The residual cancer burden index is an important quantitative measure used for assessing treatment response following neoadjuvant therapy for breast cancer. It has shown to be predictive of overall survival and is composed of two key metrics: qualitative assessment of lymph nodes and the percentage of invasive or in situ tumour cellularity (TC) in the tumour bed (TB). Currently, TC is assessed through eye-balling of routine histopathology slides estimating the proportion of tumour cells within the TB. With the advances in production of digitized slides and increasing availability of slide scanners in pathology laboratories, there is potential to measure TC using automated algorithms with greater precision and accuracy. We describe two methods for automated TC scoring: 1) a traditional approach to image analysis development whereby we mimic the pathologists' workflow, and 2) a recent development in artificial intelligence in which features are learned automatically in deep neural networks using image data alone. We show strong agreements between automated and manual analysis of digital slides. Agreements between our trained deep neural networks and experts in this study (0.82) approach the inter-rater agreements between pathologists (0.89). We also reveal properties that are captured when we apply deep neural network to whole slide images, and discuss the potential of using such visualisations to improve upon TC assessment in the future.