Surgical Planning and Optimization of Patient-Specific Fontan Grafts With Uncertain Post-Operative Boundary Conditions and Anastomosis Displacement

Abstract

Objective: Fontan surgical planning involves designing grafts to perform optimized hemodynamic performance for the patient's long-term health benefit. The uncertainty of post-operative boundary conditions (BC) and graft anastomosis displacements can significantly affect optimized graft designs and lead to undesirable outcomes, especially for hepatic flow distribution (HFD). We aim to develop a computation framework to automatically optimize patient-specific Fontan grafts with the maximized possibility of keeping post-operative results within clinical acceptable thresholds.

Methods: The uncertainties of BC and anastomosis displacements were modeled using Gaussian distributions according to prior research studies. By parameterizing the Fontan grafts, we built surrogate models of hemodynamic parameters taking the design parameters and BC as input. A two-phase reliability-based robust optimization (RBRO) strategy was developed by combining deterministic optimization (DO) and optimization under uncertainty (OUU) to reduce computational cost.

Results: We evaluated the performance of the RBRO framework by comparing it with the DO method in four cases of Fontan patients. The results showed that the surgical plans computed from the proposed method yield up to 79.2% improvement in the reliability of the HFD than those of the DO method ( ). The mean values of indexed power loss (iPL) and the percentage of non-physiologic wall shear stress (%WSS) for the optimized surgical plans met the clinically acceptable thresholds.

Conclusion: This study demonstrated the effectiveness of our RBRO framework to address the uncertainties of BC and anastomosis displacements for Fontan surgical planning.

Significance: The technique developed in this paper demonstrates a significant improvement in the reliability of the predicted post-operative outcomes for Fontan surgical planning. This planning technique is immediately applicable as a building block to enable technology for optimal long-term outcomes for pediatric Fontan patients and can also be used in other pediatric and adult cardiac surgeries.

Fig. 1.

Illustration of Fontan surgical planning. 3D reconstructed patient-specific Fontan model with various possible pathways. Deoxygenated blood flow were directed from the superior vena cava (SVC) and the inferior vena cava (IVC) to the lungs via the left pulmonary artery (LPA) and the right pulmonary artery (RPA).

Fig. 2.

Procedure of setting up uncertain BC for Fontan hemodynamic simulation. (A) Cardiovascular magnetic resonance (CMR) imaging for 3D cardiovascular structure reconstruction and blood flow rates measurement. $Q_{\text{SVC}}^{\mathrm{m}}$, $Q_{\text{IVC}}^{\mathrm{m}}$, $Q_{\text{LPA}}^{\mathrm{m}}$ and $Q_{\text{RPA}}^{\mathrm{m}}$ are time-averaged flow rates from CMR measurement. F_LPA and Q_Total represent PA flow split ratio and total inlet flow rate, respectively. (B) Q_SVC, Q_IVC, Q_LPA and Q_RPA are used for setting up deterministic Fontan hemodynamic simulation. HFD_LPA represents the hepatic flow distribution (HFD), which is the percentage of IVC flow to the LPA for the optimized graft at the planned anastomosis location. The blue graft shows the actual anastomosis location. (C) $Q_{\text{SVC}}^{\prime }$, $Q_{\text{IVC}}^{\prime }$, $Q_{\text{LPA}}^{\prime }$ and $Q_{\text{RPA}}^{\prime }$ are BC with introduced uncertainty. HFD_LPA + Δ represents the change of HFD due to the uncertainties of the post-operative BC ($Q_{\text{SVC}}^{\prime }$, $Q_{\text{IVC}}^{\prime }$, $F_{\text{LPA}}^{\prime }$) and the anastomosis displacement.

Fig. 3.

Three-dimensional representation and hemodynamic results of original Fontans. A cohort of Fontan patients (n=4) were retrospectively collected and digitally processed into 3D models for CFD simulation. The cohort consisted of 2 extracardiac-type Fontans (F₁, F₂), and 2 lateral tunnel-type Fontans (F₃, F₄). The original Fontan conduits were removed from the model for the surgical planning task. The highlighted hemodynamic parameters in red color were considered outside the thresholds.

Fig. 4.

Schematic of the RBRO computation framework. (A) The inputs of the framework include 3D Fontan models, the pre-operative BC, and the uncertainty models of BC. (B) Fontan pathway parameterization creates the design space x for automatically exploring the conduit geometry, anastomosis location and orientation. The uncertainty models of anastomosis can also be considered as the inputs of the framework. (C) The surrogate model generation involves sampling in the design space and the space of uncertain parameters, computing the hemodynamic results, and applying Gaussian process regression to learn the hemodynamic responses of different inputs. (D) The RBRO optimization process includes two optimizers. The optimizer 1 performs DO to generate warm starts for the optimizer 2. The optimizer 2 performs OUU to compute the final optimal surgical plans.

Fig. 5.

Simplified Fontan models to illustrate the translational and rotational anastomosis displacements. (A) ΔL represents the uncertainty of translational displacement. (B) α and β represent the uncertainty of rotational displacement.

Fig. 6.

Block diagram of the optimizer 1 and the optimizer 2. The optimizer 1 represents the DO process, which does not consider the uncertainty of BC and anastomosis. The optimizer 2 performs uncertainty quantification (UQ) with the uncertainty space u on each set of explored design parameters x_k to generate statistic responses of the hemodynamic parameters.

Fig. 7.

The probability of having balanced HFD P(HFD_balanced) of optimized graft designs by using deterministic optimization (DO) and optimization under uncertainty (OUU) methods for each patient. The result for the patient F₁ is shown in (A). The y-axis represents the probability of HFD_LPA within the thresholds. The x-axis represents DO design order ranking from the most reliable HFD to the least reliable HFD by using UQ. The 10 DO designs are the warm starts for OUU. The circled numbers above the bars represent the original ranking of the DO designs according to the objective function of (20). The same analysis for the patients F₂, F₃ and F₄ are shown in (B), (C), and (D), respectively. (E) illustrates how the uncertain parameters u affect HFD_LPA for the optimized designs in the groups #1 and #10 of F₄. The three rows show HFD_LPA(F_LPA, Q_IVC, x), HFD_LPA(ΔL, α, x), HFD_LPA (ΔL, β, x). x represents the graft design parameters ${\mathrm{x}}_{{\text{DO}}_1}^{\ast }$, ${\mathrm{x}}_{{\text{OUU}}_1}^{\ast }$, ${\mathrm{x}}_{{\text{DO}}_{10}}^{\ast }$, ${\mathrm{x}}_{{\text{OUU}}_{10}}^{\ast }$ in the four columns, respectively. Note: HFD_LPA > 0.6 is represented by white color, and HFD_LPA < 0.4 is represented by black color.

Fig. 8.

Illustration of the top-ranked DO and OUU designs for the 4 patients. It is hard to tell the geometrical differences between DO and OUU designs for F₃ and F₄, because the design parameters are identical for DO and OUU in F₃ (see Table I) and the graft diameter of the OUU design in F₄ is slightly larger than that of the DO design with all the other design parameters almost the same.

Fig. 9.

Effect of changing the objective function in DO as minimizing iPL. (A) HFD reliability comparison between DO designs computed from the objective functions min ∣HFD_LPA – 0.5∣ and min iPL. (B) HFD reliability comparison between OUU designs computed from the two different objective functions in DO. (C) HFD reliability comparison between DO and OUU designs. (D) $\mathbb{E}(\text{iPL})$ comparison between DO designs computed from the two different objective functions in DO. (E) $\mathbb{E}(\text{iPL})$ comparison between OUU designs computed from the two different objective functions in DO. (F) $\mathbb{E}(\text{iPL})$ comparison between DO and OUU designs. A p < 0.05 is considered statistically significant.

Fig. 10.

Illustration of Fontan grafts from manual optimization, DO and OUU. (A) Manually optimized graft duplication by minimizing the geometrical difference with the parameterized graft for the patient F₂. (B) Comparison between the top-ranked DO design and the manually optimized design for F₂. (C) Comparison between the top-ranked OUU design and the manually optimized design for F₂.

Fig. 11.

The feasibility of designing grafts with HFD_LPA = 50% under different patients’ blood flow conditions. The x-axis represents the percentage of Q_IVC in the total systemic venous flow (Q_IVC + Q_SVC). The range 0.5~0.8 represents the patient spectrum from pediatric to adult. The y-axis represents F_LPA. The HFD_LPA map is calculated by using (28). The locations of F₁, F₂, F₃, and F₄ correspond with their pre-operative BC. T₁ and T₂ represent two extreme cases with highly unbalanced PA flow splits.

Fig. 12.

Fontan graft optimization results for the patient cases T₁ and T₂ with highly unbalanced PA flow splits. Each black dot represents an optimized graft design that was computed from DO. Each red dot represents an optimized graft design that was computed from OUU with a DO solution as a warm start. The dash lines indicate that specific DO designs were used for OUU computation.