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**|**HHS Author Manuscripts**|**PMC3137757

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- Summary
- 1. Introduction
- 2. Probability Models
- 3. Decision Criteria and Trial Conduct
- 4. Application to the IA tPA Trial
- 5. Simulation Studies
- 6. Discussion
- Supplementary Material
- References

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Biometrics. Author manuscript; available in PMC 2012 December 1.

Published in final edited form as:

Published online 2011 March 14. doi: 10.1111/j.1541-0420.2011.01580.x

PMCID: PMC3137757

NIHMSID: NIHMS271750

Peter F. Thall,^{1,}^{*} Aniko Szabo,^{2} Hoang Q. Nguyen,^{1} Catherine M. Amlie-Lefond,^{3} and Osama O. Zaidat^{3}

The publisher's final edited version of this article is available at Biometrics

See other articles in PMC that cite the published article.

We consider treatment regimes in which an agent is administered continuously at a specified concentration until either a response is achieved or a predetermined maximum infusion time is reached. Response is an event defined to characterize therapeutic efficacy. A portion of the maximum planned total amount administered is given as an initial bolus. For such regimes, the amount of the agent received by the patient depends on the time to response. An additional complication when response is evaluated periodically rather than continuously is that the response time is interval censored. We address the problem of designing a clinical trial in which such response time data and a binary indicator of toxicity are used together to jointly optimize the concentration and the size of the bolus. We propose a sequentially adaptive Bayesian design that chooses the optimal treatment for successive patients by maximizing the posterior mean utility of the joint efficacy-toxicity outcome. The methodology is illustrated by a trial in which tissue plasminogen activator is infused intra-arterially as rapid treatment for acute ischemic stroke.

Many phase I/II designs that choose an optimal dose based on efficacy and toxicity have been proposed. Most of these methods characterize clinical outcomes as discrete variables (cf. Gooley, et al. 1994; Thall and Russell, 1998; O’Quigley, Hughes and Fenton, 2001; Braun, 2002; Ivanova, 2003; Thall and Cook, 2004; Bekele and Shen, 2005; Zhang, Sargent, and Mandrekar, 2006; Thall, Nguyen and Estey, 2008). Phase I/II methods also have been proposed based on two time-to-event outcomes (Yuan and Yin, 2009) and two ordinal outcomes (Houede, et al., 2010). While the problem that motivated the present paper is to optimize a 2-dimensional treatment based on efficacy and toxicity, the specific structure of our setting does not fit any of the dose-finding paradigms noted above. To explain why this is the case, we first give some background on the medical setting and treatment regime.

Acute ischemic ’stroke (AIS) is a major cause of mortality and disability in adults (Johnson, Mendis and Mathers, 2009). A new therapeutic modality for AIS is intra-arterial (IA) fibrinolytic infusion, wherein a thrombolytic agent to dissolve the clot that caused the stroke is delivered via two telescoping catheters, one supportive in the carotid artery and a smaller microcatheter within it positioned directly into the clot. The catheters are introduced to the arterial system via a sheath placed into the femoral artery. Using live X-ray fluoroscopic guidance, the catheters are moved through the carotid artery leading to the site in the brain artery where the clot leading to AIS occurred, and the agent is infused via the microcatheter. A thrombolytic agent approved by the U.S. Food and Drug Administration for intravenous (IV) treatment of AIS in adults is tissue plasminogen activator (tPA). While effects of IV tPA in adult stroke patients are well understood, optimally safe and efficacious concentrations of IA tPA have not been established. The methodology described in this paper was motivated by the desire to design a clinical trial to optimize administration of IA tPA.

The treatment regime is as follows. For a given concentration *c* in mg/kg body weight and fixed maximum volume *V*, the maximum total dose is *cV*. Since *V* is fixed, hereafter we set *V* = 1 without loss of generality. A proportion *q* of the maximum volume is given as an initial bolus at *t* = 0, followed by continuous infusion (ci) of the remaining proportion 1 – *q* at a constant rate for a maximum time period *t**. In the IA tPA trial, *t** = 2 hours. Efficacy is the time to dissolve the clot, *Y _{E}*. This includes the possibility that the clot is dissolved immediately by the bolus (

If response is observed continuously, then each patient’s outcome data consist of *Y _{T}* and either the response time

The goal is to jointly optimize (*c, q*) over a design space consisting of a rectangular grid of the eight pairs obtained from the bolus proportions *q* = 0.10 and 0.20, and the concentrations *c* = 0.20, 0.30, 0.40, and 0.50 mg/kg. These (*c, q*) combinations were chosen by two co-authors of this paper (CMA and OOZ), expert stroke, vascular, and interventional neurologists. The main problems that we address here are to (1) specify tractable probability models for *Y _{E}* and [

The probability model is given in Section 2, followed by descriptions of the utility function and design in Section 3. Application to the IA tPA trial is described in Section 4, and simulation studies are summarized in Section 5. We close with a discussion in Section 6.

While there are many parametric regression models for event time data (cf. Ibrahim, Chen and Sinha, 2001), with this treatment regime we model efficacy as a time-to-event outcome, as follows. Since efficacy is monitored on the interval 0 ≤ *t* ≤ *t**, to simplify notation we define *Y _{E}* in terms of standardized time,

and the cumulative distribution function (cdf) is

(1)

for *y* ≥ 0. In particular, *F _{E}*(0,

To apply this model, functional forms for *p*_{0} and *λ* must be specified. Any model used in sequential outcome-adaptive decision making based on small to moderate sized samples must balance flexibility to accurately reflect the observed data with tractability to facilitate computation. To allow *p*_{0} and *λ* to vary nonlinearly in both *c* and *q*, we will use *c*^{α1} and *q*^{α2} rather than *c* and *q* as arguments, where *α*_{1}, *α*_{2} > 0 are model parameters. The additional flexibility provided by *α*_{1} and *α*_{2} provides a basis for distinguishing more reliably among diffierent values of (*c, q*) in terms of their effects on *p*_{0} and *λ* when applying the adaptive decision scheme. Based on clinical experience, we assume that

*p*_{0}(*c, q,*) increases in both**α***c*and*q*, and- the clot cannot dissolve instantaneously at
*s*= 0 without a bolus infusion of some tPA, hence*p*_{0}(*c*, 0,) = 0 for all*α**c*> 0 and*p*_{0}(0,*q,*) = 0 for all 0 ≤**α***q*≤ 1.

A simple, flexible function with properties (a) and (b) is

(2)

We require the hazard function *λ*(*s, c, q, α*) for the clot dissolving during the ci to have the following properties:

*λ*(*s, c, q,*) must be continuous in**α***s*;*λ*(*s, c, q,*) must be su ciently flexible so that it may be monotone increasing, monotone decreasing, or non-monotone in**α***s*;*λ*(*s*, 0,*q,*) > 0, to allow a non-zero baseline hazard if no tPA is given,**α***c*= 0;*λ*(*s, c*, 0,) > 0 to allow the possibility that the clot is dissolved if no bolus is given;*α*- the integrated continuous hazard Λ(
*s, c, q,*) must be numerically tractable;**α** *λ*(*s, c, q,*) must increase in both**α***c*and*q*, and may be nonlinear in either*c*or*q*.

An intuitive approach to constructing a function *λ* with these properties is to define it in terms of the *cumulative delivered dose by standardized time s*, which is given by

This function increases linearly in *s* with slope *c*(1 − *q*) from minimum value *d*(0, *c, q*) = *cq* at *s* = 0 to *d*(1, *c, q*) = *c* at the last possible observation time *s* = 1 for *Y _{E}*. While a hazard function with properties (i) – (v) may be obtained by using

which may be considered the *effective cumulative delivered dose by standardized time s*. We define the hazard function to take the form

(3)

where *α _{j}* > 0 for all

Integrating (3) gives the cumulative hazard function

(4)

We define the distribution of *Y _{T}* conditional on

(5)

Under this model, *β*_{2}*c*^{β1}*q* is the effect of the bolus, *β*_{2}*c*^{β1}(1 − *q*)(*Y _{E}* ∧ 1) is the effect of the continuously infused portion,

Although treatment begins with a bolus in the IA tPA trial, if no bolus were given then *q* = 0 and the effective delivered dose at standardized time *s* would be *d*(*s*, *c*^{α1}, 0) = *c*^{α1}*s*. In this case, *α*_{2} would be dropped from the model, the hazard function (3) would simplify to

and the cumulative hazard function (4) would become

with dim(** α**) reduced from 6 to 5. The model for

Given the conditional probability *π _{T}* (

Since Pr(*Y _{T}* = 1

(6)

The first line of expression (6) is the probability that the clot is dissolved instantaneously by the bolus, the second line is the probability that the clot is dissolved during the ci, and third line is the probability that the clot is not dissolved by standardized time *s* = 1, each computed either with or without toxicity, i.e. *Y _{T}* = 1 or 0.

When *Y _{E}* is evaluated at the ends of successive intervals, as in the IA tPA trial, the likelihood must account for interval censoring. Given interval , denote

(7)

This is the relevant probability if the specific value of *Y _{E}* is not observed but, instead, it is only known that the efficacy event did not occur by time and did occur by time . In this case, infusion is stopped at the end of the interval, , and consequently the probability of toxicity is . It follows that

(8)

with *F _{E}* computed by applying formulas (1), (2), and (4) and

Let {*I*_{E,1}, …, *I _{E,M}*} be a partition of (0, 1] into all possible subintervals where

Given one of the above likelihood formulations, tailored to the trial at hand, the problem is to construct an outcome-adaptive design for determining an optimal pair (*c, q*). To do this, we take an approach similar to that of Houede, et al. (2010), who sequentially choose dose pairs in a phase I/II trial with bivariate ordinal outcomes by maximizing the posterior mean of an elicited utility. A fundamental di erence that we must address here when defining utilities is that the bivariate outcome consists of a binary toxicity and continuous time to response which may be interval censored. Denote the numerical utility of outcome ** y** by

(9)

For each new cohort of patients during the trial, we exploit the Bayesian model by adaptively selecting the (*c, q*) combination that is optimal in the sense that it maximizes the posterior mean of *u*(*c, q, θ*) based on the most recent data. Let denote the data from the first

(10)

With interval censoring due to sequential evaluation of *Y _{E}* and Pr(

A possible alternative to the utility-based approach might be to use a linear combination such as *F _{E}*(1|

It does no good to treat patients with the (*c, q*) that optimizes the posterior expected utility if all pairs being considered are either excessively toxic or inefficacious. To protect patients during the trial, we impose the following safety/futility rules. Given , a pair (*c, q*) is*unacceptable* if either it is likely to be too toxic,

(11)

or it is likely to be inefficacious,

(12)

where is the maximum allowed *π _{T}* (1,

A possible alternative to using (11) and (12) might be a single criterion based on *u*(*c, q, θ*). One might specify a fixed lower bound

Once the design parameters and model are established, given a set of (*c, q*) pairs, maximum sample size, *N*, and cohort size, the trial is conducted as follows. The first cohort is treated at a starting (*c, q*) combination chosen by the physicians, and the choice may be guided by the numerical utilities and prior means. While the usual fear in phase I where only toxicity is considered is overdosing the first few patients, in the present setting when choosing the starting (*c, q*) pair this fear may be counterbalanced by the concern that patients may be given too little tPA to dissolve their clots. A given *c* may be too low to dissolve the clot that caused the stroke but high enough to cause a variety of adverse effects not associated with observable hemmorrhage (SICH) and not easily be detected, and thus such events cannot feasibly be included in an outcome-adaptive decision making procedure. Such adverse effects include cytotoxicity, degradation of extracellular matrix, and increased permeability of the neurovascular unit with the development of cerebral edema (Kaur, et al., 2004). For each cohort after the first, if is empty then the trial is stopped early with the conclusion that all (*c, q*) pairs are unacceptable. If is not empty then the next cohort is treated at the best acceptable pair (*c, q*)^{opt}(*D _{n}*), subject to the do-not-skip rule. At the end of the trial, the (

To evaluate *Y _{E}*, the clot is imaged at the start of infusion when the bolus is given, and thereafter every 15 minutes up to the maximum infusion duration of

Elicited utilities of the joint outcomes and values of *Y*_{E} used to compute *π*_{T} (*Y*_{E}, c, q, **β**) when efficacy is monitored in 15-minute intervals. Efficacy is defined as the time to blood clot being dissolved, and the adverse event is symptomatic **...**

Up to *N* = 36 patients will be treated in cohorts of size 1, with the aim to choose the (*c, q*) pair that maximizes the posterior mean utility among the set of eight possible pairs obtained from *q* = 0.10, 0.20 and *c* = 0.20, 0.30, 0.40, 0.50. The maximum sample size was chosen based on an anticipated accrual rate of 1 patient/month/site with 15 sites participating and 5% of accrued patients both eligible and consenting. This would give .75 patients per month, so a 36-patient trial would last 48 months. The admissibility limits were specified to be and , and the probability cut-offs used in (11) and (12) were *p _{E}* =

Numerical utilities, on a scale of 0 to 100, elicited for each combination of *Y _{T}* = 0 or 1 and observation interval for

To establish priors, we used a three-step strategy. First, we elicited a large number of prior means of the probabilities *p*_{0}(*c, q, θ*),

In step 1, for each of the eight (*c, q*) combinations, we elicited the prior means of the probability of dissolving the clot immediately with the bolus, *p*_{0}(*c, q, θ*), within 60 minutes, , or within 120 minutes,

Elicited prior mean probabilities for each (*c, q*) combination studied in the IA tPA trial, and resulting parameter prior means

For the second step, the elicited values were treated like the true state of nature and used to simulate 1000 large pseudo samples, each of size 400 with exactly 50 patients for each (*c, q*) combination. Starting with a very non-informative pseudo-prior on ** θ** in which the logarithm of each entry followed a normal distribution with mean 0 and standard deviation 20, we used the pseudo data set to compute a pseudo posterior. The average of the 1000 pseudo posterior means were used as the prior means. These are given in Table 2b. The pseudo sample size of 400 was chosen to be large enough so that prior means obtained in this way would not change substantively with a larger pseudo sample size.

For the third step, we calibrated the variances of the entries of ** θ** to ensure a prior that was suitably non-informative in terms of the prior effective sample sizes (ESSs) of

The computations for each interim decision include obtaining the posterior probabilities in the admissibility criteria (11) and (12) and posterior mean utility (9) for all (*c, q*) combinations. We used Markov chain Monte Carlo (MCMC) with Gibbs sampling (Robert and Cassella, 1999) to compute all posterior quantities, based on the full conditionals. Each series of sample parameters **θ**^{(1)}, …, **θ**^{(N)} distributed proportionally to the posterior integrand was initialized at the mode using the two-level algorithm described in Braun et al. (2007). Because each sample chain was initialized at the mode, which reliably identifies the region of highest posterior probability density, we did not require any burn-in, and a single chain was used for each posterior computation. We used MCMC sample size *N* = 2, 000 for the dose assignments during the trial, and *N* = 16, 000 for the dose selection at the end of the trial. For each sample **θ**^{(i)} = (**α**^{(i)}, **β**^{(i)}), we computed *p*_{0}(*c, q*, **α**^{(i)}), then Λ(*Y _{E}, c, q, α^{(i)}), F_{E}(Y_{E}, c, q*,

Each trial was simulated 10,000 times under each of a wide variety of scenarios. Since each scenario was specified in terms of fixed values of the marginal probabilities *π _{T}*(

The simulation scenarios are given in Supplementary Tables 1 – 6. Scenario 1 uses the elicited prior means, with the utilities increasing with both *c* and *q*, so that (*c, q*) = (0.5, 0.2) is optimal. Scenarios 2 has a similar pattern, but with a larger increase as (*c, q*) goes from (0.2, 0.1) to (0.5, 0.2). In Scenario 3, the middle values *c* = 0.3 and 0.4 have the highest utilities, also with *u*(*c*, 0.1)^{true} > *u*(*c*, 0.2)^{true} so a smaller bolus is more desirable. In Scenario 4, smaller values of both *c* and *q* have higher *u*(*c, q*)^{true}. Scenario 5 is unsafe, with unacceptably high values of all *π _{T}*(

To summarize the method’s overall behavior, we used the following statistic. For each scenario, let *u ^{true,sel}* denote the true utility of the final selected (

Simulation results for the IA tPA trial. Under each scenario, *u*^{true}(*c, q*) denotes the expected utility of treating a patient with the combination (*c, q*). The value of the utility for the combination with the highest utility is highlighted in bold. Utilities **...**

We assessed sensitivity to the prior, *N*, cohort size, and *σ*, summarized in Supplementary Tables 7, 8, and 9. Supplementary Table 7 shows that, for *N* = 24 to 240, using a prior with mean 0 and *σ*^{2} = 20 for all log(*θ*_{j}) substantially degrades *R* under Scenarios 1 and 2, increases *R* under Scenarios 3 and 4, and increases the futility stopping probabilities under Scenario 6. For *N* = 36, there is no general pattern of *R* or stopping probability with cohort size 1, 2, 3, or with *σ* = 7 to 20. Sensitivity to the four interpolation methods for obtaining each scenario’s probabilities between successive evaluation times *s* = 0, 0.125, …, 1.0 is summarized in Supplementary Table 10. The stopping probabilities are insensitive to the interpolation method, but *R* may change very little or substantively, depending on the scenario and method. This is because each interpolation method gives diffierent shapes of *π _{T}*(

Table 4 gives a hypothetical case-by-case example to illustrate how a trial might play out in practice, under a scenario with the best *c* in the middle. For each patient, the treatment values and outcomes are given with the posterior mean utilities of the eight (*c, q*) combinations. To conserve space, results are given for the first 12 patients and thereafter each sixth patient. The values for all 36 patients are given in Supplementary Table 13. For this example, posterior means and standard deviations of the elements of ** θ**, and of

The methodology proposed here may be extended to oncology settings. For example, suitable modifications of the methodology may be used for a chemotherapeutic anti-cancer agent administered by ci, with a possible initial bolus, with the tumor imaged repeatedly and therapy stopped when tumor response is achieved. In such settings, the time frame likely would be much longer than the 120 minute schedule considered here, and the infusion typically would include successive cycles with interim rest periods. Additionally, toxicity might be a time-to-event variable, possibly occurring during infusion and causing treatment to be suspended or permanently stopped. Such diffierences are non-trivial, however, and would require substantive modifications of *λ*, *π _{T}*, and the decision rules.

A simpler version of the method currently is being applied to plan a similar trial of IA tPA in pediatric stroke patients. Although pediatric AIS is rare, over 75% of children with acute AIS will die or su er long-term neurological deficits (deVeber, et al., 2000). In this trial, it was decided to fix *q* 0.10 since a bolus of size *q* = 0.20 or larger was considered too risky for children. For the model, the response hazard is simplified by fixing *α*_{2} 1. The design space consists of the four concentrations {0.20, 0.30, 0.40, 0.50} and *c* is chosen based on , defined as the maximum over *c* of .

A computer program, named “CiBolus,” to implement this methodology is available from the website https://biostatistics.mdanderson.org/SoftwareDownload.

The authors thank the editor, associate editor, and a referee for their detailed and constructive comments that substantively improved the manuscript. The research of PT and HN was supported by NCI grant RO1-CA-83932.

**7. Supplementary Materials**

Supplementary Tables 1 – 14, referenced in Section 5, are available under the Paper Information link at the Biometrics website http://www.biometrics.tibs.org.

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