Assessment of death risk of breast cancer patients with joint frailty models

Breast cancer and risk factors

Breast cancer, as a metastatic type of cancer, has a higher survival rate and better prognosis with screenings for early diagnosis. However, treatment is highly difficult in the advanced stages. Breast cancer patients require follow-ups after the administration of treatment.5,6

While risk factor evaluation for breast cancer shows that it rarely develops before the age of 30, the risk increases with advancing age and in a gradual manner after menopause. In addition, genetic factors are also important, as patients with a family history of breast cancer have a higher risk of developing the disease. The risk of the development of breast cancer is also higher in patients diagnosed with colon, ovarian, and uterine cancers. Hormones play a significant role in the development of breast cancer. In addition to these factors, there are factors that can be controlled by the individual, such as smoking, alcohol consumption, lack of physical activity, and being overweight, all of which increase breast cancer risk. However, these factors may be eliminated by taking personal precautions.7,8

In recent years, there have been a great number of developments in breast cancer treatment. However, the clinical condition and disease process for each patient varies, showing that the disease is heterogeneous. Therefore, recent developments in the personalized treatments for the disease are proving useful.9

Prognostic factors are also important for the development of individualized treatments. Individual factors such as the clinical stage of the disease, treatment type, tumor number and size, hormone receptors, and the number of remaining lymph nodes have a high impact on the mortality rate.10

Joint frailty models

In cancer studies, there are generally several endpoints before death. All these different endpoints must be simultaneously evaluated in order to increase the efficacy of the clinical decision procedure. Because most breast cancer deaths are related to recurrence, patients with recurrence are expected to have a higher risk of death than patients without recurrence. Thus, patients experiencing recurrence are frailer than others. Moreover, this heterogeneity (frailty) results from differences in recurrence as well as unobserved factor effects. Frequently, there are heterogeneities among patients because of unexplained factors. As it is not possible to measure all factor effects in models, in these situations, joint frailty models should be used. These models simultaneously investigate the recurrence and death processes and consider the dependency between recurrence and death. Both events are randomly associated during the investigation. Because unobservable effects cause heterogeneity, the random effects of the source of this heterogeneity are taken into consideration within the model. The frailty term ν_i is regarded as independent and considers the heterogeneity associated with the explanatory variable that cannot be observed, the mean being “1” and the variance “θ.” The variance of the random effect is the measure of the heterogeneity among individuals. In addition, the frailty term may differ for the 2 events (recurrence and death).11,12

The researcher used the following models to obtain the risk functions for recurrence and death in time (t):

Here, i is the index individual, j is the recurrent event, Z_i is the observed explanatory variable vector of the individual, which is the same for death and recurrence events in the models above, β is the regression coefficient for recurrence, and α is the regression coefficient for death. The γ is the value which demonstrates the association between recurrence and death. Higher γ values indicate a higher association between recurrence and death. The γ coefficient can be interpreted when the θ value is statistically different from zero, namely, when the heterogeneity is statistically significant in the data set.13 The parameter estimates in the model were obtained with the maximization of the penalized log likelihood. The penalized log likelihood was expressed as below for the joint frailty models.11

Here, k₁ and k₂ are the positive smoothing parameters. The smoothing parameter for recurrence (k₁) was obtained using the shared frailty model, and the smoothing parameter for death (k₂) was obtained using Cox regression.

Dynamic prediction

The calculation of death risks is important for choosing the appropriate treatment options for the patient and for making individualized treatment decisions. In addition, it is important to estimate the risk of death for a patient using advances in personalized medicine. For this purpose, dynamic prediction provides many advantages in the estimation of the risk of death, as it considers the entire event history of the patient, and the dynamic prediction can be updated when a new event occurs. Calculations are not only instantaneous for prediction t, but also for time intervals (horizons) using patient’s history. There are 3 different estimation methods for dynamic prediction. The first estimation calculates the risk by taking into account the exact j recurrences before prediction t. In the second estimation, a recurrence of at least j in the patient’s history before the prediction t is taken into account, whereas the j^th events occurring between recurrence and prediction time are not considered. In the last estimate, the recurrence history of a patient is considered during calculations. Only the parameters obtained from the joint frailty model are used for the estimation of death risk. In other words, the recurrence history is taken into account indirectly with the parameter estimates.

The estimation t is expressed by t and the estimation window is expressed by w. The dynamic models are concerned with the estimations between t and w (horizon time). i is the death time of i. patient, X_ij is the j^th recurrence number of i. patient, Z_ij^R is the related covariates with recurrence and Z_i^D is the related covariates with death. The equations S_ij^R (t) = P(X_ij≥t) and S_i^D (t) = P(D_i≥t) express the survival functions of recurrence and death, respectively. The term ξ=(λ₀^R (.),λ₀^D(.),β, α, θ) includes all the parameters of the patient. For the predictions, the risk of death should be defined based on the patient’s history of recurrence. There are 2 different recurrence histories for this: the completed recurrence history H^j_i^,1 (t) and partial recurrence history H^j_i² (t).14

After obtaining the predictions, the Brier score, used for the accuracy of the estimates, is calculated by the mean squared error of the probability forecasts. The Brier score value is between 0 and 1. The closer the score is to 0, the greater the accuracy of the risk estimates.15