VITO: Vascular Geometry and Blood Flow Estimation Using Inverse Topology Optimization

Abstract: Computed Tomography Angiography (CTA) is widely used to reconstruct vascular geometry from projection measurements, with conventional approaches such as Filtered Back-Projection (FBP) and Iterative Reconstruction (IR) forming the clinical standard. Blood flow is subsequently estimated through Computational Fluid Dynamics (CFD) simulations, which require vascular geometry and boundary conditions to be specified a priori. Since the geometry is fixed prior to flow estimation, the recovery of unknown anatomical features (e.g., missing branches or stenoses) is precluded. In this work, we present a fluid-physics-constrained reconstruction framework that leverages topology optimization (TO) to jointly recover vascular geometry and blood velocity directly from time-resolved CTA sinograms. The formulation couples a steady incompressible flow model with a transient advection-diffusion contrast transport model, mapped to sinogram space through a differentiable projection operator. The recovered velocity fields provide hemodynamic information and can support downstream estimation of wall shear stress and flow distribution, without requiring a separate CFD pipeline. The proposed method is demonstrated on synthetic phantoms under varying sparsity and noise levels, and on representative projection data.

• The paper introduces VITO, a PDE-constrained inverse topology optimization method that jointly recovers vascular lumen geometry, velocity fields, and inlet flow parameters directly from time-resolved CE-CT sinograms.
• The method achieves Dice scores up to 0.9981 and inlet-flow scaling errors as low as approximately 0.1% on synthetic bifurcation and stenosis benchmarks using only half of the available projection angles.
• VITO preserves vascular topology under extreme sparsity and noise, including 10 projections with 25% noise, but requires validation beyond its current 2D synthetic setting with steady Newtonian flow and idealized acquisition models.

Motivation and problem statement

Conventional contrast-enhanced CT (CE-CT) pipelines reconstruct vascular geometry in two disjoint stages: image reconstruction from projection data using filtered back-projection (FBP) or iterative methods, followed by lumen segmentation. Blood flow is then estimated with computational fluid dynamics (CFD) on the segmented geometry, requiring boundary conditions specified a priori. Because geometry is fixed before flow estimation, this pipeline cannot recover unknown anatomical features—missing branches or stenoses—and is fragile under sparse-view acquisitions, where undersampling artifacts corrupt the reconstructed lumen. The paper argues that enforcing flow physics on an erroneous fixed geometry yields a "valid but unrepresentative" flow field, since hemodynamic biomarkers such as wall shear stress (WSS) and pressure drop are highly sensitive to minor boundary perturbations.

VITO addresses this by treating vascular topology as an unknown optimization variable, recovered jointly with blood velocity directly from time-resolved CE-CT sinograms through a physics-constrained inverse problem.

Methodology

The formulation is a PDE-constrained density-based topology optimization (TO). Each element of a structured mesh carries a pseudo-density $\gamma_e \in [0,1]$, with $\gamma=0$ interpreted as fluid (lumen) and $\gamma=1$ as solid tissue. Material properties are interpolated via RAMP schemes: a Brinkman penalty $\alpha(\gamma)$ suppresses velocity in solid regions within the steady incompressible Navier–Stokes equations, while a design-dependent diffusivity $D(\gamma)$ confines contrast transport to the fluid. A transient advection–diffusion equation models pulsed contrast injection, driven by the steady velocity field.

The forward model maps concentration fields to measurement space under a parallel-beam geometry: applying the Beer–Lambert law with a linear attenuation–concentration relation and log-subtraction against a pre-contrast mask yields a linear projection model, $g(t,\theta,s)=\int_{\ell(\theta,s)} c(x,y,t)\,dl$. The objective is the mean squared error between predicted and measured sinograms over sparse projection angles, minimized jointly over the pseudo-density field and inlet velocity scaling parameters using MMA. Density filtering, smoothed Heaviside projection, and continuation on penalization parameters promote binary, mesh-independent designs.

Two implementation choices are notable. First, the entire pipeline is implemented in JAX with end-to-end automatic differentiation; iterative Newton–Raphson solvers are handled via the implicit function theorem rather than unrolling solver iterations, and checkpointing manages memory for the transient transport adjoint. Second, PSPG/SUPG stabilization permits equal-order interpolation and mitigates convection-dominated oscillations. These choices yield machine-precision design sensitivities without manual adjoint derivation.

Numerical results

All experiments use synthetic time-resolved sinograms generated from prescribed ground-truth geometries on an 80×80 mesh, evaluated on a MacBook M4 Pro. Three benchmarks are considered: a bifurcated artery, a channel with a 50% Gaussian stenosis, and a stenosed carotid bifurcation with asymmetric narrowing in one daughter branch. In all cases, both geometry and inlet flow scalings are recovered simultaneously from half of the available projections (90 of 180 angles).

Benchmark	nRMSE$_\gamma$	DSC	Inlet scaling error
Bifurcated artery	0.0091	0.9981	~$10^{-3}$
50% stenosis	0.0699	0.9981	~$2\times10^{-3}$
Stenosed carotid bifurcation	0.0404	0.9933	~$8\times10^{-3}$

The bifurcation benchmark demonstrates that recovery proceeds through flow and transport physics rather than image-domain segmentation: the predicted sinogram frames match the sparse targets closely, and the associated velocity and concentration fields are produced as byproducts of the reconstruction, providing hemodynamic information without a separate CFD pipeline.

Robustness experiments on the bifurcated phantom compare three degraded settings against an FBP-plus-Otsu-segmentation baseline:

Setting	nRMSE$\gamma=0$0	DSC
18 projections, 5% noise	0.0113	0.9978
10 projections, 5% noise	0.0162	0.9965
10 projections, 25% noise	0.0583	0.9830

The FBP baseline degrades progressively with sparsity and noise, exhibiting boundary distortion, isolated artifacts, and loss of branch continuity at 10 projections with 25% noise, whereas the proposed method preserves lumen topology across all cases. This robustness follows from the fact that the optimizer searches for a geometry whose induced transport dynamics reproduce the time-resolved sinograms, rather than segmenting artifact-corrupted images. Noise is modeled as additive Gaussian—a common high-count approximation to Poisson statistics, consistent with the $\gamma=0$1 loss—rather than photon-counting noise itself.

Limitations and open questions

The paper is explicit that all validation is synthetic and two-dimensional. Several assumptions bound the current results:

• Steady, Newtonian, 2D flow: extension to 3D transient flow with non-Newtonian rheology and uncertain physical parameters remains open, as does validation against patient-specific in vivo data.
• Rigid walls: physiological arterial compliance is neglected; incorporating fluid–structure interaction is deferred.
• Prescribed inlet/outlet locations: the framework assumes known boundary placement; joint estimation of boundary locations with topology is not addressed.
• Local minima: gradient-based optimization can converge to poor local optima; no global-search or initialization strategy is provided beyond continuation schemes.
• Idealized acquisition: the parallel-beam operator treats projection angle and transport time as independent coordinates; coupling them for a rotating gantry, fan-beam geometry, or rotation-free architectures is future work.
• Deterministic output: the method returns a single best-fit design; uncertainty quantification over the recovered lumen and derived biomarkers is identified but not developed.

Conclusion

This work formulates vascular reconstruction from CE-CT sinograms as a fluid-physics-constrained topology optimization problem, jointly recovering lumen topology, velocity fields, and inlet flow scalings. On synthetic benchmarks, it achieves near-perfect structural agreement (DSC up to 0.9981) and retains topological consistency under ultra-sparse, heavily noisy measurements where FBP-based segmentation fails. The results establish hemodynamic physics as an effective geometric regularizer for tomographic reconstruction, though clinical relevance awaits extension to 3D transient flow, realistic scanner geometries, and in vivo validation.