Flow field evolution in a laboratory model of the left ventricle

J Vis DOI 10.1007/s12650-013-0179-9 R E G UL A R P A P E R S. Espa • S. Fortini • G. Querzoli • A. Cenedese Flow field evolution in a laboratory model of the left ventricle Received: 6 February 2013 / Revised: 29 May 2013 / Accepted: 29 July 2013 Ó The Visualization Society of Japan 2013 Abstract An experimental apparatus simulating the velocity field downstream of an artificial heart valve in a model of the left ventricle was developed. In particular, the cardiac cycle was reproduced by a linear motor such that both healthy and unhealthy conditions could be reproduced. In this study, a one-way, hydraulic valve has been inserted in a mitralic orifice and a strong, uniform-velocity jet which enters the ventricle is generated. This condition is very similar to the inflow generated by the natural valve. To measure the velocity field the working fluid was seeded with tracer particles and the test section, the middle plane of the ventricle model, was lit by a laser light sheet. A long time image sequence of the flow field was acquired by a high-speed camera. The Feature Tracking image analysis technique was used to quantify particle displacements and as a result a Lagrangian description of the fluid trajectories were obtained. The evolution of particle trajectories and the velocity fields during the whole cardiac cycle is presented here. The Eulerian velocity field were also evaluated to characterize the flow. Finally, the viscous shear stresses were analyzed, in order to compute the quantities that are considered to be the main cause of the blood cells damage. Keywords Left ventricle  Heart prosthetic valves  Image analysis  Feature tracking 1 Introduction Mechanical or biological heart valve prostheses are commonly used to replace the native ones, if damaged, in order to re-establish the cardiovascular system functionality. It is well known that this replacement causes major changes in the ventricular flow which, in turn, affect the functionality of the heart as a pump and possibly cause undesired side effects (Rodgers and Sabitson 1969; Garcia et al. 1996; Ismeno et al. 1999; Alemu and Bluestein 2007). Continuous improvements in surgical techniques and the variety of existing prosthetic valves (Vukicevic et al. 2012) encourage valve replacement in a wider and wider range of situations. Several models of valves have been designed to reproduce flow conditions as closely as possible to healthy cases: trying to reduce the shear stress values that are related to blood cell damage, i.e., hemolysis (Grigioni et al. 2002; Ismeno et al. 1999), to avoid a blood stagnation point (that possibly causes thromboembolism) and to ensure the correct recirculation of the flow inside the ventricle which avoids high energy losses (Kilner et al. 2000). The physical phenomena that can induce these problems are not only related to the characteristics of the mean flow, but also to the features of the vorticity field as well as to the inner structure of turbulence. As a consequence, only accurate in vitro measurements performed in laboratory S. Espa  S. Fortini (&)  A. Cenedese DICEA, Sapienza Università di Roma, via Eudossiana 18, 00184 Rome, Italy E-mail: stefania.fortini@uniroma1.it G. Querzoli DICAA, Università di Cagliari, Via Marengo 3, 09123 Cagliari, Italy S. Espa et al. Fig. 1 Non-dimensional ventricular volume DV (red line, ml) and flow-rate Q (blue line, ml/s). Q(t) is multiplied by 10 models can give an insight into a description of the effects induced by a heart valve prosthesis on the downstream flow (Reul et al. 1981). The experimental setup described here was developed to characterize and quantify these effects by simulating the behavior of a real cardiac cycle. The variation law of the ventricular volume V, DV(t) = [V(t) - V(0)]/SV (i.e., the difference between the maximum and the minimum ventricle volume, normalized by the stroke volume SV) as a function of non-dimensional time (defined as t/T, being T the cardiac cycle period) is plotted in Fig. 1 (red curve). It is the speed input-signal that controls the motor and it has been obtained by digitizing a real tracing of an echocardiograph, scaling it with the period and the stroke volume of the cardiac cycle, and finally taking the temporal derivative (Q(t), blue curve in Fig. 1). Q(t) represents also the flow rate through the valve during the cycle, namely, the flow is through the mitral valve during the diastole (0.00T–0.75T), characterized by the E and A waves, and through the aortic valve during the systole (0.75T–1.00T). A laboratory model is in similarity if the control parameters of the reproduced phenomenon, in this case Reynolds (Re) and Womersley (Wo) numbers, match with those of the real conditions: sffiffiffiffiffi UD D2 Wo ¼ Re ¼ m Tm where m is the kinematic viscosity of blood, D (the diameter of the ventricle) is the length scale and U (the peak velocity through the mitral valve) is the velocity scale. In the present experiment, the geometrical scale is the same as the real one, and the working fluid is water. The unsteady blood flows are non-Newtonian; however, in large vessels (diameter [0.5 mm) and in the cardiac chambers the relationship between the shear stress and shear strain rate can be considered linear, being the dynamic viscosity the constant of proportionality. As a matter of fact, since the shear strain rate rise to about 100 s-1, dynamic viscosity becomes about constant and the blood can be assumed as a Newtonian fluid (Kitajima and Yoganathan 2005). Almost the whole literature, in which the large vessels (aorta) and the cardiac chambers were simulated using mathematical/numerical/experimental approaches, is based on this assumption (Cenedese et al. 2005; Pedrizzetti and Domenichini 2005; Song et al. 1994). 2 Experimental setup The runs were performed in the apparatus shown in Fig. 2, already used for previous investigations and described in detail in Cenedese et al. (2005), Querzoli et al. (2010), Espa et al. (2012) and Vukicevic et al. (2012). A flexible, transparent sack made of silicone rubber simulates the left ventricle and allows for optical access. The sack is secured on a circular plate, connected by means of two Plexiglas conduits to a constant head reservoir. The conduit configuration and the valve opening degree are designed to simulate the circulatory system. A rectangular tank (A) with transparent Plexiglas walls filled with water houses the leftventricle model. The volume of the ventricle is changed by the movement of a piston (D) that is located on one side of the tank. The piston is driven by a linear motor (C), controlled by a speed-feedback servo- Flow field evolution in a laboratory model of the left ventricle Fig. 2 Experimental setup. A ventricle chamber, B laser; C motor, D piston, E pressure transducer, F fast camera control. The fluid inside the ventricle is seeded with micro-glass particles of 30 lm in diameter and illuminated by a laser light sheet (B) in the vertical plane xz. Images of the swirling motion are acquired and recorded by a high-speed camera (F, Mikrotron EoSense 1362) with a frame rate of 250 frames/s. The frame grabber (DVR-Express Core, IO Industries), connected to the camera, record directly onto a hard-disk array (about 1 Tb). Images were analyzed by an Image Analysis Technique called Feature Tracking (Cenedese et al. 2005) that allows one to track the particles’ positions during every cardiac cycle. The sparse Lagrangian velocity field obtained after trajectories reconstruction is then interpolated over a regular grid to obtain the Eulerian 2 components velocity field and derived quantities. 3 Results 3.1 Trajectories Evolution In this study, a one-way, hydraulic valve has been inserted along the pipe connected to the tank, few centimeters before the ventricle inlet. Therefore, during the diastole, a nearly uniform flow enters the ventricle. This condition is not so different from the inflow generated by the natural valve, which generates a single, regular jet, as well. In the experiments described here the flow-rate curve is plotted in Fig. 1 (blue line); the period is 6 s and the stroke volume is 64 ml in order to obtain the same values of the non-dimensional numbers, Reynolds and Womersley, as the physiological range. Figure 3 shows a snapshot of particle tracer trajectories during a cardiac cycle, obtained by tracing the particle centroids for 20 consecutive instants (spaced in time of 1/250 s): different colors represent different acquisition instants, in particular the blue one is the first instant while the red one is the last. During the diastole, in correspondence with the mitral valve maximum opening (E peak, t/T = 0.174, Fig. 3a), an axial symmetric vortex ring is generated at the edge of the jet that enters the ventricle. The vortex ring symmetry disappears in the next instants, due to the interaction of the jet with the ventricle wall which reduces its velocity with respect to the part of the vortex ring in the middle of the ventricle. This aspect is evident in Fig. 3b (t/T = 0.220, end of the E-wave) and during the diastasis when the ventricle volume is constant, i.e., Figure 3c (t/T = 0.400) and 3d (t/T = 0.500). At t/T = 0.680 (Fig. 3e), before the A-wave, the vortex ring reaches the ventricle apex, the coherent structures are broken into irregular smaller and lower structures. Figure 3f shows the systolic phase, when the fluid is ejected from the ventricle through the aortic valve. About 6,000 particles have been tracked every instant. By acquiring several cardiac cycles it was possible to perform a conditional sampling (phase resampling) over a regular grid, in order to evaluate the different order moments and the correlations at different instants of the cardiac cycle. S. Espa et al. Fig. 3 a Particle tracer trajectories at different instants of the cardiac cycle. b Particle tracer trajectories at different instants of the cardiac cycle. c Particle tracer trajectories at different instants of the cardiac cycle. d Particle tracer trajectories at different instants of the cardiac cycle. e Particle tracer trajectories at different instants of the cardiac cycle f Particle tracer trajectories at different instants of the cardiac cycle Flow field evolution in a laboratory model of the left ventricle 3.2 Velocity and vorticity fields The description of time evolution of the phenomenon with uniform flow, at 64 ml stroke volume and 6 s period, is shown in Fig. 4, where phase-averaged on a regular grid over 50 cardiac cycles in terms of velocity and vorticity fields are plotted. The nine most meaningful stages were chosen and each figure reports non-dimensional time with respect to cycle period and, at bottom left, the flow-rate plot with a red dot indicating the current instant. In Fig. 4 the first field (t/T = 0.150) corresponds to the condition just before the first diastolic peak (E-wave). The entering jet is well visible, and a vorticity layer begins to develop around the jet. At t/ T = 0.177, corresponding to the E-wave peak, the vorticity layer at the jet edges has begun to roll itself up to create a vortex ring which moves through the ventricle main axis. Two peaks there are observed in the vorticity map (A and B) and the measured vorticity reaches its maximum value. At the t/T = 0.210 the vortex B begins to interact with the ventricle wall; the non-slip condition effect creates a negative vortex layer at the wall which induces a slowing down of vortex B downwards, and an increase in the dissipating effects due to high velocity gradients. Vortex A, instead, continues freely developing through the ventricle. At the end of E-wave (t/T = 0.244), the vortex ring separates from the vorticity layer generated by the jet which follows it (the so-called pinch-off). Vortex B, which is slower, covered a shorter distance, while Fig. 4 Velocity field (black arrows) and vorticity (scale of color) in the case of 64 ml stroke volume and simulation period of 6s S. Espa et al. vortex A has now propagated till the center of the ventricle, besides becoming sensitively larger than vortex B. At t/T = 0.310 vortex A reaches the bottom wall of the ventricle in proximity of the ventricle apex, its dimension being still increased, while its maximum velocity tends to reduce; even the vortex B has come up against the wall and begins to go back upwards, while its dimensions tend to diminish. At t/T = 0.450 the field corresponds to the interval during which the ventricle volume remains constant. In this stage the structures, which have been generated during the first diastolic peak, almost completely vanish; both velocity and vorticity fields do not have meaningful features except for the presence of vortex A, extremely weak, which remains in proximity of the ventricle apex. At t/T = 0.640 there is the second diastolic peak, due to the atrium contraction (A-wave); a phenomenon similar to the previous one is observed, i.e., the beginning of forming of a second vortex ring (C and D peaks in vorticity map), which is less intense. At last, during the systolic peak (t/T = 0.847), any flow structure is destroyed and an exit flow is generated; there is still a partial memory of diastolic flow: it is possible to recognize a main circulation upwards along the walls and to the aortic valve. A similar phenomenon can even be observed at t/T = 0.954 showing the final stage of systolic wave. 3.3 Viscous shear stresses It was reported in literature that the quantities more likely related to blood cells damages are the shear stresses (Grigioni et al. 1999, 2002), calculated as (Cenedese et al. 2005): smax ¼ ðs2  s1 Þ ¼ lðe2  e1 Þ 2 where si and ei are the eigenvalues of the stress tensor and the strain velocity tensor, respectively, l is the flow viscosity and the expression is in their common, principal reference frame. To evaluate the stresses, 50 cardiac cycles acquired with the high-speed camera have been phaseaveraged. As result only one data matrix (N 9 N 9 time) representative of the cardiac cycle has been obtained. The velocity gradient is then computed from the gridded averaged velocity at each node; after the eigenvalues of its symmetric part, i.e., the strain rate tensor, are computed. Then, the maximum shear stress has been evaluated in a region immediately downstream the mitral valve and made dimensionless by means of the scale qU 2 , where q is the flow density, U is the velocity scale: U¼ Qmax 1 1  SV   A T Qmed Q is the flow rate, A is the area or the mitral orifice, T and SV the period and the stroke volume of the cardiac cycle, respectively. In Fig. 5 the spatial distribution of the shear stresses, evaluated from the averaged velocity field, is shown. The stroke volume is 64 ml, whereas the period of the cardiac cycle is 6 s. In the three plots (a, b and c) the higher shear stresses are located at the jet lateral boundary, along the vortical layer linking the mitralic orifice boundaries to the ring vortex (plots a and b). At t/T = 0.190, a high value zone begins to develop around the vortex ring. At t/T = 0.234 (c), shear stresses high values surround Fig. 5 Maximum viscous shear stresses at stroke volume 64 ml and simulation period T 6 s in presence of the hydraulic valve in mitral position Flow field evolution in a laboratory model of the left ventricle Fig. 6 Average non-dimensional shear stresses during the cardiac cycle at stroke volume 64, 80 ml and simulation period T 6s the vortical ring, even if a low intensity line is observed: this line joins the two vortexes centers. That line is visible during the whole evolution of the ring vortex, until its interaction with the ventricular boundary. The time evolution of the averaged, no dimensional shear stress is shown in Fig. 6 for two different test conditions, 64 and 80 ml stroke volumes (95 % confidence interval mean values within 2–2.2 % for the case SV 64 ml and 2.3–2.5 % with SV 80 ml). The two curves have two relative maxima at more or less the same instants: the first peak is the highest and corresponds to the E-wave, i.e., the situation described in the maps of Fig. 4 at t/T = 0.177. The second peak is related to the converging flow through the aortic valve, during the systole. The main variations in magnitude occur at the first peak. 4 Conclusions An experimental model of the left ventricle allowed to perform a detailed analysis of the flow during the cardiac cycle. The model was developed in order to guarantee the optical access within the test section. An image analysis technique, called Feature Tracking, was employed to measure the particle displacements and then the velocity fields. The uniform inflow condition was analyzed, the temporal evolution of the Lagrangian trajectories and both velocity and vorticity fields were taken into account. The instantaneous vortex structures were detected as well. In particular, for each diastolic wave, a vortex ring is generated. The vortex ring is associated to two vorticity peaks on the measure plane, moving from mitral orifice toward the ventricle apex. One peak near the boundary tends to disappear because of viscosity while the other spreads out till the entire apical zone is occupied. The viscous shear stresses have been computed, due to the fact that these quantities are assumed to be the main cause of the blood cells damage. These results show how the reorganization of the flow in a single vortex during the diastole phase arranges and directs the velocities towards the aortic outlet just before the beginning of the systole, also minimizing the volume residing inside the ventricle during the cardiac cycle. This evidence is qualitatively well described by the trajectories evolution of the flow inside the left-ventricle model. 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