Constitutive neural networks for main pulmonary arteries: Discovering the undiscovered

Key results

1. ShownWhen one network is trained jointly on all 8 samples with the fiber angle fixed at 70°, it consistently selects one isotropic and one anisotropic invariant term. The same two features, exp([I1-3]) and [I5-1]^2, recur across the unregularized and L1-regularized runs.

   The cross-sample loss covered n = 8 samples, with αreg = 0 and αreg = 0.001. The network can express 2^16 term combinations (more than 60,000 models). The unregularized result matches the behavior previously discovered for layers of healthy human aortas. Each regularization condition took about 20 minutes to train. · 2.3 Constitutive artificial neural network; 3.2 Cross-sample feature selection; Figure legends

2. ShownThe reduced two-term law (exponential I1 plus quadratic I5) fits better than the HGO model and is much less sensitive to the assumed collagen fiber angle.

   Averaged over n = 8 samples with fixed angles from 0° to 90°, the reduced model kept R^2 > 0.9 across the whole range, while HGO fit well only for 45° ≤ α ≤ 50°. On the representative sample, the reduced model gave R^2 > 0.99 (α = 50.9°). HGO gave c1 = 1.95 kPa, k1 = 0.77 kPa, k2 = 0.145 (α = 49.0°) with a lower R^2. · 3.4 Traditional model fitting; 3.5 Reduced model discovery; Figure legends

3. ShownA per-sample sweep over fixed fiber angles shows that anisotropic terms appear mainly when fibers are oriented circumferentially. The HGO model's exponential quadratic I4 term is almost never selected.

   The sweep used 11 angles from 0° to 90° for each of the n = 8 samples. Without L1, exp([I1-3]) dominates and anisotropic terms appear mostly for α > 45°. With L1 (αreg = 0.01), [I5-1]^2 is most common and I1 terms give way to I2 terms. exp([I4-1]^2) occurs in 1% of fits without L1 and 0% with L1. · 3.3 Fiber angle analysis; 3.5 Reduced model discovery; Figure legends

4. ShownOn one representative sample, unregularized discovery with a fitted angle gives a three-term model and a nearly isotropic fiber angle. L1 at αreg = 0.01 prunes this to a single I5 term and the fit gets much worse, which the authors read as over-regularization.

   Unregularized, fitted α: exp([I1-3]), exp([I5-1]) and [I5-1]^2, with R^2 > 0.99 and α = 49.8°. L1, fitted α: only [I5-1]^2, with R^2 < 0.84 and α = 49.0°. Fixed α = 70° without L1: R^2 > 0.98, with the full set of I1 terms active. Each run took about 4 minutes on a standard laptop. · 3.1 Sample-specific model discovery; Figure legends

5. SuggestedFitting the reduced law jointly to all samples converges to an almost purely circumferential fiber direction. The authors say this agrees with published collagen orientation distributions for ovine pulmonary arteries.

   The all-sample reduced fit gave α = 89.9°, with weights w1,2 = 0.424, w2,2 = 8.48 kPa, w1,15 = 0.222 and w2,15 = 0.112 kPa. The microstructural agreement rests on a cited prior study and was not measured in this work. · 3.5 Reduced model discovery; Figure legends