Ultrasensitive response explains the benefit of combination chemotherapy despite drug antagonism

Abstract

Most aggressive lymphomas are treated with combination chemotherapy, commonly as multiple cycles of concurrent drug administration. Concurrent administration is in theory optimal when combination therapies have synergistic (more than additive) drug interactions. We investigated pharmacodynamic interactions in the standard 4-drug ‘CHOP’ regimen in Peripheral T-Cell Lymphoma (PTCL) cell lines and found that CHOP consistently exhibits antagonism and not synergy. We tested whether staggered treatment schedules could improve tumor cell kill by avoiding antagonism, using in vitro models of concurrent or staggered treatments. Surprisingly, we observed that tumor cell kill is maximized by concurrent drug administration despite antagonistic drug-drug interactions. We propose that an ultrasensitive dose response, as described in radiology by the linear-quadratic (LQ) model, can reconcile these seemingly contradictory experimental observations. The LQ model describes the relationship between cell survival and dose, and in radiology has identified scenarios favoring hypofractionated radiation – the administration of fewer large doses rather than multiple smaller doses. Specifically, hypofractionated treatment can be favored when cells require an accumulation of DNA damage, rather than a ‘single hit’, in order to die. By adapting the LQ model to combination chemotherapy and accounting for tumor heterogeneity, we find that tumor cell kill is maximized by concurrent administration of multiple drugs, even when chemotherapies have antagonistic interactions. Thus, our study identifies a new mechanism by which combination chemotherapy can be clinically beneficial that is not contingent on positive drug-drug interactions.

Introduction

Combination chemotherapy is essential for the treatment of many types of cancer and often consists of multiple cycles of concurrent drug administration. Several types of Non-Hodgkin Lymphoma can be cured by combination therapies built upon the four-drug regimen ‘CHOP’, consisting of cyclophosphamide (C), doxorubicin (H), vincristine (O), and prednisone (P). For example, Diffuse Large B-Cell Lymphomas (DLBCL) are commonly treated with Rituximab plus CHOP (RCHOP), and Peripheral T-Cell Lymphomas (PTCL) are commonly treated with CHOP or CHP plus brentuximab-vedotin, depending on CD30 expression^1,2. Therapies in these CHOP-based regimens are typically administered concurrently every 2 or 3 weeks, for 6 to 8 cycles³.

Concurrent administration is in theory optimal when combination therapies have synergistic drug interactions, such that their combined effect is more than the sum of individual drugs’ efficacies. It has been recognized since the 1980s that the activity of combination therapies can depend on the timing of drug administration, with growing interest in ‘schedule-dependent synergy’ where the initial use of one drug can enhance response to another drug given some hours later^4–8. We recently observed in DLBCL that while the RCHOP regimen is overall close to additive, the cytotoxic agents ‘CHO’ exhibit antagonistic pairwise interactions, with effects that are less than the sum of individual drugs’ efficacies⁹. In line with these findings about RCHOP, a recent study has demonstrated a broader principle: therapies employing different mechanisms to induce cancer cell death often exhibit antagonism, where one death mechanism interferes with another¹⁰. Strategies to optimize synergy have been explored by adjusting timing of treatments on the scale of hours¹¹, but drug antagonism could in principle be prevented by avoiding overlapping drug exposure altogether, with administration separated by days. Here, we investigated the impact of drug antagonism on the efficacy of concurrent or sequential treatment by multiple drugs, using experiments and computational models. We selected PTCL as a model system because the CHOP regimen, with its several antagonistic interactions, remains the standard first-line treatment for many subtypes but cures fewer than half of patients².

In this study, we confirmed that the CHOP regimen has antagonistic interactions between C-H and H-O across seven PTCL cell lines. Contrary to the expected effect of antagonism, concurrently administering CHOP achieved the most cell killing as compared to sequential administration in in vitro experiments. These results reveal a benefit in dosing drugs together that overcomes the adverse effect of antagonistic drug interactions, which we propose can be explained by an ultrasensitive dose response model. The linear-quadratic (LQ) model is a method in radiation oncology that can be used to optimize radiation dose^12,13 and to understand the effects of hypofractionation or hyperfractionation: the administration of few large doses or many small doses, respectively^14,15. By adapting the LQ model to chemotherapy dosage, we found that it reproduces the experimental efficacy of different chemotherapy schedules and explains why concurrent use of multiple drugs can be more effective than sequential regimens even in the presence of antagonistic drug interaction.

Methods

Cell culture

PTCL cell lines were a gift from Dr. David M. Weinstock. Cell line identities was verified through short tandem repeat (STR) profiling by LabCorp and Cellosaurus CLASTR 1.4.4 database (Table 1). Cell lines were routinely tested for Mycoplasma and were negative (Lonza MycoAlert). Cells were cultured by standard methods in RPMI 1640 (Gibco) supplemented with Fetal Bovine Serum (Sigma) and Penicillin-Streptomycin (Gibco), at 37 °C and 5% CO₂ (Table 1). Live cell densities were counted by BioRad TC20 and trypan blue. Cells were passaged 2–3 times after thaw and before experiments at the seeding densities listed in Table 1.

Table 1:

Growing conditions for 7 T-cell lymphoma cell lines.

Cell Line	Subtype	STR Profile Match	Supplements	Seeding Density (cells / mL)	Treatment naïve?
DL-40	ALK-neg ALCL	100%	20% FBS	105; split every 3 days	Yes
KHYG-1	NK	100%	10% FBS + 100U/mL IL-2	105; split every 3 days	Yes
Ki-JK	ALK+ ALCL	100%	20% FBS	2×105; split every 2 days	Yes
MOT-N1	T-LGL	96%	20% FBS	105; split every 3 days	Yes
MTA	NK	97%	10% FBS	105; split every 3 days	Yes
SMZ-1	PTCL-NOS	N/Aa	10% FBS	105; split every 3 days	No50
SU-DHL-1	ALK+ ALCL	100%	10% FBS	105; split every 3 days	Yes

ALCL = anaplastic large cell lymphoma, ALK = anaplastic lymphoma kinase, FBS = fetal bovine serum, IL-2 = interleukin-2, NK = natural killer, NOS = not otherwise specified, STR = short tandem repeat, T-LGL = T-cell large granular lymphocytic leukemia.

^aSTR profile not in Cellosaurus CLASTR 1.4.4 database

Chemotherapy Preparation

4-hydroperoxycyclophosphamide (4-HC) was purchased from Niomech GmbH (D-18864) and all other chemotherapies were purchased from MedChemExpress. Prodrugs prednisone and cyclophosphamide were substituted with active species (prednisolone in place of prednisone and 4-HC in place of cyclophosphamide).

Measurement of drug interactions by the Bliss model

Protocols for dose response measurements were as previously described⁹ and detailed below.

Cell plating:

Cells at a density of 133,333 cells/mL were plated into 384-well black-bottomed plates (Nunc #164564) at 30 μL/well (corresponding to 4,000 cells/well), using a Thermo Fisher Multidrop Combi. After drug treatment, plates were incubated at 37 °C and 5% CO₂ (Heracell VIOS 160i) with secondary containment in plastic tubs lined with sterile wet gauze to minimize evaporation. Two separate wells in a 24-well plate were plated with 1 mL cells each at the same density, to monitor growth rate over the course of the experiment. Live cell density in one well was counted at the time of drug administration, and cell density in the second well was counted at the end of the experiment to calculate growth rate in the absence of therapy.

Drug administration:

Drugs were administered by a Tecan D300e digital drug dispenser. Wells were randomized during drug administration and re-organized during data analysis to avoid systematic spatial bias on the plates. Drugs were administered as single agents, pairwise, triplicate, and quadruplicate combinations in concentration ratios determined by their ${\mathrm{C}}_{\text{sustained}}$ values. ${\mathrm{C}}_{\text{sustained}}$ refers to a clinically relevant concentration based on measurements in patients’ serum up to 6 hours after administration¹⁶; these are: 4-HC, 15 μM^9,17; doxorubicin, 150 nM⁹; prednisolone, 5 μM^18,19; vincristine 5 nM⁹. Dose response measurements spanned concentrations from 0% to 500% ${\mathrm{C}}_{\text{sustained}}$ in log-spaced steps. Cells were incubated with drugs for 72 hours, which spans the in vivo elimination half-lives of these drugs²⁰.

Measuring relative viability:

After drug incubation, cell viability was quantified with Promega CellTiter-Glo (1:1 dilution in PBS) at 25 μL/well to visualize ATP levels by luminescence. CellTiter-Glo was administered with Multidrop Combi and plates were incubated for ten minutes. Plates were centrifuged at 200g for 5 minutes to eliminate air pockets. Luminescence was measured by BMG CLARIOstar reader using an aperture to reduce well cross-talk to below 10⁻⁴. Serial dilution of live cells confirmed a linear dynamic range over 4 orders of magnitude (100% to 0.01% relative live cell count) (Supplemental Figure 1).

Replicates:

Measurements consisted of four technical replicate wells per drug concentration per plate, with two biological replicates comprising independently propagated cultures, for a total n=8 per data point.

Analysis:

Relative viability expected by the Bliss Independence model was calculated by multiplying the relative viabilities produced by each single drug in a combination.

Measurement of drug interactions by isobologram analysis

Cell plating, drug treatment, and viability measurements were performed as above.

Drug concentrations:

For isobologram analysis, two-dimensional gradients of drug concentrations were prepared across a 11X11 grid of wells. Concentrations decreased in log-spaced steps across a 100-fold range, chosen to span a range from negligible inhibition to strong killing. Control wells with no treatment and single-agent dose responses were included on each plate.

Replicates:

Two technical replicates were repeated in each of two biological replicates for a cumulative n=4. The precision of automated liquid-handling produced high consistency between biological replicates (Supplemental Figure 1).

Analysis:

Relative viability at each concentration was an average of 4 independent experiments. A nearest-neighbor median filter was applied to relative viability across two-dimensional dose response surfaces. Isobolograms are plotted with contours highlighting 50%, 20%, and 5% relative viability.

MuSyC Analysis

To overcome the limitations of single drug interaction metrics, we also applied a model that synthesizes Bliss’ and Loewe’s models, titled Multidimensional Synergy of Combinations (MuSyC)²¹. MuSyC distinguishes between drug response curves changing in potency (A) versus efficacy (B). To apply MuSyC to combinations of 3 or more drugs, we adapted the framework to quantify changes in dose response compared with that predicted by the Bliss model. In this procedure, we applied shifts in potency (A) and efficacy (B) to the Bliss predicted response, to identify which parameters produce the best fit to the experimentally observed response, thereby quantifying how drug interactions change potency, efficacy, or both. If multiple values of A and B could produce similar best fits, we chose those with a consistent direction of change (e.g. increased potency and increased efficacy). This was necessary to exclude implausible claims of opposite interactions that cancel out and have no effect (e.g. increased potency and decreased efficacy).

Sequential versus concurrent CHOP treatments

MTA cells were plated at 200,000 cells/mL in flat-bottom, non-treated 6-well plates (Corning Costar). One plate was used per condition where each well was a replicate, n=6. In the control condition, one cycle of CHOP was administered concurrently on Day 0 at equipotent concentrations (C: 0.3 μM, H: 2.5 nM, O: 0.1 nM, P: 0.1 μM). Equipotent concentration ratios were defined by the monotherapy concentrations required for 12 days of growth suppression. Monotherapy concentrations were each lowered to 10% when constituting an equipotent 4-drug combination (full doses of each monotherapy would sterilize the flask). For sequential treatment conditions, a three-drug cocktail was administered on Day 0, and the ‘offset’ drug was administered as a single agent on Day 6, at the same concentrations as the CHOP control. Every three-days, cells were counted, washed twice by centrifugation, and resuspended in 10mL of drug-free media at 200,000 cells/mL in a fresh 6-well plate. If the cell density was below 200,00 cells/mL, the cells were not diluted. Live cell count was measured (BioRad TC20), and multiplied by the appropriate dilution factors (ie accounting for the dilution down to 200,000 cells/mL every 3 days to avoid cells overcrowding). Average live cell count was calculated from the six wells per condition (n=6).

Sequential versus concurrent C and H treatments

Monotherapy experiments were performed as described above with concentrations of full C: 3 μM, half C: 1.5 μM, quarter C: 0.75 μM, full H: 25 nM, half H: 12.5 nM, quarter H: 6.25 nM. CH combination experiments were performed as described above with doses of C: 1.5 μM and H:12.5 nM; split dosing consisted of C: 0.75 μM and H: 6.25 nM.

Adaptive Resistance

MTA cells were plated at 200,000 cells/mL in 50 mL for each of four 175 cm² flasks. One control flask received no treatment, and others were treated with single agent 4-HC (2 μM), doxorubicin (20 nM), or vincristine (0.35 nM). After three days of treatment, cells were washed twice, resuspended in drug-free media, and live cell count was measured (as described above). Cells recovered in drug-free media for 4 days, plated at 100,000 cells/mL in 50mL in 175 cm² flasks. On Day 7 post-treatment, cells were plated in 384 well plates (one plate per pre-treatment condition) and the dose response assays were performed as described in ‘Measurement of drug interactions by the Bliss model.’

Allee Effect

MTA cells were plated (150μL/well) in a black Nunc 96-well plate on a concentration gradient (in cells/mL): 400000, 300000, 200000, 100000, 33333, 11111, 3703, 1234, 411, 137, 45. N=7 per concentration. Viability was quantified with Promega CellTiter-Glo diluted 1:1 in PBS (150μL/well) three days after plating. Luminescence readouts were measured by BMG CLARIOstar reader. Five wells (1 well from 5 different plating densitites) were also quantified with Trypan blue staining to calculate a luminescence to live cell conversion rate. Luminescence readings were converted into cell counts and growth rates were calculated between days 0 and 3.

Model of ultrasensitive response and heterogeneity

Model parameters are in Table 2. The ‘Linear-Quadratic’ response equation (Eqn 1) is used in radiation oncology to describe the relationship between cell survival $\left(S\right)$ and dose $\left(D\right)$. The ‘linear’ component of dose response $\left(\mathrm{\alpha }D\right)$ corresponds to the probability that cells die from a single DNA damage event. The ‘quadratic’ component ($\mathrm{\beta }{D}^2$) corresponds to the probability that cells die from an accumulation of DNA damage. A high ratio $\mathrm{\beta }/\mathrm{\alpha }$ produces a more ultrasensitive response, which favors ‘hypofractionated’ treatment schedules. Here, we apply the LQ model to cytotoxic chemotherapies where treatment ‘fractionation’ is also relevant²².

$$S=e^{-(\alpha D+{\beta D}^2)}$$ Eqn 1

Table 2:

Parameters for the ultrasensitive drug response model. Dose in the linear-quadratic model is dimensionless such that when Dose = 1 the surviving fraction, S, is $S=e^{-{10}^x(\alpha +\beta )}$.

Parameter	Value	Units	Definition	Source
τdeath	2.5	Days	Half-life of dying cells	Measured (Supp. Fig. 8a)
τgrowth	2.5	Days	Doubling time of growing cells	Measured (Supp. Fig. 8b)
i	−0.8	Dimensionless, converted from combination index	CH drug interaction – less than additive	Measured (Supp. Fig. 8f)
μ	0	Dimensionless, dose is scaled by 10sensitivity	Sensitivity mean	Default
σ	0.42	Dimensionless, dose is scaled by 10sensitivity	Sensitivity standard deviation	Fit to experimental data (Supp. Fig. 8g)
β	0.4	Dose−2	Quadratic component LQ model	Fit to experimental data (Supp. Fig. 8g)
α	0.1	Dose−1	Linear component LQ model	Default
DC	2.6	Dose (dimensionless)	Dose of 4-H-cyclophosphamide	Measured (Supp. Fig. 8d)
DH	1.8	Dose (dimensionless)	Dose of doxorubicin	Measured (Supp. Fig. 8d)
P 0	5×105	Cells	Initial number of tumor cells	Used in experiments

A linear dose response shown is:

$$S=e^{-D}$$ Eqn 2

A ‘diminishing returns’ dose response (sub-linear) shown is:

$$S=e^{-\sqrt{D}}$$ Eqn 3

To model cellular heterogeneity in drug sensitivity, we describe a population of tumor cells with log-normally distributed drug sensitivity. That is, ‘x’ is normally distributed (μ = 0, σ = 0.42) and 10^x is the drug sensitivity parameter, which is multiplied by the dose response $\left(\mathrm{\alpha }D+\mathrm{\beta }{D}^2\right)$ to describe response in cells having greater or lesser drug sensitivity (Eqn 4).

$$S=e^{-{10}^x(\alpha D+{\beta D}^2)}$$ Eqn 4

Average survival of a cell population with $N$ subpopulations having survival $S_i$ is $\stackrel{-}{S}=\frac{1}{N}{\sum }_i{S}_i$. Average survival when quantified as ‘log-kills’ is defined as $\mathrm{\lambda }=-{\mathrm{l}\mathrm{o}\mathrm{g}}_{10}\left(\mathrm{S}\right)$. The average number of log-kills is therefore

$$\stackrel{-}{\mathrm{\lambda }}=-{log}_{10}\left(\frac{1}{N}\sum _i{10}^{-{\mathrm{\lambda }}_i}\right)$$ Eqn 5

To calculate the fraction of cells that survive drug treatment, for a population of cells with heterogeneous drug sensitivity, the dose response equation is integrated over the drug sensitivity distribution (Eqn 6).

$$\mathrm{S}=\int \frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{1}{2}{\left(\frac{x-\mu }{\sigma }\right)}^2}\left(e^{-{10}^x\left(\alpha D+\beta D^2\right)}\right)dx$$ Eqn 6

When modeling response to multiple drugs (e.g. A and B) with drug interactions, the dose term $D$ is replaced by an effective combined dose $d$ (Eqn 7) which may be greater or lesser than $D_{\mathrm{A}}+D_{\mathrm{B}}$ depending on the sign of a drug interaction term $i$:

$$d=D_{\mathrm{A}}+D_{\mathrm{B}}+i\frac{D_{\mathrm{A}}\times D_{\mathrm{B}}}{D_{\mathrm{A}}+D_{\mathrm{B}}}$$ Eqn 7

Equation 7 produces drug response ‘isoboles’ that are additive when $i=0$, synergistic when $i>0$, and antagonistic when $\mathrm{i}<0$. Conveniently, for any value of $i$ the multi-drug response surfaces produced by this equation exhibit a consistent ‘Combination Index’ ($CI$, a popular experimental metric of drug-drug interaction²³) at all magnitudes of effect. This is proven below for the case of an equipotent combination $\left(D_{\mathrm{A}}=D_{\mathrm{B}}\right)$:

Effective combined dose × Combination Index = Sum of doses

$$\left(D_{\mathrm{A}}+D_{\mathrm{B}}+i\frac{D_{\mathrm{A}}\times D_{\mathrm{B}}}{D_{\mathrm{A}}+D_{\mathrm{B}}}\right)\times CI=D_{\mathrm{A}}+D_{\mathrm{B}}$$

Substitute $D_{\mathrm{B}}=D_{\mathrm{A}}$ (drugs administered at equipotent doses):

$$\begin{gathered} \left(2D_{\mathrm{A}}+i\frac{D_{\mathrm{A}}^2}{2D_{\mathrm{A}}}\right)\times CI=2D_{\mathrm{A}} \\ \left(2+\frac{i}{2}\right)\times CI=2 \\ CI=\frac{1}{1+i/4} \end{gathered}$$ Eqn 8

Note the relationship between $CI$ and $i$ is independent of dosage, so any given value of $i$ yields a consistent Combination Index at all magnitudes of drug effect.

To model sequential treatments, it is not sufficient to multiply individually calculated values of S, as this would not track the consequence of the first cycle killing more drug sensitive cells, and thereby altering the drug sensitivity distribution. Instead, sequential dose response functions are multiplied within the integral over the sensitivity distribution:

$$\mathrm{S}=\int \frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{1}{2}{\left(\frac{x-\mu }{\sigma }\right)}^2}\left(e^{{-10}^x\left(\alpha D_1+\beta {D_1}^2\right)}\right)\left(e^{{-10}^x\left(\alpha D_2+\beta {D_2}^2\right)}\right)dx$$ Eqn 9

To model response kinetics, cells that survive drug treatment grow exponentially with a doubling time ${\mathrm{\tau }}_{\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{w}\mathrm{t}\mathrm{h}}$ (experimentally observed), named the growing population G(t) (Eqn 10). The population of cells dying from chemotherapy, H(t), are modelled as dying exponentially rather than instantly, because experimental measurements at sterilizing doses of chemotherapy reveal exponential death kinetics, with half-life ${\mathrm{\tau }}_{\mathrm{d}\mathrm{e}\mathrm{a}\mathrm{t}\mathrm{h}}$ (experimentally observed) (Eqn 11). Experimental cell counts measure both live, growing cells and also live cells that are in the process of dying. Therefore, the final model output, total live cell population, is the sum of growing and dying cell populations (Eqn 12).

$$\mathrm{G}\left(t\right)=P_0\times e^{\frac{t\mathrm{l}\mathrm{o}\mathrm{g}\left(2\right)}{{\tau }_{growth}}}\times S$$ Eqn 10

$$\mathrm{H}\left(t\right)=P_0\times e^{-\frac{t\mathrm{l}\mathrm{o}\mathrm{g}\left(2\right)}{{\tau }_{death}}}\times \left(1-S\right)$$ Eqn 11

$$\mathrm{T}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{p}\mathrm{o}\mathrm{p}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\left(\mathrm{t}\right)=\mathrm{G}\left(t\right)+\mathrm{H}\left(t\right)$$ Eqn 12

Modeling cells as either growing or dying is a simplification which was compared with a more complex model of cell states, in which increasing drug concentration decreases growth rate, then induces cytostasis, then induces an increasing death rate. This more complex model reproduces the key outcomes of the simpler model in which cells either grow or die; for clarity the simpler model is presented in main figures. Importantly, these models account for the change in sensitivity across the cell population caused by drug selection. As a consequence of repetitive treatments, drug sensitive subpopulations are the first to die, and the total cell population becomes progressively less drug sensitive. The model accounts for this phenomenon by quantifying the additional cell kill fraction from the previous treatment instead of assuming each treatment has an equivalent cell kill fraction.

When hypothetical drug pair ‘A+B’ was modeled, each single dose was modeled at D = 3. Similarly, the hypothetical single agent was modeled at D = 12 for each single dose.

Parameter regime favoring concurrent treatment

In the model, three parameters define the relative efficacy of concurrent versus sequential treatment with multiple drugs: the level of ultrasensitivity (quantified by $\mathrm{\beta }/\mathrm{\alpha }$, the ratio of quadratic and linear terms in the linear-quadratic model; Eqn 1), the strength of antagonism (quantified by Combination Index, $C$), and the strength of cell killing (quantified by the drug dose $D$, where increasing $D$ increases cytotoxic effect). Here we derive parameter regimes that define whether concurrent or sequential treatment with two drugs (drug A and drug B) has the greatest cytotoxic effect for both mutual and asymmetric drug suppression.

For mutual drug suppression, each drug decreases the potency of the other (Eqn 7). Assuming equipotent doses where $D_{\mathrm{A}}=D_{\mathrm{B}}=D$, the effective combined dose, $d$, which accounts for the effect of drug-drug interactions $i$, is:

$$d=D+D+i\frac{D\times D}{D+D}$$

Simplify:

$$d=2D+\frac{i\times D}{2}$$ Eqn 13

Solve $C=\frac{1}{1+\raisebox{1ex}{$i$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}$ (Eqn 8) for $i$:

$$i=\frac{4}{C}-4$$

Substitute into Eqn 13 and simplify:

$$d=\frac{2D}{C}$$ Eqn 14

Eqn 14 demonstrates the intended meaning of Combination Index: combining two drugs with equipotent doses $D$ produces an effective combined dose equal to $2D$ if additive ($C=1$), greater than $2D$ if synergistic ($C<1$), and less than $2D$ if antagonistic ($\mathrm{C}>1$).

For tumor cells with drug sensitivity ${10}^x$, concurrent therapy with effective combined dose $\mathit{d}$ results in a surviving cell fraction:

$$e^{{-10}^x\left(\alpha d+\beta d^2\right)}$$ Eqn 15

and sequential therapy with two individual drug doses $\mathit{D}$ results in surviving cell fraction:

$$\begin{array}{l}\left(e^{-{10}^x\left(\alpha D+\beta D^2\right)}\right)\left(e^{-{10}^x\left(\alpha D+\beta D^2\right)}\right)\\ =e^{-{10}^x\left(2\alpha D+2\beta D^2\right)}\end{array}$$ Eqn 16

Equations 15 and 16 demonstrate that irrespective of tumor drug sensitivity (${10}^x$), the relative efficacy of concurrent versus sequential therapy depends on the terms in these equations’ exponents, with concurrent therapy being more effective if:

$$\alpha d+\beta d^2>2\alpha D+2\beta D^2$$ Eqn 17

Substitute effective combined dose accounting for drug interactions (Eqn 14) into Eqn 17:

$$\alpha \left(\frac{2D}{C}\right)+\beta {\left(\frac{2D}{C}\right)}^2>2\alpha D+2\beta D^2$$

Simplify:

$$\begin{array}{c}\frac{2\alpha D}{C}+\frac{4\beta D^2}{C^2}>2\alpha D+2\beta D^2\\ \frac{\alpha }{C}+\frac{2\beta D}{C^2}>\alpha +\beta D\\ \beta \left(\frac{2D}{C^2}-D\right)>\alpha \left(1-\frac{1}{C}\right)\\ \frac{\beta }{\alpha }>\frac{C^2-C}{D\left(2-C^2\right)}\end{array}$$ Eqn 18

Thus, Eqn 18 defines parameter regimes favoring concurrent over sequential treatment in the presence of mutual drug-drug antagonism, considering the consequences of ultrasensitivity in drug response ($\mathrm{\beta }/\mathrm{\alpha }$), drug-drug interactions (Combination Index $C$), and overall strength of cell killing (based on drug dosage $D$).

To evaluate the relationship between these parameters in the presence of asymmetric antagonism (where only one drug’s potency is affected by interaction, $C$), the effective combined dose, $d$, is:

$$d=D_A+\frac{D_B}{C}$$ Eqn 19

Assuming equipotent doses where $D_{\mathrm{A}}=D_{\mathrm{B}}=D$, the equation is rewritten as:

$$d=D+\frac{D}{C}$$ Eqn 20

To compare the relative efficacy of concurrent versus sequential therapy, the same approach is used as for Eqn 17, but using Eqn 20 for the effective combined dose $d$ with asymmetric interaction:

$$\alpha (D+\frac{D}{C})+\beta {(D+\frac{D}{C})}^2>2\alpha D+2\beta D^2$$ Eqn 21

Simplify:

$$\begin{array}{c}\alpha D+\frac{\alpha D}{C}+\beta D^2+\frac{2\beta D^2}{C}+\frac{\beta D^2}{C^2}>2\alpha D+2\beta D^2\\ \frac{\alpha D}{C}+\frac{2\beta D^2}{C}+\frac{\beta D^2}{C^2}>\alpha D+\beta D^2\\ \frac{\alpha }{C}+\frac{2\beta D}{C}+\frac{\beta D}{C^2}>\alpha +\beta D\\ \frac{2\beta D}{C}+\frac{\beta D}{C^2}-\beta D>\alpha -\frac{\alpha }{C}\\ \beta \left(\frac{2D}{C}+\frac{D}{C^2}-D\right)>\alpha \left(1-\frac{1}{C}\right)\\ \frac{\beta }{\alpha }>\frac{C^2-C}{D\left(1+2C-C^2\right)}\end{array}$$ Eqn 22

Thus, Eqn 22 defines parameter regimes favoring concurrent over sequential treatment in the presence of asymmetric drug antagonism, considering the consequences of ultrasensitivity in drug response ($\mathrm{\beta }/\mathrm{\alpha }$), drug-drug interactions (Combination Index), and overall strength of cell killing (based on drug dosage $D$).

Data availability statement

Source data for all experiments are available in supplementary data.

Results

CHOP has antagonistic interactions in PTCL cells

To test if drug antagonism observed in DLCBL also occurs in PTCL, we measured pharmacodynamic drug interactions within the CHOP regimen in seven PTCL cell lines, representing four histologically diverse subtypes (Anaplastic Large Cell Lymphoma (ALCL), Natural Killer (NK), T-cell Large Granular Lymphocytic (T-LGL), and PTCL-not otherwise specified (PTCL-NOS)). Drug response was quantified by an ATP-based luminescence assay which provided a 10,000-fold dynamic range in live cell count (Supplemental Figure 1). Drug-drug interactions were analyzed by Loewe’s dose additivity model, which assesses the potency of a combination (such as concentration that elicits 90% inhibition; IC90) and is depicted by isobolograms which show contours of equal drug effect (isoboles) (Figure 1a, Supplemental Figure 2)^24,25. Two drugs are ‘additive’ when isoboles are straight lines, which indicates that the IC90 of the drug combination is achieved by adding half of the IC90 of the first drug to half the IC90 of the second drug; this also occurs when ‘combining’ a drug with itself. A drug combination is synergistic if potencies are enhanced in combination (convex isoboles) and antagonistic if potencies are diminished in combination (concave isoboles). Isobologram analysis confirmed that cyclophosphamide and doxorubicin (C-H) have interactions ranging from additive to strongly antagonistic, and doxorubicin and vincristine (H-O) are in most cases antagonistic or severely so (Figure 1a). No synergistic interactions were observed in any cell line.

Figure 1. Despite antagonistic interactions, concurrent administration of CHOP is more effective in vitro than sequential administration.

A, Isobolograms, or contour lines of equal drug effect (n=4 replicates per cell line), reveal antagonistic interactions between drugs in CHOP. Dashed line: isobole expected of an additive, or non-interacting, drug combination. Isoboles for 4-H-cyclophosphamide-doxorubicin (C-H) are graphed at IC90 for all cell lines, and isoboles for doxorubicin-vincristine (H-O) are graphed at IC90 for all lines except for KHYG-1, MOT-N1, and Ki-JK, which are graphed at IC70 because 90% inhibition was not reached by both drugs in these lines. Isoboles could not be graphed for cell line DL-40 because of insufficient drug response, and KHYG-1 was insufficiently responsive to 4-H-cyclophosphamide to produce a C-H isobole. Concentration units are normalized to each drug’s IC90 (or IC70) in the specific cell line. Complete response surfaces are in Supplemental Figure 2. B, PTCL cultures (MTA) were treated in 72-hour blocks, alternating between treatment, and recovery in drug-free media. Drugs were administered at an equipotent ratio (Methods) of C: 0.3 μM, H: 2.5 nM, O: 0.1 nM, P: 0.1 μM. Live cell count (hemocytometer and trypan blue) was measured over the span of one cycle of concurrent or sequential CHOP, indicated by colored boxes along the x-axis (n=6 independent cultures per condition, error bars are 95% confidence). Two-sided student’s t-tests were applied at Day 15, showing that sequential regimens were either significantly inferior to or the same as concurrent CHOP (Offset C: P = 6×10⁻⁵; Offset H: P = 1×10⁻⁵; Offset O: P = 0.2; Offset P: P = 4×10⁻³). C, Left) Live cell count after treatment with one full dose (3 μM), two half-doses (1.5 μM), or four quarter-doses (0.75 μM) of 4-H-cyclophosphamide. Right) Live cell count after treatment with one full dose (25 nM), two half-doses (12.5 nM), or four quarter-doses (6.25 nM) of doxorubicin. Treatment schedules are visualized underneath each time axis. On Day 15, sequential treatments were significantly inferior to their respective concurrent treatment groups (4-H-cyclophosphamide: halves P = 9×10⁻⁵, quarters P = 1×10⁻⁵; doxorubicin: halves P = 7×10⁻³, quarters P = 1×10⁻⁵). D, Live cell count after CH combination treatment using concurrent, sequential, or ‘split’ dosing (both drugs at half dose, for two sequential doses). Uppercase C and H in the schematic represent C = 1.5 μM 4-H and H= 12.5 nM doxorubicin. Lowercase c and h in the schematic represent half-doses: c = 0.75 μM 4-H-cyclophosphamide; h = 6.25 nM doxorubicin. On Day 15, the sequential and split treatments were significantly inferior to concurrent treatment (C followed by H: P =1×10⁻⁵; H followed by C: P = 2×10⁻⁴; split: P = 1×10⁻⁵).

To confirm that the pairwise antagonistic interactions between C-H and H-O negatively affect the full CHOP combination, two drug interaction metrics were used that extend to high-order interactions (combinations of two, three, and four drugs). The Bliss independence model (Supplemental Figure 3)²⁶ and the ‘Multidimensional Synergy of Combinations’ method (MuSyC) (Supplemental Figure 4)²¹ both confirmed that antagonistic interactions within CHOP significantly decrease net efficacy (P=7×10⁻³⁵; n=237 concentration points). Specifically, the number of ‘log-kills’ (90% inhibition = 1 log-kill) was approximately 20% less than expected, such that, for example, when 2 log-kills were expected, 1.6 log-kills were observed, corresponding to 2.5× more surviving tumor cells (Supplemental Figure 3).

Concurrent therapy is more effective than sequential

All metrics of drug interaction confirmed that antagonism between H-O and C-H decreases the overall efficacy of CHOP. We hypothesized that separating the administration of antagonistic drugs would increase the efficacy of the combination in vitro. We compared the long-term efficacy of concurrent or sequential therapy by measuring live cell count throughout a 12-day treatment cycle. In concurrent therapy, all four drugs in CHOP were administered at the start of the cycle; this resembles the clinical regimen. In sequential therapy, three drugs were administered at the start of the cycle and the remaining drug was given mid-cycle. Total administered dose was constant across all conditions. MTA cells were studied because their levels of drug sensitivity and drug interactions were typical of these PTCL cell lines. Concurrent therapy was observed to produce the greatest therapeutic effect, whereas sequential regimens that offset C, H, or P were much less effective, leaving 10 times more surviving lymphoma cells compared to concurrent therapy (Figure 1b). Offsetting O from the combination produced the same inhibitory effect as concurrent treatment. Thus, antagonistic drug interactions did not compromise the activity of concurrent therapy.

We next hypothesized that the dose intensity of concurrent therapy may confer an advantage that overcomes antagonistic drug interactions. First, to test this hypothesis without the confounding effect of drug interactions, we compared concurrent and sequential treatments of single agent cyclophosphamide (C) or doxorubicin (H), the drugs that had the most pronounced difference when administered sequentially. Concurrent therapy consisted of one full dose at the beginning of the cycle (day 0), and sequential therapy consisted of either two half doses (days 0 and 6) or four quarter doses (days 0, 3, 6, 9) (Figure 1c). Even as monotherapies, administering the entire dose at one time had a 10-fold or greater effect than the sequential treatments. This data shows a benefit to high intensity dosing that is independent of drug interaction. Second, to test whether this advantage applies even to an antagonistic drug combination, we compared concurrent therapy with C and H to sequential treatments of C then H, or H then C, separated by three days. Additionally, a split dosing regimen was evaluated consisting of C plus H each administered at half concentration (Figure 1d). By day 15 (9 days after the second dose), both sequential treatments and the split treatment were significantly less effective than a single concurrent administration of C and H. Thus, for both monotherapy and combination therapy, concurrent administration of the greatest possible dose intensity produces the greatest inhibitory effect, despite antagonistic drug interactions.

An inherent benefit of concurrent drug administration could be explained by three hypotheses. First, tumor cells could have physiological responses to the first dose of chemotherapy that make them more resistant to a subsequent treatment, a form of ‘collateral resistance’ that has been experimentally observed among other cancer therapies over timescales of hours to days; this is distinct from Darwinian evolution because it is rapid in onset and does not require strong selection^27–29. Second, if growth rates decline at lower population densities, an ecological principle called the Allee effect, then concurrent delivery of a strong initial cytotoxic effect may elicit more durable tumor control than sequential delivery of the same overall effects³⁰.Third, the dose response function could exhibit a form of ultrasensitivity where higher doses produce a disproportionately large increase in cytotoxic effect, thus favoring concurrent over sequential therapy. Recognizing these hypotheses are not mutually exclusive, we next tested whether adaptive resistance, the Allee effect, or ultrasensitive response can explain the benefit of concurrent therapy in the case of CHOP.

Adaptive resistance does not explain the advantage of concurrent therapy

An adaptive cellular response to an initial chemotherapy dose may diminish sensitivity to subsequent treatments, making sequential regimens less effective. Previous studies have observed that cancer cells can undergo adaptive changes within hours to days of an initial drug treatment, eliciting either collateral sensitivity or collateral resistance to subsequent treatments^27,28. To test whether such adaptive resistance diminishes drug sensitivity in sequential CHOP, we treated MTA cells with partially inhibitory doses of single agents C, H, O, or placebo, for three days, and then after four days of recovery, repeated dose response measurements. In no case were pre-treated cultures less sensitive to subsequent therapy (Figure 2a and Supplemental Figure 5). Therefore, short-term adaptive resistance does not explain the superiority of concurrent CHOP administration.

Figure 2. Adaptive resistance and the Allee effect do not explain the superiority of concurrent treatment.

A, Pre-treating lymphoma cultures with drugs in CHOP does not induce adaptive resistance to subsequent treatment by other drugs in CHOP. MTA cell cultures were treated for 3 days with either 4-H-cyclophosphamide, doxorubicin, vincristine, or no treatment (control), with drug concentrations producing 50% growth inhibition. Cultures were washed and recovered in drug-free media for 4 days. After recovery, dose responses were measured for 3-day treatments with 4-H-cyclophosphamide, doxorubicin, vincristine, prednisolone, and the CHOP combination. Drug sensitivity was quantified by area over the dose response curve (AOC; see complete dose responses in Supplemental Figure 5), normalized by AOC of controls receiving no pre-treatment. Error bars are 95% confidence intervals; n=8. B, The Allee effect does not explain the superiority of concurrent treatment because lymphoma growth rates do not decrease at low cell densities. MTA cultures were established at initial cell densities ranging from 45 to 400,000 cells/mL, and growth rates were calculated from live cell counts after 3 days (Methods). Error bars are 95% confidence intervals; n=7.

The Allee effect does not explain the advantage of concurrent therapy

The Allee effect, a positive dependence of growth rate on population density, could mean that rapidly reducing tumor population by concurrent treatment could decrease tumor cell growth rates, leading to more long-lasting tumor control than sequential treatments. We tested whether PTCL cultures exhibit an Allee effect by measuring growth rates of MTA cultures seeded at initial densities ranging from 45 to 400,000 cells/mL. Growth rates were consistently rapid at low densities (doubling every 18 hours for densities below 30,000 cells/mL), and gradually slowed at high densities, a pattern that is opposite to the Allee effect (Figure 2b). Because the lower densities reached by concurrent treatment do not result in slower growth rates, the Allee effect does not explain the superiority of concurrent CHOP administration.

Ultrasensitive response can explain the advantage of concurrent therapy

Like many chemotherapies, the agents in CHOP exhibit approximately exponential dose response functions over the first ~90% of inhibition³¹. This corresponds to a linear increase in log-kills with drug concentration (Supplementary Figure 6a), which would predict equal effect of concurrent or sequential therapy for the same total dosage. However, if this relationship is ultrasensitive (more than linear) at greater cytotoxic effect, then the higher dose-intensity of concurrent therapy could be advantageous. Ultrasensitive dose responses are a central concept in radiation oncology, where the Linear-Quadratic (LQ) model describes two components of dose response: a linear term describing the probability of cell death per single event of DNA damage ($\mathrm{\alpha }$), and a quadratic term describing the added probability of death from an accumulation of DNA damage ($\mathrm{\beta }$) (Methods, Equation 1). The LQ model describes the relative efficacy of hyperfractionated radiation (many small doses) versus hypofractionated radiation (one or few large doses) in various scenarios. When the quadratic term ($\mathrm{\beta }$) is large, the greatest cytotoxicity is achieved by a single high dose of radiation, akin to concurrent administration of many drugs. Here we apply the LQ model to cytotoxic chemotherapies that also target DNA, to test the hypothesis that ultrasensitive dose response explains the benefit of concurrent chemotherapy.

In a linear dose response, doubling the dose that produces one log-kill will produce two log-kills (Figure 3a). In principle the same effect would result from sequential administration of two such doses, because 1 log-kill followed by 1 log-kill is a net effect of 2 log-kills. Conversely, with an ultrasensitive response as described by the LQ model, concurrent administration (doubling the total dose) could produce as much as 4 log-kills, providing ‘increasing returns’ (Figure 3a). However, when examined on a log-scale, dose responses to single or combined agents in CHOP were sub-linear in each of 7 cell lines, i.e., exhibiting decreasing returns (Figure 3b and Supplemental Figure 6). In a sub-linear response, doubling the dose that produces one log-kill produces less than two log-kills; this would favor sequential therapy, in contradiction to the experimentally demonstrated superiority of concurrent therapy.

Figure 3. Ultrasensitive dose responses will be masked by cellular heterogeneity.

A, A comparison of linear and ultrasensitive dose response functions. With a linear response, doubling the concentration for 90% inhibition (1 log-kill) achieves 99% inhibition (2 log-kills). With an ultrasensitive response, doubling the dose more than doubles the number of log-kills. B, CHOP and its components exhibit ‘sub-linear’ dose responses in PTCL cell lines (n=8) (MTA shown here; see other cell lines in Supplemental Figure 6). C, Phenotypic heterogeneity, which can be genetic or stochastic in origin, can cause drug sensitivity to vary between individual cells. Variance in sensitivity to drugs in CHOP is observed to follow a log-normal distribution which is quantified by the mean (μ) and standard deviation (σ) of drug sensitivity (Supplemental Figure 7). D, Ultrasensitive dose responses with variable sensitivity illustrated by seven example clones ($viability=e^{-{10}^x*(\alpha D+{\beta D}^2)}$ see Methods). Drug sensitivity is indicated by color where red represents more sensitive, and blue represents less sensitive. The black line is the dose response of a whole population, composed of variable clones with log-normally distributed drug sensitivities. Note that the population’s average response has a different shape from individual clones, when graphed on a log-scale. This arises because the average of 1 log-kill and 3 log-kills is 1.3 log-kills (average of 10% and 0.1% is ~5%). E, Increasing cellular heterogeneity causes ultrasensitive dose responses to be obscured at the population scale. Here, all cells have ultrasensitive dose responses, but this is only evident in homogeneous populations (dark green line). Heterogeneous populations show sub-linear dose responses despite underlying ultrasensitivity of single cells (yellow line).

To resolve this seeming paradox we considered research on cell-to-cell variability, which has shown that population-level dose responses are an average of the phenotypes of individual cells or subpopulations which vary in their drug sensitivity^32,33. We hypothesized that individual cells could have ultrasensitive dose responses that appear more gradual at the population level due to cellular heterogeneity. Indeed, we recently observed variability in the sensitivities of lymphoma cells to the drugs in CHOP, with a log-normal distribution of sensitivities revealed by high-complexity clone tracing as well as CRISPR screens (Supplemental Figure 7)⁹. We therefore built a mathematical model of ultrasensitive dose response in a population of cells having log-normally distributed drug sensitivity (Figure 3c, Methods). In this model, cells that are more or less drug sensitive have the same dose response shape, but varying IC50, equivalent to shifting the function left or right along a concentration axis.

We model cells with more or less drug sensitivity by rescaling their dose response³⁴, whereby the shape of dose response is preserved but shifted left or right (respectively) along a concentration axis (Figure 3d). This model showed that when individual cells have ultrasensitive dose responses (increasing returns) but variable drug sensitivity, the population-level response can show diminishing returns (Figure 3e). Therefore an underlying ultrasensitivity would only be apparent in a homogenous population of cells. In this theory, the appearance of ‘diminishing returns’ is not because chemotherapy efficacy declines with higher dosage, but because the remaining few cells are especially hard to kill. We next tested whether this theory can reconcile three seemingly contradictory observations: that antagonistic chemotherapies, showing diminishing returns in their dose response, could be maximally effective when administered concurrently.

Given two model features – cellular heterogeneity and ultrasensitive dose response – we tested whether either or both could explain observed dose responses and dynamics of sequential or concurrent therapy. We simulated cell population dynamics in response to treatments that induce a fraction of cell death, according to dose response functions which include antagonistic interaction between C and H. In the model, cells either survive treatment and continue growing, or begin dying at an experimentally observed rate, with drug concentration defining the fraction killed (Methods and Supplemental Figure 8). We also considered a more complex model in which cells experience gradations of growth inhibition, cytostasis, or death, as a function of increasing drug concentration (Methods and Supplemental Figure 9). Both approaches had similar results, and for simplicity we present the model where cells either grow or die. All model parameters were measured except for the degree of ultrasensitivity ($\mathrm{\beta }$) and variance in drug sensitivity (σ), which were fitted to experimental data (Methods and Supplemental Figure 8).

A ‘null model’ with neither heterogeneity nor ultrasensitivity failed to reproduce any of the observed data (Figure 4). A model including only cellular heterogeneity only reproduced the diminishing returns of dose response, and a model including only ultrasensitivity only reproduced the advantage of concurrent therapy. Finally, a model including both heterogeneity and ultrasensitivity was able to reproduce both the population-level dose response as well as the observed advantage of concurrent therapy over sequential therapy, even with the model’s inclusion of antagonistic drug interaction (Figure 4).

Figure 4. Ultrasensitivity and cellular heterogeneity can together explain the observed advantage of concurrent combination chemotherapy.

Four mathematical models of tumor response to concurrent or sequential chemotherapy are compared, based on linear or ultrasensitive dose responses, and homogeneous or heterogeneous cell populations. Models were categorically judged by their ability to reproduce the observed population-level dose response (Figure 3b), and dynamics of response to concurrent or sequential use of the antagonistic CH drug pair (Figure 1d). All models include the observed antagonism between C and H (see Methods and Supplemental Figure 8 for parameters). Yellow triangles mark treatment times (concurrent: C and H both on day 0; sequential: C on day 0 and H on day 6).

The effect of drug interactions on optimal schedule

We next used this model to simulate how various drug-drug interactions affect concurrent versus sequential treatments. For drug interactions ranging from synergistic to antagonistic, we compared responses to drug pairs where the same total doses are applied but with different schedules: concurrent, sequential, or split (half doses given twice as often). In scenarios implementing synergy or additivity, the greatest tumor cell kill was achieved by the high dose intensity of concurrent therapy (Figure 5a). This simulated effect of additivity (which also applies to a drug ‘combined’ with itself)³⁵ resembles the experimentally observed responses to single agents 4-H-cyclophosphamide or doxorubicin at full dose versus sequential half doses (Figure 1c). In the presence of mild antagonism, concurrent administration is superior but to a lesser degree, as sequential therapy has some benefit from avoiding antagonism (Figure 5a). This model resembles experimental observations for the mildly antagonistic combination of 4-H-cyclophosphamide and doxorubicin (Figure 1d). With sufficiently strong antagonism, a point is reached where avoiding antagonism matches the benefit of high dose intensity, making sequential and concurrent schedules similarly effective (Figure 5a). This consequence of strong antagonism corresponds to the experimental observation that concurrent CHOP was similar in efficacy to the sequential regimen ‘CHP then O’, which avoided the strong antagonism of vincristine by doxorubicin (Figure 1a+b). Therefore, when drug antagonism is present in combination regimens, its detrimental effect can be counterbalanced by the benefit of higher dose intensity, and conversely, the benefit of avoiding antagonism by sequential regimens may be balanced by the loss of dose intensity.

Figure 5. Effect of drug interactions and dose intensity on sequential and concurrent chemotherapy.

A, Effect of drug interactions on efficacy of different treatment schedules. Simulated dynamics of response to different treatment schedules for a drug pair ‘A+B’ exhibiting synergy (i = 1.5), additivity (i = 0), mild antagonism (i = −0.8), and strong antagonism (i = −1.5). Isobolograms are shown on the left, and live cell dynamics are shown on the right, for concurrent use of both drugs (blue), sequential use of drugs (orange), or split dosing (both drugs at half dose, two sequential doses; green). B, Effect of dose response shape on efficacy of different treatment schedules. Here we consider only monotherapy to remove influence of drug interactions. Live cell dynamics are simulated for dose response functions that either produce diminishing returns ($S=e^{-\sqrt{D}}$); linear response ($S=e^{-D}$), or ultrasensitive response ($S=e^{-(\alpha D+{\beta D}^2)}$). Dose response functions are shown on the left, and live cell dynamics are shown on the right for intense treatment, consisting of the full dose on day 0 (blue), sequential use of two half-doses (orange), or sequential use of four quarter-doses (green). All models in both panels feature a consistent distribution of drug sensitivity (Methods).

High-intensity chemotherapy is only optimal with ultrasensitive dose response

We investigated how the shape of dose response affects the relative efficacy of high-intensity versus low-intensity chemotherapy schedules. We simulated single-agent treatments (avoiding drug interaction) where the same total dosage is either concentrated or spread over time. Specifically, we test administering therapy all at once (day 0), as two half doses (days 0 and 6), or as four quarter doses (days 0, 3, 6, 9) (Figure 5b). When the dose-response function is sub-linear (diminishing returns), the prolonged low-dose regimen is the most effective – opposite to experimental results (Figures 1c, 4b). With a linear dose response, efficacy depends only on total dose and all schedules have an identical final result, which is also inconsistent with experiments. Only with a supra-linear or ultrasensitive dose response is the high-intensity schedule most effective, with prolonged low-dose chemotherapy being far inferior. This is the only scenario that is consistent with the experimentally observed inferiority of lower-intensity chemotherapy (Figures 1c, 4b). Collectively these results suggest that cellular dose responses must be ultrasensitive in scenarios where high-intensity chemotherapy is superior to prolonged low-intensity chemotherapy, such as curative treatments for aggressive lymphomas.

Optimal treatment schedule defined by antagonism, ultrasensitivity, and dosage.

Our initial hypothesis was that the efficacy of an antagonistic drug combination would be improved by a sequential schedule that avoids antagonism; in vitro experiments proved this false. We next understood our experimental results with a mathematical model of multi-drug response. Here we apply this model to investigate how the interplay of (1) drug interactions, (2) ultrasensitivity, and (3) strength of cell killing affect whether concurrent or sequential regimens elicit the greatest cytotoxic effect (Figure 6). Specifically, we used the linear-quadratic equation of dose-response, with or without drug interaction, to derive conditions under which concurrent or sequential treatment elicits the greatest inhibitory effect (Methods). In the clinic, doses are limited by toxicities, and so our analysis compares the effects of using matching doses in either schedule; this question is relevant only when therapies can be safely co-administered. As expected, drugs with synergistic interaction (Combination Index < 1) are most effective when concurrently administered, but this model reveals that optimal schedule depends on multiple parameters when drugs have antagonistic interactions (Combination Index > 1). For mutual antagonism, where both drugs are weakened by interaction, an ultrasensitive dose response can cause concurrent treatment to remain optimal, but not beyond a strength of antagonism that renders two drugs inferior to one (Figure 6a). However, strong antagonism is often observed to be asymmetric, where one drug is suppressed and the other is unaffected, such as suppression of vincristine by doxorubicin (Figure 1a). We adjusted our model to describe asymmetric drug interaction (Methods), which revealed that concurrent treatment can remain optimal in the presence of stronger levels of antagonism (Combination Index up to ~2.5) (Figure 6b). Given that the strongest interactions in CHOP are asymmetric (Figure 1a), this result rationalizes the experimental and clinical efficacy of the CHOP regimen despite its antagonism. Finally, this model shows that at strong levels of cell killing, as in chemosensitive malignancies, only a small amount of ultrasensitivity is needed to overcome antagonism and favor concurrent therapy. This may rationalize the widespread utility of concurrent chemotherapy schedules in potentially curable malignancies, such as many types of Non-Hodgkin Lymphoma. The influence of net cell kill in this model also suggests that the benefits of ultrasensitive responses depend upon adequate chemosensitivity. In light of patient-to-patient variation, tumors that are initially less chemosensitive may remain in the linear region of dose response and not engage the greater benefits of ultrasensitivity; this may also result from Darwinian selection for drug resistant subpopulations over the course of therapy. Thus, we theorize that ultrasensitive dose responses could convert gradations of inter-patient differences in chemosensitivity into more divergent differences in patient outcomes.

Figure 6. Scenarios that favor concurrent or sequential use of two chemotherapies based on antagonism, ultrasensitivity, and dosage.

When total dose is preserved between concurrent or sequential treatment schedules, three parameters define whether concurrent or sequential treatment has the greatest inhibitory effect: drug interaction (Combination Index, $C$), the ultrasensitivity of dose response ($\mathrm{\beta }/\mathrm{\alpha }$), and the strength of cell killing, as defined by drug dose ($D$). Parameter regimes favoring concurrent treatment are below the blue surface and regimes favoring sequential treatment are above the surface. A, For mutual drug suppression, concurrent treatment is favored when $\frac{\beta }{\propto }>\frac{C^2-C}{D(2-C^2)}$ (derivation in Methods). B, For asymmetric drug suppression (one drug suppresses another, but not vice versa), concurrent treatment is favored when $\frac{\beta }{\propto }>\frac{C^2-C}{D(1+2C-C^2)}$ (Methods).

Discussion

Across a variety of T-cell lymphoma cell lines, the drugs comprising the clinically standard CHOP regimen exhibited antagonistic interactions, which diminish the efficacy of combination therapy. However, despite these antagonistic drug interactions, we observed that concurrent administration of all agents in CHOP produced superior or equal efficacy compared with sequential regimens. We propose that these surprising observations can be explained by an ultrasensitive dose response to chemotherapies, such that maximal efficacy results from the highest intensity regimen, whereas inferior efficacy is produced when the same total dosage is spread out over time in sequential or ‘hyperfractionated’ regimens. These findings suggest that ultrasensitive dose response is a mechanism contributing to the benefit of combination chemotherapy, distinct from synergistic drug interaction, that can overcome the detrimental effect of antagonistic drug interactions. This phenomenon may contribute to the observed clinical superiority of cycles of high-dose chemotherapy in multiple types of cancer, in contrast to the relatively scarce use of prolonged low-dose chemotherapy^36–40. Ultrasensitive dose responses may also provide a mechanistic explanation for clinical trials in which concurrent regimens have outperformed sequential regimens^41–44, although these results are also affected by tolerability.

Ultrasensitive dose responses are expected if cells become more likely to die as chemotherapy-induced damage accumulates; the same phenomenon is recognized in radiotherapy by the LQ model. We also show that heterogeneity in drug sensitivity will cause populations of cells to exhibit linear or sub-linear dose responses even when individual cells possess ultrasensitive (supra-linear) responses. This arises because some drug sensitive cells are easily killed, and remaining resistant cells are difficult to kill, such that the cytotoxic effect in a heterogeneous population shows diminishing returns with increasing dose. Indeed, the chemotherapies in CHOP exhibited sub-linear dose responses in T-cell lymphoma cultures. This challenge cannot be resolved by single-cell measurements, as dose-response functions cannot be measured in the same single cell; once a cell dies it cannot be exposed to other doses. Even a population derived from a single clone will not consist of cells with identical drug sensitivity, as phenotypic heterogeneity can arise rapidly due to stochastic fluctuations. Given the fundamental inability to directly observe this phenomenon, we instead show by simulation that without ultrasensitive dose responses, prolonged low-dose chemotherapy would be equal or superior to high-dose chemotherapy. This logically implies that in clinical scenarios where cycles of intensive chemotherapy are more effective than prolonged low-dose therapy, such as aggressive lymphomas, cancer cells must have ultrasensitive dose response functions. These results are independently supported by recent theory illustrating how convexity or concavity of dose response functions can influence optimal treatment schedules (bioRxiv 2020.10.08.331678).

The central concept of the LQ model is that cell death becomes increasingly likely as treatment-induced damage accumulates. The LQ model may plausibly apply to many cancer drugs, and is easily adapted to drug combinations when they are understood to be acting upon a shared dose response, such as DNA damage. For combinations of drugs with more varied mechanisms, this theory raises the question of whether multiple drugs jointly add to the accumulation of damage (the sum of drugs’ effects is squared), or if drugs contribute to distinct terms. In the latter scenario, even if the log-kills of two drugs are additive, the potency of their combination would be less-than-additive (Supplementary Figure 10). This occurs because a high dose of either monotherapy could reach the ultrasensitive region of response, but combining half-doses of two drugs will be less effective at reaching the ultrasensitive region of either drug’s response. The extreme instance of this idea, combining up to ten drugs at one-tenth of their effective dose, has been experimentally studied in bacteria, and indeed addition of log-kills (Bliss independence model) remains accurate while potency (Loewe model) demonstrates antagonism⁴⁵.

The insights from CHOP may apply to other cytotoxic chemotherapies, which are often administered as cycles of high doses, and often directly or indirectly damage DNA akin to the DNA-damaging effects of radiation in the LQ model. We anticipate that ultrasensitive responses are unlikely to apply to treatments that are dosed daily to achieve sustained inhibition of oncogenic signaling, such as kinase inhibitors and hormone therapies. The character of dose responses may be relevant to designing novel combination regimens, since when ultrasensitivity is present, tolerable combinations of active therapies have promise even without positive drug-drug interactions.

Tolerability is an overriding consideration in cancer treatments whose importance is emphasized by our findings. Our experiments and models held total administered dose constant and did not investigate toxicity-efficacy tradeoffs. However, ultrasensitive dose responses suggest that even though combinations of active therapies may increase efficacy when tolerable, these advantages may be seriously compromised when toxicity necessitates dose reductions. Our theory specifically suggests that loss of efficacy may be non-linear with respect to reduction in dose; therefore, it suggests that dose interruptions or decreased number of treatment cycles may better retain efficacy while managing toxicity than dose reductions. Aggressive lymphomas have a history of more intensive chemotherapy regimens failing to improve survival, possibly because more drugs did not correspond to a higher achievable sum of dose intensities⁴⁶. In PTCL, the addition of romidepsin to CHOP necessitated more frequent dose reductions of the CHOP backbone, which has been proposed to explain the lack of survival improvement^47,48. Conversely, successful combinations such as Rituximab plus CHOP for DLBCL have been well tolerated. In short, when high-dose intensity is important, tolerability is important.

The chief limitation of this study is that the dose response function of any given single cell is fundamentally unobservable, and therefore the evidence for ultrasensitive dose responses comes from their consequences for high versus low-intensity, and concurrent versus sequential regimens. The most compelling evidence for this theory is the finding that prolonged low-dose chemotherapy would hypothetically be the optimal use of chemotherapy if the dose responses of single cancer cells were sub-linear. Among the agents in CHOP, our study was limited to the cytotoxic agents CHO, because prednisolone (the active metabolite of prednisone) has negligible single agent activity in PTCL cell lines, as also reported for DLBCL^9,49. Finally, we have not investigated the mechanistic causes of antagonistic drug interactions within CHOP, as the purpose of this study was to understand the consequence of observed drug interactions on treatment schedules. Similarly, some temporal effects of drug interactions are not understood, such as our observation that H before C was more effective than C before H (Figure 1d). Since concurrent therapy was the most effective use of CHOP, this remaining question does not have clear importance.

Models to optimize treatment schedules have an established history in radiation oncology, and may also be useful for chemotherapies as discussed by McKenna et al²². The utility of cycles of high-dose chemotherapy, compared for example with daily low-dose chemotherapy, is well recognized and established in oncology. Our study found that this advantage is not obvious when considered quantitatively, especially in light of antagonistic drug interactions, but can be rationalized by heterogeneity in dose responses. This work adds to our understanding of the therapeutic efficacy of existing treatments, which can form a basis for future prospective applications. The framework presented here can be adapted to other therapies and cancer types, to understand the effects of dose intensity, treatment schedules, and drug interactions.