Understanding of the factors involved in determining the level of central arterial blood pressure in mammals has been clouded by inappropriate allometric analyses that fail to account for phylogenetic relationships among species, and require pressure to approach 0 as body size decreases. The present study analyses systolic, mean arterial, and diastolic blood pressure in 47 species of mammal with phylogenetically informed techniques applied to two-parameter equations. It also sets nonlinear, three-parameter equations to the data to remove the assumption of the two-parameter power equation that the smallest animals must have negligible blood pressure. These analyses show that blood pressure increases with body size. Nonlinear analyses show that mean blood pressure increases from 93 mmHg in a 10 g mouse to 156 mmHg in a 4 tonne elephant. The scaling exponent of blood pressure is generally lower than, though not significantly different from, the exponent predicted on the basis of the expected scaling of the vertical distance between the head and the heart. This indicates that compensation for the vertical distance above the heart is not perfect and suggests that the pressure required to perfuse the capillaries at the top of the body may decrease in larger species.
The question of whether gravity influences blood pressure, cardiac work, and heart position in the circulatory systems of vertebrate animals has been controversial (Patterson et al. 1965; Seymour and Lillywhite 1976; Badeer 1986; Hicks and Badeer 1989, 1992; Seymour et al. 1993; Pedley et al. 1996; Badeer 1997; Seymour 2000; Seymour and Blaylock 2000; Seymour and Lillywhite 2000; Gisolf et al. 2005; Hicks and Munis 2005; Mitchell et al. 2006; Gartner et al. 2010; Gartner et al. 2011; Lillywhite and Seymour 2011; Lillywhite et al. 2012). On one hand, the arterial pressure developed by the left ventricle is considered to have two conceptual components—the pressure necessary to perfuse the capillary beds of the body, plus the hydrostatic pressure generated by the vertical distance between the heart and the head (Mitchell et al. 2006). Because the total height of the blood column above the heart increases with animal size, and the hydrostatic pressure at the bottom of a column of fluid is calculated as the product of fluid density (ρ), gravitational acceleration (g), and the vertical height of the column (h), then central systemic arterial blood pressure should be positively related to body size. Consequently, the hearts of larger animals must work harder against gravity, not only to pump blood to the head, but also to pump against a higher peripheral resistance of the body that is necessary to maintain the elevated blood pressure. If, on the other hand, the siphon principle operates in the continuous, closed loops above the heart, the heart does no work against gravity, because gravity acts equally on the ascending and descending limbs of the circuit (Hicks and Badeer 1992). Assuming that the pressure necessary to perfuse the capillary beds of the body is independent of size, then, in the first case, central arterial systemic blood pressure should be influenced by the vertical distance above the heart, but in the second case, blood pressure should be size-invariant.
The hypothesis that the heart works against gravity can be tested by examining the scaling of arterial blood pressure and determining if the scaling exponent of blood pressure is significantly greater than 0. Because the aspect ratio of most mammals does not vary systematically with body mass (Roberts et al. 2010), they can be considered geometrically similar for the purposes of predicting the scaling of hydrostatic pressure. For geometrically similar objects, linear dimensions are proportional to the cube root of volume, so it can be predicted that the hydrostatic component of blood pressure (ρgh) should be proportional to mass raised to the 0.33 power. However, previous studies report scaling exponents of blood pressure of around 0.03 (Günther 1975) to 0.05 (Seymour and Blaylock 2000). This effect of size on blood pressure is regarded as small (Gartner et al. 2011), and is certainly smaller than the exponent of 0.33 expected based on variation in the vertical distance between the heart and head observed in mammals.
Previous analyses of the scaling of blood pressure have been criticized because they fail to incorporate phylogenetic information (Hicks and Munis 2005; Gartner et al. 2011). Many, but not all, traits show significant phylogenetic signal (Freckleton et al. 2002; Blomberg et al. 2003; Losos 2011), and phylogenetically informed analyses may yield differences in estimates of scaling exponents as compared with nonphylogenetic estimates (Garland et al. 2005). Thus, there is a clear need to reevaluate the scaling of mammalian blood pressure to determine if they conform to the predictions based on size-dependent changes in hydrostatic pressure. A further unidentified problem with previous analyses is that scaling exponents are estimated based on log–log transformed data for pressure (P, mmHg) and body mass (M, kg) by fitting a linear relationship of the form:
which, when untransformed, is converted to a two-parameter power equation:
A critical assumption of such an approach is that the original data conform with a power function having an intercept of 0 in a plot with arithmetic coordinates (Packard and Boardman 2008). This assumption is rarely checked and is not always met (Packard and Boardman 2009; MacLeod 2010; Packard et al. 2011). Assuming that the heart must work against resistance to perfuse tissues and potentially also against gravity, a more appropriate statistical model of the relationship between blood pressure and body mass is a three-parameter power equation:
This statistical model introduces an intercept term, c, and thereby removes the assumption of the two-parameter power equation that the smallest animals must have negligible blood pressure.
In the present study, we test the hypothesis that blood pressure is positively associated with body mass due to increased hydrostatic pressure above the heart in larger mammals. We test for a relationship between systolic, mean, and diastolic arterial blood pressure (Psystolic, Pmean, and Pdiastolic, respectively) and body mass using phylogenetic generalized least squares (PGLS; Grafen 1989; Martins and Hansen 1997; Garland and Ives 2000) to fit two-parameter power equations to log–log transformed data from 47 species, virtually doubling the sample size of the previous study that was limited to 24 because of requirements for additional data on heart size and function (Seymour and Blaylock 2000). We also use nonlinear regression to fit three-parameter power equations to untransformed data, and compare the values of the scaling exponent to those predicted on the basis of variation in hydrostatic pressure. We analyze the data with and without inclusion of the long-necked giraffe, and we run a sensitivity analysis to assess experimental errors and variability.
Published data for body mass and blood pressure of 47 species of terrestrial mammal were compiled from the literature (Psystolicn = 41 species, Pmeann = 44 species, Pdiastolic = 39 species; see Table S1) and matched to a supertree (Bininda-Emonds et al. 2007), which was then pruned to include only species for which data were available (Pdiastolic: File S1; Pmean: File S2; Psystolic: File S3). Where mean arterial pressure was not provided, it was estimated as Pdiastolic + [(Psystolic − Pdiastolic)/3], where possible (Rushmer 1976). In addition to the dated branch lengths associated with the supertree, a range of branch length transformations were also considered: star, loge, punctuated (all branches set equal and equal to one), Grafen's (Grafen 1989), Nee's (Purvis 1995), and Pagel's (Pagel 1992). Because giraffes are among the largest animals in the dataset, are abnormally shaped compared to other mammals, and have the highest blood pressure of all species in the analysis, we conducted all analyses both with and without the giraffe included. This was done to examine the generality of our conclusions and ensure that the high leverage exerted by giraffes on the relationship between blood pressure and mass did not compromise our conclusions.
PHYLOGENETIC GENERALIZED LEAST SQUARES
The relationship between log(P) and log(M) was analyzed using PGLS (Grafen 1989; Martins and Hansen 1997; Garland and Ives 2000) in the Analysis of Phylogenetics and Evolution (APE) package (Paradis et al. 2004) within R (R Development Core Team 2012) according to established procedures (Halsey et al. 2006; Duncan et al. 2007; White et al. 2009). For each of these models, a measure of phylogenetic correlation, λ (Pagel 1999; Freckleton et al. 2002), was estimated by fitting PGLS models with different values of λ and finding the value that maximizes the log likelihood. The degree to which trait evolution deviates from Brownian motion (λ = 1) was accommodated by modifying the covariance matrix using the maximum likelihood value of λ, which is a multiplier of the off-diagonal elements of the covariance matrix (i.e., those quantifying the degree of relatedness between species). All models were compared on the basis of Akaike's Information Criterion (AIC) as a measure of model fit (Burnham and Anderson 2010). The probability that any given model is actually the best fit out of those tested was measured by its Akaike weight (wi), calculated as the likelihood of the model divided by the sum of the likelihoods of all other models (Burnham and Anderson 2010).
PGLS: SENSITIVITY ANALYSIS
Blood pressure is likely to be influenced by many of the procedures employed during its measurement including chemical restraint and stress associated with handling and mechanical restraint (e.g., Peshin et al. 1991; Brøndum et al. 2009; Paterson et al. 2009). To ensure that the present study is robust to these possible influences, the sensitivity of the relationships between mass and blood pressure to the introduction of random noise was determined. Using log-transformed blood pressures, a randomly generated normal deviate with a mean of 0 and a standard deviation of 0.1 was added to the data for each species. This standard deviation is close to the observed standard deviation of the log-transformed data for all species, without accounting for the effect of size on blood pressure (Psystolic: SD = 0.089, Pmean: SD = 0.094, Pdiastolic: SD = 0.099). The relationship between log(P) and log(M) was then analyzed using PGLS as described above, and the scaling exponent of P for the best of all evolutionary models was selected on the basis of AIC. This procedure was then repeated a total of 1000 times for each of systolic, mean, and diastolic arterial pressures using a custom script written in R, and the distribution of scaling exponents was examined. The relationship between log(P) and log(M) was considered robust to the introduction of noise if less than 5% of the 1000 scaling exponents differed in sign from the scaling exponent observed for the original data.
Two- and three-parameter power equations (P = aMb and P = aMb + c, respectively) were fitted to untransformed data by iteration (Gauss–Newton algorithm) using the “nls” function in R (R Development Core Team 2012). The significance of the nonlinear relationships was assessed using likelihood ratio tests to determine if the two- and three-parameter relationships differed significantly from equivalent relationships with the scaling exponents of blood pressure fixed at either 0 or 0.33. Although PGLS is capable of modeling any relationship that can be modeled by ordinary least squares regression, including linear, polynomial, and break-point relationships (Garland and Ives 2000; Whitney et al. 2011), a phylogenetically informed implementation of nonlinear regression is not currently available (Quader et al. 2004; Cooper et al. 2008; Codron and Clauss 2010). Thus, because phylogenetic regression assumes that the residual error in the regression model is distributed according to a multivariate normal distribution with variances and covariances proportional to the historical relations of the species in the sample (Revell 2010), we tested the validity of nonphylogenetically informed nonlinear regression by testing for phylogenetic signal in the residuals of the three-parameter power equation. To do so, we calculated the value of K (Blomberg and Garland 2002; Blomberg et al. 2003) using the picante package within R (Kembel et al. 2010), and tested the significance of K using randomization tests (10,000 iterations). A sensitivity analysis was not conducted for nonlinear regression because nonlinear regression can be sensitive to the choice of starting values, and this prevents automated analyses.
PHYLOGENETIC GENERALIzED LEAST SQUARES
The best statistical model for log-transformed systolic blood pressure included log-transformed body mass (Σwi = 0.9999); the best overall model included logM and modeled evolution on a star phylogeny (Psystolic = 115M0.05 ± 0.01 [95% CI], λ = 0, wi = 0.26, Fig. 1A). Maximum likelihood values of λ were equal to 0 for all evolutionary models; the only model of all evolutionary and statistical models that included a nonsignificant effect of logM modeled evolution on a tree with dated branch lengths with λ set to 1, and was very poorly supported (wi < 0.00001). With the giraffe excluded, the best statistical model for log-transformed systolic blood pressure included log-transformed body mass (Σwi = 0.9999); the best overall model included logM and modeled evolution on a phylogeny with Nee's arbitrary branch lengths (Psystolic = 116M0.04 ± 0.02, ML λ = 0.21, wi = 0.23).
The best statistical model for log-transformed mean blood pressure included log-transformed body mass (Σwi = 0.98); the best overall model included logM and modeled evolution on a star phylogeny (Pmean = 99M0.04 ± 0.01, λ = 0, wi = 0.29, Fig. 1B). Maximum likelihood values of λ were equal to 0 for all evolutionary models. Some models with λ = 1 included nonsignificant parameter estimates for logM (dated, Grafen's, and Pagel's branch lengths), but these models were poorly supported (wi ≤ 0.002); all other statistical and evolutionary models included significant parameter estimates for logM. With the giraffe excluded, the best statistical model for log-transformed systolic blood pressure included log-transformed body mass (Σwi = 0.94); the best overall model included logM and modeled evolution on a star phylogeny (Pmean = 99M0.03 ± 0.02, λ = 0, wi = 0.23).
The best statistical model for log-transformed diastolic blood pressure included log-transformed body mass (Σwi = 0.74); the best overall model included logM and modeled evolution on a star phylogeny (Pdiastolic = 82M0.03 ± 0.01, λ = 0, wi = 0.23, Fig. 1C). Models with λ = 1 included nonsignificant parameter estimates for logM, but these models were relatively poorly supported (wi ≤ 0.06). The only model with a maximum likelihood value of λ greater than 0 (λ = 1, with evolution on tree with branch lengths equal and equal to 1) included a nonsignificant parameter estimate for logM and was also poorly supported (wi = 0.02). With the giraffe excluded, the best statistical model for log-transformed systolic blood pressure included log-transformed body mass (Σwi = 0.82); the best overall model included logM and modeled evolution on a phylogeny with Nee's arbitrary branch lengths (Pdiastolic = 84M0.03 ± 0.02, ML λ = 0.47, wi = 0.21).
PGLS: SENSITIVITY ANALYSIS
All of the PGLS relationships were robust to the addition of randomly generated noise to the data. Of the 1000 calculated exponents, 0.1% were negative, 0.3% were negative for mean blood pressure, and 3.1% were negative for diastolic blood pressure. These values are all below the 5% threshold imposed for significance (see Methods), thereby demonstrating that the addition of random variance equal in magnitude to the among-species variance in blood pressure does not alter the conclusion that the PGLS scaling exponents of blood pressure are greater than 0.
The two-parameter power relationship between Psystolic and M estimated by nonlinear regression (Psystolic = 115M0.05) was identical to that estimated by linear regression of log–log transformed data using PGLS, with a scaling exponent significantly different from both 0 (χ21 = 22.7, P < 0.001) and 0.33 (χ21 = 108.2, P < 0.001). The three-parameter relationship, on the other hand, had a higher scaling exponent (Psystolic = 21M0.18 + 95, Fig. 1A) that was significantly greater than 0 (χ22 = 24.5, P < 0.001) and not significantly different from 0.33 (χ21 = 2.07, P = 0.15). The largest value of K for the residuals from the three-parameter power equation (0.53) was for a tree with Nee's arbitrary branch lengths and was not significant (P = 0.12). With the giraffe excluded, the exponent in the three-parameter relationship decreased slightly (Psystolic = 19M0.17+98), but remained significant (χ22 = 25.1, P < 0.001) and not significantly different from 0.33 (χ21 = 1.93, P = 0.17).
The two-parameter power relationship between Pmean and M estimated by nonlinear regression (Pmean = 98M0.05) was similar to that estimated by linear regression of log–log transformed data using PGLS, with a scaling exponent that was significantly different from both 0 (χ21 = 14.6, P < 0.001) and 0.33 (χ21 = 100.6, P < 0.001). The three-parameter relationship had a higher scaling exponent (Pmean = 9.8M0.23 + 90, Fig. 1B), which was significantly different from 0 (χ22 = 17.3, P < 0.001) and not significantly different from 0.33 (χ21 = 0.66, P = 0.42). Significant phylogenetic signal for the residuals from the three-parameter power equation was detected for trees with Grafen's arbitrary branch lengths (K = 0.08, P = 0.04) and Pagel's arbitrary branch lengths (K = 0.30, P = 0.04). With the giraffe excluded, the exponent in the three-parameter relationship increased slightly (Pmean = 6.9M0.24 + 93) and was again significantly greater than 0 (χ22 = 16.9, P < 0.001) and not significantly different from 0.33 (χ21 = 0.39, P = 0.53).
The two-parameter power relationship between Pdiastolic and M estimated by nonlinear regression (Pdiastolic = 82M0.04) was similar to that estimated by linear regression of log–log transformed data using PGLS, with a scaling exponent that was significantly different from both 0 (χ21 = 7.35, P = 0.007) and 0.33 (χ21 = 85.3, P < 0.001). The three-parameter relationship had a higher scaling exponent (Pdistolic = 2.4M0.36 + 81, Fig. 1C), which was significantly greater than 0 (χ22 = 11.4, P = 0.003) and not significantly different from 0.33 (χ21 = 0.02, P = 0.90). Significant phylogenetic signal for the residuals from the three-parameter power equation was detected for trees with Grafen's arbitrary branch lengths (K = 0.08, P = 0.03) and Pagel's arbitrary branch lengths (K = 0.32, P = 0.02). With the giraffe excluded, the exponent in the three-parameter relationship increased (Pdiastolic = 0.75M0.47 + 47), was significantly greater than 0 (χ22 = 10.2, P < 0.006), and not significantly different from 0.33 (χ21 = 0.35, P = 0.55).
It is generally thought that arterial blood pressure is size independent in birds and mammals (Günther 1975; Schmidt-Nielsen 1986; Calder 1996; Dawson 2001). Empirical allometry has yielded exponents near 0.03–0.05 (Günther 1975; Seymour and Blaylock 2000), which are apparently close enough to 0 to confirm the idea in many minds. In the present study, the best two-parameter power equations fitted by PGLS modeled evolution on a star phylogeny for systolic, mean, and diastolic pressure, with scaling exponents of 0.05, 0.04, and 0.03, respectively. However, no earlier allometric analysis has considered that the regression of the data cannot reasonably pass through the origin. Fitting the standard two-parameter power equations to blood pressure data is therefore clearly wrong (Seymour and Blaylock 2000). The differences between the two equations at selected body masses illustrate the extent of the error. The two-parameter equation underestimates systolic, mean, and diastolic pressures by 10–12% for a 10 g mammal, predicts pressures to within 0.4% for a 120 kg mammal, and underestimates them by 9–17% for a 4000 kg mammal. The differences between the equations are appreciable only at small and large body sizes where the two-parameter equation results in underestimates. Therefore the three-parameter equation indicates the increase in blood pressure in larger mammals is higher than originally thought (Seymour and Blaylock 2000).
The major conclusion of the present analysis is that systemic blood pressure in mammals increases with body mass, no matter how it is analyzed. Similarly, some studies of humans have shown that blood pressure increases with body height (Voors et al. 1982; Arvedsen et al. 2012), though this finding is not ubiquitous (e.g., Kesteloot and van Houte 1974; Dyer et al. 1990). For the relationship between blood pressure and body mass in mammals, using PGLS to fit two-parameter power equations yields maximum likelihood values of λ equal to 0 in the best models, indicating no evidence of phylogenetic nonindependence in these data, with scaling exponents of 0.05 for systolic, 0.04 for mean arterial, and 0.03 for diastolic pressures. All exponents are significantly higher than 0, even when the data for the exceptional giraffe are removed.
It might be argued that arterial blood pressure increases in larger animals simply because they are longer, and according to the Poiseuille–Hagen relationship, longer distributive vessels would require a proportionately greater pressure drop across them. However, scaling of cardiac output and major blood vessel size in mammals reveals that the pressure drop should actually decrease in larger animals. Heart rate in mammals is proportional to M−0.26 (White and Kearney 2014) and stroke volume scales as M1.03 (Seymour and Blaylock 2000), so cardiac output scales as M0.77 (Stahl 1967 reports a slightly higher exponent of 0.81 for cardiac output). The radii and lengths of aortas of seven mammalian species from mice to cattle scale as M0.33 (Holt et al. 1981). Therefore, the pressure drop should scale as M−0.22 = M0.77 (M0.33)4 / M0.33. It is also well known that the major pressure drop occurs in the smallest arteries, arterioles, and capillaries of individual vascular beds, not in the major distributive arteries. Therefore, size alone cannot account for the higher arterial blood pressure in larger mammals.
The exponents from two-parameter equations are much lower than 0.33, the expected exponent if gravity were the only influence on blood pressure. However, each tissue bed requires a “perfusion pressure,” which is the difference between arterial and venous pressures that is necessary to produce an adequate flow rate against the viscous resistance in the particular bed. Assuming that this pressure is independent of body size, the three-parameter, nonlinear analysis attempts to quantify this as the constant (c) in equation (3). The resulting exponents from the gravitational component of the equation are 0.18 for systolic, 0.23 for mean arterial, and 0.36 for diastolic pressures, all of which are not significantly different from 0.33. Nevertheless, the variability is high, and it is possible that systemic tissue perfusion pressure is indeed influenced by body size. A deeper allometric analysis involving cardiac output and viscous resistance would be necessary to differentiate the two components of blood pressure. Furthermore, the presence of phylogenetic signal in the residuals from the three-parameter power relationship for mean arterial and diastolic pressures indicates that these relationships should be reassessed when phylogenetic information can be incorporated into the results.
There is in fact evidence that tissue perfusion pressure does decrease in the superior tissues of larger mammals. Given that central blood pressure must support the vertical component of the column of blood above the heart and an additional perfusion pressure to achieve blood flow at the top, we can estimate the minimum perfusion pressure at the top of the animal by subtracting the vertical component. In small mammals, where the vertical component is negligible, the intercept of the three-parameter equation in the present study is 90 mmHg (Fig. 1B), which approximates the perfusion pressure, assuming venous pressure is 0. Subtracting the vertical component in larger mammals results in lower perfusion pressures. For example, a 4000 kg African elephant stands about 3 m high at the shoulder and the heart is approximately 1.35 m below that (Laursen and Bekoff 1978). The vertical distance above the heart is equivalent to (1.35 m × 77 mmHg m−1) 104 mmHg. If systemic blood pressure were independent of body size and mean arterial pressure is 100 mmHg as often thought, there would be no perfusion pressure and no blood flow to the shoulders. However, a 4000 kg African elephant has a mean arterial blood pressure of 154 mmHg (according to our three-parameter allometric equation) or 144 mmHg (according to Honeyman et al. 1992), which gives a perfusion pressure of 40–50 mmHg at the shoulders. It is not a coincidence that this perfusion pressure is approximately the same as in a full-grown giraffe. Central arterial pressure in giraffes is about 200 mmHg (Hargens et al. 1987; Brøndum et al. 2009; Mitchell and Skinner 2009), and the head can be 2 m above the heart (Mitchell and Skinner 2009), giving a calculated perfusion pressure of 45 mmHg (200–155 mmHg). Measured cranial perfusion pressures in giraffes are similar to other mammals (Brøndum et al. 2009), including humans that also have a cranial arterial blood pressure of about 50 mmHg (Burton 1965). Therefore, it appears that blood pressure decreases with gravity as the vertical distance above the heat increases, as has been shown experimentally for giraffes (Brøndum et al. 2009), and that a positive perfusion pressure is maintained at the top of large, standing mammals. Further evidence for the role of gravity in evolution of mammalian blood pressure comes from examination of the scaling exponents of blood pressure with body mass, which are not significantly different from the expected value based on the vertical distance between the head and the heart (0.33). However, the exponents of systolic and mean arterial pressure are nonetheless somewhat lower than 0.33. Systolic and mean arterial pressures scale with exponents of 0.18 and 0.23, respectively, thereby compensating imperfectly for the increase in the vertical component of distance above the heart with size. If cranial arterial blood pressure is size-independent, then a decrease in the pressure necessary to perfuse the capillary beds of the body is predicted to be required to ensure adequate cranial perfusion. Small mammals may require higher perfusion pressures possibly associated with higher requirements for oxygen delivery associated with higher mass-specific metabolism, but this is speculation. The scaling of perfusion pressure would be interesting to examine allometrically, but it would require more detailed knowledge of vertical components of the circulatory system, resistances in vascular beds, cardiac output, and blood pressures in each species.
C. Farmer, T. Wang, and an anonymous reviewer provided comments that helped us improve an earlier version of the manuscript. CRW is an Australian Research Council QEII Research Fellow (project DP0987626).