Uncertainty quantification and sensitivity analysis of an arterial wall mechanics model for evaluation of vascular drug therapies
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Abstract
Quantification of the uncertainty in constitutive model predictions describing arterial wall mechanics is vital towards noninvasive assessment of vascular drug therapies. Therefore, we perform uncertainty quantification to determine uncertainty in mechanical characteristics describing the vessel wall response upon loading. Furthermore, a global variancebased sensitivity analysis is performed to pinpoint measurements that are most rewarding to be measured more precisely. We used previously published carotid diameter–pressure and intima–media thickness (IMT) data (measured in triplicate), and Holzapfel–Gasser–Ogden models. A virtual data set containing 5000 diastolic and systolic diameter–pressure points, and IMT values was generated by adding measurement error to the average of the measured data. The model was fitted to singleexponential curves calculated from the data, obtaining distributions of constitutive parameters and constituent load bearing parameters. Additionally, we (1) simulated vascular drug treatment to assess the relevance of model uncertainty and (2) evaluated how increasing the number of measurement repetitions influences model uncertainty. We found substantial uncertainty in constitutive parameters. Simulating vascular drug treatment predicted a 6% point reduction in collagen load bearing (\(L_\mathrm {coll}\)), approximately 50% of its uncertainty. Sensitivity analysis indicated that the uncertainty in \(L_{\mathrm {coll}}\) was primarily caused by noise in distension and IMT measurements. Spread in \(L_{\mathrm {coll}}\) could be decreased by 50% when increasing the number of measurement repetitions from 3 to 10. Model uncertainty, notably that in \(L_{\mathrm {coll}}\), could conceal effects of vascular drug therapy. However, this uncertainty could be reduced by increasing the number of measurement repetitions of distension and wall thickness measurements used for model parameterisation.
Keywords
Arterial wall mechanics Constitutive modelling Uncertainty quantification Sensitivity analysis Vascular ultrasound1 Introduction
Arterial stiffening is a major determinant of hypertension and vice versa (Humphrey et al. 2016). Treatment options for arterial stiffening can roughly be divided into two categories: (1) prescribing drugs that lower blood pressure and consequently reverse the arterial structural remodelling that occurs with hypertension, or (2) prescribing drugs that directly affect the arterial wall structure (Townsend et al. 2015). The second category includes crosslink breaking drugs that target the arterial collagen network (Kass et al. 2001; Wolffenbuttel et al. 1998). These types of drugs aim to break down the advanced glycation end products (AGE) that form crosslinks between collagen molecules (McNulty et al. 2007; Kass et al. 2001; Brownlee 1995). The desired mechanical effect of such drug therapies is to reverse pressure load bearing from a stiff, collagendominated phenotype to a less stiff, elastindominated phenotype, resulting in a decrease in material stiffness (O’Rourke and Hashimoto 2007; Wolffenbuttel et al. 1998).
In vivo assessment of the performance of vascular drugs has proved to be cumbersome (Engelen et al. 2013). Arterial stiffness is typically quantified by measuring carotid–femoral pulse wave velocity, or by local assessment of arterial distensibility (Kass et al. 2001; Wolffenbuttel et al. 1998). A limitation of these indices is their blood pressure dependence, for which an incremental change in these indices could occur solely from a change in blood pressure (Spronck et al. 2015b). In addition, arterial stiffness measurements as such do not yield insight into the effect of a drug at microstructural level, nor do they resolve whether the load bearing phenotype is collagen or elastindominated.
A potential solution to this problem lies in the use of computer models that simulate stress–strain behaviour of arteries using physical constitutive relations (Holzapfel et al. 2000). In such models, it might be possible to analyse the individual contribution of both elastin and collagen to the overall mechanical response of the vessel wall. Ex vivo studies on human carotid arteries, performed in the laboratory, reported good agreement between constitutive model computations and biaxial tensile tests (Sommer and Holzapfel 2012; Sommer et al. 2010). If such models could be parameterised using noninvasive measurements in patients, they could be used to evaluate the mechanics of the vessel wall patientspecifically. Several studies have attempted to use in vivo data to parameterise constitutive models of the arterial wall, as reviewed in Spronck et al. (2015a). Typically, diameter, pressure, and intima–media thickness measurements at the carotid artery are used to fit such models (Spronck et al. 2015a). Generally, noise in those measurements will hamper patientspecific evaluation of arterial wall mechanics.
In this study, we aim to (1) assess how measurement uncertainty propagates into uncertainty of mechanical characteristics, estimated using a model of arterial wall mechanics and (2) pinpoint the measurements responsible for the largest spread in mechanical characteristics through sensitivity.
Uncertainty quantification and sensitivity analysis are considered indispensable tools to ensure credibility of computer modelbased predictions (National Research Council 2012). We explicitly focus on two types of mechanical characteristics: (1) constitutive parameters, describing the mechanical properties of collagen and elastin, and (2) constituent load bearing parameters, describing to which extent the blood pressure load is borne per constituent. The latter parameters are considered outcome parameters, obtained by evaluating the model using the bestfit set of constitutive parameters (Fig. 1, left pane). For these purposes, a large set of virtual pressure (P), diameter (D), and intima–media thickness (IMT) measurements will be generated by sampling the measurement distributions of previously published P, D, and IMT measurements in healthy volunteers (Holtackers et al. 2016). For each DP sample, we will obtain continuous DP curves over the diastolic–systolic pressure range, using the singleexponential function introduced by Hayashi et al. (1980). Holzapfel–Gasser–Ogden constitutive models will be fitted to these data (Holzapfel et al. 2000), and uncertainty quantification will be used to quantify the spread in the estimated mechanical characteristics, given the measurement noise (Fig. 1, left pane). A sensitivity analysis will be performed to determine how noise in the individual measured variables propagates into uncertainty in the modelpredicted mechanical characteristics. The relevance of the model output uncertainty magnitude, which results from uncertainty in the model parameters, will be assessed by simulating the effect of AGEbreaker treatment on changes in collagen load bearing behaviour in our model (Fig. 1, right pane).
2 Methods
2.1 Constitutive model definition

\(c_{{\mathrm {elast}}}\): stiffness parameter of elastin, units of Pa.

\(k_1\): stress scaling parameter of collagen, units of Pa.

\(k_2\): collagen stress curve shape parameter, dimensionless.
To map from a cut, stressfree configuration of an artery, to an unloaded intact configuration, to a loaded configuration, we define two additional parameters (Fig. 2, Spronck et al. 2015a; Humphrey 2002; Holzapfel et al. 2002): the opening angle (\(\alpha \)) and the unstressed inner vessel radius (\(R_{\mathrm {i}}\)). The value for \(\alpha \) was taken from the literature (\(100^\circ \), Spronck et al. 2015a). The parameter \(R_{\mathrm {i}}\) was fitted using the constitutive model fitting routine (see below).
2.2 Parameterisation
2.2.1 Clinical measurements
The measurement protocol and data used in the present study are elaborated by Holtackers et al. (2016). Briefly, twelve apparently healthy volunteers (\(22\pm 3 \, \mathrm {years}\), 6 males, 6 females) were recruited. The study was approved by the medical ethics committee of Maastricht University Medical Centre (Maastricht, the Netherlands), and written consent was obtained from all participants prior to enrolment.
Ultrasound (US) Bmode recordings were performed at the right common carotid artery (CCA) in the anterolateral plane, and obtained in triplicate. Diastolic blood pressure (\(P_{\mathrm {d}}\)) and pulse pressure (\(P_{{{\mathrm {p}}}}\)) were measured three times during the US imaging protocol using an oscillometric device (Omron 705IT; Omron Healthcare Europe, Hoofddorp, the Netherlands).
The US recordings were analysed to determine right CCA cyclic distension (i.e. diastolic–systolic diameter change, \(\Delta D\)) and diastolic diameter (\(D_{\mathrm {d}}\)). Because the echo tracking tool used (\({}^{\mathrm{RF}}\)QAS; Esaote, Maastricht, the Netherlands) utilises the media–adventitia echoes of near and far walls, we assumed the measured diameter signal over time to reflect the CCA outer diameter (Spronck et al. 2015a). Furthermore, we obtained IMT at \(P_{\mathrm {d}}\) using an automated software tool that reported good agreement with manual IMT methods (Willekes et al. 1999).
2.2.2 Generation of data set for constitutive modelling
Overview of average values, intrasubject SDs, and uncertainty domains per measured variable
Parameter  Unit  Mean  Data of twelve subjects  

Intrasubject SD  Uncertainty domain  
\(P_{\mathrm {d}}\)  mmHg  72  3.0  [69 ; 75] 
\(P_{{\mathrm {p}}}\)  mmHg  58  3.1  [54 ; 62] 
\(D_{\mathrm {d}}\)  mm  6.37  0.22  [6.12 ; 6.62] 
\(\Delta D\)  mm  0.789  0.035  [0.750 ; 0.829] 
IMT  \(\upmu \mathrm {m}\)  539  40  [494 ; 584] 
see also Table 1. Each sample consisted of a vector \(\mathbf {M}\) containing the following variables: \(\mathbf {M}\,\)= [\(P_{\mathrm {d}}\), \(P_{{\mathrm {p}}}\), \(D_{\mathrm {d}}\), \(\Delta D\), IMT]. Here, \(\bar{M}_i\) represents the average measured value of variable \(M_i\) and \(\mathrm {SD}_{\mathrm {intra},M_i}\) represents the corresponding intrasubject standard deviation (i.e. indicating measurement error Bland and Altman 1996). In our data set, measurements were performed in triplicate (i.e. \(N_\mathrm {rep}=3)\) Holtackers et al. 2016. Samples were generated within the uncertainty domains of the measured variables, using Sobol’s low discrepancy series, implemented in the MATLAB Statistics and Machine Learning Toolbox function sobolset (MATLAB R2015a; The MathWorks, Natick, MA, USA) (Sobol 1967). This sampling method samples the uncertainty domains uniformly and was chosen as a “worstcase scenario.” To ensure adequate convergence of mechanical characteristics distributions, we generated 5000 samples. Systolic blood pressure (\(P_\mathrm {s}\)) was calculated as \(P_\mathrm {s} = P_{\mathrm {d}} + P_{{\mathrm {p}}}\); systolic diameter (\(D_\mathrm {s}\)) was calculated as \(D_\mathrm {s} = D_{\mathrm {d}} + \Delta D\). The distributions of \(D_{\mathrm {d}}\), \(D_\mathrm {s}\), \(P_{\mathrm {d}}\), and \(P_\mathrm {s}\) as well as IMT are shown in Fig. 3. The greater amount of scattering of systolic D and P data points compared to diastolic data points is caused by the fact that systolic blood pressure is defined as the sum of diastolic and pulse pressure. As diastolic and pulse pressure were assumed to be independent, their sum (systolic blood pressure) will have a larger spread than diastolic blood pressure alone. The larger spread in systolic diameter than in diastolic diameter has the same origin, as systolic diameter is defined as the sum of diastolic diameter and distension.
2.2.3 Model fitting procedure
Complete overview of lower and upper parameter bounds used for fitting the diameter–pressure data
Parameter  Unit  Lower bound  Upper bound 

\(c_{\mathrm {elast}}\)  kPa  1  400 
\(k_1\)  kPa  0.1 \(\times \,10^{3}\)  400 
\(k_2\)    0  100 
\(R_{\mathrm {i}}\)  m  0.5 \(\times \,10^{3}\)  10 \(\times \,10^{3}\) 
For each sample in the data set, we assumed the singleexponential curve to be valid within the range \(P \in \{P_\mathrm {d,sample}  15 \, \mathrm {mmHg}, \quad P_\mathrm {s,sample} + 15 \,\mathrm {mmHg}\}\) (Meinders and Hoeks 2004; Hayashi et al. 1980). In Fig. 3 (left), the distribution of \(P_\mathrm {d,sample}\) vs. \(P_\mathrm {s,sample}\) is displayed.
2.3 Simulations and analysis
2.3.1 Initial constitutive parameter estimation
The constitutive model was fitted to all 5000 samples of the generated data set using the procedure explained in the previous section. This yielded 5000 initial constitutive model realisations (i.e. termed INIT).
2.3.2 Uncertainty quantification and sensitivity analysis
All constitutive model realisations together yield insight in the distribution of the mechanical characteristics that results from the presence of measurement uncertainty. This distribution of mechanical characteristics was therefore used to quantify the uncertainty in constitutive parameters (i.e. the fitted parameters \(c_{\mathrm {elast}}\), \(k_1\), and \(k_2\)), as well as collagen load bearing parameters (i.e. the outcome parameter \(L_{\mathrm {coll}}\) at various blood pressure levels, Fig. 1, left pane). We used kernel density estimation (KDE) to visualise the distributions of constitutive parameters and load bearing parameters. KDE estimates the probability density function, which in this context implies the probability density of finding a certain value of a constitutive parameter or load bearing value (Silverman 1986). Furthermore, we quantified spread in parameters using the median and the \(25\mathrm{th}\) to \(75\mathrm{th}\) percentile confidence interval (\({\mathrm {PCI}}_{25\rightarrow 75}\)). Calculating the \({\mathrm {PCI}}_{25\rightarrow 75}\) comes at hand when assessing spread of skewed distributions.
Furthermore, \(\mathbf {M} =[M_1,M_2,...,M_{N_\mathrm {vars}}] = [P_{\mathrm {d}}, P_{{\mathrm {p}}}, D_{\mathrm {d}},\) \( \Delta D, \mathrm {IMT}]\), and \(N_\mathrm {vars}\) is the number of measured variables. Such a polynomial expansion provides a metamodel of the mechanical characteristics estimation method. After constructing the metamodel, the value of the leaveoneout crossvalidation coefficient (\(Q^2\)) was computed. Coefficient \(Q^2\), ranging between a value of 0 and 1, is a quality measure of the metamodel, indicating its predictive properties (Sudret 2015). Throughout this study, we assumed \(Q^2>0.99\) to indicate an appropriate metamodel.

Main sensitivity indices The main sensitivity index (\(S_{\mathrm {i}}\)) of measured variable \(M_i\) represents the expected reduction in uncertainty of the mechanical characteristic if \(M_i\) were known exactly. Assessment of \(S_{\mathrm {i}}\) determines which measured variables are most rewarding to be measured more accurately to reduce model output uncertainty (i.e. parameter prioritisation) (Saltelli et al. 2008).

Total sensitivity indices The total sensitivity index (\(S_\mathrm {T}\)) of \(M_i\) represents the expected uncertainty in the mechanical characteristic that would remain if all other measured variables except \(M_i\) were known exactly. Assessment of \(S_\mathrm {T}\) determines which measured variables could potentially be fixed within their uncertainty domain (i.e. parameter fixing) (Saltelli et al. 2008).
2.3.3 Simulating the effect of AGEbreaker vascular drugs
AGEbreaker treatment was simulated using the INIT realisations as takeoff points (Fig. 1, right pane). We explicitly assumed that a reduction in crosslink density can be represented in the model by reducing parameters \(k_1\) and \(k_2\). The rationale of reducing \(k_1\) and \(k_2\) is based on previous work measuring the stress–strain response of collagen tissue at multiple levels of crosslinking (Kayed et al. 2015; Fratzl 2008). In their work, it was observed that a decrease in crosslink density results in (1) a decrease in fibre stiffness at low amounts of strain and (2) a decrease in the nonlinearity of the fibre stress–strain response (Kayed et al. 2015; Fratzl 2008). In our analysis, \(k_1\) and \(k_2\) were equally reduced by 40% of their initial bestfit value yielding the \(k_1\downarrow , k_2\downarrow \) realisations (Fig. 1, right pane). All other constitutive model parameters were assumed to remain unchanged.
3 Results
3.1 Representative example of a fitted constitutive model
Main (\(S_{\mathrm {i}}\)) and total (\(S_\mathrm {T}\)) sensitivity indices for constitutive parameters \(c_{\mathrm {elast}}\), \(k_1\) and \(k_2\)
Measured variable  Symbol  Constitutive parameter  

\(c_{\mathrm {elast}}\)  \(k_1\)  \(k_2\)  
\(S_{\mathrm {i}}\)  \(S_\mathrm {T}\)  \(S_{\mathrm {i}}\)  \(S_\mathrm {T}\)  \(S_{\mathrm {i}}\)  \(S_\mathrm {T}\)  
Diastolic blood pressure  \(P_{\mathrm {d}}\)  0.012  0.014  0.00060  0.0011  0.052  0.056 
Pulse pressure  \(P_{{\mathrm {p}}}\)  0.0018  0.021  0.027  0.041  0.10  0.12 
Diastolic diameter  \(D_{\mathrm {d}}\)  0.0060  0.036  0.033  0.047  0.29  0.32 
Distension  \(\Delta D\)  0.62  0.67  0.90  0.93  0.50  0.54 
Intima–media thickness  IMT  0.31  0.31  0.0083  0.015  0.00083  0.00083 
Main (\(S_{\mathrm {i}}\)) and total (\(S_\mathrm {T}\)) sensitivity indices for collagen load bearing parameters (\(L_{\mathrm {coll}}\)) at three blood pressure levels: diastolic, mean, and systolic blood pressure (i.e. \(P_{\mathrm {d}}, P_\mathrm {m}\) and \(P_\mathrm {s}\), respectively)
Measured variable  Symbol  Load bearing parameter  

\(L_{\mathrm {coll}}\) at \(P_{\mathrm {d}}\)  \(L_{\mathrm {coll}}\) at \(P_\mathrm {m}\)  \(L_{\mathrm {coll}}\) at \(P_\mathrm {s}\)  
\(S_{\mathrm {i}}\)  \(S_\mathrm {T}\)  \(S_{\mathrm {i}}\)  \(S_\mathrm {T}\)  \(S_{\mathrm {i}}\)  \(S_\mathrm {T}\)  
Diastolic blood pressure  \(P_{\mathrm {d}}\)  0.00043  0.00098  0.00073  0.0016  0.0019  0.0027 
Pulse pressure  \(P_{{\mathrm {p}}}\)  0.021  0.035  0.023  0.035  0.052  0.054 
Diastolic diameter  \(D_{\mathrm {d}}\)  0.036  0.058  0.041  0.059  0.064  0.067 
Distension  \(\Delta D\)  0.90  0.94  0.90  0.93  0.87  0.88 
Intima–media thickness  IMT  0.0028  0.0049  0.0031  0.0046  0.0031  0.0048 
3.2 Uncertainty quantification and sensitivity analysis
The distributions of the fitted constitutive parameters are shown in Fig. 5. Bestfit parameter values for elastin stiffness (\(c_{\mathrm {elast}}\)) were 47.4 [44.8 ; 50.0] (median \([{\mathrm {PCI}}_{25\rightarrow 75}]\)). For collagen parameters, bestfit parameter values were 2.1 [0.8 ; 5.9]) kPa for \(k_1\), and 8.8 [7.9 ; 9.5]) for \(k_2\), respectively.
In Fig. 6, distributions of collagen load bearing parameters (\(L_{\mathrm {coll}}\)) are given at three blood pressure levels (\(P_{\mathrm {d}}\), \(P_\mathrm {m}\), and \(P_\mathrm {s}\), respectively). Note that \(L_{\mathrm {coll}}=0\%\) indicates that blood pressure load is fully borne by elastin, whereas \(L_{\mathrm {coll}}=100\%\) indicates that blood pressure load is fully borne by collagen. For the initial constitutive model realisations (INIT), we found \(L_{\mathrm {coll}}\) equal to 1.5% at \(P_{\mathrm {d}}\), 3.1% at \(P_\mathrm {m}\), and 16.7% at \(P_\mathrm {s}\) (medians, respectively). As shown in Fig. 6, the \(25\mathrm{th}\) to \(75\mathrm{th}\) percentile confidence interval for \(L_{\mathrm {coll}}\) at \(P_{\mathrm {d}}\) as well as \(P_\mathrm {s}\) were large ([0.6 ; 4.5]% and [12.7 ; 24.0]%, respectively), indicating large uncertainty in model predictions of constituent load bearing.
For collagen parameter \(k_2\), distension and diastolic diameter are most influential (Table 3). Moreover, pulse pressure has some influence, indicated by an \(S_{\mathrm {i}}\) of 0.10 (Table 3). For collagen load bearing (\(L_{\mathrm {coll}}\)), distension was the most important measured variable, indicated by \(S_{\mathrm {i}}\) between 0.87 and 0.90 (Table 4). The \(S_{\mathrm {i}}\)s of the other measured variables were smaller than 0.06, indicating low influence (Table 4). Reducing uncertainty of blood pressure measurements appears of negligible importance in reducing uncertainty in estimating \(c_{\mathrm {elast}}\), \(k_1\), and \(L_{\mathrm {coll}}\), indicated by an \(S_{\mathrm {i}}<0.06\) (Tables 3 and 4). Moreover, total sensitivity indices (\(S_\mathrm {T}\)) for diastolic blood pressure and pulse pressure were smaller than 0.12, suggesting these variables could be fixed in their uncertainty domain. For all measured variables, differences between main and total sensitivity indices were minor, i.e. \(S_\mathrm {T}  S_{\mathrm {i}}\) was smaller than 0.10 (Tables 3 and 4). This indicates that the contribution of interaction terms between measured variables to the total variance was negligible (Saltelli et al. 2008).
3.3 Modelbased assessment of vascular drug therapies
Figure 7 shows the effect of AGEbreaker treatment (simulated by \(k_1\downarrow ,k_2\downarrow \)) on the modelpredicted DP curve, as well as on area compliance (\(C_A\)). Here, the average DP curves, originating on the one hand from the initial bestfit constitutive parameters (INIT, black line), and on the other hand following reduction of \(k_1\) and \(k_2\) (\(k_1\downarrow , k_2\downarrow \), red dashed line), are shown. Moreover, \(C_A\) is shown in the right pane for all INIT realisations (black circles) and the \(k_1\downarrow , k_2\downarrow \) realisations (red circles). Area compliance was calculated using \(C_A = (A_\mathrm {s}  A_{\mathrm {d}})/(P_\mathrm {s}  P_{\mathrm {d}})\), with \(A_\mathrm {s}=\pi \left( D_{\mathrm {d}}+\Delta D \right) ^2/4\) and \(A_{\mathrm {d}}=\pi D_{\mathrm {d}}^2/4\), respectively. Simulating \(k_1 \downarrow , k_2 \downarrow \) caused a leftupward shift of the groupaveraged DP curve, as well as a 40% increase in \(C_A\) (Fig. 7). In Fig. 6, distributions of \(L_{\mathrm {coll}}\) for the \(k_1 \downarrow , k_2 \downarrow \) realisations (red dashed curves) are shown. We found \(L_{\mathrm {coll}}\) to equal 0.8 [0.3 ; 2.5]% at \(P_{\mathrm {d}}\), 1.7 [0.6 ; 5.0]% at \(P_\mathrm {m}\), and 10.8 [7.2 ; 17.6]% at \(P_\mathrm {s}\) (median \([{\mathrm {PCI}}_{25\rightarrow 75}]\)). As compared to the INIT realisations, spread in \(L_{\mathrm {coll}}\) was lower for \(k_1 \downarrow , k_2 \downarrow \) realisations (Fig. 6).
4 Discussion
Computational models of arterial wall mechanics could be valuable for predicting effects of vascular drug therapies on individual arterial wall constituents. The aim of this study was (1) to quantify how measurement noise propagates into uncertainty of the model predictions and (2) to pinpoint the measurements responsible for the largest spread in mechanical characteristics. The relevance of the model output uncertainty was assessed by simulating the effects of vascular drug treatment on constituent load bearing. To our knowledge, this is the first study to perform rigorous uncertainty quantification and sensitivity analysis, assessing the influence of measurement noise in clinical arterial pressure and diameter measurements on constitutive model predictions.
Effect of increasing the number of repeated clinical measurements (\(N_\mathrm {rep}\)) on collagen load bearing parameters (\(L_{\mathrm {coll}}\)) as predicted using the initial constitutive model realisations
\(N_\mathrm {rep}\)  \(L_{\mathrm {coll}}\) at \(P_{\mathrm {d}}\)  \(L_{\mathrm {coll}}\) at \(P_\mathrm {m} \)  \(L_{\mathrm {coll}}\) at \(P_\mathrm {s}\)  

Median [%]  \({\mathrm {PCI}}_{25\rightarrow 75}\) [%]  Median [%]  \({\mathrm {PCI}}_{25\rightarrow 75}\) [%]  Median [%]  \({\mathrm {PCI}}_{25\rightarrow 75}\) [%]  
3  1.5  [0.6 ; 4.5]  3.1  [1.2 ; 8.2]  16.7  [12.7 ; 24.0] 
5  1.5  [0.7 ; 3.5]  3.0  [1.4 ; 6.7]  16.5  [13.2 ; 21.8] 
10  1.5  [0.9 ; 2.7]  3.0  [1.8 ; 5.3]  16.5  [14.0 ; 19.9] 
4.1 Model fitting and parameter estimation
Model fitting was performed on singleexponential diameter–pressure curves calculated from diastolic and systolic diameters and pressures. The assumption of a markedly exponential diameter–pressure curve in the physiological (i.e. diastolic to systolic) pressure range was reported in earlier works by Hayashi et al. (1980) and Meinders and Hoeks (2004). At a wider pressure range (i.e. 40–160 mmHg), a representative constitutive model realisation suggested sigmoidal diameter–pressure behaviour (Fig. 4). Such sigmoidal behaviour was also observed in in vitro studies, performing inflation tests on human aortic segments and rat carotid arteries (Fridez et al. 2003; Langewouters et al. 1986).
In the present study, bestfit constitutive parameter values for elastin stiffness (i.e. 47.4 [44.8 ; 50.0] kPa) were in agreement to those found for cadaveric carotid arteries, reporting values between 20 and 60 kPa (Sáez et al. 2014; Sommer and Holzapfel 2012). Moreover, values for collagen parameter \(k_2\) were well within ranges found in earlier studies (Sáez et al. 2014; Sommer and Holzapfel 2012). We found a large spread for constitutive parameter \(k_1\), governing collagen stiffness, i.e. median [\({\mathrm {PCI}}_{25\rightarrow 75}\)] of 2.1 [0.8 ; 5.9] kPa. Previous in vitro studies found \(k_1\) values ranging from 2.9 to 99.9 kPa, respectively (Sommer and Holzapfel 2012). Therefore, our findings for \(k_1\) are on the low end to those found in the aforementioned studies. It has been pointed out that appropriate choices of both \(k_1\) and \(k_2\) ensure collagen to virtually not bear load at very low amounts of stretch (i.e. at subphysiological pressure loads), whereas it will become the dominant load bearer at high amounts of stretch (Holzapfel et al. 2000). In the present study, and in in vivo studies per se, diameter and pressure measurements at these very low pressures are unavailable, making robust estimation of model parameters (particularly \(k_1\)) cumbersome, as illustrated by our findings.
4.2 Sensitivity analysis
Sensitivity analysis indicated that the most important contributors to uncertainty in \(c_{\mathrm {elast}}\) are both the variables measured by ultrasound (i.e. distension and IMT), whereas uncertainty in collagen parameter \(k_1\) was primarily caused by measurement uncertainty of distension. Although our modelbased approach still requires blood pressure to be measured, improving the precision of distension and wall thickness measurements clearly appears to be most rewarding. Recent technological advances in vascular imaging, including plane wave ultrasound and imagereconstruction algorithms, could reduce the measurement noise of a single ultrasound measurement (Besson et al. 2016). More practically, uncertainty in mechanical characteristics could be reduced by increasing the number of repeated measurements (\(N_\mathrm {rep}\), Eq. 8).
4.3 Decreasing measurement uncertainty by increasing the number of repeated measurements
We evaluated to which extent increasing \(N_\mathrm {rep}\), for all measured variables displayed in Table 1, influenced uncertainty in collagen load bearing parameters. Results indicate that increasing the number of repeated measurements from 3 to 10 decreases the spread in \(L_{\mathrm {coll}}\) by \(\sim 50\%\) (i.e. reducing the 25th to 75th percentile confidence interval at systolic blood pressure from [12.7 ; 24.0]% to [14.0 ; 19.9]%, Table 5). Based on the results of our sensitivity analysis, increasing the number repetitions of ultrasound measurements appears most rewarding in reducing uncertainty in collagen load bearing parameters.
4.4 Modelbased assessment of vascular drug therapies
AGEbreaking vascular drug therapy was simulated by changing constitutive model parameters governing collagen behaviour (i.e. parameters \(k_1\) and \(k_2\)). We chose to reduce collagen stress scaling parameter \(k_1\), as well as collagen stress curve shape parameter \(k_2\) by 40% of their initial bestfit values. Consequently, the modelled collagen becomes incrementally less stiff at low amounts of strain, but will also stiffen “later” (i.e. at higher amounts of strain), compared to when the initial parameters would be used (Fig. 6). This shift in stress–strain behaviour with decreasing crosslink densities was measured also in collagenous tissue such as the pericardium (Kayed et al. 2015; Fratzl 2008). Reducing constitutive parameters \(k_1\) and \(k_2\) by 40% resulted in a modelpredicted area compliance increase from 0.15 \(\mathrm {mm}^2/\mathrm {mmHg}\) to 0.21 \(\mathrm {mm}^2/\mathrm {mmHg}\) (Fig. 7). This observed 40% increase in area compliance is in agreement with in vitro measurements performed in rat carotid arteries, following AGEbreaker treatment (Wolffenbuttel et al. 1998). Furthermore, collagen load bearing is reduced, with the load transferred to elastin instead (Figs. 6 and 7). The median reduction of collagen load bearing was 1%, 2%, and 6% at diastolic, mean, and systolic blood pressures (Fig. 6). However, the reduction in collagen load bearing was exceeded by the initial spread in collagen load bearing. This is illustrated by the reduction in collagen load bearing at systolic blood pressure (6%, respectively, Fig. 6), which is much smaller compared to the 25th to 75th percentile confidence interval ranging between 12.7 and 24.0%, respectively (Fig. 6). In other words, measurement of the benefit of a crosslink breaker on arterial compliance may be easily concealed by uncertainties in the estimation of the effects, given the impact of noise on model output.
4.5 Limitations
Unfortunately, no actual distensibility measurements, acquired before and during AGEbreaker treatment, were available in this study. Consequently, we had to resort to simulating AGEbreaker treatment using our constitutive model. This was achieved by changing constitutive parameters governing collagen behaviour, reproducing results from an earlier AGEbreaker intervention study (Wolffenbuttel et al. 1998). The HolzapfelGasserOgden model we usedneglecting active smooth muscle response and containing only three parameters to characterise elastin and collagen behaviourwas chosen as a pragmatic simplification of the actual arterial biomechanical behaviour. Furthermore, our model neglects the dispersion of collagen fibre orientation in the adventitia (Gasser et al. 2006). We are aware that more elaborate models exist describing the influence of collagen crosslinks on the stress–strain behaviour of an artery more directly. For example, in the work of Sáez et al. (2014), a crosslinking degree parameter was introduced which includes crosslinks behaviour between the main collagen fibres. However, using such a model requires estimating one extra model parameter. To ensure unique parameter values, this would require more clinical data to be measured, which might not be possible in in vivo situations. Of note, parameter values in the current study were highly similar, indicated by the fact that the bestfit constitutive parameter values—for each of the 10 random starting points—were highly similar.
In our model, not all combinations of \(c_{\mathrm {elast}}\), \(k_1\), and \(k_2\) yield physiological behaviour. However, the adjustment (fitting) of these parameters ensures that eventually their combination does yield physiological behaviour. The parameter \(c_{\mathrm {elast}}\) can be physiologically interpreted as the stiffness of the arterial elastin. All fitted values for this parameter ranged from 35 to 55 kPa, which corresponds to previous literature (Sommer and Holzapfel 2012). For \(k_1\) and \(k_2\), no separate interpretation can be made in terms of physiology. Nevertheless, the fitted combinations of \(k_1\) and \(k_2\) yielded physiologically realistic mechanical behaviour which, given that the elastin model was plausibly parameterised, corresponds to realistic collagen behaviour.
The global variancebased sensitivity analysis method used distinguishes from more commonly used local methods by taking into account the entire distribution of measured variables, being model free, and assessing interaction between measured variables (Borgonovo and Plischke 2016; Quicken et al. 2016; Sudret 2015). However, it assumes on the one hand statistical independence between measured variables and on the other hand that variance is an adequate metric for model uncertainty. The latter assumption becomes questionable for skewed distributions of parameters, i.e. as present in the distributions of parameters \(k_1\) and \(L_{\mathrm {coll}}\) at \(P_{\mathrm {d}}\) and \(P_\mathrm {m}\), respectively (Figs. 5 and 7). A solution to this problem could be to use momentindependent sensitivity methods instead (Borgonovo and Plischke 2016). To this end, Borgonovo (2007) evaluated an alternative sensitivity metric that, instead of being computed using the variance of the model uncertainty distribution, considers this distribution as a whole. In their paper, it was concluded that sensitivity indices of both methods (1) show discrepancies between influential measured variables, but (2) agree in distinguishing noninfluential from influential measured variables (Borgonovo 2007). Based on these findings, we believe that utilising the variancebased sensitivity analysis as proposed by Quicken et al. (2016) in this study is justified.
4.6 Conclusion
This study shows that in vivo assessment of arterial wall mechanics using a constitutive model is hampered by large model uncertainty. We quantified model uncertainty in constitutive parameters (i.e. \(c_{\mathrm {elast}}\), \(k_1\), and \(k_2\), respectively), and collagen load bearing parameters, at various blood pressures \((L_{\mathrm {coll}}\)). Our simulation of vascular drug therapy suggested a reduction of collagen load bearing of 6percentage points, at systolic blood pressure. This reduction is 3–4 times lower compared to its uncertainty. Therefore, model output uncertainty could conceal potential effects of vascular drugs. Sensitivity analysis revealed that estimation of mechanical characteristics would benefit most from increasing the precision of measurements of arterial diameter, distension, and wall thickness. Whereas the potential for improving the precision of, for example, a single ultrasound measure is practically limited, the effective precision of ultrasound measurements could be improved by increasing the number of repeated measurements.
Notes
Funding
This study was funded by a Kootstra Talent Fellowship awarded to M.H.G. Heusinkveld by Maastricht University Medical Centre, S. Quicken is paid by Chemelot InSciTe, R.J. Holtackers received funding from Stichting de Weijerhorst, and B. Spronck was funded by an Endeavour Research Fellowship from the Australian government.
Compliance with ethical standards
Conflict of Interest
The authors declare that they have no conflict of interest.
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