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Anatomy-Informed Neural Networks: Encoding Anatomic Priors in Loss and Architecture, with an SE(3) Formulation of Guidewire-Induced Aortoiliac Deformation
David P. Stonko
TL;DR
The paper addresses anatomically impossible predictions and poor generalization when training data are scarce. It proposes AINN, combining soft and hard anatomic priors with an SE(3)-based wire–vessel mechanics model supervised by projected angiograms. Synthetic verification supports the kinematics, loss, and projection, while mesh convergence, patient validation, and network training remain unresolved.
Problem
Deep-learning anatomy models can produce anatomically impossible outputs and generalize poorly from small surgical datasets.
Method
AINN combines soft loss penalties and hard architectural priors with an SE(3) Cosserat-rod wire–vessel model and Wasserstein-2 projected supervision.
Results
Synthetic verification supports the forward mechanics, loss, and projection, while a guidewire-stiffness sweep identifies axial pre-tension as the dominant energetic drive across the clinically plausible range.
Takeaways & Limitations
The paper provides a verified in-silico framework for anatomy-informed modeling of guidewire-induced vascular deformation under single-view supervision.
Takeaways & Limitations
No patient data are used, the residual network is not trained, and predicted displacement is not mesh-converged.
Abstract
from arXiv · showhide
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.
1 Introduction
AINN integrates anatomic knowledge through soft loss penalties and hard architectural or representational constraints. The framework targets anatomically plausible, data-efficient prediction while distinguishing guarantees from design goals and presenting a vascular demonstration that is not yet trained.
- AINN framework: AINN encodes anatomic priors softly through loss penalties and hard through architecture or state representation.Soft priors accommodate anatomical variation, whereas hard priors make certain violations impossible by construction.
- Vascular instantiation: The worked example couples an SE(3) wire and vessel model, unilateral lumen contact, tortuosity-anchored stiffness, and projected Wasserstein-2 supervision.The formulation combines geometric, contact, mobility, and data-fit components in one vascular deformation model.
- Scope: AINN is intended to improve generalization from less data and interpretability, but these are stated design goals because no AINN is trained here.The paper frames reduced anatomic hallucination as a possible benefit rather than a demonstrated result.
- Hard versus soft priors: Hard SE(3) representations guarantee valid rigid-body frames and connected reconstructed centerlines, but do not alone ensure anatomical topology or bodily containment.The exponential-map reconstruction prevents discontinuity, while branching topology, self-intersection, and containment require additional treatment.
- Prior categories: The framework organizes priors across topology, mobility and anchoring, shape, and other categories, identifying mobility and anchoring as absent from existing taxonomies consulted by the authors.Examples include connectivity and branching constraints, plus position-dependent stiffness for anatomically anchored motion.
3 The Worked Example: Geometric Infidelity in Endovascular Surgery
The worked example addresses deformation between a preoperative CT-based State 1 and an instrument-loaded intraoperative State 2, where only a single 2D angiogram commonly records the changed anatomy. It replaces a meshed-wall primitive with SE(3) centerline frames to model wire, vessel, and projection in a compact state space.
- Clinical problem: Preoperative endovascular planning treats CT anatomy as static, although stiff wires deform the aorta, iliac arteries, and branch-vessel ostia intraoperatively.This State 1 to State 2 mismatch can affect seal-zone assessment and branch-vessel cannulation.
- Clinical problem: A single 2D fluoroscopic angiogram is typically the intraoperative record of the deformed anatomy because three-dimensional re-imaging is not routine.The framework therefore uses projected supervision for a three-dimensional prediction.
- Related work: Prior finite-element studies established that guidewire-induced deformation is predictable and provided patient-specific validation against intraoperative imaging.Those approaches mesh the arterial wall and set an accuracy reference for alternative formulations.
- Geometric formulation: The proposed formulation takes the centerline and carried orientation frame as the primitive, placing vessel and wire curves in SE(3) rather than deriving centerlines from a mesh.This yields a shared state for contact geometry, strain, residual correction, and projection, with a few hundred degrees of freedom.
4 Kinematics on SE(3): Two Coupled Curves
The model represents vessel and wire as coupled curves of oriented frames in SE(3), using Lie-group kinematics to preserve valid rigid-body configurations and centerline continuity. Bishop frames provide stable orientations, while body-fixed se(3) strains describe local translation, bending, and twisting.
- Configuration representation: Both the vessel centerline and guidewire path are modeled as curves of frames in SE(3), combining spatial position with orientation.The vessel is obtained from segmented preoperative CT, while the wire represents the equilibrium configuration after insertion.
- Configuration representation: The SE(3) lift distinguishes spatially close curve segments with different orientations, mitigating crossing ambiguity in folded vessel geometries.Nearly parallel adjacent vessels with nearly identical positions and tangents remain ambiguous and require segmentation for separation.
- Frame choice: Bishop frames are used instead of Frenet frames because they remain stable near zero curvature and along nearly straight guidewire segments.For a Bishop frame, the material twist component is zero by construction, so geometric torsion must be recovered separately.
- Strain representation: Body-fixed strain in se(3) records local translation and rotation rates, with rotational components encoding bending and material twist.The Lie algebra provides the tangent-space strain representation, while the exponential map advances one frame to the next.
- Hard anatomic prior: Generating each frame from its predecessor by a rigid motion makes every integrated frame valid and makes discontinuous centerlines impossible by construction.This encodes vessel continuity as a hard prior rather than a penalty; branching topology and spatial containment require separate mechanisms.
5 The Unilateral Lumen Contact Constraint
The wire is modeled as a smaller, straighter path inside a deformable vessel, coupled through a closest-point unilateral lumen constraint. Active contact produces concentrated reactions that drive piecewise vessel deformation, while containment can be added as a second hard constraint.
- Wire–vessel geometry: Because the wire radius is smaller than the lumen radius, the wire rides inner curvature and takes chords rather than following the deformed vessel centerline.This bowstring behavior is central to modeling guidewire-induced deformation realistically.
- Constraint construction: Same-index wire–vessel comparisons fail under foreshortening because they conflate radial wall offset with axial drift between differently parameterized curves.The iliac can foreshorten under loading, so equal parameter values need not identify corresponding anatomical locations.
- Constraint construction: The imposed non-penetration condition uses the closest vessel point and bounds wire-centerline distance by R_lumen(σ*(s))−r_w.Parameterizing by wire arc length handles foreshortening and the available slack produces the bowstring path.
- Clinical readout: The enforced closest-point constraint differs from the clinically reported in-frame offset, which measures displacement in the plane orthogonal to the wire tangent.The two quantities coincide only when wire and vessel tangents are parallel, precisely when bowstringing is absent.
- Unilaterality and complementarity: Unilateral contact permits compression but not adhesion or tension: separated regions have zero reaction, while active contact has a positive normal reaction.Complementarity determines where the constraint is active along the curves.
- Deformation mechanism: Concentrated, asymmetric contact reactions at curvature apexes drive the State 1 → State 2 deformation and produce piecewise rather than uniform straightening.The coupled system allows the wire to become taut before the vessel centerline and distributes deformation according to local anatomical give.
- Containment: An anatomic envelope can be imposed as a second unilateral constraint, turning containment into a hard prior rather than relying on quadratic anchoring penalties.This extension is formulated but not implemented because the synthetic geometries lack surrounding anatomy.
6 Energy Functional and Anatomically Modulated Stiffness
The model combines wire, vessel, surrounding-tissue, anchoring, and axial energies under a unilateral lumen-contact constraint. Anatomical framing and stiffness choices determine which physical effects are represented, while several limitations constrain interpretation.
- Energy functional: The total energy combines wire, vessel, fat, anchoring, and axial terms, with equilibrium obtained by minimizing it subject to the lumen inequality.The constrained problem may have multiple local minima, so determinism belongs to the solver and initialization rather than the variational problem itself.
- Wire bending energy: Under Bishop framing, the operative guidewire stiffness reduces to the scalar bending coefficient EI because twist, stretch, and shear degrees of freedom are inert.This reduction follows from the framing and kinematic choices, but it omits twist carried by a torqueable real guidewire.
- Tortuosity modulation: Tortuosity modulation is a dimensionless normalized bending-energy covariate, not the clinical arc-to-chord tortuosity index or a nonlinear strain-stiffening law.The present quadratic form represents greater average compliance over the operative range; slack-then-taut behavior would require a superquadratic or bilinear extension.
- Extravascular resistance: Extravascular resistance models elastic energy stored in surrounding retroperitoneal tissue, paraspinal muscle, and spine when the aorta moves from its preoperative position.It is described as the most patient-specific energy term.
- Extravascular resistance: The anisotropic formulation determines deformation direction variationally, whereas bootstrap and fixed-prior formulations respectively iterate toward or prescribe the direction.Anterior displacement into retroperitoneal fat is therefore an output only of the anisotropic formulation and an input to the other two.
- Anatomic anchoring: The anchoring stiffness field encodes mobility by anatomic segment because local calcification and Hounsfield-derived stiffness do not capture shared structural tethering.The model distinguishes population-level anatomic anchoring from patient-specific Hounsfield-derived variation, but their sum is not separately identifiable from deformation data alone.
- Axial pre-tension: Axial loading is the missing mechanism needed for clinical-magnitude deformation, and the model treats intraluminal wire length as free while applying axial tension as a dead load.The earlier soft inextensibility spring was rejected as a numerical device; axial tension is assumed to range from approximately 2 to 15 N pending force-sensor calibration.
- Lumen contact: The unilateral constraint couples wire and vessel, with KKT multipliers representing active contact forces and the smooth line load λ(s)=T_axialκ(s) reported on the active set.Nodal multipliers depend on discretization, whereas the line load is treated as the physical contact reaction; bending corrections are omitted from that simplified reaction.
7 The Optimal-Transport Loss
The paper replaces pointwise comparison with Wasserstein-2 transport between measures, while acknowledging that projected supervision and the chosen approximations limit what the loss can identify.
- Wasserstein-2 formulation: Wasserstein-2 compares probability measures rather than fixed point pairs, reducing sensitivity to nuisance parameterization.The transport plan minimizes total squared movement between source and destination measures.
- Limits of the loss: W2 does not enforce branch identity or exclude anatomically impossible configurations; those roles belong to topology-aware objectives and hard priors.The framework assigns geometric validity to the SE(3) representation and lumen-contact constraint rather than to the loss alone.
- Mass and foreshortening: Normalizing measures removes total arc-length changes from W2, so the paper adds a scalar length term to supervise foreshortening.Unbalanced transport with a mass-creation penalty is noted as an alternative.
- Ground cost: The training loss uses projected measures on R2 with a Euclidean ground cost, so the sub-Riemannian SE(3) distance does not drive the training gradient.The group-intrinsic cost is reserved for settings with three-dimensional ground truth.
- Identifiability and approximation: Rotational residual degrees of freedom are weakly identified by monoplane data, while the numerical work uses a local nilpotent approximation rather than the exact Carnot–Carathéodory distance.The rotational data gradient degenerates as the residual approaches the physics prediction, and the approximation need not satisfy the global triangle inequality.
8 Single-View Projection and 2D Supervision
Because routine intraoperative supervision is a single 2D angiogram, the framework projects its 3D prediction through known C-arm geometry and compares like objects in image space.
- Motivation: Routine single-view fluoroscopy provides weak projected supervision while the model continues to predict the vascular state in three dimensions.Cone-beam CT and biplane fluoroscopy are not routinely captured at most centers.
- Projection operator: The projection extracts frame positions, transforms them by the known C-arm pose, and applies perspective pinhole projection into R2.The projection uses the acquisition metadata and π(x,y,z) = (fx/z, fy/z).
- Supervision targets: Supervision combines continuous projected wire-path transport with discrete landmark transport, matching predicted and observed like objects.The wire is radio-opaque and persistent, while the lowest renal ostium and aortic bifurcation are the most consistently identifiable landmarks.
- Limitation: Monoplane projection leaves depth unobserved, so predictions that differ along the detector-depth axis can receive the same 2D loss.The working-view geometry is known, but correctness in the image plane does not establish correctness in depth.
- Depth regularization: The depth penalty supplies a mechanical prior for the unobserved direction, but whether it outperforms alternative regularizers remains untested.The penalty is Tikhonov regularization applied specifically to the residual’s out-of-plane component.
- Model boundary: The rigid-wall coupled-inclusion model is restricted to reff < Rlumen; oversizing requires a compliant-wall formulation to represent strain rather than infeasibility.A large-bore sheath can make the rigid-wall feasible set empty in a small external iliac.
9 Architecture: a Physics Prior with an SE(3)-Equivariant Residual
The architecture places a deterministic physics simulation before a learned SE(3) residual, preserving valid frame configurations while targeting systematic simulation–reality deviations.
- Design rationale: A neural operator trained from scratch is expected to overfit the small, weakly supervised cohort, so the architecture puts physics in the model structure.The learned component is assigned only a residual correction rather than the full nonlinear mapping.
- Architecture: The simulator produces a deterministic prediction, and the network adds a small se(3)-valued residual through right composition with the matrix exponential.The objective is minimized jointly over network weights and calibration parameters.
- Hard geometric prior: Exponential composition keeps every corrected frame in SE(3), so the network can only produce physically realizable frame rotations and translations.This guarantee is per-frame and follows from closure of SE(3) under multiplication and the exponential map.
- Equivariance: Right multiplication is chosen because the residual lives in the body frame and must be invariant to global rigid motions for left-equivariance.Equivariance additionally requires the simulator itself to be equivariant.
- Architecture caveat: Independent frame corrections preserve SE(3) membership but can break chain lengths and violate the lumen inequality after composition.A sufficiently large residual can push the wire through the wall, so the hard contact constraint is not fully preserved by the learned correction.
- Smoothness: Residual smoothness regularization is added so high-frequency corrections do not introduce non-physical kinks into the composed curve.The penalty acts on arc-length derivatives or adjacent-frame differences.
- Interpretability: The physics component exposes contact forces, energy budgets, and term contributions, while a large residual can identify anatomy or mechanics missing from the simulator.Examples include adventitial tethering and other under-modeled physical effects.
- Equivariance: The framework is intended to support rotation-consistent predictions without requiring the network to observe every patient orientation during training.This depends on using an SE(3)-equivariant implementation and preserving simulator equivariance.
10 Numerical Experiments
The numerical experiments verify the SE(2)/SE(3) kinematics, contact formulation, Wasserstein-2 loss, and C-arm projection on controlled toy geometries. The mechanics solver reproduces bowstring deformation and contact patterns, but displacement magnitude is not mesh-converged and mechanics verification remains internal.
- Phase 1 planar forward model: The Phase 1 model minimizes coupled wire–vessel energies with a hard closest-point lumen-contact inequality on an SE(2)-lifted planar toy.The model uses a single-period sinusoidal iliac and a Lunderquist guidewire, with contact enforced through sequential quadratic programming.
- Phase 2 three-dimensional forward model: The Phase 2 model lifts the mechanics to SE(3) and adds a uniform extravascular Winkler term for a helical iliac.Rodrigues maps and Bishop parallel transport maintain smooth framing on near-straight segments.
- Phase 2 three-dimensional forward model: 4.41 mm peak and 2.90 mm mean displacement occur in Phase 2, with 43 of 60 nodes in active contact and an 11.77 N integrated wall reaction.The three-dimensional contact forms one extended patch along the helical turn, and the wire shortens from 177.1 mm to 158.5 mm.
- Sensitivity to wire bending stiffness: A 94× change in wire stiffness changes peak displacement by only 35% in two dimensions and 44% in three dimensions.In three dimensions, the endpoint values are 4.41 and 6.33 mm; the sweep supports qualitative insensitivity, not a mesh-converged ratio.
- Mesh convergence and verification: Peak displacement falls from 13.10 to 6.89 mm as mesh resolution quadruples, indicating that predicted displacement magnitude is not mesh-converged.Node-wise contact permits corner-cutting between constraint points; segment-wise non-penetration is identified as the key numerical future work.
- Verification: The toy program verifies kinematics, contact, Wasserstein-2 loss, and C-arm projection against controlled ground truth, while the anchored mechanics is checked only against its own optimality conditions.The C-arm tests include isocenter mapping, magnification, and a 90° oblique image-axis swap.
11 Discussion
The discussion frames AINN as a combination of soft anatomic loss terms and hard architectural or state-representation priors, applied here to a compact vascular mechanics formulation. The study remains an in-silico prototype: no patient data or trained network are used, clinical validation is pending, and important numerical and modeling limitations remain.
- AINN paradigm: AINN combines soft anatomic priors in the loss with hard priors carried by the state representation, making the prior-design choice explicit.The vascular example couples geometric, mobility, symmetry, and contact priors in one formulation.
- Novelty: The anatomically structured mobility and anchoring field is presented as the worked example’s apparently novel component.The position-dependent stiffness encodes fixed arch regions and mobile external iliacs, and it is absent from the cited prior taxonomies.
- Scope and computational role: The SE(3) formulation offers a smaller problem in clinically relevant coordinates, but its toy solves take 30 and 11 seconds and are too slow for a training loop.Finite-element approaches are described as more mature and patient-validated, whereas this formulation is not.
- Scope boundary: No patient data are used, no measurements validate the models, and the specified residual network is not yet trained.The experiments verify implementation components on synthetic geometries constructed by the authors.
- Numerical limitations: Mesh-convergence failure must be resolved before any displacement magnitude is compared with measurement.The study also leaves open whether an interior-point solver or a structure-preserving Lie-group integrator is needed for stiff assemblies.
- Modeling limitations: The anchoring profile and effective wire–device stiffness require calibration against data, while the rigid-wall assumption excludes the large-sheath regime.The toys also do not exercise bifurcation-related topological priors.
- Future evaluation: Clinical evaluation is proposed using roughly 25 paired CT–angiogram cases, with two-dimensional distances and a held-out subset carrying completion cone-beam CT.The cited target is leave-one-out cross-validation appropriate to the cohort size, with comparison against reported millimeter-scale errors.
12 Conclusion
The paper introduces AINN and applies it to guidewire-induced aortoiliac deformation using an SE(3)-based coupled mechanics formulation. In silico verification supports the formulation, but absolute displacement is not yet mesh-converged and comparative predictive performance remains untested.
- AINN paradigm: AINN encodes anatomic structure through soft loss priors and hard architectural or state-representation priors.The paper organizes priors across topology, mobility, symmetry, shape, atlas-relative position, and contact, identifying mobility and anchoring as underserved.
- Vascular formulation: The vascular test case represents vessel centerlines and moving frames in SE(3), coupling a Cosserat rod to an anatomically anchored vessel with unilateral lumen contact.This formulation is paired with an optimal-transport loss and a projection operator for supervising 3D predictions from single-view angiograms.
- In silico verification: 3.64 to 9.97 mm: correcting an earlier formulation increased predicted peak displacement in the planar toy and removed nonphysical energy terms.Across clinically plausible guidewire stiffness, axial pre-tension supplied the dominant energetic drive except near nominal solid-rod stiffness, while deformation was largely stiffness-insensitive.
- Limitations: Mesh refinement preserved qualitative behavior and energy ordering, but absolute displacement magnitude remained unconverged.The authors identify this numerical issue as the most pressing future-work item.
- Scope: Whether the representation improves predictions over prior meshed finite-element models remains an empirical question, and AINN’s generality requires testing in further anatomic domains.The vascular example is presented as the first domain in which the framework is tested.
Scope and Status of This Preprint
This preprint establishes a mathematical formulation and its in silico verification, not a clinically evaluated neural-network predictor.
- Status: No neural network was trained, no patient data or clinical images were used, and no predictive accuracy on patients is claimed.The architecture described in the paper is a specification rather than a trained clinical model.
Data and Code Availability
The paper’s toy models, core computational primitives, and figure-generation scripts are implemented in MATLAB and available from the author on reasonable request.
- Code: The implemented materials include two toy models, Lie-group, optimal-transport, and projection primitives, plus scripts generating Figures 6–10.The toy geometries are fully specified by Table 2.
- Computing environment: Results were produced in MATLAB R2025b using Optimization Toolbox solvers for constrained equilibria and the exact Kantorovich program.The runs used a single 2020 MacBook Pro with an Apple M1 processor, eight cores, and 16 GB of unified memory.
Declaration of Generative AI Use
The author used Claude during preparation for schematic figures, technical auditing, formatting, and text editing, while retaining responsibility for final anatomic and mechanical judgment.
- Generative AI use: Claude assisted with Figures 1–5, an adversarial technical audit, document formatting, and clarity editing.Figure 2 was corrected and redrawn by the author, who determined its final content.
Ethics
The work reports mathematical development and in silico verification using synthetic geometries, without human, animal, or patient data. No funding or competing interests were reported.
- The study used only synthetic geometries and involved no human subjects, animal subjects, or patient data.The broader research program is approved by the Johns Hopkins Medicine Institutional Review Board under IRB00562871.
- No funding was used, and the author declared no competing interests.