# Joint models with time to progression

Consider a RCT (Randomized Controlled Trial) which aims at assessing the efficacy of a drug in patients suffering from a given cancer. In this trial, $$p$$ individuals are observed at several time points. At a certain point in time, some individuals are given a placebo and others are given the drug. Assume (out of simplicity) that each patient only has one tumor. The efficacy of the treatment (for a given individual $$i$$) can be assessed by measuring the tumor size $$M_i(t)$$. Longitudinal modeling can be used to gain insights on the way the tumor size evolves over time.

The true time of disease progression, denoted by $$P_i$$ $$(1 \leq i \leq p$$), can be defined as the earliest time at which some criteria is met. In the following, we define progression time to be the time at which $$M_i(t) > c M_i(0)$$ (for some $$c > 1$$). This progression time is usually not observed as it may occur between two consecutive visits. We can define $$P_{i}^{\mathrm{obs}}$$ the earliest visit at which the criteria is met. In addition to this, this progression time can be right-censored as individuals may dropout before progression is actually observed. Let $$C_i$$ denote the censoring time. We actually observe $$T_i = \min(C_i, P_{i}^{\mathrm{obs}})$$ and the tumor sizes at each visit $$\mathbf{m}_i = (m_{i,j})_{1 \leq j \leq n_i}$$. In this situation, the time-to-event is censored but the observations are censored too. An individual who reaches "progression" is removed from the RCT.

Joint models allow to combine survival analysis (here, the "time-to-event" is the time to progression) and longitudinal data analysis (modeling of tumor growth). As presented in this book by D. Rizopoulos), a joint model consists of a Cox proportional hazard model for the survival part and a Linear Mixed Effects (LME) model for the longitudinal part. Given that, for each individual, the observations are: $$(T_i, \delta_i, \mathbf{m}_i)$$ with $$\delta_i = \mathbb{1}_{P_{i}^{\mathrm{obs}} \leq C_i}$$, one can write the likelihood $$p(T_i, \delta_i, \mathbf{m}_i \mid \theta)$$ as:

$$p(T_i, \delta_i, \mathbf{m}_i \mid \theta) = \int p(T_i, \delta_i, \mathbf{m}_i \mid \mathbf{b}_i, \theta) p(\mathbf{b}_i \mid \theta) \, d\mathbf{b}_i,$$

where $$\mathbf{b}_i$$ denote the random effects of the LME model and $$\theta$$ denote the model parameters. A key assumption is the following: conditionally on $$\mathbf{b}_i$$, the survival part and longitudinal part are independent. That is:

$$p(T_i, \delta_i, \mathbf{m}_i \mid \mathbf{b}_i, \theta) = p(T_i, \delta_i \mid \mathbf{b}_i, \theta) p(\mathbf{m}_i \mid \mathbf{b}_i, \theta).$$

I am wondering whether this assumption (the conditional independence) actually holds in the situation I presented above. In many examples [used to illustrate joint models], there is no explicit relationship between the progression of the measurement and the time-to-event. For instance, there is no explicit relationship between time to death and the number of CD4 cells in patients suffering from AIDS. Still, in the present case, the relationship between the time-to-event (progression) and the longitudinal trajectory (tumor size) is explicit.

How can I include this explicit relationship between the time-to-event and the longitudinal trajectory in a joint model? More specifically, should the survival part of the joint model depend on the threshold $$c$$?

It seems to me that the only information is in the tumor size process $$M_i(t)$$, $$i = 1, \ldots, n$$. Either

• you treat this as a longitudinal outcome evaluated at some follow-up times $$t_{ij}$$, $$j = 1, \ldots, n_i$$, and you fit an appropriate mixed effects model describing the average longitudinal evolutions;
• or you are interested in the time until $$M_i(t) > c M_i(0)$$ that should be interval censored data, and you fit an appropriate survival model for it.

It is not evident why you want to consider the same process twice in a joint model.

At the point of model convergence, you will have (Pearson / Schoenfeld) residuals for the mixed and Cox models. Plot the residuals against each other. Inspect any possible trend. If the residuals show a trend, there are likely time-dependent effects of treatment on tumor size/status and a more sophisticated treatment should be handled such as time varying covariates.

• I think I understand your question better. Basically, you are concerned because the time-to-PD is "explicitly" (completely) dependent on the tumor size, that simply adjusting for treatment assignment doesn't work, right? Are you using RECIST to measure PD? If so, then the only case that PD isn't a function of target or non-target lesion growth is the appearance of new lesions. Fit that as a separate event time. Does that answer your question? – AdamO Nov 22 '19 at 16:45
• Yes, time-to-PD depends on the tumor size. Indeed, "progression" is defined as the time at which the tumor becomes "too big". In a joint model, the survival part assumes that: $$h_i(t \mid M_i(t), w_i) = h_0(t) \exp\left( \gamma^{\top}w_i + \alpha m_i(t) \right),$$ and $$S_i(t) = \mathbb{P}\left( P_i^{\mathrm{obs}} > t \right) = \exp\left( \int_{0}^{+\infty} h_i(s) \, ds \right),$$ with $w_i$ some baseline covariates. I was under the impression that this could be modified to make the relationship between time-to-pd and tumor size explicit. – Pouteri Nov 22 '19 at 17:05
• @Pouteri could you clarify if disease progression is RECIST or not? – AdamO Nov 22 '19 at 17:08
• It could be another progression criteria than RECIST. We just assume that the progression time can be explicitly obtained from the time-varying covariates (RECIST is an example). – Pouteri Nov 22 '19 at 17:12
• @Pouteri basically, response assessment (change in tumor volume or area from baseline) is precisely what is used to determine whether PD occurred. So conditional on the longitudinal volume of a single-lesion tumor, there is 0 information added by the designation of "PD". – AdamO Nov 22 '19 at 17:19