Introduction

Therapeutic resistance is well recognized as one of the major contributors to treatment failure and poor outcomes in patients (Housman et al. 2014; Frieri et al. 2017; Vasan et al. 2019; Zaman et al. 2017). Traditionally, resistance is thought to emerge from the selection of subpopulations of cells that have acquired one or more resistance-conferring mutations (Hanahan and Weinberg 2011). However, recent studies also show the importance of phenotypic plasticity as a mechanism for mediating resistance, e.g., by cells entering a polyaneuploid, mesenchymal, inflammatory, or stem-like state (Amend et al. 2019; Chin and Lim 2019; Gonzalez et al. 2022; Katsuno et al. 2019; Li et al. 2019; Pienta et al. 2020; Pienta et al. 2021; Weng et al. 2019; Yuan et al. 2022; Zhou et al. 2020).

In order to develop novel therapeutic strategies for bacterial and cancer extinction or control, a deep understanding of their ecological (population), evolutionary (resistance), and demographic (state) dynamics is required. To this end, many mathematical models have been constructed and analyzed to elucidate various aspects of therapeutic resistance via plasticity and clonal evolution. However, most models consider only one of these mechanisms (Bukkuri and Brown 2021; Dhawan et al. 2016; Forouzannia et al. 2020; Tonekaboni et al. 2017). For the few models that incorporate both (Bukkuri et al. 2022a; Bukkuri et al. 2022b; Cunningham et al. 2021), plastic transitions are usually assumed to occur within a discrete state space.

In this paper, we use integral projection models (IPMs) to develop a method to model the ecological, evolutionary, and demographic (hereafter referred to as eco-evo-demo) dynamics of populations that evolve resistance in the face of therapy via continuous phenotypic transitions and clonal evolution. IPMs developed as an offshoot from matrix population models (Bukkuri and Brown 2023; Caswell 2001; Caswell 2012; Caswell 2019; de Vries et al. 2020) as a way to deal with population structure of a continuous nature, relaxing the requirement that organisms in a population must be classified into a set of discrete states (Easterling et al. 2000; Rees et al. 2014). These models allow ecologists and demographers to understand how continuous state structure impacts a population’s eco-evolutionary dynamics (Coulson 2012; Coulson et al. 2010). IPMs are typically constructed from regression models that infer how an organism’s state and environment impacts its survival, growth, and reproduction (Adler et al. 2010; Dahlgren and Ehrlen 2009; Dalgleish et al. 2011; Merow et al. 2014; Nicolè et al. 2011). Such models have been used in a variety of ecological contexts, to help us understand and control invasive species (Erickson et al. 2018; Goodsman et al. 2018), the adaptation of insects and plants to climate change (Schwartz et al. 2016), herbivore–plant interactions (Hegland et al. 2010; Miller et al. 2009; Rose et al. 2005; Williams et al. 2010), optimal flowering size in plants (Hesse et al. 2008; Metcalf et al. 2003; Miller et al. 2012; Rees and Rose 2002; Williams et al. 2009), and host–parasite dynamics (Metcalf et al. 2016). We port over a more qualitative version of this framework to the cellular level and investigate the eco-evo-demo dynamics in populations subjected to environmental stress. We also investigate two facultative mechanisms by which cell populations may modulate their stress-induced responses depending on their condition in the environment: stress-induced mutagenesis (Badyaev 2005; Bjedov et al. 2003; Foster 2007; Fitzgerald et al. 2017) and facultative plasticity (Aiello et al. 2018; Amend et al. 2019; Fisher et al. 2017; Zañudo 2019; Liau et al. 2017; Pienta et al. 2020; Pienta et al. 2021; Pienta et al. 2020a; Pienta et al. 2020b; Rabé et al. 2020; Sharma et al. 2010; Shen and Clairambault 2020). Using this knowledge, we then examine the efficacy of intermittent and adaptive therapy strategies (Gatenby et al. 2009; West et al. 2020).

Methods

Analytical framework

We construct our modeling framework using IPMs. First, we outline the ecological dynamics for the most simple case in the form of an integrodifference equation:

$$n_{{\bar{u},t + 1}} (z^{\prime}) = \int {K_{{\bar{u},t}} (z,z^{\prime})n_{{\bar{u},t}} (z){\text{d}}z} {\text{ }}$$
(1)

where n is the population distribution, u is the drug resistance trait, \(z \in Z=[0,1]\) is the current state of the cell, and \(z'\) is state at the next time step. The states \(z \in Z\) may represent the state of cells on epithelial to mesenchymal, adrenergic (ADRN) to mesenchymal (MES), or aneuploidy continua, for example. We assume that population density is \(\delta\) peaked in u space around the mean drug resistance value, \(\bar{u}\), so the ecological, demographic, and evolutionary dynamics are all dependent on solely \(\bar{u}\). The kernel K can be decomposed into survival, plasticity, birth, and fecundity as follows:

$$\begin{aligned} {\begin{matrix} {K_{\bar{u},t}(z,z')} &{}= \underbrace{{s_{\bar{u},t}(z)}}_\text {Survival}\overbrace{{G}(z,z')}^\text {{Growth} Kernel}+\underbrace{b(z)}_\text {Birth}\overbrace{R(z,z')}^\text {{Fecundity} Kernel}\\ \end{matrix}} \end{aligned}$$
(2)

The first term is the growth kernel, often denoted by \({G_{\bar{u}}(z,z')}\), and the second term is the fecundity kernel, \(R(z',z)\). The survival and birth functions, \({s_{\bar{u},t}(z)}\) and b(z), are called individual components, whereas the growth and fecundity kernels, \(G(z,z')\) and \(R(z,z')\), are state-redistribution components. Under this formulation, we assume that drug resistance, u, only impacts the survival of the cells—in the future sections, we will relax this assumption. The specific forms for each of these components are

$$\begin{array}{*{20}c} {G(z,z^{\prime})} & \sim N(z,\sigma _{G} ^{2} ) \\ {R(z,z^{\prime})} & \sim N(z,\sigma _{R}^{2}) \\ {s_{{\overline{{u,t}} }} (z)} & { = \max \left( {0,1 - {\text{d}}k - \frac{{m(t)}}{{{\text{exp}}(\phi z) + \beta \bar{u}}}} \right)} \\ {b(z)} & { = b_{1} {\text{exp}}(b_{2} z)} \\ \end{array} {\text{ }}$$
(3)

First, consider the state-redistribution components, \(G(z,z')\) and \(R(z,z')\). We approximate the movement of cells through state space and the production of progeny using a Gaussian distribution. Although this is a fairly standard assumption in integral projection modeling (Easterling et al. 2000; Ellner and Rees 2006; Merow et al. 2014; Rees et al. 2014) (and indeed, G function (Bukkuri and Brown 2021; Vincent et al. 2005) and adaptive dynamics (Diekmann 2004; Kisdi and Geritz 2010) modeling), the distribution can be tailored to specific situations or fitted to data if required. In our case, the normal distribution is appropriate since cells tend to produce progeny very similar to themselves with most mutations having a small phenotypic effect. Similarly, cells tend to stay within similar cell states and only rarely drastically shift to the tail ends of their distribution.

Next, we turn our attention to the individual components. Consider the survival function, \({s_{\bar{u}}(z)}\). Bounded below by zero, the survival of cells is reduced by two factors. The first term, dk, captures the natural death rate and assumes a linear cost of evolvability, k. In other words, we assume that the natural death rate of a cell increases as its evolvability increases, e.g., due to mutational meltdown (Bukkuri 2022), and enforce \(d>0\) and \(k\ge 0\). The second term, \(\frac{m}{exp(\phi z)+\beta \bar{u}}\), captures death due the effect of drug via a Michaelis–Menten-like formulation (Bukkuri and Brown 2021; Bukkuri et al. 2022a; Bukkuri et al. 2022b), where \(m(t)\ge 0\) represents drug dosage at time t, \(\phi >0\) is the impact of the cell state on survival, and \(\beta >0\) represents the impact of resistance on survival. Finally, consider the birth function, b(z). The maximal birth rate of the cell is captured by \(b_1\). This proliferation rate is tempered by the cell’s state with the multiplicative exponential term. Namely, we assume that higher cell states lead to lower birth rates (enforced by \(b_2<0\)). This survival-birth trade-off is worth stressing. Higher cell states (e.g., mesenchymal, inflammatory, or high ploidy) attain higher survival under therapy than lower cell states (e.g., epithelial, adrenergic, proliferative, or low ploidy). But this comes at a cost: a lower birth rate. This trade-off, visualized in the heatmap below (Fig. 1), is experimentally observed in a multitude of contexts across bacterial and cancer evolution (Amend et al. 2019; Chin and Lim 2019; Katsuno et al. 2019; Liau et al. 2017; Shun et al. 2019; Sánchez-Danés et al. 2018; Shen et al. 2019; Zhou et al. 2020).

Fig. 1
figure 1

Survival-Birth Trade-off: higher cell states (z) confer a greater survival under therapy but come with a lower birth rate. Assuming no cost of resistance, resistance levels (u) allow for higher survival under therapy and do not impact birth rates

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From these ecological dynamics, we can then derive the evolutionary dynamics of resistance via Fisher’s fundamental theorem of natural selection (Li 1967; Basener and Sanford 2018; Frank and Slatkin 1992; Lessard 1997), sensu the G function framework (Bukkuri and Brown 2021; Vincent et al. 2005). This approach states that evolution is the product of evolvability (the capacity to generate heritable variation upon which natural selection can act), and the selection gradient. Namely, for the discrete-time case, we have

$$\begin{array}{*{20}c} {\Delta \bar{u}} & { = \frac{k}{{\bar{\lambda }}}\frac{{\partial \lambda }}{{\partial u}}} \\ {} & { = \frac{k}{{\bar{\lambda }}}\left( {\int {\int {\frac{{\partial \lambda }}{{\partial K}}} } \frac{{\partial K}}{{\partial s}}\frac{{\partial s}}{{\partial u}}{\text{d}}z{\text{d}}z^{\prime}} \right)} \\ {} & { = \frac{k}{{\bar{\lambda }}}\left( {\int {\int {{\mathbf{s}}({\mathbf{u}},{\mathbf{z}},{\mathbf{z^{\prime}}})} } P(z,z^{\prime})\frac{{\beta m}}{{(\beta u + {\text{exp}}(\phi z))^{2} }}{\text{d}}z{\text{d}}z^{\prime}} \right)} \\ \end{array}$$
(4)

where \(\bar{u}\) represents the mean drug resistance value in the population, \(\lambda\) is the per-capita growth rate or fitness function calculated as the spectral radius of the kernel operator (Bukkuri and Brown 2023), and \(\mathbf {s(u,z,z')}\) is the sensitivity of the fitness function to perturbations in the kernel operator, calculated as the normalized inner product of the reproductive and stable state distribution eigenvectors. Note that if drug resistance levels impact components other than just survival, this equation will need to be modified to incorporate the other terms via a basic application of the chain rule as shown above. Under this framework, we allow cells to respond to therapeutic stressors via both plasticity (changing cell state in a continuous and reversible fashion) and genetic evolution (evolving higher levels of resistance, u).

Numerical implementation

There are two technical points to note concerning the numerical implementation of our model. The discretized IPM kernel used in the simulations is the result of overlaying a mesh over the operator, treating it as a large finite-dimensional matrix. Thus, rather than dealing with the integrodifference equation in Equation 1 directly, we treat it as a high-dimensional, discrete-time matrix population model (Bukkuri and Brown 2023; Caswell 2001) and simply project it forward in time using matrix multiplication. The choice of mesh size (and thereby dimensionality of the resulting matrix) is a trade-off between accuracy and computational cost: The smaller the mesh size, the more accurate the numerical results, but the greater computational effort. In our case, we used a 100x100 matrix. Another aspect to consider are the boundary conditions. Since we defined a bounded interval for our state distribution, Z, we must correct for eviction, whereby individuals diffuse outside these boundaries due to the Gaussian nature of our size-redistribution kernels. To do this, we return all cells that occupy states outside Z to the boundary from which they were evicted (Merow et al. 2014) (see (Williams et al. 2012) for alternative corrections).

Another consequence of our modeling assumptions is the existence of a small subpopulation of highly resistant cells occupying high cell states prior to drug exposure. Due to the Gaussian nature of our kernels, these cells can indeed become or generate drug-sensitive cells and vice versa. This observation has found experimental support, e.g., Shaffer et al. identify a rare population of melanoma cells that transiently display high expression of AXL in the absence of therapy. These cells are resistant to anti-BRAF therapy and have been found to stochastically give rise to drug-sensitive cells (Shaffer et al. 2017).

Results

We ran a series of simulations capturing the eco-evo-demo dynamics of populations under therapy. The parameter values used in the simulations can be found in Table 1. These parameters were chosen to be biologically plausible, numerically convenient for simulation purposes, and to clearly show differences among stress-responses and therapeutic outcomes. Although many of these parameters can quantitatively alter results (e.g., a higher birth rate and evolvability or a lower drug dosage and natural death rate can promote the survival of the cell population under therapy), the same broad qualitative trends hold.

In all simulations, the population is initialized with densities of 0.5 across all cell states for a total population size of 50 cells. This choice of initial condition is relatively arbitrary and can be customized for specific applications. Here, we assume that at time 0, the cancer is initiated as a speciation event from normal somatic cells. As such, it is far from a demographic equilibrium (Pienta et al. 2020a; Pienta et al. 2020b). Therapy will be administered as a constant dose through time once the population reaches one million cells and the simulation will be stopped once the population recovers back to the one million cell threshold or once the population size is less than 10 (an arbitrary threshold). Note that, when the cancer is detected and treatment is administered, the population has yet to reach its demographic equilibrium. This is in accordance with clinical practice and ongoing experimental studies that suggest that cancer cell populations often are not near their demographic equilibrium when treatment is given. Key summary statistics are provided in Table 2: final drug resistance level, minimum of population, time of therapeutic application, time to progression (TTP: number of time steps between the start of therapy and when the population reaches one million cells again), and time to treatment failure (TTF: number of time steps between the start of therapy and when the population starts increasing in size).

Table 1 Parameter definitions and baseline values used in simulations
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Model exploration

First, we run a series of simulations to explore the dynamics of our model. We consider the following cases: baseline (parameters given in Table 1), no evolvability (\(k=0\)), high evolvability (\(k=4\)), low growth variance \(\sigma _G=0.01\), and high growth variance (\(\sigma _G=0.1\)) (Fig. 2 and Table 2). First, note the similarities among the simulations. Since cells in lower cell states proliferate more rapidly than those in higher cell states, the average cell state in the population decreases before therapy is administered. Once the population size passes the one million cell threshold and therapy is administered, the population’s demography shifts toward higher cell states, allowing the cells to minimize the effects of therapy at the cost of proliferating more slowly. This demographic transition buys time for adaptive genetic evolution to occur (Diamond and Martin 2016; Fox et al. 2019; Merilä and Hendry 2014). As resistance develops and treatment begins to fail, the population shifts toward pre-therapy demography, leading to higher proliferation rates and the eventual progression of the tumor (Table 2). However, if cells are unable to evolve resistance (i.e., \(k=0\)), the population remains in a high cell state. For the parameter values chosen here, this shift in demography is not enough for the population to avoid extinction.

There are also key differences among our simulations as a result of differences in evolvability and variance. Starting with the former, note that due to the cost of evolvability, it takes more evolvable populations longer to surpass the one million cells threshold, leading to a delayed start of therapy (Table 2). Although highly evolvable populations still shift their demography to higher cell states upon administration of therapy, this demographic transition is less extreme due to the rapid evolution of resistance. Finally, as one might expect, the rapid evolution of resistance leads to a higher minimum of the population and a shorter TTP and TTF than other cases (Table 2).

Now, consider the impact of variance. For biological realism and simplicity, we fix the variance in the fecundity kernel and solely consider changes to the growth kernel. An increase in the variance of the fecundity kernel can be considered a form of saltational evolution (Willis 1923), sensu Goldschmidt’s “hopeful monster" macromutations (Goldschmidt 1933) (a rather controversial theory (Dietrich 2003), particularly at the cellular level). Combining this with the fact that the fecundity kernel produces variance across states, not the resistance trait, we choose to focus solely on the impacts of variance in the growth kernel. However, due to the structure of our model, results for fecundity variance are qualitatively similar to those of growth variance, with quantitative differences arising from the magnitude of the difference between survival and birth probabilities and the interplay with evolutionary dynamics (survival is dependent on strategy but birth is not).

We see that lower variance kernels move closer to the tails of their distributions more effectively than higher variance kernels, a phenomenon that is paralleled in the pre-therapy demographic dynamics. This allows them to attain a lower average cell state and higher population size when therapy is administered. After therapy is given, the high growth variance population is able to shift its demography toward high cell states the most rapidly, reaching the highest average cell state among the evolvable populations. These factors dampen the selective pressure on the high growth variance population to evolve resistance. Thus, although this population progressed and reached treatment failure the quickest, it also evolved the least resistance (Table 2).

Fig. 2
figure 2

Impact of evolvability and growth variance on eco-evo-demo dynamics. Blue, black, red, green, and brown lines capture dynamics of populations with no evolvability, baseline conditions, low growth variance, high growth variance, and high evolvability, respectively. Higher growth variances and evolvabilities lead to higher population minima and faster times to progression and treatment failure (color figure online)

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Table 2 Model exploration summary
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Facultative stress-induced responses

With a basic understanding of how variance and evolvability impact eco-evo-demo dynamics, we now examine facultative stress-induced responses that populations may undergo. Throughout this paper, we have discussed two mechanisms by which cells can adapt to a therapeutic stressor: by changing their cell state (plasticity) and by evolving higher resistance levels (genetic evolution). However, both of these adaptations come with a cost. A shift in demography toward higher cell states leads to lower proliferation rates, and a faster rate of evolution (via a high evolvability) leads to a higher death rate. Thus, it is natural to ask: What if populations could modulate their degree of plasticity and evolvability depending on their condition? In other words, what if cells could increase their plasticity and evolvability when they are stressed and decrease them when they are stable? In this section, we consider how facultative plasticity and adaptive mutagenesis impact the eco-evo-demo dynamics of populations under therapy (Fig. 3 and Table 3).

Plasticity, the process by which cells transiently adopt a distinct phenotypic identity in a non-genetic fashion, is a commonly observed stress response in cancer and bacteria. Critically, these plastic transitions are completely reversible. In bacteria, a slow-cycling drug-tolerant persister (DTP) state plays such a role, providing a stepping-stone toward more permanent genetic resistance (Sharma et al. 2010). A similar DTP state was detected in non-small cell lung cancer, melanoma, and glioblastoma (Biehs et al. 2018; Liau et al. 2017; Marion et al. 2020; Rambow et al. 2018; Sánchez-Danés et al. 2018; Shen et al. 2019). In addition, the ADRN–MES transition in neuroblastoma, epithelial-to-mesenchymal transition, and polyaneuploid transition (PAT) have all been shown to contribute to therapeutic resistance in cancer (Amend et al. 2019; Chin and Lim 2019; Katsuno et al. 2019; Shun et al. 2019; Pienta et al. 2020; Pienta et al. 2021; Weng et al. 2019; Xiaojun et al. 2022; Zhou et al. 2020). To implement facultative plasticity, we use the same parameter values as in Table 1, but let the variance in the growth kernel depend on the cell’s condition: \(\sigma _G=.01+.15exp\left( -\frac{.3}{d_{drug}}\right)\) where \(d_{drug}=\frac{m}{exp(\phi z)+\beta u}\) represents the drug-induced death rate for a cell in state z with resistance level u. In addition to the eco-evo-demo dynamics, we plot the change in average growth variance over time (Fig. 3).

Adaptive mutagenesis is a widely recognized phenomenon in bacterial (Bjedov et al. 2003; Foster 2007; Fitzgerald et al. 2017; Rosenberg et al. 2012), yeast (Galhardo et al. 2009; Hastings et al. 2004; Layton and Foster 2003; Lombardo et al. 2004; Lovett 2006; Petrosino et al. 2009), and (albeit to a lesser extent) cancer (Hall 1995; Finch and Goodman 1997; Rosenberg et al. 1994; Strauss 1986) evolution. Also known as stress-induced mutations, adaptive mutations describe the generation of heritable variation that occurs in response to the environment (Rosenberg 2001). To implement adaptive mutagenesis, we use the same parameters in Table 1, but let evolvability be a function of the cell’s condition: \(k=5exp\left( -\frac{.1}{d_{drug}}\right)\). In addition to the eco-evo-demo dynamics, we plot the change in average evolvability over time (Fig. 3).

Before drug exposure, the variance in growth kernel and mutation rate is low. This allows the population to reach lower average cell states and proliferate more quickly or minimize natural cell death, respectively. Once therapy is applied, the population quickly increases its growth variance or evolvability. In the former case, facultative plasticity allows cells to quickly move toward higher states, whereas in the latter case, adaptive mutagenesis allows for rapid evolution of resistance. In both, the population drop is less severe (Table 3). Once the initial burst of adaptation or evolution has occurred, the population decreases its growth variance or evolvability. This allows for continued modest evolution of resistance, while allowing for higher proliferation rates or lower natural death rates. These aspects contribute to a faster rate of evolution and a shorter TTP and TTF.

Fig. 3
figure 3

Impact of facultative stress-induced responses on eco-evo-demo dynamics. The ability of a population to modulate the extent of its plasticity or nature of its mutations in response to its condition leads to faster proliferation/higher survival pre-therapy, higher minima, and shorter times to progression and treatment failure

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Table 3 Facultative plasticity and stress-induced mutagenesis results summary
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Therapeutic strategies

Throughout this paper, we have seen how the potent combination of plasticity and genetic evolution allows populations to evolve and adapt to stressful conditions. This leads to the natural question: How can we develop therapeutic strategies to effectively control or drive populations to extinction given the complex interplay among ecological, evolutionary, and demographic dynamics? As done in our past work and as suggested by others, one approach is to use a life-history enlightened approach (Bukkuri et al. 2022a; Bukkuri et al. 2022b; Shen and Clairambault 2020) by attempting to block plastic transitions toward drug-resistant phenotypes. Such approaches, at the epigenetic, intracellular pathway, and microenvironmental levels, have shown promise in a variety of cancers, including glioblastoma, breast cancer, melanoma, and prostate cancer (Guler et al. 2017; Liau et al. 2017; Roswall et al. 2018; Rusan et al. 2018; Sánchez-Danés et al. 2018; Straussman et al. 2012; Wallner et al. 2006). Similarly, efforts have been made to develop drugs to target cell states that are resistant to traditional therapies (Boshuizen et al. 2018; Hangauer et al. 2017; Rambow et al. 2018; Tsoi et al. 2018; Viswanathan et al. 2017). In this section, we focus on therapeutic strategies assuming that neither of these options exist.

Specifically, we will investigate the efficacy of intermittent and adaptive therapy protocols when cells can adapt to the stressor in both plastic and genetic ways. For simplicity, we will implement therapy solely on the baseline case (see Table 1) here. Under both protocols, we let therapy begin once the population size exceeds one million cells. We focus on three statistics: minimum of population, TTF, and TTP. If extinction is the goal, the minimum of the population is the metric of most importance. If the physician is aiming at control and only has one drug, TTP is most relevant. However, if the goal is control, several drugs are available, TTF is key.

First, we begin with the intermittent therapy. Under this protocol, therapy follows a fixed administration schedule, with both treatment and drug holiday times set constant for the entirety of the simulation. Once again, we set the drug dosage to be constant for the entirety of the treatment periods. Using the parameter values from Table 2, we simulate a sample intermittent protocol with 8 time steps on therapy and 4 time steps off, arbitrarily chosen (Fig. 4). The effects of the intermittent nature of the protocol can clearly be seen reflected in the ecological, evolutionary, and demographic dynamics of the population. Before progression, during periods of therapy, the population size decreases, resistance evolves, and the population shifts toward lower cell states. As cells gain resistance, however, each of these trends becomes less severe. Once progression occurs, the population is able to proliferate and reduce their cell state even in the presence of therapy (albeit in a less effective manner).

Fig. 4
figure 4

Intermittent therapy protocol. Longer holidays/shorter treatment times lead to higher minima, a shorter time to progression, but a longer time to treatment failure

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To better understand the effects of intermittent therapy, we simulate a range of treatment times and drug holidays and summarize key statistics in Fig. 5. Greater drug holiday durations and lower treatment times lead to higher minima as the therapy is not given enough time to reduce the population down to its minimum. Longer holidays and shorter treatment durations also lead to shorter TTP (since the population recovers to higher levels) and a longer TTF (since resistance evolves less quickly). Thus, assuming no cost of resistance, continuous therapy is more effective for extinction and control purposes when only one drug is available. On the other hand, including drug holidays may be more effective for multi-drug control.

Fig. 5
figure 5

Intermittent therapy: minimum of population, time to progression, and time to failure

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Next, we turn to our adaptive therapy protocol. Under this protocol, we administer therapy when the population increases past a threshold and remove therapy when the population decreases below another threshold. We first create a sample adaptive therapy protocol by setting the upper threshold to 800,000 cells and the lower threshold to 400,000 cells, arbitrarily chosen (Fig. 6). As before, we can clearly see the impacts of treatment times and treatment holidays on the eco-evo-demo dynamics of the population. Due to the adaptive nature of the dosing, we note that switching treatment on and off is more frequent at the beginning of therapy, when cells are relatively more sensitive to the drug. As cells gain resistance, therapy must be given for longer periods of time to push the population below the lower threshold. As before, drug dosage is held constant during treatment periods.

Fig. 6
figure 6

Adaptive therapy protocol. Higher thresholds for starting and stopping therapy leads to a shorter time to progression and a longer time to treatment failure

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To gain a deeper understanding on how the thresholds we set impact TTP and TTF, we simulate adaptive therapy protocols over a range of thresholds (Fig. 7). As we can see, higher thresholds for starting and stopping therapy lead to a shorter TTP and a longer TTF. High thresholds induce more therapy cycles than lower thresholds because less therapy is required to reduce the population down to the threshold bounds. This prolongs the time required to gain resistance; however, since the population is cycling at high levels, progression is not far once the cells escape.

Fig. 7
figure 7

Impact of thresholds on adaptive therapy protocols: time to progression and time to failure

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Under this model formulation, we assume no cost of resistance. This leads to an infinite improvement model of evolution, in which u is monotonically increasing. While this is a valid assumption in some cases, many cancers exhibit such costs. If we instead include a cost of resistance (as exists in many cancers (Gallaher et al. 2018; Jensen et al. 2015; Kam et al. 2015; Smalley et al. 2019)), we expect u to decrease during times of drug holidays and become re-sensitized, at least to some degree, to therapy upon the next administration. This could conceivably prolong TTP and TTF and may change our qualitative observations.

To test this hypothesis, we let \(b({u,}z)=b_1exp(b_2z+b_3u)\) where \(b_3=-0.2\). This induces a proliferation-resistance trade-off. For biological realism, we force \(u\ge 0\) to prevent negative resistance values pre-therapy. Using a drug dosage of \(m=0.7\), we simulate continuous therapy, intermittent therapy, and adaptive therapy protocols (Fig.  8). For intermittent therapy, therapy is on for 8 time steps, then turned off for 4 time steps. For adaptive therapy, we use an upper threshold of 800,000 cells and a lower bound of 400,000 cells.

Due to the cost of resistance, during times of therapy, u increases, but during times of drug holiday, u decreases, leading to resensitization of therapy. In all cases, this leads to lower overall levels of resistance and prolongs TTF and TTP (Table 4). However, even with this cost, we notice the same general trends as before: Continuous therapy is more effective at producing a low minimum of the population and promotes the longest TTP. On the other hand, intermittent and adaptive therapies are effective at keeping resistance levels low in the population, promoting a longer TTF.

Fig. 8
figure 8

Impact of cost of resistance on continuous, intermittent, and adaptive therapy protocols. The cost of resistance leads to higher times to progression and failure and lower resistance levels across the board. However, the same qualitative trends hold, with continuous therapy leading to a lower population minimum and higher time to progression, whereas intermittent and adaptive therapies lead to higher times to failure

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Table 4 Continuous, intermittent, and adaptive therapy with cost of resistance summary
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Conclusion

In this paper, we developed a novel theoretical framework using IPMs to examine stress-induced responses in bacterial and cancer cell populations. This framework allowed us to simultaneously consider the roles of plasticity (in a continuous, rather than discrete sense) and gradual genetic evolution in a population’s ability to respond to stressful environments. Currently, few methods exist to model both mutationally driven resistance and plasticity apart from our prior work on matrix population modeling (Bukkuri and Brown 2023; Bukkuri et al. 2022a; Bukkuri et al. 2022b). The key contribution of this paper is to add to this literature of modeling tools for eco-evo-demo dynamics. We found that faster evolving (i.e., highly evolvable) and more plastic populations (i.e., those with a high growth variance) are able to respond to stress effectively, avoiding extinction and causing treatment failure and progression rapidly. The interplay between plasticity and evolvability is a complex and practically relevant one: faster evolving species shift cell states less dramatically, and highly plastic populations evolve less mutationally-driven resistance. This observation may have implications for therapeutic regimens aimed at long-term control and life-history enlightened strategies (Bukkuri et al. 2022a). We branched into facultative stress-induced responses, in which plasticity and evolvability were condition-dependent. We found that these mechanisms were even more effective at promoting evolutionary rescue in the population and led to faster proliferation pre-therapy, higher minima, and shorter times to progression and treatment failure. We then considered intermittent and adaptive therapy regimens. Although continuous therapy is generally more effective if the goal is extinction or control with a single drug, introducing drug holidays can prolong TTF, an important aspect if the goal is multi-drug control.

Our study has several limitations. Being theoretical in nature, our parameter values were not informed by data. Future work could seek to calibrate and compare models with time-series RNA sequencing (at bulk or single-cell resolution) and population count data pre-therapy, during therapy, and after therapy. Furthermore, complementary methods such as agent-based modeling techniques could be employed for finer resolution and stochasticity. Different forms of the cost of evolvability and resistance, biased kernels for adaptive mutations and directed plastic transitions, and density dependence are all things future work can consider.

Theoretically, we hope to expand our modeling framework to include several continuous states. For instance, we could model the stochastic reversal of evolutionary resistance (a plausible outcome when a resistant population is left without therapy for extended periods of time) by treating resistance as a second state. We also hope to include hybrid discrete and continuous states, a notion particularly important when incorporating a polyploid state (Bukkuri et al. 2022b; Bukkuri et al. 2022a) or considering primary and metastatic tumors or habitat heterogeneity within a tumor (Cunningham et al. 2021). Finally, we hope that future work will apply this framework to the specific problems mentioned here: the epithelial-to-mesenchymal transition, ADRN–MES transition in neuroblastoma, and the PAT transition, as well as additional biological conundrums such as the stem-cell paradigm in wound healing and macrophage polarization.