1. Introduction
Cardiovascular disease (CVD) remains the leading cause of death worldwide and contributes substantially to morbidity, disability and death in patients [
1]. Besides continued efforts to find more effective treatment, identifying subjects at risk of CVD—especially among those categorized as having a low risk according to classical risk factors (age, gender, blood pressure, lipid levels, smoking status…)—has received a lot of attention in pre-clinical research. A particular aim of such studies is to prevent the development of CVD at a young age [
2,
3]. There is compelling evidence that arterial stiffness is a biomarker that carries prognostic power above and beyond these classical risk factors [
4].
The clinical “gold standard” method for evaluating arterial stiffness is carotid-femoral pulse wave velocity (PWV
c-f), calculated as the ratio of the distance between the carotid and femoral measuring sites, and the time taken for the arterial pulse to propagate from the carotid to the femoral artery [
5,
6,
7]. Different devices are available to measure these pulse waveforms, but current methods have methodological issues (there is no unequivocal path between the carotid and femoral artery, and the measurement basically excludes the most distensible proximal aorta). Furthermore, they are relatively demanding to use (e.g., arterial applanation tonometry, ultrasound), requiring trained operators and, in the case of magnetic resonance imaging, are associated with high costs [
8,
9]. These limitations justify the continued search for an accurate, low-cost and non-contact measurement device for PWV assessment that can easily be used in daily clinical practice. In the early stages of atherosclerotic disease, alteration of the arterial elastic properties starts in localized regions to then become more widespread. Thus, a local PWV value represents a more precise indicator of the ongoing arterial stiffening compared with the regional PWV value calculated by the most common commercial devices [
8]. Therefore, an easy to use device which allows arterial PWV assessment locally is needed.
Within this context, we are exploring laser Doppler vibrometry (LDV) as a method to locally measure PWV in the common carotid artery (CCA), a superficial large elastic artery known to be prone to remodel and stiffen with age and disease [
10,
11,
12]. LDV allows optical measurement of out-of-plane skin displacements as a function of time, in which the mechanical perturbation caused by the arterial pressure pulse can be identified. This measurement is then performed in at least two different spatial locations along the carotid artery (CA) and PWV can consequently be calculated as the ratio of the distance between the laser beams and the time shift of the recorded displacement signals. Due to the limited distance between the laser beams and the total length of the CCA (a few centimeters), LDV provides a local estimation of PWV compared to the carotid-femoral PWV measurement.
The technical feasibility of LDV as a technique for PWV assessment has been explored in some in vitro and in vivo studies [
13,
14,
15,
16,
17]. Recent studies have also shown good correspondence between PWV evaluated with LDV and ultrasound-based pulse wave imaging (PWI) in a phantom, and demonstrated the potential of LDV to detect arterial wall stiffening [
18]. Even though previous studies have shown promising results for LDV as screening tool for CVD, the understanding of the generated wave propagation mechanics between arterial wall and skin surface is still limited. Indeed, the waves are subjected to reflections, attenuation and interaction with the surrounding structures and organs, potentially leading to wave interference or wave mode conversions, complicating the relationship between the arterial pressure pulse and the corresponding skin displacement.
A computer modelling environment able to reproduce the complex wave phenomena associated with the propagation of the mechanical perturbation caused by the arterial pulse provides a flexible way to perform a complete analysis of the wave propagation. The goal of this study is to provide deeper insights into the naturally occurring wave physics in non-stenotic carotid vessels using numerical simulations. These insights can ultimately improve current LDV analyses for PWV assessment and pinpoint limitations of the LDV technique.
In this work, a model representing the CCA embedded within soft tissues was created using the finite element method (FEM). Its realism in accurately representing wave propagation was assessed through modelling specific modes of bulk and surface waves, where a known analytical solution is at hand. Finally, we used this model to perform a parametric study investigating the correlation between arterial pressure and displacement pattern at the skin surface. The effect of the following parameters was examined: (i) pressure conditions within the artery, (ii) artery depth below the skin surface and (iii) arterial geometry (patient-specific carotid bifurcation).
5. Discussion
In the last decade, few studies have investigated the potential of LDV-based methods to assess local PWV estimation from skin vibrations generated by the pressure pulse traveling inside the underlying artery. LDV measurements have been performed with a two-beam LDV both in vitro (phantom set up mimicking arterial pulse propagation), and in vivo (CCA in healthy volunteers) [
16,
17]. Preliminary results have assessed the technical feasibility of LDV to accurately measure the displacement waveform at two skin locations, but also emphasized that the complexity of the skin motion prevented accurate estimation of the pulse transit time between the two measurement sites, resulting in an uncertainty in the PWV results. The transit–time error depends on many factors: shape differences between the two recorded pulses, perturbations of the LDV outputs due to a non–optimal or too weak a degree of light reflection, the existence of spurious reflections, artifacts due to secondary sources of motion (e.g., swallowing, any involuntary body movement, movement of the LDV and vibrations coming from the environment) as well as limitations of the algorithm used to estimate pulse transit time [
16]. An in vivo study on 48 patients [
27] has shown that signal analysis is still the most difficult challenge to validate local PWV through an LDV-based method. As expected pulse transit times are small (ranging from 2.5 to 1 ms for a beam separation of 10 mm when assuming a PWV between 4–10 m/s). Thus, a small inaccuracy in the pulse transit time estimation can lead to a considerable error in PWV estimation. Several algorithms are currently available to determine the pulse transit time. Some of them use the first or second derivative of the displacement signal (e.g., second derivative method, tangent-intersect method) but these methods are sensitive to noise [
28]. The cross-correlation method measures the similarity of two waveforms as a function of a time-lag applied to one of them but fails when comparing signals with shapes that are too different [
17]. Common algorithms to estimate the pulse transit time from in vivo measurements are based on the identification of a reference point, typically the dicrotic notch or the systolic foot [
29]. However, the identification of the reference point can often not be assessed unambiguously in a substantial portion of the data [
27]. Further studies have tried to improve local PWV estimation by increasing the number of measuring points from two to four with 15 mm spacing between them. PWV obtained by such LDV measurements in an in vitro set-up consisting of an elastic tube embedded in tissue-mimicking materials, showed a good agreement with measurements performed by the more established ultrasound-based pulse wave imaging [
18]. The LDV technique was also able to distinguish phantoms with greater stiffness (higher PWV) but not to determine the precise location of a stiffer inclusion due to the low spatial resolution of the measurements.
In this work, we have presented a finite element model to gain a more fundamental understanding of wave propagation from mechanical perturbations initiated by arterial pressure pulses in a healthy carotid artery model, to the skin surface (measured by the LDV-technique). As explained in
Section 3.1.2, the choice of numerical settings such as time increment size, numerical damping and mesh size, together with the material parameters (e.g., Poisson’s ratio) highly influenced the computational efficiency and the reliability of the wave propagation simulations. Therefore, the appropriateness of the chosen numerical settings was initially verified by comparing the simulated displacement patterns of waves where a theoretical solution was at hand (shear and Rayleigh waves, as described in
Section 2). It should be noted that these analytical solutions assume an infinite (shear waves) and semi-infinite (Rayleigh waves) isotropic linear elastic material, whereas our final model did not meet these geometrical requirements. Despite this limitation, the simulation results showed that the chosen numerical settings made it possible to reproduce a shear wave propagation speed with a deviation of −2.0% from the analytical value, whereas Rayleigh wave speed propagation deviated only 0.9%. Additionally, a good match between the spatial and temporal displacement characteristics of simulation and theory was observed (see
Figure 3). The geometry of the patient-specific model introduced a higher degree of complexity which considerably complicated the test of the computational settings. Only shear waves were simulated in this set-up, resulting in a shear wave speed propagation that differed by −2.0% from the analytical value.
Consecutively, a parametric study was set up to study the impact of different arterial pressure loads and geometrical conditions on the induced skin displacement in a controlled environment. Tracked displacement patterns showed the coexistence of a primary (higher amplitude) and secondary wave front travelling with different velocities both at the boundary of the lumen and the skin surface.
Figure 11 summarizes the estimated wave velocities for the primary (V
1) and secondary (V
2) wave fronts at the skin level for all simulated cases.
Initially, a localized, non-propagating short pressure impulse was used to induce free wave propagation in the soft-tissue mimicking material surrounding the CCA model. Tracked displacement patterns in the CCA model showed a main wave front propagating with a velocity V
1 of 4.66 m/s at the lumen surface and a similar wave speed was found for the first secondary wave front at the skin level (V
2 = 4.79 m/s). These values approached the analytical value of shear (4.82 m/s) and Rayleigh (4.59 m/s) wave speed, however,
Figure 4 and
Figure 5 also clearly showed a complex wave pattern with multiple wave fronts due to phenomena such as wave recombination, refraction and reflection, and therefore the relation between the observed wave patterns and the ideal free wave propagation modes is unclear. Additionally, it should be noted that we are only analyzing one displacement component in the TOF-plots (component relevant for LDV-analyses), whereas the traveling waves will have displacement components in multiple directions.
When considering a traveling impulse (wave speeds varying within the physiological range 4–10 m/s), it can be observed that for imposed propagation speeds of 7 and 10 m/s, all tested conditions in the CCA model yielded an estimated propagation speed V
1 close to the imposed value (differences −5.0 to −0.4;
Figure 11). However, the lower imposed wave speed of 4 m/s is overestimated at the skin surface by 4%–33.5% in the model, in which the deviation increased with increasing arterial depth. It should be noted that the selected model’s geometry and boundary conditions play a role in reaching a steady state regime for wave pattern development at the skin surface. However, since the tracked skin speeds for the higher imposed PWV corresponded fairly well with the imposed value for all considered arterial depths, we assume that the present transition phenomena in current modeling environment are minimal. As such, these results suggest that observed changes in wave velocity at the skin level may not fully reflect the effective changes in the arterial pulse wave velocity, as shown in
Figure 11.
For the patient-specific CA model, the estimated lumen wave speed was composed of a split main wave front characterized by different velocities along the length of the model, as shown in
Figure 9. This split wave front is caused by the differences in length and orientation of the upper wall compared to that of the centerline at x = 0 mm. The estimated wave speed corresponded best with the imposed PWV (1.3%) in the proximal part of the model (z < 20 mm), since the geometry of the upper wall and centerline is then very similar (see
Figure 1b). At the skin level, almost no effect of this split wave front was observed and traveling wave fronts V
1 and V
2 were tracked by a single line. Similarly to the CCA model, estimated wave speed at the skin corresponded best with the imposed V
z when considering a PWV of 7 m/s (−3.1%). The deviation was higher for a V
z of 4 m/s (47.0%) and a V
z of 10 m/s (36.0%).
Figure 11 shows a large range in estimated speeds for the secondary waves in all simulation cases. It should be noted that in some cases, i.e., the CCA model with shock impulse loading condition and the patient-specific CA model with imposed PWV of 10 m/s, the speed of the secondary wave corresponded better with the expected or imposed velocity than the velocity of the main wave at the skin surface. These results suggest that the main wave front at the lumen level not necessarily translates into a main wave front at the skin level. However, these results again need to be interpreted with care as the modeled geometry and boundary conditions also have an effect. In other simulation cases, the nature of these secondary waves might be related to the propagation of the free vibrations of the surrounding tissue that is excited during the pulse transmission; However future work should include a more in-depth study of all displacement components and the application of advanced wave analysis techniques to better understand the nature of these secondary waves.
Results supported the conclusion that the skin displacement is not a simple overlapping of free and induced waves, but the result of a combination of multiple reflections and re-combinations which iterate along the model’s length in time, and are thus hard to predict and interpret. Therefore, we can conclude that the transmission of a single pulse creates a complex skin displacement pattern that manifests with changes in displacement amplitude, spatial lag and propagation velocities (V1 and V2).
Our results can be considered as an initial step towards a better understanding of the LDV-measured skin vibration signals by studying the wave propagation physics between the lumen and the skin. The model showed that the correspondence between the actual PWV in the lumen and the PWV measured at the skin surface depended on the magnitude of the PWV in the lumen and the studied subject (arterial depth and geometry). To the best of our knowledge, this work represents the first computational study that has investigated in detail the induced skin displacement in a structural model of a healthy artery for the estimation of pulse wave propagation speed. Indeed, until now in this context, a fluid-structure finite element approach has been adopted only to study the sound generated in a stenosed artery and transmitted through biological tissues to the skin, to investigate the possibility of using the LDV technique for arterial stenosis detection [
30,
31,
32].
Although the developed numerical model has allowed us to study the propagation of a single traveling impulse under prescribed conditions in a simplified environment, it rests on several assumptions. For instance, the artery was approximated as a straight cylinder in the CCA model without any wall thickness. The absence of modeling the arterial wall did not allow the control of arterial stiffness in a direct way (e.g., increasing the Young’s modulus of the material), thus arterial stiffening was indirectly simulated by varying the speed of the propagating pulse. Moreover, the effect of the surrounding biological structures (e.g., muscle, fat, bones) was not reproduced and the soft tissue was modeled as an isotropic homogenous viscoelastic medium without any biological discontinuities. It is generally known that soft tissue is more viscoelastic than PVA-material and this will have an extra damping effect on the wave patterns reducing all high frequency patterns. Also, the boundary conditions to minimize wave reflections did not work optimally due to the nature of the infinite elements (see
Figure 1). The loading conditions did not replicate the shape of a physiological pulse nor its periodic course, therefore the actual mechanical interaction of the pressure pulse and the lumen wall was not realistically simulated. In addition, we did not account for the fluid–structure interaction or any vibrations that may arise from instabilities in the flow field, propagating to the wall and skin. It should be kept in mind that the accuracy of the presented results depends on the chosen finite element mesh size, interpolation grids in Matlab and the projection angle of the Radon transform. This is also related to the available computer power and performance requirements, which were very demanding for the FEM simulations of the patient-specific model (72 h on a Linux cluster using a single computing node containing two Intel E5–2670 processors each with 8 cores, 2.60 GHz processor base frequency, 64 GB of memory and Fourteen Data Rate InfiniBand). The main limitation of the current modeling environment is the lack of validation. However, we believe that the good correspondence between simulation and theoretical results for shear and surface wave propagation (see
Table 1 and
Figure 3) lends confidence to the current modeling results.
The developed finite element model offers the advantage of visualizing and investigating the within-tissue wave propagation mechanics between lumen and skin, generated by the traveling pulse wave, which is difficult to obtain from experiments. Furthermore, it allows one to analyze multiple components of the displacements in 3D and gives access to other variables such as stresses. Compared to the LDV technique, the simulations provided a higher spatial resolution and adjustable temporal resolution (10 kHz and 0.2 mm over a 50 mm vs. 500 kHz and 15 mm over 45 mm [
18]) and are not contaminated with any measurement noise. This high spatial resolution allowed us to employ an alternative method to the traditional LDV algorithms to estimate PWV, i.e., the Radon transform, an approach which is commonly used in the field of elastography to estimate wave speeds [
33]. In contrast to the available methods for the LDV technique, this method provides a robust way of estimating wave speeds by yielding a linear wave trajectory based on the maximal Radon-projected displacements generated by the propagating wave. As this method requires a higher spatial and temporal resolution in combination with more measurement points, it might be more productive in the context of the recently developed LDV system consisting of 6-beams spaced 5 mm apart (total length 25 mm) by Li et al. [
34]. Nevertheless, the developed simulation framework in this study can act as a validation tool for newly proposed LDV-processing techniques, since the simulated displacement data are not contaminated by any of the complexities present in an actual experiment (noise, limited spatial and temporal resolution, etc.) and it offers direct access to the actual PWV in the lumen.
In addition to patient-specific intrinsic factors affecting the wave propagation from lumen to skin (e.g., carotid depth—as considered in this study), there are also some technical and operator-dependent factors influencing the LDV measured PWV value such as laser signal strength and the localization of the measurement points. The latter issue might be improved by providing a real-time data visualization of the spatio-temporal plots to the operator that demonstrates wave propagation at the skin, as shown in the results section of our model. Indeed, this real-time tool may help the user to identify LDV positions where the LDV device is correctly aligned with the artery’s axis and without any signal contamination caused by other vibrating structures (e.g., jugular vein). Combining this real-time tool together with palpation can thus improve positioning and alignment of the LDV device with the artery, potentially resulting in a more robust and accurate PWV estimation.