Blood flow induced wall stress in the left ventricle of the heart

Transcription

1 Blood flow induced wall stress in the left ventricle of the heart A. K. Macpherson 1, S. Neti 1, J. A. Mannisi 2 & P. A. Macpherson 3 1 Institute for Biomedical Engineering and Mathematical Biology, Lehigh University, USA 2 Lehigh Valley Cardiologists, Bethlehem, USA 3 Department of Applied Technology, Rogers State University, USA Abstract Mechanical stresses on the walls (endocardium) of the heart (myocardium), play an important role in the development of heart disease. One tool used by cardiologists in making a diagnosis is the echocardiogram. An echocardiogram uses ultrasonic pulses to produce images of the changes of shape of the left ventricle of the heart in two dimensions over time. The present research involved the development of software to automatically calculate wall stress maps and eventually provide a graphical representation as a diagnostic tool for a cardiologist. The fluid mechanics program [1] has been described previously. The program uses the shapes of the ventricle at various times as well as the three dimensional data to calculate the velocities of the blood on a Cartesian grid. The calculations are undertaken at the macroscopic scale as well as the microscopic scale. The stress applied to the surface is resolved into two components: a shear force parallel to the surface and another normal to it. The later is important for the penetration of antagonists into the myocardium. The flow forces can be calculated from the blood flow patterns by considering the velocity gradient and the dynamic head close to the wall. Six patients were studied including both healthy and diseased hearts. It was found that the maximum shear forces are similar in both healthy patients and patients suffering from heart disease. The difference between the groups is the shear stress gradients which are known to produce gene modification in the endothelium. The peak stress is greatest near the apex of the heart. The low level stress is commonly located around the entire perimeter of the left ventricle and is much greater in diseased hearts than healthy ones. It is proposed that the long-term exposure to low stress is a major factor in gene reprogramming.

2 322 Advances in Fluid Mechanics V 1 Introduction Mechanical stresses on the walls (endocardium) of the heart (myocardium), play an important role in the development of heart disease. It is considered to be the trigger inducing a growth response in the overloaded myocardium [1]. The effect of shear stress gradients on the lining of arteries and veins (vascular endothelium cells) has been shown to modulate endothelial gene expression [2]. It is known that gene upregulation in heart cells is responsible for heart disease due to angiotensin II [3]. The specific genes that are modified are NF-κB, Erg-1, c-jun and c-fos. These genes are known to be modified in heart disease. It is reasonable to assume that a similar mechanism will occur in the heart enthdothelium. One important tool used by cardiologists in making a diagnosis is the echocardiogram. An echocardiogram uses ultrasonic pulses to produce a sequence of images to depict the changes of shape of left ventricle of the heart in two dimensions over time. Other data produced are the velocities at various locations within the heart as well as views in several directions. It would be possible to undertake an exact calculation of the flow within the left ventricle using available computational power. The aim of the research is to produce stress maps automatically without human intervention. The results must be produced in a reasonable time period to be useful to the cardiologist. Thus it is necessary to sacrifice accuracy as well as some details obtainable from the echocardiographs in order to automate the process. For example in table 1, only values which could be machine read are used in the calculations. With such calculations of the change of the left ventricle shape over time, the stress on the heart walls can be calculated for a given ventricular shape. This will provide a cardiologist with additional quantitative information to consider in the patient diagnosis. 2 Method of calculation The general method of calculation used here has been described previously [4]. In the solution, the blood flow into the left atrium is simulated by a source distributed throughout the atrium. In order to conserve mass, sinks are distributed around the periphery of the integration domain. The change in shape is obtained from the echocardiograms and used as boundary conditions for the flow. The source strength has to match the change in volume of the ventricle. The valves have to be modelled as thicker than in reality as Lagrangian integration must go around both sides of the valve. The Navier-Stokes equations are then solved with a predictor corrector scheme [4]. The Navier-Stokes equations defined on an x-y Cartesian co-ordinate system for an incompressible fluid are u ˆ ρ + uˆ uˆ + p = µ 2 uˆ Fˆ t + (1) u ˆ = 0 (2)

3 Advances in Fluid Mechanics V 323 where û is the velocity vector, ρ is the density, t is the time, p is the pressure and the viscosity is µ. The boundary force Fˆ arising from the heart muscles is L Fˆ ( xˆ, t) = fˆ ( s, t) δ ( xˆ Xˆ ( s, t) )ds (3) 0 Here fˆ is the force on the boundary element at the point s defined on a n Lagrangian system where xˆ is defined on the Cartesian system and Xˆ is the nth point on the Lagrangian system The flow velocities and pressures can be used to calculate the stresses on the surface of the heart walls. These forces can then be used to examine the microscopic interaction with the cells in the heart wall (endocardium). The first step in the solution is to obtain the shape of the ventricle at various times. This is often difficult as echocardiogram images are sometimes indistinct. Following a method often used by echocardiographers, only four images in a cardiac cycle were selected at specific points: when the valves were closed, when the valves were fully open, just before the atrium starts to contract and finally at the end of the ventricle filling (diastole) stage. A linear time variation was assumed between each frame. It was also assumed that the motion of the wall would be normal to the surface. As described below, the required times for valve opening and atrium contraction can be obtained from Doppler measurements of the velocity through the mitral valve. The shape derived from the echocardiogram contained many irregularities which required special processing. The echocardiogram tracing was obtained first as a digital image. A least squares curve fit was undertaken to quantify the ventricular shapes at various times using quadratic forms where the constants were polynomials fitted at each digital point. If the source was allowed to start while the valves were closed then the program would fail due to excessive pressure. Similarly the wall could not be allowed to move until the source also started. Thus an initial short period was required without source or wall motion to allow the valves to start opening (these events are independent of fluid motion are dependent on cardiac electrical signals). The second step required the simulation of the atrium. The atrium changes shape during the diastole stage and thus changes the pressure. However the use of a source in place of the correct inflow pattern to the atrium was an artifice which made the actual atrium shape unimportant. The atrium shape was fixed at near hemispherical shape with valves in the closed and early open positions. After some time the atrium contracts for a period before the mitral valve closed. The shape was expanded and contracted as required for the different sized mitral valves. The source strength was increased slowly as the valves opened in accordance with the increase in volume of the ventricle. Once the calculation of the flow velocities and pressures was completed, the stresses at the walls could then be obtained. In accordance with the aim of the research, the evaluation of wall stresses at the boundary layer had to be modelled properly. Two points were chosen as close to the wall as possible along a line

4 324 Advances in Fluid Mechanics V normal to the surface. A finite difference method was used to obtain the derivative of the velocity along this line. Similarly the velocity normal to the wall was calculated along the same line. As only pressure gradients are used in the calculations, an arbitrary constant was added to the pressure to make it relative to atmospheric pressure. The microscopic calculation of the blood involving the effects of angiotensin II contained in the blood and on the cells of the myocardium will not be discussed here. It is necessary to have an appropriate length scale between the continuum calculation and the above microscopic scale. This is undertaken using a Monte Carlo method. These details are presented elsewhere [4]. The basic process of describing the effects of ang II starts with the Landau equation which for the test particle takes the form below and has been described as a generalized diffusion equation in velocity space by Chandrasekhar [5]. Expressed in a non-dimensional form it becomes [6] φ τ = v ( Fr strs)φ (4) r where φ is the velocity distribution, the v r differentiation is with respect to nondimensional velocity v/2kt, subscript τ is differentiation with respect to the non-dimensional time defined below. The solution is obtained in terms of the drag force F r and a random forcet rs. F r = 8v 1 G( v) v r (5) 1 3 T rs = 2v H ( v) δ rs + 2v E( v) vrvs (6) The movement of the blood components assumes they are sufficiently far apart so that collisions between the components will not occur. This is the usual assumption made for the application of the Landau equation. Under these circumstances the force on an ion will consist of a drag due to G(v) and a random force due to H(v). 3 Results Tracings from Doppler measurements of velocity through the mitral valve as functions of time (in the cardiac cycle) are shown in Figures 1 and 2. The labelling of the patients indicates male (M) or female (F) and a number indicating the age. As discussed above, the only values used are those that can be machine read automatically. Many features of the tracings that could be used in an extensive calculation and improve the accuracy are considered not to be machine readable. The timings of the various events are some of the most important differences between hearts in different patients. The values obtained from the echocardiographs are discussed in detail to show how the values vary between healthy and diseased patients. The eventual use of the procedure is for the diagnoses of patients before extensive heart disease has occurred. Thus the scheme does not work well on patients who are obviously ill. Table 1 shows the times from the start of diastole used in the calculations. The tracings in Figure 1 for patient M53 indicate slow pulse rate, rapid rise of velocity due to the valve

5 Advances in Fluid Mechanics V 325 opening (early diastolic velocity E) with regular shapes of peaks. The increase in velocity due to contraction of the atrium (atrium reversal velocity A) is less than E velocity (associated with mitral valve opening) and the patient overall appears to be in good health. These results (in Figure 1) are used as a standard to analyze the other results from Doppler measurements. Due to space limitations, Figure 2 only shows the results for the two patients with heart trouble. Patient F39 has an E velocity slightly higher than patient M53 and a significantly lower A peak velocity. This is a healthy feature. Patient M78 has been diagnosed to have a slightly enlarged heart. The E velocity through the mitral valve was slightly lower than patient M53 and the filling rate was slower than that for patient M53. This is possibly due to the stiffening of the myocardium. The initial valve opening velocities (A peak) for patient F73 are normal though the E peak velocity is slightly lower than that of patient M53. The pressure to be exerted by the atrium for patient F73 is a higher than that for patient M53. The patient is known to have a leaking mitral valve. The Doppler tracing of patient F63 was generated just after congestive heart failure. The velocity tracing pattern is very irregular with an E peak higher than the A peak. The atrium is required to provide much more pumping effort than the initial inflow. Patient M72 underwent cardiovascular surgery shortly after the echocardiogram below was taken. For M72, obviously the atrium is generating much of the pumping force. Table 1: Time from the start of the diastole to the events shown in Figures 1 and 2. Patient End of Diastole Secs First Vel. Peak E secs Start atrium Contraction A secs Second Vel. Peak A secs Time between Beats secs M F M F F M The time till the mitral valve is fully open is very important as a rapidly opening valve generates strong vortices as shown in Figure 3a. This time is related to the time to the E peak and it was found that a value of 1.5 times the E time appeared to produce correct results. It can be seen from table 1 that the mitral valve of the very ill patient M72 took a long time to open. The atrium contraction occurs till the peak value at A is reached and is shorter in patients with diseased hearts. The flow pattern for M53 just after atrium contraction is shown in Figure 3b. It can be seen that the blood floods both the atrium and the ventricle without forming strong vortices. This feature and the significance of the last column will be discussed below.

6 326 Advances in Fluid Mechanics V Figure 1: Doppler velocity measurements through the mitral valve for patient M53. Patient F63 Patient M72 Figure 2: Doppler velocity measurements as in Figure 1 for two patients. Axes captions are as in Figure 1 and dimensional scale as in table 1. The calculated mitral valve velocities corresponding for patient M53 and F63 are shown in Figure 4. The peak velocity is a function of the source strength and could be adjusted for particular case if necessary. The general shapes are in accordance with Figures 1 and 2. One feature that is significantly different is that the A peak in the case of F63 is too low and E peak possibly too high. It can be seen from Figure 3 that the profiles are irregular as the next beat is higher than the first and has an irregular top. The lower A peak is likely to be due to the neglect of the change in the mitral valve ring as the atrium contracts. The

7 Advances in Fluid Mechanics V 327 contraction of the ring will be difficult to machine read. As shown below the important period is between the A and E peaks. It is intended to investigate the possibility of machine reading the ring contraction. The results for patient M72 were not satisfactory. The shape is very different from the general profile. However the aim of the research is to predict outcomes for apparently healthy patients or patients with limited symptoms. (a) (b) Figure 3: Flow vectors (a) As the valves are opening (b) As the atrium contracts. PatientM53. M53 M63 Figure 4: Mitral velocity calculated for two patients M53 and F63.

8 328 Advances in Fluid Mechanics V The shear stress is at a maximum during the valve opening in all cases. The magnitude is similar for both healthy and diseased hearts. At the time between the E peak and the start of the atrium contraction the shear stress level is low. Shown in Figure 5 are the shear stress values for the test sample cases M53 and F63. Although the mitral velocity flow simulation for M72 did not match the measured data the shear stress results are also shown. In all cases, the shear stress gradients are high although much higher in the diseased hearts than in the healthy heart. In the case of the patient M72, who underwent surgery, the shear stress gradients are almost four times larger than for M53. M53 F63 M72 Figure 5: Shear stress at time between A and E for patients M53, F63 and M72.

9 Advances in Fluid Mechanics V 329 M53 F63 Figure 6: The normal velocity near the ventricular wall - Patients M53 and F63. Figure 7: The docking of drug molecules with convective velocity 1cm/sec. The velocity normal to the wall is shown in Figure 6. This is important as the Ang II must penetrate to the receptors on the endothelium to start the gene reprogramming. If the wall was completely smooth then the normal velocity at the wall would be zero. However the wall of the heart is covered with ridges (trabeculae) so the present results are at the two locations nearest the wall.

10 330 Advances in Fluid Mechanics V Using the Monte Carlo method described above and a normal velocity of 1 cm/sec the interaction of the drug molecules with the endothelium can be calculated as shown in Figure 7. There are two methods for the drugs to reach the surface diffusion and convection. At low convective velocities of 0.1cm/sec the diffusion dominates and at high velocities 10cm/sec convection dominates. Also shown in Figure 7 is the rate that molecules are removed from the Monte Carlo zone due to interaction with the surface receptors. 4 Conclusions Although the present work only involves 6 patients, the shear stress gradients were found to be higher in patients with heart irregularities than normal patients. As our intention is to use the techniques as an automated diagnostic tool, individual hand corrections cannot be utilized to fine tune the results. However, using a more extensive group, a data base of echocardiograms types could be established and a two pass procedure can be developed to refine the calculations. Once an initial tracing is performed, the data base would be consulted to locate similar patterns and correct anomalies in the results. In addition, it would provide the physician with an immediate bracketing of the information with patients of similar attributes and medical histories to aid in a more accurate diagnosis. References [1] Ruwhof C, van der Laarse A. Mechanical stress-induced cardiac hypertrophy: mechanisms and signal transduction pathways Cardiovascular Research ,2000. [2] Nagel T, Resnik N, Forbes Dewey.C, Gimbrone M.A. Jr Vascular Endothelial Cells Respond to Spatial Gradients in Flid Shear Stress by Enhanced Activation of Transcription Factors, Arteriosler Thromb Vasc. Biol. 19; , [3] Kim S, Iwao H Molecular and cellular mechanisms of angiotensin IImediated cardiovascular and renal diseases Pharmacological Revs. 52,2,11-34,2000. [4] Macpherson A.K. and Neti S, The effect of Angiotensin II on heart blood flow and hypertension, Advances Fluid Mechanics IV, eds M. Rahman, R.Verhoeven, C.A. Brebbla, WIT press, Southampton, 1-12, 2002 [5] Chandrasekhar, S. Principles of Stellar Dynamics, Uni. Of Chicago Press, Chicago, [6] Balesu, R. Equilibrium and Nonequilibrium Statistical Mechanics, Wiley, New York, 1975.