Physics-Informed Operator Learning for Pulsatile Milk Flow in Distal Generations of a Bifurcated Mammary Duct Network

Pulsatile milk transport through the lactating mammary ductal tree involves complex interactions between pressure gradients, wall compliance, and non-Newtonian rheology across spatial scales that span nearly two orders of magnitude in lumen radius. Direct experimental characterization of flow in distal ductal generations remains infeasible due to their sub-millimeter caliber, leaving the hemodynamic environment of the secretory ductules largely unknown. We present a two-stage physics-informed operator-learning framework that extends validated flow predictions from three instrumented duct generations to twenty generations of a bifurcated mammary network. A Physics-Informed Neural Network (PINN) trained against particle image velocimetry measurements across seven ducts achieved R^2 = 0.924-0.997. A Deep Operator Network (DeepONet) distilled from the PINN and refined through physics-constrained training on the governing one-dimensional fluid-structure interaction equations achieved R^2(u) = 0.857-0.985 across all validated ducts, with predictions for Generations 4-20 obtained by supplying Murray's Law geometry and mass-conservation-scaled boundary conditions to the frozen operator. Three biophysically significant findings emerge: a mean velocity plateau of 0.14-0.18 m/s across Generations 4-13 produced by Cross shear-thinning compensation offsetting Murray-branching deceleration; a non-monotonic pulsatility index that declines from 0.048 at Generation 1 to a minimum of 0.039 at Generation 5 before rising monotonically to 1.37 at Generation 20 as progressive wall stiffening drives the most distal ductules into a microcirculation-like hemodynamic regime; and a brief elastic-recoil transition zone at Generations 4-5 where mean axial pressure drop reverses sign. To the authors' knowledge, these results provide the first quantitative characterization of pulsatile milk flow across the full hierarchy of a bifurcated mammary ductal tree using a physics-informed operator-learning framework with implications for ductal mechanobiology, milk ejection mechanics, and mastitis pathogenesis.