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

Problem Statement

Cardiovascular disease (CVD) assessment relies heavily on hemodynamic descriptors such as Wall Shear Stress (WSS) and Fractional Flow Reserve (FFR), which are derived from patient-specific vascular geometries. Standard clinical workflows utilize Contrast-Enhanced Computed Tomography (CE-CT) to reconstruct these geometries. However, conventional pipelines—typically involving Filtered Back-Projection (FBP) or iterative reconstruction followed by segmentation—struggle with time-resolved acquisitions characterized by sparse and noisy measurements due to radiation dose constraints and physiological motion artifacts. These limitations lead to undersampling artifacts, resulting in geometric inaccuracies like missing branches or artificial stenoses.

Furthermore, existing physics-constrained inverse methods for hemodynamics often assume a fixed, pre-segmented geometry. Enforcing flow physics on an erroneous fixed geometry yields valid but unrepresentative flow fields, failing to capture critical hemodynamic parameters. While some recent works have attempted to reconstruct geometry and flow jointly, they often rely on measured velocity images (e.g., from MRI) rather than recovering geometry directly from raw CT projection data (sinograms). There remains a gap in frameworks capable of jointly recovering vascular topology and flow parameters directly from sparse, noisy CE-CT sinograms without a priori geometric assumptions.

Methodology

The authors propose VITO, a physics-constrained reconstruction framework that formulates vascular geometry recovery as a density-based Topology Optimization (TO) problem. Instead of treating geometry as a fixed input, the vascular lumen is treated as an unknown design variable.

Core Framework

The method solves an inverse problem where the objective is to find a fluid-solid distribution that, when subjected to governing physical laws, produces simulated projection data matching the acquired sparse sinograms.

1. Design Representation:

   • The computational domain is discretized into mesh elements, each assigned a pseudo-density variable $\gamma_e \in [0, 1]$.
   • $\gamma = 0$ represents fluid (vascular lumen), and $\gamma = 1$ represents solid (tissue/walls).
   • Material properties are interpolated using the Rational Approximation of Material Properties (RAMP) scheme.
     • Brinkman Penalty ($\alpha$): A large penalty in solid regions suppresses fluid velocity, effectively enforcing no-slip conditions without explicit boundary tracking.
     • Diffusivity ($D$): Limits contrast agent transport in solid regions to prevent artificial diffusion.

2. Governing Physics (Forward Model):

   • Steady Flow: The incompressible Navier-Stokes equations are solved using the Brinkman penalization term to model the fluid flow field $\mathbf{u}$.
   • Transient Transport: A transient advection-diffusion equation models the transport of the contrast agent concentration $c$, driven by the steady velocity field $\mathbf{u}$ and time-varying inlet boundary conditions.
   • Forward Projection: The resulting time-resolved concentration field is mapped to the measurement space using a differentiable parallel-beam projection operator based on the Beer-Lambert law. This generates predicted sinograms $\hat{g}$.

3. Optimization Loop:

   • Objective Function: The Mean Squared Error (MSE) between the predicted sinograms and the acquired sparse/noisy sinogram data ($g_{data}$).
   • Design Variables: The pseudo-density field $\gamma$ (geometry) and inlet velocity scaling parameters $a$ (flow conditions).
   • Sensitivity Analysis: The framework utilizes the JAX ecosystem to implement an end-to-end differentiable pipeline.
     • Implicit Function Theorem (IFT): Used to compute sensitivities for the steady-state Navier-Stokes solver, avoiding the memory cost of unrolling iterative Newton-Raphson steps.
     • Checkpointing: Employed for the transient advection-diffusion solver to manage memory usage during reverse-mode automatic differentiation (AD) across time steps.
   • Optimizer: The Method of Moving Asymptotes (MMA) is used to update design variables. Continuation strategies (gradually increasing penalization and projection sharpness) are employed to drive the solution toward a binary fluid-solid topology and avoid local minima.

Key Contributions

• Joint Inverse Reconstruction: The paper introduces a unified framework that recovers vascular topology and inlet flow parameters simultaneously from raw CT sinograms, eliminating the need for a pre-segmented geometry.
• Physics-Driven Regularization: By coupling the inverse problem with steady Navier-Stokes and transient advection-diffusion equations, the method regularizes the reconstruction using hemodynamic physics rather than image priors alone.
• Differentiable Multiphysics Pipeline: The implementation leverages automatic differentiation (JAX) to handle the complex sensitivities of coupled fluid-transport-projection systems, including the use of IFT and checkpointing to make the optimization computationally tractable.
• Robustness to Data Limitations: The framework is explicitly designed to handle sparse-view and noisy acquisition conditions typical of low-dose or fast-acquisition clinical CT.

Experimental Results

The method was validated on synthetic phantoms representing three vascular scenarios:

1. Bifurcated Artery: Successfully recovered the bifurcation topology and inlet flow scalings from sparse sinograms.
2. Stenosed Artery: Accurately reconstructed a localized 50% lumen narrowing (stenosis) and flow parameters.
3. Stenosed Carotid Bifurcation: Recovered a complex geometry combining a bifurcation and an asymmetric stenosis.

Performance Metrics:

• Accuracy: The reconstructed geometries achieved high Dice Similarity Coefficients (DSC > 0.99) and low normalized Root Mean Square Errors (nRMSE) compared to ground truth across all benchmarks.
• Noise and Sparsity Robustness: In experiments involving ultra-sparse data (10 projections) and high noise levels (25% relative Gaussian noise), the proposed method maintained topological consistency (DSC = 0.9830). In contrast, a baseline FBP approach with thresholding suffered from boundary distortion, artifacts, and loss of branch continuity under similar conditions.
• Flow Estimation: The framework successfully recovered inlet flow scaling parameters close to unity (ground truth), demonstrating its ability to infer boundary conditions alongside geometry.

Significance and Claims

The paper claims that integrating underlying hemodynamics into the reconstruction process provides a robust geometric regularization, enabling the recovery of complex vascular features (bifurcations, stenoses) directly from sparse clinical projection data. This approach addresses the critical limitation of current workflows where geometric errors propagate into hemodynamic analysis.

The authors position this work as a significant step toward reliable vascular analysis from sparse clinical data. They explicitly state that the current study is limited to 2D steady flow and rigid boundaries. Future work is identified as necessary to extend the framework to 3D transient flow, incorporate fluid-structure interaction (FSI) for elastic walls, treat inlet/outlet locations as unknowns, quantify uncertainty, and adapt the projection operator to specific scanner geometries (e.g., fan-beam CT). The paper does not claim immediate clinical deployment but rather establishes the mathematical and computational foundation for such applications.