A review of mechanistic learning in mathematical oncology

Abstract

Mechanistic learning refers to the synergistic combination of mechanistic mathematical modeling and data-driven machine or deep learning. This emerging field finds increasing applications in (mathematical) oncology. This review aims to capture the current state of the field and provides a perspective on how mechanistic learning may progress in the oncology domain. We highlight the synergistic potential of mechanistic learning and point out similarities and differences between purely data-driven and mechanistic approaches concerning model complexity, data requirements, outputs generated, and interpretability of the algorithms and their results. Four categories of mechanistic learning (sequential, parallel, extrinsic, intrinsic) of mechanistic learning are presented with specific examples. We discuss a range of techniques including physics-informed neural networks, surrogate model learning, and digital twins. Example applications address complex problems predominantly from the domain of oncology research such as longitudinal tumor response predictions or time-to-event modeling. As the field of mechanistic learning advances, we aim for this review and proposed categorization framework to foster additional collaboration between the data- and knowledge-driven modeling fields. Further collaboration will help address difficult issues in oncology such as limited data availability, requirements of model transparency, and complex input data which are embraced in a mechanistic learning framework.

Keywords: ODE (ordinary differential equation); deep learning; machine learning; mathematical modeling; mechanistic learning.

Figure 1

Examples of mechanistic learning structured in four combinations: Parallel combinations (top left) with examples of surrogate models and neural ordinary differential equations (ODEs). Data- and knowledge-driven models act as alternatives to complement each other for the same objective. Sequential combinations (bottom left) apply data- and knowledge-driven models in sequence to ease the calibration and validation steps. Extrinsic combinations (top right) combine knowledge-driven and data-driven modeling at a higher level. For example, mathematical analysis of data-driven models and their results or as complementary tasks for digital twins. Intrinsic combinations (bottom right), like physics- and biology-informed neural networks include the knowledge-driven models into the data-driven approaches. Knowledge is included in the architecture of a data-driven model or as a regularizer to influence the learned weights.

Figure 2

The mechanistic learning landscape shows room for the combination of data-driven and knowledge-driven modeling. We suggest that purely data-driven or purely knowledge-driven models represent the extremes of a data-knowledge surface with ample room for combinations in different degrees of synergism. Further, in the left-bottom corner with almost no data nor knowledge, any modeling or learning technique is limited.