Probabilistic model discovery reveals distinct constitutive behavior of kidney cortex and medulla

The kidney is a critical soft-tissue organ responsible for blood filtration. Accurate constitutive models of the kidney are essential to predict tissue deformation and stress, yet existing models prescribe the strain energy function a priori, largely treat tissue variability as deterministic, and do not distinguish between cortex and medulla. Here we use Gaussian constitutive artificial neural networks to discover probabilistic strain energy functions for kidney cortex and medulla from tension, compression, and shear experiments. The medulla is approximately twice as stiff as the cortex across all three modes: The effective Young's moduli are 3.41, 5.04, and 4.48 kPa for the medulla and 1.43, 2.32, and 2.87 kPa for the cortex in tension, compression, and shear. Both regions display pronounced tension--compression asymmetry. When trained on all three modes, the network discovers strain energy functions in the second invariant alone over the tested deformation range. The cortex model selects two exponential I2 terms, while the medulla model selects an additional linear I2 term. Gaussian external weights propagate specimen-to-specimen variability into closed-form probabilistic stress predictions. Our region-specific probabilistic models capture the mechanical heterogeneity and variability of the kidney and enable more realistic simulations of kidney deformation and stress for surgical planning, needle interventions, and renal trauma.