Data from transplant patients has many unique characteristics that can cause problems with statistical modeling. The patient's underlying disease / health trajectory is known to affect both longitudinal biomarker values and the probability of both death and transplant. In liver transplant patients, biomarker values show a sharp exponential increase in the days preceding death or transplant. Patients who receive transplants show an immediate drop in biomarker values post-transplant, followed by an exponential decrease. Patients' survival probabilities also change post-transplant, with dependencies on pre-transplant biomarker values. To properly incorporate these clinical features, we developed a joint longitudinal-survival model that links an exponential growth-decay longitudinal model to a modified cure survival model. This allows us to evaluate patient biomarker trajectories and survival times both pre- and post-transplant. The models are linked by patient-level shared random effects that appear in the biomarker trajectories and the frailties of the survival functions. Estimates are obtained via the EM algorithm, with random effects integrated out of the complete data likelihood function using adaptive quadrature techniques. Simulations show our model performs reasonably well under a variety of conditions. We demonstrate our methods using liver transplant data from the United Network of Organ Sharing (UNOS). We use total serum bilirubin as our longitudinal outcome, with age at waitlisting and gender as linear covariates. Gender is used as a covariate in the survival model both pre- and post-transplant.