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Submitted to Manufacturing & Service Operations Managementmanuscript

Clinical Trials for New Drug Development:Optimal Investment and Application

Panos KouvelisWashington University in St. Louis, [email protected]

Joseph MilnerUniversity of Toronto, [email protected]

Zhili TianFlorida 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

Kouvelis, Milner, and Tian: Clinical Trials for New Drug Developmen2 Article submitted to Manufacturing & Service Operations Management; manuscript no.

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

Kouvelis, Milner, and Tian: Clinical Trials for New Drug DevelopmenArticle submitted to Manufacturing & Service Operations Management; manuscript no. 3

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 test-

ing 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 enroll-

ment 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


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,


N the (initial) sample size required for completion of the drug trialK(t) number of patients enrolled and treated by time tu(t) the enrollment target rate at time tc(u) the cost rate associated with enrollment rate uτ the stopping time of the enrollment processζ the number of (non-zero) enrollment levels (cost function breakpoints)ui, i= 1, . . . , ζ patient enrollment level iθi, i= 1, . . . , ζ marginal cost at investment level iT 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


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 σ2 be the infinitesimal variance of the process. Letting w be standard Brownian motion, we

assume

K(t) =

∫ t

0

u(s)ds+

∫ t

0

σ√u(s)dw (1)

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 = infs : (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


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−1e−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∑j=1

θj(uj −uj−1) + θi(u−ui−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. Optimal Investment for a Clinical Study Without Interim Analysis4.1. Derivation and Characterization of the Solution

We determine the optimal patient enrollment rate u(t) to maximize the present value of the cumu-

lative profit. We begin by defining the optimization problem.


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

E[∫ Tt|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

J(t,K(t);u) =E

[−∫ τ

t

e−r(s−t)c(u)ds+ e−r(τ−t)Γ(τ,N)− e−r(T−t)Ψ(T )χτ=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)

(PI) : G(t,K(t)) = supu∈At

J(t,K(t);u) (2a)

s.t. dK(t) = u(t)dt+σ√u(t)dw(t) (2b)

K(0) = 0. (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 σ2ux2 ≥ υx2 for x ∈R. To prove the uniqueness of the solution,

we assume there exists a small ε > 0, redefine u0 = ε and require u∈Uε ≡ [ε, u]. We also redefine At

to be Uε valued. (The value ε 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:

G(t,K(t)) = supu∈At

E[−∫ v

t

e−r(s−t)c(u(s))ds+ e−r(v−t)G(v,K(v))]

(3a)

s.t. dK(t) = u(t)dt+σ√u(t)dw(t) (3b)

K(0) = 0. (3c)

From result §13.3 in Sethi and Thompson (2000), (3a) implies

G(t,K(t)) = maxu∈Uε

E[−c(u)dt+ (1 + rdt)−1G(t+ dt,K + dK)]


From Ito’s Lemma, we have

G(t+ dt,K + dK) =G(t,K) +∂G

∂KdK +

1

2

∂2G

∂K2dK2 +

∂G

∂tdt+ o(dt)

where o(dt) represents terms that go to zero faster than dt. Substituting in (3a) and noting

dK2 = σ2u(t)dt+ 2u(t)dtσ√u(t)dw+ o(dt), we obtain

G(t,K) =

maxu∈Uε

E

[−cdt+

G(t,K) + ∂G∂Kudt+ uσ2

2∂2G∂K2 dt+ ∂2G

∂K2u(t)dtσ√u(t)dw+ ∂G

∂tdt+ ∂G

∂Kσ√udw+ o(t)

(1 + r)dt

].

Using (1 + rdt)−1 = (1− rdt+ o(t)) and E[ dw] = 0, we have

G(t,K) = maxu∈Uε

[−cdt+G(t,K) +

∂G

∂Kudt+

uσ2

2

∂2G

∂K2dt+

∂G

∂tdt− rG(t,K)dt+ o(t)

].

Dividing both sides by dt, letting it go to 0, and rearranging terms, we have

∂G

∂t− rG+ max

u∈Uε

[uσ2

2

∂2G

∂K2+u

∂G

∂K− c(u)

]= 0. (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 C1,2(Q) be the set of functions of G with ∂G/∂t, ∂G/∂K, and ∂2G/∂K2 continuous on Q; let

C(Q) be the set of continuous functions on Q, the closure of Q. We have:

Proposition 2 Equations (4)–(6) have a unique solution G∈C1,2 ∩C(Q).

Next we solve the maximization problem in (4) to obtain u∗(t). Let K∗0 = 0 and let K∗i (t) satisfy

σ2

2

∂2G(t,K∗i )

∂K2+∂G(t,K∗i )

∂K= θi

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=σ2

2

∂2G

∂K2+∂G

∂K.

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

and if K(t)>K∗ζ (t), u∗(t) = uζ.


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,K∗i ) = θi for i= 1, . . . , ζ, (7c)

G(T,K(T )) =−Ψ(T ), (7d)

G(t,K) is continuous at K∗i (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 K∗1 (t) and K∗2 (t) dividing the space. (We define S1(K) as the inverse function of K∗1 (t)

and S2(K) as the inverse function of K∗2 (t) below). In region Ω0, there is no investment and the

project is suspended (u∗ = ε); 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


Lower Boundary

Upper Boundary

t

K(t)

N

T0 S2(K) S1(K)

S2(N) S1(N)

K

K2*

K1*

W1

W0

W2

Figure 1 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

Γ(τ,N) =

∫ ∞0

E[Λ](1− e−rT )λe−λT dT


=E[Λ]r

λ+ r.

The expected profit given policy u at time t is then

J(t,K;u) =E

[−∫ τ

t

e−r(s−t)c(u)ds+ e−r(t−τ)E[Λ]r

λ+ r

]=E

[−∫ v

0

e−rsc(u)ds+ e−rvE[Λ]r

λ+ r

]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 K∗i

independent of t such that if K∗i−1 ≤K ≤K∗i 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 (8a)

subject to the boundary conditions

G(N) =E[Λ]r

λ+ r, (8b)

LG(K∗i ) = θi for i= 1, . . . , ζ, (8c)

G(K) is continuous at K∗i (t) for i= 1, . . . , ζ. (8d)

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

γ(u) =u−√u2 + 2ruσ2

uσ2.

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

Proposition 4

K∗i =K∗i+1−1

γ(ui)ln

[θiui

rG(K∗i+1, ui+1) + c(ui)

]for i= 1, · · · , ζ (9)

G(K;ui) = exp((K∗i+1−K)γ(ui))

(G(K∗i+1, ui+1) +

c(ui)

r

)− c(ui)

rfor i= 0,1, . . . , ζ (10)

and G(K) =G(K;ui−1) if K∗i−1 <K ≤K∗i for i= 1, · · · , ζ and G(K) =G(K;uζ) if K >K∗ζ .

In Proposition 4, we notice that K∗i is independent of t. Thus we find the constant K∗i that

define the initial investment rates.

Proposition 5 For i= 1, · · · , ζ, K∗i is decreasing in E[κ], E[π] and σ2, and increasing in θi.

The comparative statics given in Proposition 5 are intuitive. As K∗i 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 K∗i . Similarly, increasing the cost increases K∗i .


10 20 30 40 50 60Σ

6812

6814

6816

6818

K2

*

0.2 0.4 0.6 0.8E@YD

5000

10000

15000

20000

25000

K2*

0.2 0.4 0.6 0.8E@YD

-20000

-15000

-10000

-5000

5000

K1*

Figure 2 Thresholds K∗2 and K∗1 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

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= 120×52

= 0.00096

1week

. Using these values we find K∗1 =−11694 and K∗2 = 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 K∗1 < 0, the

firm would never abandon the study, and would increase the enrollment rate to u2 when K >K∗2 .

In Figure 2 we show how the thresholds vary with σ, holding E[Y ] = 0.54, and how they vary

with the expectation of quality, holding σ= 11. As σ increases from 2 to 60 (√

patients/week), we

observe K∗2 is a decreasing function of σ. (We do not show K∗1 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 K∗1 vs. E[Y ], if E[Y ] < 0.17, the optimal initial investment rate is ε, 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 σ2, and decreases with

θi.


20 40 60 80 100Σ

400

410

420

430

GH0L

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8E@YD

200

400

600

800

1000

GH0L

Figure 3 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(K∗i ;ui) is a constant since K∗i is a constant.

Proposition 7

G(K∗i ;ui) =θiuir− c(ui)

r

for i= 1, · · · , ζ.

Using Proposition 7, we simplify the expressions of K∗i and G(K;ui).

Corollary 1

K∗i =K∗i+1−1

γ(ui)ln

(θiθi+1

)for i= 1, · · · , ζ − 1, and

G(K;ui) = exp[(K∗i+1−K)γ(ui)]θi+1uir− c(ui)

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.


4.3. 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 K∗i (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 K∗i (t), i.e., Si(K∗i (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 expec-

tation of the discounted values-to-go, G(t+x,K+ 1) for x= 1, . . . , T − t and G(T,K), subtracting

off the cost of operating at level ui. Let τaν,ς be the stopping time defined by the minimum of T and

the time when a additional patients are enrolled. That is, τaν,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

of τaν,ς is

fa(x;ν, ς) =a

ς√

2πx3exp

(−(a− νx)2

2ς2x

), x > 0, ν > 0.

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

F a(x;ν, ς) = Φ(

(λ/x)1/2

(−1 +x/µ))

+ e2λ/µΦ(− (λ/x)

1/2(1 +x/µ)

)where µ= a/ν and λ= a2/ς2 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.


The discounted value-to-go is

J(t,K,ui) =

∫ T−t

x=0

exp(−rx)(G(t+x,K + 1)− c(ui)x)f1(x;ui)dx

+ exp(−r(T − t))(G(T,K)− c(ui)(T − t))P (τ 1ui

= T ).

We approximate this as

J(t,K,ui)≈T−t∑x=0

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

= T )

−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)

r

[1− ekγ(u)F k(T − t;u, (2ς2r+u2)1/2))− e−r(T−t)F k(T − t;u, ς)

].

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.


0"

100"

200"

300"

400"

500"

600"

0" 100" 200" 300" 400" 500" 600"

Enrollm

ent)

Time)(weeks))

σ)=)4.7)

S2"

S1"

Ω2"

Ω0"

Ω1""

0"100"200"300"400"500"600"

0" 100" 200" 300" 400" 500" 600"

Enrollm

ent)

Time)(weeks))

σ=)9.5)

S2"

S1"

Ω2"

Ω0"

Ω1""

0"100"200"300"400"500"600"

0" 100" 200" 300" 400" 500" 600"

Enrollm

ent)

Time)(weeks))

σ=)19)

S2"

S1"

Ω2"

Ω0"

Ω1""

0"100"200"300"400"500"600"

0" 100" 200" 300" 400" 500" 600"

Enrollm

ent)

Time)(weeks))

σ)=)28.5)

S2"

S1"

Ω2"

Ω0"

Ω1""

Figure 4 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

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 (ε). 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


10 15 20 25Σ

450

500

550

600

GH0L

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8E@YD

200

400

600

800

1000

GH0L

Figure 5 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


T . Paralleling the development in Section 4, let O1 = (−∞, n1) and Q1 = [0, T )×O1. As before, let

K(0) = 0. Then

τ1 = infs : s≥ 0, (s,K(s)) 6∈Q1

Similarly, let O2 = (−∞,N) and Q2 = [τ1, T )×O2. Then

τ2 = infs : 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 ξ′is. 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 a′1 >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−1e−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) = κ0Y1(ξ1) where Y1(ξ1) be the con-

ditional 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


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.)

Jc(t,K(t),ξ1;u) =E

[−∫ τ2

t

e−r(s−t)c(u)ds+ e−r(τ2−t)Γ2(τ2|ξ1)− e−r(T−t)Ψ(T ) ·χτ2=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

(PI)c : Gc(t,K(t),ξ1) = supu∈At

Jc(t,K(t),ξ1;u)

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 (PI)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).

Suppose at time τ1 the firm can earn Ψ(τ1) by abandoning the study and selling the patent to

another firm. Based on the firm’s evaluation of the drug’s market potential, this may be worth

more than continuing the study. Let a′1 = minξ1g(ξ1) subject to Γc(τ1|ξ1)≥ Ψ(τ1). That is, a′1 is

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 a′1 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 a′′1 = maxa1, a′1, the

firm would abandon the study if g(ξ1)<a′′1 . 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.


Summarizing, the value of the dynamic program at the end of the first interval given ξ1 is

Γ1(τ1|ξ1) =

Ψ(τ1) g(ξ1)≤ a′′1Γc(τ1|ξ1) a′′1 < 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

J1(t,K(t);u) =E

[−∫ τ1

t

e−r(s−t)c(u)ds+ e−r(τ1−t)Γ1(τ1)− e−r(T−t)Ψ(T ) ·χτ1=T

]where the expectation is taken over τ1. Finally we can define the control problem in the first interval

for t≥ 0

(PI)1 : G1(t,K(t)) = supu∈At

J1(t,K(t);u),

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

Γ2(τ2|ξ1) =E[Λ1|ξ1]

∫ ∞0

(1− e−rT )λe−λT dT =E[Λ1|ξ1]r

λ+ r.


If the firm abandons the study after the interim, the firm will earn Ψ(τ1). We assume Ψ(τ1) is

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

Jc(t,K(t),ξ1;u) =E

[−∫ τ2

t

e−r(s−t)c(u)ds+ e−r(τ2−t)E[Λ1|ξ1]r

λ+ r

].

We solve Problem (PI)c with the above objective function using the method that we developed

in Section 4.2. Let K∗2,i be the threshold for investment at rate ui in the second period. We define

K∗2,ζ+1 =N and G(K∗2,ζ+1;uζ) =E[Λ1|ξ1]r/(λ+ r). By Proposition 4, Gc(K,ξ1) =Gc(K,ξ1;ui−1) if

K∗2,i−1 <K ≤K∗2,i for i= 1, · · · , ζ and Gc(K,ξ1) =Gc(K,ξ1;uζ) if K >K∗2,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((K∗2,i+1−K)γ(ui))

(G(K∗2,i+1,ξ1;ui+1) +

c(ui)

r

)− c(ui)

r(11)

Similarly, one can calculate the value of K∗2,i iteratively. According to Proposition 4, the thresholds

for the different enrollment rates are

K∗2,i =K∗2,i+1−1

γ(ui)ln

[θiui

rG(K∗2,i+1,ξ1;ui+1) + c(ui)

], for i= 1, . . . , ζ. (12)

Proposition 10 K∗2,i decreases and Gc(K,ξ1;ui) increases as g(ξ1) increases if π is a linear func-

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

Γp(τ1|ξ1) =E[Λ1|ξ1]

∫ ∞0

(1− e−rT )λe−λT dT

=E[Λ1|ξ1]r

λ+ r.

We next solve Problem (PI)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 K∗2,i < n1 ≤ K∗2,i+1 for i = 0,1, · · · , ζ and use uζ if n1 > K∗2,ζ . By

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


optimal policy defined in (12) switches from ui−1 to ui in the continuing policy. We denote such

z′s as zi for i= 1, · · · , ζ. That is, the zi can be found by searching over [a′′1 , b1] such that

exp[(N −n1)γ(uζ)] [rΓ2(τ2) + c(uζ)] = θζuζ , for i= ζ, (13)

exp[(K∗2,i+1−n1)γ(ui)] =θiθi+1

, for i= 1, · · · , ζ − 1, (14)

where Γ2(τ2) and K∗2,i+1 are functions of the sufficient statistic z. The expected value of the second

period profit is given by

Γ1(τ1) =EY0

[∫ a′′1

0

Ψ(τ1)φ(ξ1)dz+

∫ z1

a′′1

Gc(n1,ξ1;u0)φ(ξ1)dz+

∫ z2

z1

Gc(n1,ξ1;u1)φ(ξ1)dz+ · · ·

+

∫ b1

zζ

Gc(n1,ξ1;uζ)φ(ξ1)dz+

∫ z

b1

Γp(τ1|ξ1)φ(ξ1)dz

]. (15)

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 (PI)1 becomes an infinite horizon dynamic program. Thus we can similarly solve

(PI)1 using the method in Section 4.2. We defineK∗1,ζ+1 = n1 andG1(K∗1,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 K∗1,i−1 <

K ≤K∗1,i and G1(K) =G1(K;uζ) if K >K∗1,ζ , where G1(K;ui) is given by

G1(K;ui) = exp((K∗1,i+1−K)γ(ui))

(rG1(K∗1,i+1;ui+1) +

c(ui)

r

)− c(ui)

r, for i= 0,1, . . . , ζ. (16)

One can determine the optimal policy by comparing LG1 with θi. According to Proposition 4, the

thresholds using different levels of enrollment rates are

K∗1,i =K∗1,i+1−1

γ(ui)ln

[θiui

rG1(K∗1,i+1;ui+1) + c(ui)

], for i= 1, . . . , ζ. (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′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


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−1c(uζ)/Γp)−1.

1. If the prior distribution, H0, is Beta, and the test statistic g(ξ1) has a binomial distribution:

For n1 <a′′1 , G1(0) is a decreasing function of n1; for n1 ≥ a

′′1 , 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 n1ξ < a′′1, G1(0) is a decreasing function of n1; for n1ξ ≥ a′′1, G1(0) is a decreasing function

of n1 if −γ(uζ)(ς2/ς2

0 )>% 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

(PI)c for each realization of g(ξ) and then, by backwards induction, Algorithm 1 can again be


used to solve Problem (PI)1. That is, for each a′′1 < 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 (PI)c can be found for all realization of τ1 by solving one instance of

Problem (PI)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 clin-

ical study value first decreases and then increases as σ increases from 4.7 to 28.5√patients/week.

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.


Max. Study Required Annual Opt.Length Sample Revenue Expected Approval Cost Enrollment Exp.T Size κ Quality Prob. Parameter Rate Value

Drug (Weeks) N ($ M) Y0 π θ1 θ2 u1 u2 ($ M)Vfend 546 564 237 0.56 0.80 0.463 0.834 2.55 4.64 434Ellence 520 716 280 0.55 0.80 0.136 0.587 2.30 3.44 467

Xanax XR 286 492 331 0.50 0.80 0.323 0.903 5.36 9.66 827Chantix/Champix 650 2052 867 0.51 0.75 0.055 0.235 17.76 26.64 3490

Table 2 Data for Pfizer’s New Drug Development and the Optimal Expected Discounted Value.

6. 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 K∗1 (t) and K∗2 (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.


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 K∗2 , 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 K∗1 and K∗2 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 K∗1 (t) and K∗2 (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


0"

100"

200"

300"

400"

500"

600"

0" 100" 200" 300" 400" 500" 600"

Enrollm

ent)

Time)(Weeks))

Vfend)

S2"S1"

Ω2" Ω1"

Ω0"

T"

N"

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

0" 100" 200" 300" 400" 500" 600"

Enrollm

ent)

Time)(weeks))

Ellence)

S2"

S1"Ω2"

Ω0"

Ω1""

T"

N"

0"

100"

200"

300"

400"

500"

0" 50" 100" 150" 200" 250" 300"

Enrollm

ent)

Time)(Weeks))

Xanax)XR)

S2"

S1"

Ω1"

Ω2"

Ω0"

T"

N"

0"

500"

1000"

1500"

2000"

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

Enrollm

ent)

Time)(weeks))

Chan4x/Champix)

S2"

S1"Ω1"Ω2"

Ω0"

T"

N"

Figure 6 Optimal Threshold Policies for Vfend, Ellence, Xanax XR, and Chantix

0"

200"

400"

600"

800"

0" 200" 400" 600"

Enrollm

ent)

Time)(weeks))

Quality)=)0.10)

S2"

S1"Ω0"

Ω1"

0"

200"

400"

600"

800"

0" 200" 400" 600"

Enrollm

ent)

Time)(weeks))

Quality)=)0.20)

S2"

S1"Ω0"

Ω1"

0"

200"

400"

600"

800"

0" 200" 400" 600"

Enrollm

ent)

Time)(weeks))

Quality)=)0.40)

S2"

S1"

Ω2"

Ω0"Ω1"

0"

200"

400"

600"

800"

0" 200" 400" 600"

Enrollm

ent)

Time)(weeks))

Quality)=)0.80)

S2"

S1"

Ω2"

Ω0"Ω1"

Figure 7 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


-

100

200

300

400

500

600

700

0.00 0.20 0.40 0.60 0.80 1.00O

pti

mal

Val

ue

($M

M)

n1/N

Optimal Value vs. Interim Sample Size

Continuation Abandonment & Early Termination

$130 MM

Figure 8 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 K∗1 (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.


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 eco-

nomics 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.


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 ∆ym1be the length

of interval m1 on [y, y] and ∆zm2be the length of interval m2 on [z, z]. We note that ∆zi is a function of

∆yi.

Step 1. Initialize ∆ym1for 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= PrY0 ≤ y0,m1

−PrY0 ≤ y0,m1−1.

Step 4. Initialize ∆zm2for 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= Prg(ξ1)≤ zm2

−Prg(ξ1)≤ zm2−1.

Step 7. If g(ξ1)≤ a′′1 , 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 (PI)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 (PI)1 using Algorithm 1.

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Appendix. Proofs

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

cally decreasing in τ . By d2Γdτ2

=−E[Λ]r2e−r(T−τ) < 0, Γ(τ) is strictly concave in τ .

Proof of Proposition 2. For any u∈Uε, the HJB equation (4)

∂G

∂t− rG+ max

u∈Uε

[σ2

2u∂2G

∂K2+u

∂G

∂K− c(u)

]= 0

is uniformly parabolic. We now verify that condition (4.1) of Theorem IV.4.1 in ? holds forO=OM = [−M,N ]

where M is an arbitrary positive real number. Since the set Uε 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

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= maxu∈Uε

[LG− θ1]u+ θ1u0− c(u0) +∂G

∂t, for u0 ≤ u≤ u1 (18a)

. . .

rG= maxu∈Uε

[LG− θi]u+ θiui−1− c(ui−1) +∂G

∂t, for ui−1 <u≤ ui (18b)

rG= maxu∈Uε

[LG− θi+1]u+ θi+1ui− c(ui) +∂G

∂t, for ui <u≤ ui+1. (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 K∗i (t) that are given by

LG(t,K∗i ) = 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

K∗i−1(t)<K(t)≤K∗i (t) for i= 1,2, . . . , ζ.

We next prove thatK∗i (t) 6=K∗i+1(t), ∀t∈ [0, T ), by contradiction. Assume that the two boundaries intersect

at some time s, K∗i (s) =K∗i+1(s) =K∗(s), ∀t∈ [0, T ). By (20), we have

σ2

2

∂2G(s,K∗i )

∂K2+∂G(s,K∗i )

∂K= θi

σ2

2

∂2G(s,K∗i+1)

∂K2+∂G(s,K∗i+1)

∂K= θi+1

The above two equations contradict that θi < θi+1. This contradiction establishes that K∗i (t) 6=K∗i+1(t).


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

0.5σ2uiz2 +uiz− r= 0 for i= 0,1, · · · , ζ. (21)

Solving (21), we get z1(ui) =−ui+√u2i+2ruiσ2

uiσ2 and z2(ui) =−ui−√u2i+2ruiσ2

uiσ2 . We next determine a particular

solution to (21). Since c(ui) is not a function of K, a particular solution to (21) is GP (K;ui) = − c(ui)

r.

Combining the above results, we have the general solution in the following form,

G(K;ui) =Ciez1K +Die

z2K − c(ui)

rfor i= 0,1, · · · , ζ. (22)

We next determine the coefficient Di. For a constant control ui, we express J(K;ui) in terms of state

variable K as J(K;ui) = E[Λ]r

λ+r+ c(ui)

rE[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)(

Λr(α+r)

+ c(ui)

r

)− c(ui)

r.

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 K∗i by backward induction. By (8b), we have G(K;uζ) =

Cζez1N − c(ui)

r= E[Λ]r

r+λ. Solving for Cζ , we get Cζ = e−z1NE[Λ]r

r+λ+

c(uζ)

r. Substituting in (22) and using

γ(uζ) =−z1(uζ), we obtain G(K;uζ) = exp((N −K)γ(uζ))E[Λ]r

r+λ+

c(uζ)

r− c(uζ)

r. To determine K∗ζ we need

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

=−γ(uζ) exp[(N −K)γ(uζ)]E[Λ]r

r+λ+

c(cζ)

r,

∂2G∂K2 = γ2(uζ) exp[(N−K)γ(uζ)]E[Λ]r

r+λ+

c(cζ)

r. Substituting in (8c), we obtain [0.5σ2γ2(uζ)−γ(uζ)] exp[(N−

K)γ(uζ)]E[Λ]r

r+λ+

c(cζ)

r= θζ . Using 0.5σ2uζγ

2(uζ)−uζγ(uζ)− r= 0, we get the free boundary K∗ζ

K∗ζ =N − 1

γ(uζ)ln

[θζuζ

E[Λ]r2/(λ+ r) + c(uζ)

]=N − 1

γ(uζ)ln

[θζuζ

rG(K∗ζ+1, uζ+1) + c(uζ+1)

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

c(ui+1)

r− c(ui+1)

r. By (8c), we get

G(K∗i+1;ui) =Ciez1K

∗i+1 − c(ui)

r=G(K∗i+1;ui+1). Solving for Ci, we get Ci = e−z1K

∗i+1 [G(K∗i+1;ui+1) + c(ui)

r].

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

c(ui)

r− c(ui)

r. To determine K∗i we need the derivatives of G(K;ui). Differentiating G(K;ui), we get ∂G

∂K=

−γ(ui) exp[(K∗i+1 − K)γ(ui)]G(K∗i+1;ui+1) + c(ci)

r, ∂2G

∂K2 = γ2(ui) exp[(K∗i+1 − K)γ(ui)]G(K∗i+1;ui+1) +

c(ci)

r. Substituting in (8c), we obtain [0.5σ2γ2(ui)−γ(ui)] exp[(K∗i+1−K)γ(ui)]G(K∗i+1;ui+1)+ c(ci)

r= θi.

Using 0.5σ2uiγ2(ui)−uiγ(ui)− r= 0, we get the free boundary K∗i

K∗i =K∗i+1−1

γ(ui)ln

[θiui

rG(K∗i+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. We prove these propositions by induction. We first prove that

1) K∗ζ decreases as E[κ], E[π], and σ2 increase and increases in θζ ; 2) G(K;uζ) increases as E[κ], E[π], and

σ2 increase and decreases as θζ increases. The derivatives of K∗ζ with respect to E[κ], E[π], σ2, and θζ are

∂K∗ζ∂E[κ]

=1

γ(uζ)

E[Λ]r2/(α+ r) + c(uζ)

θζuζ

θζuζ

[E[Λ]r2/(λ+ r) + c(uζ)]2

∂E[Λ]

∂E[κ],


∂K∗ζ∂E[π]

=1

γ(uζ)

E[Λ]r2/(α+ r) + c(uζ)

θζuζ

θζuζ

[E[Λ]r2/(λ+ r) + c(uζ)]2

∂E[Λ]

∂E[π],

∂K∗ζ∂σ2

=1

γ2(uζ)ln

[θζuζ

E[Λ]r2/(λ+ r) + c(uζ)

]dγ(uζ)

dσ2,

∂K∗ζ∂θζ

=− 1

γ(uζ)

E[Λ]r2/(α+ r) + c(uζ)

θζuζ

uζE[Λ]r2/(α+ r) + c(uζ)

> 0.

Since ∂E[Λ]

∂E[κ]> 0, ∂E[Λ]

∂E[π]> 0,

dγ(uζ)

dσ2 > 0 and ln[

θζuζ

E[Λ]r2/(λ+r)+c(uζ)

]< 0 we have

∂K∗ζ∂E[κ]

< 0,∂K∗ζ∂E[π]

< 0, and∂K∗ζ∂σ2 < 0.

We now prove the results for G(K;uζ). Rearranging terms in (10), we have G(K;uζ) =

exp[(N −K)γ(uζ)]r

λ+rE[Λ]− c(uζ)

r1−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−1e−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

σ2, we obtain∂G(K,uζ)

∂σ2 = (N −K)exp[(N −K)γ(uζ)](r

λ+rE[Λ] +

c(uζ)

r)dγ(uζ)

dσ2 . Bydγ(uζ)

dσ2 > 0, we prove that∂G(K,uζ)

∂σ2 > 0. Differentiating G(K;uζ) with respect to θζ , we get∂G(K,uζ)

∂θζ=−uζ

r1− 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)∂K∗i+1

∂E[κ]< 0,

∂K∗i+1

∂E[π]< 0,

∂K∗i+1

∂σ2 < 0, and∂K∗i+1

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

E[π], and σ2, but decreases as θi+1 increases. Differentiating K∗i with respect to E[κ], E[π], σ2, and θi, we

get

∂K∗i∂E[κ]

=∂K∗i+1

∂E[κ]+

1

γ(ui)

rG(K∗i+1, ui+1) + c(ui)

θiui

θiui[rG(K∗i+1;ui+1) + c(ui)

]2 ∂G(K∗i+1;ui+1)

∂E[κ],

∂K∗i∂E[π]

=∂K∗i+1

∂E[π]+

1

γ(ui)

rG(K∗i+1, ui+1) + c(ui)

θiui

θiui[rG(K∗i+1, ui+1) + c(ui)

]2 G(K∗i+1;ui+1)

∂E[π],

∂K∗i∂σ2

=∂K∗i+1

∂σ2+

1

γ2(ui)ln

[θiui

rG(K∗i+1;ui+1) + c(ui)

]dγ(ui)

dσ2

+1

γ(ui)

rG(K∗i+1, ui+1) + c(ui)

θiui

θiui[rG(K∗i+1, ui+1) + c(ui)

]2 G(K∗i+1;ui+1)

∂σ2,

∂K∗i∂θi

=∂K∗i+1

∂θi+1

∂θi+1

∂θi− 1

γ(ui)

rG(K∗i+1, ui+1) + c(ui)

θiui

uirG(K∗i+1, ui+1) + c(ui)

> 0.

By the induction hypothesis, ln[

θiuirG(K∗

i+1;ui+1)+c(ui)

]< 0, dγ(ui)

dσ2 > 0, and∂θi+1

∂θi> 0, we prove that

∂K∗i∂E[κ]

< 0,∂K∗i∂E[π]

< 0, and∂K∗i∂σ2 < 0.

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

K)γ(ui)]G(K∗i+1, ui+1) − c(ui)

r1 − exp[(K∗i+1 −K)γ(ui))]. Differentiating G(K;ui) with respect to E[κ],

E[π], σ2, and θi, we have

∂G(K;ui)

∂E[κ]= γ(ui)e

[(K∗i+1−K)γ(ui)]∂K∗i+1

∂E[κ][G(K∗i+1, ui) +

c(ui)

r]

+ e[(K∗i+1−K)γ(ui)]∂G(K∗i+1, ui+1)

∂E[κ]

∂G(K;ui)

∂E[π]= γ(ui)e

[(K∗i+1−K)γ(ui)]∂K∗i+1

∂E[π][G(K∗i+1, ui) +

c(ui)

r] + e[(K∗i+1−K)γ(ui)]

∂G(K∗i+1, ui+1)

∂E[π]

∂G(K;ui)

∂σ2= γ(ui)e

[(K∗i+1−K)γ(ui)]∂K∗i+1

∂σ2[G(K∗i+1, ui) +

c(ui)

r]

+ e[(K∗i+1−K)γ(ui)]∂G(K∗i+1, ui+1)

∂σ2+ (K∗i+1−K)e[(K∗i+1−K)γ(ui)]

∂γ(ui)

∂σ2[G(K∗i+1, ui) +

c(ui)

r]


∂G(K;ui)

∂θi= γ(ui)e

[(K∗i+1−K)γ(ui)]∂K∗i+1

∂θi[G(K∗i+1, ui) +

c(ui)

r] + e[(K∗i+1−K)γ(ui)]

∂G(K∗i+1, ui+1)

∂θi

− uir1− exp[(K∗i+1−K)γ(ui)]

By the results, K∗i and induction hypothesis, and∂θi+1

∂θi, we prove that ∂G(K;ui)

∂E[κ]> 0, ∂G(K;ui)

∂E[π]> 0, ∂G(K;ui)

∂σ2 > 0

and ∂G(K;ui)

∂θi< 0.

Proof of Proposition 7 and Corollary 1. Replacing K with K∗ζ in (10), we get

G(K∗ζ ;uζ) = exp((K∗ζ+1−K∗ζ )γ(uζ))

(G(K∗ζ+1, uζ+1) +

c(uζ)

r

)− c(uζ)

r.

We recall that K∗ζ = N − 1γ(uζ)

ln[

θζuζ

E[Λ]r2/(λ+r)+c(uζ)

]= K∗ζ+1 − 1

γ(uζ)ln[

θζuζ

rG(K∗ζ+1

,uζ+1)+c(uζ)

]. Substituting

K∗ζ+1 − K∗ζ = 1γ(uζ)

ln[

θζuζ

rG(K∗ζ+1

,uζ+1)+c(uζ)

]in the expression for G(K∗ζ ;uζ) and cancel terms, we obtain

G(K∗ζ ;uζ) =θζuζ

r− c(uζ)

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

K∗ζ − 1γ(ui)

ln[θζ−1uζ−1

θζuζ−c(uζ)+c(uζ−1)] =K∗ζ − 1

γ(ui)ln[

θζ−1uζ−1

θζuζ−1] =K∗ζ − 1

γ(ui)ln[

θζ−1

θζ].

We now prove that G(K∗i ;ui) = θiuir− c(ui)

rfor i= 0,1, · · · , ζ − 1. Replacing K with K∗i in (10), we get

G(K∗i ;ui) = exp((K∗i+1−K∗i )γ(ui))

(G(K∗i+1, ui+1) +

c(ui)

r

)− c(ui)

r.

From (9), we get K∗i+1−K∗i = 1γ(ui)

ln[

θiuirG(K∗

i+1,ui+1)+c(ui)

]. Substituting K∗i+1−K∗i into the expression for

G(K∗i ;ui) and simplifying the expression, we have G(K∗i ;ui) = θiuir− c(ui)

r. Substituting G(K∗i+1, ui+1) in

(9) and simplify the result, we show that K∗i =K∗i+1− 1γ(ui)

ln[

θiuiθi+1ui+1−c(ui+1)+c(ui)

]=K∗i+1− 1

γ(ui)ln[

θiθi+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, τ1ui, ui) = c(ui)

r[1−exp(−r(τ1

ui− t)] for τ1

ui<T and C(t, τ1

ui, ui) = c(ui)

r[1−exp(−r(T − t)] for τ1

ui= T . The

expectation of C(t, τ1ui, ui) is given by C(t, ui) = c(ui)

r1−E[exp(−r(τ1

ui−t))χτ1ui<T ]−exp(−r(T −t))χτ1ui≥T.

The result follows as E[exp(−r(τ1ui− t))χτ1ui<T ] = exp(kγ(ui))F

k(T − t;u, (2ς2r + u2)1/2) and exp(−r(T −

t))χτ1ui≥T= exp(−r(T − t))F k

(T − t;u, ς).

Proof of Proposition 9. We first show the following:

Lemma 1 If J(t,K;ui)≤ J(t,K;ui−1), then ∂J(t,K;ui)

∂t− ∂J(t,K;ui)

∂ti−1< 0 for i= 1,2, . . . , ζ.

Proof of Lemma 1. J(t,K;ui) and J(t,K;ui−1) satisfy (7a). From (7a) we have ∂J(t,K;ui)

∂t=

−LJ(t,K;ui)ui + rJ(t,K;ui) + c(ui) and∂J(t,K;ui−1)

∂t= −LJ(t,K;ui−1)ui + rJ(t,K;ui−1) + c(ui−1). Sub-

tracting the second equation from the first one, we get ∂J(t,K;ui)

∂t− ∂J(t,K;ui−1)

∂t= LJ(t,K;ui−1)ui−1 −

LJ(t,K;ui)ui + rJ(t,K;ui)− rJ(t,K;ui−1) + c(ui)− c(ui−1). Since LJ(t,K;ui) = θi and LJ(t,K;ui−1) =

θi−1, we rewrite the above equation as follows ∂J(t,K;ui)

∂t− ∂J(t,K;ui−1)

∂t= θi−1ui−1 − θiui + rJ(t,K;ui) −

rJ(t,K;ui−1) + c(ui)− c(ui−1). By c(ui)− c(ui−1) = θi(ui− ui−1), we have ∂J(t,K;ui)

∂t− ∂J(t,K;ui−1)

∂t= (θi−1−

θi)ui−1 + rJ(t,K;ui) − rJ(t,K;ui−1). By the condition given in the lemma and θi−1 < θi, we prove the

lemma.


We first prove Part 1. At time t= T , J(T,K;u0) = Ψ(T,K(T )). We consider an interval [s1, T ], in which

J(t,K;u1) ≤ J(t,K;u0). By Lemma 1, ∂[J(t,K;u1)−J(t,K;u0)]

∂t< 0. As t decreases from T to s1, J(t,K;u1)−

J(t,K;u0) increases. By the continuity of J(t,K;u1) and J(t,K;u0) in t, there exists a t such that

J(S1,K;u1)−J(S1,K;u0) = 0. We now consider another interval [s2, S1], in which J(t,K;u2)≤ J(t,K;u1).

By Lemma 6, ∂[J(t,K;u2)−J(t,K;u1)]

∂t< 0. As t decreases from S1 to s2, J(t,K;u2)− J(t,K;u1) increases. By

the continuity of J(t,K;u2) and J(t,K;u1) in t, there exists a t such that J(S2,K;u2)− J(S2,K;u1) = 0.

We also have S2 <S1 since S2 is in [s2, S1].

Assume that there exist S1, S2, · · · , Si such that J(Sk,K;uk) = J(Sk,K;uk−1) for k ≤ i. We consider an

interval [si+1, Si], in which J(t,K;ui+1)≤ J(t,K;ui). By Lemma 6,∂[J(t,K;ui+1)−J(t,K;ui)]

∂t< 0. As t decreases

from Si to si+1, J(t,K;ui+1) − J(t,K;ui) increases. By the continuity of J(t,K;ui+1) and J(t,K;ui) in

t, there exists a t such that J(Si+1,K;ui+1)− J(Si+1,K;ui) = 0. We also have Si+1 < Si since Si+1 is in

[si+1, Si]. By induction on i, we prove Part 1.

Part 2 follows from Part 1 and Proposition 3.

Proof of Proposition 10. We require the following two lemmas:

Lemma 2 If Y0 follows Beta distribution with parameters α and β, and the likelihood function is the product

of the pdfs of Bernoulli variables ξi for i= 1, · · · , n1, then E[Y1|ξ1] and E[Y 21 |ξ1] increase as g(ξ1) increases.

Proof of Lemma 2. From the lemma’s statement Y1 follows a Beta distribution with the following

parameters α′ = α + g(ξ1) and β′ = β + n1 − g(ξ1) where g(ξ1) =∑n1

1 ξi. Therefore, E[Y1|ξ1] = α′

α′+β′=

α+g(ξ1)

α+β+n1and V ar(Y1|ξ1) = α′β′

(α′+β′)2(α′+β′+1)= (α+g(ξ1))(β+n1−g(ξ1))

(α+β+n1)2(α+β+n1+1). Hence, E[Y1|ξ1] increases as g(ξ1)

increases. The second moment of Y1 is E[Y 21 |ξ1] = var(Y 2

1 |ξ1) + (E[Y1|ξ1])2 = (α+g)(β+n1−g)(α+β+n1)2(α+β+n1+1)

+

( α+gα+β+n1

)2 = (α+g)(β+n1−g)+(α+g)2(α+β+n1+1)

(α+β+n1)2(α+β+n1+1). Combining terms in the numerator, we have E[Y 2

1 |ξ1] =(α+g)(α+β+n1)+(α+g)2(α+β+n1)

(α+β+n1)2(α+β+n1+1). Thus, we prove that E[Y 2

1 |ξ1] increases in g(ξ1).

Lemma 3 If Y0 follows Normal distribution with mean µ0 precision ς20 and the likelihood function is the

product of the pdfs of Normal variables ξi with mean Y0 and precisions ς2 for i = 1, · · · , n1, then E[Y1|ξ1]

and E[Y 21 |ξ1] increase as g(ξ1) increases.

Proof of Lemma 3. From the lemma’s statement Y1 follows a Normal distribution with the mean

E[Y1|ξ1] =µ0ς

20+g(ξ1)ς2

ς20+n1ς2and precision, ς21 = ς20 + n1ς

2. In these expressions, g(ξ1) =∑n1

1 ξi. Since E[Y1|ξ1] is

a linear function of g(ξ1), we prove that E[Y1|ξ1] increases as g(ξ1) increases. The second moment of Y1 is

E[Y 21 |ξ1] = var(Y 2

1 |ξ1) + (E[Y1|ξ1])2 = 1ς20+n1ς2

+

(µ0ς

20+g(ξ1)ς2

ς20+n1ς2

)2

. Thus, we prove that E[Y 21 |ξ1] increases in

g(ξ1).

We now prove the proposition. The derivative of K∗2,ζ with respect to g(ξ1) is

∂K∗2,ζ∂g(ξ1)

=1

γ(uζ)

E[Λ1|ξ1]r2(α+ r)−1 + c(uζ)

θζuζ

θζuζ[E[Λ1|ξ1]r2(α+ r)−1 + c(uζ)]2

∂E[Λ1|ξ1]

∂g(ξ1)

∂Gc(K,ξ1;uζ)

∂g(ξ1)= exp[(N −K)γ(uζ)]

∂E[Λ1|ξ1]

∂g(ξ1)

r

λ+ r


By π is a linear function of y1, π = dy1, we differentiate E[Λ1|ξ1] with respect to g(ξ1) to get ∂E[Λ1|ξ1]

∂g=

κd∂E[Y 2

1 (ξ1)|ξ1]

∂ge−r(lr+La). By Lemmas 2 and 3, E[Y1(ξ1)|ξ1] increases as g(ξ1) increases. We prove that

∂E[Λ1|ξ1]

∂g(ξ1)> 0. By this result and γ(uζ)< 0, we prove that K∗2,ζ decreases as g(ξ1) increases.

Since exp((N−K)γ(uζ))> 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 K∗2,i+1 decreases and Gc(K,ξ1;ui+1) increases as g(ξ1) increases. The derivative of K∗2,i

with respect to g(ξ1) is

∂K∗2,i∂g(ξ1)

=∂K∗2,i+1

∂g+

1

γ(ui)

rG(K∗i+1, ξ1;ui+1) + c(ui)

θiui

θiui[rGc(K∗i+1, ξ1;ui+1) + c(ui)]2

∂Gc(K∗i+1, ξ1;ui+1)

∂g.

By the induction hypothesis, we show that∂K∗2,i∂g(ξ1)

< 0. The derivative of Gc(K,ξ1;ui) with respect to g(ξ1) is

∂Gc(K,ξ1;ui)

∂g(ξ1)= γ(ui) exp((K∗2,i+1−K)γ(ui))

∂K∗2,i+1

∂g

(Gc(K

∗i+1, ξ1;ui+1) +

c(ui)

r

)+ exp((K∗2,i+1−K)γ(ui))

∂Gc(K,ξ1;ui+1)

∂g(ξ1)

By the induction hypothesis, we show that ∂Gc(K,ξ1;ui)

∂g(ξ1)> 0. By induction, we prove the proposition.

Proof of Proposition 11 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

following definitions, zB = (n1 +1)y0 and zN =√n2

1y20 +n1ς−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

Lemma 5, φ(z,n1) is the pdf of z. The following property of z is used in the proof: z = g(ξ1) =∑n1

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 ≥ zBand ∆φ(z)< 0 for 0≤ z < zB.

Using φ(z,n1) =(n1

z

)yz0(1 − y0)n1−z and ∆φ(z) = φ(z,n1 + 1) − φ(z,n1) = (n1)!

z!(n1−z)!yz0(1 −

y0)n1−z −(n1+1)y0+z

n1+1−z , we prove Lemma 4.

Lemma 5 If z = g(ξ1) follows normal distribution with mean ρ′ = n1y0 and variance (V )2 = n1ς−2, then

∂φ(z)

∂n1≥ 0 for z ≥ zN and ∂φ(z)

∂n1< 0 for 0≤ z < zN .

Using z ≥ 0 and ∂φ(z)

∂n1= 1√

2π(V )2exp[− (z−ρ′)2

2(V )2][− dVdn1

+ z−ρ′V

dρ′

dn1+ ( z−ρ

′

V)2 dVdn1

], we prove Lemma 5.

By Corollary 1, we simplify the value function of Problem (PI)1 in Section 5.2 as follows,

G1(K) =G1(K;ui) = e((K∗1,i+1−K)γ(ui))θi+1uir− c(ui)

r, if K∗1,i <K ≤K∗1,i+1, for i= 0,1, · · · , ζ − 1,

G1(K) =G1(K;uζ) = exp[(n1−K)γ(uζ)][Γ1(τ1) + r−1c(uζ)

]− r−1c(uζ), if K >K∗1,ζ .

We similarly simplify the thresholds for using different enrollment rates as follows,

K∗1,i =K∗1,i+1−1

γ(ui)ln

(θiθi+1

)for i= 1, · · · , ζ − 1,

K∗1,ζ = n1−1

γ(uζ)ln

[θζuζ

rΓ1(τ1) + c(uζ)

]if K >K∗1,ζ .


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 dis-

tribution of Y0. Using the Envelope Theorem in ?, we differentiate G1(K;ui) = exp[(K∗1,i+1−K)γ(ui)]θi+1uir−

c(ui)

rwith respect to n1 as,

∂G1(K;ui)

∂n1

= exp((K∗1,i+1−K)γ(ui))θi+1uir

[γ(ui) +

γ(ui)

γ(uζ)

r∂Γ1(τ1)/∂n1

rΓ1(τ1) + c(uζ)

](23)

We determine how the sign of the summation in the bracket changes with n1 as exp((K∗1,i+1−K)γ(ui))θ1uir>

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)

dn1is positive for n1 <n1B and negative for n1 >n1B;

2) for z following normal distribution, there exits an n1N such that dΓ1(τ1)

dn1is positive for n1 < n1N and

negative for n1 > n1N . To simplify notation in the following proof, we use ∂φ(z)

∂n1to 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),

dΓ1(τ1)

dn1

=EY0

[∫ a′′1

0

(Ψ(τ1)

∂φ(z)

∂n1

)dz+

∫ z1

a′′1

(∂Gc(n1,ξ1;u0)

∂n1

φ(z) +Gc(n1,ξ1;u0)∂φ(z)

∂n1

)dz

+

∫ z2

z1

(∂Gc(n1,ξ1;u1)

∂n1

φ(z) +Gc(n1,ξ1;u1)∂φ(z)

∂n1

)dz+ · · ·

+

∫ b1

zζ

(∂Gc(n1,ξ1;uζ)

∂n1

φ(z) +Gc(n1,ξ1;uζ)∂φ(z)

∂n1

)dz

+

∫ n1ξ

b1

(∂Γp(τ1|ξ1)

∂n1

φ(z) + Γp(τ1|ξ1)∂φ(z)

∂n1

)dz+

db1dn1

(Γp(τ1|ξ1)−Gc(n1,ξ1;uζ)

)φ(z)|z=b1

]In the above expression, Gc(n1,ξ1;ui) and K∗2,i are given in (11) and (12) for i = 0, 1, · · · , ζ − 1. By

Γp(τ1|ξ1)>Gc(n1,ξ1;uζ) and db1dn1

> 0, the last term in the bracket is positive.

Using the expression of dΓ1(τ1)

dn1and 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 ∂Γ1(τ1)

∂n1= 0 and ∂G1(K;ui)

∂n1< 0 for any n1 ≤ a′′1. For any n1 > a′′1,

we have 1) ∂G1(K;ui)

∂n1< 0 if the proposition condition holds (which implies that γ(ui)

γ(uζ)

r∂Γ1(τ1)/∂n1

rΓ1(τ1)+c(uζ)<−γ(uζ)),

and 2) we have ∂G1(K;ui)

∂n1is positive for n1 <n

∗1B and is negative for n1 >n

∗1B, otherwise. For g(ξ1) following

normal distribution, we have ∂Γ1(τ1)

∂n1= 0 and ∂G1(K;ui)

∂n1< 0 for any n1 ≤ a′′1/ξ. For any n1 > a′′1/ξ, we have

1) ∂G1(K;ui)

∂n1< 0 if the proposition condition holds (which implies that γ(ui)

γ(uζ)

r∂Γ1(τ1)/∂n1

rΓ1(τ1)+c(uζ)<−γ(uζ)), and 2)

we have ∂G1(K;ui)

∂n1is positive for n1 <n

∗1N and is negative for n1 >n

∗1N , otherwise.

Proof of Proposition 13. Conditioning on ξ1, it is immediate that Problem (PI)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 (PI)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.

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