Statistical models of breast cancer tumour growth and spread

Abstract

In this thesis, we develop statistical methods for studying tumour growth and metastatic lymph node spread in breast cancer. The methods can be used for analysing breast cancer disease progression before diagnosis. They may be used to answer questions such as: For how long does a tumour grow inside of the body before it is detected? How much will the tumour metastasise in the lymph nodes before detection? Or, which women have a high risk of missing breast cancer at mammography screening? These questions are important for studying the effects of mammography screening at an individual level. We work in a framework called continuous growth models. This is an alternative to Markov models, which is the most commonly used approach for modelling breast cancer disease progression. Standard Markov models assume that all women in each disease state are identical, making the model easy to implement and practically useful. Unfortunately, women with breast cancer are not identical, and relaxing this assumption quickly increases the Markov model’s complexity. Continuous growth models are instead more complex at the outset. However, as the number of clinical factors increase, continuous growth models become more flexible and less complex than Markov models.

In Study I, we focus on a continuous growth process used for modelling tumour volume at diagnosis. We provide a detailed description of the so called Stable Disease Assumptions that are used for continuous growth modelling. We use them to derive new theoretical results for the model. These are then integrated into our growth model, which helps to simplify and reduce computational complexity of the model. With these results, we were able to greatly reduce the computational time needed for estimating growth parameters. The theoretical results derived in Study I are further used in Studies II and III.

In Study II, we extend our continuous growth model to also include a sub-model for metastatic lymph node spread. The result is a joint model of tumour volume and number of lymph node metastases at diagnosis, conditional on mammography screening history and mammographic density. When applied on empirical data on 1860 incident invasive breast cancer cases, our model provides a dramatically better fit than other models in current use. Furthermore, we show that our sub-model of lymph node spread can be estimated independently of the tumour growth process. This property forms the first part of the theoretical basis for study IV. In Study II, we use the property to validate our lymph node spread model on an independent data set, consisting of 3961 women diagnosed with invasive breast cancer.

In Study III, we show how to study the effect of other clinical factors on metastatic lymph node spread. Our approach is to regress clinical factors on the proportionality constant in our model for lymph node spread. We illustrate our method by studying the association between hormone replacement therapy (HRT) and tumour growth rate and rate of lymph node spread. Using data from 1631 women diagnosed with breast cancer, we estimate that women using HRT have a 36% lower rate of lymph node spread than non-users (95% confidence interval: 58% to 8%). This can be contrasted with the effect of HRT on tumour growth rate. We estimate growth rates to be 15% slower in HRT users (p = 0.16). We also derive theoretical distributions for metastatic lymph node spread at future points in time. We use them to illustrate the potential consequences of false negative screens, in terms of lymph node spread.

In Study IV, we use the method developed in Study III to study the association between clinical factors and rate of breast cancer lymph node spread. We use data on 10950 women to study the associations with grade, estrogen receptor (ER) status, progesteron receptor (PR) status, molecular subtype, and a polygenic risk score. We found that grade 2 and 3 tumours, respectively, were associated with 1.63 and 2.17 times faster rates of lymph node spread than grade 1 tumours (p < 10^-16). ER negative breast cancer was associated with a 1.25 times faster spread than ER positive and PR negative breast cancer was associated with a 1.19 times faster spread than PR positive cancer (p = 0.0011, p = 0.0012, respectively). Her2-enriched breast cancer was associated with a 1.53 times faster spread than luminal A cancer (p = 0.00072).