Abstract

Biological growth and remodeling are commonly described through the multiplicative decomposition of the deformation gradient into an active (growth) and an elastic contribution. While the forward morphoelastic problem — predicting the equilibrium configuration generated by a prescribed active deformation $F_a$ — is well understood, the inverse problem of identifying $F_a$ from observed displacements remains comparatively unexplored. Building on our previous finding that naively reversing the active deformation fails to recover the observed configuration, this work reformulates the inverse problem as a PDE-constrained optimization: $F_a$ is treated as an unknown parameter, identified by minimizing the mismatch between predicted and observed displacement fields under the governing finite-strain equilibrium equations. Unlike our earlier infinitesimal-strain treatment, the entire analysis presented here — both the forward and the inverse problem — is formulated directly within the finite-deformation framework, consistently with the multiplicative decomposition itself and with growth amplitudes representative of real biological tissues. Four algebraic classes of active tensor (isotropic, anisotropic symmetric, symmetric shear, and skew-symmetric) are investigated on a two-dimensional finite-element model, each over twenty independent synthetic experiments. The proposed identification recovers all four classes with near machine-precision accuracy, outperforming the naive inversion baseline by four to six orders of magnitude. A notable exception concerns the skew-symmetric class: only the magnitude of its rotational parameter is identifiable, its sign remaining fundamentally unresolved by displacement observations alone, an outcome supported by both a closed-form algebraic argument and one order of magnitude greater robustness to observation noise for the corresponding solution. The framework further extends, without loss of accuracy, to observation noise up to $10\%$ and to spatially heterogeneous, multilobed active deformations beyond the four homogeneous classes. These results clarify which components of an active deformation tensor are mechanically observable, with direct implications for the design of growth-identification experiments in computational biomechanics.

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