Discrete spring–mass system inversion of lung stiffness from 4DCT with adjoint regularization and graph convolutional network
Embargo Date
2029-01-31
OA Version
Citation
Abstract
We present a physics informed framework for estimating regional lung stiffness from four dimensional CT (4DCT) without requiring elastography labels. At its core is a discrete spring–mass system (SMS) formulation that approximates small strain linear elasticity on a tetrahedral mesh. Each respiratory phase is modeled as a quasi static equilibrium state, and an axis aligned quadratic SMS energy leads to a single sparse symmetric linear system per phase. We prove that this discrete energy is mathematically equivalent to the classical continuum linear elastic model under suitable parameter identification, and verify the equivalence numerically on a heterogeneous “drooped cone” phantom, where SMS and finite elements (FEM) match displacements within about 4% and allow recovery of a two region stiffness field to within a few percent.Given observed motion from deformable registration of 4DCT, we formulate stiffness recovery as a PDE constrained inverse problem that minimizes the mismatch between simulated and observed displacements, regularized by Charbonnier smoothed TV penalties on both displacement and stiffness. Using an adjoint state derivation, we obtain analytic gradients of the data misfit and both regularizers with respect to element wise stiffness; all gradient evaluations reduce to linear systems with the same structure as the forward SMS solve.
To stabilize this ill posed inverse problem on patient data, we introduce a mesh based graph convolutional network (GCN) as a learned prior. Per tetrahedron Hounsfield units are first mapped to a physiology guided baseline stiffness via a piecewise linear HU to modulus relation. A tetrahedral GCN operating on face adjacency then predicts small log space corrections to this baseline. Its parameters are trained end to end by back propagating the simulation loss through a differentiable GPU accelerated SMS layer. On a public 4DCT lung dataset, the resulting stiffness maps exhibit reproducible spatial patterns: emphysematous regions appear consistently softer than surrounding parenchyma, and lower lobes are stiffer than upper lobes, in line with expected COPD pathophysiology.
Description
2026
License
Attribution-NonCommercial 4.0 International