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11 Department of Cardiovascular Medicine, The Cleveland Clinic Foundation, Cleveland, OH, USA
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Abstract |
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(Received 18 May 2005;
accepted after revision 5 July 2005;
first published online 7 July 2005)
Corresponding author J. D. Thomas: Department of Cardiology, Desk F-15, The Cleveland Clinic Foundation, 9500 Euclid Avenue, Cleveland, OH 44195, USA. Email: thomasj{at}ccf.org
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Introduction |
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Scaling in biology is usually allometric, i.e. the size of an organ size or its functional characteristics may be expressed as a power function of body mass or weight (Lindstedt & Schaeffer, 2002). In general, equations of the type Y = kMß express the allometric relationship. For example, RR interval duration scales allometrically to the body size with the power of 0.25 (Noujaim et al. 2004), while the metabolic rate and cardiac output scale with the power of 0.75 (West et al. 1997). It has been proposed that the scaling with the power of one-quarter, or of its multiples (such as 0.75, 1, etc.), reflects a fundamental fractal-like branching structure of living organisms (West et al. 1997). Since the heart weight of all mammals keeps a constant proportion to their body mass, the value of the allometric scaling exponent of any particular heart function index would be the same versus either the body or the heart size. In our laboratory, we have the opportunity to collect and analyse data using the same equipment in a variety of animal models that differ dramatically in size. We have observed that, in contrast to what was originally assumed, there is a wide variability of reconstructed myocardial tissue velocities among different species. This has not been previously assessed systematically because of technical issues involved in myocardial tissue velocity analysis by standard pulsed-wave Doppler imaging in the setting of rapid heart rate, and problems induced by anaesthesia of the animals (Rottman et al. 2003) and lack of awareness of the problem. At the same time, assessment of myocardial tissue velocities is becoming increasingly popular for the noninvasive assessment of left ventricular (LV) function in small-animal models of heart disease (Derumeaux et al. 2002; Askari et al. 2003). The primary aim of this study, therefore, was to determine whether long-axis myocardial tissue velocities scale to heart size. The secondary aim was to quantify the scaling exponents of myocardial tissue velocities and their time-integral (i.e. long-axis tissue displacement).
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Methods |
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The line describing the systolic mitral annulus (MA) velocity as a function of time (Fig. 1) has characteristic ascending and descending portions, which together with the horizontal (time) axis comprise a triangular profile. We begin, therefore, by simplifying the shape of a systolic portion of the myocardial Doppler tissue velocity profile to a triangle. As seen from Fig. 1, the base equals LV ejection time, the height (h) corresponds to the peak systolic velocity, and the shaded area of the triangle (i.e. the time integral of the velocity) equals the long-axis LV displacement.
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It should be pointed out that the triangular model, as illustrated in Fig. 1, is a simplification. The deviations observed in a real-life heart require direct measurements and evaluation of all three systolic parameters (peak velocity, ejection time and displacement) in order to make conclusions about the applicability of a particular scaling relationship.
Study samples
The data from human subjects used and analysed for this study were reported in part in a previous publication (Sun et al. 2004). The studies in human subjects were performed in accordance with the Declaration of Helsinki. The Institutional Review Board approved that study, and all participants gave informed consent. Normal volunteers free of any known cardiovascular disease were recruited. All subjects underwent a thorough history, physical examination, and had a normal electrocardiogram. None of them reported taking any kind of cardio-active medication (except aspirin). From the initial study group (Sun et al. 2004), we selected 38 subjects whose age was between 21 and 45 years.
The animal data were collected during echocardiographic evaluation of control (normal) animals used in several different studies in the period of 2001–2004. Although these studies may be viewed technically as retrospective to our current report, the data collection in each of them was prospective and specified by the respective study protocols. Importantly, the evaluation of the reported echocardiographic data was performed in each study and animal by using compatible equipment and techniques.
The animal studies were approved by the Institutional Animal Research Committee and were in compliance with the ‘Guide for the Care and Use of Laboratory Animals’ published by the National Institutes of Health.
Procedures
All subjects (humans and animals) were studied in left lateral decubitus position. Due to the specificity of the animals' behaviour, the procedures differed slightly among the species. Eight mongrel dogs were trained to lie down calmly for echocardiography. Six white New Zealand rabbits were sedated with a subanaesthetic dose of 85 mg kg–1 of ketamine intramuscularly, with the handler keeping the head covered with soft cloth in a dark room. Fifteen Lewis rats were sedated with a subanaesthetic dose of 85 mg kg–1 of ketamine intraperitoneally. After ketamine administration, both rats and rabbits responded to transducer application to the chest wall, and they had to be gently restrained by the sonographer and/or handler. Finally, 10 C57/BL6 mice were assessed in a conscious state, with each one of them having at least one previous echocardiography session with the same handler.
Data collection
Echocardiography was performed using either Vivid 5, or Vivid 7 echocardiography machine (GE Medical, Milwaukee, WI, USA). M-mode echocardiography, two-dimensional echocardiography, and two-dimensional colour tissue Doppler echocardiography data were collected using a dual harmonic 1.7/3.4 MHz or 2.0/4.2 MHz sector transducer (humans, dogs), 6 MHz or 11.5 MHz paediatric sector transducer (rabbits, rats) and 14 MHz epicardial linear transducer (mice). The minimal frame rates acquired during standard two-dimensional echocardiography in humans, dogs, rabbits, rats and mice were 30, 50, 70, 90 and 160 frames s–1, respectively. The minimal frame rates acquired during colour tissue Doppler two-dimensional echocardiography in humans, dogs, rabbits, rats and mice were 125, 125, 189, 200 and 208 frames s–1, respectively. Data were digitized in a proprietary format for further analysis.
Data analysis
Data were analysed using Echopac PC (GE Medical). LV ejection time was measured directly from the digital pulsed-wave Doppler tracing of the LV outflow tract with the sweep speed set to 200 mm s–1. Short-axis displacement was calculated as:
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We reconstructed the velocity of the septal point of the MA using an apical four-chamber view recorded in transthoracic Doppler echocardiography (TDE) mode. The regions of interest for individual species were set to 8 x 8, 6 x 6, 4 x 4, 2 x 1 and 1 x 1 mm, respectively. For the analysis of human, dog and rabbit data, at least three beats were measured. For small rodents, at least six beats (for rats) or nine beats (for mice) were averaged. In rats and mice, MA diastolic E- and A-wave velocities were frequently fused. In these cases, the highest diastolic velocity observed after isovolumic relaxation interval was taken to represent the E-wave velocity. To obtain end-systolic displacement, MA velocities over at least three beats were averaged for humans, dogs and rabbits, while at least six or nine beats were averaged for rats and mice, respectively. The data were then integrated from the mitral valve closure time to aortic valve closure time to obtain end-systolic MA displacement. Long-axis–short-axis displacement ratio (L/Sh) was calculated by dividing end-diastolic displacement by short-axis displacement.
Intra- and interobserver variability of small rodent Doppler tissue imaging data
Intra- and interobserver variability of reconstructed myocardial tissue velocity data have been extensively studied in humans (Popovic et al. 2002; Sun et al. 2004). Here, we present intra- and interobserver variability of reconstructed myocardial tissue velocity data obtained in small rodents. Fourteen data sets obtained in rats that were not included in this report were measured twice after a time interval of >1 month by the same observer, and at that time also by a second observer blinded to the measurements of the first observer. In an identical manner, eight data sets obtained in mice that were not included in this paper were measured. Inter- and intraobserver variability was then quantified as absolute and relative difference between the two measurements.
Statistical methods
Allometric scaling is assumed to follow the power function (Zar, 1984):
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To fit the data, we used a nonlinear regression method with Levenberg Marquardt algorithm (SPSS 10.0, SPSS, Inc., Chicago, IL, USA). Confidence intervals (95%) for exponent ß were calculated using asymptotic error of estimate. A single-sample t statistic was used to compare the exponent ß with a priori values of 0 and 0.25. A t test statistic was also used to compare the values of exponent ß (Dawson-Saunders & Trapp, 1990). Data are presented as means ± S.D., or as exponent ß ± standard error of estimate (S.E.E). A P value < 0.05 was considered significant.
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Results |
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The RR interval scaling to LV end-diastolic volume and LV mass was compatible with the value of exponent of 0.25 (exponent ßEDV = 0.307 ± 0.030, and ßLVmass = 0.315 ± 0.034, P > 0.05 when compared with the value of 0.25 for both). The LV ejection time also scaled to LV end-diastolic volume and LV mass, in concordance with the value of exponent of 0.25 (exponent ßEDV = 0.247 ± 0.017, and ßLVmass = 0.267 ± 0.018, P > 0.1 when compared with the value of 0.25 for both) (Fig. 4A and B).
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MA end-systolic displacement scaled with powers of ßEDV = 0.358 ± 0.047, and ßLVmass = 0.390 ± 0.051. Both were larger than the corresponding scaling exponents of LV ejection time (P = 0.021, and P = 0.023, respectively; Fig. 5A and B). In fact, the displacement exponents ßEDV and ßLVmass were close to the Euclidean exponent of 0.33. We have shown in the Methods (Background) that in such cases the peak systolic velocity should not remain constant, but should scale with a power of ß = 0.0833.
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Short-axis function and long-to-short axis displacement relationship
LV short-axis displacement scaled to LV end-diastolic volume with the power of ßEDV = 0.308 ± 0.033, and to LV mass with the power of ßLVmass = 0.311 ± 0.034. These values were not significantly different from the scaling power for the LV ejection time (P > 0.1) (Fig. 7A and B). As a result, the L/Sh axis displacement ratio showed significant scaling exponents (ßEDV = 0.077 ± 0.017 and ßLVmass = 0.086 ± 0.019, P < 0.0001 for both when compared with 0) (Fig. 8A and B).
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Intra- and interobserver variability in small rodents
Intra- and interobserver variability of reconstructed myocardial tissue velocity data is presented in Table 2. While, expectedly, relative variability of myocardial tissue displacement, and S- and E-wave velocities, was higher in mice than in rats, it was still in a satisfactory range.
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Discussion |
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Previous studies
Schaefer et al. (2003) and Watson et al. (2004) noted that, when compared with humans, myocardial tissue velocities are smaller in mice and rats, respectively. However, no scaling of these data to ventricular size was attempted. Also, anaesthesia may have affected their measurement (Rottman et al. 2003). Interestingly, a recent paper noted that myocardial tissue velocities increase with cardiac growth in children (Eidem et al. 2004). While no quantitative relationship was shown between tissue velocities and either body or heart weight, it has been previously shown that, during individual growth, organ function is allometrically scaled to body size (Beunen et al. 1997). Thus, ample evidence corroborates that size-dependent scaling of myocardial tissue velocities occurs in mammals.
Significance of a scaling exponent
Many biological phenomena, including heart and circulation variables, scale with a power of 0.25 or its multiples versus the body mass. For example, the electrocardiogram RR interval scales with the power of 0.25 (West et al. 1997). Cardiac output and metabolism scale according to power of 0.75 (West et al. 1997; Lindstedt & Schaeffer, 2002), while LV contractility, SV and LV mass scale to animal size according to the power of 1 (West et al. 1997; Georgakopoulos et al. 1998; Lindstedt & Schaeffer, 2002). West et al. (1997) have hypothesized that the quarter-power scaling may be explained by the interplay of physical and geometric constraints that are implicit in the organ architecture, such as invariance of the smallest units (e.g. myocytes), tendency to minimize energy spent on the transport of resources, and fractal-like behaviour of the energy-supply branching network (e.g. circulatory system). Our data show that RR intervals and LV ejection time exhibit a scaling exponent compatible with the value of 0.25. In contrast, the scaling exponent for MA displacement was larger than that for LV ejection time, which led to the new observation that long-axis myocardial tissue velocities show scaling to the heart size. Our data are concordant with our second proposed model that predicts that if the ejection time scales with the exponent of 0.25, and if the displacement scales with the Euclidean exponent of 0.33, the resulting velocity should scale to the heart size with exponent of 0.083. We showed that scaling occurs with both systolic and diastolic velocities, although the observed values were slightly larger than the predicted by the model. Such discrepancy could be explained by our novel observation that exponents of short- and long-axis displacement scaling were different, which unexpectedly resulted in a significant scaling between the heart size and long-to-short axis displacement ratio.
Scaling, ventricular structure and function
It is safe to propose that ventricular function should reflect its structure. A ventricle composed of circumferentially orientated fibres would show predominant short-axis displacement. And, vice versa, a ventricle composed exclusively of longitudinally orientated fibres would display a piston-like behaviour, with predominant long-axis displacement. The fact that the normal human ventricle shows a complex and changing transmural fibre orientation (Rademakers et al. 1994), is logically reflected in its function by a significant amount of both long- and short-axis displacement during systolic contraction (Henein & Gibson, 1999; Aurigemma et al. 1995). Furthermore, since a large number of fibres are orientated neither parallel nor perpendicular to the long axis of the heart, a torsional component of the contraction is readily observed (Knudtson et al. 1997).
As previous work has shown that the degree and type of transmural change of fibre orientation is quantitatively very similar among species of different size, one could assume that the long-axis, short-axis and torsional components of contraction should show a constant proportion across all species (Grimm et al. 1976; Costa et al. 1997). In contrast, a recent paper reported that left ventricle of a mouse heart displays a total twist that is 10 times smaller than the one observed in humans (Henson et al. 2000). Since twist is an expression of fibre orientation (Ashikaga et al. 2004), one has to conclude that functional characteristics that reflect LV structure may be affected by the heart size. Furthermore, although not systematically assessed previously, it is well known that fractional shortening (short-axis displacement divided by initial short-axis diameter) is larger in smaller animals. For example, two independent groups reported average fractional shortening values of 52 and 56% in normal mice (Gardin et al. 1995; Yang et al. 1999), while the normal fractional shortening in man is only 33 ± 3% (Felner & Schlant, 1976). In contrast, a recent report that used three-dimensional imaging techniques has shown a mouse ejection fraction of 63 ± 5% (Dawson et al. 2004), which is very comparable to normal values of 67 ± 8% in humans. Thus, it seems obvious that relative contribution of long- versus short-axis function to overall ejection fraction markedly differs between the species. Our present results further extend these observations by demonstrating that heart size affects the contribution of long- versus short-axis displacement to total ventricular output in a predictable manner. The origin of interspecies differences in myocardial function that are influenced by heart size can only be speculated. McMahon analysed the fact that the ratio of length to diameter in major skeletal muscles is smaller in large mammals (McMahon, 1975). He conjectured that this type of scaling is due to elastic constraints imposed on a skeletal muscle, which, as a result, leads to specific scaling of the body shape to the weight of the organism. Some unknown elastic constraints might be affecting long-to-short axis function scaling of cardiac muscle. Another possible factor could be that LV fibres are rope-shaped structures built by myocytes, with basically invariant dimensions in different species (Torrent-Guasp et al. 2001). It is possible that the shape of this structure, which is essentially a one-dimensional object embedded in a three-dimensional matrix, exhibits scaling to heart size in order to accommodate the invariant myocyte dimensions.
Practical implications
For a long time, studies in comparative physiology have stressed similarities of the heart structure and function among various mammalian species. This is important as it enables translation of experimental data into the clinical domain. We have shown that while these assumptions still hold true qualitatively, there are quantitative differences that become important when data are compared among species that dramatically differ in size. Furthermore, since the velocities are lower in the very small mammalian hearts, they may be more difficult to measure accurately.
Limitations
First, different animal subjects were handled slightly differently. Sedation may have decreased LV systolic function in rabbits and rats (Rottman et al. 2003), while handling the nonanaesthetized mice may have brought catecholamine surge. On the other hand, mouse systolic function operates close to its contractility maximum (Reyes et al. 2003), and thus an additional contractility increase induced by the stress of the echocardiography examination would be small. Also, the RR intervals closely followed the predicted and previously described quarter-power allometric equation, implying that our data were collected in generally normal conditions. Second, limited resolution of the images obtained in mice was the underlying cause of the pronounced increase in intra- and interobserver variability reported in Table 2. Furthermore, it should be acknowledged that similar increases in intra- and interobserver variability might be expected for the other cardiac parameters measured directly (or indirectly) in mice versus the other species. Third, LV mass was not directly measured, but was instead estimated by the bullet equation. However, our supplemental data confirm that the measurement errors were within acceptable limits. Of note, there was no systematic measurement bias in the echo estimates, while the 95% confidence intervals in mice, although wider than in other animal species, were within acceptable limits and consistent with the previously published validation study (Collins et al. 2001). Finally, some methodological limitations of applying power function in biological sciences have been raised (Smith, 1984).
In conclusion, we have shown that long-axis myocardial tissue velocities express allometric scaling to the heart size. This observation can be mechanistically linked to observed differences in the scaling exponents of long-axis displacement and ejection time. Interestingly, scaling exponents for long- and short-axis displacement were also different. This reduced the long- versus short-axis contribution to heart function in very small mammals.
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Supplemental material |
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This material can also be found as part of the full-text HTML version available from http://www.blackwell-synergy.com
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References |
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Askari AT, Unzek S, Popovic ZB, Goldman CK, Forudi F, Kiedrowski M, Rovner A, Ellis SG, Thomas JD, DiCorleto PE, Topol EJ & Penn MS (2003). Effect of stromal-cell-derived factor 1 on stem-cell homing and tissue regeneration in ischaemic cardiomyopathy. Lancet 362, 697–703.[CrossRef][Medline]
Aurigemma GP, Silver KH, Priest MA & Gaasch WH (1995). Geometric changes allow normal ejection fraction despite depressed myocardial shortening in hypertensive left ventricular hypertrophy. J Am Coll Cardiol 26, 195–202.[Abstract]
Beunen GP, Rogers DM, Woynarowska B & Malina RM (1997). Longitudinal study of ontogenetic allometry of oxygen uptake in boys and girls grouped by maturity status. Ann Hum Biol 24, 33–43.[CrossRef][Medline]
Collins
KA, Korcarz
CE, Shroff
SG, Bednarz
JE, Fentzke
RC, Lin
H, Leiden
JM
&
Lang
RM (2001). Accuracy of echocardiographic estimates of left ventricular mass in mice. Am J Physiol Heart Circ Physiol
280, H1954–1962.
Costa
KD, May- Newman
K, Farr
D, O'Dell
WG, McCulloch
AD
&
Omens
JH (1997). Three-dimensional residual strain in midanterior canine left ventricle. Am J Physiol Heart Circ Physiol
273, H1968–1976.
Dawson
D, Lygate
CA, Saunders
J, Schneider
JEYeX, Hulbert
K, Noble
JA
&
Neubauer
S (2004). Quantitative 3-dimensional echocardiography for accurate and rapid cardiac phenotype characterization in mice. Circulation
110, 1632–1637.
Dawson-Saunders B & Trapp RG (1990). Basic and Clinical Biostatistics. Appleton and Lange, East Norwalk, CT.
Derumeaux
G, Mulder
P, Richard
V, Chagraoui
A, Nafeh
C, Bauer
F, Henry
JP
&
Thuillez
C (2002). Tissue Doppler imaging differentiates physiological from pathological pressure-overload left ventricular hypertrophy in rats. Circulation
105, 1602–1608.
Eidem BW, McMahon CJ, Cohen RR, Wu J, Finkelshteyn I, Kovalchin JP, Ayres NA, Bezold LI, O'Brian Smith E & Pignatelli RH (2004). Impact of cardiac growth on Doppler tissue imaging velocities: a study in healthy children. J Am Soc Echocardiogr 17, 212–221.[CrossRef][Medline]
Felner JM & Schlant RC (1976). Echocardiography: a Teaching Atlas. Grune and Stratton, New York.
Gardin
JM, Siri
FM, Kitsis
RN, Edwards
JG
&
Leinwand
LA (1995). Echocardiographic assessment of left ventricular mass and systolic function in mice. Circ Res
76, 907–914.
Georgakopoulos
D, Mitzner
WA, Chen
CH, Byrne
BJ, Millar
HD, Hare
JM
&
Kass
DA (1998). In vivo murine left ventricular pressure–volume relations by miniaturized conductance micromanometry. Am J Physiol Heart Circ Physiol
274, H1416–1422.
Grimm AF, Katele KV & Lin HL (1976). Fiber bundle direction in the mammalian heart. An extension of the ‘nested shells’ model. Basic Res Cardiol 71, 381–388.[CrossRef][Medline]
Henein
MY
&
Gibson
DG (1999). Long axis function in disease. Heart
81, 229–231.
Henson
RE, Song
SK, Pastorek
JS, Ackerman
JJ
&
Lorenz
CH (2000). Left ventricular torsion is equal in mice and humans. Am J Physiol Heart Circ Physiol
278, H1117–1123.
Knudtson
ML, Galbraith
PD, Hildebrand
KL, Tyberg
JV
&
Beyar
R (1997). Dynamics of left ventricular apex rotation during angioplasty: a sensitive index of ischemic dysfunction. Circulation
96, 801–808.
Lindstedt
L
&
Schaeffer
PJ (2002). Use of allometry in predicting anatomical and physiological parameters of mammals. Lab Anim
36, 1–19.
McMahon
TA (1975). Using body size to understand the structural design of animals: quadrupedal locomotion. J Appl Physiol
39, 619–627.
Noujaim
SF, Lucca
E, Munoz
V, Persaud
D, Berenfeld
O, Meijler
FL
&
Jalife
J (2004). From mouse to whale: a universal scaling relation for the PR interval of the electrocardiogram of mammals. Circulation
110, 2802–2808.
Plehn
JF, Foster
E, Grice
WN, Huntington-Coats
M
&
Apstein
CS (1993). Echocardiographic assessment of LV mass in rabbits: models of pressure and Volume overload hypertrophy. Am J Physiol Heart Circ Physiol
265, H2066–2072.
Pombo
JF, Troy
BL
&
Russell
RO
Jr (1971). Left ventricular volumes and ejection fraction by echocardiography. Circulation
43, 480–490.
Popovic ZB, Grimm RA, Perlic G, Chinchoy E, Geraci M, Sun JP, Donal E, Xu XF, Greenberg NL, Wilkoff BL & Thomas JD (2002). Noninvasive assessment of cardiac resynchronization therapy for congestive heart failure using myocardial strain and left ventricular peak power as parameters of myocardial synchrony and function. J Cardiovasc Electrophysiol 13, 1203–1208.[CrossRef][Medline]
Prothero J (1979). Heart weight as a function of body weight in mammals. Growth 43, 139–150.[Medline]
Rademakers
FE, Rogers
WJ, Guier
WH, Hutchins
GM, Siu
CO, Weisfeldt
ML, Weiss
JL
&
Shapiro
EP (1994). Relation of regional cross-fiber shortening to wall thickening in the intact heart. Three-dimensional strain analysis by NMR tagging. Circulation
89, 1174–1182.
Reichek
N, Helak
J, Plappert
T, Sutton
MS
&
Weber
KT (1983). Anatomic validation of left ventricular mass estimates from clinical two-dimensional echocardiography: initial results. Circulation
67, 348–352.
Reyes
M, Freeman
GL, Escobedo
D, Lee
S, Steinhelper
ME
&
Feldman
MD (2003). Enhancement of contractility with sustained afterload in the intact murine heart: blunting of length-dependent activation. Circulation
107, 2962–2968.
Rottman JN, Ni G, Khoo M, Wang Z, Zhang W, Anderson ME & Madu EC (2003). Temporal changes in ventricular function assessed echocardiographically in conscious and anesthetized mice. J Am Soc Echocardiogr 16, 1150–1157.[CrossRef][Medline]
Schaefer A, Klein G, Brand B, Lippolt P, Drexler H & Meyer GP (2003). Evaluation of left ventricular diastolic function by pulsed Doppler tissue imaging in mice. J Am Soc Echocardiogr 16, 1144–1149.[CrossRef][Medline]
Smith
RJ (1984). Allometric scaling in comparative biology: problems of concept and method. Am J Physiol Regul Integr Comp Physiol
246, R152–160.
Sun JP, Popovic ZB, Greenberg NL, Xu XF, Asher CR, Stewart WJ & Thomas JD (2004). Noninvasive quantification of regional myocardial function using Doppler-derived velocity, displacement, strain rate, and strain in healthy volunteers: effects of aging. J Am Soc Echocardiogr 17, 132–138.[CrossRef][Medline]
Torrent-Guasp F, Buckberg GD, Clemente C, Cox JL, Coghlan HC & Gharib M (2001). The structure and function of the helical heart and its buttress wrapping. I. The normal macroscopic structure of the heart. Semin Thorac Cardiovasc Surg 13, 301–319.[Medline]
Watson LE, Sheth M, Denyer RF & Dostal DE (2004). Baseline echocardiographic values for adult male rats. J Am Soc Echocardiogr 17, 161–167.[CrossRef][Medline]
West
GB, Brown
JH
&
Enquist
BJ (1997). A general model for the origin of allometric scaling laws in biology. Science
276, 122–126.
Wyatt
HL, Heng
MK, Meerbaum
S, Hestenes
JD, Cobo
JM, Davidson
RM
&
Corday
E (1979). Cross-sectional echocardiography. I. Analysis of mathematic models for quantifying mass of the left ventricle in dogs. Circulation
60, 1104–1113.
Yang
XP, Liu
YH, Rhaleb
NE, Kurihara
N, Kim
HE
&
Carretero
OA (1999). Echocardiographic assessment of cardiac function in conscious and anesthetized mice. Am J Physiol Heart Circ Physiol
277, H1967–1974.
Zar JH (1984). Biostatistical Analysis. Prentice Hall, Englewood Cliffs, NJ.
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, mice; 
, rats; 
, rabbits; 
