/s11538-024-01278-0.