Joint models of tumour size and lymph node spread for incident breast cancer cases in the presence of screening

Abstract

Continuous growth models show great potential for analysing cancer screening data. We recently described such a model for studying breast cancer tumour growth based on modelling tumour size at diagnosis, as a function of screening history, detection mode, and relevant patient characteristics. In this article, we describe how the approach can be extended to jointly model tumour size and number of lymph node metastases at diagnosis. We propose a new class of lymph node spread models which are biologically motivated and describe how they can be extended to incorporate random effects to allow for heterogeneity in underlying rates of spread. Our final model provides a dramatically better fit to empirical data on 1860 incident breast cancer cases than models in current use. We validate our lymph node spread model on an independent data set consisting of 3961 women diagnosed with invasive breast cancer.

Keywords: Breast cancer; joint modelling; lymph node spread; random effects modelling; screening; tumour growth model.

Figure 1.

Model-based estimates of expected lymph node spread as a function of tumour size, based on CAHRES. Circles and bars represent averages and 95% confidence intervals of numbers of lymph nodes affected within each tumour size interval. Model A (dotted) produces excessive spread at large tumour sizes, while model B (solid) underestimates spread at large tumour sizes.

Figure 2.

Model-based estimates of expected lymph node spread as a function of tumour size (CAHRES). Circles and bars represent averages and 95% confidence intervals of numbers of lymph nodes affected within each tumour size interval. The spread component of Model A (dotted) produces excessive spread in large tumours, whereas in terms of expected numbers of affected lymph nodes the spread model with k = 5 (solid) fits at all tumour sizes.

Figure 3.

Model-based estimates of expected lymph node spread as a function of tumour size (CAHRES). Circles and bars represent averages and 95% confidence intervals of numbers of lymph nodes affected within each tumour size interval. The spread component of Model A (dotted) is plotted alongside the random effects spread model with k = 4 (solid).

Figure 4.

Observed and predicted numbers of affected lymph nodes (CAHRES). The bars represent the observed numbers of affected lymph nodes, within tumour size interval 10–15 mm (left) and 35–45 mm (right), in the CAHRES dataset. Circles represent predicted probabilities from the Poisson model with k = 5, estimated on the CAHRES data set, and dots represent predicted probabilities from the random effects Poisson model with k = 4, also estimated on the CAHRES data set.

Figure 5.

Model-based estimates of expected lymph node spread as a function of tumour size (CAHRES). To the left, circles and bars represent averages and 95% confidence intervals of numbers of lymph nodes affected within each tumour size interval for screen detected cancers, and to the right the corresponding quantities for symptomatically detected cancers. On both figures, the spread component of Model A (dotted) is plotted alongside the random effects spread model with k = 4 (solid).

Figure 6.

Model-based estimates of expected lymph node spread as a function of tumour size based on the random effects Poisson model (k = 4), estimated on CAHRES (dotted line) and Libro-1 (solid line), along with 95% confidence intervals of average lymph node spread obtained from Libro-1.

Figure 7.

Observed and predicted numbers of affected lymph nodes (Libro-1). The bars represent the observed numbers of affected lymph nodes, within tumour size interval 10–15 mm (left) and 35–45 mm (right), in the Libro-1 dataset. Circles represent predicted probabilities from the Poisson model with k = 5, estimated on the CAHRES data set, dots represent predicted probabilities from the random effects Poisson model with k = 4, also estimated on the CAHRES data set, and crosses represent estimated probabilities from the random effects Poisson model with k = 4, estimated on the Libro-1 data set.

Figure 8.

Model-based estimates of the inverse growth rate distribution (left) and screening sensitivity (right), based on simulated data (dotted). Solid lines represent the same distributions based on the true parameter values.