Anatomy-Informed Neural Networks: Encoding Anatomic Priors in Loss and Architecture, with an SE(3) Formulation of Guidewire-Induced Aortoiliac Deformation

David P. Stonko

cs.AIAug 21, 2026

Anatomy-Informed Neural Networks: Encoding Anatomic Priors in Loss and Architecture, with an SE(3) Formulation of Guidewire-Induced Aortoiliac Deformation

Authors: David P. Stonko

Organizations: Division of Vascular Surgery and Endovascular Therapy, Department of Surgery, The Johns Hopkins Hospital, Baltimore, MD, USA.

Abstract

Deep-learning models of anatomy can be numerically plausible yet anatomically impossible, and they generalize poorly when data are scarce. We introduce Anatomy-Informed Neural Networks (AINN), in which soft anatomic priors enter as penalty terms in the loss (e.g., a branching penalty that treats a renal transplant artery off the iliac instead of the aorta as unexpected rather than impossible), in direct analogy to a physics-informed neural network, and hard anatomic priors (e.g., continuity of the vessel) are built into the architecture and state representation, making such invalid predictions impossible by construction wherever the prior admits architectural enforcement. We develop it on a clinical test case with limited data: how the aortoiliac tree deforms when a stiff wire is introduced endoluminally. This is important to contemporary aortic surgery and will matter to autonomous endovascular navigation. We lift the vessel centerline and the wire path from R^3 to curves of frames in the Lie group SE(3), and couple a Cosserat-rod wire to a tortuosity-modulated, anatomically anchored vessel through a unilateral lumen-contact inequality. The prediction is a constrained minimizer of the coupled elastic energy, with contact forces as its Lagrange multipliers. Supervision is a Wasserstein-2 optimal-transport loss between the predicted projection through the C-arm geometry and the observed angiogram, so a 2D angiogram can train a 3D prediction. The kinematics, loss and projection are verified against known ground truth; the mechanics solver only against its own optimality conditions, and predicted displacement is not yet mesh-converged. Here, no network is trained. Future work will transfer this in silico model to real CT scans and test whether it improves predictive accuracy and reduces the training data required.

Figures & tables

Figure 1: The AINN conceptual model. A neural network maps an input anatomy (I, preoperative State 1) to an output prediction (O, intraoperative State 2), and anatomy enters in two ways. Hard priors are built into the representation and architecture, so every frame is a point in $\mathrm{SE}(3)$ and every predicted frame is a valid rigid-body configuration. That per-frame guarantee does not by itself exclude every anatomically invalid configuration, as § 4 sets out. Soft priors enter through the training loss, which adds an anatomy error to the data-fit error, and the lower right panel shows the general form such an anatomy term can take. In the vascular worked example the anatomic content is carried almost entirely by hard priors and the soft $\mathcal{L}_{\mathrm{anatomy}}$ slot is specified but not exercised, the Wasserstein-2 machinery there serving the data term that compares predicted projection to observed angiogram. ANN, artificial neural network. $\mathrm{SE}(3)$ , special Euclidean group.

Figure 2: Aortoiliac deformation under stiff-wire loading, State 1 $\rightarrow$ State 2 (schematic; not patient imaging). (A) Preoperative anatomy, with native tortuosity in the iliac arteries and the aorta and no instruments in place. (B) The intraoperative configuration after bilateral stiff guidewires are advanced from femoral access. Each wire is a single smooth, near-chord curve rather than a copy of the vessel centerline, because it is by far the stiffest member of the system. Orange arrows mark the contact force of the wire on the wall and gray arrows the resulting ostial displacement. The dashed black line traces the State 1 aortic contour and the dashed gray outline the State 1 iliac contour, each reproduced from panel A for comparison. The wires converge and cross at the aortic bifurcation and then run parallel up the aorta. The worked example (§ 4 –§ 10 ) models the single dominant wire and its bowstring contact (Figure 4 ), of which panel B is the clinical superposition.

Figure 3: Lifting a vessel centerline from $\mathbb{R}^{3}$ into $\mathrm{SE}(3)$ . Attaching an orthonormal frame at each centerline point makes position plus orientation one element $g(s)$ of $\mathrm{SE}(3)$ , with the incremental twist between adjacent frames and the exponential map that inverts it shown for one step.

Figure 4: The unilateral lumen contact constraint, or bowstring effect. The wire’s offset from the centerline cannot exceed $R_{\mathrm{lumen}}(s)-r_{w}$ , shown here with the resulting chord-taking path and the apex contacts where the constraint binds.

Parameter	Mathematical role	Clinical proxy
$K_{w}$ (or $K_{\mathrm{eff}}$ )	Bending stiffness of wire, or of the wire-and-device assembly, as a quadratic form on $\mathfrak{se}(3)$	Lunderquist versus Amplatz versus Glidewire; device deployment state
$C_{v}(s)$	Local anisotropic vessel stiffness tensor	Calcification score, wall quality
$k_{0},\alpha$	Coefficients of the Hounsfield-to-stiffness ramp feeding $k_{\mathrm{HU}}$	BMI, periaortic fat density
$k_{\mathrm{anat}}(s)$	Anatomic-position anchoring stiffness	Arch, visceral, retroperitoneal and iliac segments
$T_{\mathrm{axial}}$	Dead load driving the wire toward its chord	Surgical push and pull; intraprocedural force sensing
Reporting covariate, not a model parameter

Table 1: Parameter families of the forward model.

T_{\mathrm{tort}}

is listed separately because § 6.2 demotes it from a model parameter to a per-case reporting covariate.

Figure 5: End-to-end AINN architecture for the worked example. Preoperative CT feeds a 3D preprocessing pipeline producing the State 1 representation and the patient-specific physical parameters. The § 6 physics simulator gives a deterministic forward prediction by minimizing the total energy under the lumen-contact constraint, and an $\mathrm{SE}(3)$ -equivariant residual network adds a small Lie-algebra-valued correction composed through the matrix exponential. The result is projected to 2D and supervised against the angiogram through the optimal-transport loss, with gradients flowing to the network weights and calibration parameters by differentiable simulation. Note that the loss box is labeled Lie-Wasserstein in the schematic; as § 7.2 sets out, the loss actually used for training acts on projected positions with a Euclidean ground cost, and the group-intrinsic cost becomes operative only where three-dimensional ground truth exists.

Parameter	Symbol	Phase 1 (2D)	Phase 2 (3D)	Rationale
Lumen diameter	$D_{\mathrm{lumen}}$	12 mm	12 mm	common iliac luminal diameter
Wire outer radius	$r_{w}$	0.4445 mm	0.4445 mm	Lunderquist $0.035^{\prime\prime}$
Wire bending modulus	$EI_{w}$	65 N $\cdot$ mm 2	65 N $\cdot$ mm 2	swept, see § 10.4
Vessel bending modulus	$EI_{v}$	200 N $\cdot$ mm 2	200 N $\cdot$ mm 2	assumed
Axial pre-tension	$T_{\mathrm{axial}}$	10 N	10 N	assumed surgical pull
Anatomic anchoring	$k_{\mathrm{anat}}$	0.03 N/mm	0.03 N/mm	loose tethering, $>0$ required

Table 2: Canonical parameters for the two toy models. Phase 2 parameters are reported explicitly, since the two models share mechanics but not geometry. Here

EI_{v}

denotes the isotropic scalar reduction of the stiffness tensor

C_{v}(s)

of § 6.2 used in the toys, and

k_{\mathrm{fat}}

the uniform toy value of the field

k_{\mathrm{HU}}

of § 6.3 .

Figure 6: Phase 1 planar toy at equilibrium. (A) State 1 geometry, lumen wall and centerline. (B) State 1 $\rightarrow$ State 2 overlay with the equilibrium wire and the active wall contacts. (C) Contact line load $\lambda(s)=T\kappa(s)$ over the active set. The shaded trace is median-filtered over three nodes for display, because a single node’s discrete curvature can spike at an active-set boundary where the local Voronoi length collapses; the peak and integral in the panel title are the raw solver values. (D) Energy budget. The inextensibility term is identically zero by construction (§ 10.1 ).

Figure 7: Phase 2 three-dimensional synthetic toy at equilibrium , a Lunderquist wire in a helical iliac. (A) State 1 geometry, lumen-wall tube and centerline. (B) State 1 $\rightarrow$ State 2 overlay with equilibrium wire and active contacts, the State 1 lumen translucent behind the deformed State 2 lumen. (C) Contact line load along arc length. (D) Energy budget.

Phase 1, planar, $N=80$
$EI_{w}$ (N $\cdot$ mm 2 )	peak (mm)	mean (mm)	$J_{\mathrm{wire}}$	$J_{\mathrm{axial}}$	$J_{\mathrm{anchor}}$	$J_{\mathrm{fat}}$	$J_{\mathrm{vessel}}$
65	9.97	5.43	0.7%	49.9%	46.3%	—	3.1%
200	10.13	5.49	1.9%	47.9%	46.9%	—	3.2%
500	10.44	5.63	4.2%	44.2%	48.1%	—	3.4%
1000	10.87	5.85	7.0%	39.3%	50.1%	—	3.6%
2000	11.56	6.24	11.0%	31.8%	53.5%	—	3.7%

Table 3: Wire-stiffness sweep. Energy shares are percentages of

J_{\mathrm{total}}

. The extravascular term is absent from the planar toy by construction.

Figure 8: Sensitivity of the planar (Phase 1) equilibrium to wire bending stiffness. The swept interval spans the effective stiffness commonly assumed for a Lunderquist through the nominal solid-rod value implied by $E\pi r_{w}^{4}/4$ . (A) Peak vessel displacement. (B) Composition of the energy budget.

$N$	peak disp. (mm)	$L^{\mathrm{arc}}_{w}$ (mm)	$J_{\mathrm{wire}}$	$J_{\mathrm{vessel}}$	$J_{\mathrm{anchor}}$	$J_{\mathrm{axial}}$	$J_{\mathrm{wire}}$ share
40	13.10	152.41	0.350	3.671	44.27	24.10	0.48%
60	11.30	153.93	0.553	3.449	48.62	39.27	0.60%
80	9.97	155.31	0.739	3.285	49.21	53.07	0.70%
120	8.10	157.54	1.023	2.842	46.94	75.40	0.81%
160	6.89	159.18	1.231	2.474	43.83	91.84	0.88%

Table 4: Phase 1 mesh refinement at

EI_{w}=65

\cdot

mm 2 . Energies in N

\cdot

mm.

Figure 9: Optimal-transport loss on the $\mathrm{SE}(3)$ -lifted wire. (A) Transport between the simulator wire measure and a synthetic target. (B) Transport plan. (C) $W_{2}^{2}$ against perturbation amplitude. (D) Nilpotent sub-Riemannian ground cost against the Riemannian cost.

Figure 10: C-arm projection. (A) Three-dimensional scene with lumen wall, centerline, equilibrium wire, wall-contact forces, and the beam axis. (B) Synthetic anteroposterior angiogram and (C) synthetic $30^{\circ}$ oblique angiogram of the Phase 2 toy, with the wire rendered radio-opaque against the projected centerline.

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Learning Unsteady Aneurysm Hemodynamics with Physics-Informed DeepONets

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Clinically actionable, patient-specific hemodynamic assessment, specifically wall shear stress, vortex structure and pressure distributions, is critical for determining risky or unfavorable evolution in Abdominal Aortic Aneurysms (AAA). While Physics-Informed Deep Operator Networks (PI-DeepONets) show promising results in complementing established 5 tools such as Computational Fluid Dynamics (CFD), a persistent architectural challenge remains for complex 3D flows. In this direction, we propose a Modified Multi-Input Multi-Output PI-DeepONets (M3PI-DeepONet) designed for predicting unsteady flows in an idealized AAA geometry. Central to our model is the Aggregated Injection strategy, where latent representations from multiple input branches are fused prior to trunk injection, allowing the coordinate basis to adapt to multiple physical constraints. To the best of our knowledge, this is the first architecture to combine the layer-wise gating mechanism with a multi-branch operator-network topology, yielding an input-adaptive trunk basis. Additionally, we integrate the 3D Navier-Stokes equations as governing physical laws, so the model is trained based on physics-informed residuals, initial and boundary conditions, and only 0.3% of the labeled internal data together with the selected branch-conditioning signals. The M3PI-DeepONet simultaneously predicts unsteady 3D flow velocity and pressure fields with an average relative L2 velocity error below 4% and pressure error around 5% while achieving a conservative retained-cycle inference speedup of approximately 36x compared to reference CFD simulations once the branch inputs used for conditioning are available. This work advances the application of deep learning in cardiovascular disease modeling, marking step toward real-time, non-invasive clinical diagnostics.

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