A natural history and copula-based joint model for regional and distant breast cancer metastasis

Abstract

The few existing statistical models of breast cancer recurrence and progression to distant metastasis are predominantly based on multi-state modelling. While useful for summarising the risk of recurrence, these provide limited insight into the underlying biological mechanisms and have limited use for understanding the implications of population-level interventions. We develop an alternative, novel, and parsimonious approach for modelling latent tumour growth and spread to local and distant metastasis, based on a natural history model with biologically inspired components. We include marginal sub-models for local and distant breast cancer metastasis, jointly modelled using a copula function. Different formulations (and correlation shapes) are allowed, thus we can incorporate and directly model the correlation between local and distant metastasis flexibly and efficiently. Submodels for the latent cancer growth, the detection process, and screening sensitivity, together with random effects to account for between-patients heterogeneity, are included. Although relying on several parametric assumptions, the joint copula model can be useful for understanding - potentially latent - disease dynamics, obtaining patient-specific, model-based predictions, and studying interventions at a population level, for example, using microsimulation. We illustrate this approach using data from a Swedish population-based case-control study of postmenopausal breast cancer, including examples of useful model-based predictions.

Keywords: Natural history model; breast cancer; copula; microsimulation; survival analysis.

Figure 1.

Fitted model coefficients under each copula specification, application to Cancer And Hormone REplacement Study (CAHRES) data. Confidence intervals for the model assuming a Clayton copula are omitted, as the fitted Hessian at the optimum was found to be numerically unstable.

Figure 2.

Marginal survival curves (in black) for time to diagnosis of distant metastasis for subjects with zero, one, and two affected lymph nodes at diagnosis. Curves are shown for both the model with a Frank copula and the model with a product (independence) copula and averaged over the observed covariates’ distribution in the Cancer And Hormone REplacement Study (CAHRES). Kaplan-Meier curves (in grey) for relevant subgroups of the study population are included for comparison.

Figure 3.

Five-year risks of distant metastasis for women with 15  mm wide tumours, calculated at diagnosis and with risks inferred from scenarios representing early diagnosis, by one, two and three years (Time). Probabilities are estimated using a simulation approach. We simulated 10 million tumours based on the joint model parameter estimates described in Table 3. The naive estimates (in grey) are affected by lead-time, while the lead-time corrected estimates (in black) are directly comparable to the risk estimates based on detection at 15  mm. Results are presented for tumours with 0, 1 and 2 lymph nodes and by tertiles of inverse growth rates. Each plot includes annotations with average tumour size (as diameter, in mm) at each detection time considered in this microsimulation study.

Figure 4.

Five-year risks of distant metastasis for women with 15  mm wide tumours, calculated at diagnosis and with risks inferred from scenarios representing early diagnosis at diameters of 10, 5 and 1  mm (Size). Probabilities are estimated using a simulation approach. We simulated 10 million tumours based on the joint model parameter estimates described in Table 3. The naive estimates (in grey) are affected by lead-time, while the lead-time corrected estimates (in black) are directly comparable to risk estimates based on detection at 15  mm. Results are presented for tumours with 0, 1 and 2 lymph nodes and by tertiles of inverse growth rates. Each plot includes annotations with the average time (in years) it would take to detect tumours at the smaller sizes.