A General Biphasic Bodyweight Model for Scaling Basal Metabolic Rate, Glomerular Filtration Rate, and Drug Clearance from Birth to Adulthood

Abstract

The objective of this study is to propose a unified, continuous, and bodyweight-only equation to quantify the changes of human basal metabolic rate (BMR), glomerular filtration rate (GFR), and drug clearance (CL) from infancy to adulthood. The BMR datasets were retrieved from a comprehensive historical database of male and female subjects (0.02 to 64 years). The CL datasets for 17 drugs and the GFR dataset were generated from published maturation and growth models with reported parameter values. A statistical approach was used to simulate the model-generated CL and GFR data for a hypothetical population with 26 age groups (from 0 to 20 years). A biphasic equation with two power-law functions of bodyweight was proposed and evaluated as a general model using nonlinear regression and dimensionless analysis. All datasets universally reveal biphasic curves with two distinct linear segments on log–log plots. The biphasic equation consists of two reciprocal allometric terms that asymptotically determine the overall curvature. The fitting results show a superlinear scaling phase (asymptotic exponent >1; ca. 1.5–3.5) and a sublinear scaling phase (asymptotic exponent <1; ca. 0.5–0.7), which are separated at the phase transition bodyweight ranging from 5 to 20 kg with a mean value of 10 kg (corresponding to 1 year of age). The dimensionless analysis generalizes and offers quantitative realization of the maturation and growth process. In conclusion, the proposed mixed-allometry equation is a generic model that quantitatively describes the phase transition in the human maturation process of diverse human functions.

Graphical Abstract

Appendices

APPENDIX

Characteristics of the Biphasic Model

This study proposes a mixed-power function (the biphasic allometry model) (Equation 4) to scale human BMR, GFR, and drug CL from adults to neonates. In fact, this study was inspired by allometric scaling of whole-plant respiration from small seedlings to giant trees (12). Equation 4 states that the reciprocal of a physiological or metabolic parameter (P) is equal to the sum of reciprocals of each of two individual power-law bodyweight functions (i.e., allometric functions). To understand the key features of the model, a simulation was conducted to characterize the mixed-power function. The following hypothetical parameters were used: A = 0.1, B = 10, α= 2.5, and β= 0.5. By plugging the parameter values into Equation 4, a set of hypothetical (P, W) data was generated for a simulated bodyweight range of 3–70 kg (at a 1-kg interval). Figure 10A shows the simulated data and two asymptotic lines on log–log coordinates. Let P₁ and P₂ be the first and second asymptotic functions, respectively, and then the asymptotic bodyweight functions are

$$P_1=AW^\alpha$$

(23)

$$P_2=BW^\beta$$

(24)

and Equation 4 can be expressed as

$$\frac{1}{P}=\frac{1}{P_1}+\frac{1}{P_2}$$

(25)

when P₁ ≪ P₂, P ≈ P₁, which is the first asymptote near the low bodyweight region of W ≪ (B/A)^{1/(α − β)} (by solving the inequality relationship). In contrast, when P₁ ≫ P₂, P ≈ P₂, the second asymptote occurs at W ≫ (B/A)^{1/(α − β)}. Thus, the characteristic bodyweight ($\overset{\sim }{w}$) is revealed as

$$\widetilde w=\left(B/A\right)^{1/\left(\alpha-\beta\right)}$$

(26)

Mathematically, $\overset{\sim }{w}$ is exactly the bodyweight at which the two asymptotic lines intersect (P₁ = P₂ and $A{\overset{\sim }{w}}^{\alpha }=B{\overset{\sim }{w}}^{\beta }$). Biologically, it is considered here as the critical bodyweight at which the body undergoes transition from maturation phase to growth phase (i.e., the phase transition bodyweight). By taking the derivative of Equation 4, the following expression for the slope (S) of the biphasic curve is obtained:

$$\mathit{S}=\frac{d\log P}{d\log W}=\beta \left(\frac{P_1}{P_1+{P}_2}\right)+\alpha \left(\frac{P_2}{P_1+{P}_2}\right)$$

(27)

Therefore, according to Equation 27, the slope of the biphasic model can be estimated at any given bodyweight. Figure 10B shows that the slope decreases with increasing bodyweight. The slope curve is indeed flanked by two asymptotic lines: slope ≈ α, at the low body weight region where $W\ll \overset{\sim }{w}$ or when P₁ ≪ P₂; whereas slope ≈ β, at $W\gg \overset{\sim }{w}$ or when P₁ ≫ P₂. Note that the characteristic slope (S*) at the phase transition bodyweight is exactly the mean of the two asymptotic exponents, since at $W=\overset{\sim }{w}$, P₁ = P₂, Equation 27 is reduced to

$${S}^{\ast }=\left(\alpha +\beta \right)/2$$

(28)

In sum, Fig. 10 captures the essence of the proposed mixed-allometry model by revealing all the parameter values used for the simulation: two asymptotic exponents of 2.5 and 0.5 (exactly α and β), two asymptotic allometric coefficients (A = 0.1, B = 10), and a critical bodyweight of 10 kg (i.e.,$\overset{\sim }{w}={\left(B/A\right)}^{1/\left(\alpha -\beta \right)}={\left(10/0.1\right)}^{1/2}$) where the slope of the biphasic curve at this critical point is exactly the mean of the two exponents (i.e., 1.5).

Fig. 10

Characteristics of the mixed-power function. A Simulated data of physiological quantity (P) vs. bodyweight (W). The lines represent the two asymptotic power-law functions with respective y-intercepts (i.e., 0.1 and 10, at W = 1 kg) and an interception point which defines the critical point of phase transition (at W = 10 kg). B Slope as a function of bodyweight. The plot highlights the two allometric exponents (α and β) and the corresponding mean value at the point of phase transition