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.

Lymph node metastases in breast cancer: Investigating associations with tumor characteristics, molecular subtypes and polygenic risk score using a continuous growth model

“…They differed most in terms of grade, with a 13% difference in number of Grade 3 cases, and ER status, with a 7% difference in number of ER-positive cases. Both cohorts were analyzed on a stand-alone basis, and as one big data set, using a novel continuous growth model that adjusts for tumor volume Though our model fits data better than previously suggested lymph node spread models from the literature, [7][8][9] as well as…”

“…Under our modeling assumptions, the number of affected lymph nodes can be expressed as a direct function of current tumor characteristics. The approach was developed by Isheden et al 8 and was recently extended to include covariate effects in the rate of lymph node spread 10 . A detailed description of the modeling approach is given in the Appendix S1.…”

“…Where s * is a gamma distributed random effect, D(t,r) is the number of times the cells in the tumor has divided and D 0 (t,r) is the rate of cell division in the tumor-both at time t, assuming an inverse growth rate r. In Isheden et al, 8 it was shown that these models lead to there being, at any time point, a negative binomial distribution, for the number of affected lymph nodes N, such that the probability of n affected lymph nodes, conditional on current tumor volume V, follows the functional form…”

“…7 The model assumes that tumor volume follows an exponential Gompertz function with decelerating doubling time, individually assigned doubling times, and that the instantaneous rate of lymph node spread at time t is equal to λ(t) = b 1 + b 2 V(t) + b 3 V 0 (t), where V(t) is tumor volume at time t, V 0 (t) is the rate of growth at time t, and b 1 , b 2 and b 3 are constants. In Isheden et al, 8 it was shown that the model of Shwarz suffers from two weaknesses: firstly, it implies that slow growing tumors have a higher degree of lymph node spread compared to fast growing tumors, and, secondly, the model implies either an unrealistically high degree of lymph node spread for large tumors or an unrealistically low degree of lymph node spread for small tumors. Based on two independent data sets, it was shown that lymph node spread following an inhomogeneous Poisson process with rate λ(t) proportional to the number of times the tumor cells have divided, D(t), to the power four, and the rate of cell division in the tumor D 0 (t), that is, λ(t) = σD(t) 4 D 0 (t), combined with a gamma distributed random effect for individual spread σ, gives a significantly better model fit compared to the model of Shwartz, 7 and the lymph node spread model of Hanin and Yakovlev.…”

“…Based on two independent data sets, it was shown that lymph node spread following an inhomogeneous Poisson process with rate λ(t) proportional to the number of times the tumor cells have divided, D(t), to the power four, and the rate of cell division in the tumor D 0 (t), that is, λ(t) = σD(t) 4 D 0 (t), combined with a gamma distributed random effect for individual spread σ, gives a significantly better model fit compared to the model of Shwartz, 7 and the lymph node spread model of Hanin and Yakovlev. 9 Here, we base our analyzes on the lymph node metastases modeling approach of Isheden et al 8 and a recent extension of the model to include a covariate effect on the rate of lymph node spread. 10 In this article, we use a natural history lymph node spread regression model to quantify the rate of lymph node spread based on grade, ER status, progesteron receptor (PR) status, molecular subtype and polygenic risk score (PRS).…”