INTRODUCTION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

THE PHYSIOLOGICAL STUDY OF developing circulation remains technically challenging (for a review, see Ref. 37). The mouse is the primary model of mammalian development, but the assessment of developmental cardiovascular physiology, critical to the study of functional genomics, has lagged behind the recent advances in molecular cardiology. We recently (1, 36, 40) demonstrated that quantitative in utero study of mouse embryonic cardiovascular physiology is feasible using 40- to 50-MHz ultrasound biomicroscopy (UBM) and 43-MHz spectral pulsed-wave (PW) Doppler. UBM image-guided PW Doppler, in particular, is a powerful tool for the quantitative and noninvasive investigation of early mouse circulatory development, allowing for more physiologically relevant data than previously obtained (36).

Precise characterization of developing embryonic circulatory physiology requires information on cardiac output and volumetric blood flow, allowing for insights into blood flow distribution and oxygen delivery. Volumetric blood flow (Q) in a given blood vessel may be calculated as

Overall, current analysis of Doppler signals to provide volumetric blood flow information is very poorly developed (13). In Doppler assessment of cardiovascular physiology, several assumptions are required to allow transformation of Doppler frequency shifts to blood flow velocities and, finally, to blood flow volume (5, 12, 16, 21). These may be conveniently categorized because those assumptions are related to the physical properties of the fluid (blood) and tube (blood vessel) and characteristics of Doppler beam transmission, reception, and processing. Of the blood and blood vessels, we generally assume the following: 1) blood is an incompressible, homogeneous, and Newtonian fluid, 2) the concentration of the erythrocytes across the vessel lumen is uniform, or the rheological properties of blood are negligible, 3) the flow is laminar, 4) the blood vessel does not taper; that is, it is of uniform cross-sectional area, 5) the blood vessel is a straight tube, 6) the cross-sectional shape of the blood vessel is circular, 7) there is no entrance effect; that is, there is enough tube length to allow for fully developed flow, and 8) the blood vessel is a rigid (nonpulsatile) tube. Such assumptions may not, in fact, hold true for the small arteries of the embryonic mouse.

Several assumptions are made of the Doppler signal itself, including exclusion of extraneous Doppler signals, uniformity and accuracy of Doppler beam transmission and reception, and accurate weighting of the Doppler power spectrum.¹ In the final analysis, the reflected Doppler signal must represent only the velocities of the erythrocytes/fluid of interest in a specific blood vessel and not other signals (such as wall motion, nearby sources of blood flow, or flow at a suboptimal incident angle). Such assumptions are particularly relevant when a Doppler processor yields "mean" velocities because the weighting of the Doppler power spectrum will be influenced by nonrepresentative sources of flow (12). These assumptions, required in both experimental Doppler-derived phasic mean velocities (10, 29) and clinical spectral Doppler, are fraught with error (16).

The assumption that a Doppler power spectrum accurately represents mean spatial fluid velocity in a vessel can be circumvented by the use of the peak spectral velocity. If the spatial velocity profile is geometrically simple (flat or parabolic profile, for instance), then one can readily calculate the mean velocity from the peak spectral velocity, based on the assumed geometry of the profile. We sought to develop a reasonable first-order approximation model of the spatial velocity profile in the early mouse embryonic aorta, with the help of newly available UBM-Doppler imaging data. We hypothesized that peak spectral Doppler velocities may be related to mean velocities by a simple model of spatial velocity profile. Such a model would validate and/or circumvent many of the assumptions currently in use and allow more accurate calculation of volumetric blood flow than is currently available.

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

It is important to note that there are currently no noninvasive methods that allow direct, accurate measurement of velocity profiles in small blood vessels. Therefore, we indirectly modeled the spatial velocity profile in the embryonic circulation using a combination of fluid biomechanical theory, published data, and our own UBM-Doppler data from middorsal aorta in normal midgestation Swiss-Webster mouse embryos. Thus RESULTS includes, in addition to experimental data from our laboratory, discussions of biomechanical theory and published results as appropriate.

Details of the preparation and imaging of mouse embryos, including UBM image-guided Doppler interrogation and analysis, have been previously described (1, 36, 40). In brief, timed pregnant Swiss-Webster mice (Taconic; Germantown, NY) were anesthetized with pentobarbital sodium (6 mg/100 g body wt ip), with added MgSO₄ · 7 H₂O (14 mg/100 g body wt) to reduce the spontaneous uterine contractions that interfere with image acquisition. In staging embryos, gestational day 0.5 (E0.5) was defined as noon of the day a vaginal plug was found after overnight mating. Preparation and imaging of the pregnant mouse occurred in a closely regulated thermal environment to maintain mouse and embryo temperatures at 37 ± 1°C (36). A focused 40-MHz UBM transducer was used, with a measured lateral resolution of 90 µm and axial resolution of 30 µm at 6 dB, and depth of penetration of 7-10 mm.

The UBM system was used to determine the position of the embryo, to identify the heart and dorsal aorta, and to allow measurements of the dorsal aorta as described below. To obtain the sharpest possible images and the best possible orientation for measurement of aortic curvature, aortic diameter, and inlet length (see below), embryos were imaged in "semi-invasive" fashion (27, 35) (Fig. 1). After the pregnant mouse was sedated, the maternal abdomen was incised. The entire uterus was then gently pulled out of the abdomen, allowing both sides of the uterine horn to be examined and the embryos to be counted and identified. The embryos were then gently placed back into the abdomen. A thin 35-mm-diameter rubber membrane with a 5- to 15-mm central slit was stretched over a central 25-mm hole in a petri dish, and the petri dish was mounted over the pregnant mouse. Part of the uterus containing one or two embryos was carefully pulled through the slit into Dulbecco's phosphate-buffered saline (37 ± 1°C; Sigma; St. Louis, MO) that filled the petri dish. Thus embryonal-placental continuity was maintained, and the embryo remained in situ, encased in its uterine sac and amniotic fluid. We have shown previously that this semi-invasive approach yields physiologically relevant data (35).

View larger version (34K): [in this window] [in a new window]   Fig. 1.   Diagram of the ultrasound bimicroscopy (UBM) Doppler system showing "semi-invasive" embryos in utero. The embryos are bathed in 37°C phosphate-buffered saline (PBS), which acts as both a normothermic environment as well as an acoustic coupling medium. The pregnant mouse is positioned on a heating pad within the two-level holding stage. Both the PBS and the maternal rectal temperatures are maintained at 37 ± 1°C.

Figures 2 and 3 show various measurements made using the UBM system. Aortic dimension was measured at the middorsal aorta, where we recorded Doppler signals (36), as well as at the proximal dorsal aorta (just beyond the aortic arch). Blood imaged at 40 MHz is highly echogenic, imparting a speckle movement in the heart and blood vessels. Therefore, the diameter of the aorta was judged by the width of the blood column or by the clear demarcation of the aortic wall typically present at older stages (E13.5-E14.5). Aortic curvature was determined using a best-circle fit method. A UBM image of the dorsal aorta was digitized from a VHS videotape using DV Rex Video 2.52 (Canopus; San Jose, CA), compressed into MPEG format using Amber version 2.0 (Canopus), and then imported into PowerPoint (Microsoft; Redmond, WA). The radius of curvature of the aorta was determined offline by superimposing a circle with a curvature that best fit the aortic curvature onto the UBM image, and by measuring its radius. To obtain an accurate radius of curvature, only dorsal aortas imaged in a strictly sagittal plane were analyzed. "Inlet length," the distance required for Newtonian flow to become fully developed (see Inlet length and fully developed flow), was measured from the proximal dorsal aorta just beyond the aortic arch to the middorsal aorta. Finally, the cross-sectional aortic geometry was assessed by imaging a cross section of the embryo perpendicular to its long axis.

View larger version (106K): [in this window] [in a new window]   Fig. 2.   Representative UBM image of the embryonic dorsal aorta at 13.5 days of gestation (E13.5). Each small tick on the right-hand scale of the UBM image represents 0.10 mm, so that the distance between two large bars corresponds to 1 mm.

View larger version (13K): [in this window] [in a new window]   Fig. 3.   Diagram of the dorsal aorta with flow depicted in the middorsal aorta and several of the measurements made. Radius of curvature (r) was determined by superimposing a circle with a curvature that best fit the aortic curvature onto the UBM image and by measuring its radius. L refers to the inlet length measured in this study from the proximal dorsal aorta just beyond the aortic arch to the middorsal aorta (although the true inlet of the aorta is at the valve, as indicated). R refers to the aortic radius.

The nonfocused PW Doppler transducer operates at a center frequency of 43 MHz, with a lateral beam width of 1.24 mm. The sample volume of the PW Doppler was placed over the region of interest (middorsal aorta) using UBM guidance. The Doppler frequency shifts were processed and displayed within 5-10 s using a LabView virtual instrument (National Instruments; Austin, TX); the complex Doppler signal was analyzed via a short-time window, Fourier transform approach, resulting in a spectrogram representation of the Doppler waveform (Fig. 4). The Doppler signals were recorded on digital audiotape for detailed offline analysis. Although velocities were not corrected for Doppler incident angle, the optimal angle and highest velocities in the middorsal embryonic aorta were always sought (36).

View larger version (60K): [in this window] [in a new window]   Fig. 4.   Spectrogram of the Doppler waveform in the middorsal aorta of an E13.5 mouse embryo. Amplitude is represented as 8-bit grayscale. The bar indicates peak velocity. The velocity time integral (units = mm) was measured by tracing the outer edge of the waveform for one cardiac cycle.

It should be noted that measurements of dorsal aortic curvature, aortic diameter, and inlet length were performed in semi-invasively imaged embryos, whereas blood flow velocity and heart rate data were obtained noninvasively in a previous study (36). Although it is not an ideal situation, we felt that parameters such as aortic diameter, curvature, and inlet length could be more precisely measured semi-invasively, whereas velocity and heart rate are more physiological when measured noninvasively. Thus the calculations of fluid mechanical indexes such as the Reynolds number (Re), Womersley parameter (), and Dean number (N_d) are derived from pooled data and not the averages of data from individual embryos. However, as will be seen, the values fall well within a range that allows for unambiguous interpretation.

All animals used in this study were maintained according to protocols approved by the Institutional Animal Care and Use Committee at New York University School of Medicine.

Statistics. All results are expressed as means ± SD. Differences between the diameters of the proximal and middorsal aorta were analyzed using a paired t-test.

	RESULTS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

Embryonic blood characteristics and rheology. Plasma behaves as a Newtonian fluid: homogeneous and incompressible (32). Blood, however, is a suspension of cells (primarily erythrocytes) in plasma and is a non-Newtonian fluid, although under certain circumstances it could be treated essentially as a Newtonian fluid (for reviews, see Refs. 14 and 32). Embryonic erythrocyte counts, and therefore hematocrit, increase progressively with maturation. Published erythrocyte counts in normal mouse embryos are ~4 × 10⁵ per mm³ at E12 and 6-10 × 10⁵ per mm³ at E14 compared with ~4-5 × 10⁶ per mm³ in the very early newborn period (38). Erythrocyte counts are therefore ~10% of early postnatal levels at E12, increasing to ~20% of early postnatal levels at E14-E15. Because hematocrit is the product of mean corpuscular volume and erythrocyte count, and because neonatal mean corpuscular volume is ~100 µm³, then the hematocrit of a young newborn mouse is 0.50 (38). Because erythrocyte sizes are similar in embryos, the hematocrit at our embryonic stages is therefore 10-20% of early postnatal levels, or 0.05-0.10. At these hematocrits, blood viscosity is 1.2-1.3 times that of plasma (3), so that kinematic viscosity (, defined as µ/ = viscosity/density) is 0.016 stokes at E11.5 and 0.017 stokes at E14.5.

Is flow laminar? Whether flow is laminar depends on the nondimensional Re for steady flows and the nondimensional  for oscillating flows (14, 31, 32, 44). Although strictly applicable only to steady flows, Re, given by Refs. 14, 32, and 44, is
and is often used by investigators to determine arterial flow characteristics, but mean velocity (V_mean) is substituted with peak velocity. D refers to tube diameter. When Re is calculated using this peak velocity, the assumption is that this Re represents the "worst case scenario," although turbulence under pulsatile conditions occurs during the deceleration of flow (44). Peak blood flow velocity in the dorsal aorta increases with gestation and averages ~75 mm/s at E11.5 and ~160 mm/s at E14.5, but velocities as high as 226 mm/s were obtained at E14.5 (Table 1). Using these velocities and our measured dorsal aortic diameters (Table 1), we obtain a maximal Re of only 452, at E14.5 (Table 2). These values are far lower than the critical Re (typically >1,000-2,000) needed for development of turbulent flow (2, 32, 44).

Does aortic tapering affect flow? Although taper would impact flow velocities and pressure gradients (mean velocity increases as dorsal aorta tapers) and impedance (wave reflection node locations change), taper by itself would not impact the spatial velocity profile geometry significantly (32), although it may exert indirect effects by increasing or decreasing the velocity of the fluid column. Regardless, UBM imaging of the proximal and middorsal aorta reveals no measurable or visible taper (Fig. 2). Pooled data from 65 embryos aged E11.5 through E14.5 revealed no significant difference between proximal and middorsal aortic diameters (P > 0.290).

Is the cross-sectional shape of the aorta circular? Unless there is extrinsic compression of the dorsal aorta, the shape of the aortic cross section should be circular under the influence of an internally distending and pulsating fluid column. We were able to qualitatively determine the cross-sectional shape of the dorsal aorta in 45 of 68 embryos (66%); in the remaining embryos, image quality was inadequate to judge arterial geometry. In every case, and at all stages from E11.5 through E14.5, the aorta appeared circular in cross section (Fig. 5).

View larger version (127K): [in this window] [in a new window]   Fig. 5.   Cross-sectional view of the embryo with the dorsal aorta shown in the transverse plane; it is circular in cross section.

Effect of aortic curvature. Tube curvature generates secondary flow phenomena (vortices) and skews the spatial velocity profile (2, 8, 9, 34). The magnitude of both the secondary flow and skewing of an axisymmetric spatial velocity profile is a function of the nondimensional N_d given by (8, 9)
where R refers to tube (aortic) radius and r is the radius of curvature, which was calculated by UBM using a best circle-fit method. r is large, so that R/r is low and decreases with gestational age, being maximal at E11.5 (Table 2). Thus N_d is ~40-50 at all stages, with a maximal N_d of only ~100 (Table 2). Empirical data show that for N_d = 1,140 and R/r = 0.10, there is only mild skewing of the velocity profile that occurs only after the tube has curved >90°; secondary flow phenomena are considered small (8). Because N_d and R/r in the mouse embryo are much lower than these values, and because dorsal aortic curvature is <90° (Fig. 2), our results indicate a minimally skewed axisymmetric spatial velocity profile. Thus the flow behaves as if in a straight tube.

Inlet length and fully developed flow. For flow to become fully developed, the inlet length (L) needs to be adequate enough to dissipate entrance effects that flatten the spatial velocity profile (14, 32). L is given by (30)
where k is an experimentally derived constant. For k = 0.06 (5), and using averaged peak Doppler velocities from Table 2, L = 1.3 mm at E11.5 and 2.4 mm at E14.5. Because the parameter V in the above equation should refer to the time- and space-averaged velocity, use of peak velocity would introduce error by overestimating the inlet length. Blood flow has its true "entrance" at the aortic valve and then develops as it courses around the arch. Therefore, flow is already partially developed at the proximal dorsal aorta. UBM measurements from the proximal dorsal aorta to the middorsal aorta [where velocities are obtained (36)] show average distances of >2.0 mm (Table 2). Because the estimated inlet lengths should be overestimated, our measurements suggest that the flow should be fully developed.

Is the tube (aorta) rigid? Aortic wall pulsations would complicate the calculation of volumetric flow, because vessel cross-sectional area would then be a dynamically changing quantity. Although we saw no evidence of aortic pulsations by UBM even over several cardiac cycles, the axial resolution (see MATERIALS AND METHODS) and low frame rate of UBM (8/s) may reduce its sensitivity to small and/or rapid changes in cardiac and vascular geometry. However, it should be noted that even in embryos exhibiting heart rates of 100-120 beats/min (<2 beats/s), we failed to observe any aortic pulsations. Moreover, previous observations also confirm that the embryonic aorta (10, 45) and small arteries (32) do not pulsate. We believe, therefore, that the embryonic mouse dorsal aorta can be treated essentially as a rigid tube.

Spatial velocity profiles in small arteries and arterioles: experimental evidence. No experimental data on spatial velocity profiles exist in a system such as the embryonic mouse dorsal aorta. However, results of related experiments may be relevant to the embryonic dorsal aorta.

Estimation of cardiac output in dorsal aorta: preliminary analysis. The above results indicate that flow in the early mouse embryonic dorsal aorta is laminar and that the spatial velocity profile is parabolic. For any parabolic flow profile, the mean velocity is one-half the peak (centerline) velocity, so that Q is given by

or


In this study, UBM-Doppler-derived velocity-time integrals at E13.5 averaged 10.1 ± 1.8 mm (n = 11) and at E14.5 averaged 11.7 ± 2.5 mm (n = 9) in noninvasively imaged (transabdominally imaged) embryos. The estimated average stroke volumes are then 0.52 mm³ (for average middorsal aortic diameter of 350 µm) and 0.53 mm³ (for average middorsal aortic diameter of 340 µm), respectively. These average values likely underestimate the true stroke volumes because the Doppler incident angles are not always optimal (36); indeed, on review of Doppler incident angles from UBM images, we estimate that ~60-65% of the time, the incident angles were <30° (thus velocity error due to angle <14%) and 10-15% of the time, the Doppler incident angles could not be accurately determined due to embryonic lie within the maternal abdomen. However, this error may be offset by intrinsic spectral broadening (see DISCUSSION). Still, our estimated stroke volumes are remarkably consistent with, and what we would expect based on, volumetric flows in the mouse obtained from video planimetry of the combined embryonic ventricular outputs (0.56-0.63 mm³) (24) and Doppler interrogation of umbilical arteries (0.37-0.42 mm³) (29). They are also remarkably consistent with values obtained in the chick dorsal aorta (~0.50 mm³) at similar stages of development (10), when chick dorsal aortic dimension and embryonic weight are comparable to those of the mouse (23, 29).

	DISCUSSION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

Our study, although basic in terms of fluid mechanics, strongly suggests that blood flow in the mouse embryonic dorsal aorta can be regarded as essentially Poiseuille flow in a straight, rigid tube, with a parabolic spatial velocity profile. This simple model appears to be an excellent first approximation, allowing for a reasonable determination of cardiac output in the middorsal aorta using peak spectral Doppler velocities. The strength of this study lies in the assessment of arterial geometry and flow characteristics in the in utero early developing embryo, with preservation of intrinsic geometry and physiology of the cardiovascular system that would affect flow profiles. Such data have become available only through the relatively recent development of UBM and UBM image-guided PW Doppler and their application to the developing mammalian cardiovascular system in our laboratory (1, 36, 40). We believe ours is the first study to examine the physics of flow in the early developing embryonic circulation, where the dimensions and flow parameters are substantially different from the mature animal.

Experimental evidence using peak versus mean Doppler velocities. Our model is useful because only the peak (centerline) velocity, obtainable by spectral Doppler noninvasively (1, 36), is required to determine volumetric blood flow. Confounders of phasic mean velocity determinations are irrelevant. Several investigations (10, 23, 24) have processed the Doppler frequency shifts into phasic mean velocities, with the argument that such velocities may be easily converted into volumetric blood flow by multiplying by the cross-sectional area of the blood vessel. Importantly, such results rely on an accurate determination of the true mean blood flow velocity, however, which depends on proper weighting of the Doppler signal¹ and assumptions that the reflected Doppler signal truly represents the velocities of the individual particles in the blood vessel. This approach is inherently inaccurate, because signal processing, extra-arterial motion, and nonuniform attenuation of the Doppler signal (due to depth, reflector characteristics, etc.) all contribute to errors in the Doppler-derived mean velocity (16, 32). Calibration of the system (10) does not necessarily obviate these confounding problems in determinations of phasic mean velocities because the phantoms are in isolation, not in the in vivo embryonic milieu where extra-arterial signals and nonuniform signal attenuation may occur. Moreover, in prior experiments (10, 22), the phantom was of substantially different caliber (3.0 mm diameter) than the aorta imaged (0.29-0.41 mm diameter), with a Doppler sample volume length of 1-2 mm (10, 22). This discrepancy between the phantom and the actual vessel sizes may also result in an inaccurate Doppler power spectrum for the blood vessel. Indeed, Hartley et al. (22) have stated that, ideally, such Doppler probes should be calibrated on the same vessel in which the flow is measured.

Limitations and directions for future work. We recognize that there is no currently available approach for fully validating the results of this study. Moreover, many of the parameters discussed (such as Re, , N_d, and r) are close approximations and cannot be measured or calculated precisely. More research will also be necessary to determine whether red blood cell aggregation confounds this model in the embryonic mouse aorta. Nevertheless, we believe the combined theoretical and empirical analysis above argues strongly for Poiseuille flow in the mouse embryonic dorsal aorta.