Rapid estimation of left ventricular contractility with a physics-informed neural network inverse modeling approach

Abstract

Physics-based computer models based on numerical solutions of the governing equations generally cannot make rapid predictions, which in turn limits their applications in the clinic. To address this issue, we developed a physics-informed neural network (PINN) model that encodes the physics of a closed-loop blood circulation system embedding a left ventricle (LV). The PINN model is trained to satisfy a system of ordinary differential equations (ODEs) associated with a lumped parameter description of the circulatory system. The model predictions have a maximum error of less than 5% when compared to those obtained by solving the ODEs numerically. An inverse modeling approach using the PINN model is also developed to rapidly estimate model parameters (in ∼ 3 min) from single-beat LV pressure and volume waveforms. Using synthetic LV pressure and volume waveforms generated by the PINN model with different model parameter values, we show that the inverse modeling approach can recover the corresponding ground truth values for LV contractility indexed by the end-systolic elastance E_es with a 1% error, which suggests that this parameter is unique. The estimated E_es is about 58% to 284% higher for the data associated with dobutamine compared to those without, which implies that this approach can be used to estimate LV contractility using single-beat measurements. The PINN inverse modeling can potentially be used in the clinic to simultaneously estimate LV contractility and other physiological parameters from single-beat measurements.

Keywords: Cardiac contractility; Lumped parameter model; Parameter estimation; Patient-specific modeling; Physics-informed neural network; Sensitivity analysis.

Figure 7:

Comparison of the ReLU and softplus functions (left) and their derivatives (right) over the range of common input values

Figure 8:

Comparison of the LV volume waveform for the baseline case, calculated using the exact ReLU function in equations Eqs. (2d) and (2e), versus those calculated using softplus function as their approximation in Eq. 3. The RMAE is 0.02.

Figure 9:

Volume, pressure, and flow rate in different components of the closed-loop blood circulation system, obtained from the numerical solution of the system of equations (Eq. 1) and the trained PINN model for the baseline case

Figure 10:

A few examples of PV loops that can be captured by the model

Figure 11:

First order Sobol indices associated with $V_{lv}$ and $P_{lv}$ for each input parameter

Figure 12:

Second order Sobol indices associated with $V_{lv}$ and $P_{lv}$ for each input parameter

Figure 13:

Total order Sobol indices associated with $V_{ao}$, $V_{art}$, $V_{vc}$, and $V_{la}$ for each input parameter

Figure 14:

Comparison of the PV loops from inverse modeling with the corresponding measurements from the swine models (a) PV loops for eight swine subjects at their baseline (b) PV loops for three swine subjects before (top panel) and after (bottom) Dobutamine administration

Figure 1:

The electrical equivalent diagram of the closed-loop blood circulation. lv, left ventricle; ao, aorta; art, peripheral artery; vc, vena cava; la, left atrium; av, aortic valve; mv, mitral valve.

Figure 2:

Model architecture. The input layer consists of 22 nodes, which include the first 12 terms from the Fourier series and 10 physiological input parameters, denoted as $𝒫$. Four separate neural networks are utilized, each characterized by its unique set of weights and biases, represented as ${\theta }_i$. These networks are used to model the 4 volume state variables, namely, ${\widehat{V}}_{lv}$, ${\widehat{V}}_{ao}$, ${\widehat{V}}_{art}$, and ${\widehat{V}}_{la}$.

Figure 3:

Results of the PINN training (a) Training and test loss curves. (b) RMAE between the PINN model and numerical results in the 1000 evaluation cases

Figure 4:

Total order Sobol indices associated with $V_{lv}$ and $P_{lv}$ for each input parameters

Figure 5:

RMAE between the estimated input parameters and their ground truth with noise for the 100 synthetic cases

Figure 6:

Comparison of the fitted LV pressure and volume waveforms from inverse modeling with the corresponding measurements from the swine models (a) Eight swine subjects at baseline (b) Three swine subjects before (top panel) and after (bottom) administration of dobutamine