A Three-Dimensional Analytical (Rheological) Model of the Human Left Ventricle in Passive-Active States: Nontraumatic Determination of the In Vivo Values of the Rheological Parameters

In this paper a three-dimensional continuum model of a mammalian left ventricle is formulated. The stresses in the model satisfy the conditions of zero stress on the outer (epicardial surface-representing) boundary. The strains of the model are obtained from the actual dynamic geometry measurements (obtained from cineangiocardiography). Since the left ventricular muscle is incompressible, the dilatational strain is zero and hence the (three-dimensional) deviatric stress components are related to the corresponding strain components by Maxwell and Voigt rheological model analogues of one-dimensional systems; the parameters of the model are series and parallel elastic (SE, PE) elements and the contractile element (CE) (representing the sarcomere). The incorporation of the rheological features of the cardiac muscle into the three-dimensional constitutive equations (for the three-dimensional continuum model of the left ventricle) is a feature of this paper. A procedure is presented to determine the parameters of the constitutive equations (i.e., the SE, PE, and the parameters of the force-velocity relation for the CE) for the left ventricle of a subject from data on the dimensions and chamber pressure of the left ventricle. The values of these parameters characterize the rheology of the left ventricular muscle of the subject. In order to demonstrate clinical application of the analyses, in vivo data of the subjects' left ventricular pressure and dimensions are obtained, and the analyses are applied to the data to determine (for each subject) the values and characteristics of the elastic elements and CEs.     

Numerical simulation of blood and interstitial flow through a solid tumor

A theoretical framework is presented for describing blood flow through the irregular vasculature of a solid tumor. The tumor capillary bed is modeled as a capillary tree of bifurcating segments whose geometrical construction involves deterministic and random parameters. Blood flow along the individual capillaries accounts for plasma leakage through the capillary walls due to the transmural pressure according to Sterling’s law. The extravasation flow into the interstitium is described by Darcy’s law for a biological porous medium. The pressure field developing in the interstitium is computed by solving Laplace’s equation subject to derived boundary conditions at the capillary vessel walls. Given the arterial, venous, and tumor surface pressures, the problem is formulated as a coupled system of integral and differential equations arising from the interstitium and capillary flow transport equations. Numerical discretization yields a system of linear algebraic equations for the interstitial and capillary segment pressures whose solution is found by iterative methods. Results of numerical computations document the effect of the interstitial hydraulic and vascular permeability on the fractional plasma leakage. Given the material properties, the fractional leakage reaches a maximum at a particular grade of the bifurcating vascular tree.     

Numerical simulation of blood flow in the left ventricle and aortic sinus using magnetic resonance imaging and computational fluid dynamics

Understanding cardiac blood flow patterns has many applications in analysing haemodynamics and for the clinical assessment of heart function. In this study, numerical simulations of blood flow in a patient-specific anatomical model of the left ventricle (LV) and the aortic sinus are presented. The realistic 3D geometry of both LV and aortic sinus is extracted from the processing of magnetic resonance imaging (MRI). Furthermore, motion of inner walls of LV and aortic sinus is obtained from cine-MR image analysis and is used as a constraint to a numerical computational fluid dynamics (CFD) model based on the moving boundary approach. Arbitrary Lagrangian–Eulerian finite element method formulation is used for the numerical solution of the transient dynamic equations of the fluid domain. Simulation results include detailed flow characteristics such as velocity, pressure and wall shear stress for the whole domain. The aortic outflow is compared with data obtained by phase-contrast MRI. Good agreement was found between simulation results and these measurements.     

Heterogeneous mechanics of the mouse pulmonary arterial network

Individualized modeling and simulation of blood flow mechanics find applications in both animal research and patient care. Individual animal or patient models for blood vessel mechanics are based on combining measured vascular geometry with a fluid structure model coupling formulations describing dynamics of the fluid and mechanics of the wall. For example, one-dimensional fluid flow modeling requires a constitutive law relating vessel cross-sectional deformation to pressure in the lumen. To investigate means of identifying appropriate constitutive relationships, an automated segmentation algorithm was applied to micro-computerized tomography images from a mouse lung obtained at four different static pressures to identify the static pressure–radius relationship for four generations of vessels in the pulmonary arterial network. A shape-fitting function was parameterized for each vessel in the network to characterize the nonlinear and heterogeneous nature of vessel distensibility in the pulmonary arteries. These data on morphometric and mechanical properties were used to simulate pressure and flow velocity propagation in the network using one-dimensional representations of fluid and vessel wall mechanics. Moreover, wave intensity analysis was used to study effects of wall mechanics on generation and propagation of pressure wave reflections. Simulations were conducted to investigate the role of linear versus nonlinear formulations of wall elasticity and homogeneous versus heterogeneous treatments of vessel wall properties. Accounting for heterogeneity, by parameterizing the pressure/distention equation of state individually for each vessel segment, was found to have little effect on the predicted pressure profiles and wave propagation compared to a homogeneous parameterization based on average behavior. However, substantially different results were obtained using a linear elastic thin-shell model than were obtained using a nonlinear model that has a more physiologically realistic pressure versus radius relationship.     

Pulsatile pressure and flow in the skeletal muscle microcirculation

Although blood flow in the microcirculation of the rat skeletal muscle has negligible inertia forces with very low Reynolds number and Womersley parameter, time-dependent pressure and flow variations can be observed. Such phenomena include, for example, arterial flow overshoot following a step arterial pressure, a gradual arterial pressure reduction for a step flow, or hysteresis between pressure and flow when a pulsatile pressure is applied. Arterial and venous flows do not follow the same time course during such transients. A theoretical analysis is presented for these phenomena using a microvessel with distensible viscoelastic walls and purely viscous flow subject to time variant arterial pressures. The results indicate that the vessel distensibility plays an important role in such time-dependent microvascular flow and the effects are of central physiological importance during normal muscle perfusion. In-vivo whole organ pressure-flow data in the dilated rat gracilis muscle agree in the time course with the theoretical predictions. Hemodynamic impedances of the skeletal muscle microcirculation are investigated for small arterial and venous pressure amplitudes superimposed on an initial steady flow and pressure drop along the vessel.     

Identification of in vivo material and geometric parameters of a human aorta: toward patient-specific modeling of abdominal aortic aneurysm

Recent advances in computational modeling of vascular adaptations and the need for their extension to patient-specific modeling have introduced new challenges to the path toward abdominal aortic aneurysm modeling. First, the fundamental assumption in adaptation models, namely the existence of vascular homeostasis in normal vessels, is not easy to implement in a vessel model built from medical images. Second, subjecting the vessel wall model to the normal pressure often makes the configuration deviate from the original geometry obtained from medical images. To address those technical challenges, in this work, we propose a two-step optimization approach; first, we estimate constitutive parameters of a healthy human aorta intrinsic to the material by using biaxial test data and a weighted nonlinear least-squares parameter estimation method; second, we estimate the distributions of wall thickness and anisotropy using a 2-D parameterization of the vessel wall surface and a global approximation scheme integrated within an optimization routine. A direct search method is implemented to solve the optimization problem. The numerical optimization method results in a considerable improvement in both satisfying homeostatic condition and minimizing the deviation of geometry from the original shape based on in vivo images. Finally, the utility of the proposed technique for patient-specific modeling is demonstrated in a simulation of an abdominal aortic aneurysm enlargement.     

Numerical investigation of the non-Newtonian blood flow in a bifurcation model with a non-planar branch

The non-Newtonian fluid flow in a bifurcation model with a non-planar daughter branch is investigated by using finite element method to solve the three-dimensional Navier–Stokes equations coupled with a non-Newtonian constitutive model, in which the shear thinning behavior of the blood fluid is incorporated by the Carreau–Yasuda model. The objective of this study is to investigate the influence of the non-Newtonian property of fluid as well as of curvature and out-of-plane geometry in the non-planar daughter vessel on wall shear stress (WSS) and flow phenomena. In the non-planar daughter vessel, the flows are typified by the skewing of the velocity profile towards the outer wall, creating a relatively low WSS at the inner wall. In the downstream of the bifurcation, the velocity profiles are shifted towards the flow divider. The low WSS is found at the inner walls of the curvature and the lateral walls of the bifurcation. Secondary flow patterns that swirl fluid from the inner wall of curvature to the outer wall in the middle of the vessel are also well documented for the curved and bifurcating vessels. The numerical results for the non-Newtonian fluid and the Newtonian fluid with original Reynolds number and the corresponding rescaled Reynolds number are presented. Significant difference between the non-Newtonian flow and the Newtonian flow is revealed; however, reasonable agreement between the non-Newtonian flow and the rescaled Newtonian flow is found. Results of this study support the view that the non-planarity of blood vessels and the non-Newtonian properties of blood are an important factor in hemodynamics and may play a significant role in vascular biology and pathophysiology.     

A constitutive theory for biomembranes: application to epicardial mechanics.

We present a new theoretically motivated experimental approach for identifying the functional form of a constitutive relation for any nonlinear, anisotropic pseudoelastic biological membrane. The utility of this approach is illustrated by identifying, from biaxial data, a new constitutive relation for excised ventricular epicardium. Values of the associated material parameters are calculated and compared for right and left ventricular specimens. Based on our findings, we suggest that there are no significant differences in the biomechanical behavior of epicardium excised from the right and left ventricular free walls of canine hearts.     

Experimental validation of a time-domain-based wave propagation model of blood flow in viscoelastic vessels

Time-domain-based one-dimensional wave propagation models of the arterial system are preferable over one-dimensional wave propagation models in the frequency domain since the latter neglect the non-linear convection forces present in the physiological situation, especially when the vessel is tapered. Moreover, one-dimensional wave propagation models of the arterial system can be used to provide boundary conditions for fully three-dimensional fluid-structure interaction computations that are usually defined in the time domain. In this study, a time-domain-based one-dimensional wave propagation model in a cross-sectional area, flow and pressure (A,q,p)-formulation is developed. Using this formulation, a constitutive law that includes viscoelasticity based on the mechanical behaviour of a Kelvin body, is introduced. The resulting pressure and flow waves travelling through a straight and tapered vessel are compared to experimental data obtained from measurements in an in vitro setup. The model presented shows to be well suited to predict wave propagation through these straight and tapered vessels with viscoelastic wall properties and hereto can serve as a time-domain-based method to model wave propagation in the human arterial system.     

Why is the subendocardium more vulnerable to ischemia? A new paradigm

Myocardial ischemia is transmurally heterogeneous where the subendocardium is at higher risk. Stenosis induces reduced perfusion pressure, blood flow redistribution away from the subendocardium, and consequent subendocardial vulnerability. We propose that the flow redistribution stems from the higher compliance of the subendocardial vasculature. This new paradigm was tested using network flow simulation based on measured coronary anatomy, vessel flow and mechanics, and myocardium-vessel interactions. Flow redistribution was quantified by the relative change in the subendocardial-to-subepicardial perfusion ratio under a 60-mmHg perfusion pressure reduction. Myocardial contraction was found to induce the following: 1) more compressive loading and subsequent lower transvascular pressure in deeper vessels, 2) consequent higher compliance of the subendocardial vasculature, and 3) substantial flow redistribution, i.e., a 20% drop in the subendocardial-to-subepicardial flow ratio under the prescribed reduction in perfusion pressure. This flow redistribution was found to occur primarily because the vessel compliance is nonlinear (pressure dependent). The observed thinner subendocardial vessel walls were predicted to induce a higher compliance of the subendocardial vasculature and greater flow redistribution. Subendocardial perfusion was predicted to improve with a reduction of either heart rate or left ventricular pressure under low perfusion pressure. In conclusion, subendocardial vulnerability to a acute reduction in perfusion pressure stems primarily from differences in vascular compliance induced by transmural differences in both extravascular loading and vessel wall thickness. Subendocardial ischemia can be improved by a reduction of heart rate and left ventricular pressure.     

Estimation of contrast agent concentration in intra- and extra-vascular spaces of brain tissue

This article presents a new method for estimating the leakage of a contrast agent out of a vessel. The proposed method is developed based on tissue homogeneity (TH) model, modified Patlak model, and Monte Carlo simulation. The analytical methods published in the literature estimate the contrast agent leakage by solving the coupled differential equations associated with the TH model under adiabatic conditions. These methods employ unrealistic simplifying assumptions and become intractable in their applications to the vessels that have a non-uniform permeability. Without making any unrealistic assumptions, our approach simply tracks the passage of the contrast agent through the capillary and its crossing of the vessel walls based on the blood flow in the vessel, the vessel's permeability, and the condition of the blood-brain barrier (BBB). These are treated as statistical processes that can be modeled reasonably well using the Monte Carlo method. In the proposed approach, the intra- and extra-vascular spaces are divided into multiple compartments, similar to the Patlak model. A real, measured arterial input function (AIF) is used as the capillary input and the concentration of the contrast agent is found as a function of time and distance, inside and outside of the capillary. This is done for normal and abnormal capillaries with uniform and non-uniform permeability. The proposed method generates concentration curves similar to those of the analytical method for simple AIF models. It also generates reasonable concentration curves for a real AIF. The proposed method does not fit a mathematical function to the measured AIF and does not make unrealistic simplifying assumptions. It is not therefore prone to the fitting errors and generates more realistic and more accurate results than the analytical methods.     

Power output estimation in vertical jump performed by young male soccer players

The purpose of the study was (a) to assess the accuracy of the regression equations available in the literature to estimate the actual peak power (PPac) of the countermovement jump (CMJ) executed by young male soccer players, (b) to develop new regression equations from this population, and (c) to verify whether regression equations obtained from age-based subgroups could increase the accuracy of the estimation (PPes) of PPac. In all, 117 young players (age: 13.6 ± 2.4 years) were enrolled in the study. Each subject performed 5 CMJs on a force platform. The new regression equations were obtained from the entire experimental sample (G1) and 3 age-based subsamples (G2 = prepubertal, G3 = peripubertal, G4 = postpubertal) using 2 different approaches: the best jump and the mean values achieved by each subject. All the equations in the literature underestimated the peak power (p < 0.00005) in all the groups. The approach based on the mean values was more accurate (adjusted R = 0.925, SEE = 302.9 W) than the one based on the best jump (adjusted R = 0.892; SEE = 360.8 W). Moreover, calculating the regression equations from the 3 age-based subsamples, SEE resulted improved (15.5% in G2, 5.6% in G3 and 0.9% in G4). Regression equations must be derived from homogeneous populations, in terms of gender, sports practice, and age. The approach based on the mean values for each subject was more accurate than the approach used in the literature up to now. In practical applications, regression equation estimates cannot be used to assess the performance of a single subject, because errors may exceed 50%, whereas they may be useful for group comparisons.     

Biaxial elastic material properties of porcine coronary media and adventitia

The importance of mechanical stresses and strains has become well recognized in vascular physiology and pathology. To compute the stress and strain on the various components of the vessel wall, we must know the constitutive equations for the different layers of the vessel wall. The objective of the present study is to determine the constitutive equation of the coronary artery treated as a two-layer composite: intima-media and adventitial layers. Twelve hearts were obtained from a local slaughterhouse, and the right coronary artery and left anterior descending artery were dissected free from the myocardium. The vessel wall was initially mechanically tested biaxially (inflation and axial extension) as a whole (intact wall) and subsequently as intima-media or adventitial layer. A Fung-type exponential strain energy function was used to curve fit the experimental data for the intact wall and individual layers for the right coronary artery and left anterior descending artery. Two methods were used for the determination of material constants, including the Marquardt-Levenberg nonlinear least squares method and the genetic algorithm method. Our results show that there were no statistically significant differences in the material constants obtained from the two methods and that either set of elastic constants results in good fit of the data. Furthermore, at an in vivo value of axial stretch ratio, we find that the stiffness is as follows: intima-media > intact > adventitia. These results underscore the composite nature of coronary arteries with different material properties in each layer. The present results are necessary for analysis of coronary artery mechanics and to provide a fundamental understanding of vessel physiology.     

Physical properties of resistance vessel wall in peripheral blood flow regulation--I. Mathematical model

A mathematical model is introduced to investigate the influence of the physical properties of the resistance vessel wall on the metabolic and myogenic mechanisms. The resistance vessel wall is assumed to have an elastic property and the elastic modulus to be a function of pressure (myogenic) and flow (metabolic). Blood is Poiseuille's flow. The resulting mathematical equations for pressure-flow, pressure-diameter, pressure-wall tension and pressure-wall elastic modulus relationships introduced obey Laplace's law. Poiseuille's law and Hooke's law. In comparison with the experimental data (pressure diameter), the mathematical model is confirmed to explain well the dynamic behavior of the resistance vessel wall in vivo.     

Determination of material parameters of the two-dimensional Holzapfel–Weizsäcker type model based on uniaxial extension data of arterial walls

Soft tissues are anisotropic materials yet a majority of mechanical property tests have been uniaxial, which often failed to recapitulate the tensile response in other directions. This paper aims to study the feasibility of determining material parameters of anisotropic tissues by uniaxial extension with a minimal loss of anisotropic information. We assumed that by preselecting a certain constitutive model, we could give the constitutive parameters based on uniaxial extension data from orthogonal strip samples. In our study, the Holzapfel–Weizsäcker type strain energy density function (H–W model) was used to determine the material parameters of arterial walls from two fresh donation bodies. The key points we applied were the relationships between strain components in uniaxial tensile tests and the methods of stochastic optimisation. Further numerical experiments were taken. The estimate–effect ratio, defined by the number of data with the precision of estimation less than 0.5% over whole size of data, was calculated to demonstrate the feasibility of our method. The material parameters for Chinese aorta and pulmonary artery were given with the maximum root mean square (RMS) errors 0.042, and the minimal estimate–effect ratio in numerical experiments was 90.79%. Our results suggest that the constitutive parameters of arterial walls can be determined from uniaxial extension data, given the passive mechanical behaviour governed by H–W model. This method may apply to other tissues using different constitutive models.     

Cat lung hemodynamics: comparison of experimental results and model predictions

Commonly, attempts have been made to learn about the structure and function of the pulmonary vascular bed from measurements of arterial and venous pressures and blood flow rate under steady-state conditions (e.g., from pressure vs. flow data) or dynamic conditions (e.g., from vascular occlusion data). Zhuang et al. (J. Appl. Physiol. 55: 1341-1348, 1983) have presented a detailed model of steady-state cat lung hemodynamics based on direct measurements of anatomical and elasticity data. This model provides an opportunity to better understand the information content of the hemodynamic data. Therefore, in the present study we carried out a series of steady-state and dynamic experiments on isolated cat lungs. We then compared the results with those predicted by the model. We found that the model provided a good fit to the steady-state data. However, to fit the dynamic data, some modifications were necessary to account for the viscous behavior of the vessel walls and to move the first moment of the distribution of vascular resistance toward the arterial end of the vascular bed relative to that of the distribution of vascular compliance. Due to the sensitivity of the vascular resistance to small changes in vessel diameters and branching ratio, the modifications in morphometry represent small changes in morphometric data and are probably within the range of uncertainty in such data. The modifications had little effect on the steady-state model simulations but substantially improved the dynamic model simulations, suggesting that the dynamic data are quite sensitive to small changes in the relative distributions of vessel diameters and elasticity.     

Hydraulic properties of individual xylem vessels of Fraxinus americana

Studies of the hydraulic properties of xylem vessels have been limited to measurements of whole plant or whole stem segments. This approach allows the longitudinal transport properties of the ensemble of vessels within a stem to be determined, but provides little information on radial transport. Here the xylem of Fraxinus americana L. has been examined using a new method that allows the transport properties of individual vessels to be examined. One goal of this study was to quantify transport parameters relevant to embolism repair. The longitudinal conductivity of vessel segments open at both ends (i.e. no end walls) agreed with values predicted by the Poiseuille equation. Radial specific conductance (conductance per unit area) was approximately six orders of magnitude lower than the longitudinal conductance of the vessel segment normalized by the cross-sectional area of the vessel lumen. There was a step increase in the radial specific conductance of previously gas-filled vessels when the delivery pressure exceeded 0.4 MPa. This is consistent with the idea that positive pressure, required for embolism repair, can be compartmentalized within a vessel if the bordered pit chambers are gas-filled. The diffusion coefficient for the movement of gas from a pressurized air-filled vessel was of the same order of magnitude as that for air diffusing through water (1.95 e(-9) m(2) s(-1)). Estimates of the time needed to displace all of the gas from an air-filled vessel were in the order of 20 min, suggesting that gas removal may not be a major limitation in embolism repair.