Submitted to Manufacturing & Service Operations Management manuscript

Clinical Trials for New Drug Development: Optimal Investment and Application Panos Kouvelis Washington University in St. Louis, [email protected]

Joseph Milner University of Toronto, [email protected]

Zhili Tian Florida International University, [email protected]

Firms conduct Phase III drug trials by enrolling and treating hundreds or thousands of patients that meet a defined set of conditions. Finding these patients is expensive (up to 35% of the R&D cost of a drug) and time consuming (up to 3 yrs), with a great deal of uncertainty around these. We developed an effective dynamic investment policy in Phase III clinical testing of new drugs, that accounts for available information on drug quality, the current success in enrolling qualified patients, costs of intensified clinical testing efforts, and FDA approval and potential market size. We consider cases with and without interim analysis of the collected clinical data of the drug as the trial progresses. We develop structural results and numerical algorithms, and provide conditions for accelerating or suspending a clinical study. We offer managerial insights on how a drug’s expected market revenue and quality affect clinical investments to it over time. Key words : pharmaceutical drug development, new drug development management, clinical trial, R&D project management, optimal investment History : This paper has yet to be submitted

1.

Introduction

The development of new drugs for treatment of disease has been central to the improvement of health care over the past 40 years. Annually, worldwide pharmaceutical firms spend $100 billion in research and development (R&D) of new drugs according to the International Federation of Pharmaceutical Manufacturers and Associations[IFPMA (2011)]. Pharmaceutical Research and Manufacturers of America [PhRMA (2011)] reports that the average cost of development of a new drug is $1.3 billion in 2005 and can take 10 to 15 years. As part of this R&D process, and in order to gain FDA approval of a new drug, firms must engage in clinical testing on human subjects to evaluate the efficacy, safety, and tolerability of a drug. Consisting of three phases, with increasing 1

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cohort sizes, the total duration of such testing is on average 72-84 months and may account for $740 million (62%) of the total cost of development according to PhRMA (2011). Our research focuses on Phase III clinical testing. In Phase III, a firm tests the efficacy of the drug with a large sample of several hundred to several thousand patients to generate statistically significant data for a new drug application (NDA) submission. Upon successful completion of Phase III, a firm submits an NDA to the US FDA for evaluation and approval to market the drug. Phase III typically accounts for 30.5 months or roughly 40% of the testing phase and 60% of the costs of clinical testing (PhRMA 2011). A clinical study is complete when the firm has enrolled and treated as many patients as it specified in its clinical trial protocol. In order to increase enrollment, firms can open more clinical testing centers, engage in broad community education, and relate promotional efforts to better inform potential patients, their families, and doctors. However, the cost of doing so may significantly outweigh any potential revenues for some new drugs. Thus, it is relevant to ask the question of what is the dynamically optimal investment rate in clinical testing for such drugs based on observed enrollment and future revenue estimates. As part of the clinical trial protocol firms may specify when they will view and analyze the data during the Phase III study. This interim analysis, which must be approved by the FDA, can be used by the firm to test hypotheses and to reassess the funding level they put on a project. Further, if warranted by the interim analysis, the firm may terminate the study early in order to file the NDA. Alternatively if the results of the interim analysis show no efficacy or indicate a safety concern the firm may abandon the study at that time. Thus, the question arises as to what the value of such interim analysis is and when it should be conducted. The success rate of new drug development is very low. Over the ten year period 1993-2003, 60% of drugs passed from Phase I to Phase II, 39% from Phase II to Phase III, and 60% from Phase III to NDA submission. Once submitted, 85% were approve, leading to about a 1 in 8 chance of gaining FDA approval for a drug starting in Phase I (TCSDD 2010). In addition, only two of 10 marketed drugs generate sufficient total revenue to recover the development costs (PhRMA 2011). While some have limited markets with unknown potential beneficiaries, others are copied by generic drugs in the years after patent protection expires. Because US patent law promotes the filing of patents on new molecules prior to clinical testing, uncertainty in the preclinical and clinical phases leads to uncertainty over the duration of the market exclusivity period (MEP), the

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time when drugs can generate significant profits with little competition (Eisenberg 2003). Thus, in addition to the uncertainty of the approval process, there is considerable uncertainty around the sales revenues that may be generated during the lifetime of a drug. We study the development of an effective dynamic investment policy in Phase III clinical testing of new drugs. The goal is to find an effective policy that provides simple and implementable guidance on when to slow down or even abandon clinical testing and when to intensify such efforts taking into account observed relevant information. Our policy defines dynamically targeted patient enrollment rates. The targets explicitly account for available time to market for the new drug, demonstrated success and remaining uncertainties in patient enrollment, and the forecast for approval success and potential market size of the drug. To achieve the targeted patient enrollment rates requires operating the right number of clinical testing facilities and spending at the required levels on community education and promotional efforts and activities. These drive the realized expenditures (or investments) in the clinical testing for the new drug. Our initial analysis is performed under the assumption that all collected data will be analyzed at the end of the clinical testing (i.e., without an interim analysis of the data). We subsequently consider the performance of an interim analysis on collected clinical testing data. The remainder of the paper is as follows. Section 2 provides a literature review. We give basic model definitions in Section 3. Section 4 discusses the model when no interim analysis is conducted. In Section 5 we discuss the case where such an analysis may be made. We apply our model to a case study for a new drug development in Section 6 and provide conclusions in Section 7.

2.

Literature Review

Previous work that relates to our study comes from two streams: (a) efforts to value uncertain product development, and (b) dynamic investments in projects with real options in the presence of various uncertainties. Our work provides ways to estimate the value of drug development projects and thus has similarities with stream (a). At the same time, the most important contribution of our work is to suggest dynamic investment rates in clinical studies of new drugs fully capturing their operational details, costs, and uncertainties, thus encompassing the real options, dynamic execution flavor of stream (b). Representative work of stream (a) is Kellogg and Charnes (2000), Jacob and Kwak (2003), and Girotra et al. (2007). Kellogg and Charnes (2000) use a binomial lattice method to evaluate a biotechnology company whose value equals the sum of the values of the new drug development

4

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projects. Jacob and Kwak (2003) suggest that a real options approach to new drug development evaluation is better than a net present value or discounted cash flow approach, because active management and operational flexibility bring significant value to a project. Girotra et al. (2007) derive the value of a Phase III new drug development project by measuring the change in the market value of a firm when clinical trials fail. Representative work of stream (b) is Schwartz (2003), Lucas (1971), McDonald and Daniel (1986), Pindyck (1993), and Dutta (1997). Schwartz (2003) develops a simulation approach to determine the option value of patent protected R&D projects in the pharmaceutical industry. He models the uncertainty in the cost to complete a project using a controlled diffusion process and the uncertainty in the cash flow with a geometric Brownian motion and demonstrates how the abandonment option affects the project’s value. In contrast to Schwartz, we determine the dynamic investment rate for the Phase III clinical studies considering operational details. Lucas (1971) develops an optimal control model for an R&D project which includes a fixed return, discounted to the time of completion, and variable development costs incurred during project. The completion time is a random variable which decreases if the project owner increases the investment. McDonald and Daniel (1986) investigate the ideal timing of an irreversible investment when benefits and costs follow geometric Brownian motions. They derive thresholds for the ratio of the project value to the investment cost for the firm. Pindyck (1993) considers how technical and input cost uncertainties raise the value of investment for a project. He shows that it is optimal to invest only if the expected cost is less than a threshold. Dutta (1997) optimally allocates resources among several stages of an R&D project. He shows that when profits are realized only after all stages are successfully completed, the optimal strategy is to spread the resources evenly among the stages. Our model differs in that information on the completion of the project is updated as patients are enrolled in the trial. Also, the pharmaceutical firm may file the NDA early if the drug passes the hypothesis test in an interim analysis. Because of this possibility, our policy does not evenly allocate resources among different intervals of the study. We offer a different angle in evaluating new drug development projects. The duration of a clinical study determines the available remaining time under patent protection and so affects the cumulative drug revenue. Our work tries to control it through optimizing the investment in the clinical study, a factor not previously considered in drug development project valuation methodologies. At the same time, the inclusion of the operational details of the clinical study (success in enrolling patients,

Kouvelis, Milner, and Tian: Clinical Trials for New Drug Developmen Article submitted to Manufacturing & Service Operations Management; manuscript no.

N K(t) u(t) c(u) τ ζ ui , i = 1, . . . , ζ θi , i = 1, . . . , ζ T

5

the (initial) sample size required for completion of the drug trial number of patients enrolled and treated by time t the enrollment target rate at time t the cost rate associated with enrollment rate u the stopping time of the enrollment process the number of (non-zero) enrollment levels (cost function breakpoints) patient enrollment level i marginal cost at investment level i the maximum allowable duration of the Phase III study Table 1

Notation

use of interim data analysis to assess drug efficacy and quality, etc.) allows us to suggest more precise targets for intensifying or aborting clinical studies, and, through our modeling framework, to enhance and more accurately value such options in drug development projects. It is the ability of our work to model in detail the operational investment decisions during Phase III clinical studies that sharpens our contributions to both of the above literature streams.

3.

Model Definitions

In this section we define the stochastic patient enrollment process, the cost model, and the revenue model. A summary of the parameters and variables is given in Table 1. Other terms will be defined as needed. We assume that the enrollment process begins at time t = 0 and that the required sample size for the clinical study, N , is given. Typically the sample size is determined by the test hypotheses specified for approval of the drug. We let T be the time by which the firm must complete the study in order to successfully market the drug. This usually reflects the time until patent expiration, but if competition is unlikely to develop immediately at that time, T could be some greater time. However, because of the time required for NDA approval as well as time for market development, in practice the clinical trials must be completed long before T . Two other useful notations are: la , the expected length of the approval process and lr , the expected length of the early phase of drug market entry with low (or for our model purposes, no profit). The typical expected FDA approval process length is around 10 months (TCSDD 2010). Following Lai (2001) we assume that the patient enrollment in a clinical trial is a controllable random process. The firm can influence the patient enrollment by increasing the number of testing centers and efforts to reach out to qualified patients. Of those patients reached, some may not be suitable, and others may choose not to continue after initiating participation. Therefore the number of patients enrolled and treated by time t, K(t), is considered a random variable which for

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simplification of analysis is treated as continuous (except in our numerical algorithms as detailed below). To ease our discussion we write of the “enrollment” when more formally we should write “enrollment and treatment”. We let u(t) be the infinitesimal drift (the mean enrollment rate) and let σ 2 be the infinitesimal variance of the process. Letting w be standard Brownian motion, we assume Z K(t) =

t

Z u(s) ds +

0

t

σ

p

u(s) dw

(1)

0

where K(0) = 0. In (1), the first term is the targeted number of patients enrolled by time t and the second term is the deviation from this target. We assume u(t) ∈ U , which is a decision variable on U = [0, u ¯] giving the instantaneous targeted enrollment rate. In the Appendix we show K(t) is uniquely defined by (1). We assume the firm knows the enrollment history K(s), for s ∈ [0, t) when it chooses u(t) at time t. Because of the Markovian nature of the Brownian motion, we consider the control process based on the patient enrollment up to time t. Enrollment is completed at time τ , when either N participants have been recruited, treated, and followed-up or there is no time remaining (t = T ). To clarify, let O = (−∞, N ) and Q = [0, T ) × O. For (t, K(t)) ∈ Q, τ = inf {s : (s, K(s)) 6∈ Q}. Thus τ is a stopping time; if the firm cannot enroll N patients by time T , τ = T and the firm abandons the study. (Note we define the state-space O on (−∞, N ) rather than on the more natural [0, N ) to avoid introducing the considerable mechanics of a reflecting boundary at 0. Because we expect the process to be controlled away from 0, doing so is both practical and of little loss in modeling fidelity.) Upon successful completion of a Phase III study, a firm will file an NDA for FDA approval. Let random variable Λ express the discounted value of the sales of an approved drug over an infinite sales horizon at the time of completion of the study. The value of Λ depends on the likelihood of FDA approval, the market size, the quality of the drug, and the discount rate. The actual revenue received, in turn, depends on Λ and the effective revenue-positive time remaining in the MEP given by T − (τ + la + lr ). Let π be a Bernoulli random variable equal to 1 if the drug is approved and 0 otherwise. We assume there is a market size κ that defines the contribution rate in $/unit-time, for an approved drug. We assume π and κ may depend on the quality of the drug. Let Y be a random variable with support on [y, y¯], CDF H0 (y), pdf h0 (y), representing the drug’s quality. We assume that the distribution H0 is based on the Phase II clinical study and any other information the firm has at the start of the Phase III study. Quality is an overall measure of how the business manager expects the drug to perform in the market, relative to the nominal market size. As such

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it depends on perceptions of the drug’s safety, efficacy, and tolerability. It is not purely a measure of a drug’s medical effectiveness which is being tested in the trial. However, based on observation of this effectiveness, the business manager may (and should) update his/her expectations of the drug’s financial performance. Finally, let r be the discount rate. Then, Λ = π(Y )κ(Y )r−1 e−r(la +lr ) . (Note the revenue is discounted by e−r(la +lr ) because of the delay la + lr time units after the study completion before revenues begin.) The expected value at time τ of the total contribution over a drug’s lifetime is Γ(τ, N ) = E[Λ](1 − e−r(T −τ ) ). The expectation is taken over both π and Y . We assume if τ = T (the firm does not complete the study), it incurs a cost Ψ(T ). (To be consistent with our Λ definition, Ψ(T ) has also been discounted by e−r(la +lr ) .) Proposition 1 The expected total contribution Γ(τ, N ) is decreasing and strictly concave in τ . (All proofs appear in Appendix B.) We assume that the cost of recruiting participants is convex, increasing in the targeted enrollment rate. Let c(u) be the cost rate given u. We approximate it by a piecewise linear convex function of u: We assume that the function is defined by breakpoints at ζ enrollment levels, not including 0. Let u0 < u1 < · · · < uζ define the endpoints of the segments where u0 = 0 and uζ = u ¯. Let θi be the marginal cost of investment on segment (ui−1 , ui ], i ∈ [1, . . . , ζ]. Thus for u ∈ [ui−1 , ui ], for some i ∈ [1, ζ], c(u) =

i−1 X

θj (uj − uj−1 ) + θi (u − ui−1 ).

j=1

The use of a piecewise linear cost function may be natural as in many cases the cost per patient will depend on the number of test centers opened. As the number increases, the marginal cost would increase as the least costly, most effective centers would be opened first.

4. 4.1.

Optimal Investment for a Clinical Study Without Interim Analysis Derivation and Characterization of the Solution

We determine the optimal patient enrollment rate u(t) to maximize the present value of the cumulative profit. We begin by defining the optimization problem.

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For t ∈ [0, T ) let (Ω, F , P ) be the probability space defining the uncertainty on the enrollment and let {Fs } be an increasing family of σ-algebras with Fs ⊂ F for all s ∈ [t, T ]. We let At denote the collection of all Fs -progressively measurable and U -valued processes u(·) on (t, T ) which satisfy i hR T E t |u(s)|m ds < ∞ for m = 1, 2, . . .. Let χA be the indicator function where χA = 1 if A is true and 0 otherwise. The expected profit J given t and K(t) for rate function u(·) ∈ At is  Z τ  −r(s−t) −r(τ −t) −r(T −t) J(t, K(t); u) = E − e c(u) ds + e Γ(τ, N ) − e Ψ(T )χτ =T t

where (t, K(t)) ∈ Q, the initial state of the system, define the current time and enrollment, and the expectation is taken over τ . We let G(t, K(t)) be the optimal value of the investment in the Phase III clinical study at time t. We determine the optimal target enrollment rate u(t) by solving problem (PI) (P I) :

G(t, K(t)) = sup J(t, K(t); u) u∈At p s.t. dK(t) = u(t)dt + σ u(t) dw(t) K(0) = 0.

(2a) (2b) (2c)

Here, (2b) expresses the differential equation form of (1). We now derive the Hamilton-Jacobi-Bellman (HJB) dynamic programming equation for (PI) along with its boundary and terminal conditions. First note that (PI) is a singular stochastic control problem since u(s) can be zero for some s ∈ [0, T ]. That is, for u ∈ [0, u ¯] there does not exist a constant υ > 0 such that σ 2 ux2 ≥ υx2 for x ∈ R. To prove the uniqueness of the solution, we assume there exists a small  > 0, redefine u0 =  and require u ∈ U ≡ [, u ¯]. We also redefine At to be U valued. (The value  may represent the small ongoing administrative cost of a suspended clinical study prior to patent expiration.) Let v > t such that v − t is small. We can express (PI) as the following (approximate) dynamic program:   Z v   −r(s−t) −r(v−t) G(t, K(t)) = sup E − e c(u(s)) ds + e G(v, K(v)) u∈At t p s.t. dK(t) = u(t) dt + σ u(t) dw(t) K(0) = 0.

(3a) (3b) (3c)

From result §13.3 in Sethi and Thompson (2000), (3a) implies G(t, K(t)) = max E[−c(u) dt + (1 + r dt)−1 G(t + dt, K + dK)] u∈U

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From Itˆo’s Lemma, we have G(t + dt, K + dK) = G(t, K) +

∂G ∂G 1 ∂2G dK 2 + dK + dt + o( dt) 2 ∂K 2 ∂K ∂t

where o( dt) represents terms that go to zero faster than dt. Substituting in (3a) and noting p dK 2 = σ 2 u(t) dt + 2u(t) dtσ u(t)dw + o( dt), we obtain G(t, K) = " maxE −c dt +

∂G G(t, K) + ∂K u dt + uσ2

2

∂2G ∂K 2

u∈U

# p √ ∂2G ∂G dt + ∂K u(t) dw + ∂G u dw + o(t) dt + σ 2 u(t)dtσ ∂t ∂K . (1 + r) dt

Using (1 + r dt)−1 = (1 − r dt + o(t)) and E[ dw] = 0, we have   ∂G uσ 2 ∂ 2 G ∂G G(t, K) = max −c dt + G(t, K) + u dt + dt + dt − rG(t, K) dt + o(t) . u∈U ∂K 2 ∂K 2 ∂t Dividing both sides by dt, letting it go to 0, and rearranging terms, we have  2 2  ∂G uσ ∂ G ∂G − rG + max +u − c(u) = 0. u∈U ∂t 2 ∂K 2 ∂K

(4)

Letting t = τ < T and t = T in (PI), provides the boundary and terminal conditions G(τ, N ) = Γ(τ, N ) for τ ∈ [0, T ),

(5)

G(T, K) = −Ψ(T ) for (T, K) ∈ {T } × O.

(6)

Let C 1,2 (Q) be the set of functions of G with ∂G/∂t, ∂G/∂K, and ∂ 2 G/∂K 2 continuous on Q; let ¯ be the set of continuous functions on Q, ¯ the closure of Q. We have: C(Q) ¯ Proposition 2 Equations (4)–(6) have a unique solution G ∈ C 1,2 ∩ C(Q). Next we solve the maximization problem in (4) to obtain u∗ (t). Let K0∗ = 0 and let Ki∗ (t) satisfy σ 2 ∂ 2 G(t, Ki∗ ) ∂G(t, Ki∗ ) + = θi 2 ∂K 2 ∂K for i = 1, 2, . . . , ζ. To simplify the notation in the proof of the following proposition and in the rest of the paper, we let the differential operator L be defined on function G as LG =

σ 2 ∂ 2 G ∂G + . 2 ∂K 2 ∂K

∗ ∗ Proposition 3 For i = 1, 2, . . . , ζ, (1) Ki−1 (t) < Ki∗ (t); (2) if Ki−1 (t) < K(t) ≤ Ki∗ (t), u∗ (t) = ui−1 ,

and if K(t) > Kζ∗ (t), u∗ (t) = uζ .

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Kouvelis, Milner, and Tian: Clinical Trials for New Drug Developmen Article submitted to Manufacturing & Service Operations Management; manuscript no.

Substituting the value of u given by Proposition 3 into (4) and rearranging terms, the optimal value function G solves the following differential equations: ∂G/∂t = −LGui + rG + c(ui ) for i = 0, 1, . . . , ζ

(7a)

subject to G(τ, N ) = Γ(τ, N ) for all τ ∈ (0, T ),

(7b)

LG(t, Ki∗ ) = θi for i = 1, . . . , ζ,

(7c)

G(T, K(T )) = −Ψ(T ),

(7d)

G(t, K) is continuous at Ki∗ (t) for i = 1, . . . , ζ.

(7e)

In Figure 1 we present an example of an optimal policy for the case ζ = 2. The regions Ω0 , Ω1 and Ω2 define the investment policy dependent on the time and current enrollment with the boundaries K1∗ (t) and K2∗ (t) dividing the space. (We define S1 (K) as the inverse function of K1∗ (t) and S2 (K) as the inverse function of K2∗ (t) below). In region Ω0 , there is no investment and the project is suspended (u∗ = ); in region Ω1 , there is limited investment (u∗ = u1 ); in region Ω2 , there is high investment (u∗ = u2 ). Consider the three sample paths illustrated leading to enrollment level K. The left-most path shows the case where the firm does not have any difficulty in finding qualified patients and invests at rate u2 at level K. The middle path presents the case where the firm continues to invest at rate u1 . The value of increasing the investment rate to reduce the clinical study time is not justified by the current results. For the right-most sample path, at level K, there is insufficient time remaining to acquire the required number of test subjects, and if not yet suspended, the project should be. In practice, firms typically set a target time for reaching a particular enrollment. They determine such time by rules of thumb on the expectations of drug profitability. Suppose that the firm chooses S2 (K) as the target time. If the firm fails to achieve this target, it will increase efforts to obtain more subjects through opening additional test centers. If the firm achieves the target, it will continue with the current recruiting effort. Our method suggests the policy should take into account the enrollment just prior to the decision time, expected future revenue, and the cost of recruiting patients. For the first case (shown by the middle path), our method suggests that the firm should still use the intermediate enrollment rate to obtain more information on patient enrollment process. For the second case (shown by the left path), our method suggests that the firm should increase its effort in recruiting patients to complete the clinical study as quickly as possible. The current

Kouvelis, Milner, and Tian: Clinical Trials for New Drug Developmen Article submitted to Manufacturing & Service Operations Management; manuscript no.

K(t) N

K

11

Upper Boundary

K2*

S2(N)

S1(N)

W1

W2

K 1*

W0 0

S2(K)

S1(K)

Lower Boundary Figure 1

T

t

Example of Investment Regions

enrollment indicates that the firm does not have difficulty in finding qualified patients, and thus increased investment is warranted. 4.2.

Special Case: Exponentially Distributed T − τ

In this section we derive a closed form solution for the case where the duration of the revenue generating period (T − τ ) is exponentially distributed. This assumption is equivalent to assuming the period is of infinite length but an accelerated discounting rate applies to the perpetuity Λ. We argue this may be more applicable to low revenue drugs, those earning less than $100 million per year. Such drugs represent 70% of approved drugs. They differ from higher revenue drugs in several ways. First, consistent with the infinite perpetuity, they tend to have longer market exclusivity periods (MEPs) (15 years vs. 10 years, on average). Second, the number of generic entrants tend to be lower (2–3 vs. 4–7). (See Grabowski and Kyle (2007), Grabowski et al. (2013).) This implies there is greater variability around the duration of the MEP. Also, the limited competition implies the revenue may not drop quickly after generic introduction. For higher revenue drugs, such drops lead to a truncation of the lifetime distribution in ways not applicable to lower revenue drugs. Thus, while an exponential distribution is clearly a simplification necessary for solving for a closed-form result, it does provide some insight for this set of more niche-oriented, lower cumulative revenue drugs. Let T¯ = T − τ and assume T¯ is exponentially distributed with parameter λ. The expected revenue at time τ is Z Γ(τ, N ) = 0

∞

¯ ¯ E[Λ](1 − e−rT )λe−λT dT¯

Kouvelis, Milner, and Tian: Clinical Trials for New Drug Developmen Article submitted to Manufacturing & Service Operations Management; manuscript no.

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= E[Λ]

r . λ+r

The expected profit given policy u at time t is then  Z τ  r −r(s−t) −r(t−τ ) J(t, K; u) = E − e c(u) ds + e E[Λ] λ+r  Zt v  r =E − e−rs c(u) ds + e−rv E[Λ] λ+r 0 where v = τ − t. Noting the expected profit depends only on the duration τ − t, J(K; u) ≡ J(t, K; u) is independent of t and so must be the optimal policy. That is, we claim there exist thresholds Ki∗ ∗ independent of t such that if Ki−1 ≤ K ≤ Ki∗ then the investment rate is ui−1 . Observing ∂G/∂t = 0

and substituting into (7a)–(7e), we have G solving: LG(K)u − rG(K) − c(u) = 0 for i = 0, 1, . . . , ζ

subject to the boundary conditions r G(N ) = E[Λ] , λ+r LG(Ki∗ ) = θi for i = 1, . . . , ζ, G(K) is continuous at Ki∗ (t) for i = 1, . . . , ζ.

(8a)

(8b) (8c) (8d)

The thresholds Ki∗ , i = 1, · · · , ζ satisfy (8c). Let

√ u − u2 + 2ruσ 2 γ(u) = . uσ 2

∗ ∗ and note γ(u) < 0. We define Kζ+1 = N and G(Kζ+1 ; uζ+1 ) = E[Λ]r/(λ + r).

Proposition 4   1 θi ui = − ln for i = 1, · · · , ζ (9) ∗ γ(ui ) rG(Ki+1 , ui+1 ) + c(ui )   c(ui ) c(ui ) ∗ ∗ − for i = 0, 1, . . . , ζ (10) G(K; ui ) = exp((Ki+1 − K)γ(ui )) G(Ki+1 , ui+1 ) + r r ∗ and G(K) = G(K; ui−1 ) if Ki−1 < K ≤ Ki∗ for i = 1, · · · , ζ and G(K) = G(K; uζ ) if K > Kζ∗ . Ki∗

∗ Ki+1

In Proposition 4, we notice that Ki∗ is independent of t. Thus we find the constant Ki∗ that define the initial investment rates. Proposition 5 For i = 1, · · · , ζ, Ki∗ is decreasing in E[κ], E[π] and σ 2 , and increasing in θi . The comparative statics given in Proposition 5 are intuitive. As Ki∗ is the threshold for recruiting patients at rate ui , decreasing the threshold implies targeting a higher enrollment rate. Therefore we would expect higher market size, greater likelihood of FDA approval and more volatile enrollment to decrease Ki∗ . Similarly, increasing the cost increases Ki∗ .

Kouvelis, Milner, and Tian: Clinical Trials for New Drug Developmen Article submitted to Manufacturing & Service Operations Management; manuscript no.

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K2* 6818

K2* 25 000

6816

K1* 5000

20 000

0.2

15 000

6814

10 000 6812

Figure 2

20

30

40

50

60

Σ

0.6

0.8

E@YD

-10 000

5000 10

0.4

-5000

-15 000 0.2

0.4

0.6

0.8

E@YD

-20 000

Thresholds K2∗ and K1∗ vs. Enrollment Volatility σ and Drug Quality E[Y ].

Example Consider a clinical study for testing a drug which treats migraines. Suppose N = 7153 and the two target enrollment rates under consideration are 101 and 131 patients per week. Here c(u) has two marginal costs, θ1 = $0.0303M/week and θ2 = $0.0425M/week. We consider how the p policy changes as the variability of the enrollment rate, initially set to σ = 11 ( patients/week), and the quality, Y , defined by the percentage of patients whose symptoms are relieved by the drug, initially set to E[Y ] = 0.54, change. (We assume Λ is multiplicative in π, κ and Y , and let r = 10% annually, π = 0.8, κ = $352M , and la = 2 × 52 weeks and lr = 2.5 × 52 weeks so that E[Λ] = $1038M . Details for how these numbers were derived are given in Section 6.) We assume λ = 1 . week

1 20×52

= 0.00096

Using these values we find K1∗ = −11694 and K2∗ = 6818. Thus the firm would initiate testing

at a targeted rate of u1 patients per week. This may be accomplished by opening u1 /η test sites where η is the average number of patients recruited per test site per unit time. Because K1∗ < 0, the firm would never abandon the study, and would increase the enrollment rate to u2 when K > K2∗ . In Figure 2 we show how the thresholds vary with σ, holding E[Y ] = 0.54, and how they vary p with the expectation of quality, holding σ = 11. As σ increases from 2 to 60 ( patients/week), we observe K2∗ is a decreasing function of σ. (We do not show K1∗ vs σ since it is negative throughout.) Note, however, the threshold value is relatively constant on the studied region. This implies that the optimal enrollment rate is relatively insensitive to the volatility of the enrollment process. Similarly, as the expectation of Y increases the firm is more likely to recruit patients at a higher rate. In particular, for E[Y ] > 0.78 the firm would initiate enrollment at rate u2 . Note that from the graph of K1∗ vs. E[Y ], if E[Y ] < 0.17, the optimal initial investment rate is , i.e., the firm would not invest in the drug because its expected efficacy would be too low. We also have the following comparative statics on the value of the clinical study. Proposition 6 For i = 0, 1, · · · , ζ, G(K; ui ) increases with E[κ], E[π], and σ 2 , and decreases with θi .

Kouvelis, Milner, and Tian: Clinical Trials for New Drug Developmen Article submitted to Manufacturing & Service Operations Management; manuscript no.

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Figure 3

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Clinical Study Value vs. Enrollment Volatility and Drug Quality with No Interim Analysis (Solid Line) and One Interim Analysis (Dashed Line) for Exponentially Distributed MEP.

According to Proposition 6, the value of a clinical study, G(0), increases with market size of the drug, the likelihood of FDA approval and the uncertainty of enrollment, and decreases with the marginal cost of conducting the study. In Figure 3 we display G(0) as a function of σ holding E[Y ] = 0.54 and as a function of E[Y ] holding σ = 11. (The figure also displays G(0) with an interim analysis - see Section 5.) As volatility increases, the value of the clinical study increases because it results in a higher option value for suspending the study if the enrollment is low. We also observe that the value of the clinical study increases with the expected quality. Next we observe G(Ki∗ ; ui ) is a constant since Ki∗ is a constant. Proposition 7 G(Ki∗ ; ui ) =

θi ui c(ui ) − r r

for i = 1, · · · , ζ. Using Proposition 7, we simplify the expressions of Ki∗ and G(K; ui ). Corollary 1 Ki∗

=

∗ Ki+1

  θi 1 − ln γ(ui ) θi+1

for i = 1, · · · , ζ − 1, and ∗ G(K; ui ) = exp[(Ki+1 − K)γ(ui )]

θi+1 ui c(ui ) − r r

for i = 0, · · · , ζ − 1. By Corollary 1, and noting the definition of γ(u) given above, the volatility of the enrollment and the marginal cost of conducting a clinical study determine the boundaries between the optimal enrollment rates.

Kouvelis, Milner, and Tian: Clinical Trials for New Drug Developmen Article submitted to Manufacturing & Service Operations Management; manuscript no.

4.3.

15

Algorithmic Solution to the General Case

In this section we develop a dynamic programming algorithm to solve (4) subject to the boundary and terminal conditions. The solution provides the optimal value-to-go function G(t, k) and the free boundary Ki∗ (t) that separate the policy regions. Further, we characterize these boundaries. Numerical solution is required for the case where T is a fixed constant because the revenue depends on the time T − τ . This case expresses the experience of firms when patent expiration has a significant impact on the revenue generated by a drug. Typically, for drugs that earn over $500 million per year, the end of the MEP also marks the end of a drug’s ability to generate significant revenues. For example, the annual revenue for Difulcan, an anti-fungal medication, decreased from $1 billion to $400 million after its patent expired in 2004 according to Pfizer (2006). The problem is solved through backwards recursion. We discretize the state space Q so that t ∈ {0, 1, . . . , T } and K ∈ {0, 1, . . . , N }. The terminal value is given for time t = T , G(T, K) = −Ψ(T ), and the boundary value is given for K = N , G(τ, N ) = Γ(τ, N ) for τ ∈ {0, . . . , T − 1}. Let

Si (K) ∈ {0, . . . , T } be the inverse function of Ki∗ (t), i.e., Si (Ki∗ (t)) = t. Si (K) is the switching time between policies of letting u = ui and u = ui−1 if K patients are enrolled. We solve for Si (K) by letting iteratively decreasing K from N − 1 to 0 and, for each K, iteratively decreasing t from T to 0, comparing J(t, K; ui ) versus J(t, K, ui−1 ). This simultaneously produces G(t, K) and Si (K) for K ∈ {0, . . . , N } and for i = 1, . . . , ζ. The algorithm terminates when K = 0 and t = 0. To evaluate J(t, K; ui ), we calculate the expected value-to-go for the state by taking the expectation of the discounted values-to-go, G(t + x, K + 1) for x = 1, . . . , T − t and G(T, K), subtracting a off the cost of operating at level ui . Let τν,ς be the stopping time defined by the minimum of T and a the time when a additional patients are enrolled. That is, τν,ς is the hitting time of the Brownian

motion starting from 0 to level a with drift ν and instantaneous variance ς. The well-known density a of τν,ς is

  a (a − νx)2 f (x; ν, ς) = √ exp − , 2ς 2 x ς 2πx3 a

x > 0, ν > 0.

a (Karlin and Taylor 1975). The CDF of τν,ς is given by

    1/2 1/2 F a (x; ν, ς) = Φ (λ/x) (−1 + x/µ) + e2λ/µ Φ − (λ/x) (1 + x/µ) where µ = a/ν and λ = a2 /ς 2 and Φ(x) is the CDF of the standard normal distribution (Folks and √ Chhikara 1978). Let F¯ (x) = 1 − F (x). Throughout the following ς = σ u and we suppress it for clarity.

Kouvelis, Milner, and Tian: Clinical Trials for New Drug Developmen Article submitted to Manufacturing & Service Operations Management; manuscript no.

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The discounted value-to-go is Z

T −t

exp(−rx)(G(t + x, K + 1) − c(ui )x)f 1 (x; ui ) dx

J(t, K, ui ) = x=0

+ exp(−r(T − t))(G(T, K) − c(ui )(T − t))P (τu1i = T ). We approximate this as J(t, K, ui ) ≈

T −t X

exp(−rx)G(t + x, K + 1)pui (x) + exp(−r(T − t))G(T, K)P (τu1i = T )

x=0

− C(t, ui )

where pui (x) = F 1 (x; ui ) − F 1 (x − 1; ui ) is the probability of enrolling one individual at time x and C(t, u) is the cost of enrolling patients at the target rate u from time t to T . That is, we approximate the reward in state (t, K) by finding the approximate reward in the next state less the exact operating cost for enrollment rate u until T . We show: Proposition 8 Let k = N − K where K is the enrollment at time t. Then C(t, u) =

 c(u)  1 − ekγ(u) F k (T − t; u, (2ς 2 r + u2 )1/2 )) − e−r(T −t) F¯ k (T − t; u, ς) . r

The algorithm is given in the following. Algorithm 1 for solving Problem (PI): Step 1. Compute G(t, N ) = Γ(t, N ) ∀ t ∈ {0, · · · , T } and let G(T, K(T )) = −Ψ(T ). Step 2. Set k = 1. Step 3. Set K = N − k. Step 4. Set t = 1 and i = ζ. Step 5. Compute J(t, K; ui ) and J(t, K; ui−1 ). Step 6. If J(t, K; ui ) < J(t, K; ui−1 ), set Si = t and G(t, K) = J(t, K; ui−1 ), and go to Step 7; else set G(t, K) = J(t, K; ui ), and if t < T set t = t + 1 and go to Step 5, otherwise go to Step 8. Step 7. If i > 0, set i = i − 1 and go to Step 5; else compute J(s, K; ui ) and set G(s, K) = J(s, K; ui ) for t ≤ s ≤ T . Step 8. If k ≤ N , set k = k + 1 and go to Step 3; else stop. We have the following structural results for the solution provided by the algorithm.

Kouvelis, Milner, and Tian: Clinical Trials for New Drug Developmen Article submitted to Manufacturing & Service Operations Management; manuscript no.

σ)=)4.7)

σ=)9.5) 600"

500" 400"

Ω1"

Ω2"

300"

"

S2"

200"

S1"

Ω0"

100" 0" 0"

Enrollment)

Enrollment)

600"

500" 400"

"

S2"

200" 100" 0"

100" 200" 300" 400" 500" 600"

S1"

Ω0"

0"

100" 200" 300" 400" 500" 600"

Time)(weeks))

σ=)19)

σ)=)28.5)

600"

600"

500" 400"

Ω1"

Ω2"

300"

"

S2"

200" 100"

Ω0"

0" 0"

100" 200" 300" 400" 500" 600"

Time)(weeks))

S1"

Enrollment)

Enrollment)

Ω1"

Ω2"

300"

Time)(weeks))

Figure 4

17

500" 400"

Ω1"

Ω2"

300"

"

S2"

200" 100"

Ω0"

0" 0"

S1"

100" 200" 300" 400" 500" 600"

Time)(weeks))

Enrollment Thresholds Curves for Various σ’s for Finite MEP with T = 10.5.

Proposition 9 For i = 1, . . . , ζ, (1) there exists Si (K) > 0 such that J(Si , K, ui ) = J(Si , K, ui−1 ) for K = 0, . . . , N − 1 and (2) ∂Si /∂K ≥ 0. The result implies that there is a region where the firm will operate at ui and will switch to ui−1 if there is insufficient enrollment. Further, the proposition implies that as the firm enrolls more patients, the switching time to a lower enrollment rate increases, i.e., successful recruiting implies the firm is further away from switching to a lower recruiting rate. Example We consider now a higher value drug than that in Section 4.2. Here the market exclusionary period is bounded by T = 10 years of patent protection. (We let E[Y ] = 0.56, σ = 9.5, annual interest rate r = 0.08. Other parameter values are given in Table 2 under Vfend). Figure 4 shows the switching times at different levels of enrollment for four different σ’s of the enrollment p process. As σ increases from 4.7 to 28.5 ( patients/week), the curve of S2 moves to the left while the curve of S1 stays relatively stationary. This leads to a growth in the size of region Ω1 . Recall region Ω1 represents the region where the intermediate investment rate is used. Thus for low uncertainty in the enrollment rate, there are essentially two regions – high investment at u ¯ or no investment (). As the uncertainty grows, the firm would use the intermediate investment rate, u1 , if there is insufficient enrollment. We display the value of the clinical study as a function of σ and E[Y ] in Figure 5. We observe that the value is a convex function in σ achieving a minimum at σ = 12. That the value is minimized for a given σ indicates there are two effects of the demand uncertainty. For values less than the

Kouvelis, Milner, and Tian: Clinical Trials for New Drug Developmen Article submitted to Manufacturing & Service Operations Management; manuscript no.

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550

400

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200 450 10

Figure 5

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Clinical Study Value vs. Enrollment Volatility and Drug Quality for Finite MEP with T = 10.5 with No Interim Analysis (Solid Line) and One Interim Analysis (Dashed Line).

minimum, a reduction in the variability allows for greater certainty that the program will terminate successfully, raising the value of the study. For values higher than the minimum, the firm is more likely to exercise its option to abandon the study earlier, again raising the value of the study.

5.

Optimal Investment with Interim Analysis

In this section we consider the investment problem for a Phase III clinical trial where an interim analysis of a drug’s quality can be made. At the interim, the firm uses updated information on quality to determine whether to suspend the study, submit the NDA earlier, or continue the study. We focus on a study with one such analysis, noting that it is possible in some clinical studies for a firm to test the hypothesis several times. The methodologies we develop would apply in such cases, but for ease of presentation we consider just one. In this section we first formulate the optimal investment problem. We then develop a solution for the case with exponentially distributed revenue generating periods. Finally we present an algorithmic solution to the general case. 5.1.

Model

We model the problem as a two-stage dynamic program where in each stage we solve a continuous dynamic program analogous to that in Section 4. In the first stage the firm recruits patients until n1 are enrolled (where n1 is defined by the clinical hypothesis). At that time an observation is made about the quality of the drug. If there is sufficient positive evidence of such the firm will terminate the trial and file the NDA early. Similarly, if the observed quality is sufficiently low, the firm will abandon the clinical study. Otherwise the firm proceeds with the study until N patients are enrolled or time T , which ever comes first, as in Section 4. However, the firm should now use updated information based on the observation. We divide the horizon [0,T] into two intervals. The first interval starts at time 0 and ends at (random) time τ1 , when n1 patients are enrolled (unless τ1 = T ). The second interval is from τ1 to

Kouvelis, Milner, and Tian: Clinical Trials for New Drug Developmen Article submitted to Manufacturing & Service Operations Management; manuscript no.

19

T . Paralleling the development in Section 4, let O1 = (−∞, n1 ) and Q1 = [0, T ) × O1 . As before, let K(0) = 0. Then τ1 = inf {s : s ≥ 0, (s, K(s)) 6∈ Q1 } Similarly, let O2 = (−∞, N ) and Q2 = [τ1 , T ) × O2 . Then τ2 = inf {s : s ≥ τ1 , (s, K(s)) 6∈ Q2 }. If τ1 < T , let ξi , for i ∈ {1, . . . , n1 }, be the observations on the quality of the drug for the first n1 patients made at time τ1 and let ξ1 be vector of ξi0 s. We introduce a test statistic for the drug’s quality given as g(ξ1 ) for some function g(·). We assume there exists some b1 such that if g(ξ1 ) ≥ b1 the firm will terminate the Phase III trial and file the NDA early. Similarly we assume there is a value a1 such that if g(ξ1 ) ≤ a1 , the trial is abandoned and no NDA is filed. The values of a1 and b1 are typically defined by the test statistics. Because the suspension of the clinical trial is a financial decision, even if the drug surpasses the minimum clinical performance indicated by a1 , it is possible the firm will suspend the clinical trial based on the outcome of the interim analysis. That is, there may also exist an a01 > a1 , for which the firm would suspend the trial because of its beliefs about the drug’s performance in the market. We develop these ideas below. Suppose the firm continues the trial at τ1 < T . It would face the same problem as that in Section 4, however, now defined on the interval [τ1 , T ] starting with n1 observations and the updated distribution H1 (y |ξ1 ). If subsequently the firm completes the study at time τ2 > τ1 , let Γ2 (τ2 |ξ1 ) be the present value at τ2 of the contribution received going forward from that point:

Γ2 (τ2 |ξ1 ) = E[Λ1 |ξ1 ](1 − e−r(T −τ2 ) ), where conditional random variable Λ1 |ξ1 = κ(ξ1 )π(ξ1 )r−1 e−r(lr +la ) . Here we let π(ξ1 ) be the conditional probability of approval and κ(ξ1 ) be the (stochastic) market size given ξ1 . Without loss of generality, we assume that κ(ξ1 ) = κ0 Y1 (ξ1 ) where Y1 (ξ1 ) be the conditional random variable for the drug’s market quality given ξ1 and κ0 is a constant. In particular, h1 (y |ξ1 ), the pdf for Y1 (ξ1 ) is given by Bayes’ Theorem. As above, the firm would incur cost at rate c(u) while conducting the study. Similarly if it failed to complete the study by time T , it would incur cost Ψ(T ). Let the expected present value of the

Kouvelis, Milner, and Tian: Clinical Trials for New Drug Developmen Article submitted to Manufacturing & Service Operations Management; manuscript no.

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profit of continuing the study at time t ≥ τ1 be Jc (t, K(t), ξ1 ; u) for u(·) ∈ At . (Here the subscript ‘c’ stands for continue.) τ2

 Z Jc (t, K(t), ξ1 ; u) = E −

e

−r(s−t)

c(u) ds + e

−r(τ2 −t)

−r(T −t)

Γ2 (τ2 |ξ1 ) − e

 Ψ(T ) · χτ2 =T

t

where the expectation is taken over τ2 . Then the optimal investment when continuing the study at time t for t ≥ τ1 is given by the solution to the control problem (P I)c :

Gc (t, K(t), ξ1 ) = sup Jc (t, K(t), ξ1 ; u) u∈At p s.t. dK(t) = u(t)dt + σ u(t) dw(t), K(τ1 ) = n1 .

Note that the patient enrollment process in the second interval is still a Brownian motion by the Strong Markov Property (Karatzas and Shreve (2000)). We can solve (P I)c using the methods of Section 4. Thus at time τ1 , the present value of the profit for continuing the clinical trial is given by Γc (τ1 |ξ1 ) = Gc (τ1 , n1 , ξ1 ). ˜ 1 ) by abandoning the study and selling the patent to Suppose at time τ1 the firm can earn Ψ(τ another firm. Based on the firm’s evaluation of the drug’s market potential, this may be worth ˜ 1 ). That is, a01 is more than continuing the study. Let a01 = minξ1 g(ξ1 ) subject to Γc (τ1 |ξ1 ) ≥ Ψ(τ the smallest value of the test statistic of the interim analysis for which the firm should continue the trial based on the economic value of the trial. In comparison, the value a1 , defined above, is the smallest value of the statistic for which the firm would continue the trial based solely on the clinical effectiveness of the drug. The difference between a1 and a01 highlights the idea behind our research: the decision regarding whether to continue a drug trial should depend on both the clinical effectiveness of the drug and the economic value of the drug. Letting a001 = max {a1 , a01 }, the firm would abandon the study if g(ξ1 ) < a001 . The threshold b1 for early termination depends on the quality of the drug used in the existing treatment. Hence, it is usually an increasing function of n1 . Following the development in Section 4, we assume that if the firm terminates the trial early because g(ξ1 ) ≥ b1 , it receives the expected present value of the contribution from time τ1 to T at τ1 given by Γp (τ1 |ξ1 ) = E[Λ1 |ξ1 ](1 − e−r(T −τ1 ) ). In the above equation, the subscript ‘p’ stands for passing the hypothesis test.

Kouvelis, Milner, and Tian: Clinical Trials for New Drug Developmen Article submitted to Manufacturing & Service Operations Management; manuscript no.

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Summarizing, the value of the dynamic program at the end of the first interval given ξ1 is  ˜ 1)  Ψ(τ g(ξ1 ) ≤ a001 Γ1 (τ1 |ξ1 ) = Γc (τ1 |ξ1 ) a001 < g(ξ1 ) < b1  Γp (τ1 |ξ1 ) g(ξ1 ) ≥ b1 Then we define the expected value of the second period profit at time τ1 as Γ1 (τ1 )   Γ1 (τ1 ) = EY0 Eξ1 |Y0 [Γ1 (τ1 |ξ1 )] where we explicitly take the expectation with respect to the prior distribution of Y0 and the conditional distribution of ξ1 |Y0 . Next we find the optimal investment in the first interval. As before, for a progressively measured process u(·) we let  Z J1 (t, K(t); u) = E −

τ1

e

−r(s−t)

c(u) ds + e

−r(τ1 −t)

Γ1 (τ1 ) − e

−r(T −t)

 Ψ(T ) · χτ1 =T

t

where the expectation is taken over τ1 . Finally we can define the control problem in the first interval for t ≥ 0 (P I)1 :

G1 (t, K(t)) = sup J1 (t, K(t); u), u∈At p s.t. dK(t) = u(t)dt + σ u(t) dw(t), K(0) = 0.

5.2.

Special Case: Exponentially Distributed Revenue Generating Period

Recently, pharmaceutical firms, facing a thinner pipeline of breakthrough drugs and the expiration of patents for older ones, have turned to developing multiple lower revenue drugs. Because of the potential loss from conducting complete clinical studies for these drugs, performing interim analyses for such drugs is increasingly important to ensure financial success. We consider such cases in this section, recalling for these drugs that the end of the horizon may be described as exponentially distributed. Following the case of a lower revenue drug without interim analysis, we let T¯ = T − τ2 and assume that T¯ is exponentially distributed with parameter λ. If the firm continues the study after the interim analysis, the expected revenue at time τ2 is Z Γ2 (τ2 |ξ1 ) = E[Λ1 |ξ1 ] 0

∞

r ¯ ¯ (1 − e−rT )λe−λT dT¯ = E[Λ1 |ξ1 ] . λ+r

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˜ 1 ). We assume Ψ(τ ˜ 1 ) is If the firm abandons the study after the interim, the firm will earn Ψ(τ independent of τ1 because T − τ1 is also exponentially distributed. Because of the infinite horizon, the expected profit for continuing the study, given policy u, at time t ≥ τ1 becomes τ2

 Z Jc (t, K(t), ξ1 ; u) = E −

−r(s−t)

e

c(u) ds + e

−r(τ2 −t)

t

 r E[Λ1 |ξ1 ] . λ+r

We solve Problem (P I)c with the above objective function using the method that we developed ∗ in Section 4.2. Let K2,i be the threshold for investment at rate ui in the second period. We define ∗ ∗ K2,ζ+1 = N and G(K2,ζ+1 ; uζ ) = E[Λ1 |ξ1 ]r/(λ + r). By Proposition 4, Gc (K, ξ1 ) = Gc (K, ξ1 ; ui−1 ) if ∗ ∗ ∗ K2,i−1 < K ≤ K2,i for i = 1, · · · , ζ and Gc (K, ξ1 ) = Gc (K, ξ1 ; uζ ) if K > K2,ζ . In the above expres-

sions, Gc (K, ξ1 ; ui ) is the value of continuing the study at enrollment rate ui and is given by Gc (K, ξ1 ; ui ) =

∗ exp((K2,i+1

 − K)γ(ui ))

∗ G(K2,i+1 , ξ1 ; ui+1 ) +

c(ui ) r

 −

c(ui ) r

(11)

∗ iteratively. According to Proposition 4, the thresholds Similarly, one can calculate the value of K2,i

for the different enrollment rates are ∗ ∗ K2,i = K2,i+1 −

  1 θi ui ln , for i = 1, . . . , ζ. ∗ γ(ui ) rG(K2,i+1 , ξ1 ; ui+1 ) + c(ui )

(12)

∗ decreases and Gc (K, ξ1 ; ui ) increases as g(ξ1 ) increases if π is a linear funcProposition 10 K2,i

tion of y1 and if the prior distribution, H0 is Beta and the test statistic g(ξ1 ) has a binomial distribution, or if H0 and g(ξ1 ) have normal distributions. The proposition implies that as the observed quality of the drug increases, the optimal investment rate increases as well. If the firm terminates and submits the NDA after the interim analysis, it earns revenue in an exponentially distributed time interval T − τ1 , which is given by Z Γp (τ1 |ξ1 ) = E[Λ1 |ξ1 ]

∞

¯ ¯ (1 − e−rT )λe−λT dT¯

0

= E[Λ1 |ξ1 ]

r . λ+r

We next solve Problem (P I)1 to determine the investment policy prior to the interim analysis. Let φ(z |y0 ) be the probability mass or density function of the sufficient statistic z = g(ξ |y0 ). Conditional ∗ ∗ ∗ on z, the firm will use ui if K2,i < n1 ≤ K2,i+1 for i = 0, 1, · · · , ζ and use uζ if n1 > K2,ζ . By

Proposition 10, the above inequalities uniquely determine the threshold values of z where the

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optimal policy defined in (12) switches from ui−1 to ui in the continuing policy. We denote such z 0 s as zi for i = 1, · · · , ζ. That is, the zi can be found by searching over [a001 , b1 ] such that exp[(N − n1 )γ(uζ )] [rΓ2 (τ2 ) + c(uζ )] = θζ uζ , for i = ζ,

(13)

θi , for i = 1, · · · , ζ − 1, θi+1

(14)

∗ exp[(K2,i+1 − n1 )γ(ui )] =

∗ where Γ2 (τ2 ) and K2,i+1 are functions of the sufficient statistic z. The expected value of the second

period profit is given by " Z 00 Z a1 ˜ 1 )φ(ξ1 ) dz + Γ1 (τ1 ) = EY Ψ(τ 0

b1

+

Z

z2

Gc (n1 , ξ1 ; u1 )φ(ξ1 ) dz + · · ·

Gc (n1 , ξ1 ; u0 )φ(ξ1 ) dz +

a00 1

0

Z

z1

Z

z1

Γp (τ1 |ξ1 )φ(ξ1 ) dz .

Gc (n1 , ξ1 ; uζ )φ(ξ1 ) dz + zζ

#

z¯

(15)

b1

where z¯ is the maximum value g(ξ1 ) can attain. Proposition 11 For the case of an exponentially distributed revenue period, Γ1 (τ ) does not depend on τ1 . Hence, Problem (P I)1 becomes an infinite horizon dynamic program. Thus we can similarly solve ∗ ∗ (P I)1 using the method in Section 4.2. We define K1,ζ+1 = n1 and G1 (K1,ζ+1 ; uζ+1 ) = Γ1 (τ )r/(λ+r). ∗ < According to Proposition 4, the value function of this problem is G1 (K) = G1 (K; ui−1 ) if K1,i−1 ∗ ∗ K ≤ K1,i and G1 (K) = G1 (K; uζ ) if K > K1,ζ , where G1 (K; ui ) is given by

G1 (K; ui ) =

∗ exp((K1,i+1

 − K)γ(ui ))

∗ rG1 (K1,i+1 ; ui+1 ) +

c(ui ) r

 −

c(ui ) , for i = 0, 1, . . . , ζ. (16) r

One can determine the optimal policy by comparing LG1 with θi . According to Proposition 4, the thresholds using different levels of enrollment rates are ∗ K1,i

=

∗ K1,i+1

  1 θi ui , for i = 1, . . . , ζ. − ln ∗ γ(ui ) rG1 (K1,i+1 ; ui+1 ) + c(ui )

(17)

Lastly, we determine when a firm should conduct an interim analysis. Recall the firm has to specify in the clinical study protocol the number of enrollees, n1 , at which it will conduct the interim analysis. This protocol must be approved by the FDA prior to enrolling patients in the clinical study. Hence, the firm wants to choose an n1 so that the value of developing the drug, G1 (0), is maximized. We let [0, ξ] be the support of ξ 0 s. (E.g., if g(ξ1 ) has a binomial distribution, ξ = 1.) We need the following notation to introduce the condition for inclusion of an interim analysis. Let

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Γp be the upper bound of Γp for all the realizations of ξ1 . For Y0 with beta distribution, we let α and β be the distribution parameters; for Y0 and ξi with normal distributions, we let ς and ς0 be the precisions of the Y0 and ξi , i = 1, · · · , ζ. Assuming that the FDA approval rate, π, does not depend on ξ1 , we have the following results which the pharmaceutical firms can determine whether an interim is needed in their clinical testing. Proposition 12 Let % = (1 + r−1 c(uζ )/Γp )−1 . 1. If the prior distribution, H0 , is Beta, and the test statistic g(ξ1 ) has a binomial distribution: 00

00

For n1 < a1 , G1 (0) is a decreasing function of n1 ; for n1 ≥ a1 , G1 (0) is a decreasing function of n1 if −γ(uζ )(α + β) > % and is a unimodal function otherwise. 2. If the prior distribution, H0 and the test statistic g(ξ1 ) have normal distributions: For n1 ξ < a00 1 , G1 (0) is a decreasing function of n1 ; for n1 ξ ≥ a00 1 , G1 (0) is a decreasing function of n1 if −γ(uζ )(ς 2 /ς02 ) > % and is a unimodal function n1 otherwise. The proposition implies an interim analysis is more valuable for drugs with higher enrollment volatility and/or higher quality. Proposition 12 also indicates that there is an optimal sample size for the interim analysis. We find the optimal n1 through a bi-section search on the appropriate interval defined by the proposition. The resulting value need only be compared to the case where n1 = 0, i.e., the case without an interim analysis. (We use the assumption that π is independent of ξ1 to facilitate the proof. We observe that the value of a clinical study is a unimodal function of n1 for the case where π depends on ξ1 in a numerical example in Section 6.) Example Continuing the example introduced in Section 4.2, we now allow the firm to conduct one interim analysis. Figure 3 shows how the optimal value of the clinical study, G1 (0), changes with σ and E[Y0 ]. As expected, the value with an interim analysis is at least as great as without. In agreement with Proposition 12, it is monotonically increasing and convex in σ and E[Y0 ]. We note that for E[Y0 ] around 0.4 it has little value (any value stems from the ability to sell the patent early). For E[Y0 ] between 0.4 and 0.6, we note (but do not show) that the interim analysis should be conducted when around 50% of required patients are enrolled and treated. For E[Y0 ] above 0.6, the number is around 20% of the required patients, leading to an increase in the drug’s value. 5.3.

Algorithmic Solution to the General Case with an Interim Analysis

The general case can be solved by applying Algorithm 1 in the second period to solve Problem (P I)c for each realization of g(ξ) and then, by backwards induction, Algorithm 1 can again be

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used to solve Problem (P I)1 . That is, for each a001 < g(ξ1 ) < b1 , we need to find the value of Γ(τ1 |ξ1 ) = Gc (τ1 , n1 , ξ1 ). We observe in the following proposition, that we only need to execute Algorithm 1 once for each ξ1 – doing so will provide Γ(τ1 |ξ1 ) for all τ1 . This considerably reduces the dimensionality of the problem. Because in a typical clinical drug study there would only be one or two interim analyses, backwards induction is very feasible with respect to run-times. Proposition 13 Problem (P I)c can be found for all realization of τ1 by solving one instance of Problem (P I)c . The details of Algorithm 2 that solves the general case are given in Appendix A. We observe that there exists an n1 that maximizes G1 (0, 0), the value of the clinical study. We compute the optimal n1 through a bi-section method. As noted, firms have to specify the point at which to conduct an interim analysis in the clinical study protocol for testing a new drug in Phase III prior to beginning the clinical testing, and this plan must be approved by the FDA. Currently, firms are able to determine a wide window during which it would be advantageous to conduct an interim analysis based on statistical methods related to the hypotheses being tested on a drug’s efficacy. However, through our method, a firm can determine the optimal point for an interim analysis taking into account of the uncertainty of the patient enrollment process, clinical study cost, and future revenue. Our method provides a refined decision aid from a method taking into account of only the statistical validity of the clinical results. Example We continue the example with a finite MEP with T = 10.5 years as in Section 4.3, now including an interim analysis. We present the clinical study value for different levels of σ in the left panel in Figure 5. We observe a similar pattern as in the case without an interim analysis. The clinp ical study value first decreases and then increases as σ increases from 4.7 to 28.5 patients/week. p We observe a minimum value at σ = 11 patients/week. We note that adding an interim analysis has lowered the value of σ that defines the minimum point from above. The introduction of an interim analysis raises the option value to terminate a study early (either by submitting the NDA or by abandoning the study). As we noted the function is convex because for lower values of σ, the study’s value derives from the greater certainty completion of the study, whereas there is greater option value with greater uncertainty. As the latter increases, the minimizing value of σ naturally decreases. We also display how the clinical study value changes with the quality of the drug (E[Y0 ]) in the right panel in Figure 5. For the case with a finite MEP, we observe there is always a positive value for allowing an interim analysis.

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Max. Study Required Annual Length Sample Revenue Expected Approval Cost Enrollment T Size κ Quality Prob. Parameter Rate Drug (Weeks) N ($ M) Y0 π θ1 θ2 u1 u2 Vfend 546 564 237 0.56 0.80 0.463 0.834 2.55 4.64 Ellence 520 716 280 0.55 0.80 0.136 0.587 2.30 3.44 Xanax XR 286 492 331 0.50 0.80 0.323 0.903 5.36 9.66 Chantix/ 650 2052 867 0.51 0.75 0.055 0.235 17.76 26.64 Champix Table 2

6.

Opt. Exp. Value ($ M) 434 467 827 3490

Data for Pfizer’s New Drug Development and the Optimal Expected Discounted Value.

Application to Pfizer’s New Drug Development

In this section we apply our policies to data for several drugs recently developed by Pfizer. We consider Ellence which treats breast cancer, Xanax XR which treats panic disorder, Vfend, an anti-fungal drug, and Chantix/Champix which helps adults stop smoking. To do so, we assume ζ = 2, i.e., there are two non-zero enrollment rates, and estimate appropriate enrollment rates in patients per week and their associated costs for these two levels in $M/patient enrolled. For each drug, we use data on revenues and patent protection terms as given in Pfizer’s annual financial reports, and patient enrollment, start and completion times, treatment arms, clinical study results, and approval dates of drugs provided by publicly accessible FDA databases. We estimate the costs of the clinical studies using data on the R&D expenditures for projects in various stages at Pfizer as given in their 10-K filings from 1995 to 2010, and by using estimates of R&D expenditures for clinical studies in pharmaceutical industry presented by the Congressional Budget Office (CBO 2006). Table 2 presents the parameters estimated for the drugs. For all cases we assume the annual interest rate r = 0.08. While not provided by Pfizer, subsequent discussions with Pfizer executives indicated that these are reasonable estimates of their true operational costs and quality estimates. 6.1.

Optimal Switching Policy without Interim Analysis

Using Algorithm 1 presented in Section 4 we determine the optimal value and threshold enrollment policies for each drug. The value of the drugs, in increasing order, are presented in Table 2. The threshold policies are presented in Figure 6. As before Ω2 designates the region for the higher enrollment rate, Ω1 , the lower rate, and Ω0 the region where investment in further enrollment is not justified. We also indicate the values of N and T , and thresholds K1∗ (t) and K2∗ (t). For example, for Ellence, we observe that the higher enrollment rate of 3.44 patients per week is justified initially and maintained until either 716 patients are enrolled or the process enters Ω1 , e.g., by recruiting only 200 patients by week 320. The lower enrollment rate of 2.3 would then be used. As discussed this would typically be achieved by closing test sites.

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We observe several behaviors of the optimal policy. First, as the potential market value of the drug increases we observe the size of region Ω1 decreases. This indicates that with increased market value, there is little value in maintaining a test below the maximum enrollment rate. Similarly we observe the slope of the thresholds becomes more vertical as the value of the drug increases. Defining m(t), the relative slope of K2∗ , as ( dK(t)/N )/( dt/T ), we observe that approximate values, mVfend = 2.56, mEllence = 2.77, mXanaxXR = 2.95, and mChantix = 8.44. The higher numbers for the more valuable drugs indicate that there is essentially a time barrier between the region where further testing is profitable and that where the study should be discontinued. This is in contrast to low value drugs (those less than $100 M per year in expected revenues) where as discussed in Section 4.2 as modeled by an exponentially distributed revenue generating period, the optimal policy is invariant in time. That is, the slope of the threshold is zero, and switching to a higher (or lower) enrollment depends on the amount of enrollment completed. The intermediate value drugs (Vfend and Ellence) show a slope that is at an intermediate value. Finally we observe that the thresholds K1∗ and K2∗ and the terminal time approach each other at N . This combined with the previous observations indicates that the firm should suspend the clinical study of a more valuable drug later than that of a less valuable one, ceteris paribus, as is intuitive. We conclude from these observations that for high value drugs, firms should never change the investment rate, just stop when time runs out. In contrast, for low value drugs (with exponentially long exclusionary periods), firms should continue to invest, just at a higher rate as a study nears completion. For intermediate-valued drugs, the general optimal policy is of greater interest as these simple rules of thumb will fail. Pharmaceutical firms can use Algorithm 1 to compute the thresholds that determine the number of test sites to use to reach the next milestone of an on-going clinical study. Next we consider how drug quality affects the nature of the optimal policy. To demonstrate, we vary E[Y ] for Ellence, setting it equal to 0.1, 0.2, 0.4 and 0.8. The results are shown in Figure 7. We observe with very low quality (e.g., E[Y]=0.1), investment at level u2 is not justified. With increasing quality the area in region 2 increases as does the slope of thresholds K1∗ (t) and K2∗ (t). Thus the firm can justify increasing the initial number of test centers with increasing quality, and maintain those test centers longer in the face of low enrollment. For a drug with intermediate quality (e.g., E[Y] = 0.2) the firm should initiate investment at a nominal level (sufficient to recruit patients at rate u1 ), and increase it if enough patients have been recruited with adequate time

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Vfend)

Ω1"

Ω2"

400"

Ellence) S2" S1"

300" 200"

800" N" 700" 600" 500" 400" 300" 200" 100" 0" 0"

Enrollment)

Enrollment)

600" N" 500"

Ω0"

100" 0" 0"

100"

200"

300"

400"

T" 500"

600"

S2"

Ω2"

S2" S1"

Ω2"

200" 100"

400"

T" 500"

600"

100"

150"

200"

1500"

S2"

Ω2"

1000"

S1"

Ω1"

500"

Ω0" 50"

Enrollment)

Enrollment)

Ω1"

300"

Ω0"

0"

250" T" 300"

0"

Time)(Weeks))

100"

200"

300"

400"

500"

600"

Time)(weeks))

T"

700"

Optimal Threshold Policies for Vfend, Ellence, Xanax XR, and Chantix

Quality)=)0.10)

Quality)=)0.20) 800"

600"

Ω1"

400"

S2"

Ω0"

200"

S1"

Enrollment)

Enrollment)

800"

0" 0"

200"

400"

600"

Ω1"

400"

S1"

0"

600"

0"

200"

400"

600"

Time)(weeks))

Quality)=)0.40)

Quality)=)0.80)

800"

800"

600"

Ω2"

400"

S2"

200"

S1"

Ω1" Ω 0"

0" 0"

200"

400"

600"

Enrollment)

Enrollment)

S2"

Ω0"

200"

Time)(weeks))

600"

Ω2"

400"

S2"

200" 0" 0"

Time)(weeks))

Figure 7

300"

Chan4x/Champix)

N"

0"

Figure 6

200"

2000"

400"

0"

S1"

Time)(weeks))

Xanax)XR)

N"

"

Ω0" 100"

Time)(Weeks))

500"

Ω1"

200"

Ω1"

400"

S1"

Ω0" 600"

Time)(weeks))

Optimal Threshold Policies for Ellence as Drug Quality Changes

remaining until the study’s completion. Moderate quality reduces the expected value of a drug so that the firm should initially invest conservatively. 6.2.

Optimal Use and Value of Interim Analysis

Next we investigate the point during a clinical study at which the firm should implement an interim analysis, if it is to do so. As discussed, this point is given as a pre-set number of patients, n1 , that must be enrolled when an analysis of the quality of the drug would be undertaken. To illustrate how n1 may be chosen, and the associated value of doing so, we consider the drug Vfend. For the case of a clinical study without an interim analysis, we found the expected value

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Optimal Value vs. Interim Sample Size

Optimal Value ($MM)

700

$130 MM

600 500 400 300 200 100

0.00

0.20

0.40

0.60

0.80

1.00

n1/N Continuation

Figure 8

Abandonment & Early Termination

Optimal Values of Clinical Study for Vfend with an Interim Analysis

of the optimal policy to be $434M. However, if Pfizer conducts an interim analysis, based on the finding, the firm could either abandon the study, terminate the study and file an NDA early, or continue with an updated estimate of the drug quality. In Figure 8 we present the total value as a function of the n1 . Alternatively, suppose that Pfizer conducts an interim analysis after observing n1 patients and updates the quality of the drug, but does not terminate the study early based on the results. Rather, suppose it proceeds with an appropriate level of investment based on the updated quality (which could include suspending the study if the recruitment level were below the threshold K1∗ (t).) We show the value of this policy, referred to as “continuation” in the figure using the dashed line. We observe the case allowing early termination provides additional value as n1 increases, reaching a maximum at approximately 37% of the sample size with an increase of $130M over the continued investment policy. This represents the option value derived from both the significant savings in conducting the clinical trial and by increasing the MEP, in this case by 1.5 years on average. This shows that such interim analyses are of significant value. According to FDA (2000), without considering the market value and drug quality as we do in this work, the firm would generally conduct an interim analysis after observing between 30% and 70% of the total sample N . While choosing 30% would have lead to a small expected reduction in value compared with the optimal, choosing to wait until 70% was observed would have resulted in the loss of nearly all of the option value of early termination. This demonstrates the contribution of considering the market and drug quality in choosing the interim analysis terms.

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7.

Conclusions

Currently pharmaceutical firms determine how much to invest in on-going, Phase III clinical studies using rules of thumb focused on achieving the targeted sample size by a stated date. In this paper, we develop methods to control such investment based on the progress of a clinical trial in attracting subjects, the results of any interim analyses made, and the likely market value of a drug if it were to be approved. We show that the optimal policy specifies targeted patient enrollment rates defined by time dependent thresholds. We obtain closed-form expressions for the thresholds for the case where revenues are generated during an exponentially distributed period. We argue such is the case for drugs with low annual revenue. For the general case, we develop an algorithm for solving the optimal investment problem. We also provide results on when an interim analysis of the collected data is useful. These methods can help pharmaceutical firms optimize their investment in late-stage, new drug development, and, in turn, reduce the drug development cost. Through extensive numerical experiments using data from clinical studies conducted by Pfizer, we generate managerial insights into the optimal investment rate for clinical studies. We emphasize the difference in the investment policy for high versus moderate and low expected revenue drugs. For the latter, our analysis recommends a conservative approach, setting the initial investment at a nominal level and increasing the level when substantial success in patient enrollment has occurred well before the study’s completion time. In contrast, a high expected-revenue drug requires an aggressive clinical investment early. Typically, this will continue until the successful completion of the study or the study is abandoned, in contrast to continuing the study at some intermediate investment level. Increasing quality levels lead to similar implications for threshold levels of the optimal policy: greater quality implies high initial investment, for a longer time, with low enrollment leading to a study’s abandonment rather than reduced investment. Our work clearly illustrates the value of an interim data analysis informed by the market economics of a drug. We calculate the desired time for an interim analysis and accurately quantify its option value. The option value, derived from expediting an NDA or abandoning earlier a failing study, can be significant (over $100 million in our example). In summary, our methodologies can help a pharmaceutical to better design a clinical study both in terms of allocating resources to attract qualified patients and on the appropriate use of interim data analysis.

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Appendix. Algorithm 2 Let M1 and M2 be the numbers of intervals that divide the supports of Y0 and φ(ξ1 ). In order to reduce the computation time we divide the supports into the intervals with different lengths. Let ∆ym1 be the length of interval m1 on [y, y] and ∆zm2 be the length of interval m2 on [z, z]. We note that ∆zi is a function of ∆yi . Step 1. Initialize ∆ym1 for m1 = 1, . . . , M1 and set m1 = 0 and y0,m1 = y. Step 2. Set m1 = m1 + 1 and y0,m1 = y0,m1 −1 + ∆ym1 . Step 3. Compute p1,m1 = P r{Y0 ≤ y0,m1 } − P r{Y0 ≤ y0,m1 −1 }. Step 4. Initialize ∆zm2 for m2 = 1, . . . , M2 and set m2 = 0 and zm2 = z. Step 5. Set m2 = m2 + 1 and zm2 = zm2 −1 + ∆zm2 . Step 6. Compute p2,m2 = P r{g(ξ1 ) ≤ zm2 } − P r{g(ξ1 ) ≤ zm2 −1 }. 00

Step 7. If g(ξ1 ) ≤ a1 , then set Γ1 (τ1 |ξ1 ) = −Ψ(τ1 ) and go to Step 5. Step 8. If g(ξ1 ) ≥ b1 , then set Γ1 (τ1 |ξ1 ) = Γ(τ1 |ξ1 ) and go to Step 5. Step 9. Compute E[Λ1 |ξ1 ], solve Problem (P I)c using Algorithm 1, and set Γ1 (τ1 |ξ1 ) = Gc (τ1 , n1 , ξ1 ). Step 10. If m2 ≤ M2 go to Step 5. Step 11. If m1 ≤ M1 go to Step 2. Step 12. Compute E[Γ1 (τ1 |ξ1 )] and solve Problem (P I)1 using Algorithm 1.

References CBO. 2006. A CBO Study: Research and Development in the Pharmaceutical Industry. Congressional Budget Office. Dutta, P. K. 1997. Optimal management of an R&D budget. Journal of Economic Dynamics and Control 21 575–602. Eisenberg, R. S. 2003. Patents, product exclusivity, and information dissemination: How law directs biopharmaceutical research and development. Fordham L. Rev. 72 477. FDA. 2000. Statistical evaluation and review: Application number, 21-266 and 21-267. Federal Food and Drug Administration . Folks, J. L., R. S. Chhikara. 1978. The inverse gaussian distribution and its statistical application-a review. J. R. Statist. Soc. B 40(3) 263–289. Girotra, K., C. Terwiesch, K. T. Ulrich. 2007. Valuing R&D projects in a portfolio: evidence from the pharmaceutical industry. Management Science 50(9) 1452–1466. Grabowski, H., G. Long, R. Mortimer. 2013. Recent trends in brand-name and generic drug competition. Journal of Medical Economics 1–8.

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Grabowski, H. G., M. Kyle. 2007. Generic competition and market exclusivity periods in pharmaceuticals. Manage. Decis. Econ. 28(4) 491–502. IFPMA. 2011. The pharmaceutical industry and global health: Facts and figures. International Federation of Pharmaceutical Manufacturers and Associations . Jacob, W. F., Y. H. Kwak. 2003. In search of innovative techniques to evaluate pharmaceutical R&D projects. Technovation 23 291–296. Karatzas, I., S. Shreve. 2000. Brownian Motion and Stochastic Calculus. Springer. Karlin, S., H. E. Taylor. 1975. A first course in stochastic processes. Academic Press. Kellogg, D., J. M. Charnes. 2000. Real-options valuation for a biotechnology company. Financial Analysts Journal May/June 76–84. Lai, Dejian. 2001. Brownian motion and long-term clinical trial recruitment. Journal of Statistical Planning and Inference 93(1-2) 239–246. Lucas, R. E. 1971. Optimal management of a research and development project. Management Science 17(11) 679–697. McDonald, R., S. Daniel. 1986. The value of waiting to invest. The Quarterly Journal of Economics 17(4) 707–728. Pfizer. 2006. 2005 financial report. PFIZER INC . PhRMA. 2011. The pharmaceutical industry profile. Tech. rep., Pharmaceutical Research and Manufacturers of America. Pindyck, R. S. 1993. Investments of uncertain cost. Journal of Financial Economics 34(1) 511–530. Schwartz, E. S. 2003. Patent and R&D as real options. National Bureau of Economic Research . Sethi, S., G. Thompson. 2000. Optimal Control Theory: Applications to Management Science and Economics. Kluwer Academic Publishers. TCSDD. 2010. Outlook 2010. Tech. rep., Tufts Center for the Study of Drug Development.

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Appendix. Proofs Proof of Proposition 1. cally decreasing in τ . By Proof of Proposition 2.

Γ(τ ) = E[Λ](1 − e−r(T −τ ) ). By dΓ/ dτ = −E[Λ]re−r(T −τ < 0, Γ(τ ) is monotoni2

d Γ dτ 2

= −E[Λ]r2 e−r(T −τ ) < 0, Γ(τ ) is strictly concave in τ .



For any u ∈ U , the HJB equation (4)  2  ∂G ∂G σ ∂2G + u − rG + max u − c(u) =0 u∈U ∂t 2 ∂K 2 ∂K

is uniformly parabolic. We now verify that condition (4.1) of Theorem IV.4.1 in ? holds for O = OM = [−M, N ] where M is an arbitrary positive real number. Since the set U is a closed interval on the real line, it is compact. By the definition of OM , it is bounded with ∂OM , which is a manifold of class C (3) . In the state p transition equation dK(t) = u(t) dt + σ u(t) dw(t), both the coefficient of dt and that of dw have continuous partial derivatives with respect to t. In addition, both Γ and Ψ are three times differentiable with respect to K on the closure of [0, T ) ∈ R. Hence, condition (4.1) of Theorem IV.4.1 holds. By Theorem IV.4.1, we prove Proposition 2. Since M is an arbitrary real number, the proposition is still true when M goes to ∞.  Proof of Proposition 3.

We obtain u∗ (t) by solving the maximization problem on the left hand side of

(4). Substituting c(u) into (4) and rearranging terms, we obtain rG = max [LG − θ1 ] u + θ1 u0 − c(u0 ) + u∈U

∂G , for u0 ≤ u ≤ u1 ∂t

(18a)

... ∂G , for ui−1 < u ≤ ui ∂t ∂G rG = max [LG − θi+1 ] u + θi+1 ui − c(ui ) + , for ui < u ≤ ui+1 . u∈U ∂t rG = max [LG − θi ] u + θi ui−1 − c(ui−1 ) + u∈U

(18b) (18c)

We solve the linear program on the right hand sides of (18a) – (18c) to obtain, u∗ (t) = u0 , if LG ≤ θ1 ,

(19a)

u∗ (t) = ui , if θi < LG ≤ θi+1 , for i = 1, · · · , ζ − 1,

(19b)

u∗ (t) = uζ , if LG > θζ .

(19c)

Equation (18a) – (18c) have free boundaries at Ki∗ (t) that are given by LG(t, Ki∗ ) = θi , for i = 1, 2, . . . , ζ.

(20)

We rewrite conditions (19a) and (19b) in terms of the thresholds of patient enrollment as: u∗ (t) = ui if ∗ ∗ Ki− 1 (t) < K(t) ≤ Ki (t) for i = 1, 2, . . . , ζ.

We next prove that Ki∗ (t) 6= Ki∗+1 (t), ∀t ∈ [0, T ), by contradiction. Assume that the two boundaries intersect at some time s, Ki∗ (s) = Ki∗+1 (s) = K ∗ (s), ∀t ∈ [0, T ). By (20), we have σ 2 ∂ 2 G(s, Ki∗ ) ∂G(s, Ki∗ ) + = θi 2 ∂K 2 ∂K σ 2 ∂ 2 G(s, Ki∗+1 ) ∂G(s, Ki∗+1 ) + = θi+1 2 ∂K 2 ∂K The above two equations contradict that θi < θi+1 . This contradiction establishes that Ki∗ (t) 6= Ki∗+1 (t).



Kouvelis, Milner, and Tian: Clinical Trials for New Drug Developmen Article submitted to Manufacturing & Service Operations Management; manuscript no.

34 Proof of Proposition 4.

We first obtain the complementary solution to (8a). Letting G(K) = ezK , we

write the characteristic equation for the homogenous equation of (8a) as 0.5σ 2 ui z 2 + ui z − r = 0 for i = 0, 1, · · · , ζ. (21) √ √ 2 −ui − u2 +2rui σ 2 −ui + ui +2rui σ 2 i and z2 (ui ) = . We next determine a particular Solving (21), we get z1 (ui ) = ui σ 2 ui σ 2 solution to (21). Since c(ui ) is not a function of K, a particular solution to (21) is GP (K; ui ) = − c(ur i ) . Combining the above results, we have the general solution in the following form, G(K; ui ) = Ci ez1 K + Di ez2 K −

c(ui ) for i = 0, 1, · · · , ζ. r

(22)

We next determine the coefficient Di . For a constant control ui , we express J(K; ui ) in terms of state r + variable K as J(K; ui ) = { Eλ[Λ] +r

c(ui ) r

}E[e−r(τ −t) ] −

c(ui ) r

. The expectation, E[e−r(τ −t) ], is the moment

generating function of the passage time τ − t of Brownian motion (2b) given that K(t) patients are enrolled   up to time t. Using this result, we express J(K; ui ) as J(K; ui ) = exp(−(N − K)z1 ) (αΛ+rr) + c(ur i ) − c(ur i ) . The solution G(K; ui ) to (8a) corresponds to constant control ui and, hence, is in the same function form as J(K; ui ). This result implies that Di = 0. Lastly, we determine the coefficients Ci and Ki∗ by backward induction. By (8b), we have G(K; uζ ) = Cζ ez1 N −

c(ui ) r

=

E [Λ]r r +λ

γ(uζ ) = −z1 (uζ ), we obtain G(K; uζ ) = exp((N − K)γ(uζ )){

E [Λ]r r +λ

the derivatives of G(K; uζ ). Differentiating G(K; uζ ), we get 2

∂ G ∂K 2

= γ 2 (uζ ) exp[(N −K)γ(uζ )]{

r K)γ(uζ )]{ Er[Λ] + +λ

c(uζ ) r c(uζ ) r

r . Solving for Cζ , we get Cζ = e−z1 N { Er[Λ] + +λ

E [Λ]r r +λ

+

c(cζ ) r

∂G ∂K

+

}. Substituting in (22) and using

}−

c(uζ ) r

. To determine Kζ∗ we need c(cζ ) r

r = −γ(uζ ) exp[(N − K)γ(uζ )]{ Er[Λ] + +λ

},

}. Substituting in (8c), we obtain [0.5σ 2 γ 2 (uζ )−γ(uζ )] exp[(N −

c(cζ ) r

} = θζ . Using 0.5σ 2 uζ γ 2 (uζ ) − uζ γ(uζ ) − r = 0, we get the free boundary Kζ∗     1 θζ uζ 1 θζ uζ ∗ Kζ = N − ln =N − ln γ(uζ ) E[Λ]r2 /(λ + r) + c(uζ ) γ(uζ ) rG(Kζ∗+1 , uζ +1 ) + c(uζ +1 )

We assume that G(K; ui+1 ) = exp((Ki∗+2 − K)γ(ui+1 )){rG(Ki∗+2 ; ui+2 ) + G(K

∗ i+1

; ui ) = Ci e

∗ z1 Ki+1

−

c(ui ) r

= G(K

∗ i+1

c(ui+1 ) r

; ui+1 ). Solving for Ci , we get Ci = e

}−

∗ −z1 Ki+1

c(ui+1 ) r

. By (8c), we get

∗ i+1

[G(K

; ui+1 ) + c(ur i ) ].

Substituting in (22) and using γ(ui ) = −z1 (ui ), we obtain G(K; ui ) = exp((Ki∗+1 − K)γ(ui )){G(Ki∗+1 ; ui+1 ) + c(ui ) r

}−

c(ui ) r

. To determine Ki∗ we need the derivatives of G(K; ui ). Differentiating G(K; ui ), we get ∗ i+1

−γ(ui ) exp[(K c(ci ) r

− K)γ(ui )]{G(K

∗ i+1

; ui+1 ) + 2 2

c(ci ) r

},

2

∂ G ∂K 2

2

= γ (ui ) exp[(K ∗ i+1

}. Substituting in (8c), we obtain [0.5σ γ (ui ) − γ(ui )] exp[(K

∗ i+1

∗ i+1

− K)γ(ui )]{G(K

− K)γ(ui )]{G(K

∗ i+1

; ui+1 ) +

∂G ∂K

=

; ui+1 ) +

c(ci ) r

} = θi .

Using 0.5σ 2 ui γ 2 (ui ) − ui γ(ui ) − r = 0, we get the free boundary Ki∗   1 θi ui ∗ ∗ Ki = Ki+1 − ln . γ(ui ) rG(Ki∗+1 ; ui+1 ) + c(ui ) By induction, we prove the first part of Proposition 4. By Prop. 3, we prove the second part. Proof of Proposition 5 and Proposition 6. ∗ ζ



We prove these propositions by induction. We first prove that

2

1) K decreases as E[κ], E[π], and σ increase and increases in θζ ; 2) G(K; uζ ) increases as E[κ], E[π], and σ 2 increase and decreases as θζ increases. The derivatives of Kζ∗ with respect to E[κ], E[π], σ 2 , and θζ are ∂Kζ∗ 1 E[Λ]r2 /(α + r) + c(uζ ) θζ uζ ∂E[Λ] = , 2 2 ∂E[κ] γ(uζ ) θζ uζ [E[Λ]r /(λ + r) + c(uζ )] ∂E[κ]

Kouvelis, Milner, and Tian: Clinical Trials for New Drug Developmen Article submitted to Manufacturing & Service Operations Management; manuscript no.

Since

∂E [Λ] ∂E [κ]

35

∂Kζ∗ 1 E[Λ]r2 /(α + r) + c(uζ ) θζ uζ ∂E[Λ] = , 2 2 ∂E[π] γ(uζ ) θζ uζ [E[Λ]r /(λ + r) + c(uζ )] ∂E[π]   ∂Kζ∗ 1 θζ uζ dγ(uζ ) = 2 , ln ∂σ 2 γ (uζ ) E[Λ]r2 /(λ + r) + c(uζ ) dσ 2 ∂Kζ∗ 1 E[Λ]r2 /(α + r) + c(uζ ) uζ =− > 0. ∂θζ γ(uζ ) θζ uζ E[Λ]r2 /(α + r) + c(uζ ) h i ∂K ∗ ∂K ∗ θζ uζ dγ (uζ ) [Λ] > 0 and ln < 0 we have ∂E [κζ ] < 0, ∂E [πζ ] < 0, and > 0, ∂E > 0, 2 2 ∂E [π ] dσ E [Λ]r /(λ+r )+c(uζ )

∂Kζ∗ ∂σ 2

< 0.

We now prove the results for G(K; uζ ). Rearranging terms in (10), we have G(K; uζ ) = r exp[(N − K)γ(uζ )] λ+ E[Λ] − r

c(uζ ) r

{1 − exp[(N − K)γ(uζ )]}. Since the coefficient of E[Λ] is positive, G(K, uζ )

increases with E[Λ]. By this result and E[Λ] = E[π]E[κ]r−1 e−r(la +lr ) , we prove that G(K, uζ ) increases with E[κ] and E[π]. (We use the fact that κ is independent of π.) Differentiating G(K, uζ ) with respect to σ 2 , we obtain ∂G(K,uζ ) ∂σ 2

∂G(K,uζ ) ∂σ 2

c(uζ ) dγ (uζ ) r dσ 2 ∂G(K,uζ ) uζ ∂θζ r

r = (N − K)exp[(N − K)γ(uζ )]( λ+ E[Λ] + r

> 0. Differentiating G(K; uζ ) with respect to θζ , we get

)

. By

=−

dγ (uζ ) dσ 2

> 0, we prove that

{1 − exp[(N − K)γ(uζ )]}. By

1 − exp[(N − K)γ(uζ )] > 0, we prove that G(K; uζ ) decreases with θζ . We assume that 1)

∗ ∂Ki+1

∂E [κ]

< 0,

∗ ∂Ki+1

∂E [π ]

< 0,

∗ ∂Ki+1

∂σ 2

< 0, and

∗ ∂Ki+1

∂θi+1

> 0, 2) G(K, ui+1 ) increases with E[κ],

E[π], and σ , but decreases as θi+1 increases. Differentiating K with respect to E[κ], E[π], σ 2 , and θi , we 2

∗ i

get ∂Ki∗+1 ∂G(Ki∗+1 ; ui+1 ) ∂Ki∗ 1 rG(Ki∗+1 , ui+1 ) + c(ui ) θi ui = + ,   2 ∂E[κ] ∂E[κ] γ(ui ) θi ui ∂E[κ] rG(Ki∗+1 ; ui+1 ) + c(ui ) ∂Ki∗+1 G(Ki∗+1 ; ui+1 ) ∂Ki∗ 1 rG(Ki∗+1 , ui+1 ) + c(ui ) θi ui = + ,   2 ∂E[π] ∂E[π] γ(ui ) θi ui ∂E[π] rG(Ki∗+1 , ui+1 ) + c(ui )   θi ui ∂Ki∗ ∂Ki∗+1 1 dγ(ui ) ln = + 2 ∗ 2 2 ∂σ ∂σ γ (ui ) rG(Ki+1 ; ui+1 ) + c(ui ) dσ 2 ∗ G(Ki∗+1 ; ui+1 ) 1 rG(Ki+1 , ui+1 ) + c(ui ) θi ui + ,   2 γ(ui ) θ i ui ∂σ 2 rG(Ki∗+1 , ui+1 ) + c(ui ) ∂Ki∗ ∂Ki∗+1 ∂θi+1 1 rG(Ki∗+1 , ui+1 ) + c(ui ) ui > 0. = − ∂θi ∂θi+1 ∂θi γ(ui ) θi ui rG(Ki∗+1 , ui+1 ) + c(ui ) i h ∂θ ui By the induction hypothesis, ln rG(K ∗ ;θui i+1 < 0, dγdσ(u2i ) > 0, and ∂θi+1 > 0, we prove that )+c(ui ) i ∂Ki∗ ∂E [π ]

< 0, and

∂Ki∗ ∂σ 2

i+1

∂Ki∗ ∂E [κ]

< 0,

< 0.

We now prove the results for G(K, ui ). Rearranging terms in (10), we have G(K; ui ) = exp[(Ki∗+1 − K)γ(ui )]G(Ki∗+1 , ui+1 ) −

c(ui ) r

{1 − exp[(Ki∗+1 − K)γ(ui ))]}. Differentiating G(K; ui ) with respect to E[κ],

E[π], σ 2 , and θi , we have ∂Ki∗+1 c(ui ) ∂G(K; ui ) ∗ = γ(ui )e[(Ki+1 −K )γ (ui )] [G(Ki∗+1 , ui ) + ] ∂E[κ] ∂E[κ] r ∂G(Ki∗+1 , ui+1 ) ∗ + e[(Ki+1 −K )γ (ui )] ∂E[κ] ∗ ∂K ∂G(Ki∗+1 , ui+1 ) ∂G(K; ui ) c(ui ) ∗ ∗ i+1 = γ(ui )e[(Ki+1 −K )γ (ui )] [G(Ki∗+1 , ui ) + ] + e[(Ki+1 −K )γ (ui )] ∂E[π] ∂E[π] r ∂E[π] ∗ ∂Ki+1 ∂G(K; ui ) c(ui ) ∗ = γ(ui )e[(Ki+1 −K )γ (ui )] [G(Ki∗+1 , ui ) + ] ∂σ 2 ∂σ 2 r ∂G(Ki∗+1 , ui+1 ) ∂γ(ui ) c(ui ) ∗ ∗ + e[(Ki+1 −K )γ (ui )] + (Ki∗+1 − K)e[(Ki+1 −K )γ (ui )] [G(Ki∗+1 , ui ) + ] ∂σ 2 ∂σ 2 r

Kouvelis, Milner, and Tian: Clinical Trials for New Drug Developmen Article submitted to Manufacturing & Service Operations Management; manuscript no.

36

∂Ki∗+1 ∂G(Ki∗+1 , ui+1 ) ∂G(K; ui ) c(ui ) ∗ ∗ = γ(ui )e[(Ki+1 −K )γ (ui )] [G(Ki∗+1 , ui ) + ] + e[(Ki+1 −K )γ (ui )] ∂θi ∂θi r ∂θi ui ∗ − {1 − exp[(Ki+1 − K)γ(ui )]} r By the results, Ki∗ and induction hypothesis, and and

∂G(K ;ui ) ∂θi

∂θi+1 ∂θi

, we prove that

∂G(K ;ui ) ∂E [κ]

> 0,

∂G(K ;ui ) ∂E [π ]

> 0,

∂G(K ;ui ) ∂σ 2

>0

< 0.

 ∗ ζ

Proof of Proposition 7 and Corollary 1.

Replacing K with K in (10), we get   c(uζ ) c(uζ ) ∗ ∗ ∗ ∗ − G(Kζ ; uζ ) = exp((Kζ +1 − Kζ )γ(uζ )) G(Kζ +1 , uζ +1 ) + . r r h i h i θ u θ uζ . Substituting We recall that Kζ∗ = N − γ (u1 ζ ) ln E [Λ]r2 /(λζ+ζr)+c(uζ ) = Kζ∗+1 − γ (u1 ζ ) ln rG(K ∗ ,uζζ+1 )+ c ( u ) ζ ζ+1 i h θ uζ Kζ∗+1 − Kζ∗ = γ (u1 ζ ) ln rG(K ∗ ,uζζ+1 in the expression for G(Kζ∗ ; uζ ) and cancel terms, we obtain )+c(uζ ) ζ+1

G(Kζ∗ ; uζ ) =

c(uζ ) r θζ−1 uζ−1 θζ uζ −c(uζ )+c(uζ−1 )

θζ uζ r

Kζ∗ − γ (1ui ) ln[

∗ . Substituting G(Kζ∗ ; uζ ) in (9) and simplifying the expression, we get Kζ− 1 =

−

∗ i

] = Kζ∗ − γ (1ui ) ln[ θi ui r

θζ−1 uζ−1 θζ uζ−1

] = Kζ∗ − γ (1ui ) ln[

θζ−1 θζ

].

c(ui ) r

for i = 0, 1, · · · , ζ − 1. Replacing K with Ki∗ in (10), we get   c(ui ) c(ui ) ∗ ∗ ∗ ∗ G(Ki ; ui ) = exp((Ki+1 − Ki )γ(ui )) G(Ki+1 , ui+1 ) + − . r r h i θi ui From (9), we get Ki∗+1 − Ki∗ = γ (1ui ) ln rG(K ∗ ,u . Substituting Ki∗+1 − Ki∗ into the expression for )+ c ( u ) i i+1 We now prove that G(K ; ui ) =

−

i+1

G(Ki∗ ; ui ) and simplifying the expression, we have G(Ki∗ ; ui ) = θirui − c(ur i ) . Substituting G(Ki∗+1 , ui+1 ) in h i h i θi (9) and simplify the result, we show that Ki∗ = Ki∗+1 − γ (1ui ) ln θi+1 ui+1 −cθi(uuii+1 )+c(ui ) = Ki∗+1 − γ (1ui ) ln θi+1 .  Proof of Proposition 8

For any τ greater than T , the firm must stop the clinical study since the drug

patent expires at time T . For a constant u(s), ∀τ ∈ [0, T ], we compute the cumulative cost as follows, C(t, τu1i , ui ) =

c(ui ) r

[1 − exp(−r(τu1i − t)] for τu1i < T and C(t, τu1i , ui ) =

c(ui ) r

[1 − exp(−r(T − t)] for τu1i = T . The

expectation of C(t, τu1i , ui ) is given by C(t, ui ) =

c(ui ) r

The result follows as E[exp(−r(τu1i − t))χτu1

] = exp(kγ(ui ))F k (T − t; u, (2ς 2 r + u2 )1/2 ) and exp(−r(T −

t))χτu1

i

0 and E[Λ1 |ξ1 ] increases as g(ξ1 ) increases, we show that Gc (K, ξ1 ; uζ ) increases as g(ξ1 ) increases. We assume that K2∗,i+1 decreases and Gc (K, ξ1 ; ui+1 ) increases as g(ξ1 ) increases. The derivative of K2∗,i with respect to g(ξ1 ) is ∂K2∗,i ∂K2∗,i+1 ∂Gc (Ki∗+1 , ξ1 ; ui+1 ) 1 rG(Ki∗+1 , ξ1 ; ui+1 ) + c(ui ) θ i ui = + . ∂g(ξ1 ) ∂g γ(ui ) θi ui [rGc (Ki∗+1 , ξ1 ; ui+1 ) + c(ui )]2 ∂g By the induction hypothesis, we show that

∗ ∂K2,i

< 0. The derivative of Gc (K, ξ1 ; ui ) with respect to g(ξ1 ) is   ∂K2∗,i+1 c(ui ) ∂Gc (K, ξ1 ; ui ) = γ(ui ) exp((K2∗,i+1 − K)γ(ui )) Gc (Ki∗+1 , ξ1 ; ui+1 ) + ∂g(ξ1 ) ∂g r ∂G (K, ξ ; u ) c 1 i+1 + exp((K2∗,i+1 − K)γ(ui )) ∂g(ξ1 )

By the induction hypothesis, we show that Proof of Proposition 11

∂g (ξ1 )

∂Gc (K,ξ1 ;ui ) ∂g (ξ1 )

> 0. By induction, we prove the proposition.

 ˜ From (14) we see that zi , i = 1, · · · , ζ, is not a function of τ1 . In addition, Ψ(τ1 )

and Gc (K, ξ1 ; ui ) are not functions of τ1 . Thus Γ1 (τ1 ) is not a function of τ1 .



Proof of Proposition 12

The proof requires the following two lemmas. To simplify notation, we use the p following definitions, zB = (n1 + 1)y0 and zN = n21 y02 + n1 ς −2 . In Lemma 4, φ(z, n1 ) is the probability mass function (pmf) of z; let ∆φ(z) = φ(z, n1 + 1) − φ(z, n1 ) be the differential of the pmf with respect to n1 . In Pn Lemma 5, φ(z, n1 ) is the pdf of z. The following property of z is used in the proof: z = g(ξ1 ) = 1 1 ξj is a non-decreasing function of n1 . Lemma 4 If z = g(ξ1 ) follows binomial distribution with parameters n1 and y0 , then ∆φ(z) ≥ 0 for z ≥ zB and ∆φ(z) < 0 for 0 ≤ z < zB . Using φ(z, n1 ) = n1 −z −(n1 +1)y0 +z 0 n1 +1−z

y )

n1 z




y0z (1 − y0 )n1 −z

and ∆φ(z) = φ(z, n1 + 1) − φ(z, n1 ) =

(n1 )! z !(n1 −z )!

y0z (1 −

, we prove Lemma 4.

Lemma 5 If z = g(ξ1 ) follows normal distribution with mean ρ0 = n1 y0 and variance (V )2 = n1 ς −2 , then ∂φ(z ) ∂n1

≥ 0 for z ≥ zN and

Using z ≥ 0 and

∂φ(z ) ∂n1

∂φ(z ) ∂n1

< 0 for 0 ≤ z < zN . h i z−ρ0 )2 z−ρ0 dρ0 z−ρ0 2 dV dV = √2π1(V )2 exp[− (2( ] − + + ( ) , we prove Lemma 5. V )2 dn1 V dn1 V dn1

By Corollary 1, we simplify the value function of Problem (P I)1 in Section 5.2 as follows, θi+1 ui c(ui ) ∗ G1 (K) = G1 (K; ui ) = e((K1,i+1 −K )γ (ui )) − , if K1∗,i < K ≤ K1∗,i+1 , for i = 0, 1, · · · , ζ − 1, r r  G1 (K) = G1 (K; uζ ) = exp[(n1 − K)γ(uζ )] Γ1 (τ1 ) + r−1 c(uζ ) − r−1 c(uζ ), if K > K1∗,ζ . We similarly simplify the thresholds for using different enrollment rates as follows, ! 1 θi ∗ ∗ ln for i = 1, · · · , ζ − 1, K1,i = K1,i+1 − γ(ui ) θi+1 " # θ ζ uζ 1 ∗ K1,ζ = n1 − ln if K > K1∗,ζ . γ(uζ ) rΓ1 (τ1 ) + c(uζ )

Kouvelis, Milner, and Tian: Clinical Trials for New Drug Developmen Article submitted to Manufacturing & Service Operations Management; manuscript no.

39

To prove this proposition, we need to prove a general case, i.e., G1 (0; ui ) is a unimodal or non-increasing function of n1 because which enrollment rate is used at the beginning of the clinical study depends on the distribution of Y0 . Using the Envelope Theorem in ?, we differentiate G1 (K; ui ) = exp[(K1∗,i+1 −K)γ(ui )] c(ui ) r

θi+1 ui r

−

with respect to n1 as, " # ∂G1 (K; ui ) γ(u ) r∂Γ (τ )/∂n θ u i 1 1 1 i +1 i γ(ui ) + = exp((K1∗,i+1 − K)γ(ui )) ∂n1 r γ(uζ ) rΓ1 (τ1 ) + c(uζ )

(23)

We determine how the sign of the summation in the bracket changes with n1 as exp((K1∗,i+1 − K)γ(ui )) θ1rui > 0. Since γ(u) is an increasing function of u and γ(u) negative, we have

γ (ui ) γ (uζ )

> 1. To prove that G1 (K; ui )

is a unimodal or non-increasing function of n1 , we need to prove the following two claims: 1) for z following binomial distribution, there exits an n1B such that

dΓ1 (τ1 ) d n1

is positive for n1 < n1B and negative for n1 > n1B ;

2) for z following normal distribution, there exits an n1N such that

dΓ1 (τ1 ) is dn1 ∂φ(z )

negative for n1 > n1N . To simplify notation in the following proof, we use

∂n1

positive for n1 < n1N and to represent ∆φ(z) when z

follows binomial distribution. In the proofs of the above two claims, we need the derivative of Γ1 (τ1 ) that is given in (15), " Z 00 ! ! Z z1 a 1 dΓ1 (τ1 ) ∂φ(z) ∂φ(z) ∂Gc (n1 , ξ1 ; u0 ) ˜ = EY0 Ψ(τ1 ) dz + φ(z) + Gc (n1 , ξ1 ; u0 ) dz dn1 ∂n1 ∂n1 ∂n1 0 a00 1 ! Z z2 ∂φ(z) ∂Gc (n1 , ξ1 ; u1 ) + φ(z) + Gc (n1 , ξ1 ; u1 ) dz + · · · ∂n1 ∂n1 z1 ! Z b1 ∂Gc (n1 , ξ1 ; uζ ) ∂φ(z) + dz φ(z) + Gc (n1 , ξ1 ; uζ ) ∂n1 ∂n1 zζ ! # Z n1 ξ  ∂Γp (τ1 |ξ1 ) db1  ∂φ(z) + dz + φ(z) + Γp (τ1 |ξ1 ) Γp (τ1 |ξ1 ) − Gc (n1 , ξ1 ; uζ ) φ(z)|z=b1 ∂n1 ∂n1 dn1 b1 In the above expression, Gc (n1 , ξ1 ; ui ) and K2∗,i are given in (11) and (12) for i = 0, 1, · · · , ζ − 1. By Γp (τ1 |ξ1 ) > Gc (n1 , ξ1 ; uζ ) and Using the expression of

db1 dn1

dΓ1 (τ1 ) dn1

> 0, the last term in the bracket is positive. and the above result, we prove the two claims.

The first term in the bracket in the right hand side of (23) is negative while the second is positive. For g(ξ1 ) following binomial distribution, we have we have 1)

∂G1 (K ;ui ) ∂n1

and 2) we have ∂G1 (K ;ui ) ∂n1

we have

= 0 and

∂G1 (K ;ui ) ∂n1

< 0 for any n1 ≤ a00 1 . For any n1 > a00 1 ,

< 0 if the proposition condition holds (which implies that

∂G1 (K ;ui ) ∂n1

∂ Γ1 (τ1 ) ∂n1

= 0 and

∂G1 (K ;ui ) ∂n1

< −γ(uζ )),

< 0 for any n1 ≤ a00 1 /ξ. For any n1 > a00 1 /ξ, we have

< 0 if the proposition condition holds (which implies that

∂G1 (K ;ui ) ∂n1

γ (ui ) r∂ Γ1 (τ1 )/∂n1 γ (uζ ) r Γ1 (τ1 )+c(uζ )

is positive for n1 < n∗1B and is negative for n1 > n∗1B , otherwise. For g(ξ1 ) following

normal distribution, we have 1)

∂ Γ1 (τ1 ) ∂n1

∗

∗

γ (ui ) r∂ Γ1 (τ1 )/∂n1 γ (uζ ) r Γ1 (τ1 )+c(uζ )

is positive for n1 < n1N and is negative for n1 > n1N , otherwise.

Proof of Proposition 13.

< −γ(uζ )), and 2) 

Conditioning on ξ1 , it is immediate that Problem (P I)c is identical to the

initial conditions for the case without an interim analysis with suitable transformations of the parameters. Therefore, we can use Algorithm 1 to solve Problem (P I)c with the second constraint replaced by K(0) = 0. The algorithm generates all the optimal values for (n1 , t1 ) for t1 ∈ (0, T ) so that for each ξ1 , the algorithm need only be run once.