We combine recent work on modeling cardiac mechanics using a finite volume method with the insight that heart wall myofiber orientations exhibit a particular volumetric geometry. In our finite vol- ume mechanical simulation we use Maurer-Cartan one-forms to add a geometrical consistency term to control the rate at which myofiber ori- entation changes in the direction perpendicular to the heart wall. This allows us to estimate material properties related to both the passive and active parameters in our model. We have obtained preliminary results on the 4 canine datasets of the 2014 mechanics challenge using the FEBio software suite. In ongoing work we are validating and improving the model using rat heart (ex-vivo DTI and in-vivo tagging) MRI datasets, from which we have estimated strain tensors.