Evaluation and translation of combination therapies in oncology – A quantitative approach

Quantitative techniques improve our understanding of tumor volume data for combination treatments and its translation across in vivo models and species. The focus of this paper is therefore on understanding in vivo data, highlighting key structural elements of pharmacodynamic tumor models, and challenging these methods from a translational point of view. We introduce the concept of Tumor Static Exposure (TSE) both for single and multiple combined anticancer agents. The TSE curve separates all possible exposure combinations into regions of tumor growth and tumor shrinkage. Moreover, the degree of curvature of the TSE curve indicates the degree of synergy or antagonism. We demonstrate the TSE approach by two case studies. The ﬁ rst examines a combination of the drugs cetuximab and cisplatin. The TSE curve associated with this combination reveals a weak synergistic e ﬀ ect, suggesting only modest gains from combination therapy. The second case study examines combinations of ionizing radiation and a radiosensitizing agent. In this case, the TSE curve exhibits a pronounced curvature, indicating a strong synergistic e ﬀ ect; tumor regression can be achieved at signi ﬁ cantly lower exposure levels and/or radiation doses. Finally, an allometric approach to human dose prediction demonstrates the translational power of the model and the TSE concept. We conclude that the TSE approach, which embodies model-based measures of both drug (potency) and target properties (tumor growth rate), has a strong potential for ranking of compounds, supporting compound selection, and translating preclinical ﬁ ndings to humans.


Introduction
Pharmacodynamic models are commonly used to aid decisionmaking in drug discovery and development.In oncology, tumor growth inhibition models have been used extensively in the past decades (Evans et al., 2014;Ribba et al., 2014;Simeoni et al., 2004;Tate et al., 2014).For a recent review, see Mould et al. (2015).Some growth models are empirical (Simeoni et al., 2004), whereas others have a more mechanistic foundation (Jumbe et al., 2010;Wendling et al., 2016).After being calibrated to experimental data, the models are used to evaluate, predict, and guide future experimental designs (Hutchinson et al., 2016).Models are also used in translational efforts to predict first-inman doses, using techniques such as allometric scaling (Boxenbaum, 1982(Boxenbaum, , 1984)).
Tumor growth inhibition models also exist for combination therapies (Goteti et al., 2010;Rocchetti et al., 2013), with the objective of evaluating and comparing different dose combinations with respect to efficacy and drug synergies.Many techniques have been proposed to evaluate drug interactions (Foucquier and Guedj, 2015).Of these, perhaps the most commonly used approach is the isobologram, which plots iso-effective dose combinations and determines synergy or antagonism based on the shape of the resulting curve (Grabovsky and Tallarida, 2004;Tallarida, 2006).Until recently, no isobologram-like techniques had been explored for different exposure-based models, including tumor models.Moreover, tumor models and techniques have been studied primarily for chemical interventions, thereby excluding an important pillar of modern oncology treatmentradiation therapy (Baskar et al., 2012).
This paper focuses on two questions: Can quantitative techniques provide a better understanding of tumor volume data when combinations of chemicals and radiation are used?and What translational potential does such an approach have?Our aim is to chisel out relevant model structures that enhance both the underlying mechanisms of action and the predictive power of the model.The methodology is demonstrated by in vivo data from two previous studies (Cardilin et al., 2017(Cardilin et al., , 2018)).Case study I introduces the Tumor Static Concentration (TSC) for both a single compound and multiple compounds given in combination.The analysis focuses specifically on deriving useful quantitative tools and then implementing graphical ways to view additivity, synergy or antagonism based on exposure metrics.Case study II expands the methods to also include radiation therapy as a co-variate.The combination of drug concentrations and radiation is then called Tumor Static Exposure (TSE).Case study II examines combination therapy with ionizing radiation and a radiosensitizing agent that exhibit a substantial level of synergy, manifesting as a more pronounced curvature of the TSE curve.We also thoroughly describe the two semi-mechanistic tumor models used to describe the experimental data in the case studies.Finally, we discuss how TSE can be used in translational efforts, be implemented into the drug discovery process, and for communication with a biologically-oriented audience.

Background
The concept of a threshold concentration has been applied to various tumor models and for different anti-cancer agents (Jumbe et al., 2010;Magni et al., 2006;Simeoni et al., 2004).Jumbe et al. (2010) derived the TSC for a single compound.Koch et al. (2009) derived an equation for threshold concentrations for a combination of two compounds.Later, the TSC curve was introduced as a graphical tool contrasting the isobologram (Cardilin et al., 2017;Gabrielsson et al., 2016).In particular, Cardilin et al. (2017) discussed how the shape of the TSC curve was associated with the combined effect (synergy or antagonism) of the combination treatment.Moreover, a TSC-like curve was also proposed by Miao et al. (2016) and for more general combination effect terms by Koch et al. (2016) under the name half maximal-effect curve.Most recently, Cardilin et al. (2016Cardilin et al. ( , 2018) ) extended the TSC concept to combinations involving ionizing radiation by introducing the Tumor Static Exposure (TSE) curve.In this section, we provide a brief intuitive description of the TSE concept.We refer to Appendix A for a more technical treatment of the subject.
The tumor is exposed to a single compound A during monotherapy (Fig. 1a), where C A denotes the plasma concentration of drug.If the drug is sufficiently potent, at some concentrations maintained over an extended period of time, the tumor will start to shrink.At concentrations below the TSC, which are insufficient for tumor regression, the drug will at best slow down the tumor growth.In particular, there will also be a value of C A in between that will keep the tumor in stasis, which we call the TSC value (Fig. 1a).In the case of radiation therapy, or when plasma concentrations cannot be measured, we may prefer to use the dose.For example, we may define a Tumor Static Dose to be the minimum daily radiation dose that will keep the tumor in stasis.This will lead to different values depending on administration schedule.In order to cover both cases we generally refer to the concept as Tumor Static Exposure (TSE).
The TSE concept can be generalized to the case where two compounds are given simultaneously as combination therapy.Let A and B denote the compounds and let C A , and C B be their plasma concentra- tions.For each fixed value of C A , we may find a value of C B such that maintained simultaneous exposure to both compounds leads to tumor stasis.Hence, we obtain a collection of concentration pairs (C A , C B ) that all lead to tumor stasis.These pairs form a curve in the plane formed with C A along the x-axis and C B along the y-axis (Fig. 1b).We refer to this curve as the TSE curve for the compounds A and B. Note that this curve separates regions of concentration pairs corresponding to tumor growth (red) and tumor shrinkage (green).
This procedure can be extended to combinations involving three or more compounds, leading to a hypersurface in higher-dimensional concentration space.The scenario with three compounds A, B, and C with plasma concentrations C A along the x-axis, C B along the y-axis, and C C along the z-axis is illustrated in Fig. 1c.

Problem formulation
This case study is based on results that were reported in a previous publication (Cardilin et al., 2017).Cetuximab is an epidermal growth factor receptor (EGFR) inhibitor used clinically to treat certain colorectal and head-and-neck cancers, either as monotherapy or as combination therapy together with, e.g., irinotecan, cisplatin and other platinum-based chemotherapy, or radiation (Bou-Assaly and Mukherji, 2010; Reynolds and Wagstaff, 2004).This in vivo study was aimed at investigating combination therapy with cetuximab and cisplatin to treat non-small cell lung cancer (NSCLC).Our goal was to develop a tumor model to describe the experimental data and use the model for evaluation with respect to synergy and overall efficacy using TSE curves.

Experimental data
Forty patient-derived xenograft (PDx) models in female mice were used in the study.The animals were divided into four treatment arms with = N 10 animals in each of the following arms: (A) vehicle, (B) cetuximab monotherapy (30 mg/kg per dose), (C) cisplatin mono- therapy (5 mg/kg per dose), and (D) combination therapy with cetuximab (30 mg/kg per dose) and cisplatin (5 mg/kg per dose).The mice were given doses twice a week for two weeks.Tumor length (L) and width (W ) were measured twice a week for 28 days.Tumor volumes were estimated using the formula: = × V L W /2 2 .The details of this study were reported previously (Amendt et al., 2014;Cardilin et al., 2017).Experimental and model-predicted time courses for each of the four treatment arms are shown in Fig. 2.

Pharmacodynamic model analysis
Standard one-and two-compartment pharmacokinetic models were used to describe plasma exposure of cetuximab and cisplatin, respectively.The models, as well as the parameter estimates, were taken from the literature.Further details can be found in Cardilin et al. (2017).
A pharmacodynamic tumor model was developed to describe tumor evolution over time.The model consists of a main compartment of proliferating cells, and a set of damage compartments that each dying cell must traverse before disappearing.The drug action of cetuximab is described as an inhibition of the proliferation rate according to an inhibitory E max -function ( = I 1 max ), whereas the drug action of cisplatin is described as a linear stimulation of the natural death rate (Fig. 3).
The pharmacodynamic model is defined by the following system of differential equations (1 ) where V 1 is the main compartment and V V , , 2 3 and V 4 are the damage compartments, V total is the total tumor volume, k g and k k are the growth and kill rates, IC 50 is the concentration of cetuximab needed to achieve 50% inhibition of the proliferation, and b is a pharmacodynamic parameter associated with cisplatin.The initial conditions for each tumor volume compartment are given by where V 0 is the initial volume of the main compartment.The parameter estimates for the TGI model were, as reported in the original article, =

Tumor static exposure for cetuximab and cisplatin
For the model in Eq. ( 9), an expression for the TSE curve can be found using the following reasoning: The main compartment, V 1 , is the only compartment where cells proliferate.Hence, if V 1 is eradicated the entire tumor will eventually be eradicated as well.Therefore, if dV 1 / dt < 0 the tumor will eventually be eradicated.Since V 1 will not be zero unless the tumor is already eradicated, this implies that Eq. ( 11) must hold.
The TSE curve, which separates tumor growth from tumor shrinkage, will therefore consist of all concentration pairs C C ( , ) cetuximab cisplatin such that the left-hand side in Eq. ( 11) is equal to zero.Solving for C cisplatin gives Insertion of the pharmacodynamic parameter estimates for the tumor model into Eq.( 12) produces the TSE curve shown in blue in Fig. 4. The region of concentration pairs that will lead to tumor eradication is shown in green, whereas the region that will not eradicate  The slight curvature indicates that there is only a slight advantage from combining the two compounds with respect to tumor regression.However, a complete assessment of the combination would also require taking into account the toxicity of both compounds when given as single agent as well as any toxicological interactions.

Translation of tumor static exposure to humans
Translation of the tumor model for cetuximab and cisplatin therapy and the associated TSE curve was considered by means of allometric scaling.The primary candidates for scaling were the rate constants k g and k k .Scaling is generally performed based on the relative body weight (BW) of the animals (mouse and man) with a typical value for the exponent of − 0.25 (West et al., 2002).Hence, the following para- meter scalings were applied: where typical body weights of human and mouse were assumed to be 70 kg and 25 g, respectively.
The TSE curve in Eq. ( 12) depends on the rate parameters only through their quotient, which will remain the same if the same scaling factor is used for k g and k k .Hence, from the perspective of allometric scaling, the TSE curve should be the same for mouse and human and Fig. 5 can be used for predictions in humans.Note, however, that realizations of the tumor model will look different depending on the allometric scaling used, but the qualitative behavior is the same (i.e., growing or shrinking tumor).The TSE curve demonstrated a weak synergistic effect from combining cetuximab and cisplatin.An allometric scaling approach indicated that the predicted TSE curve may be the same across species.

Problem formulation
This case study is based on results from a previous publication (Cardilin et al., 2018).Radiation therapy is commonly applied in cancer treatment for a wide range of tumors (Shaue and McBride, 2015).Radiation damages the DNA of cells, e.g., through single-or double-strand breaks, and can lead to cell death or loss of the cell's proliferative capabilities (Eriksson and Stigbrand, 2010;Ross, 1999).The benefits of combining radiation therapy with different doses of a radiosensitizing agent were investigated.The goal was to develop a tumor model that describes experimental data of radiation and radiosensitizer treatment and to apply the model for evaluation of combinations with respect to synergy and overall efficacy using TSE curves.In addition, the translational aspects of the model were demonstrated for the human situation.

Experimental data
Eighty FaDu xenograft mouse models were used in the study.The mice were divided into eight treatment arms with = N 10 mice in each arm as follows: (A) vehicle, (B) radiation (2 Gy per dose), (C-E) monotherapy with a radiosensitizing compound (10, 50, or 200 mg/kg per dose), and (F-H) combination therapy (2 Gy and 10, 50, or 200 mg/ kg per dose).Mice were given daily treatments for five days.Some tumor volume-time courses representing different treatment arms are

Pharmacodynamic model analysis
Exposure to the radiosensitizing compound was described using a standard one-compartment model.To describe the tumor growth data, a TGI model was modified to account for radiation therapy.The modified model consists of a main compartment V 1 of proliferating cancer cells following logistic growth, and three damage compartments, V V , 2 3 and V 4 , which cells must traverse before dying.Irradiated cells are in- stantaneously transferred to a separate compartment where they are allowed at most one more cell division before dying.Drug action of the radiosensitizing agent is described as separate stimulations of both the natural-and the radiation-induced cell deaths.The full model is shown in Fig. 6.The tumor model was mathematically described using the following system of differential equations where k g is the growth rate, k k the kill rate, K the tumor capacity parameter, R is the rate function for radiation, corresponding to a se- quence of instantaneous cell transfers (mathematically described as a series of Dirac delta distributions), and S 1 is a linear stimulatory function of the radiosensitizer given by where a is a pharmacodynamic parameter associated with monotherapy with the radiosensitizer.The system in Eq. ( 15) has the initial conditions , 1, 2, 3, 4 and (0) 0, 1, 2 Finally, the fraction of lethally irradiated cells given a radiation dose D and concurrent plasma concentration C is given by where α and β are the linear and quadratic radiation parameters, respectively, and b is a pharmacodynamic parameter associated with stimulation of the radiation effect.Following a nonlinear-mixed effects approach, population parameter mean estimates were given by =

Tumor Static Exposure (TSE) of radiation and radiosensitizer
The main compartment V 1 dominates the tumor model in the same sense as in Case study 1, i.e., if V 1 is eradicated, the entire tumor will eventually be eradicated.In this case, cell proliferation is not exclusive to V 2 , but is also included in the transfer from U 1 to U 2 .However, the irradiated cells are only allowed to proliferate once (the resulting daughter cells are non-proliferating) and can therefore not sustain the tumor by themselves.Moreover, given that the treatment goal is tumor regression and ultimately eradication, we can safely ignore the saturation effect that occurs for large tumor volumes.The differential equation for V 1 in Eq. ( 15) therefore essentially reads ).The latter effect is only present in individuals treated with combination therapy.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)Upon irradiation at time = t 0, a fraction SF D C ( , (0)) of the cells survive.This fraction will then continue to grow according to Eq. ( 19) until the time T of the next irradiation.The fraction of surviving cells, for times between the first and second instance of radiation, will be given by the main compartment will be in approximate stasis, and if the main compartment will have shrunk.If such dosing is repeated, the main compartment will continue to shrink and eventually be eradicated.By using Eqs.( 18) and ( 20), it is possible to derive an expression for the TSE curve (see Cardilin et al., 2018 for details).The main compartment will be in approximate stasis if where C ̅ is the average plasma concentration of the compound, and k e is its elimination rate according to the pharmacokinetic model.The region of all average plasma concentrations and radiation doses that result in tumor shrinkage will therefore consist of those for which the left-hand side in Eq. ( 21) becomes negative.If we view Eq. ( 21) as a quadratic equation with respect to the radiation dose D we can use standard formulas to solve for D and find where D is the daily radiation dose, C denotes the average plasma concentration of the radiosensitizing compound over the period T [0, ], and the function G is given by Inserting the population parameter values into Eq.( 22) produces the TSE curve in Fig. 7.The curve exhibits a large curvature due to strong synergy between the treatments.In particular, the (relatively) large value of the interaction parameter b represents this synergy.The shrinkage, or eradication area (green) is therefore increased significantly compared to the case = b 0 (black, dashed curve).This means that tumor regression will be possible at much lower combination exposures.
In particular, we find the individual TSE values to be approximately 2.17 Gy for radiation and 8.26 µg/mL for the radiosensitizer, corre- sponding to monotherapy with either treatment.

Translation of tumor static exposure to humans
An allometric scaling approach was used to translate the tumor model and the associated TSE curve to humans (Boxenbaum, 1984;Kang and Lee, 2011).The rate constants k g , k k , and k e , were scaled using a standard allometric exponent b = − 0.25 (Savage et al., 2008).Assuming body weights of 25 g and 70 kg for the mouse and man, respectively, the rate constants in man were decreased to approximately one seventh of their values in the mouse.Specifically, the rates were scaled as Moreover, the Dirac delta distribution, which describes the transfer of cells during irradiation, has a unit of 1/day and was therefore scaled in the same manner.Based on the allometrically scaled parameters, a prediction of TSE in man can be produced.Fig. 8 (left) shows the TSE curves for mouse (blue) and man (red), respectively.Fig. 8 (right) shows the resulting tumor growth curves following treatment with three different exposure levels (A, B, and C).Similarly to the TSC curve for cetuximab and cisplatin from case study I, the TSE values for monotherapy with either compound are the same for mouse and man.However, due to slower clearance in man, the corresponding TSE curve has less curvature.This follows because slower clearance makes the difference between peak and average plasma concentration smaller.Although the TSE curve exhibits less curvature, lower human clearance of the radiosensitizer suggests that less drug is needed to achieve a plasma concentration exceeding the TSE.

Conclusions from case study II
Case study II taught us to analyze tumor volume data obtained from different combinations of chemical and radiation interventions, and how the TSE curve was constructed and displayed for a relevant administration schedule.We also extended the underlying tumor model with a semi-mechanistic approach to show how intervention from chemicals and radiation jointly act upon the tissue growth/kill.The calibrated model showed good predictive capabilities for all treatment arms.The TSE curve associated with radiation and radiosensitizer combinations demonstrated a strong synergistic effect from combination therapy treatment.An allometric scaling approach showed differences in TSE across species due to differences in drug clearance.

Discussion
This paper was focused on two questions: Can quantitative techniques provide a better understanding of tumor volume data when combinations of chemicals and radiation are used?and What The TSE curve has significant cur- vature due to a pronounced synergistic effect between the treatments.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)translational potential does such an approach have?We shall discuss these two questions in the rest of this section.

Quantitative tumor models
Quantitative techniques play an important role in modern pharmacology in order to improve our understanding of dose-exposure-response relationships (Standing, 2017;Gabrielsson and Green, 2009).In particular, pharmacokinetic and pharmacodynamic models have proven useful in describing tumor growth after administration of anticancer agents, displaying good predictive capabilities (Li et al., 2013;Zhu et al., 2015).In the case of combination therapy with multiple compounds, quantitative models have also been useful to identify and quantify potential drug synergies (Choo et al., 2013;Frances et al., 2011).We presented two models of combination therapies in oncology (Cardilin et al., 2017(Cardilin et al., , 2018) ) and two case studies that involved combinations of two agents.Our results indicate that TSE can be used to assess and quantify synergy between different compounds, and that a quantitative approach provides a better understanding of tumor growth and how to intervene with combinations of drugs and/or therapies.
The first model, used to describe cetuximab and cisplatin, shares many similarities with previous tumor growth models (e.g., Goteti et al., 2010;Rocchetti et al., 2013) but there are a few important differences.Firstly, our model incorporates a natural rate of cell death, meaning that even in the absence of treatment, some cells will be in the process of dying.This captures some of the heterogeneity of cancer cells and consequently provides a somewhat more biologically reasonable model.The inclusion of a natural death rate also meant that the initial distribution of cancer cells had to be revised (Cardilin et al., 2017).
The second model, used to describe radiation therapy combined with radiosensitizing treatment, has several novel aspects.One the one hand, the model can be seen as a generalization of the first model in that it includes the common structure of proliferating cells and a chain of transit compartments representing increasingly damaged cells (Simeoni et al., 2004).On the other hand, since we wanted to describe the effects of radiation therapy, we took some inspiration from the linear-quadratic theory of radiobiology (Brenner, 2008;Brenner et al., 1998).By combining these two perspectives, we arrived at a model that, while novel, is still founded on practical and empirical evidence.This was further consolidated when the model was used to predict the outcome following a prolonged administration schedule (Cardilin et al., 2018).
For both models we introduce the concept of TSE for combinations of anticancer agents.The TSE concept is a generalization of the Tumor Static Concentration value discussed by Jumbe et al. ( 2010) among others.TSE provides insight about drug efficacy and the in vivo sensitivity of tumor cells.In particular, TSE identifies drug exposure that is sufficient for tumor regression for monotherapies as well as combination therapies involving two or more compounds.Most importantly, TSE provides guidance for clinical dose regimen selection, where dosing interval troughs should exceed these concentrations for the effective tumor kill rate to exceed tumor growth rates for net antitumor activity.In other words, a quantitative approach provides a better understanding of tumor growth and how to intervene with combinations of drugs and/or therapies.
The two case studies we presented both involve combinations of two agents.The resulting TSE curves separate exposure pairs yielding tumor regression (green regions) from those that yield tumor growth (red regions).The curvature of the TSE curve is associated with the strength of synergy between the treatments.In case study I (cetuximab and cisplatin) the associated TSE curve (Fig. 4) indicated a weak synergistic effect, whereas in case study II (radiation and a radiosensitizer), the TSE curve (Fig. 7) indicated a strong synergy.We conclude that TSE can also be used to assess and quantify synergy between different compounds.

Translational potential
Since mechanism-based pharmacodynamic tumor models are based on measured exposure to drugs and complete volume-time curves, in contrast to the isobologram used for different dose combinations aiming at equal efficacies (Tallarida, 2006), the proposed concepts have a much stronger potential for translation of doses to systemic exposures, and observations in animal species to human plasma target concentrations.In an attempt to translate TSE to humans, we used an allometric scaling approach (Boxenbaum, 1982).We described how physiological or system parameters (i.e., k g and k k ) and drug rate parameters (k e ) may be allometrically scalable across different strains and species, including prediction of human doses.This approach is consistent with what has been tried for other tumor models (Herman et al., 2011;Lindauer et al., 2017).However, our approach relies on an assumption that drug-related parameters, e.g., EC 50 , are approximately the same across species.This is consistent with what others have studied regarding allometric scaling of pharmacodynamic parameters (e.g., Lui et al., 2016;Mager et al., 2009;Zuideveld et al., 2007).In particular, Mager et al. (2009) note that physiological turnover rates are often allometrically scalable across species, whereas parameters related to capacity and sensitivity tend to be similar.For the second model, we chose to scale the fraction lethally irradiated cells.One reason for this was reasonabilityhad we chosen not to scale this fraction, the necessary radiation doses tumor shrinkage would have been much lower than what is reasonable.Another argument is that the fraction of lethally irradiated tumor cells is formally described by a Dirac delta distribution with unit − days 1 and is therefore in a sense si- milar to a rate parameter.It is worth keeping in mind that allometry is not perfect and has not always agreed with empirical observations, and alternative approaches have been discussed, see for instance Agutter and Tuszynski (2011).When we applied allometric scaling to the model in case study I, the TSE curve remained the same across species.In contrast, when we applied allometric scaling to case study II, differences in drug clearance resulted in different curvatures of the TSE curves.
Finally, we believe that TSE is also applicable to different cell-lines since both growth (k g ) and natural cell kill (k k ) appear as intrinsic properties of the models.The models are not limited to a net growth expression, but instead consider separate growth and (natural) death rates.Moreover, the models encompass the utility of multiple drug (or radiation) interventions in their basic structure.

Conclusions
Our analysis focused specifically on deriving necessary mathematical tools and then implementing graphical ways to visualize additivity, synergy or antagonism based on exposure metrics.A quantitative mathematical/analytical approach strongly demonstrated its feasibility in pharmacology.The TSE analysis can easily be implemented in the drug discovery process and used as an independent approach to select or de-select compounds.The TSE concept has a unique potential for ranking of compounds and support the compound selection process in drug discovery, since it embodies model-based metrics of both drug (EC 50 ) and target properties (k g and k k ), where all experimental data (tumor volume time courses) are taken into account.The graphical TSE method is a valuable way to communicate complicated processes to non-quantitative experts.The translational potential was shown in two case studies.
is the vector of compartment volumes, = is the vector of the initial volumes of each compartment at time = t 0, and E the exposure to compound A. Typically, we will have = E C, i.e., the (plasma) con- centration of the compound.However, in the case of radiation therapy for example, Dose-Response-Time modelling will need to be considered.Hence, = E D, the (radiation) dose.Theoretically, there could also be other ways of expressing the exposure to certain anti-cancer agents.We shall first consider the case where E is constant and then move on to the case where E varies depending on the administration schedule.We would like to find all exposure levels of A that result in tumor eradication, i.e., we are interested in the set If we further assume that anti-tumor activity is monotonically increasing with exposure, we are particularly interested in the smallest exposure that leads to tumor eradication, which we define as the tumor static exposure (TSE) value of compound A.
The TSE value will then have the property that for any > E TSE A the tumor will be eradicated following maintained exposure.When = E C we shall sometimes also refer to the TSE value as the tumor static concentration (TSC).An illustration of the TSE value is given in Fig. 1a.The basic guideline would then be the following: choose an administration schedule for compound A such that the exposure level is kept above TSE A .This will ensure tumor regression and eventual eradication.

TSE for combination therapy with two anti-cancer agents
For combination therapy with two compounds, denoted A 1 and A 2 , we consider the same tumor model (Eq.(A1)) with exposure vector , where E 1 represents the exposure to compound A 1 and E 2 represents the exposure to compound A 2 .Analogously to the case with a single compound, we define the set S of simultaneous exposures E that yield tumor eradication V t E V ( , ) 0 : lim ( ; ) 0, 0 The set S is a region in the positive exposure plane.We define the TSE curve to be the boundary of this set

= ∂ TSE S
A A ( , ) 1 2 (A5) The TSE curve divides the first quadrant of the exposure plane into two regions: one which will lead to tumor eradication, and another in which the tumor will persist.An illustration of a TSE curve is given in Fig. 1b.The points where the TSE curve intersects the coordinate axes will be the TSE values of the respective compounds.The general guideline remains the same: choose administration schedules for A 1 and A 2 such that the exposure pair E is kept above the TSE curve, i.e., in the green region in Fig. 1b.

TSE for combination therapy with three or more anti-cancer agents
The TSE concept can be generalized to an arbitrary number of anti-cancer compounds,  ( , , ) In particular, for three compounds, the TSE set will be a surface in three-dimensional exposure space.An illustration of a TSE surface is given in Fig. 1c.

TSE for fixed administration schedules
An important generalization is to allow E to vary according to a given administration schedule.We shall consider administration schedules that are repeated.The simplest example being daily bolus dosing, but also, e.g., biweekly dosing belongs to this category.The primary case is = E D, but another possibility would be to use the resulting average plasma concentration C when applicable.We can once more consider the tumor model in Eq. ( A1), where we think of the administration schedule as being incorporated into f .We can now pose questions such as this: For which values of E, repeated according to the given administration, will the tumor eventually be eradicated?We then define the set S and the TSE value as in Eq. (A3).This construction can be performed for an arbitrary number of compounds in the same way as presented above.
Further generalizations of the TSE concept are possible.In particular, it is possible to consider different administration schedules.This means that E will be a time-varying function belonging to some vector space.The corresponding shrinkage set S will then consist of all, say L 2 , functions E for which the tumor will be eradicated.Consequently, S will be a subset of an infinite-dimensional vector space of functions.This is the same perspective that is used in Optimal Control theory, which has been applied to different tumor models in order to optimize the treatment effect (Schättler and Ledzewics, 2015;Sbeity and Younes, 2015) However, because the optimal administration schedule is often difficult to determine, one has to settle for describing certain properties that the optimal solution possesses.Moreover, the solutions can be sensitive with respect to perturbations in the parameter estimates (Sbeity and Younes, 2015).

Declaration of interests
None.

Fig. 1 .
Fig. 1. a) TSE value of an anti-cancer agent (black).Exposure lower than the TSE value (red) results in tumor growth, whereas exposure greater than the TSE value (green) results in tumor shrinkage.b) TSE curve for combinations of two anti-cancer agents with concentrations C 1 and C 2 .Exposure pairs below the curve (red) yield tumor regression, whereas exposure pairs above the curve (green) yields tumor shrinkage.c) TSE surface for combinations of three anti-cancer agents with concentrations C 1 , C 2 , and C 3 .Exposure triples below the TSE surface (red) result in tumor growth, whereas exposure triples above the curve (green) result in tumor regression.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 3 .
Fig. 3. Tumor growth inhibition TGI model for combination therapy with independent drug action.The model consists of a main compartment V 1 and three damage compartments, V 2 , V 3 , and V 4 that cells pass through before dying.Cetuximab (A) inhibits cell proliferation with inhibitory function I C ( ) cetuximab , whereas cisplatin (B) stimulates cell apoptosis with stimulatory function S C ( ) cisplatin .The k g and k k parameters represent the natural cell kill and death rates.
3.6.Conclusions from case study IThe case study taught us to analyze tumor volume data obtained from different combinations of two different compounds, and how the TSE curve could be constructed and displayed.We did simulations of the time courses of tumor growth curves with different dose combinations of the two compounds.The resulting predictions showed the tumor volume-time behavior when we focused on chemical compound combinations targeting tumor shrinkage, tumor stasis or tumor growth.

Fig. 4 .
Fig. 4. The Tumor Static Concentration (TSE) curve for combinations of cetuximab and cisplatin plasma concentrations (blue, bold) separating regions of tumor growth (light red) and tumor shrinkage (light green).Reference straight line connecting the individual TSE values (black, dashed).(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) 008 Gy 2 , = a 0.09 mL/µg, and = b 0.45 mL/µg.

Fig. 6 .
Fig.6.Tumor model describing the radiation and radiosensitizing treatments.The vehicle model, shown in blue, consists of a main compartment V 1 of proliferating cancer cells with growth rate k g and natural kill rate k k .Before dying, cells must traverse a set of damage compartments V 2 , V 3 , and V 4 .Radiation induces an instantaneous transfer of cells between V 1 and U 1 , after which cells are allowed up to one more cell division before dying.The radiation-induced mass transfer is shown in red.The radiosensitizer stimulates both the natural death process (shown in green, S C ( ) 1) and the radiation-induced cell death (also shown in green, S C ( ) 2

Fig. 7 .
Fig. 7.The Tumor Static Exposure (TSE) curve as a function of radiation dose and radiosenzitizer concentration.The set of all concentration-dose pairs that yield tumor shrinkage upon daily administration is shown in green.Concentration pairs outside this region lead to tumor growth and are shown in red.The TSE curve is shown in blue and can be compared to the case with no synergy = b ( 0), shown in black (dashed).The TSE curve has significant cur- vature due to a pronounced synergistic effect between the treatments.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 8 .
Fig. 8. (left) Predicted TSE curves for mouse (blue) and man (red), and (right) simulated tumor growth in mouse (blue) and man (red) following average exposures of A: (0.25 Gy, 1 µg/mL), B: (0.5Gy, 2 µg/mL), and C: (0.75Gy, 3 µg/mL).(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)