Abstract
Although many methods have been proposed on the overall survival estimation in randomized trials permitting treatment switching after the progressive disease (PD), the cured subgroup of patients within these trials has not been fully considered. These cured patients would never experience PD and subsequent risk of treatment switching, yet they may suffer death hazard similar to those without the disease. Due to the mix of the cured subgroup, existing methods may yield biased effect estimation for the uncured patients between treatment groups. To address this limitation, we propose a multistate transition model that integrates multi-states of the cure, PD, treatment switching, and death during trials. In this model, the cure probability for all the patients and the death hazard of the cured subgroup are modeled separately. Meanwhile, the semi-competing risks model is used for the treatment effect evaluation on the uncured patients through transitional hazards between states of PD, treatment switching, and death. The particle swarm optimization algorithm is employed to estimate the model parameters. Extensive simulation studies have been conducted to assess the performance of the proposed multistate model in comparison with existing treatment switching adjustment methods. The results show that the treatment effect estimations of our proposed model are more accurate across all scenarios. Moreover, the illustration based on a simulated diffuse large B-cell lymphoma trial demonstrates the applicability and advantages of the proposed model. The robustness of the proposed multistate transition model enables it to accurately estimate the treatment effect in trials that involve a cured subgroup and the treatment switching after PD.
Supplementary Information
The online version contains supplementary material available at 10.1186/s12874-025-02623-0.
Keywords: Multistate model, Treatment switching, Cured subgroup, Semi-competing risks, Effect estimation
Background
In randomized controlled trials (RCT), progression-free survival (PFS) is often adopted as a surrogate endpoint of overall survival (OS) due to shorter follow-up requirements and well prediction of OS. For the ethical and practice concerns [1, 2], some trials with primary endpoint of PFS permit patients in the control group switching to the experimental group after progressive disease (PD) if the treatment efficacy has been evaluated. It is ethically recommended to provide patients with early access to the new treatment to obtain potential benefits. Additionally, allowing treatment switching has the merit of facilitating trial recruitment without impacting on the analysis of short-term endpoint (i.e., PFS), since the balance between groups remains intact before treatment switching. Therefore, the treatment switching is becoming increasingly common, especially in cancer clinical trials [3].
However, when it comes to estimating the long-term survival benefit (i.e., OS), which is critical for health technology assessment (HTA) decision-making and a necessary endpoint in certain therapeutic areas according to health authorities’ requirements, the treatment switching contributes to considerable confounding [4]. Numerous statistical methods have been developed to adjust for treatment switching effects. For example, Robins and Tsiatis [5] proposed the rank preserving structural failure time model (RPSFTM) to estimate the treatment effect in trials with treatment switching. Using accelerated failure time (AFT) models, this approach calculates the counterfactual latent failure time of switchers and the grid search is then employed to obtain the accelerated factor (AF) estimation based on the randomization. Regarding the calculation of counterfactual failure time, White et al. [6] elucidated the necessity of re-censoring to sever the dependence between censoring time and treatment. Further, Branson and Whitehead [7] proposed an iterative parameter estimation (IPE) algorithm to adjust for treatment crossover. Unlike the rank-test approach in RPSFTM, the IPE algorithm used likelihood-based analysis, which greatly expedites the computation of effect estimation. It is worth noting that both the RPSFTM and IPE methods are based on the key assumption of uniform treatment effect (CTE) between the control group switchers and the original experimental patients. However, in practical, if the control group patients suffering PD and consequently switching to the experimental group, their treatment effect may be different from patients initially randomized to the experimental treatment. In such cases, Latimer et al. [8] proposed a two-stage estimation (TSE) method to address the switching effect between the survival of switchers and non-switchers in control group after the PD. This method assumes that, when patients experience PD, the baseline time-dependent prognostic factors for mortality that independently predict treatment switching between groups are assumed remains balanced,, which is referred as “the second baseline” period. The similar assumption is also applied to the inverse probability of censoring weights (IPCW) method [5]. However, a major flaw of these treatment switching adjustment methods [1, 8–13] is that they fail to distinguish the treatment effect on patients with or without PD but adopt a single parameter to summarize the treatment effect between groups.
To describe the effect of different trajectories to death, multistate models have been proposed in the literatures. Specifically, Zeng et al. [14] proposed to model the observed PD and death times through a semi-competing risks model. The logistic model was used to model the occurrence of PD, and then semi-parametric hazard models are used for the transitions among randomization to death, randomization to PD, and PD to death. Zhang et al. [15] extended Zeng’s method by incorporating the gamma frailty model to account for the dependence of survival time on the PD time. Specifically, the model posits that the higher the risk of PD, the greater the death hazard, and the later the onset of PD, the longer the patient’s survival time. Subsequently, a Bayesian procedure was adopted to estimate the treatment effect. Huang et al. [16] adopted copula models to establish the joint distribution of PD and death times. Ristl [17] used weighted log-rank tests to model non-proportional hazards, taking into account delayed treatment effects, treatment switching and heterogeneous patient populations in oncology RCTs. Chen [18] extended Zeng’s method to allow for two-way time-varying switching. With design-based treatment switching in the context of recurrent events data, Chen [19] proposed a semiparametric frailty modeling approach to estimate time-varying effects.
However, with the development of medical science, many diseases, such as early-stage cancers [20], could be clinically curable, enabling patients to have a life expectancy comparable to that of healthy individuals. The cured patients are no longer at risk of developing PD and subsequent treatment switching. Therefore, the cured subgroup exhibits a distinct death trajectory compared to the uncured subgroup, indicating different mortality risks. If the cured subgroup is lumped together with uncured subgroup in semi-competing risks model for the treatment effect estimation, it may introduce bias between groups. In a trial for acute graft-versus-host disease (aGVHD), Lee et al. [21] proposed a novel multistate model taking into account the cured patients insusceptible PD. However, this model did not consider treatment switching. Zhang et al. [22] have also considered the cured fraction and adopted a cure rate model for the PD time with the joint modeling for longitudinal biomarkers and the survival data, but their concept of treatment switching referred to both the treatment and control groups switching to next-line therapies. This is more akin to patients dropping out of the trial the trial rather than the within-group crossover scenario, in which patients from the control group switch to the treatment group.
In this paper, we propose a novel multistate transition model for the treatment effect estimation on OS, which includes multi-states of the cure, PD, treatment switching and death during the course of trials. Within the hypothetical strategy of estimand framework, logistic models are adopted for the patient cured or uncured heterogeneity and their treatment switching choices. Additionally, a semi-competing risks model with patient-specific shared frailty is used to capture the correlation of PD time and survival time. With the particle swarm optimization (PSO) algorithm, the treatment effects on cure rate, PD, and death hazard can be measured by the coefficients of the covariate “group” in each sub-model so that the PD and OS outcomes for both the cured and uncured subgroups can be elucidated accurately.
Methods
Multistate transition model
We considered four states transitions, including cured, uncured, PD and death, in cancer clinical trials as shown in Fig. 1. Dotted lines represent the patient transitioning to a specific state with a certain probability (indicated by the characters on the dotted line). Initially, the study population consists of two heterogeneous subgroups as uncured and latent cured states with the cure probability of π. For a specific patient, his cure status can only be one of the two types, which will not change thereafter nor exist in an intermediate state. Patients who are cured will be no longer at risk of PD and subsequent treatment switching. They face a death hazard (shown as 
) similar to the general healthy population. For the uncured subgroup, they are exposed to a mortality risk as 
, or progress to PD 
and then to death (
). The patients randomized to the control group can switch to the experimental group with a probability of p, and those who experience later PD are assumed with a lower probability of switching [23]. When estimating the long-term treatment effect (i.e., OS), we adopted hypothetical strategy within estimand framework and proposed a novel multistate transition model which is a continuous-time discrete-state Markov process in essence [24].
Fig. 1.
Proposed multistate transition model in which the cured fraction (bold) is innovatively considered
Specifically, let 
or 0 representing the latent cured patient or uncured patient, respectively. The probability of being cured (denoted by π) is expressed as
 |
1 |
where 
or 0 denotes the experimental treatment or the control treatment.
For the cured subgroup, their baseline hazard aligns with the general healthy population, while situation of the uncured subgroup is more complicated. Suppose 
to be the state of PD, treatment switching and death when these indicators = 1, and 
be the time from randomization to PD, to death, and the time from PD to death, respectively. For patients who have experienced PD, there is 
. We convert 
to an ordinal categorical variable 
according to the four quantiles of 
for convenience. A logistic model can be employed to establish the probability of treatment switching (denoted by p) as
 |
2 |
Let 
denote the patient-specific shared frail and follow a gamma distribution with mean one and variance θ, i.e., 
[25], which reflects the correlation between PD and the death hazards for each patient. That is, patients with higher PD risk tend to face greater death hazard, and those who experience later PD would have longer survival time. Based on similar studies [14, 21], we elucidate the transition hazards between states 1–4 as follows.
 |
3 |
Therefore, the first equation 
expresses just the baseline hazard similar to the general healthy population. The second and third equations describe the risk of PD (
) and death for the uncured subgroup (
), respectively. While the last equation 
indicates the hazard of death after PD for the uncured subgroup with the coefficient β34,1 and β34,2 for the treatment effects of non-switchers and switchers, respectively. The main parameter estimates and their interpretation are listed as Table 1.
Table 1.
The main parameter estimated in the multistate transition model
| Parameters | Interpretation |
|---|---|
 |
The probability of being cured in the  group |
 |
The probability of treatment switching among patients in the control group |
 |
The state of PD, treatment switching and death when these indicators = 1, and vice versa |
 |
The time from randomization to PD, to death, and the time from PD to death |
 |
The patient-specific shared frailty indicator |
| λ | The baseline hazards for each subpopulation with different state by the respective equation in Eq. (3) |
| β23 | The treatment effect on risk of PD for the uncured subgroup |
| β24 | The treatment effect on risk of death without PD for the uncured subgroup |
 34
|
The treatment effect on the hazard of death after PD for the uncured subgroup (with β34,1 for non-switchers and β34,2 for the switchers) |
Maximum likelihood estimation (MLE) of the treatment effect
Suppose a trial with N patients, the possible observed status scenarios for each patient can be listed as Table 2. Patients with observed PD are undeniably uncured as in statuses A-D (
), while patients without observed PD could belong to either the latent cured or the uncured subgroup. Since patients in the control group are allowed to switch to the experimental group only after the occurrence of PD, the switching indicator is not applicable in statuses E and F.
Table 2.
Observed status indicator and likelihood contribution corresponding to six possible scenarios during the trial
| Status | Description | Observed data 
|
Likelihood |
|---|---|---|---|
| A | Observed PD, switching, and death | (1, 1, 1) | f1 |
| B | Observed PD, switching, and censored death | (1, 1, 0) | f2 |
| C | Observed PD and death without switching | (1, 0, 1) | f3 |
| D | Observed PD and censored death without switching | (1, 0, 0) | f4 |
| E | Observed death without PD | (0, -, 1) | f5 |
| F | Censored PD and censored death | (0, -, 0) | f6 |
Based on the observed data, the MLE method can be employed to estimate the parameters. Firstly, the likelihood functions 
are the marginal distributions with respect to the distribution of shared frailty (
), i.e., 
. Moreover, the likelihood functions 
should be marginalized with respect to the distribution of c. Let 
and 
denote the certain state probability of cured and treatment switching, and 
be the indicator function for the treatment group or control group, where the values and meanings are consistent with the definitions above. For example,
represents the uncured probability, 
represents the probability of the uncured control group patients switching to the treatment group. Consequently, the corresponding conditional distributions can be calculated as follows,
 |
 |
 |
 |
 |
 |
where 
, 
,
, and 
.
Assume the transitional baseline hazards 
are constant, with marginal distributions of the conditional distributions above, the observed data likelihood for 
is
 |
4 |
To maximize 
in Eq. (4), let
 |
 |
5 |
Then, the goal is to find 
that maximize 
in Eq. (5).
Particle swarm optimization (PSO) algorithm
Although Laplace transformations-based approaches has be utilized by some researchers [26], we employed a simple and easily implemented PSO method in this study. The PSO algorithm is a population-based stochastic optimization technique developed by Kennedy and Eberhart [27] with the operational principle shown as Fig. 2. The globally optimal likelihood value is obtained via stochastic searching and optimization of candidate parameter estimate (denoted by particles). Once maximum number of iterations is reached or the change of the estimated likelihood falls below a certain threshold, the algorithm converges. Therefore, it is easily applied without complex formula derivations and widely applicable for optimization problems without the requirement of the concavity property of objective functions [28]. Given the simple iterations with low computational complexity, the PSO algorithm is theoretically fast -converging to meet the practical requirement. The step-by-step descriptions of PSO and more details of the initial particle setup are provided in Appendix 1.
Fig. 2.
Flow chart of the PSO algorithm [29]
Simulation study
Study design
We simulated independent datasets based on an RCT with randomization ratio of 1:1 to the experimental treatment and control groups and a certain cured proportion in both groups. For cured patients, the PD would not occur, and their risk of death was low and not related to the treatment. Uncured patients in the control group are allowed to switch to the treatment group by the time elapsed until PD. Assuming the transition hazards between states are known, the actual values of the parameters are as follows for the basic scenario,
 |
 |
 |
Specifically, the cure rates in treatment and control groups are 30% and 15%, respectively, the values of 
and 
are calculated accordingly. With the switching proportion of 30% and the value of 
and 
could be computed via iterative processes. Under the setting of 
and
, approximately 20% of uncured patients would die without PD. The treatment therapy has the same treatment effect on the uncured patients regardless of risk to PD, death or switchers, that is, 
All the simulations were performed in SAS 9.4 (SAS Institute Inc., Cary, NC, USA).
Comparison methods for switching adjustment
To evaluate the performance of the proposed multistate transition model, we also present several reference methods for comparison, encompassing include simple treatment switching adjustment methods and multistate model methods as follow:
intention to treat (ITT) analysis which ignores the treatment switching. It can also be interpreted as treatment strategy within the estimand framework, which ignores the intercurrent events such as PD and switching and just compares treatment effects on OS among patients in their initially assigned groups;
per-protocol analysis which censors the switchers at the PD time (PPcen). It aligns with the while-on strategy within the estimand framework, which evaluates treatment effects only during the period when patients adhere to their initially assigned treatment;
per-protocol analysis which excludes the switchers from the analysis dataset (PPexc);
RPSFTM method estimates the treatment effect via grid searching (GE) [5];
RPSFTM method estimates the treatment effect via iteration parameter estimation (IPE) [7].
It is noted that re-censoring is considered in GE and IPE methods.
Investigated scenarios
To evaluate the robustness of the proposed multistate transition model compared with other methods, the parameters are configured as follows across scenarios.
Cure rate: Assume that the cure rate in the control group remains at 15%, while the cure rates in the experimental group are 15%, 30% and 45%.
Switching proportion: 10% (low), 30% (moderate), and 50% (high).
- Treatment effect assumption:
-
① CTE satisfied: the treatment effect on OS is the same for control group switchers and patients initially randomized to treatment group, regardless of whether the PD occurs, that is, i.e.,
.When CTE is not satisfied, two scenarios are considered as follows: - ② The treatment effects on OS are different for patients with or without PD and we assume that the treatment has a greater protective effect on patients without PD, i.e.,
. - ③ The control group switchers benefit less from the treatment compared to patients initially in the treatment group, i.e.,
.
-
Sample size: 500, 1000, and 2000.
Performance measures
The main purpose of this paper is to estimate the treatment effect on OS with a proportion of cured patients in both groups and allowance for treatment switching after PD in control group. To evaluate the performance of proposed methods for the treatment effects estimation of multi-states, we assess the accuracy of 
, and β34,1 by the percentage bias (PB(%)), mean squared error (MSE), and empirical standard error (SE) of the estimate. Specifically, PB(%) is calculated as 
, where 
refers to the true value of parameters and 
denotes the mean of the parameter estimates. MSE is calculated as 
, where 
is the number of the simulation replicates.
It should be emphasized that since some simple treatment switching adjustment methods (i.e., ITT, PPcen, PPexc, GE, IPE, and TSE) ignored the cured or uncured subgroups’ different impacts on PD, they estimate the average treatment effect across all patients and states. To compare with these methods, we evaluate them from two perspectives. Firstly, the treatment effect on death without PD (
) the target parameter, and we use it as the true value for comparison. Second, under each scenario, we simulate data for 1 000 000 patients without incorporating treatment switching and estimate the average hazard ratio (
) on OS. The logarithm of the average HR, denoted as 
, serves as the true value to evaluate the performance of the simple treatment switching adjustment methods. For GE, IPE, and TSE methods, the survival time data adjusted by the estimated AF are used to estimate the HR via a proportional hazard model [6].
Results
In preliminary simulations, we had examined the impact of the number of simulations on result accuracy and stability under some specific scenarios. With the estimation PB(%), MSE and SE, results from 100 simulations showed minor variation and stable outcomes, indicating acceptable accuracy and robustness. Given the large number of parameters, diverse simulation scenarios, and the computationally intensive nature of the process, we simulated 100 datasets for each scenario in subsequent broader simulations to balance computational efficiency with result reliability.
The simulation results are displayed in two parts. In the first part, the multistate models are compared, specifically the proposed multistate transition model with or without shared frailty against Zeng’s semi-competing risks model. As shown in Table 3, the proposed multistate transition model could provide an almost unbiased estimation of the parameters via the PSO algorithm, especially for the baseline hazards of transitions between states (i.e., 
, and 
). For most parameters, excluding 
, the estimation bias is less than 4% of the true value. However, when the proposed model is applied without considering shared frailty (middle section of Table 3), the estimation bias increases significantly. It is observed that the baseline hazards (i.e., 
, and 
) are underestimated uniformly. The treatment effects are also underestimated from the perspective of HR (i.e., 
and 
), while the treatment effect for switchers (i.e., 
) is overestimated. The probabilities of cure and treatment switching are also overestimated. As shown in the last four columns in Table 3, the semi-competing risks model without considering the cured fraction and shared frailty exhibits substantial estimation biases, particularly in the effect estimate on the death without PD (i.e., 
). These findings underscore the importance of incorporating shared frailty and accounting for the cured fraction in multistate models to achieve accurate and reliable parameter estimates.
Table 3.
Parameter estimation performances of the proposed multistate transition model with or without shared frailty and Zeng’s semi-competing risks model under the basic scenario, N = 2000
| Parameter | True value | Proposed multistate transition model | Multistate transition model without considering shared frailty | Semi-competing risks model | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Est | PB (%) | SE | MSE | Est | PB (%) | SE | MSE | Est | PB(%) | SE | MSE | ||
| a0 | −1.730 | −1.770 | −2.338 | 0.201 | 0.041 | −1.174 | 32.147 | 0.069 | 0.314 | ||||
| a1 | 0.890 | 0.917 | 3.065 | 0.190 | 0.036 | 0.933 | 4.848 | 0.104 | 0.013 | ||||
| b0 | −3.870 | −3.917 | −1.224 | 0.299 | 0.091 | −3.012 | 22.169 | 0.301 | 0.826 | ||||
| b1 | 1.099 | 1.112 | 1.223 | 0.095 | 0.009 | 2.118 | 92.770 | 0.102 | 1.049 | ||||
| λ14 | 0.0003 | 0.0003 | −7.285 | 0.001† | 0.001† | 0.001† | −20.342 | 0.001† | 0.001† | ||||
| λ23 | 0.020 | 0.020 | −0.913 | 0.002 | 0.001† | 0.013 | −35.298 | 0.001 | 0.001† | 0.016 | −21.880 | 0.001 | 0.001† |
| β23 | −0.916 | −0.913 | 0.371 | 0.117 | 0.014 | −0.536 | 41.490 | 0.075 | 0.150 | −0.534 | 41.773 | 0.073 | 0.152 |
| λ24 | 0.005 | 0.005 | 1.722 | 0.001† | 0.001† | 0.003 | −36.624 | 0.001† | 0.001† | 0.002 | −62.394 | 0.001† | 0.001† |
| β24 | −0.916 | −0.917 | −0.118 | 0.146 | 0.021 | −0.566 | 38.236 | 0.129 | 0.139 | −0.786 | 14.181 | 0.112 | 0.029 |
| λ34 | 0.030 | 0.030 | −0.877 | 0.002 | 0.001† | 0.019 | −36.511 | 0.001 | 0.001† | 0.019 | −36.790 | 0.001 | 0.001† |
| β34,1 | −0.916 | −0.896 | 2.216 | 0.114 | 0.013 | −0.654 | 28.644 | 0.098 | 0.078 | −0.678 | 26.016 | 0.107 | 0.068 |
| β34,2 | −0.916 | −0.908 | 0.950 | 0.134 | 0.018 | −1.084 | −18.354 | 0.128 | 0.045 | −1.127 | −23.001 | 0.121 | 0.059 |
| θ | 1.000 | 0.991 | −0.922 | 0.125 | 0.016 | ||||||||
Est parameter estimate, PB(%) percentage bias, SE empirical standard error, MSE mean squared error. γ†: values less than 0.001
Results comparing the three multistate model methods under various scenarios are shown in Tables S1-S8 of the Appendix 2 in Additional file. On the whole, the performances of the three methods across scenarios with different cure rates, switching proportions, TE assumptions, and sample sizes aligns with the findings in the basic scenario. The estimation bias from the proposed multistate transition model consistently demonstrates small estimation biases across most scenarios, except in cases with a small sample size (N = 500, see Table S8). This limitation arises because a sample size of 500 may be insufficient to reliably estimate so much parameters simultaneously. because sample size of 500 might be too small to estimate 13 parameters at once. Additionally, the estimation of the parameter 
is less satisfactory in some scenarios due to the influence of extreme estimates. This is likely because the true value of 
(i.e., 0.0003) is extremely small to be particularly susceptible to inaccuracies. But we can also find that the parameter estimation biases are larger when the shared frailty is not considered in the proposed multistate transition model. In contrast, Zeng’s semi-competing risks model exhibits substantial biases across all scenarios. Specifically, as the cure rate ratio increases, the parameter 
is markedly overestimated (see Tables S1-S2 and Table 3). This occurs because the increased cured patients in the experimental group greatly decrease the overall death hazard, thereby amplifying the treatment effect of the experimental therapy. Furthermore, with the increase of the switching proportion, the estimation biases of the parameters 
and 
decrease while the estimation bias of the parameter 
increases (see Tables S3-S4 and Table 2). With the decrease of the sample size, the SE and MSE of the parameter estimate increas for all three methods (see Tables S7-S8 and Table 3).
In the second part, the performances of simple treatment switching adjustment methods are investigated. Figure 3 shows PB(%) of the parameter estimations under scenarios with varying cure rates (Fig. 3 A1-A2), switching proportions (Fig. 3 B1-B2), TE assumptions (Fig. 3 C1-C2), and sample sizes (Fig. 3 D1-D2). The logarithm of HR for death without PD (i.e., 
) is served as the true value as shown in the left column in Fig. 3. For simple treatment switching adjustment methods, the estimation performance is also evaluated using the logarithm of average HR for death across all patients, as shown in the right column in Fig. 3.
Fig. 3.
PB(%) of the estimate using the proposed model and contrast methods under different scenarios. *: Only for control methods (i.e., homogeneity assumption-based methods). TE: treatment effect. In C1-C2, scenario 1 is the case that the CTE assumption is satisfied with 
; scenario 2 is the case that the CTE assumption is not satisfied with 
; scenario 3 is the case that the CTE assumption is not satisfied with 
N: sample size
As shown in Fig. 3, the performance of the proposed multistate transition model demonstrates robust performance with scenarios of varying cure rates, switching proportions, and TE assumptions. In cases of small sample sizes, the estimation bias increases but remains still acceptable (less than 10%). In contrast, the simple treatment switching adjustment methods exhibit significant estimation biases for 
. Moreover, the bias is positive in most scenarios, which means the protective treatment effect of the experimental therapy is underestimated. This is because the treatment switching after PD in the control group narrows the gap in death hazard between groups. And the presence of the cured patients dilutes the experimental effect. In comparison with the left column, the estimation biases of simple methods are notably smaller in the right column in Fig. 3. Because these methods measure the average treatment effect in essence, which is consistent with the true value in the right column. Nevertheless, the performances of ITT and PPcen methods remains suboptimal. When estimating the average treatment effect with simple treatment switching adjustment methods, the cure rates and switching proportion significantly influence the estimation performance (see Fig. 3A2 and B2). Specifically, as the cure rate in the experimental group increases, the average death hazard in the experimental group decreases and naturally the bias for the average treatment effect decreases.
The details of the estimation bias variations with the cure rate, switching proportion, TE assumption, and sample size for simple treatment switching adjustment methods and the proposed multistate transition model are as follows.
The simulations in Fig. 3A1 were conducted under the scenario of N=2000, in which CTE assumption is satisfied with switching proportion of 30%. It is observed that the bias of the proposed multistate transition model remains small (less than 2%) across different cure rate scenarios. In the scenario of 15% vs 15% (i.e., the cure rates are equal in the experimental and control groups), the biases of 
from simple methods are extremely large. As the cure rate in the experimental group increases to 30% vs 15% and further to 45% vs 15%, the underestimation of the treatment effect gradually diminishes and the bias of 
decreased accordingly. Notably, under the scenario of 45% vs 15%, the larger cure rate in the experimental group leads to an overestimation of the treatment effect for PPexc, GE, IPE, and TSE methods.
Figure 3B1 illustrates that as the switching proportion increases, the estimation bias for all methods also increases. But the bias of the proposed multistate transition model is much smaller than that of the simple methods. With a sample size of 2000, the bias of the former remains below 1%, while the biases of other methods exceed 20%. The estimation biases of ITT and PPcen methods are particularly sensitive to the switching proportion.
In Fig. 3C1, the CTE assumption is satisfied in scenario 1, where the treatment effect on OS is a constant before and after progression across all uncured patients, but not in the other two scenarios. It is observed that the proposed multistate transition model performs well in all scenarios. The bias of simple methods increases in scenario 2. Because in scenario 2, the treatment effect in progressed patients and switchers is smaller, leading to a decrease in the average treatment effect and exacerbating the underestimation. Compared to scenario 1, the treatment effect in switchers is smaller in scenario 3, i.e., the switchers benefit less in comparison with the initially experimental patients. The estimation biases of ITT, GE, IPE, and TSE methods decrease in scenario 3. This is because the decreased switching effect lessens the impact of treatment switching on OS. Additionally, the estimation biases of PPcen and PPexc methods remain unchanged in scenario 3 compared to that in scenario 1, as the analysis excludes observations after treatment switching, thus the switching effect does not influence these methods.
Figure 3D1 shows the estimation bias relative to the sample size. It is observed that the proposed multistate transition model is more sensitive to the sample size due to much more parameters it requires to estimate. The estimation bias increases as the sample size decreases. In contrast, the performance of simple methods is less affected by the sample size variations. Nevertheless, even with smaller sample sizes, the estimation bias of the proposed multistate transition model remains lower than that of simple treatment switching adjustment methods.
Furthermore, the robustness of the proposed multistate transition model was evaluated under the assumption that the survival time follows a Weibull distribution. The simulation results confirm that the model maintains its robustness (see Appendix 3).
Case study
Diffuse large B-cell lymphoma (DLBCL) is a curable lymphoma [30]. The standard treatment for DLBCL consists of cyclophosphamide, doxorubicin, vincristine, and prednisone (CHOP) chemotherapy with anti‐CD20 antibody rituximab (R) [31]. Evidence indicates that approximately 45% of DLBCL patients would be cured with R-CHOP therapy [32]. However, patients experiencing treatment failure often face poor prognoses. Therefore, it is reasonable to allow patients who progressed after R-CHOP treatment to switch onto the experimental arm in RCT testing the effectiveness of a new treatment for DLBCL.
In our simulation, we posited a novel treatment for DLBCL with a cure rate of 50% and an HR of 0.6 (
) relative to R-CHOP therapy. We generated simulation datasets of the RCT comparing the new treatment and R-CHOP, allowing for crossover from the R-CHOP arm to the experimental arm upon disease progression. The simulation parameter were based on Coiffer’s research [32], with further details elaborated in Appendix 4 of the Additional file. With the switching proportion of 10%, 30%, and 50%, the sample size of 2000, 1000, and 500, the bias of the HR estimates of the new treatment obtained by the proposed multistate transitional model, Zeng’s semi-competing risks model, ITT, PPcen, PPexc, GE, IPE, and TSE methods. Figure 4 shows that based on a single dataset, the proposed model exhibits substantially lower estimation bias compared to the other methods. As the switching proportion escalates, the estimation bias of comparator methods increases, whereas the proposed model maintains robust performance. Under the scenario of N = 1000, the estimation bias of both the proposed model and comparator methods is heightened, potentially attributable to sampling error within the simulated dataset. This observation further emphasizes the proposed model’s relatively higher dependence on sample size, likely due to its incorporation of a larger number of parameters.
Fig. 4.
HR estimation bias of the proposed model and contrast methods based on the DLBCL dataset
Discussion
To estimate the treatment effect on OS in RCTs with treatment switching permit in the presence of a cured subgroup, we propose a novel multistate transition model which integrates the latent cure, PD, treatment switching, and death into a unified framework. The proposed model separately measures the treatment effect on the cure rate, PD hazard, and death hazard with or without progression with distinct parameters. It delineates three trajectories to death in the model, including death among cured patients, death without PD among uncured patients, and death after PD among uncured patients. Simulation studies under various scenarios show the good estimation performance and robustness. The proposed model can provide accurate estimation on OS in the presence of heterogeneous cured subpopulation, as well as multi-state transitions such as PD and treatment switching.
In oncology trials, some traditional methods for treatment effect estimation on OS includes ITT, PPcen, PPexc survival hazard using rank-test approaches, GE or IPE with RPSFT methods, and TSE methods. However, these methods fail to account for the multiple trajectories to death in patients, such as the disease cured, disease progress and treatment switching and assume the uniform treatment effect across all patients and all disease stages. As shown in the simulation study, when the goal is to estimate the treatment effect on the hazard of a specific transition, such as the death hazard in patients without PD, these traditional methods can provide just the average treatment effect estimation and introduce substantial biases. The magnitude of these biases varies with the cure rate, switching proportion, treatment effect, and switching effect. For example, the PPcen method produces the largest bias when estimating the average treatment effect while the GE, IPE, and TSE methods perform relatively better. On the contrast, our method provides methodological support for precise treatment effect estimations on transition hazards between multiple states in oncology research. From the perspective of clinical practice, the parameter estimations are easily interpretable for clinicians, enabling them to develop individualized treatment strategies for patients in each specific state, which may address challenges encountered in the clinical practice.
In this study, Zeng’s semi-competing risks [14] model was applied as another multistate model for comparison in the simulation stud. Unlike Zeng’s method, our proposed model incorporates the cured fraction and patient-specific frailty, while assuming constant transitional baseline hazards to simplify model construction. The simulation results reveal that failing to account for the cured fraction and patient-specific frailty leads to significant estimation biases. Consequently, the proposed model is superior, as the inclusion of the cured fraction and frailty indicator enhances its scientific rigor, methodological soundness, and practical applicability. We also note that Ristl’s method [17] may be, to some extent, comparable to our proposed model. Ristl used weighted log-rank tests to model non-proportional hazards with delayed treatment effects, treatment switching and different biomarker-defined patient populations in oncology trials. Although he also considered the impact of population heterogeneity on the treatment effect, we focus on the mortality risk and treatment effect estimation of uncured patients who switch treatment groups after PD, especially distinguishing them from cured patients who would not experience PD. Ristl simulates survival data through piecewise constant hazards and utilizes the weighted log-rank tests for non-proportional hazards, which can only provide the hypothesis testing results of the superiority of the experimental treatment over control. In contrast, we conducted the simulation with different probabilities for various subpopulation with multiple states, enabling the accurate estimation of treatment effect on OS for the heterogeneous cured subpopulation, as well as transitions related to PD and treatment switching. Therefore, the detail multi-states and their trajectories to death in this study were quite different from Ristl’s method. Consequently, the specific parameters and testing methods differ significantly between the two models.
This study had some highlights. The proposed multistate transition model innovatively takes the cured fraction of patients into consideration. By delineating different trajectories to death within the proposed model, the treatment effects for patients in various states are quantified by different parameters so that more detail information and treatment strategies can be provided for patients in different disease states. When the treatment effect estimates align in the same direction, it means that the experimental treatment benefits all patients. However, when the treatment effect estimates are in inconsistent across states, scientific rationale analysis and further confirmation research are recommended. If such inconsistencies are confirmed, it may suggest that the experimental treatment benefits only a subset of patients, highlighting the need for personalized treatment strategies.
However, the model remains somewhat complex due to the necessity of estimating numerous parameters across multiple states transitions, which may limit its practicality. In this study, we have made efforts to simplify the model’s formulation as much as possible, with detailed derivations provided in the appendix. To address the computational challenges, the PSO algorithm was employed to estimate all parameters in a single run through straightforward iterations. While it does not directly provide confidence intervals (CIs), bootstrap methods may be considered for CIs estimation in the future work. For optimizing parameter setting, we recommend that the inertia weight may be started with a higher value (e.g.,0.9) to enhance global exploration to avoid premature convergence, and then reduce it (e.g., 0.4) to improve local refinement and ensure precise parameter estimation. The accelerating factors can remain the classic setting as 
. The number of particles significantly impacts both iteration frequency and time required for convergence. We recommend using particles be at 5–10 times the number of parameters being estimated and at least 30 particles to ensure sufficient parameter space coverage. Naturally, the number of particles directly affects computation efficiency. In our simulation, a standard i5 computer may require 10–20 min with 100 particles. Fortunately, increased sample sizes of the trials do not substantially prolong computation time, while significantly enhancing estimation accuracy. Additionally, if concerning the parameter sensitivity, the global best fitness curve across iterations may be applied to assess convergence behavior under different parameter settings for convergence monitoring and optimal progress.
The proposed model was tested on data generated using the same underlying assumptions in the simulation study, which raises concerns about potential overfitting and inflated performance. The multistate transition model necessitates precise longitudinal data with supportive variables, particularly regarding the time-to-events information, such as PFS, treatment switching and survival status. It is challenging to identify a real trial with data encompassing the multiple disease states described in our research process, because accurately determining whether a patient is cured remains difficult in practice. The collection of real-world data in routine clinical environment often present significant challenges for data management. The interval censoring, incomplete or inaccurately recorded event timing may substantially affect both the accuracy validity of model estimation. Additionally, the model proposed adopts a piecewise-constant baseline hazard structure and has been tested for robustness under exponential and Weibull distributions. Potential biases or performance degradation in more complex real-world scenarios (such as non-proportional hazards) should be fully addressed. Furthermore, our estimand framework primarily employs a hypothetical strategy for intercurrent events, which implicitly assumes full compliance with switching probabilities. It does not account for random or informative treatment switching, protocol deviations, or other non-random switching patterns. For these intercurrent events in real-world practice, alternative strategies should be considered to avoid introducing bias. While, the issue of multi-state transitions indeed stems from clinical practice, and the proposed model may address the urgent requirements encountered by oncology clinician for decision-making strategies. Therefore, future work needs to expand on this topic to substantially strengthen the credibility and generalizability of the model.
Conclusions
The accurate and robustness of the proposed multistate transition model qualify it to estimate the treatment effect on OS in oncology trials involving cured subgroup and treatment switching permit after PD. As a theoretical exploration, this model provides a certain methodology for survival estimation in multistate transition.
Supplementary Information
Additional file 1. Additional file.pdf contains the details of the PSO algorithm and the initial particle setups (Appendix 1), simulation results of the proposed multistate transitional model and Zeng’s semi-competing risks model under other scenarios (Appendix 2), simulation results of the proposed model when the exponential distribution extends to the Weibull distribution (Appendix 3), and the parameter setting consideration for the simulated DLBCL dataset used in the case study (Appendix 4).
Acknowledgements
The authors would like to thank Dr. Xueqian Wang in Tsinghua University for his helpful suggestions which significantly improved thequality of the paper.
Abbreviations
- AF
Accelerated factor
- AFT
Accelerated failure time
- aGVHD
Acute graft-versus-host disease
- CTE
Common treatment effect
- DLBCL
Diffuse large B-cell lymphoma
- EM
Expectation-maximization
- GE
Grid estimation
- HR
Hazard ratio
- HTA
Health technology assessment
- IPCW
Inverse probability censoring weights
- IPE
Iterative parameter estimation
- ITT
Intention-to-treat
- MSE
Mean squared error
- OS
Overall survival
- PB
Percentage bias
- PD
Progressive disease
- PFS
Progression-free survival
- PP
Per-protocol
- PSO
Particle swarm optimization
- R-CHOP
Rituximab, cyclophosphamide, doxorubicin, vincristine, and prednisone
- RCT
Randomized controlled trial
- RPSFTM
Rank preserving structural failure time model
- SE
Standard error
- TE
Treatment effect
Authors’ contributions
HH and CL proposed the conception and designed the work. HH, CL and LW performed the data analysis and drafted the manuscript. JX, KW and WG participated in the results interpretation and manuscript revision. All the authors reviewed and approved the final manuscript.
Funding
This work was supported by the National Nature Science Foundation of China (Grant No. 82273728, 82273729 and 82373680). The funding bodies played no role in the design of the study and collection, analysis, and interpretation of data and in writing the manuscript.
Data availability
No datasets were generated or analysed during the current study.
Declarations
Ethics approval and consent to participate
The text is based on methodological simulation research. The case study is a hypothetical scenario, which does not involve actual research on humans, and hence exempt from approval by ethics committees and informed consent.
Consent for publication
Not applicable.
Competing interests
The authors declare no competing interests.
Footnotes
Chen Li is the primary corresponding author.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Contributor Information
Ling Wang, Email: wl.medstat@qq.com.
Chen Li, Email: lc.biosta@qq.com.
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Associated Data
This section collects any data citations, data availability statements, or supplementary materials included in this article.
Supplementary Materials
Additional file 1. Additional file.pdf contains the details of the PSO algorithm and the initial particle setups (Appendix 1), simulation results of the proposed multistate transitional model and Zeng’s semi-competing risks model under other scenarios (Appendix 2), simulation results of the proposed model when the exponential distribution extends to the Weibull distribution (Appendix 3), and the parameter setting consideration for the simulated DLBCL dataset used in the case study (Appendix 4).
Data Availability Statement
No datasets were generated or analysed during the current study.