Table 1.
Formulae to compute ST, where f0 and Var represent the mean and variance of the outputs respectively, as defined in Eqs (19) and (20).
l runs from 1 to N for the number of model samples.
Fig 1.
The two electrical analougue models utilised in this work.
(A) is a nine parameter representation of the systemic circulation originally presented by Bjordalsbakke et al. [51]. (B) is a twenty parameter representation of the systemic circulation originally presented by Shi et al. [52].
Table 2.
Input parameters for the 1 chamber model.
Each input parameter’s unit is stated alongside a chosen initial value for the 9 parameter, 1-chamber model. τ is the cardiac cycle length and is fixed such that τ = 1s. The ventricular shift parameter Eshift = 0 s as no atrium is present in this model.
Table 3.
Input parameters for the 2 chambers model.
Each input parameter’s unit is stated alongside a chosen initial value for the 20 parameter, 2-chamber model. τ is the cardiac cycle length and is fixed such that τ = 1s. The ventricular shift parameter Eshift = 0.92 s as an atrium is present in this advanced 20 parameters model.
Fig 2.
Workflow to compute sensitivity indices within Julia.
Full code utilising these steps can be found at https://github.com/H-Sax/Orthgonality-SA.
Fig 3.
Time series solutions for the 1 and 2 chamber cardiovascular models investigated in this work.
The solutions shown are the ones which are utilised in the investigation.
Fig 4.
Convergence and uncertainty of indices associated with the minimum ventricular elastance Emin.
Fig A displays the convergence and uncertainty of the Sobol indices ST calculated on discrete measurements for the 1-chamber model against increasing sample size. Here, the vertical line signifies the chosen sample size for the 1-chamber model at N = 10, 000. Fig B presents the continuous Sobol indices with uncertainty bounds, calculated at a sample size N = 10, 000, on continuous measurements over a single cardiac cycle, for the 1-chamber model. Fig C displays the convergence and uncertainty of ST calculated on discrete measurements for the 2-chamber model against increasing sample size. Again, the vertical line signifies the chosen sample size for this model, at N = 20, 000. Fig D shows the continuous Sobol indices with uncertainty bounds for N = 20, 000, on continuous measurements over a single cardiac cycle, for the 2-chamber model. The measurements shown in blue, yellow and green denote the left ventricular pressure, the systemic arterial pressure and the left ventricular volume, respectively. In the discrete settings (i.e., A and C), the measurements are the mean left ventricular pressure, the maximum systemic arterial pressure and the maximum left ventricular volume.
Fig 5.
Total order Sobol indices ST of the arterial compliance Csa for the 1-chamber model with continuous measurements.
Panels A—T show ST of Csa, for 3 continuous measurements—left ventricular pressure, systemic arterial pressure and the left ventricular volume (represented in blue, yellow and green curves, respectively), over a single cardiac cycle with differing estimators and sampling methodologies. Measurements are evaluated with N = 10, 000 samples, using B = 1000 bootstrapped samples to evaluate the uncertainty of the estimate. The bands represent 95% confidence intervals associated with specific indices displayed as solid curves.
Fig 6.
Orthogonality distributions of input parameters for the 1-chamber model with continuous measurements—Histograms A-T show the distribution of orthogonality returned from examinations of the sensitivity vectors, calculated from continuous measurements.
Here, an orthogonality score of 1 represents total independence of input parameters, whereas 0 represents total dependence. Each individual diagram denotes a specific combination of sampling methodology and estimator type. The frequency of each histogram is normalised such that it is comparable between plots, i.e., the larger the frequency of a bin, the larger the number of orthogonality scores calculated from the original sensitivity vectors.
Table 4.
Input parameter ranking for the 1-chamber model with continuous measurements—Here, input parameters are ranked based on the averaged orthogonality score returned from the calculated total order sensitivity matrix.
In addition, the ranking is stratified by both sampling and estimator types.
Table 5.
The ranges of input parameters across 5 sampling types for a specific estimator for the 1-chamber model with continuous measurements.
Table 6.
The ranges of input parameters across 4 estimator types for a specific sampling method for the 1-chamber model with continuous measurements.
Fig 7.
Total order Sobol indices ST of the mitral valve resistance Rmv for the 1-chamber model with discrete measurements.
Panels A—T show ST of Rmv, for 3 discrete measurements: mean left ventricular pressure, maximum systemic arterial pressure and maximum left ventricular volume (represented in blue, yellow and green, respectively), evaluated at increasing sample sizes (N ∈ [2000, 40000] using B = 1000 bootstrapped samples), with differing estimators and sampling methodologies. The bands represent 95% confidence intervals associated with specific indices displayed as solid curves. The red solid vertical lines represent the point (N = 10, 000) at which the sample size is taken.
Fig 8.
Orthogonality distributions of input parameters for the 1-chamber model with discrete measurements—Histograms A-T show the distribution of orthogonality returned from examinations of the sensitivity vectors, calculated from continuous measurements.
Here, an orthogonality score of 1 represents total independence of input parameters, whereas 0 represents total dependence. Each individual diagram denotes a specific combination of sampling methodology and estimator type. The frequency of each histogram is normalised such that it is comparable between plots, i.e., the larger the frequency of a bin, the larger the number of orthogonality scores calculated from the original sensitivity vectors.
Table 7.
Input parameter ranking for the 1-chamber model with discrete measurements—Again, input parameters are ranked based on the averaged orthogonality score returned from the calculated total order sensitivity matrix.
The ranking is also stratified by both sampling and estimator types.
Table 8.
The range of parameter ranking across 5 sampling types for a specific estimator for the 1-chamber model with discrete measurements.
Table 9.
The ranges of input parameters across 4 estimator types for a specific sampling method for the single ventricle model with discrete measurements.
Fig 9.
Total order Sobol indices ST of the maximal left ventricular elastance for the 2-chamber model with continuous measurements.
Panels A—T show ST of , for 3 continuous measurements—left ventricular pressure, systemic arterial pressure and the left ventricular volume (represented in blue, yellow and green curves, respectively), over a single cardiac cycle with differing estimators and sampling methodologies. Measurements are evaluated with N = 20, 000 samples, using B = 1000 bootstrapped samples to evaluate the uncertainty of the estimate. The bands represent 95% confidence intervals associated with specific indices displayed as solid curves.
Fig 10.
Orthogonality distributions of input parameters for the 2-chamber model with continuous measurements—Histograms A-T show the distribution of orthogonality returned from examinations of the sensitivity vectors, calculated from continuous measurements.
Here, an orthogonality score of 1 represents total independence of input parameters, whereas 0 represents total dependence. Each individual diagram denotes a specific combination of sampling methodology and estimator type. The frequency of each histogram is normalised such that it is comparable between plots, i.e., the larger the frequency of a bin, the larger the number of orthogonality scores calculated from the original sensitivity vectors.
Table 10.
Input parameter ranking for the 2-chamber model with continuous measurements—Here, input parameters are ranked based on the averaged orthogonality score returned from the calculated total order sensitivity matrix.
In addition, the ranking is stratified by both sampling and estimator types.
Table 11.
The ranges of input parameters across 5 sampling types for a specific estimator for the 2-chamber model with continuous measurements.
Table 12.
The ranges of input parameters across 4 estimator types for a specific sampling method for the 2-chamber model with continuous measurements.
Fig 11.
Total order Sobol indices ST of the venous compliance Csvn for the 2-chamber model with discrete measurements.
Panels A—T show ST of Csvn, for 3 discrete measurements: mean left ventricular pressure, maximum systemic arterial pressure and maximum left ventricular volume (represented in blue, yellow and green, respectively), evaluated at increasing sample sizes (N ∈ [10000, 30000] using B = 1000 bootstrapped samples), with differing estimators and sampling methodologies. The bands represent 95% confidence intervals associated with specific indices displayed as solid curves. The red solid vertical lines represent the point (N = 20, 000) at which the sample size is taken.
Fig 12.
Orthogonality distributions of input parameters for the 2-chamber model with discrete measurements—Histograms A-T show the distribution of orthogonality returned from examinations of the sensitivity vectors, calculated from continuous measurements.
Here, an orthogonality score of 1 represents total independence of input parameters, whereas 0 represents total dependence. Each individual diagram denotes a specific combination of sampling methodology and estimator type. The frequency of each histogram is normalised such that it is comparable between plots, i.e., the larger the frequency of a bin, the larger the number of orthogonality scores calculated from the original sensitivity vectors.
Table 13.
Input parameter ranking for the 2-chamber model with discrete measurements—Here, input parameters are ranked based on the averaged orthogonality score returned from the calculated total order sensitivity matrix.
In addition, the ranking is stratified by both sampling and estimator types.
Table 14.
The ranges of input parameters across 5 sampling types for a specific estimator for the systemic circulation model with discrete measurements.
Table 15.
The ranges of input parameters across 4 estimator types for a specific sampling method for the 2-chamber model with discrete measurements.
Fig 13.
Estimator comparisons with larger samples for the 2-chamber model with discrete measurements—The Sobol and Homma estimators results are based on 100k samples, compared to the Jansen and Janon estimators using 40k samples both with 95% confidence.
The input parameter effect is displayed against the maximum left ventricular volume as an example here.