Computationally-efficient constrained constructive optimisation algorithm for patient-specific coronary microvascular network generation

Technical Summary: Computationally-Efficient Constrained Constructive Optimisation for Patient-Specific Coronary Microvascular Network Generation

Problem Statement

Accurate patient-specific analysis of myocardial perfusion requires realistic reconstructions of the coronary microvascular network. However, direct in vivo measurement of these microstructures is extremely challenging. While Constrained Constructive Optimisation (CCO) is a established method for generating vascular trees based on physiological optimality principles (e.g., minimizing total energy or volume), existing implementations face significant limitations:

1. Computational Cost: Standard CCO algorithms require global re-optimization of the entire tree after every vessel insertion and evaluate connections against all existing segments, leading to exponential growth in computational time ($O(n_t^2)$ or worse) as the network size increases. This makes generating networks with millions of vessels (typical of the coronary microvasculature) computationally prohibitive.
2. Lack of Patient-Specific Integration: Traditional CCO typically initiates from a single inlet vessel. It does not natively support extending existing, patient-specific epicardial arterial trees (reconstructed from clinical imaging) into the downstream microvasculature without altering the original anatomy.
3. Anatomical Fidelity: Standard CCO lacks specific constraints to enforce the unique topological organization of the coronary tree, such as the relationship between principal path lengths and myocardial wall thickness, often resulting in unrealistic geometries when applied to complex, non-convex domains like the left ventricle (LV).

Methodology

The authors propose a novel framework integrating a computationally efficient CCO variant (eCCO) with coronary-specific constraints to generate microvascular networks from existing patient-specific epicardial trees.

1. Computationally Efficient CCO (eCCO)
The authors introduce a modified global-local optimization strategy to reduce computational complexity:

• Coarse Global Search: Instead of performing $n$ local iterations for each of the $N_{near}$ (typically 20) nearest candidate vessels, the algorithm first evaluates a single representative bifurcation location (the centroid of the triangular bifurcation plane) for all candidates.
• Refined Local Search: Only the candidate vessel yielding the minimum cost at this coarse level undergoes a full, refined local optimization (discretizing the plane into $n$ points).
• Complexity Reduction: This restructuring reduces the number of cost function evaluations per terminal insertion from $20n$ to $20 + n$, significantly improving scalability while preserving the structural characteristics of the generated tree.

2. Integration with Patient-Specific Epicardial Trees
The framework modifies the standard CCO initialization and connection logic:

• Initial Condition: The algorithm starts with a segmented, patient-specific coronary arterial tree (derived from clinical imaging) rather than a single root vessel.
• Connection Strategy: New terminal vessels connect directly to the endpoints of existing tree segments or to previously generated micro-vessels.
  • Fixed Parent Radii: Unlike standard CCO where parent radii are adjusted, the radii of the existing segmented tree remain fixed.
  • Radius Updates: New vessel radii are calculated based on flow balance and reduced hydraulic resistance.
  • Physiological Validation: At junctions between the fixed epicardial tree and new branches, the bifurcation exponent ($\gamma$) is computed. Connections are accepted only if $2.5 \leq \gamma \leq 3.5$, ensuring physiological consistency.
• Domain Constraints: For non-convex LV geometries, a mesh-based approach using Gaussian quadrature points selects terminal candidates. An equipotential function (solving the Laplace equation) ensures all generated vessels remain within the myocardial wall.

3. Coronary-Specific Topological Constraints
To preserve the characteristic transmural organization of coronary arteries, a new constraint is applied:

• Principal Path Length: For any new subtree originating from an epicardial vessel, the length of the principal path ($l_p$) is computed.
• Rejection Criterion: If $l_p > \alpha \cdot m_t$ (where $m_t$ is the local myocardial wall thickness and $\alpha$ is a scaling constant, set to 1.5 in this study), the candidate vessel is rejected. This prevents the algorithm from generating excessively long, unrealistic subtrees that violate the anatomical limits of the myocardial wall.

Key Results

The proposed framework was evaluated in three stages:

1. Spherical Domain Benchmark:

   • Generated 10,000 terminal vessels in a spherical domain.
   • Efficiency: The eCCO algorithm was approximately 13 times faster than the standard CCO algorithm.
   • Accuracy: The objective function values (total vascular volume) and the 3D distributions of bifurcation angles ($\Psi_1, \Psi_2, \theta_1, \theta_2$) were comparable between eCCO and standard CCO, confirming that efficiency gains did not compromise geometric fidelity.

2. Human Left Ventricle (LV) Reconstruction:

   • The framework was applied to a human LV using a high-resolution cryomicrotome dataset (Schwarz et al., 2020) as the reference.
   • Initial Condition: Only large epicardial vessels (Strahler Order > 6) were used as the starting tree.
   • Impact of Constraints: Without the principal path constraint, the algorithm produced terminal vessels with a mean radius of $1.4 \pm 1.2$ $\mu$m (significantly smaller than the reference). With the constraint, the mean terminal radius increased to $29.58 \pm 10.9$ $\mu$m, closely matching the reference value of $32.16 \pm 16.44$ $\mu$m.
   • Geometric Agreement: The 3D bifurcation angle distributions of the generated network showed close agreement with the reference dataset, both qualitatively and quantitatively.

3. Porcine LV Application:

   • The framework was successfully applied to a porcine heart dataset to generate a microvascular network with 70,000 terminal vessels.
   • The resulting bifurcation angle distributions were similar to those of the human model, demonstrating the framework's applicability across different species and imaging modalities.

Significance and Claims

The authors claim that this work provides the first application of the CCO framework to generate coronary microvasculature in a patient-specific geometry starting from an existing coronary arterial tree.

• Computational Scalability: The eCCO algorithm enables the practical generation of large-scale microvascular networks (millions of vessels) suitable for patient-specific hemodynamic modeling, overcoming the computational bottlenecks of previous methods.
• Patient-Specific Integration: The method allows for the seamless extension of clinical imaging data (epicardial trees) into the microcirculation without modifying the original patient anatomy, bridging the gap between non-invasive imaging and microvascular analysis.
• Physiological Realism: By incorporating coronary-specific topological constraints (principal path length vs. wall thickness), the framework generates networks that accurately reproduce the terminal vessel radii and 3D branching characteristics observed in high-resolution human data.

The authors note that while the current study focuses on geometric reconstruction, the framework is designed to support future large-scale personalized perfusion modeling and can be extended to other vascularized organs (e.g., liver, brain) by incorporating organ-specific constraints.