Prediction of peak-to-peak pressure gradient in patients with aortic coarctation using Physics-Informed Neural Networks from Magnetic Resonance Images

Presenting Author(s)

• 

  Julio Sotelo

  Assistant Professor
  Departamento de Informática, Universidad Técnica Federico Santa María
  Santiago, Region Metropolitana, Chile

Co-Author(s)

• SJ

  Sebastián Jara

  PhD Candidate in Informatics Engineering
  Universidad Técnica Federico Santa María, Santiago, Chile, Chile
• FG

  Felipe Galarce, PhD

  Assistant Professor
  Pontificia Universidad Católica de Valparaíso, Chile
• HM

  Hernan Mella, PhD

  Assistant Professor
  Pontificia Universidad Católica de Valparaíso, Chile
• RÃ

  Ricardo Ñanculef, PhD

  Professor of Computer Science
  Universidad Técnica Federico Santa María, Chile
• RS

  Rodrigo Salas, PhD

  Professor of Computer Science
  Universidad de Valparaíso, Chile
• 

  Philipp Beerbaum, MD

  Professor
  German Centre for Cardiovascular Research; University Children’s Hospital, Hannover Medical School
  Hannover, Niedersachsen, Germany
• 

  Heynric B. Grotenhuis, MD, PhD

  Pediatric Cardiologist
  Wilhelmina Children's Hospital / University Medical Center Utrecht
  Bilthoven, Utrecht, Netherlands
• DM

  David Marlevi, PhD

  Postdoctoral Fellow
  Karolinska Institutet, Massachusetts, Sweden
• 

  Israel Valverde, MD, PhD

  Pediatric Cardiologist
  The Hospital for Sick Children
  SickKids
  Toronto, Ontario, Canada
• SU

  Sergio Uribe, PhD

  Professor
  Monash University
  Melbourne, Victoria, Australia

Primary Author(s)

• 

  Julio Sotelo

  Assistant Professor
  Departamento de Informática, Universidad Técnica Federico Santa María
  Santiago, Region Metropolitana, Chile

Background: Aortic coarctation (CoAo) accounts for 5-8% of congenital heart disease [1]. For the follow-up evaluation of patients with aortic coarctation (CoAo) in inconclusive cases, the use of invasive catheterization may be required to assess the peak-to-peak pressure gradients (PGpp) across the coarctation. Surgical intervention is required when the PGpp are greater than 20 mmHg at rest [2]. Cardiac magnetic resonance imaging (CMR) allows for the evaluation of both the anatomy and blood flow in different positions of the aorta. The application of Physics-Informed Neural Networks (PINNs) [3] in patients with CoAo may offer a promising non-invasive solution for estimating GPpp, thereby reducing the need for invasive procedures.

Methods: We analyzed 11 patients with CoAo who underwent both CMR and invasive catheterization at rest and under pharmacological stress [4]. From 2D cine phase-contrast CMR at the ascending aorta (AAo) and diaphragmatic aorta (Dia), cross-sectional area and velocity were extracted. Based on these data, two PINN models were trained, one for the AAo and one for the Dia (see Figure 1-A). During training, the PINN models are constrained to satisfy the equations of the 1D model described in Figure 2. This allows the prediction of the mean pressure in these sections (see Figure 1-H). The resulting pressure curves (P) for AAo and Dia (see Figure 1-J) allow for estimation of the PGpp by calculating the difference between the maximum values of each curve. To assess significant differences between the actual measurements and the model’s predictions, the nonparametric Wilcoxon signed-rank test was used. The proposed methodology is summarized in Figure 1. In addition, the pressure gradient was also estimated using the simplified Bernoulli equation [5], using as referente the velocity in after the CoAo.

Results: The predictions of the PGpp obtained using the PINNs approach for eleven patients are summarized in Figure 3. The PGpp values obtained at rest and under stress are in good agreement with those obtained by catheterization (p-value = 0.5312 for Rest, p-value = 0.4351 for Stress). For PGpp at rest, the median and range of absolute error between the catheterization measurements and the proposed PINNs approach was 1 [0.3 10.8] mmHg. Under stress, the median and range of absolute error for PGpp was 4.8 [0.5 17.1] mmHg, see Figure 3. Patient 12 and patient 2 present the largest differences between the real and predicted values of PGpp (Patient 12 = 10.8 mmHg, patient 2 = 17.1 mmHg) at rest and stress respectively. Simplified Bernoulli's equation has more discrepancies with respect to the invasive measurement, compared to PINNs.

Conclusion: PINNs enable accurate, non-invasive estimation of PGpp in patients with CoAo, showing comparable performance to catheterization. This approach has potential to reduce the need for invasive procedures in clinical management of CoAo, representing a promising step toward integrating physics-informed machine learning into cardiovascular care.

Figure 1. (A) 3D reconstruction of the aorta. (B) 2D Cine PC-MRI in the ascending aorta (AAo) and diaphragmatic aorta (Dia). (C) Time series of velocity and area. (D) Spatiotemporal domain of the neural network. (E) Loss function: a reduced model of the Navier-Stokes differential equations. (F) Feed-forward neural network with 4 hidden layers, each containing 60 neurons. (G) Adam optimization algorithm. (H) Prediction of average velocity, area, and mean pressure. (I) Invasive catheterization (Gold Standard). (J) Comparison of pressure curves between the clinical measurement and the prediction of the PINNs.
Figure 2. 1D Model for Blood Flow in Arteries: (A) Representation of a distensible vessel segment. A(x,t), u(x,t)and p(x,t) represent the area, mean velocity and mean pressure over a cross section, respectively. (B) Reduced 1D model of the Navier-Stokes equations for an incompressible fluid [6]. KR is a friction parameter, beta represents the vessel elasticity and depends on the vessel wall thickness and Young's modulus. pext and A0 are usually set to the diastolic pressure and the cross-sectional area in diastolic phase, respectively. Finally, rho is the density of blood which will be assumed equal to 1060 Kg/m3.
Figure 3. (A) Peak-to-peak pressure gradient results under rest conditions. (B) Peak-to-peak pressure gradient results under stress conditions. In blue, peak-to-peak gradient measurements made by invasive catheterization. In orange, the simplified Bernoulli method. In green, the prediction made by the proposed PINNs approach.