Costing ‘the’ MTD … in 2-D

Background I have previously evaluated the efficiency of one-size-fits-all dosing for single agents in oncology (Norris 2017b). By means of a generic argument based on an Emax-type dose-response model, I showed that one-size-fits-all dosing may roughly halve a drug’s value to society. Since much of the past decade’s ‘innovation’ in oncology dose-finding methodology has involved the development of special methods for combination therapies, a generalization of my earlier investigations to combination dosing seems called-for. Methods Fundamental to my earlier work was the premise that optimal dose is a characteristic of each individual patient, distributed across the population like any other physiologic characteristic such as height. I generalize that principle here to the 2-dimensional setting of combination dosing with drugs A and B, using a copula to build a bivariate joint distribution of (MTDi,A, MTDi,B) from single-agent marginal densities of MTDi,A and MTDi,B, and interpolating ‘toxicity isocontours’ in the (a, b)-plane between the respective monotherapy intercepts. Within this framework, three distinct notional toxicities are elaborated: one specific to drug A, a second specific to drug B, and a third ‘nonspecific’ toxicity clinically attributable to either drug. The dose-response model of (Norris 2017b) is also generalized to this 2-D scenario, with the addition of an interaction term to provide for a complementary effect from combination dosing. A population of 1,000 patients is simulated, and used as a basis to evaluate population-level efficacy of two pragmatic dose-finding designs: a dose-titration method that maximizes dose-intensity subject to tolerability, and the well-known POCRM method for 1-size-fits-all combination-dose finding. Hypothetical ‘oracular’ methods are also evaluated, to define theoretical upper limits of performance for individualized and 1-size-fits-all dosing respectively. Results In our simulation, pragmatic titration attains 89% efficiency relative to theoretically optimal individualized dosing, whereas POCRM attains only 55% efficiency. The passage from oracular individualized dosing to oracular 1-size-fits-all dosing incurs an efficiency loss of 33%, while the parallel passage (within the ‘pragmatic’ realm) from titration to POCRM incurs a loss of 38%. Conclusions In light of the 33% figure above, the greater part of POCRM’s 38% efficiency loss relative to titration appears attributable to POCRM’s 1-size-fits-all nature, rather than to any pragmatic difficulties it confronts. Thus, appeals to pragmatic considerations would seem neither to justify the decision to use 1-size-fits-all dose-finding designs, nor to excuse their inefficiencies


INTRODUCTION
Until now, I have advanced my criticism of 1-size-fits-all dose-finding methods strictly in the realm of monotherapy. (Norris 2017a) advocated a philosophy of adaptive dose individualization, under the heading 'dose titration algorithm tuning' (DTAT), and illustrated this concretely through simulated neutrophil-nadir-targeted titration of a chemotherapy agent to an individualized 'MTD i ' that varies from one patient to another. (Norris 2017b) examined '1-size-fits-all' specifically as a dose-finding constraint-and estimated the social costs of this constraint as a function of CV(MTD i ), the coefficient of variation of MTD i in the population. (Norris 2017c) elaborated a new coherence principle in dose finding, precautionary coherence (PC), and employed it to motivate a pragmatic '3+3/PC' dose-titration design that beats any conceivable 1-size-fits-all design on both ethical and utilitarian grounds. The present paper begins a process of extending these earlier arguments into the realm of combination therapy.

TOXICITY ISOCONTOURS FOR 2-AGENT THERAPY
It has been recognized for at least a quarter-century (Korn and Simon 1993) that the combinationtherapy setting spoils the determinacy of dosing identified on a maximum-tolerated-dose (MTD) heuristic. Formally speaking, in combination dosing with n agents, the set of tolerable doses will in general be bounded by an (n − 1)-dimensional manifold of just-barely-tolerable dose combinations. Specifically in the case of 2 agents A and B, this is a curve in the (a × b)-plane.
Like so many concepts in dose finding, the problem of combination dose finding yields up fresh conceptual clarity when we consider it from an individual-patient perspective. Considered as belonging to individual patients, the abovementioned curves arise naturally as isocontours of the individual's disutility U i (a, b) for the joint toxicities of a dose. Operating on the MTD heuristic amounts to identifying a disutility U * i that defines a just-barely-tolerable dose for individual i, and seeking a combination dose somewhere on the toxicity isocontour {(a, b)|U i (a, b) = U * i }. It would introduce a distracting level of detail into the present discussion, however, to model such utilities explicitly here. Instead, I obtain individual-level toxicity isocontours in a manner that more directly generalizes my earlier single-agent methodological investigations (Norris 2017b). Specifically, I suppose that for each specific toxicity tox, the associated single-agent MTD's MTD tox i,A and MTD tox i,B are marginally Gamma-distributed: , with their joint distribution constructable using the Clayton copula: From this joint distribution, we may draw pairs (MTD tox i,A , MTD tox i,A ) as in Figure 1. These yield the endpoints (MTD tox i,A , 0) and (0, MTD tox i,B ) of the toxicity isocontours, which we interpolate in the (a, b)-plane using solutions of With γ tox ∈ (0, ∞), Equation 2 can model a range of toxicity interactions, as shown in Figure 2. At γ tox = 1, we obtain a straight line consistent with a purely additive toxicity, such as would occur when A and B produce the toxicity by a shared mechanism. 1 When γ tox < 1 we have by comparison a relatively synergistic (i.e., more-than-additive) toxicity, while for γ tox > 1 drugs A and B produce the toxicity relatively independently. As γ tox → ∞, the toxicities of A and B tend toward a complete independence.

THREE TOXICITIES, WITH INTER-INDIVIDUAL HETEROGENEITY
A perennial failing in the biostatistical dose-finding literature is its abnegation of clinical knowledge and biological understanding of toxicity. With respect to combination dose finding, in particular, this manifests most conspicuously as the presumption that specific toxicities cannot be clinically attributed to one of the combination agents. (This literature does contain isolated forays into toxicity attribution; see e.g. (Jimenez, Tighiouart, and Gasparini 2018) and the sparse prior efforts surveyed therein.) To provide a concrete and clinically reasonable basis for considering toxicity attribution, I posit 3 notionally distinct toxicities: one specific to drug A, another to drug B, and a third 'nonspecific' toxicity clinically attributable to either A or B. See Table 1. To implement this concept within the toxicity-isocontour framework set forth above, we treat each toxicity as defining its own MTD tox i,D , D ∈ {A, B}, tox ∈ {neu, dia, fat}, where (e.g.) the attributability of neutropenia and diarrhea to drugs A and B respectively emerges because while the nonspecific nature of fatigue reflects that Indeed, if we render the doses of drugs A and B commensurate by rescaling them to unity in the ranges where their characteristic toxicities occur: then the clinical logic of Table 1 translates roughly to Assuming that the disutility of multiple distinct toxicities is determined by the disutility of the worst toxicity 2 , then these ideas lend themselves to straightforward treatment in a simple planar diagram. Each toxicity divides the (A, B) dose plane into two regions: tolerable and intolerable. The intersection of the tolerable regions with respect to each separate toxicity yields the set of dose combinations tolerable with respect to all toxicities. These tolerable regions are the white areas close to the origin in Figure 3.
2 That is, I suppose the several toxicity-disutilities at any given dose combine via ), rather than additively or even multiplicatively as one might expect at least with simultaneous toxicities of comparable subjective severity. This assumption goes hand-in-hand, however, with the binary (all-or-nothing) toxicity conception which I employ throughout this paper, and which constitutes the true 'original sin' here. See the later section titled, 'Navigating 2-Dimensional Titration'.  Table 1, for 24 different patients. The regions of (a, b)-dose space rendered intolerable by the 3 toxicities are filled with distinct colors. The attributability of neutropenia and diarrhea to drugs A and B corresponds to the verticality and horizontality of the respective toxicity isocontours. Likewise, the inattributability of fatigue reflects that its isocontours (dashed) typically have intercepts on both A and B axes. Within the (white) tolerable regions are plotted level curves of the probability of remission P r (a, b) from Equation 3. The black dot in each panel shows the tolerable combination dose that yields optimal efficacy for each individual. For half of the individuals shown, the optimal dose occurs at the intersection of the neutropenia and diarrhea contours (look for the half-magenta, half-blue 'bowties'), meaning that each of these toxicities is barely-tolerable at the optimal dose. For id1, it is neutropenia and fatigue which jointly constrain the optimal dose. For id18, the optimal dose produces dose-limiting diarrhea and fatigue. For id6, id9, id17 and id20, fatigue alone constrains the optimal dose, while for id10 only diarrhea constrains the optimal dose.

DOSE-RESPONSE MODEL
The remission probability level curves in the tolerable regions of Figure 3 are derived from which generalizes the dose-response model of (Norris 2017b, Equation 1) to the case of 2-drug combination therapy. 3 The choice of φ = 0.85 > 0 models a complementary (more-than-additive) effect such as we would typically seek from combination therapy.

THE COST OF THEORETICALLY OPTIMAL 1-SIZE-FITS-ALL DOSING
The scenario of Figure 3 provides a setting within which the performance of combination dosefinding methods can be examined and compared. Before undertaking (in the next section) to examine the performance of an actual 1-size-fits-all dose-finding method, I wish first to characterize the losses attributable to 1-size-fits-all dosing itself, apart from the vagaries of any given method. In this section, therefore, we examine the performance of an 'oracular' 1-size-fits-all dosing regime that maximizes the population-rate of remission. As in (Norris 2017b), we define this as the single dose which, when given to all patients who can tolerate it, achieves the highest population rate of remission.
By this definition, the optimal 1-size-fits-all dose for 1000 simulated patients (of whom Figure 3 shows just the first 24) turns out to be (a = 0.44, b = 0.42). At this dose, 9.5% of patients have a DLT, and the population rate of remission is 0.396; see Figure 4. Relative to the population rate of 0.586 achieved under 'oracular' individualized dosing, this represents a 32.5% loss of efficacy.
The shared toxicity introduced here complicates a direct comparison with the monotherapy setting of (Norris 2017b). Nevertheless, with all MTD tox i,X in our simulation sharing the same shape parameter α = 8, I find it hard to resist substituting this into Equation 3 of that previous paper. This yields a value ofP r (α) = 0.346, corresponding to an efficacy loss of 30.9% relative to the population rate of 0.5 which (Norris 2017b) posited for optimal, individualized dosing.
The close proximity of these efficacy losses need not surprise us too much, especially in light of the generic set-up of (Norris 2017b) (requiring no special appeal to a single toxicity), and the robustness of its findings to the several sensitivity analyses performed therein. As the next section will demonstrate, the transition to 'higher dimensions' in dose finding most profoundly affects the very issue that in this section we have assumed away by taking our optima for granted.

PERFORMANCE OF POCRM
To implement the practical search for a 1-size-fits-all dose, we select the well-known POCRM approach (Wages, Conaway, and O'Quigley 2011). This method builds on the CRM of (O'Quigley, Pepe, and Fisher 1990), far and away the most popular model-based method for 1-size-fits-all dose finding in monotherapy (Iasonos and O'Quigley 2014). The novel aspect of POCRM is its employment of partial ordering (PO) notions to address the loss of monotonicity which characterizes dose-response in the combination-therapy setting.
I implement POCRM on the 4 × 4 grid of doses {d 1 , ..., d 16 } shown in Figure 5, following the lead of (Wages, Conaway, and O'Quigley 2011) as to the grid dimensions. In order to reflect the intention to seek truly combination dosing, I however drop doses d 4 and d 13 from further consideration. 4 Also following (Wages, Conaway, and O'Quigley 2011), I posit 3 possible toxicity orderings (models) over these dose levels: {m 1 , m 2 , m 3 }. The first, m 1 , is the true ordering, derived from the toxicity rates observed in our 1000 simulated patients. The latter two, m 2 and m 3 , are obtained by systematically distorting these probabilities. See Figure 6 and Table 2.   Figure 3). Alternative models m 2 and m 3 are obtained by gently distorting these DLT rates, as shown in the center and right panels. In this simulation, we find that 1-size-fits-all dosing at 'the' MTD selected by POCRM achieves a population-level remission rate with expectation 0.323. This represents a loss of 18.2% relative to the optimal 1-size-fits-all remission rate of 0.396. Considering that this reflects both the loss of 'oracular' knowledge about outcomes, and the coarse grid of available doses to search over, POCRM seems to have performed quite well as a 1-size-fits-all dose-finding method in this scenario.

9/12 NAVIGATING 2-DIMENSIONAL TITRATION
Any realistic description of a dose-titration design requires a graded conception of DLT, enabling us to approach toxicity boundaries (such as the toxicity isocontours of Figure 3) without frequently crossing them. Because I have retained a binary (all-or-nothing) toxicity concept in this paper, I make no attempt here to specify a combination-dose titration algorithm explicitly.
Instead, I simply appeal to the characteristic toxicities of drugs A and B, together with the dose-intensity concept of (Simon and Korn 1990), as 'navigational aids' that make dose-titration in the (a, b)-plane conceivable. With synergy in mind, we posit a dose-intensity measure:  Figure 7. Efficacy losses attributable to several constraints, relative to 'oracular' optimal, individualized dosing on a continuum (top, left corner). In the specific scenario simulated in this paper, the 1-size-fits-all constraint (bottom row) appears substantially more costly than the 'pragmatism' of dosing on a coarse grid (right column). Remarkably, 1-size-fits-all dosing proves more costly here than the loss of 'oracular' knowledge about outcomes. Figure 7 summarizes the performance results obtained here, placing the efficacy of two pragmatic methods (POCRM, titration) in context with corresponding 'oracular' methods. Clearly, in the scenario considere here, efficacy losses incurred in 1-size-fits-all dosing substantially exceed those due to pragmatism. Thus, considerations of pragmatism seem neither to justify the decision to use 1-size-fits-all dose-finding designs, nor to excuse their inefficiencies.

DISCUSSION
Of course, the results above derive from a single scenario. But to characterize the performance of POCRM (or any other 1-size-fits-all dose-finding method) across a full range of pharmacological scenarios would be as pointless as it is impossible. The essential point of my efforts here is to place POCRM and similar 1-size-fits-all dose-finding methods into a setting imbued with some relevant pharmacological realism. In this setting, the stark error of the past 30 years of 'innovation' in model-based 1-size-fits-all dose-finding methodology becomes readily apparent-at least to a person with clinical sensibilities, or sound pharmacological instincts.
At this point in my campaign against 1-size-fits-all dose finding, I feel less inclined to address my argument to the 1-size-fits-all dose-finding methodologists than to their clinical collaborators. 5 To clinicians, the reality which Figure 3 palpably approximates must seem unrecognizable in the barren artifice of Table 2. Considering that the latter fully suffices to provide the inputs needed for a POCRM simulation, POCRM's abnegation of clinical realism becomes evident. I continue to hope that self-respecting clinicians, who may have spent decades cultivating their clinical and pharmacological intuitions, will demand more respect from their trial-methodologist collaborators. On this point, one can hardly do better than to recall the long-neglected advice which (Sheiner 1991) offered under the heading 'Reasserting Epistemologic Authority': If I am right about what the illness is, then the cure is simple: clinicians must regain control over clinical trials to assure that the important questions are addressed. Here is my prescription for action: "If you don't understand it, don't do it." If a study design seems clinically inappropriate, wasteful, or unlike real medicine, the clinical scientist should ask why, and neither be satisfied nor acquiesce until it makes sense to him or her.
-Lewis B. Sheiner, MD (1940-2004 CONCLUSIONS I have advanced a visual idiom for appreciating multiple, heterogeneous dose-limiting toxicities in the combination-therapy setting. In a simulation scenario, I have examined the populationlevel efficacy of two pragmatic dose-finding designs: a dose-titration method that maximizes dose-intensity subject to tolerability, and the well-known POCRM method for 1-size-fits-all combination-dose finding. Hypothetical 'oracular' methods serve to define theoretical upper limits of performance for individualized and 1-size-fits-all dosing, enabling separate consideration of the losses due to 'pragmatism' (i.e., the lack of 'oracular' knowledge of outcomes) and '1-size-fits-all-ness'. In the scenario considered here, 1-size-fits-all-ness makes a poor showing.

DATA AVAILABILITY
Open Science Framework: Code for reproducing all of this paper's Figures and analyses may be found at doi:10.17605/osf.io/yn2q3.