Computer simulation of intraventricular flow and pressure gradients during diastole

A two-dimensional axisymmetric computer model is developed for the simulation of the filling flow in the left ventricle (LV). The computed results show that vortices are formed during the acceleration phases of the filling waves. During the deceleration phases these are amplified and convected into the ventricle. The ratio of the maximal blood velocity at the mitral valve (peak E velocity) to the flow wave propagation velocity (WPV) of the filling wave is larger than 1. This hemodynamic behavior is also observed in experiments in vitro (Steen and Steen, 1994, Cardiovasc. Res., 28, pp. 1821-1827) and in measurements in vivo with color M-mode Doppler echocardiography (Stugaard et al., 1994, J. Am. Coll. Cardiol., 24, 663-670). Computed intraventricular pressure profiles are similar to observed profiles in a dog heart (Courtois et al., 1988, Circulation, 78, pp. 661-671). The long-term goal of the computer model is to study the predictive value of noninvasive parameters (e.g., velocities measured with Doppler echocardiography) on invasive parameters (e.g., pressures, stiffness of cardiac wall, time constant of relaxation). Here, we show that higher LV stiffness results in a smaller WPV for a given peak E velocity. This result may indicate an inverse relationship between WPV and LV stiffness, suggesting that WPV may be an important noninvasive index to assess LV diastolic stiffness, LV diastolic pressure and thus atrial pressure (preload).     

Microcomputer analysis of the intraventricular pressure curve of the isolated rat heart

The authors evolved an automatic system for the measurement and analysis of the course of pressure in the left ventricle of the isolated laboratory rat heart perfused in vitro after Langendorff. Attached to it is a microcomputer which automatically samples the course of the left ventricular pressure curve over a segment comprising five cardiac cycles and, within 30 s, evaluates the mean systolic and diastolic pressure, the maximum rate of the increase and decrease in pressure, the contractility index, the mean integral pressure and the heart rate. The apparatus shows a standard error of less than 1% for pressure and of up to 2% for derivation.     

Systems engineering analysis of aortic root blood pressure

This paper describes the aortic blood pressure as a function of aortic blood flow and the parameters of the blood and circulatory system. The method of performance involves the analogue of a multi-branched electrical to hydraulic transmission line applying graphical convolution to the blood flow-transform impedance relationship resulting in a theoretical pressure curve for the infinite aorta. The difference between the single pressure pulse and the computed adjusted infinite aorta pressure curve is described as the reflected wave. This reflected wave is then shown to be of reasonable configuration in time and velocity. The blood pressure is thus finally described completely by the physical parameters of the blood and the circulatory system and the blood flow.     

Mouth pressure curve on abrupt interruption of airflow during forced expiration

An attempt was made to investigate how the mouth pressure curve represents the process of air flowing into the collapsed segment downstream to the choke point when the airflow is abruptly interrupted at the mouth during forced expiration. Immediately after the interruption of airflow, the mouth pressure suddenly increased (phase 1), followed by a slower rise in pressure (phase 2) within approximately 100 ms until the pressure reached the alveolar pressure. The pleural and alveolar pressures remained constant during this process. The first phase of the abrupt rise represented the pressure induced by the instantaneous interruption of the airflow itself. Analysis of the supramaximal flow (Vsupramax) observed after resumption of the airflow suggested that the choke point remained constant during the second phase of the mouth pressure after interruption of maximal flow (Vmax). From these results, examination of the second phase of the mouth pressure curve may provide useful information about the downstream segment of the airway.     

Mathematical model analysis of pressure pulse propagation of the aortic wall

A mathematical blood vessel mode is established of a semi-infinite, fluid filled, cylindrical, elastic tube where a pressure wave is propagated in the fluid. This model has assigned to it physical values of the aorta and is subject to a pressure pulse at the origin of the same configuration and magnitude as that observed in the human. The resulting mathematical solution demonstrates wall displacement in agreement with observed phenomena of (a) a high frequency component analogous to cardiovascular sounds; (b) a low frequency component resembling the pressure pulse analogous to the pulsatile movement of the vessel wall.     

An elongation model of left ventricle deformation in diastole

A numerical method of the left ventricle (LV) deformation, an elongation model, was put forth for the study of LV fluid mechanics in diastole. The LV elongated only along the apical axis, and the motion was controlled by the intraventricular flow rate. Two other LV models, a fixed control volume model and a dilation model, were also used for model comparison and the study of LV fluid mechanics. For clinical sphere indices (SIs, between 1.0 and 2.0), the three models showed little difference in pressure and velocity distributions along the apical axis at E-peak. The energy dissipation was lower at a larger SI in that the jet and vortex development was less limited by the LV cavity in the apical direction. LV deformation of apical elongation may represent the primary feature of LV deformation in comparison with the secondary radial expansion. The elongation model of the LV deformation with an appropriate SI is a reasonable, simple method to study LV fluid mechanics in diastole.     

An analysis of aortic pressure curves taking into account visco-elastic properties of the aorta and variations in peripheral resistance

The aortic pressure curve necessarily reveals the mechanical properties of the aorta and peripheral resistance as well as of the dynamics of blood flow. The present study uses a reasonable model of visco-elastic properties of the aorta, a reasonable form for variations in peripheral resistance and blood flow to predict an aortic pressure tracing. Numerical values of constants measured experimentally were available in the published literature. These were used in the nonlinear differential equations of motion of the system under analysis. The equations yielded to piece-wise solution, giving the aortic circumference and the aortic pressure as functions of time. The form of both curves resembles clinical tracings, but numerical values of circumference were higher and of pressure lower thanin vivo. The discrepancies between predicted and clinical curves may reveal certain inadequacies in published measurements on visco-elastic constants. These measurements have been made on longitudinal rather than circumferential strips often containing dead rather than living muscle. The discrepancies, therefore, indicate specific gaps in our knowledge of aortic behaviorin vitro. The suggested model of the system aided in the design of experiments which could supply data necessary to substantiate or to revise the model.     

Determinants of the occurrence of vortex rings in the left ventricle during diastole

This study employs classical inviscid fluid dynamics theory to investigate whether LV diastolic inflow volume and the size of the LV play a role in vortex ring formation. Fluid injection across an orifice into a large container results in the generation of a vortex ring having a constant size and speed. Relations between the vortex size and speed and the injection were obtained by applying conservation laws regarding kinetic energy, impulse and vorticity; the initial state was computed using a bolus injection model, and the final state by using the Kelvin vortex model. An important parameter in the equations is the relative injection length, i.e., the ratio of the length of the injected bolus and the radius of the orifice (L/R). Its estimated highest value in man, L/R = 15, produces a rather thick vortex ring (relative thickness 0.77). Comparable results following from the Hill vortex model convinced us that the Kelvin vortex model can be applied in the whole range of injection lengths in the human left ventricle. In an in vitro model it is shown experimentally that vortex rings can be generated for L/R in the range from 2 to 16. The measured traveling speed of the vortex ring is in fair agreement with the theory, as well as the ring radius for large injections. A vortex ring located in a narrow channel cannot reach its proper traveling speed. The method of images is used to estimate the speed reduction of vortex rings within a cylinder. It turns out that propagation of vortex rings is possible when the ratio of orifice to cylinder radius is less than about 0.5.(ABSTRACT TRUNCATED AT 250 WORDS)     

Structural finite deformation model of the left ventricle during diastole and systole

A model of left ventricular function is developed based on morphological characteristics of the myocardial tissue. The passive response of the three-dimensional collagen network and the active contribution of the muscle fibers are integrated to yield the overall response of the left ventricle which is considered to be a thick wall cylinder. The deformation field and the distributions of stress and pressure are determined at each point in the cardiac cycle by numerically solving three equations of equilibrium. Simulated results in terms of the ventricular deformation during ejection and isovolumic cycles are shown to be in good qualitative agreement with experimental data. It is shown that the collagen network in the heart has considerable effect on the pressure-volume loops. The particular pattern of spatial orientation of the collagen determines the ventricular recoil properties in early diastole. The material properties (myocardial stiffness and contractility) are shown to affect both the pressure-volume loop and the deformation pattern of the ventricle. The results indicate that microstructural consideration offer a realistic representation of the left ventricle mechanics.     

Mathematical analysis of the heart rate performance curve during incremental exercise testing

In this study we performed laboratory treadmill protocols of increasing load. Heart rate was continuously recorded and blood lactate concentration was measured for determination of lactate threshold by means of LTD-max and LT4.0 methods.Our results indicate that the shape of heart rate performance curve (HRPC) during incremental testing depends on the applied exercise protocol (change of initial speed and the step of running speed increase, with the constant stage duration). Depending on the applied protocol, the HRPC can be described by linear, polynomial (S-shaped), and exponential mathematical expression.We presented mathematical procedure for estimation of heart rate threshold points at the level of LTD-max and LT4.0, by means of exponential curve and its relative deflection from the initial trend line (tangent line to exponential curve at the point of starting heart rate). The relative deflection of exponential curve from the initial trend line at the level of LTD-max and/or LT4.0 can be defined, based on the slope of the initial trend line. Using originally developed software that allows mathematical analysis of heart rate-load relation, LTD-max and/or LT4.0 can be estimated without direct measurement of blood lactate concentration.     

Mean aortic pressure is the geometric mean of systolic and diastolic aortic pressure in resting humans.

The aim of our study was twofold: 1) to establish a mathematical link between mean aortic pressure (MAP) and systolic (SAP) and diastolic aortic pressures (DAP) by testing the hypothesis that either the geometric mean or the harmonic mean of SAP and DAP were reliable MAP estimates; and 2) to critically evaluate three empirical formulas recently proposed to estimate MAP. High-fidelity pressures were recorded at rest at the aortic root level in controls (n = 31) and in subjects with various forms of cardiovascular diseases (n = 108). The time-averaged MAP and the pulse pressure (PP = SAP - DAP) were calculated. The MAP ranged from 66 to 160 mmHg [mean = 107.9 mmHg (SD 18.2)]. The geometric mean, i.e., the square root of the product of SAP and DAP, furnished a reliable estimate of MAP [mean bias = 0.3 mmHg (SD 2.7)]. The harmonic mean was inaccurate. The following MAP formulas were also tested: DAP + 0.412 PP (Meaney E, Alva F, Meaney A, Alva J, and Webel R. Heart 84: 64, 2000), DAP + 0.33 PP + 5 mmHg [Chemla D, Hébert JL, Aptecar E, Mazoit JX, Zamani K, Frank R, Fontaine G, Nitenberg A, and Lecarpentier Y. Clin Sci (Lond) 103: 7-13, 2002], and DAP + [0.33 + (heart rate x 0.0012)] PP (Razminia M, Trivedi A, Molnar J, Elbzour M, Guerrero M, Salem Y, Ahmed A, Khosla S, Lubell DL. Catheter Cardiovasc Interv 63: 419-425, 2004). They all provided accurate and precise estimates of MAP [mean bias = -0.2 (SD 2.9), -0.3 (SD 2.7), and 0.1 mmHg (SD 2.9), respectively]. The implications of the geometric mean pressure strictly pertained to the central (not peripheral) level. It was demonstrated that the fractional systolic (SAP/MAP) and diastolic (DAP/MAP) pressures were reciprocal estimates of aortic pulsatility and that the SAP times DAP product matched the total peripheral resistance times cardiac power product. In conclusion, although the previously described thumb-rules applied, the "geometric MAP" appears more valuable as it established a simple mathematical link between the steady and pulsatile component of aortic pressure.