From basic science to translation, modern biomedical research demands computational models which integrate several interacting physical systems. This paper describes the infrastructural framework for generic multiphysics integration implemented in the software $\boldsymbol{\mathcal{C}}\mathbf{Heart}$, a finite-element code for biomedical research. To generalize the coupling of physics systems, we introduce a framework in which the geometric and operator relationships between the constituent systems are rigorously defined. We then introduce the notion of topological interfaces and define specific operators encompassing many common model coupling requirements. These interfaces enable the evaluation of weak form integrals between mesh subregions of arbitrary finite-element bases' orders, types, and spatial dimensions. Equation maps are introduced which provide abstract representations of the individual physics systems that can be automatically combined to permit a monolithic matrix assembly. Flexible solution strategies for the resulting coupled systems are implemented, permitting fine-tuning of solution updates during fixed point iterations, and subgrouping where several problems are being solved together. Partitioning of coupled mesh domains for optimal load balancing is also supported, taking into account the per-processor cost of the entire coupled problem within the graph problem. The demonstration of the performance is illustrated through important real-world multiphysics problems relevant to cardiac physiology.