Following the volumetric growth theory introduced by Rodriguez et al.¹, the deformation gradient is multiplicatively split such that

For a more thorough review see here². The growth deformation gradient takes the form

where each θ dictates growth in either the fiber direction (eccentric growth) or the sheet and sheet normal directions (concentric growth).

where s is a function taking on some stimulus value (varies with position) and s₀ is the stimulus value reached after a steady state simulation. F_g is updated and the weak form solved in the absence of any active stress or external load, and F_e restores compatibility. The reference mesh is updated such that the nodes take on the positions obtained by applying the displacement solved for, thus relieving residual stresses.

Animations of preliminary concentric growth are shown below. The active stress at end-systole is used as a stimulus, obtained from the steady state cycle of the ellipsoid LV demo. Ten “growth steps” are taken to grow in the sheet and sheet-normal directions, and the LV filled to the same previous end-diastolic volume:

and then the 5 beat simulation is run again. Initial simulation is shown on the left, and the grown mesh simulation shown on the right with the corresponding PV loops below:

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