Disease progression trajectory curves to estimate saved time in Alzheimer’s disease trials

Abstract

With the recent successful disease-modifying therapies against Alzheimer’s disease (AD), there have been discussions on easily interpretable measures for treatment effects. Among them, saved time for patients treated with a new drug as compared to patients randomized to the placebo group offers easier interpretation than the reduced percentage in outcome decline at last visit which were commonly used in AD trials. The existing method to calculate saved time utilized the disease progression trajectory of the placebo group and the treatment effect at the last visit. We propose to develop two new methods that use the disease progression trajectories of both groups: (1) slope adjusted method; and (2) area under the curve method. We used data from the two donanemab trials and the donepezil trial to illustrate the application of the proposed methods and conducted simulation studies to compare these methods. When a drug has a constant treatment effect over time or early and middle difference in the disease progression, the second new method often has the saved time being longer than the existing method. When the treatment effect is an increasing function of time before the last visit as observed in disease-modifying therapy trials, the slope adjusted method could have a larger saved time as compared to the existing method. In many cases, the area under the curve method often has the smallest standard deviation of saved time.

1. Introduction

In clinical trials for Alzheimer’s disease (AD), the percentage change of the primary outcome measure at the pre-specified follow-up visit from baseline is often chosen as the primary outcome [1, 2, 3, 4, 5]. The primary outcome measure could be a composite score (e.g., the integrated AD rating scale (iADRS)) or a cognitive and functional outcome such as clinical dementia rating - sum of boxes (CDR-SB) [6, 7, 8, 9]. The treatment effect is the estimated primary outcome difference between the drug group and the placebo group at the pre-specified follow-up visit which can be illustrated as the vertical line between the two disease progression trajectories at that visit. When individual level data are available, several disease progression models were developed [10] including the Cox proportional hazard cognitive disease progression model by Wang et al. [11] and the progression models for repeated measures with time as a continuous covariate by Raket [12]. However, it is a challenge to obtain the individual level data. In many cases, one may be able to obtain the summary statistics (e.g., mean and standard deviation (SD) of the primary outcome at each visit for each group) from published articles and other reliable resources.

The saved time concept was introduced by using the summary statistics without individual level data to calculate saved time for patients treated by a new drug as compared to patients in the placebo group in a pre-specified follow-up time window. It can be estimated as the length of the horizontal line starting from the last data point of the drug group to the intersection of that line and the placebo disease progression trajectory. That approach has been used in several AD trials [13, 14]. The saved time approach can be used in both symptomatic drug trials and disease-modifying therapy (DMT) trials. In symptomatic drug trials, the new drug group and the placebo group may have a similar rate of decline in cognition and function [15]. The homogeneous decline rate is one of the key assumptions in the saved time calculation. That assumption is often satisfied in symptomatic drug trials [3]. However, patients in the DMT drug group may reduce the cognitive and functional decline as compared to patients in the placebo group [16, 17], which could violate the homogeneous decline rate assumption in the saved time method [18].

When the homogeneity of the disease progression rate is not satisfied due to the accumulative effects from a drug, it is necessary to adjust the saved time by comparing the disease progression rate between the two groups.

By adding the disease progression trajectory of the drug group, we may be able to provide a decline rate adjusted estimate for saved time. If the drug group has a slower disease progression rate than the placebo group, it is as expected that the saved time would be longer than the time-saving from the existing method, as it takes additional time to decline the same magnitude in the primary outcome for patients treated with a drug. Otherwise, the adjusted saved time should be shorter than that from the existing method.

Another statistical challenge was how to better use the disease progression trajectory of the placebo group. Instead of the treatment effect at the last visit, we may consider applying the restricted mean survival time (RMST) method to estimate the average disease progression over the study period time by using data from disease progression trajectories. The RMST is a new statistical tool for survival analysis that has been increasingly used in clinical trials, including cancer trials [19, 20, 21, 22, 23]. The RMST can be easily interpreted as the average survival time of patients from baseline to the pre-specified follow-up time [24, 25]. We propose to use the RMST method to calculate the area above the disease progression trajectory curve when we assume it is a downward trajectory for the outcome. The area above the disease progression trajectory can be interpreted as an overall disease progression scale over the course of a study.

2. Methods

We first introduce the existing method based on the treatment effect at the last visit and the disease progression trajectory of the placebo group. Then, we propose two new methods to estimate saved time by using the disease progression trajectories of both groups.

2.1. Existing method

The existing method calculated saved time by first calculating the time needed in the placebo group to have the same degree of decline as that in the drug group from baseline to the pre-specified follow-up time (e.g., $T_f=76$ weeks). This method projects the change in the drug group to the placebo disease progression (PDP) trajectory, and we refer this existing method as the PDP method. We used the primary outcome data from the phase 2 donanemab trial to illustrate the application of the saved time calculation in Figure 1. That was a phase 2 randomized, placebo-controlled study with a total sample of 257 patients [26].

Figure 1:

Saved times by using the existing PDP method and the new adjusted PDP method after adding the disease progression trajectory of the drug group, based on the iADRS score change from the phase 2 donanemab trial.

Let $T_p$ be the time at the intersection between the placebo trajectory curve and the horizontal line from the score change at $T_f$ in the drug group, see Figure 1. Then, saved time is computed as the difference between $T_f$ and $T_p$:

$$T_{PDP}=T_f-T_p.$$

In this donanemab trial, the pre-specified follow-up time was $T_f=76$ weeks. The time of $T_p$ was calculated as 53.17 weeks. Then, saved time based on the PDP method was $T_{\text{PDP}}=22.83$ weeks which is 5.25 months [13].

2.2. Trajectory adjusted PDP method

In the calculation of $T_{PDP}$, the disease progression trajectory curve of the placebo group and the treatment effect at the last follow-up visit (${\mathrm{\Delta }}_f$) were used. However, the drug group’s disease progression trajectory was not fully utilized, except the treatment effect at the last visit. In Figure 1, between $T_p=53.17$ weeks and $T_f=76$ weeks, the estimated linear slopes were −1.68 and −1.92 point changes in iADRS every 12 weeks for the placebo group and the donanemab group, respectively. It can be seen that between 53.17 weeks and 76 weeks, patients in the donanemab group declined faster than the placebo group by 14% with regards to the primary outcome.

In the PDP method, the equal decline rate between $T_p$ and $T_f$ is one of the key assumptions. When the decline rate is higher in the drug group than the placebo group for the time between $T_p$ and $T_f$, the actual saved time should be less than $T_{PDP}$ by an adjusted time $T_A$. The value of $T_A$ is calculated from the estimated drug disease progression trajectory and the difference between the treatment effect at $T_p$ and that at $T_f$. Suppose the treatment effects at $T_p$ and $T_f$ are ${\mathrm{\Delta }}_p$ and ${\mathrm{\Delta }}_f$. Their absolute difference is ${\mathrm{\Delta }}_d=\left|{\mathrm{\Delta }}_p-{\mathrm{\Delta }}_f\right|$. Given the treatment slope of $\beta$ (e.g., $\beta =-1.92/12$ per week in the donanemab group), the adjusted time is computed as $T_A=\left|{\mathrm{\Delta }}_d/\beta \right|$. Then, the new drug slope adjusted PDP (PDPa) saved time is

$$T_{PDPa}=T_{PDP}+I\left(\text{drug}\text{declines}\text{slower}\text{than}\text{placebo}\right)\times T_A,$$

where $\mathrm{I}(.)$ is an index function with the value of 1 when the drug group declines slower than the placebo group between $T_p$ and $T_f$, 0 when they have the same decline rate, and −1 otherwise.

In the two components of $T_{\mathit{PDPa}}$, the existing $T_{\mathit{PDP}}$ saved times uses the placebo disease progression trajectory, and the second part $T_A$ partially utilizes the disease progression trajectory from the drug group. When the drug group and the placebo group have a similar slope in the last few visits, the new $T_{\mathit{PDPa}}$ should be very close to $T_{\mathit{PDP}}$.

2.3. Restricted mean disease progression scale

In the existing PDP method and the new PDPa method, it is often the case that the disease progression trajectory at early visits was not fully used in the saved time calculation. We propose to develop a new measure based on the RMST approach, which can be considered as an alternative measure of the treatment effect at $T_f$. We refer this new measure to be as the restricted mean disease progression (RMDP) scale which is illustrated with the shaded area in the color of blue for the drug group in Figure 2 with data mimicking the CDR-SB score change from the lecanemab trial [2]. The RMDP is a measure of disease progression from time 0 to a pre-specified time $T_f$ (see, Figure 2), specifically,

$${\mu }_g\left(T_f\right)={\int }_0^{T_f}{\mathrm{\Delta }}_g\left(t\right)dt,$$ (1)

where ${\mathrm{\Delta }}_g\left(t\right)$ is the disease progression function for the drug group $(g=\mathrm{D})$ and the placebo group $(g=\mathrm{P})$. The disease progression function is defined as the outcome change at time $t$ from baseline. For example, the function of ${\mathrm{\Delta }}_g\left(t\right)$ may be viewed as the CDR-SB change at time $t$ from baseline. The value of ${\mu }_g\left(T_f\right)/T_f$ can be interpreted as the average outcome change from time 0 to $T_f$ in group $g$. In a typical phase 3 AD trial, the value of $T_f$ may be chosen as 18 months. A low value of RMDP shows less disease progression from baseline to time $T_f$.

Figure 2:

Saved time by using the new RMDP method with data mimicking the CDR-SB score change from the lecanemab trial.

We propose to develop a new saved time by using the RMDP scale in Equation (1). The area above the disease progression curve for the drug arm from baseline to $T_f=18$ months is firstly calculated: ${\mu }_D\left(T_f=18\right)$. The main idea was to determine the time $T_q$ such that the estimated RMDP for the placebo group from time 0 to time $T_q$ equals to ${\mu }_D\left(T_f=18\right)$, specifically,

$${\mu }_P\left(T_q\right)={\mu }_D\left(T_f=18\right).$$

The bisection method was used to determine the $T_q$ value as ${\mu }_P\left(T_q\right)$ is an increasing function of $T_q$. The RMDP values for the drug group and the placebo group were presented as the shaded areas in the color of blue and green in Figure 2. Then, the saved time in months based on the new RMDP scale is computed as

$$T_{RMDP}=T_f-T_q.$$

In the calculation of the RMDP scale, the disease progression trajectories of both groups were largely used although the trajectory of the control group at later visits may not be fully included in the RMDP calculation.

From the mimicking CDR-SB score change data from the lecanemab trial as in Figure 2, the estimated RMDP scales from baseline to 18 months for the drug arm and the placebo arm were ${\mu }_D\left(T_f=18\right)=11.16$ and ${\mu }_P\left(T_f=18\right)=15.53$, respectively. By using the RMDP approach, we calculated the value of $T_q=15.17$ months. Then, saved time was estimated as 18–15.17=2.83 months.

3. Results

We first used data from two donanemab trials [26, 4] and the donepezil trial [1] to illustrate the application of the three methods to calculate saved time in the first two subsections. Summary data from these trials were available from their published with their articles. In the last subsection, data of the placebo group from the donepezil trial [1] was used in conjunction with several treatment effect patterns to simulate a trial with 36 months to compare mean and standard deviation (SD) of saved time through extensive simulation studies.

3.1. Phase 2 and 3 donanemab trials

In the phase 2 donanemab trial for patients with early symptomatic AD [26], there were 131 and 126 patients being assigned to the donanemab group and the placebo group. The primary outcome measure was the composite score of iADRS having the range from 0 to 144 which is a simple sum of the two scores: AD Cooperative Study - instrumental activities of daily living (ADCS-iADL) and the transformed ADAS-Cog13 (85 - the score of ADAS-Cog13). The primary outcome was the change of iADRS at 76 weeks from baseline. Secondary outcome measures included CDR-SB, ADAS-Cog13, ADCS-iADL, and MMSE.

We presented the saved times based on the three methods using these 5 outcomes in Table 1. For the PDP method, the range of saved time was between 2.38 months and 5.96 months with SD of 1.43 months. The new RMDP method’s saved time had the range from 2.17 months to 4.54 months with SD=0.95 months. We noticed that the PDPa method had a wider range than the other two methods for this study, from 1.72 months to 6.17 months with SD=1.70 months. The decline rate in ADAS-Cog13 of the donanemab group was much slower than that in the placebo group during the last few visits. Due to that reason, the saved time by using the PDPa method was much longer than other methods based on the ADAS-Cog13. As compared to other outcome measures, MMSE always had the smallest saved time for each method.

Table 1:

Saved times in months of the existing PDP method, and the two new methods (PDPa and RMDP) by using the five outcomes from the phase 2 donanemab trial with the pre-specified follow-up time of 76 weeks.

Method	iADRS	CDR-SB	ADAS-Cog13	ADCS-iADL	MMSE
PDP	5.25	5.29	3.92	5.96	2.38
PDPa	4.61	3.38	6.17	2.89	1.72
RMDP	4.43	3.88	3.81	4.54	2.17

In the phase 3 donanemab trial [4], there were a total of 1736 patients enrolled in that parallel randomized study. The five outcomes from the phase 2 donanemab trial were also measured in this phase 3 trial. We simulated data mimicking the phase 3 donanemab data to estimate the saved time by using the three methods with the saved times presented in Table 2. The saved time based on CDR-SB was the highest as compared to the saved time based on other measures for each method. The decline rate of CDR-SB in the donanemab group was slower than that in the placebo group which leads to a longer saved time by using the PDPa method as compared to the PDP method: 6.26 months vs 5.21 months. From baseline to the week of 76, the treatment effect based on the CDR-SB was an increasing function of time. For that reason, their difference at early visits was small that leads to a short saved time by using the RMDP method as 3.40 months. Among these five outcome measures, ADAS-Cog13 was the one measure that had early and middle difference by comparing donanemab and placebo. For that reason, saved time based on the RMDP method for ADAS-Cog13 was almost 1 month longer than that of other two methods.

Table 2:

Saved times in months of the existing PDP method, and the two new methods (PDPa and RMDP) by using the five outcomes from the phase 3 donanemab trial with the pre-specified follow-up time of 76 weeks.

Method	iADRS	CDR-SB	ADAS-Cog13	ADCS-iADL	MMSE
PDP	2.52	5.21	1.92	3.65	1.80
PDPa	2.93	6.26	1.92	4.96	2.15
RMDP	2.47	3.40	2.89	2.52	1.62

3.2. Donepezil clinical trial

In the donepezil trial [1], mild cognitive impairment (MCI) patients were randomized to receive either donepezil, vitamin E or placebo in a 36-month trial with the primary outcome as the conversion time from MCI to AD. We used the five secondary outcomes in this article to compare the saved time: CDR-SB, ADAS-Cog11, ADCS-iADL, MMSE, and global deterioration scale (GDS). The GDS measure was developed to assess the stages of cognitive functions by the evaluation results from caregivers [27].

We utilized the estimated score changes from baseline of the 5 cognitive and functional outcomes at month 6 to month 36 from the article by Petersen et al. [1] to calculate saved times based on the PDP method, the PDPa method, and the RMDP method in Table 3. Saved time based on the new RMDP method is much longer than the PDP method and the PDPa method, more than double in many cases. From the disease progression trajectories, the treatment effect was larger in the first few visits as compared to that in later visits. In such cases, the existing PDP method relies too much on the treatment effect on the last visit. With a faded treatment effect over time, the PDP method would have a short saved time. However, the RMDP method captures the treatment effect over the study period that is able to include the early treatment effect in the saved time comparison. We noticed that the disease progression trajectories of the denepezil group and the placebo group based on GDS had a crossing between the last two visits. We further discussed that topic in Section 4.

Table 3:

Saved times in months of the existing PDP method, and the two new methods (PDPa and RMDP) by using the five outcomes from the donepezil trial with the pre-specified follow-up time of 3 years.

Method	CDR-SB	ADAS-Cog11	ADCS-iADL	MMSE	GDS
PDP	0.63	0.47	0.56	2.69
PDPa	0.59	0.37	0.43	3.14
RMDP	2.46	2.87	2.36	6.62	3.06

The disease progression trajectories of the two groups based on Global Deterioration Scale (GDS) had a late crossing which leads to the missing estimate of the saved time by using the PDP method and the PDPa method.

We also presented saved time for the first four measures by using data up to month $12,18,\dots ,36$ in Figure S1. The highest saved time was observed between month 18 and month 24 based on CDR-SB, ADAS-Cog13, and MMSE. For ADCS-iADL, there was a quick decline in the donepezil group between month 6 and month 12 which leads to a very low saved time for the PDP method and the PDPa method at month 12. The saved time variation of the PDPa method was large for this measure due to the high variation of treatment effect in early visits. Overall, the RMDP method had a small variance of saved time as compared to other methods. For this trial using data from baseline to month 18, MMSE had the largest saved time than others.

3.3. Simulation studies

We then conducted simulation studies to compare saved times among the PDP method, the PDPa method, and the RMDP method. We evaluated the effect of the treatment trajectory as compared to the placebo trajectory on the saved time estimates. The following 4 treatment effect configurations were investigated: (1) constant treatment effect: ${\mathrm{\Delta }}_D\left(t\right)-{\mathrm{\Delta }}_P\left(t\right)=\mathrm{c}$, (2) early difference: ${\mathrm{\Delta }}_D\left(t\right)-{\mathrm{\Delta }}_P\left(t\right)=w\times \left(\mathrm{0.26,0.22,0.18,0.14,0.1}\right)$ is an non-increasing function of $t$, (3) proportionally reduced decline in the drug group: $1-{\mathrm{\Delta }}_D\left(t\right)/{\mathrm{\Delta }}_P\left(t\right)=r$, and (4) constant delaying: ${\mathrm{\Delta }}_D\left(t+t_c\right)={\mathrm{\Delta }}_P\left(t\right)$, where $c,w,r$, and $t_c$ are pre-specified constant values, and ${\mathrm{\Delta }}_g\left(t\right)$ is the score change at time $t$ from baseline in the group $g$, see examples in Figure S2. These treatment effects are common in symptomatic drug and DMT trials [12, 28].

We used the estimated CDR-SB score change of the placebo group in the donepezil trial [1] for a study with a follow-up time of 36 months. The average CDR-SB score change values at the months 6, 12, 18, 24, 30, and 36 from baseline were ( 0.14, 0.39, 0.65, 0.85, 1.05, 1.47). The estimated variance and covariance matrix were estimated in Equation (2).

We simulated studies with the sample sizes of 100, 300, 600, and 900 per group with a number of 1,000 simulations to calculate the mean and SD of saved time by using the three methods. In Figure 3, we showed

$$\left(\begin{array}{llllll}0.80& 0.77& 0.74& 1.05& 1.08& 1.26\\ 0.77& 1.56& 1.45& 1.66& 1.89& 2.47\\ 0.74& 1.45& 2.36& 2.00& 2.42& 3.03\\ 1.05& 1.66& 2.00& 3.04& 3.43& 4.23\\ 1.08& 1.89& 2.42& 3.43& 5.12& 5.80\\ 1.26& 2.47& 3.03& 4.23& 5.80& 7.93\end{array}\right)$$ (2)

saved times for the first case with constant treatment effect over time. The PDPa method had a longer saved time than the existing PDP method, but their difference became smaller as the sample size per group goes up from 100 to 900. The RMDP method often had the largest saved time than the other two methods when sample sizes were 300 or above. The results for n=300 per group were similar to those for larger sample sizes. For that reason, we only presented the results with sample sizes up to 600 in the next few figures. The results for n=100 could be different from others with the PDPa method having the largest saved time. The RMDP method had the lowest SD, followed by the PDP method and the PDPa method. When sample size is small (e.g., 100), the disease progression trajectory’s variation could be high which leads to a very large SD of saved time by using the PDPa method.

Figure 3:

Mean and SD of saved time under the first case with constant treatment effect: $\mathbf{c}={\mathrm{\Delta }}_D\left(t\right)-{\mathrm{\Delta }}_P\left(t\right)$, with 1000 simulations for a study with the sample sizes of 100, 300, 600, and 900 per group.

For a study with early difference as Figure S3, the RMDP method had substantial gain in saved time as compared to the other two methods. The estimated SD of saved time by using the RMDP method was similar to that of the PDP method, and both methods had lower SD than the PDPa method in many cases. When the treatment effect is an increasing function of time as in Figure S4, the disease progression rate of the drug group is slower than that of the placebo group. For that reason, the new PDPa method had a larger saved time than the PDP method. The SD of saved time by using the PDPa method was also higher than others. In such cases, the RMDP method had the lowest saved time. When a new drug has a constant delayed effect as compared to placebo, saved time comparisons were presented in Figure S5. The PDPa method had the largest saved time, followed by the PDP method and the RMDP method. Their SD of saved time followed the same pattern. Their difference became bigger as the value of constant delayed effect was getting larger.

4. Conclusions

When a study has the follow-up time being longer than 18 months, saved times based on CDR-SB, ADCS-iADL, and MMSE may become shorter as seen in Figure S1. For ADAS-Cog13, the saved time started to decrease after month 24. When the maximum treatment effect occurs between month 12 and month 24 (see left plot in Figure S6), the new RMDP method is still able to provide meaningful difference when the pre-specified follow-up time is beyond 24 months, while other methods are more likely to have a short saved time due to the reduced the treatment effect at the last visit.

We noticed that the disease progression trajectories of the denepezil group and the placebo group based on the GDS had a crossing between the last two visits, as seen in Figure S6. When this crossing occurs, the estimated saved time for the drug group would be negative by using the existing method. The equal decline rate assumption of the existing model is not satisfied any more. In addition to that, the PDP method needs to assume a linear or non-linear disease progression rate of the placebo group beyond the last visit. The PDP method does not account for the treatment effect with the improvement from the drug between baseline and the visit before trajectory crossing. In such cases, the new RMDP method can still provide a reasonable saved time estimate, and this may be the only option to estimate time saving. The RMDP method is recommended to be used in trials having early difference, constant treatment effect, and crossing disease progression curves. The PDP method and the PDPa method may be used for trials with constant delaying and proportionally reduced decline. It should be noted that the PDPa method may have a large variance of saved time as compared to the PDP method. The software R code may be found in the supplementary file section. We also developed an R shiny app to calculate saved time: https://adaptivedesignstrial.shinyapps.io/savedtimermst/.

The PDP method and the new PDPa method often have larger variance of saved time than the RMDP method in many cases. For the five measures in the donanemab phase 2 trial, the variance of saved time using the PDP method was 50% larger than that of the RMDP method. In the donanemab phase 3 trial with more patients, the variance of the saved time was similar to that from the phase 2 data for the PDP method or the PDPa method, while the RMDP method reduced the variance of saved time by around 30% as compared to its variance using the phase 2 data. We also calculated the variance of saved time using data from the donepezil trial with the assumed follow-up times of month 12 to month 36 as presented in Figure S1. The findings were similar to these from the donanemab trials with regard to the variance of saved time. From the variance covariance matrix of the CDR-SB change in the donepezil trial, the estimated variance of the score change increases as time goes on. That trend was also noticed in the ADAS-Cog score change in the ADNI study [12]. It is common in AD trials that the variance of the outcome measure increases during the follow-up visits. When time saving is used in other diseases, it is possible that the variance may stay constant over time. For that reason, we presented the constant treatment effect cases with equal variance of 2 and covariance of 1.5 in Figure S7. The average time saving based on the RMDP method is longer than the other two methods, and the PDP method and the PDPa method often have similar estimated time savings. The SD of saved time using the RMDP method is larger than the SD values of the other two methods for these cases.

With an increased variance, the statistical power to detect treatment effect could be reduced. These variance values were calculated by using data from the placebo group. In AD trials for new drugs, we may not have accurate variance estimate for the primary outcome in the drug group. Meta-analysis would be useful to obtain a reasonable variance estimate from multiple early phase trials to inform future large trials [29]. Therefore, it is important to consider the possible increased variance of score change in the power analysis to increase the success rate of AD trials.

In AD trials, we often have several important outcome measures, such as the primary outcome(s) and the secondary outcomes. These outcomes can all be used to calculate saved time comparing the disease progression curve of the drug group and that of the placebo group. Dickson et al. [30] suggested to use a global statistical test to combine multiple outcome measures to estimate saved time. We would consider the time saving methods in conjunction with the global test to be used for AD trials with multiple outcomes. In addition, we would like to extend these methods for a study comparing multiple treatments in the future. When comparing saved time from different studies, we would use the Wald-type test to evaluate the equality of time saving estimates. For saved times based on outcomes from one study, these saved times are correlated and the test statistics for correlated data would be used for comparison.