Wave propagation models of blood flow and blood pressure in arteries play an important role in cardiovascular research. For application of these models in patient-specific simulations a number of model parameters, that are inherently subject to uncertainties, are required. The goal of this study is to identify with a global sensitivity analysis the model parameters that influence the output the most. The improvement of the measurement accuracy of these parameters has largest consequences for the output statistics. A patient specific model is set up for the major arteries of the arm. In a Monte-Carlo study, 10 model parameters and the input blood volume flow (BVF) waveform are varied randomly within their uncertainty ranges over 3000 runs. The sensitivity in the output for each system parameter was evaluated with the linear Pearson and ranked Spearman correlation coefficients. The results show that model parameter and input BVF uncertainties induce large variations in output variables and that most output variables are significantly influenced by more than one system parameter. Overall, the Young's modulus appears to have the largest influence and arterial length the smallest. Only small differences were obtained between Spearman's and Pearson's tests, suggesting that a high monotonic association given by Spearman's test is associated with a high linear corelation between the inputs and output parameters given by Pearson's test.
In the last few decades, lumped parameter [1], [2], [3], [4], [5] and [6] and one dimensional wave propagation models [7], [8], [9], [10] and [11] of blood flow and blood pressure in arteries are increasingly used in cardiovascular research. Experimental and clinical validation of wave propagation models demonstrated the ability to qualitatively describe blood pressure and blood volume flow waveforms [7], [12] and [13]. Furthermore, wave propagation models were employed to estimate arterial properties [14] or predict the effects of (surgical) interventions [15].
One dimensional wave propagation models of an arterial tree require an input blood volume flow (BVF) waveform and many system parameters to define geometry, boundary conditions and mechanical behavior of the vessel wall. For patient-specific simulations, input BVF and system parameters should be obtained from in vivo measurements employing high resolution techniques such as ultrasound, magnetic resonance imaging or tonometry. However, these measurements are exposed to uncertainties which can significantly influence the output obtained, e.g. vessel distension, blood velocity, and blood pressure waveforms at arbitrary locations. Moreover, the number of measurements for each patient is constrained by the duration of the measurement session. To optimize the measurement protocol, it would be of great interest to identify the input parameters that influence the output the most. Improved measurement of those parameters is the more beneficial [16] and [17].
The effect of system parameter uncertainty can be studied with sensitivity analysis (SA) methods. Local SA is the standard method; it evaluates the elementary effect of each parameter at a specific initial state of the model [18] and [19]. However, the results obtained are only valid at the initial state chosen, and can not be extrapolated to other points in the input space. Global SA quantifies the effect of each parameter within the entire input space and can be performed by a Monte-Carlo study based on multiple runs of the model. The simulated data-set corresponds to a random sampling of the system parameters within their uncertainty ranges. The global sensitivity of the system parameters follows then from appropriate statistical tests. As shown by local SA studies [14] and [18], blood pressure wave propagation models exhibit a complex relationship between the system parameters and the output. However, a global sensitivity analysis that considers in vivo measurement uncertainties in a wave propagation madel, has not been performed previously.
The goal of this study is to identify with a global sensitivity analysis the input properties and model parameters that influence the output the most. We focus on a model of the arteries of the arm because these arteries are frequently subject of medical investigations [20], [21] and [22]. Furthermore, the local systolic and diastolic blood pressure can be measured directly in the brachial artery and ultrasound measurements can be performed in the main arteries. In the first part of this paper, the subject-specific model of the arm, system parameters and their uncertainty ranges are described. Then, a global sensitivity analysis, based on a Monte-Carlo study, is performed. The results of the model evaluations are quantitatively illustrated employing Cobweb's graphical method. Further, the linear Pearson and ranked Spearman correlation coefficients are calculated to evaluate quantitatively the sensitivity of the model output to system parameters. Subsequently, the derived correlation coefficients are classified and discussed.
In summary, a Monte-carlo study has been performed on a wave propagation model of the arm. The linear Pearson and ranked Spearman correlation coefficients were obtained to determine the global sensitivity index of the system parameters and input blood volume flow. It was found that the majority of the input parameters are significantly influencing the output, with the Young modulus (E) having the largest influence and the arterial lengths the lowest. The results suggest that imaging techniques with high temporal and spacial resolution should be preferred.
