An in-silico analysis of experimental designs to study ventricular function: A focus on the right ventricle

Abstract

In-vivo studies of pulmonary vascular disease and pulmonary hypertension (PH) have provided key insight into the progression of right ventricular (RV) dysfunction. Additional in-silico experiments using multiscale computational models have provided further details into biventricular mechanics and hemodynamic function in the presence of PH, yet few have assessed whether model parameters are practically identifiable prior to data collection. Moreover, none have used modeling to devise synergistic experimental designs. To address this knowledge gap, we conduct a practical identifiability analysis of a multiscale cardiovascular model across four simulated experimental designs. We determine a set of parameters using a combination of Morris screening and local sensitivity analysis, and test for practical identifiability using profile likelihood-based confidence intervals. We employ Markov chain Monte Carlo (MCMC) techniques to quantify parameter and model forecast uncertainty in the presence of noise corrupted data. Our results show that model calibration to only RV pressure suffers from practical identifiability issues and suffers from large forecast uncertainty in output space. In contrast, parameter and model forecast uncertainty is substantially reduced once additional left ventricular (LV) pressure and volume data is included. A comparison between single point systolic and diastolic LV data and continuous, time-dependent LV pressure-volume data reveals that at least some quantitative data from both ventricles should be included for future experimental studies.

Fig 1. Model schematic.

The computational model here consists of a lower order simulator of sarcomere dynamics within the left atrium (LA), left ventricle (LV), right atrium (RA), right ventricle (RV), and septum (S). The LV, RV, and S are simulated using the TriSeg model [8], and account for biventricular interaction. Lastly, a circuit model is used to describe the systemic arteries (SA) and veins (SV), as well as the pulmonary arteries (PA) and veins (PV).

Fig 2. Nominal simulations and noise corrupted data.

The nominal simulations are generated to match the data range reported by Philip et al. [23] in sham mice. Noise corrupted data is generated by adding additive, Gaussian errors with mean zero and a variance of 1.

Fig 3. Workflow schematic.

The initial set of 49 parameters is reduced to 38 due to apriori parameter fixing. Morris screening is used to confirm which parameters are on average the most influential on the four model outputs. This reduces the parameter set from 38 to 17 parameters. A local sensitivity analysis using the different experimental designs as the quantities of interest is used to determine if any parameters show local interdependence in their sensitivities, which suggests possible practical non-identifiability. If the Fisher information matrix constructed from the model sensitivity is ill-condition, the least influential parameters of the subset are fixed. This reduces the parameter subset from 17 parameters to a set of 13 parameters. Lastly, the parameters and experimental designs are subjected to profile likelihood analysis and MCMC to test for practical identifiability.

Fig 4. Sensitivity results from the Morris screening algorithm.

Parameter ranking is based on the index $M_{i,f}=\sqrt{{{\mu }_{i,f}^{*}}^2+{\sigma }_{i,f}^2}$. The dotted line in each plot denotes the average model sensitivity for each output.

Fig 5. Profile likelihood confidence intervals.

Confidence intervals are constructed by fixing one parameter and inferring all others over a range of values. Each row corresponds to a different experimental design. Note that the minimally informative experimental designs (f₁ and f₂) have non-identifiable parameters, indicated by infinite or one-sided confidence bounds. In contrast, inclusion of LV data (f₃ and f₄) remedy the issue of non-identifiable parameters in the set. Large deviations in the profile likelihood correspond to local minima and parameter sets that are incompatible for the system of DAE’s.

Fig 6. Chain iterations and marginal posteriors after MCMC for the TriSeg parameters.

The model parameters indicative of the TriSeg geometry (wall volume, V_wall, and reference mid-wall area, A_m,ref) are shown for each experimental design, corresponding to each column. The true, data generating parameters corresponding to the outputs in Fig 2 are shown as red lines. Three of the twelve initializations of MCMC are shown in different shades of gray. The marginal posterior distributions for the simplest experimental design (f₁) are much wider than the subsequent more informed experimental designs, suggesting an improvement in practical identifiability.

Fig 7. Chain iterations and marginal posteriors after MCMC for the sarcomere parameters.

Similar to Fig 6, three of the twelve MCMC instances are provided for the sarcomere parameters important for the rise, decay, and length of fiber shortening (τ_rise,v, τ_decay,v, and τ_sys,v, respectively), and maximal active force generation (σ_act,v). Note that all four experimental designs (given by each column) provide sufficient information to the likelihood so that the true data generating parameters (in red) are within the marginal posteriors.

Fig 8. Chain iterations and marginal posteriors after MCMC for the hemodynamic compartment parameters.

As in Figs 6 and 7, three of the twelve MCMC instances are provided for systemic vascular resistance, R_sys, pulmonary vascular resistance, R_pul, and pulmonary arterial compliance, C_pa. Though the marginal posteriors do contain the true parameters (in red) within the marginal posteriors for the simplest design (f₁, first column), additional data in the other designs substantially reduce posterior uncertainty.

Fig 9. Output uncertainty in RV and LV pressures and volumes for each experimental design.

The average model response (red) as well as ± one standard deviation (Std., gray) are provided along with the data (black circles) for each experimental design, corresponding to each column. In the first design, f₁, only RV pressure is used in the likelihood, hence the uncertainty in RV volume and LV forecasts are substantially larger than that of the RV pressure. As more data is included, uncertainty in model forecasts is reduced. Note that differences between f₃ and f₄ are less pronounced.

Fig 10. Output uncertainty in cardiac pressure-volume loops.

Realizations in forecasts of chamber pressure-volume loops in the LA (first row), LV (second row), RA (third row), and RV (fourth row). The simplest design (f₁) has the largest uncertainty in simulated pressure-volume loops, except for RV pressure, which is accounted for in the likelihood. Subsequent experimental designs substantially reduce uncertainty bounds in the RV and RA (f₂) and eventually in the LV and LA (f₃ and f₄).

Fig 11. Forecast uncertainty in LV, RV, and S wall strain.

Realizations in the LV, RV, and S engineering strain, along with the mean and ± standard derivation (Std), obtained from the posterior distributions. For designs only including RV dynamics (f₁ and f₂), S engineering strain has a large uncertainty in the direction of strain (i.e., leftward or rightward). Designs including LV data (f₃ and f₄) reduce the range of S strains, with the design f₄ ensuring that S strain is in the same direction as the LV. LV and RV strain have substantially less uncertainty than that of S, which shrinks with more informative designs.

Fig 12. Simulated output quantities that are typically recorded when studying PH.

Histogram plots of outputs typically recorded during in-vivo studies of PH progression are generated using the same 600 samples from the posterior that were used in Figs 9, 10 and 11. These include mean pulmonary artery pressure (${\overline{p}}_{sa}$), RV stroke volume (SV_RV, defined as difference between maximum and minimum RV volumes), pulmonary arterial elastance (Ea_pa, defined the difference between ${\overline{p}}_{sa}$ and mean LA pressure divided by SV_RV), RV end systolic elastance (Ees_RV, defined as the end systolic ratio of RV pressure and RV volume), and ventricular-vascular coupling (VVC, defined as the ratio Ees_RV/Ea_pa). Differences in the experimental design had little effect on ${\overline{p}}_{pa}$. As expected, SV_RV was more accurately captured with additional RV volume data. The wide variability in values of Ea_pa, Ees_RV, and VVC using the design f₁ is remedied once additional volume data is included in the design. Note that output values of VVC are made substantially more precise with additional LV data in f₃ and f₄.