Linear and nonlinear viscoelastic modeling of aorta and carotid pressure-area dynamics under in vivo and ex vivo conditions

Modeling Cerebral Blood Flow Velocity During Orthostatic Stress

Cerebral autoregulation refers to the physiological process that maintains stable cerebral blood flow (CBF) during changes in arterial blood pressure (ABP). In this study, we propose a simple, nonlinear quantitative model with only four parameters that can predict CBF velocity as a function of ABP. The model was motivated by the viscoelastic-like behavior observed in the data collected during postural change from sitting to standing. Qualitative testing of the model involved analysis of dynamic responses to step-changes in pressure both within and outside the autoregulatory range, while quantitative testing was used to show that the model can fit dynamics observed in data measured from a healthy young and a healthy elderly subject. The latter involved analysis of structural and practical identifiability, sensitivity analysis, and parameter estimation. Results showed that the model is able to reproduce observed overshoot and adaptation and predict the different responses in the healthy young and the healthy elderly subject. For the healthy young subject, the overshoot was significantly more pronounced than for the elderly subject, but the recovery time was longer for the young subject. These differences resulted in different parameter values estimated using the two datasets.

Viscoelastic Deterioration of the Carotid Artery Vascular Wall is a Possible Predictor of Coronary Artery Disease

AIM

The viscoelastic properties of the artery are known to be altered in patients with vascular diseases. However, few studies have evaluated the viscoelasticity of the vascular wall in humans. We sought to investigate the degree of viscoelastic deterioration of the carotid artery and assess its clinical implications.

METHODS

Between January 2011 and June 2013, patients in whom the toe-brachial index was measured at the vascular laboratory were included in this single-institute retrospective observational study. I(＊), a parameter of viscoelastic deterioration, was computed using a non-invasive ultrasonic Doppler effect sensor on the carotid artery. I(＊) is a non-dimensional value, and I(＊)>0 is considered abnormal. Other patient characteristics were identified and tested for correlations with I(＊).

RESULTS

The study included 383 patients. The mean I(＊) value was 0.13 ± 0.22 with a normal distribution. Factors that increased the I(＊) value were a female sex (0.18 ± 0.23 vs. 0.10 ± 0.21, P<0.001), age ≥ 60 (0.14 ± 0.22 vs. 0.06 ± 0.23, P<0.05) and systolic blood pressure of >140 (0.15 ± 0.22 vs. 0.10 ± 0.22, P<0.05). I(＊) abnormality was a significant risk factor for coronary artery disease (OR 2.20, 95% CI 1.00-4.80, P<0.05) in a univariate analysis. In the multivariate analysis, I(＊) abnormality was also found to be an independent risk factor for coronary artery disease (OR 4.56, 95% CI 1.21-30.1, P<0.05).

CONCLUSIONS

I(＊) may reflect the degree of atherosclerotic changes in the arterial wall and could possibly be used to predict coronary artery disease.

Modelling and subject-specific validation of the heart-arterial tree system

A modeling approach integrated with a novel subject-specific characterization is here proposed for the assessment of hemodynamic values of the arterial tree. A 1D model is adopted to characterize large-to-medium arteries, while the left ventricle, aortic valve and distal micro-circulation sectors are described by lumped submodels. A new velocity profile and a new formulation of the non-linear viscoelastic constitutive relation suitable for the {Q, A} modeling are also proposed. The model is firstly verified semi-quantitatively against literature data. A simple but effective procedure for obtaining subject-specific model characterization from non-invasive measurements is then designed. A detailed subject-specific validation against in vivo measurements from a population of six healthy young men is also performed. Several key quantities of heart dynamics-mean ejected flow, ejection fraction, and left-ventricular end-diastolic, end-systolic and stroke volumes-and the pressure waveforms (at the central, radial, brachial, femoral, and posterior tibial sites) are compared with measured data. Mean errors around 5 and 8%, obtained for the heart and arterial quantities, respectively, testify the effectiveness of the model and its subject-specific characterization.

Structural correlation method for model reduction and practical estimation of patient specific parameters illustrated on heart rate regulation

We consider the inverse and patient specific problem of short term (seconds to minutes) heart rate regulation specified by a system of nonlinear ODEs and corresponding data. We show how a recent method termed the structural correlation method (SCM) can be used for model reduction and for obtaining a set of practically identifiable parameters. The structural correlation method includes two steps: sensitivity and correlation analysis. When combined with an optimization step, it is possible to estimate model parameters, enabling the model to fit dynamics observed in data. This method is illustrated in detail on a model predicting baroreflex regulation of heart rate and applied to analysis of data from a rat and healthy humans. Numerous mathematical models have been proposed for prediction of baroreflex regulation of heart rate, yet most of these have been designed to provide qualitative predictions of the phenomena though some recent models have been developed to fit observed data. In this study we show that the model put forward by Bugenhagen et al. can be simplified without loss of its ability to predict measured data and to be interpreted physiologically. Moreover, we show that with minimal changes in nominal parameter values the simplified model can be adapted to predict observations from both rats and humans. The use of these methods make the model suitable for estimation of parameters from individuals, allowing it to be adopted for diagnostic procedures.

Engineering of arteries in vitro

This review will focus on two elements that are essential for functional arterial regeneration in vitro: the mechanical environment and the bioreactors used for tissue growth. The importance of the mechanical environment to embryological development, vascular functionality, and vascular graft regeneration will be discussed. Bioreactors generate mechanical stimuli to simulate biomechanical environment of arterial system. This system has been used to reconstruct arterial grafts with appropriate mechanical strength for implantation by controlling the chemical and mechanical environments in which the grafts are grown. Bioreactors are powerful tools to study the effect of mechanical stimuli on extracellular matrix architecture and mechanical properties of engineered vessels. Hence, biomimetic systems enable us to optimize chemo-biomechanical culture conditions to regenerate engineered vessels with physiological properties similar to those of native arteries. In addition, this article reviews various bioreactors designed especially to apply axial loading to engineered arteries. This review will also introduce and examine different approaches and techniques that have been used to engineer biologically based vascular grafts, including collagen-based grafts, fibrin-gel grafts, cell sheet engineering, biodegradable polymers, and decellularization of native vessels.

Novel wave intensity analysis of arterial pulse wave propagation accounting for peripheral reflections

We present a novel analysis of arterial pulse wave propagation that combines traditional wave intensity analysis with identification of Windkessel pressures to account for the effect on the pressure waveform of peripheral wave reflections. Using haemodynamic data measured in vivo in the rabbit or generated numerically in models of human compliant vessels, we show that traditional wave intensity analysis identifies the timing, direction and magnitude of the predominant waves that shape aortic pressure and flow waveforms in systole, but fails to identify the effect of peripheral reflections. These reflections persist for several cardiac cycles and make up most of the pressure waveform, especially in diastole and early systole. Ignoring peripheral reflections leads to an erroneous indication of a reflection-free period in early systole and additional error in the estimates of (i) pulse wave velocity at the ascending aorta given by the PU-loop method (9.5% error) and (ii) transit time to a dominant reflection site calculated from the wave intensity profile (27% error). These errors decreased to 1.3% and 10%, respectively, when accounting for peripheral reflections. Using our new analysis, we investigate the effect of vessel compliance and peripheral resistance on wave intensity, peripheral reflections and reflections originating in previous cardiac cycles.

The effect of static stretch on elastin degradation in arteries

Previously we have shown that gradual changes in the structure of elastin during an elastase treatment can lead to important transition stages in the mechanical behavior of arteries. However, in vivo arteries are constantly being loaded due to systolic and diastolic pressures and so understanding the effects of loading on the enzymatic degradation of elastin in arteries is important. With biaxial tensile testing, we measured the mechanical behavior of porcine thoracic aortas digested with a mild solution of purified elastase (5 U/mL) in the presence of a static stretch. Arterial mechanical properties and biochemical composition were analyzed to assess the effects of mechanical stretch on elastin degradation. As elastin is being removed, the dimensions of the artery increase by more than 20% in both the longitude and circumference directions. Elastin assays indicate a faster rate of degradation when stretch was present during the digestion. A simple exponential decay fitting confirms the time constant for digestion with stretch (0.11 ± 0.04 h(-1)) is almost twice that of digestion without stretch (0.069 ± 0.028 h(-1)). The transition from J-shaped to S-shaped stress vs. strain behavior in the longitudinal direction generally occurs when elastin content is reduced by about 60%. Multiphoton image analysis confirms the removal/fragmentation of elastin and also shows that the collagen fibers are closely intertwined with the elastin lamellae in the medial layer. After removal of elastin, the collagen fibers are no longer constrained and become disordered. Release of amorphous elastin during the fragmentation of the lamellae layers is observed and provides insights into the process of elastin degradation. Overall this study reveals several interesting microstructural changes in the extracellular matrix that could explain the resulting mechanical behavior of arteries with elastin degradation.

In vivo determination of elastic properties of the human aorta based on 4D ultrasound data

Computational analysis of the biomechanics of the vascular system aims at a better understanding of its physiology and pathophysiology. To be of clinical use, however, these models and thus their predictions, have to be patient specific regarding geometry, boundary conditions and material. In this paper we present an approach to determine individual material properties of human aortae based on a new type of in vivo full field displacement data acquired by dimensional time resolved three dimensional ultrasound (4D-US) imaging. We developed a nested iterative Finite Element Updating method to solve two coupled inverse problems: The prestrains that are present in the imaged diastolic configuration of the aortic wall are determined. The solution of this problem is integrated in an iterative method to identify the nonlinear hyperelastic anisotropic material response of the aorta to physiologic deformation states. The method was applied to 4D-US data sets of the abdominal aorta of five healthy volunteers and verified by a numerical experiment. This non-invasive in vivo technique can be regarded as a first step to determine patient individual material properties of the human aorta.

A mathematical model for analyzing the elasticity, viscosity, and failure of soft tissue: comparison of native and decellularized porcine cardiac extracellular matrix for tissue engineering

The clinical success of tissue-engineered constructs commonly requires mechanical properties that closely mimic those of the human tissue. Determining the viscoelastic properties of such biomaterials and the factors governing their failure profiles, however, has proven challenging, although collecting extensive data regarding their tensile behavior is straightforward. The easily calculated Young's modulus remains the most reported mechanical measure, regardless of its limitations, even though single-relaxation-time (SRT) models can provide much more information, which remain scarce due to a lack of manageable tools for implementing these models. We developed an easy-to-use algorithm for applying the Zener SRT model and determining the elastic moduli, viscosity, and failure profiles of materials under different mechanical tests in a user-independent manner. The algorithm was validated on the data resulting from tensile tests on native and decellularized porcine cardiac tissue, previously suggested as a promising scaffold material for cardiac tissue engineering. This analysis yields new and more accurate measurements such as the elastic moduli and viscosity, the model's relaxation time, and information on the factors governing the materials' failure profiles. These measurements indicate that the viscoelasticity and strength of the decellularized acellular extracellular matrix (ECM) are similar to those of native tissue, although its elasticity and apparent viscosity are higher. Nonetheless, reseeding and culturing the ECM with mesenchymal stem cells was shown to partially restore the mechanical properties lost after decellularization. We propose this algorithm as a platform for soft-tissue analysis that can provide comparable and unbiased measures for characterizing viscoelastic biomaterials commonly used in tissue engineering.

Progressive structural and biomechanical changes in elastin degraded aorta

Aortic aneurysm is an important clinical condition characterized by common structural changes such as the degradation of elastin, loss of smooth muscle cells, and increased deposition of fibrillary collagen. With the goal of investigating the relationship between the mechanical behavior and the structural/biochemical composition of an artery, this study used a simple chemical degradation model of aneurysm and investigated the progressive changes in mechanical properties. Porcine thoracic aortas were digested in a mild solution of purified elastase (5 U/mL) for 6, 12, 24, 48, and 96 h. Initial size measurements show that disruption of the elastin structure leads to increased artery dilation in the absence of periodic loading. The mechanical properties of the digested arteries, measured with a biaxial tensile testing device, progress through four distinct stages termed (1) initial-softening, (2) elastomer-like, (3) extensible-but-stiff, and (4) collagen-scaffold-like. While stages 1, 3, and 4 are expected as a result of elastin degradation, the S-shaped stress versus strain behavior of the aorta resulting from enzyme digestion has not been reported previously. Our results suggest that gradual changes in the structure of elastin in the artery can lead to a progression through different mechanical properties and thus reveal the potential existence of an important transition stage that could contribute to artery dilation during aneurysm formation.

Pulse wave propagation in a model human arterial network: Assessment of 1-D visco-elastic simulations against in vitro measurements

The accuracy of the nonlinear one-dimensional (1-D) equations of pressure and flow wave propagation in Voigt-type visco-elastic arteries was tested against measurements in a well-defined experimental 1:1 replica of the 37 largest conduit arteries in the human systemic circulation. The parameters required by the numerical algorithm were directly measured in the in vitro setup and no data fitting was involved. The inclusion of wall visco-elasticity in the numerical model reduced the underdamped high-frequency oscillations obtained using a purely elastic tube law, especially in peripheral vessels, which was previously reported in this paper [Matthys et al., 2007. Pulse wave propagation in a model human arterial network: Assessment of 1-D numerical simulations against in vitro measurements. J. Biomech. 40, 3476–3486]. In comparison to the purely elastic model, visco-elasticity significantly reduced the average relative root-mean-square errors between numerical and experimental waveforms over the 70 locations measured in the in vitro model: from 3.0% to 2.5% (p<0.012) for pressure and from 15.7% to 10.8% (p<0.002) for the flow rate. In the frequency domain, average relative errors between numerical and experimental amplitudes from the 5th to the 20th harmonic decreased from 0.7% to 0.5% (p<0.107) for pressure and from 7.0% to 3.3% (p<10−6) for the flow rate. These results provide additional support for the use of 1-D reduced modelling to accurately simulate clinically relevant problems at a reasonable computational cost.