Abstract
Despite major strides in the treatment of cancer, the development of drug resistance remains a major hurdle. One strategy which has been proposed to address this is the sequential application of drug therapies where resistance to one drug induces sensitivity to another drug, a concept called collateral sensitivity. The optimal timing of drug switching in these situations, however, remains unknown.
To study this, we developed a dynamical model of sequential therapy on heterogeneous tumors comprised of resistant and sensitive cells. A pair of drugs (DrugA, DrugB) are utilized and are periodically switched during therapy. Assuming resistant cells to one drug are collaterally sensitive to the opposing drug, we classified cancer cells into two groups, AR and BR, each of which is a subpopulation of cells resistant to the indicated drug and concurrently sensitive to the other, and we subsequently explored the resulting population dynamics.
Specifically, based on a system of ordinary differential equations for AR and BR, we determined that the optimal treatment strategy consists of two stages: an initial stage in which a chosen effective drug is utilized until a specific time point, T, and a second stage in which drugs are switched repeatedly, during which each drug is used for a relative duration (i.e. fΔt-long for DrugA and (1 – f) Δt-long for DrugB with 0 ≤ f ≤ 1 and Δt ≥ 0). We prove that the optimal duration of the initial stage, in which the first drug is administered, T, is shorter than the period in which it remains effective in decreasing the total population, contrary to current clinical intuition.
We further analyzed the relationship between population makeup, , and the effect of each drug. We determine a critical ratio, which we term , at which the two drugs are equally effective. As the first stage of the optimal strategy is applied, changes monotonically to and then, during the second stage, remains at thereafter.
Beyond our analytic results, we explored an individual based stochastic model and presented the distribution of extinction times for the classes of solutions found. Taken together, our results suggest opportunities to improve therapy scheduling in clinical oncology.
1 Introduction
Drug resistance is observed in many patients after exposure to cancer therapy, and is a major hurdle in cancer therapy [1]. In most cases, treatment with appropriate chemo- or targeted therapy reliably reduces tumor burden upon initiation. However, in the majority of cases, resistance inevitably arises, and the disease relapses [2]. The observation of relapse is typically accomplished during surveillance through imaging, or in some cases a blood based marker [3, 4]. Disease recurrence is observed, at the earliest, when the disease burden reaches some threshold of detection, at which point the first line therapy is deemed to have failed and a second line drug is used to control the disease (see Figure 1 (a)). We argue herein that a redesign of treatment should start earlier than this time point, not only because the detection threshold is higher than the minimum disease burden, but also because the first drug could become less efficient as the duration of therapy reaches Tmax. In this research, we focus on the latter reason and figure out how much earlier we should switch drug in advance of Tmax, assuming that the former reason is less important (tDT – to, ≈ Tmax).
While for many years it was assumed that tumors were simply collections of clonal cells, it is now accepted that tumor heterogeneity is the rule [5]. The simplest manifestation of this hetero-geneity can be represented by considering the existence of both therapy resistant and sensitive cell types co-existing prior to therapy [6], with the future cellular composition shaped by the choice of drugs (illustrated in Figure 1 (b)). Beyond simple selection for resistant cells, cells can also become altered toward a resistant state during treatment, either by (i) genetic mutations [7, 8] or (ii) phenotypic plasticity and resulting epigenetic modifications [9, 10, 11].
To combat resistance, many strategies have been attempted, including multi-drug therapies targetting more than one cell-type at a time. While multi-drug therapy has enjoyed successes in many cancers, especially pediatric ones, the resulting combinations can often be very toxic. Further, recent work has suggested that the success of multi-drug therapy at the population level is likely overstated in individuals, given intra-patient heterogeneity [12]. Recently, researchers have sought specific sequential single drug applications that induce sensitivity, a concept is called collateral sensitivity [13, 14, 15, 16]. In some cases, several drugs used sequentially can complete a collateral sensitivity cycle [15, 14], and corresponding periodic drug sequence can be used in the prescription of long term therapies – though the continued efficacy of this cycle is not guaranteed [17]. In this research, we focus on a drug cycle comprised of just two drugs, each of which can be used as a targeted therapy against cells that have evolved resistance to the previous drug (illustrated in Figure 1 (b)).
The underlying dynamics of resistance development has previously been studied using cell populations consisting of treatment sensitive and resistant types, using either genotypic or phenotypic classifications [18]. Additionally, others have justified their choices of detailed cellular heterogeneities using: (i) stages in evolutionary structures [19, 20], (ii) phases of cell cycle [21, 22, 23, 24], or (iii) spatial distribution of irregular therapy effect [25, 26]. Among these, researchers (including [18, 22, 23, 27, 28]) have studied the effect of a pair of collaterally sensitive drugs as we propose here, using the Goldie-Coldman model or its variations [19, 28, 29, 30]. These models utilize a population structure consisting of four compartments, each of which represents a subpopulation that is either (i) sensitive to the both drugs, (ii) and (iii) resistant to one drug respectively, or (iv) resistant to both.
In this manuscript we propose a modeling approach which is the minimal model sufficient to study the effects of two populations of cells and two collaterally sensitive drugs. The model’s simplicity facilitates exact mathematical derivations of useful concepts and quantities, and illustrates several novel concepts relevant to adaptive therapy. The remainder of the manuscript is structured as follows. In Section 2, we outline the model and define terms. In Section 3 we present analysis of drug switch timing and duration. In Section 4 we relax several assumptions in our analytic model and study extinction times in a stochastic formulation, which agrees well with analysis in the mean field. In Section 5 we conclude and present work for future directions.
2 Modeling setup
2.1 Basic cell population dynamics under a single drug administration
Based on the sensitivity and resistance to a therapy, the cell population can be split into two groups. We refer to the population sizes of sensitive cells and resistant cells as CS and CR respectively, and then use the total cell population size, CP := CS + CR, to measure disease burden and drug effect. We account for three dynamical events in our model: proliferation of sensitive (s) and resistant cells (r), and transition between these cell types (g). Here, net proliferation rate represents combined birth and death rate, which can be positive if the birth rate is higher than the death rate or negative otherwise. It is reasonable to assume that, in the presence of drug, the sensitive cell population size declines (s < 0), resistant cell population size increases (r > 0), and that g > 0. Therefore, for the remainder of the work we consider only conditions in which s < 0, r > 0 and g > 0.
Figure 2 illustrates the population dynamics, and the system of ordinary differential equations that {CS, CR} obey. The solution of the system (1) is where . By (2), total population is
CP(t) is a positive function comprised of a linear combination of exponential growth (er t) and exponential decay e−(g–s)t) with positive coefficients. Despite the limitations of simple exponential growth models [31], we feel it is a reasonable place to start, since the relapse of tumor size starts when it is much smaller than its carrying capacity which results in almost exponential growth.
CP has one and only one minimum point in {–∞, ∞}, after which CP increases monotonically. If , the drug is inefficient (CP(t) is increasing on t ≥ 0, see an example on Figure 3 (a)). Otherwise, if , the drug is effective in reducing tumor burden at the beginning, although it will eventually regrow (due to drug resistance; see example in Figure 3 (b)).
2.2 Cell population dynamics with a pair of collateral sensitivity drugs
Here we describe the effect of sequential therapy with two drugs switched in turn, by extending the model for a single-drug administration (System (1)). Assuming that the drugs are collaterally sensitive to each other, cell population is classified into just two groups reacting to the two types of drugs in opposite ways. Depending on which drug is administered, cells in the two groups will have different proliferation rates and direction of cell-type transition (see Figure 4). That is, the population dynamics of the two groups follow a piecewise continuous differential system consisting of a series of the system (1), each of which is assigned to a time slot bounded by drug-switching times.
In summary, we assume that:
There is a pair of collaterally sensitive drugs, DrugA and DrugB, which are characterized by their own model parameters: pA = {sA,rA,gA} and pB = {sB,rB,gB} respectively,
A modeled tumor can be characterized entirely by two subpopulations, AR – resistant to DrugA and simultaneously sensitive to DrugB, and BR – resistant to DrugB and simultaneously sensitive to DrugA.
Three factors determine the dynamical patterns, (i) drug parameters, {pA, pB}, (ii) the initial population sizes, {AR(0), BR(0)}, and (iii) the drug switching schedule.
An example of {AR, BR, AR + BR} histories is shown in Figure 5.
3 Analysis of therapy scheduling
3.1 Drug-switch timing
To begin exploring the possible strategies of drug switching and timing within our model, we first tested an idea based on clinical intuition. As we discussed, the norm in the clinic is to change drugs when failure is observed either radiographically or through a bio-marker. We know, however, that the true failure occurs somewhat before this, yet at that time it is below the threshold of detection. To model drug switching at the point of ‘true failure’, the intuitive (yet unobservable) time point when the tumor population begins to rebound, we switch the drugs at the global minimum point of tumor size which we term Tmax (see Figure 1a), which was shown to exist uniquely in the previous section if and only if CR(0)/CS(0) < –s/r. The expression for Tmax derived from our model, is with (R/S)0 := CR(0)/CS(0). (See Appendix A.1 for this derivation.)
We see that the quantity Tmax depends only on (i) the parameters of the drug being administered, and (ii) the initial population makeup. In the DrugA-based therapy, it is , and in the DrugB-based therapy, it is , where .
In addition to Tmax, another important time point is Tmin, explained below. Since the rate of population decrease is almost zero around Tmax, with no switch (see the black curve of Figure 6), we seek to find a way to extend the high rate of population decrease by switching drugs before Tmax. To decide how much earlier to do so, we compared the derivative of CP under constant selective pressure (no switch) at an arbitrary time point, t1, and compared it to the right derivative of CP at t1 with the drug-switch assigned to t1.
For example, if the first drug is DrugA and the follow-up drug is DrugB (illustrated in Figure 6), we compare from (3). This comparison reveals that the two derivatives are equal iff t1 is a specific point (Tmin(see the yellow curve in Figure 6)) The derivative when the drugs are switched is lower (decreasing faster) iff t1 > Tmin (see the blue and green curves in Figure 6), and the derivative when the drugs are not switched is lower iff t1 < Tmin (see the red curve in Figure 6).
The general form of Tmin depends on the parameters of the “pre-switch” drug {s1, r1 g1} and for the “post-switch” drug {s2,r2}, as well as the initial population ratio between resistant cells and sensitive cells to the “pre-switch” drug, (See Appendix A.1 for details derivation). Here, the transition parameter in the second drug (g2), and the respective values of the two populations are unnecessary in the evaluation of Tmin, which is found to be
In the DrugA-to-DrugB switch, it is Tmin(pA,pB, (A/B)0), and in the DrugB-to-DrugA switch, it is Tmin(pB,pA, 1/(A/B)0)), where ((A/B)0) = AR(0)/BR(0).
It is important to note that the population curve with a single drug-switch after Tmin (and before Tmax, assuming that Tmin < Tmax) is not guaranteed to be lower than that of a single drug-switch switch at Tmax over the entire time range. As an example, as illustrated in Figure 6, the green curve relevant to the switch at (Tmin + Tmax)/2 and the blue curve relevant to the switch at Tmax intersect at t ≈ 58 and the blue curve is lower after the time of this intersection. However, sequential drug switches starting between Tmin and Tmax create the possibility of finding a better drug schedule than the Tmax–based strategy. Figure 7 shows possible choices of follow up switches (green and black curves) which achieve better results than a Tmax–switch (red curves), unlike the drug-switches starting before Tmin, which remain less effective (magenta curve).
The optimal drug switching scheme will be discussed in detail in Section 4.2. The optimal scheduling for the example shown in Figure 5 starts by using the first drug until Tmin (blue curve for 0 < t ≤ Tmin) followed by a rapid exchange of the two drugs afterwards (black curve for t > Tmin)- Switching before Tmax that is, before the drug has had its full effect, goes somewhat against clinical intuition, and is therefore an opportunity for unrealized clinical improvement based on a rationally scheduled switch at Tmin. In order to realize this however, there are conditions about the order of Tmax and Tmin which must be satisfied. In particular:
In our analysis and simulations, we will deal with the cases mostly satisfying r1r2 < s1s2, as otherwise the choice of drugs is not powerful to reduce the cell population (explained in detail in the next section and Figure 8).
This window of opportunity, where the clinical gains could be made, which we will term Tgap, is the difference between Tmin and Tmax. This relationship allows us to compare Tmin and Tmax using different parameters.
We analyze sensitivity of Tgap over a reasonable space of non-dimentionalized drug parameters in Appendix B. As expected, as the proliferation rates under the second drug increases (r2 ↑ and/or s2↑), the optimal time to switch to the second drug is delayed (Tmin ↑ and Tgap ↓). As r1 increases, both Tmin and Tmax decrease. However, Tmax decreases more than Tmin does, so overall Tgap decreases, s1 and Tgap do not have a monotonic relationship. As s1 increases, Tgap increases for a while (when s1 is relatively low), and then decreases afterward (when s1 is relatively high).
3.2 Population makeup and drug effect
In the previous section, the derived time points (Tmin, Tmax) are dependent on the initial population makeup () from Equations (4)-(5), but not on explicit size of the total population or subpopulations. This makes sense, since absolute population size plays a role by scaling overall behavior of populations from (2)), and Tmin and Tmax are both defined by derivatives at the time points (i.e., CP(Tmax) = 0, and from (5)). In this section, we seek to clarify the relationships between population makeup and therapy effects defined using , and roles of Tmin and Tmax in these relationships. We first define functions of the ratio between the two cell subpopulations:
We further define functions measuring drug effectiveness as the relative rate of population change depending only on and drug parameters:
In the case where we classify cells as AR and BR, we similarly define their population makeup as:
Then at Tmin, using a DrugA-to-DrugB switch , and A/B, using a DrugB-to-DrugA switch , are equivalent:
At Tmax with DrugA , and with DrugB , we have and further, as s < and r > 0, values of are all positive. We give a more thorough description of (9) and (10) in Appendix A.1.
The effects of DrugA (specified by pA) and DrugB (specified by pB), both defined by (8), are equivalent at Tmin, that is . The effect of DrugA is larger if , since the DrugA resistant cell population is relatively smaller than the population of the other cell type, otherwise, DrugB has a more beneficial effect. When t = and therefore when , DrugA has no effect on population reduction (i.e. Ef(–(sA/rA|pA) = 0). If is getting smaller, DrugA becomes effective. Furthermore, the smaller is, the better the effect DrugA has. Similarly the effect of Drug B is zero when t = and and increases as increases above it (see Figure 8).
The population makeup changes in the opposite direction as DrugA (or DrugB) therapy continues, therefore continues to increase (or decrease). Therefore, if DrugA (or DrugB) is given too long, it goes through a period of no or almost no effect around = –sA/rA (or around = –rB/sB), but once the drug is switched after that, there will be a higher therapy effect with DrugB (or with DrugA). These two opposite aspects are balanced by switching the drug when the population makeup reaches , which is applied to the optimal therapy regimen described in the next section.
Depending on condition (6), the order of the three population makeups at Tmin, and changes. In particular, if rArB < sAsB, there exists an interval (–rB/sB,–sA/rA) in in which both drugs are effective in decreasing the population size. Otherwise, if rArB < sAsB, no drug is effective when ∈ –sA/rA. –rB/sB). These results are illustrated in Figure 8.
3.3 Optimal scheduling and its clinical implementation
In this section, we describe a drug-switching schedule design to achieve the best effect possible with a pair of collaterally sensitive drugs. The area under the curve of the total population simulated under an assigned treatment strategy is utilized to measure the aggregate effect of the strategy. The smaller the area, the better the corresponding strategy. The numerically determined optimal strategy consists of two stages:
Stage 1: Treat with first drug until reaching the population makeup where the effects of each drug are balanced (), that is until the Tmin of the first drug.
Stage 2: Begin switching drugs with a specific temporal ratio (represented by k or k’, see Figure 9) determining the difference in the treatment duration of each drug, and switching frequently (represented by Δt ≈ 0). Both conditions are used to keep close to constant near .
We represent the relative durations of DrugA compared to the duration of DrugB in Stage by k and k’. The explicit formulation of k can be derived from the solution of the differential equations (2). To do so we (i) evaluate the level of after Δt time has passed during DrugA therapy, when starting with , that is , and then (ii) by measuring the time period taken to regain (A/B)* from through therapy with DrugB, denoted by At’, and finally (iii) taking the ratio between the two therapy periods, which is k := Δt/Δt’, k depends on the frequency of drug switching and model parameters:
This k is consistent with k’ = k’(Δt, pA, pB), which is the ratio similarly evaluated with DrugB as the first therapy and DrugA as the follow-up therapy, in the optimal case of instantaneous switching:
For a more detailed derivation of k*, see Appendix A.1. We further studied how sensitive k* (or f* = k*/( 1 + k*)) is over a reasonable range of non-dimentionalized {pA,pB} (see Appendix B for details). k* (or f*) increases, as rA and/or sB decreases as sA and/or rB increases.
Figure 10 shows examples of population curves with the optimal strategy (Tmin switch) and one non-optimal strategy (Tmax switch) using the same choice of parameters/conditions. Visual comparison of total population curves (Figure 10 (a)) reveals that the predicted optimal strategy outperforms the intuitive strategy. To quantitatively compare the efficacy of each strategy, we can use area between the two population curves. This area is:
With a choice of upper limit large enough to include most treatment schedules, x = 100 (days), we used sensitivity analysis of the integral (13) (See Appendix B for the details). The advantage of the optimal treatment strategy is demonstrated by the lower population sizes in all cases. And the evaluations of the areas under the population curves from t = 0 to a range at the upper limit of integration (Figure 10 (b)) confirms the superior effect of the optimal strategy over time. Figure 10 (c) shows the typical pattern of in the optimal therapy compared to the other, which is monotonically changing toward in the first stage and constant in the second stage.
While our theory predicts optimality with “instantaneous drug switching”, we realize this is not clinically feasible. Therefore, the instantaneous drug switching in Stage 2 could be approximated by a high frequency switching stratgey with Δt ≳ along with the corresponding k(Δt) from (11), or k* (12) independent from Δt. As expected, the smaller Δt is chosen, the closer the population follows the ideal case with Δt = 0 (see Appendix C for the details), but improvements can still be made over non-strategic switching, if the temporal ratio is followed.
We have proved that the effect of instantaneous drug switching, with an arbitrary ratio in duration between two drugs (k), is consistent with the effect of a mixed drug with a relative dosage ratio, which is also k (Theorem A.8 in Appendix A.2). The theorem is used in the derivation of a differential system/solution of the optimal strategy (Theorem A.11 in Appendix A.3). According to these results, in Stage 2 of optimal regimen, all types of populations, AR, BR and AR + BR, change with the same constant proliferation rate:
While not clinical proof, these theoretical results suggest a method of application of two drugs in sequence, which would approximate multi-drug therapy in efficacy, but which could be free of the increase in side effects from the combination.
4 Studying extinction time with a stochastic formulation
In the previous sections we utilized an entirely deterministic model of heterogeneous tumor growth. Cancers, however, are not deterministic, and without stochasticity in our system we could not model an important part of cancer treatment: extinction. We therefore constructed a simple individual based model using a Gillespie algorithm [32] to study this critical aspect of therapy that is not limited by the assumptions we were required to make for purposes of analytic tractability.
Our stochastic model depends not only on net proliferation rates (s, r, see Equation (1)) but also on the combination of birth rates (bS, bR) and death rates (dS, dR) where s = bS – dS and r = bR – dR. These five parameters (bs, br, ds, dr, g) govern the probabilities of events occurring. The time at which one of these events occurs is determined by an exponential probability distribution, and we represent the algorithm as pseudo-code thus:
(Step 1) Initialize .
(Step 2)
(Step 3) t ⟵ t + dt and repeat (Step 2) until a set time has passed or extinction has occurred.
We expanded the stochastic process for a single drug to treatment with two drugs being switched in turn, as in our ODE system (See Appendix D, for the details of the computational code). Figure 11 (a) shows the consistency between the mean field behavior of the stochastic model and the ODE system.
Despite the generally similar patterns of population curves simulated with same {s, r, g}-type parameters and initial conditions, we observe differences in terms of elimination time if birth/death combinations are different. To quantify these differences we directly studied the elimination times (defined as the distribution of times to the absorbing state of total population = 0) simulated with different combinations of birth/death rates, with a choice of fixed proliferation rates (as well as other fixed transition rates and initial condition). We defined an index to represent different levels of birth and death rate combinations: where I indicates a type of sensitivity or resistance and J does a type of drug. Given a specific net proliferation rate (bI,J – dI,J), the larger the index, the larger both birth (bI,J) and death (dI,J) rates are.
Increased Istoch result in larger fluctuations, these fluctuations then increase the probability of reaching the absorbing state which is extinction (tumor cure). The relationship between Istoch and extinction time is shown in Figure 11 (b). The relationship is approximated by a linear model with slope, −93.68 (days2), p-value of the slope, p < 0.05, and squared residual of regression, r2 = 0.1726.
5 Conclusions and discussion
The emergence of resistance to the best current cancer therapies is an almost universal clinical problem, and the solution to this represents one of the greatest unmet needs in oncology. While much effort has been put into novel drug discovery to combat this, there is also a growing interest in determining the optimal sequences, or cycles of drugs that promote collateral sensitivity. To study this second paradigm, we proposed a simple dynamical systems model of tumor evolution in a heterogeneous tumor composed of two cell phenotypes. While in reality, cell phenotype can be defined in many ways, here we completely describe it by considering only sensitivity (or resistance) to a pair of collaterally sensitive drugs, which is encoded in their differential growth rates in specific conditions. While the resulting mathematical model conveys only simple, but essential, features of cell population dynamics, it does yield analytical solutions that more complex models cannot.
Our original motivation was to consider more complicated sequences, or cycles of drug therapy, however, the model presented herein is difficult to apply for an expanded system of more than two drugs. On the other hand, the cell classification used by others [18, 19, 28, 29, 33] considers sensitivity and resistance independently, or even specifically to a given, abstracted, genotype [34, 35]. Therefore, in the case of 2 drugs, there are 22 = 4 groups, (i) sensitive to both drugs, (ii) and (iii) resistant to only one drug, and (iv) resistant to both drugs. This formulation could be expanded and applied to more than two drugs [18, 33]. Also, in other earlier researches, cell populations are divided by more specific criteria for the choices of cancers and drugs (e.g., level of protein expression, enzyme inhibitors, or growth factors [10, 11, 8]). We will consider both of the general and specific approaches of population classification in future work.
The simplicity of our exponential growth/decay model arises from the assumption of a constant growth rate. Use of exponential growth is likely not overly inappropriate, as we are most interested in the development of resistance – and resistance is typically thought to begin when the tumor burden is much smaller than the carrying capacity. However, the assumption might have oversimplified patterns of cell growth, which is assumed to be non-exponential by others (e.g. logistic growth [31, 36, 37]), due to the limited space and resources of the human body for tumor growth, as well as increasing levels of resistance (increasing growth rates) in the face of continued selective pressure [38]. We will consider the concept of changing growth rates in terms of time and population density, and explore its effect on our analytical results (such as Tgap, , k* and etc.) in future work.
We provided a strategy for drug-switching which yields the best possible effect in this model system, i.e. the fastest decrease in cell population. The strategy is defined explicitly in terms of parameters determined by the drugs that are used, therefore the applicability of our model relies on the availability of drug parameters. Drug parameters for several drugs are known based on in vitro experiment or clinical studies [39, 40]. However, these parameters are not available for all drugs, and even the usefulness of in vitro results may change from one patient to the next. Because of this, we propose focusing our future work on learning to parameterize models of this type from individual patient response data. Examples of parameterizing patient response from imaging [41] as well as blood based markers [42] already exist, suggesting this is a reasonable goal in the near future.
In our optimized treatment regimen we must first apply DrugA (if DrugA is better at the initial time, i.e., , see Figure 8). Surprisingly the ideal treatment course switches to DrugB while DrugA is still effective at reducing the total population. Since treatment should ideally switch before the tumor relapses our study justifies the search for techniques that either identify or predict resistance mechanisms early. Our study also argues against the opposite extreme, wherein resistant cells are targeted at the beginning of treatment. The preponderance of cells sensitive to the standard of care makes this treatment initially ideal, and does not preclude eventual success in our model. Further, the rapid tumor size reduction, associated with targeting the larger sensitive population first, could be clinically meaningful.
Our stochastic model allowed us to explore the contributions of cell birth and death separately, as opposed to the ODE which could only consider the net growth rate. These parameters can be altered in cancer since cancer treatments have various cytostatic and cytotoxic effects, and therefore different treatments can have different effects on death and birth. In our model, increasing the total birth and death rate (as opposed to the net growth rate) caused, on average, extinction earlier in time (Figure 11 (b). This can be explained by the fact that extinction is the only absorbing state in our model, and therefore higher death rates determine when extinction occurs, even when birth rates are also higher. Our stochastic model therefore suggests that highly cytotoxic drugs (even those with correspondingly minimal cytostatic effects) are more effective at eliminating tumors, at least when the tumor population is small.
In summary, we have presented a simple model of a heterogeneous, two phenotype tumor, with evolution occurring between resistant and sensitive states. We derive exact analytic solutions for tumor response in temporally changing drug conditions and find an optimal regimen which involves drug switching after a specific, critical time point which occurs before resistance would normally be clinically evident. While our model is highly simplified, we have identified several opportunities to improve our understanding and treatment of drug resistance, and also future opportunities for new modeling endeavors.
Appendix A Derivations of explicit expressions
A.1 Details of Equations (4), (5), (6), (7), (9), (10) and (12)
1. Tmax: Equation (4)
Tmax is a minimum point of Cp(t) (from (3)). Therefore,
2. Tmin: Equation (5)
Let us consider the case of drug switch with DrugA being the “pre-switch” drug and DrugB being the “post-switch” drug. If, at a specific time point t1, cell population is decreasing faster 5u by continuing DrugA-therapy than by changing drug to DrugB, from Equation (3) where evaluated from Equation (2). Then, Similarly, iff the population is dropping faster using DrugB than by continuing to use DrugA, and iff the population is dropping at an equal rate with either drug.
The general form of Tmin is where the parameters of “pre-switch” and “post-switch” drugs are {s1, r1, g1 and {s2, r2. g2} respectively, and initial population makeup, , is the resistant cell population divided by the sensitive cell population for the “pre-switch” drug.
3. Tgap: Equation (6) – (7) And, Similarly Tgap = 0 iff r1 r2 = s1 s2, and Tgap < 0 iff r1 r2 > s1 s2.
4. at Tmax and Tmin: Equation (9) – (10).
It is clear that and by the expressions of AR(t), BR(t), Tmax and Tmin from Equations (2), (5) and (4).
Otherwise, it can be proved more simply using the concept of Tmin and Tmax. Since + , from the differential system (1), the derivatives of AR(t) + BR(t) are sA BR(t) + rA AR(t) and sB AR(t) + rB BR(t) under DrugA and DrugB respectively. At Tmin (whether it is or ) the derivatives of total populations are equivalent either under DrugA or under DrugB. Then,
Therefore,
Under DrugA at . Therefore,
Similarly, .
5. k*: Equation (12) The sizes of the subpopulations after Δt-long therapy with DrugA started from initial population makeup of are derived from Equation (2), with some constant K scaling population size. Then the population makeup at the Δt and its derivative in terms of Δt are
The time taken from t = Δt to reach back to the time of given DrugB is from Equation (5).
Then the relative ratio between the periods of DrugA and DrugB, k’, illustrated in Figure 9, and its limit, k*, can be derived using:
A.2 Differential system of 552 instantaneous drug switch
The goal of this section is to derive the simple differential equations of V = {AR, BR} under instantaneous drug switch (Theorem A.8). For the sake of convenience, we want to use matrix operations and equations based on the vectors and matrices defined below.
Proposition A.1. Using Drug A therapy:
Using Drug B therapy:
Proposition A.2. Both AR and BR are monotonic functions under either therapy. In the presence of Drug A, AR is increasing, and BR is decreasing. And, in the presence of Drug B, AR is decreasing, and BR is increasing.
Proposition A.3.
Proposition A.4.
Lemma A.5.
Proof.
Lemma A.6. for any positive integer, n, and for all 0 ≤ f ≤ 1
Proof. Let and . Then, we need to prove that F(n) = L for n = 1,2,3,…
If n =1,
Otherwise, if n ≥ 2 and F(m) = L for all 1 ≤ m ≤ n – 1, Therefore, proved.
Lemma A.7. for any positive integer, n, and for all 0 ≤ f ≤ 1
Proof. Using mathematical induction, if n = 1,
If n ≥ 2, and the equality works for all integers 1 ≤ m ≤ n – 1, Therefore, proved.
Theorem A.8. If Drug A and Drug B are prescribed in turn with a relative intensity of f and 1 – f, and are switched instantaneously, V obeys
Proof. For any time point t0, let us define Vϵ(t) as a vector-valued function of AR(t) and BR(t) describing the cell population dynamics under a periodic therapy starting at t0 with DrugA assigned at t0 + m ϵ ≤ t < t0 + (m + f)ϵ and DrugB at t0 + m ϵ ≤ t < t0 + (m + 1)ϵ for m = 0,1,2, 3,…. Then, by Proposition A.1 and the definitions of 픸 and 픹, where . And, V0(t) represents instantaneous drug switching.
For any Δt > 0 and any positive integer n, there exists ϵ =ϵ(n, Δt) such that Then by the squeeze theorem,
For such Δt, n and ϵ(n, Δt), Vϵ(t0 + Δt) is bounded, since local extrema can occur only when drugs are switched by Proposition A.2. That is,
Also,
And,
Similar to (*4), By (*4) – (*6), Then, by (*3), (*7) and 585 the squeeze theorem, Therefore,
A.3 Population dynamics with the optimal regimen
In this section, we want to write the differential equations of V = {AR,BR} under the optimal control strategy described in Section 3.3. Based on Appendix A.2 and a couple of lemma/theorem, we will reach to a concise form of a differential system described at Theorem A.11.
Lemma A.9. is an eigen pair of with (A/B)* and defined by Equations (9) and (12).
Proof. Let and . Then, , where along with
Since UTV =0 where V = ((rB – sA)=(rA – sB), 1)T, (λ, V) is an eigen pair of 픻*.
Theorem A.10. In Stage 2 of the optimal strategy, both AR and BR change with a constant net-proliferation rate,
Proof. Without a loss of generality, let us prove it only when A/B(0) < (A/B)*.
If A/B(0) < (A/B)*, DrugA has a better effect initially. So following the optimal therapy scheduling, DrugA is assigned alone at the beginning as long as (Stage 1), and then Stage 2 starts at with initial condition where
By Theorem A.8, in Stage 2, V (t) obeys
By Lemma A.9, V () is an eigenvector of 픻* with the corresponding eigenvalue, λ. Then, the solution of (**2) with the initial value (**1) is
Theorem A.11. With optimal therapy utilizing DrugA and DrugB, V obeys the following equations and solutions.
If A/B(0) < (A/B)*,
Similarly if A/B(0) ≥ (A/B)*,
Proof. Straightforward, by Theorem A.10
Appendix B Sensitivity analysis on optimal scheduling
The two determinant quantities of optimal control scheduling are (i) the duration of the first stage , and (ii) the relative intensity between two drugs in the second stage (k*). Here, we show sensitivity analysis on the quantities related to them, Tgap and f*, over a range of (scaled) model parameters. Additionally over the same range, we studied how much our Tmin-based optimal scheme is better than the Tmax-based scheme evaluated by the integral in equation (13).
1. Sensitivity analysis of Tgap Using g1, we non-dimentionalize all the values, like then,
In general, cells mutate slower than they proliferate, so we ran sensitivity analysis on Tgap for all a 1 for . Figure 12 shows Tgap over the range of 20 < . So, under the assumption that g1 min{–s1, –s2, r1 r2}, which approximates the contour curves of Figure 12.
2. Sensitivity analysis of f*
Regarding the regulated intensities among the two drugs, k*, we assumed that g1 ~ g2 := g, similarly assuming that they are both much smaller than {– s1, –s2, r1 r2). Then we normalized all the parameters with the unit of g, like k* can be rewritten in terms of the dimensionless parameters.
In this sensitivity analysis, we use which represents intensity fraction of the initially better drug out of the total therapy. We evaluated f* over the same ranges of {s1, s2,r1,r2}, like the previous exercise (see Figure 13) over the range max{g1, g2} ≪ min{–s1, – s2, r1, r2}, so k* and f* can be approximated by the simpler forms:
3. Sensitivity analysis of Integral (13)
To study the sensitivity of the advantage of using the optimal control defined by Integral (13), we assumed that g1 ≈ g2 ≈ g = 0.001. Then similar to the previous studies, we explored the sensitivity of the normalized parameters in terms of g, that is:
Appendix C Clinical implementation of instantaneous switch in the optimal strategy
In clinical practice, the instantaneous drug-switch which we suggest in the second stage of the optimal treatment scheduling is not implementable. Therefore, we compared similar schedules to the optimal case. In the “similar” schedules, the first stage, using an initial drug, remained the same as the optimal schedule. However the second part, where we previously used an instantaneous switch (with Δt = 0), was modified to use a fast switch (Δt > 0). Figure 15 (a) and (b) shows how instantaneous switching (Δt = 0) and fast switching (multiple choices of Δt > 0) compare in terms of population size using different drug parameters. As expected, the smaller Δt is, the closer to the ideal case. And, a choice of a reasonably small Δt (like 1 day or 3 days) results in an outcome quite close to the optimal scenario.
We repeated this exercise with k* (from equation (12)) instead of k(Δt) modulated by Δt (Figure 15 (c) and (d)). Only small differences are observed between Figure 15 (a) and (b) and Figure 15 (c) and (d), which justifies the general usefulness of k* independent of Δt.
Appendix D Stochastic simulation codes
The computational code written in Python will be provided at Github (https://github.com/nryoon12/Optimal-Therapy-Scheduling-Based-on-a-Pair-of-Collaterally-Sensitive-Drugs).