Three-dimensional display of calculated velocity profiles for physiological flow waveforms

Transcription

1 Three-dimensional display of calculated velocity profiles for physiological flow waveforms Ramez E. N. Shehada, MSc, Richard S. C. Cobbold, Phi), K. Wayne Johnston, MD, FRCS(C), and Rene Aarnink, MSc, Toronto, Ontario, Canada Purpose: To improve the understanding of the nature of pulsatile flow, three-dimensional idealized velocity profiles corresponding to measured physiological mean flow velocity waveforms were displayed at selected instants throughout the flow cycle. Methods: The Fourier harmonics for each waveform were determined, and their corresponding velocity profiles at each instant of time were calculated with the Womersley equations. Velocity profiles were calculated by summing the contributions from each harmonic. Results: Calculated profiles were displayed in a three-dimensional perspective for both normal carotid and femoral arteries and for simple sinusoidal flow with a superimposed steady component. Conclusion: The potential value of such displays is discussed in terms of gaining an improved understanding of the nature of pulsatile flow and clarifying the interpretation of Doppler ultrasound recordings. (J VASC SURG 1993;17:656-6.) In arterial flow significant changes in the shape of the velocity profile occur during the cardiac cycle. Usually such profiles are displayed in two dimensions, showing the velocity as a function of radial distance from the vessel axis at various instants throughout the cardiac cycle. Indeed, it appears that the first such calculations of velocity profiles are those reported by Hale et al.1 in which the measured pressure gradient in the f~moral artery of a dog was used to calculate the harmonic amplitudes and phases, from which the velocity profiles were calculated. These results have been widely quoted and are often used as a basis for discussing the nature and significance of pulsatile flow. 2,3 More recently, Evans et al.4 have displayed two-dimensional profiles calculated from the measured mean velocity waveforms from human common carotid and femoral arteries of a normal volunteer. Because of the complex nature of the velocity From the Institute of Biomedical Engineering and the Department of Surgery, University of Toronto, Ontario. Supported by the Heart and Stroke Foundations of Canada. Reprint requests: K. Wayne Johnston, MD, Toronto General Hospital, Eaton Wing 9-217, 2 Elizabeth St., Toronto, Ontario, Canada M5G 2CA. Copyright 1993 by The Society for Vascular Surgery and International Society for Cardiovascular Surgery, North American Chapter /93/$ /1/ profile, we recognized that even in situations in which the profile is axisymmetric there might be significant advantages in displaying the profiles in a three-dimensional manner. Such a display migh~t provide a clearer physical understanding of the changes occurring over time and, in addition, a better basis for interpreting the results of Doppler ultrasound flow measurements. Specifically, when pulsed Doppler ultrasound recordings are made with a small sample volume positioned within the arterial lumen, it can be difficult to interpret the Doppler spectral recording because of the complex three-dimensional nature of the velocity profile. Similarly, color Doppler flow maps may appear complex even for normal pulsatile flow. We believe that a three-dimensional display of the calculated ideal velocity profiles can provide a better basis for understanding the relation between a velocity profile and the corresponding Doppler spectrum or ~ color-flow Doppler image. The purpose of this study is simply to display three-dimensional idealized velocity profiles corresponding to measured physiologic mean flow velocity waveforms at selected instants throughout the flow cycle and to demonstrate the potential of such displays both in gaining an improved understanding of the nature of pulsatile flow and for clarifying the interpretation of Doppler recordings.

2 JOURNAL OF VASCULAR SURGERY Volume 17, Number 4 Shehada et al 657 %- 3O ~ (a) if) 3 (b) ~ 2 2 ~ 1 4--' Cl2 O O -1-2'! ' I I f f " f c~ o ks_ 1-1 I -2 T ~ v ' 1 "~ t O (c) 2! ~: _ ~ -1i -2 " [ f ' T ' I ' l l Fig. 1. Spatially averaged flow waveforms assumed for calculating flow velocity profiles. Time-averaged mean is indicated by broken line. a, Sinusoidal with a steady flow offset; b, common femoral artery; c, common carotid artery. Waveforms in b and c are based on unscaled mean velocity waveforms given by Evans et al. 4 Table I. Parameters used in calculating velocity profiles Vessel diameter IGnematic viscosity Mean flow Peak flow Frequency Wavefo~n type (mm) (cstokes) (ml/sec) (ml/sec) a* (Hz) Sinusoid~d Common. femoral Common carotid *For the nonsinusoidal waveforrns the value of a is given for the fundamental frequency. METHODS The mean flow velocity waveforms measured by Doppler ujtrasound for the common carotid and femoral arreries reported by Evans et al.4 were used for our calculations. On the basis of a qualitative comparison of these waveforms with those of our own experience, we concluded that they are typical of those seen with normal subjects. Because the waveforms were: unscaled with respect to their velocity and time axes, it was necessary to obtain representative scaling factors. To scale the time axis, we assumed a heart rate of 7/m;m, corresponding to a frequency of 1.17 Hz. To convert the mean velocity waveforms into mean flow waveforms, we made use of published data, together with the assumption that the vessel diameter is constant throughout the flow cycle. Specifically, from the data ofuematsu et al.5 obtained from 35 normal subjects (aged 21 to 61 years and older), according to a special two-transducer Doppler

3 JOURNAL OF VASCULAR SURGERY ~ 658 Shehada et al. April ~ 24 6 ~ 12 3 ~ 15 ~ 27* o. 8 9o. 8 2oo. 9 ~ Fig. 2. Three-dimensional velocity profiles for sinusoidal waveform of Fig. 1, a. Instant of flow cycle is indicated by position of waveform cursor in miniature version of flow waveform. Phase increment between each panel is 3 degrees. technique, we took the average internal diameter of the common carotid artery to be 7.31 mm with a time-averaged flow of 7.46 ml/sec. For the common femoral artery, Lewis et al. 6 reported that for 51 normal subjects (aged 18 to 75 years) the average internal diameter was 9. mm and the time-averaged flow was 5.83 ml/sec. We first digitized the unscaled femoral and carotid waveforms. To determine the appropriate scaling factors, we calculated the normalized timeaveraged flow velocity for each digitized waveform and compared these results to those calculated from the experimental data referenced above. A Fourier analysis was then performed on the scaled waveforms to determine the relative amplitudes and phases of the various harmonics. It was found that harmonics up to the eleventh were more than sufficient for a nearly exact synthesis of the original waveform. To justify calculation of the velocity profiles with the equations developed by Womersley, 2'r's a number Fig. 3. Three-dimensional velocity profiles for common femoral waveform of Fig. 1, b. Instant of flow cycle is indicated by position of waveform cursor in miniatti_,~e version of flow waveform. Phase increment for first nine panels is 15 degrees. of simplifying assumptions are needed. As carefully discussed by Milnor, 2 these include (1) blood is homogeneous and behaves as an incompressible newtonian fluid, (2) the flow is laminar and fully developed, (3) the vessel is cylindric and the walls are rigid, and (4) no slip conditions are present at the walls. The Womersley equations were used to calculate the velocity profile for each harmonic at any time instant of the flow cycle. Assuming linearity, thc-~ shape of the velocity profile was calculated b~ ~ summing the contributions for each velocity profile arising from each harmonic. Mathematic details are well described by Evans et al. 4 A menu-driven computer program was written in Pascal with three-dimensional plotting software (Quinn-Curtis, Needham, Mass.) for use on personal computers. The program enables any periodic waveform to be analyzed whose waveform is specified as a list of sampled values at regular intervals during the

4 JOURNAL OF VASCULAR SURGERY Volume 17, Number 4 Shehada et al. 659 flowcycle. If the number of sample values is less than 124, a cubic spline-fitting program is automatically used to generate 124 sample values. The program allows the user to input the vessel diameter and hematocrit value (from which the kinematic viscosity can be calculated), as well as to select the times in the flow cycle at which it is desired to display and print the profile. ~ 6*~ 12 ~,'~ ~ i,~i RESULTS For all the results presented, we assumed the,anematic viscosity v (the ratio of the absolute viscosity to the density) to be 3.6 cstokes. This is within the "normal" high-shear rate range quoted 9 for human blood at 37 C with a hematocrit value of 45%. Three types of waveform were used as shown in Fig. 1, the parameters for which are given in Table I. The sinusoidal waveform has a frequency (f) of 1.17 Hz (7/min) and a superimposed steady flow of 6.6 ml/sec. As indicated in Table I, the Womersley parameter* for this waveform was chosen to be 5.7, corresponding to a vessel diameter (D) of 8 mm. This cx value is close to the first harmonic values for the other waveforms. Also the peak and time-averaged mean flow-rate values for the sinusoidal waveform were chosen so that the mean and peak Reynolds numbers correspond closely to those for the other waveforrns (Table I). Figs. 2, 3, and 4 show the three-dimensional velocity profiles at various instants of time throughout the flow cycle for the three waveforms shown in Fig. 1. At: the top left of each panel a cursor is shown superimposed on the flow waveform to indicate the time at which the profile is obtained. It is important to note that in Fig. 2 the time intervals between each panel are equally spaced, whereas in Figs. 3 and 4 the first nine panels are at 15-degree intervals to show more clearly the rapid profile changes that occur during systole and early diastole. DISCUSSION Throughout the entire flow cycle, the flow velocity profile for sinusoidal flow (Fig. 2) is characterized by a central core that tags the outer regions. Because of the higher viscous drag in the wall region, the fluid momentum in the outer regions is small compared with that of the central core where the viscous drag is less. As a result, the outer regions respond more rapidly to changes in the pressure gradient that cause the flow. If the frequency of the *~ = ~/~ Fig. 4. Three-dimensional velocity profiles for common carotid waveform of Fig. 1, c. Instant of flow cycle is indicated by position of waveform cursor in miniature version of flow waveform. Phase increment for first nine panels is 15 degrees. waveform is decreased (corresponding to a decreased heart rate in the case of a physiologic waveform), the Womersley parameter a would also decrease, causing the lag in the central core to be reduced until eventually the profile becomes close to parabolic throughout the flow cycle. In examining the carotid profiles (Fig. 4) it will be noted that the rapid systolic increase causes the central core to lag behind the outer regions, thereby producing a relatively flat profile. However, once peak systole is reached, because the flow decreases relatively slowly beyond this point, the lagging central core has time to ;~catch up" with the outer regions and a more nearly parabolic flow profile occurs throughout the remainder of the flow- cycle. On the other hand, during the quite rapid deceleration phase of the femoral waveform (Fig. 3) leading into flow reversal, the effect of the lagging central core remains significant. We believe that the above features of the flow-

5 66 Shehada et al. JOURNAL OF VASCULAR SURGERY April 1993 velocity profile are recognized more easily in the three-dimensional display. In addition, this representation enables the influence of the profile on the Doppler spectra recorded with either pulsed or continuous-wave systems or by color Doppler flow mapping to be more readily appreciated. For example, the intersection of a pulsed beam with the profiles at a given angle enables the resulting Doppler spectra to be estimated in a qualitative manner. More specifically, if a small sample volume is placed at the center of the vessel for the carotid waveform, from Fig. 4 it can be readily seen that the exact positioning of the sample is not critical up to peak systole but after that it becomes a significant factor in determining the Doppler spectrum. As a second example, consider color-flow mapping of the femoral artery. From Fig. 3 the color-flow map at a time corresponding to a phase of 12 degrees should contain both forward and reverse colors; such a pattern is most readily interpreted with the help of the three-dimensional profile display. In principle, a quantitative spectral estimate could be provided by extending the computer program. The user could be provided with the facility of adjusting the beam direction and point of intersection interactively on the display console and to produce spectra corresponding to various sample volume locations and lengths. Finally, it should be noted that the calculated three-dimensional profiles are based on a number of simplifying assumptions that may result in significant differences compared with those present in real vessels. Nonetheless, they are useful for gaining better insight of the nature of pulsatile flow and can be a valuable initial means for interpreting Doppler ultrasound spectra and images. We thank Professors C. Ross Ethier and Matadial Ojha and Mr. Peter Bascom for advice and help. REFERENCES 1. Hale JF, McDonald DA, Womersley JR. Velocity profiles of oscillating arterial flow, with some calculations of viscous drag and the Reynolds number. J Physiol 1955;128: ~-'~ 2. Milnor WR. Hemodynamics. Baltimore: Williams & Wilkins, 1982: McDonald DA. Blood flow in arteries. London: Edward Arnold, 196: Evans DH, McDicken WN, Skidmore R, Woodcock J-P. Doppler ultrasound. New York: Wiley & Sons, 1989: Uematsu S, Yang A, Preziosi TJ, Kouba R, Toung TJK. Measurement of carotid blood flow in man and its clinical application. Stroke 1983;14: Lewis P, Psaila JV, Davies WT, McCarty K, Woodcock JP. Measurement of volume flow in the human common femoral artery using a duplex ultrasound system. Ultrasound Med Biol 1986;12: Womersley JR. Oscillatory motion of a viscous liquid in a thin-walled elastic tube. I: The linear approximation for long waves. Philosophical Magazine 1955;46: Nichols WW, O'Rourke MF. McDonald's blood flow in arteries. 3rd ed. Philadelphia: Lea & Febiger, 199: Milnor WR. Hemodynamics. Baltimore: Williams & Wilkins, 1982:51. Submitted Jan. 31, 1992; accepted May 27, 1992.