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. 2020 Jul 31;239(1):242–250. doi: 10.1111/joa.13286

Is relative growth by the mammalian heart biphasic or monophasic?

Gary C Packard 1,✉
PMCID: PMC8197952  PMID: 32735739

Abstract

I re‐examined data for relative growth by the heart in four species of mammal to reconcile divergent reports that appear in the literature. Raw data for heart and body mass for Horro sheep, humans, gray kangaroos, and tammar wallabies were studied by linear and nonlinear regression, thereby enabling me to avoid the confounding effects of logarithmic transformation and to evaluate multiple statistical models for describing pattern in each set of observations. My analyses indicate that relative growth by the heart is monophasic in all four species and either isometric or near isometric on the arithmetic scale. The heart in these mammals consequently grows in mass in approximate proportion to growth in mass by the body. The appearance of biphasic allometric growth in prior studies was an artifact resulting from logarithmic transformation. Although parturition in sheep and humans is accompanied by a change in the distribution of blood out of the heart and into pulmonary and systemic circuits, the challenge is met without marked increases in absolute or relative size of the heart.

Keywords: allometry, growth, heart, mammal, scaling

Relative growth by the heart of Horro sheep has been reported to be biphasic and discontinuous on the logarithmic scale, with a substantial increase in both relative and absolute mass of the heart occurring shortly after birth. However, the discontinuity was an artifact of logarithmic transformation. Both the logarithmic distribution and the arithmetic distribution are well described by straight lines.

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1. INTRODUCTION

A recent study appearing on the pages of this journal described a remarkable and previously undetected pattern of allometric growth by the heart of a mammal (Snelling et al., 2019). In brief, mass of the heart and mass of the body were measured in an ontogenetic series of Horro sheep (Ovis aries) ranging in age from fetus to near‐adult. When the observations were displayed on a bivariate graph with logarithmic coordinates, the heart appeared to experience an abrupt increase in mass immediately after parturition, but without a concomitant increase in body size. Thus, both absolute and relative size of the heart seemingly increased rapidly in newborns, presumably in response to new demands imposed on the heart while perfusing the lungs and delivering blood at higher pressure to the systemic circulation. Biphasic growth by the heart has been reported for other mammals, but the pattern has generally entailed a putative change in the rate of relative growth coincident with parturition rather than an abrupt increase in both absolute and relative size.

The study of sheep has apparent implications with respect to developmental processes, physiological integration, and evolution. However, the investigation was performed on logarithmic transformations, which are notoriously difficult to interpret (Menge et al., 2018), so the findings beg validation and interpretation on the original arithmetic scale (Finney, 1989). Moreover, elements of the statistical analysis raise questions about the protocol. For example, the authors say that they used a broken stick (=piecewise) regression analysis to confirm the presence of a breakpoint in the logarithmic distribution corresponding to the time of birth (Snelling et al., 2019). However, algorithms for piecewise regression fit straight lines to observations in adjacent, non‐overlapping ranges for the X‐variable (Nickerson, Facey, & Grossman, 1989; Ryan & Porth, 2007; Toms & Lesperance, 2003; Yeager & Ultsch, 1989). Close examination of figure 2b in Snelling et al. reveals an overlap in the distributions for the X‐variable for the fitted lines, so the analysis could not have been performed as stated. The location of the breakpoint needs to be resolved because that location supposedly provided the rationale for comparing prenatal animals with postnatal ones in subsequent analyses.

I accordingly undertook a new analysis of the data on Horro sheep using linear and nonlinear regression to study untransformed observations. Data from three other mammals also were examined to better establish the patterns of relative growth by the heart in different species. I find that relative growth by the heart is essentially isometric in all species for which data were recovered and that the concept of biphasic growth by the heart is an artifact of logarithmic transformation.

2. METHODS AND RESULTS

Data for Horro sheep were recovered from a file lodged online by Snelling (https://www.researchgate.net/profile/Edward_Snelling). However, the original data from studies of humans (Homo sapiens; Hirokawa, 1972), gray kangaroos (Macropus fuliginosus; Snelling, Taggart, Maloney, Farrell, & Seymour, 2015), and tammar wallabies (Macropus eugenii; Hulbert, Mantaj, & Janssens, 1991) were not readily available, so I used WebPlotDigitizer (https://automeris.io/WebPlotDigitizer) to capture data on heart (or ventricle) mass and body mass from graphs in the respective papers. Reverse engineering doubtless introduced some additional measurement error into the data, but such error is both small and random. Graphs based on data that I recovered by reverse engineering compare favorably with graphs in the original publications, and statistical models fitted to different sets of logarithmic transformations are virtually identical to those reported in the original papers.

Data for all four species were submitted to essentially the same analytical protocol, as described below. However, I begin with the treatment of Horro sheep because the potential import of the study necessitates a more detailed treatment than is required for the other species.

2.1. Horro sheep

I transformed data for the 10 fetuses and 11 postnatal animals to natural logarithms, displayed the transformations on a bivariate graph, and fitted a straight line to the distribution by ordinary least squares (Figure 1, 2). The straight line is a good visual fit to the transformations, and it accounts for 96% of the variation in the response variable (which is ln(Y), not untransformed Y). The slope for the line cannot be distinguished from 1 by the Wald confidence interval (0.96–1.29), but it slightly exceeds 1 by the likelihood ratio confidence interval (1.02–1.23). Standardized residuals are randomly distributed with respect to predictions from the equation (Figure 1B), and none of the residuals is so extreme as to mark it as an outlier. In short, the straight line is a good fit to the transformations, and the slope for the line is close to 1 (which would point to isometric growth).

Figure 1.

Figure 1

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(A) Logarithmic transformations for heart mass and body mass for 21 Horro sheep (Ovis aries) are well described by a straight line. The prediction interval shown by the dashed lines defines the range of values for ln(heart mass) in which 95% of future observations are expected to occur. (B) Standardized residuals from the fit of the straight line (Figure a) are randomly distributed with respect to predicted values, and none of the residuals qualifies as an outlier. (C) The piecewise regression fitted to logarithmic transformations of heart and body mass reveals a breakpoint at 1.68 along the X‐axis. AICc for the piecewise model is 5.1 higher than AICc for the straight line, so the straight line is the better descriptor for pattern in the observations (Burnham & Anderson, 2002)

Figure 2.

Figure 2

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(A) A straight line with intercept and a two‐parameter power equation are statistically equivalent functions for describing pattern in untransformed data for heart mass and body mass in pre‐ and postnatal sheep (Table 1). (B) Standardized residuals from the fit of the straight line to untransformed data for pre‐ and postnatal sheep (Figure A) are randomly distributed with respect to predicted values for heart mass. The computational algorithm in SAS has reversed the sign for each of the residuals. (C) Separate straight lines have been fitted to logarithmic transformations of data for pre‐ and postnatal sheep. The lines imply that two different power equations are needed to describe the original, untransformed distribution. (D) Separate power equations formed by back‐transforming the equations fitted to logarithms (Figure C)

SigmaPlot 10 (Systat Software, Inc.) was then used to fit a two‐segment, piecewise regression to the transformations (Ryan & Porth, 2007; Toms & Lesperance, 2003). A breakpoint was identified in the distribution, but the point of inflexion does not coincide with parturition (Figure 1C). The model accounts for 96% of the variation in the response variable (Figure 1c), which is the same as for the fit of the straight line (Figure 1A). The change in slope at the point of inflexion is not pronounced (Figure 1C), so I ranked and compared the piecewise regression with the straight line using Akaike’s Information Criterion for small samples, or AICc (Burnham & Anderson, 2002). In general, an AICc that differs from the best (i.e., the lowest) AICc by no more than two identifies a model that is equivalent to the best model; an AICc that differs from the reference by 3–7 identifies a plausible alternative to the best model; if the difference in AICc is 8–14, the model in question is only weakly supported; and if the difference is ≥15, the model has no support relative to the best model in the pool of candidates (Burnham, Anderson, & Huyvaert, 2011). Inasmuch as AICc for the piecewise model is 5.1 higher than AICc for the straight line (∆AICc = 0), it is apparent that the extra parameters in the piecewise model did little to improve the fit. The straight line consequently is the preferred fit.

The quality of the straight line fitted to logarithms, and the absence of a useful (“significant”) breakpoint in the logarithmic distribution, led me to examine the fit to the original observations of the two‐parameter power function that is formed by back‐transformation and to compare that power function with other possible models. I therefore fitted straight lines and power equations, with and without explicit intercepts, directly to the original data. Each functional equation was modeled with three different forms for random error: normal and homoscedastic error, normal and heteroscedastic, and lognormal and heteroscedastic (Packard, 2020). The models were fitted using the Model Procedure in SAS 9.4 (SAS Institute Inc.); script for the routine can be accessed via an online supplement to Packard (2019). One of the equations, of course, was a two‐parameter power function with lognormal error that is identical in every important respect to the two‐parameter power equation estimated indirectly via logarithms (Packard, 2015). The 12 models were ranked and compared using AICc (Burnham & Anderson, 2002).

Generally speaking, models with normal, homoscedastic error and normal, heteroscedastic error were not as good as corresponding models with lognormal, heteroscedastic residuals for capturing information in the data. I therefore focus on the models with lognormal error, and especially, the two‐parameter power equation because of its identity with the equation estimated by back‐transformation from the logarithmic domain.

All four of the models with lognormal error are good candidates for describing the data (Burnham et al., 2011), but the two‐parameter power equation is best and the straight line with an intercept is its equivalent (Table 1). However, the intercept for the straight line does not differ appreciably from zero (p = 0.22), and the exponent for the power equation is indistinguishable from or only marginally >1 (confidence intervals are the same as for the slope of the straight line fitted to logarithms). Both models capture the dominant pattern in the data (Figure 2A); both are approximations to straight lines passing through the origin; and both reveal a continuous increase in relative size of the heart (i.e., no abrupt increase at any point in development). Standardized residuals for the straight line are balanced (Figure 2B), and so too are those for the power equation (not shown). Either of the functions could be used to describe pattern in the bivariate distribution.

Table 1.

Summary of regression analyses on untransformed data for scaling of heart (or ventricular) mass versus body mass in four species of mammal

Species (n) Model functional form Log likelihood AICc ∆AICc
Horro sheep (21) Y = 6.32 X −64.04 132.8 2.7
Y = −0.92 + 6.97 X −62.21 131.8 1.7
Y = 5.59 X 1.12 −61.34 130.1 0
Y = −0.02 + 5.62 X 1.12 −61.34 133.2 3.1
Human (43) Y = 3.40 X −144.55 293.4 0.3
Y = 0.32 + 3.33 X −144.43 295.5 2.4
Y = 3.82 X 0.94 −143.39 293.4 0.3
Y = −2.18 + 5.29 X 0.84 −142.02 293.1 0
Kangaroo (29) Y = 6.57 X −97.18 198.8 6.8
Y = −0.18 + 6.83 X −92.50 192.0 0
Y = 6.23 X 1.03 −95.25 197.5 5.5
Y = −0.20 + 6.95 X 0.99 −92.43 194.5 2.5
Wallaby (27) Y = 5.46 X 23.92 −43.3 0
Y = 3.2e–5 + 5.46 X 23.92 −40.8 2.5
Y = 5.45 X 1.00 23.95 −40.9 2.4
Y  = 5.1e–4 + 5.44 X1.00 23.99 −38.2 5.1
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Random error is assumed to be lognormal and heteroscedastic. AICc = Akaike’s Information Criterion for small samples.

Finally, it is instructive to look at the patterns of relative growth exhibited by prenatal and postnatal animals separately. I therefore fitted straight lines to logarithmic transformations of observations for animals in the two groups and plotted the results on a bivariate graph (Figure 2C). The line for postnatal sheep clearly is elevated relative to the line for prenatal animals. This difference in elevation was the basis for the conclusion by Snelling et al. that a sudden spurt in absolute and relative size of the heart occurs immediately following parturition. However, logarithmic transformation can be quite deceptive (Menge et al., 2018) because it creates a new mathematical distribution that is compressed (relative to the original distribution) at the high end of the size range and expanded at the low end (Osborne, 2002; Smith, 1993). Thus, overall differences tend to be magnified at the low end of the size range and to be minimized (or masked) at the high end.

The straight lines fitted to logarithms in Figure 2C were back‐transformed, and the resulting two‐parameter power functions were displayed with the original data on a bivariate plot (Figure 2D). The curves were extended to the axes to facilitate comparison of the functions, which would be expected by many investigators to follow parallel courses (Glazier, 2020; Niklas & Hammond, 2019). The curves actually follow very different trajectories despite the parallel trajectories that the equivalent functions follow in the logarithmic plot (Figure 2C). The allometric exponents for the equations are identical, whereas it is the allometric coefficients that differ (Figure 2D). This finding confirms that graphical descriptions for patterns of relative growth depend as much on the allometric coefficients as on the allometric exponents (Packard, 2018).

Three points concerning the back‐transformed equations bear mention (Figure 2D). First, both the lines converge on the origin of the graph, as they must because the intercepts are implicitly 0. Second, the line for the postnatal animals affords a reasonable description for the combined sample, although it overshoots some of the observations for prenatal animals. And third, the line based on the equation for prenatal animals follows a shallow trajectory that seems to have been determined largely by two data points for the largest fetuses in the sample, that is, by two data points that are influential in the context of the sample of only 10 fetuses (see Figure 3 in Anscombe, 1973). The two points in question fall below the lowest values for postnatal sheep (Figure 2D) but are not as divergent as might be expected from a consideration of just the logarithmic transformations (Figure 2C). Indeed, the difference between the largest of the prenatal sheep and the smallest of the postnatal ones is little more than a reflection of the randomness that occurs in any biological sample (Figure 2A,B).

I suggest, therefore, that random variation in a small sample of sheep (n = 21), combined with transformation and the subsequent misidentification of the breakpoint in the logarithmic distribution, conspired to mislead the original authors into believing that a substantial, discontinuous increase in absolute and relative mass of the heart occurred immediately after parturition. Had the authors begun their investigation by graphing the observations on the arithmetic scale, they very likely would have arrived at a different conclusion concerning allometric growth by the heart. The original observations are well described by a straight line with an intercept near 0 and by a two‐parameter power equation with an exponent near 1, both of which point to a smooth, linear or quasi‐linear increase in mass of the heart as animals increase in body size (Figure 2A). The pattern of increase is arguably isometric. Although major changes occur in the circulation at the time of birth, the changes are not facilitated or supported by a spurt in absolute or relative growth by the heart.

2.2. Humans

I recovered measurements of ventricular mass and body mass for 13 newborn humans and 30 infants/adults from Figure 3 in Hirokawa (1972). His sample of infants/adults numbered 31, but one of the values apparently was obscured by an overlying data point and went undiscovered. I omitted from my compilation the data for newborns that died of pulmonary disease because cardiac development in these individuals may have been affected (Hirokawa, 1972).

Hirokawa plotted his data on a bivariate graph with logarithmic scaling and then fitted different straight lines to observations for newborns and for infants/adults. The slope for the line fitted to data for newborns was higher than the slope for the line for infants/adults, thereby giving the impression that relative growth by the ventricle is higher in newborns than in older individuals. However, Hirokawa may have been predisposed to expect different patterns of allometric growth by newborns and older individuals. My plot of logarithmic transformations reveals that the full distribution is described quite well by a straight line (Figure 3A). Standardized residuals are satisfactory (Figure 3B), so parameterization for the straight line is acceptable.

Figure 3.

Figure 3

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(A) Logarithmic transformations for ventricular mass and body mass for 43 humans (Homo sapiens) are well described by a straight line. Hirokawa (1972) reported 31 observations for infants/adults without cardiopulmonary disease, but I was able to identify only 30 observations in his figure 3. The prediction interval shown by dashed lines defines the range of values for ln(ventricle mass) in which 95% of future observations are expected to occur. (B) Standardized residuals from the straight line fitted to transformations (Figure 3A) are randomly distributed with respect to predictions and none is marked as an outlier. (C) A piecewise regression fitted to the log‐log distribution is statistically equivalent to the straight line (∆AICc = 0.5). The breakpoint consequently is of dubious importance. (D) All four of the models fitted to untransformed data provide good descriptions for pattern in the bivariate distribution (Table 1). The straight line with no intercept and the two‐parameter power equation are shown here because the former is the simplest function in the pool of candidates and the latter is the function that is produced by back‐transforming the equation fitted to logarithms

I next used SigmaPlot to fit a piecewise regression to the transformations (Figure 3C). The breakpoint does not distinguish newborns from older individuals, and the point of inflexion is hard to identify with the naked eye (Figure 3C). Evaluation of the piecewise fit by AICc indicates that this model is no better than the straight line for capturing pattern in the data (∆AICc = 0.5), so there is no need to invoke biphasic allometric growth by the ventricle of postnatal humans.

Because a breakpoint was not demonstrable, I proceeded to consider different statistical models for describing untransformed observations. All four models with lognormal error were acceptable fits (Table 1). However, the straight line with no intercept is the simplest model, and the two‐parameter power function is the same equation that is estimated by back‐transforming the straight line fitted to logarithms. Both these functions capture the dominant pattern in the arithmetic distribution (Figure 3D). The Wald and likelihood confidence intervals for the exponent in the power equation include 1, which is the exponent in the equation for a straight line (Huxley, 1932). Thus, allometric growth by the ventricle (and presumably the entire heart) of postnatal humans is isometric because both the models fitted to untransformed data are for straight lines passing through the origin.

2.3. Kangaroo

Snelling et al. (2015) measured mass of the heart and mass of the body for 29 gray kangaroos, 11 of which were joeys that had yet to emerge from their mother’s pouch. The data were displayed on a graph with logarithmic coordinates, thereby revealing a breakpoint in the distribution that seemed to correspond with emergence of joeys from the pouch (Snelling et al., 2015). The rate of allometric growth for joeys (in pouch) appeared to be greater than that for out‐of‐pouch juveniles and adults. Thus, relative growth by the heart seems to conform to a pattern of biphasic, loglinear allometry.

I fitted a straight line (Figure 4a) and a piecewise regression (Figure 4b) to logarithmic transformations to replicate steps followed in the original study. The straight line provides a good fit that agrees with the fit reported by Snelling et al. (2015). However, the piecewise regression is much better than the straight line for describing pattern in the observations (∆AICc = 23.8 for the straight line). The breakpoint separates joeys from older animals in the sample (Figure 4B). This finding is consistent with the finding of the original study and is the basis for their conclusion that allometric growth by the heart changes at the time joeys emerge from the pouch.

Figure 4.

Figure 4

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(A) A straight line has been fitted to logarithmic transformations of heart mass and body mass in a sample of kangaroos (Macropus fuliginosus) comprised 11 joeys and 18 juveniles and adults. The two smallest juveniles are relatively high‐leverage observations (Studentized Deleted Residuals = 2.2 and 3.2), but the fitted line nonetheless explains 99% of the variation in the response variable. The prediction interval shown by dashed lines defines the range of values for ln(heart mass) in which 95% of future observations are expected to occur. (B) The piecewise regression model explains only slightly more of the variation in the response variable, but AICc for the piecewise fit indicates that it is substantially better than the straight line (∆AICc for the straight line = 23.8). (C) The best model in the pool of candidates is the straight line with intercept (Table 1). Note that observations for the two smallest juveniles lie closer to the fitted line than do observations for larger individuals in the sample, which is counter to perceptions based on the logarithmic distribution (Figure 4A). (D) Standardized residuals for the two smallest juveniles deviate some from other residuals, but neither of them is so extreme as to be branded an outlier. Note again that the computational algorithm in SAS has reversed the sign for each of the residuals

The appearance of biphasic, loglinear allometry is arguably an artifact resulting from unwitting misspecification of the functional form for an equation describing pattern in the original, untransformed data (Sartori & Ball, 2009). The unidentified function in a situation like this typically has an explicit, non‐zero intercept. When data for the kangaroos are expressed and graphed on the arithmetic scale, they can be described quite well by the equation for a straight line with intercept that explains 97% of the variation in heart mass (Figure 4C). The three‐parameter power equation is statistically equivalent to the straight line (Table 1), and the fitted parameters for the power equation are virtually identical to those for the straight line. In neither case does the intercept differ significantly from 0 (p ≥ 0.16), so the fitted equations pass very close to the origin. Standardized residuals for the straight line are acceptable, although the two lowest values may be of some concern (Figure 4d). These two residuals are for the two smallest juveniles and probably were the cause for non‐linearity in log domain. Note the locations for these two observations along the X‐axis for the graphs with logarithmic (Figure 4A) and linear scaling (Figure 4C). Because the X‐axis has been compressed at the high end of the size distribution for logarithms, the observations in question have a greater influence on the statistical analysis of logarithms than they do on the analysis of arithmetic data. Thus, the appearance of biphasic growth allometry may be nothing more than an illusion resulting from logarithmic transformation of a relatively small data set. The pattern of allometric growth by the heart appears, again, to approximate isometry.

2.4. Wallaby

Hulbert et al. (1991) measured masses of heart and body in 27 wallabies ranging in age from early pouch stage to adult. I recovered data from their Figure 2A, transformed them to natural logarithms, and plotted them on a bivariate graph (Figure 5A). Animals weighing <400 g were assumed to be joeys that had not emerged from the pouch (Hulbert et al., 1991). A straight line fitted to the log‐log distribution is an excellent fit (Figure 5A), and standardized residuals are well balanced (Figure 5B). The model points to a pattern of isometric growth by the heart, but alternative models should be considered before arriving at this conclusion.

Figure 5.

Figure 5

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(A) A straight line has been fitted to logarithmic transformations of heart mass and body mass in a sample of wallabies (Macropus eugenii) comprised approximately 15 joeys and 12 juveniles and adults. The number of joeys was estimated at 15 by using 400 g as the body mass for animals that have just emerged from their mother’s pouch for the first time (Hulbert et al., 1991). (B) Standardized residuals from the log‐log fit are balanced and well behaved. (C) The straight line with no intercept is the best fit to the original data and the simplest function (Table 1)

I therefore fitted four different equations (with lognormal error) directly to the untransformed observations (Table 1). All four of the equations are arguably for straight lines, but the fit of the straight line without an intercept is both best and simplest (Table 1). The straight line is an excellent fit (Figure 5C), thereby confirming conclusions of isometric growth by the heart (Hulbert et al., 1991).

3. DISCUSSION

Three different patterns of relative growth by the heart have been reported in the literature: monophasic, loglinear (isometric) growth in wallabies; biphasic, continuous, loglinear growth in humans and kangaroos; and biphasic, discontinuous, loglinear growth in sheep. My new analyses confirm the pattern of isometric growth in wallabies but lead to different conclusions concerning relative growth by the heart in sheep, humans, and kangaroos. Relative growth by the heart in the latter three species is described quite well on the arithmetic scale by a straight line passing through, or near, the origin, or by a two‐parameter power equation that approximates a straight line. In other words, relative growth by the heart in all the species approximates isometry. In any event, the notion of biphasic growth is not supported.

Misidentification by the original authors of the pattern of relative growth in three of the species can be traced to logarithmic transformations, which altered the bivariate distributions for the data in ways that were neither obvious nor desirable. The problem was then confounded by failure to validate the fitted models on the original arithmetic scale. This unfortunate situation can be traced to the widespread embrace by students of allometry of a research protocol promoted by Julian Huxley in his monograph on Problems of Relative Growth (Huxley, 1932), that is, fitting a straight line (or lines) to logarithmic transformations and then interpreting the outcome in the context of the original, untransformed observations. Huxley’s protocol never worked all that well (Smith, 1984)—witness his treatment of relative growth by the crusher claw in male fiddler crabs (Packard, 2012)—but it was the best protocol that was available in his time for fitting a simple power equation to data following a curvilinear path on the arithmetic scale. Because of ease of application, the protocol was widely adopted, and even now it is the method of choice. However, Huxley’s approach has been superseded by newer methods: the simple power equation (and other models) now can readily be fitted to untransformed observations by nonlinear regression. It always is preferable to work with arithmetic data because the arithmetic scale is the scale on which interpretations are made.

CONFLICT OF INTEREST

The author declares that he has no conflict of interest, financial, or otherwise.

ACKNOWLEDGEMENTS

I thank Craig White and an anonymous reviewer of the manuscript for constructive suggestions that helped me to improve the presentation. This investigation was funded by the author out‐of‐pocket and received no financial support from any agency in the governmental, academic, or private sector.

Packard GC. Is relative growth by the mammalian heart biphasic or monophasic?. J. Anat.2021;239:242–250. 10.1111/joa.13286

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