Secrets of Kleiber’s and Maximum Metabolic Rate Allometries Revealed with a Link to Oxygen-Deficient Combustion Engineering

Abstract

The biology literature addresses two puzzles: (i) the increase in specific metabolic rate of organs (SOrMR, W/kg of organ) with a decrease in body mass (M_B) of biological species (BS), and (ii) how the organs recognize they are in a smaller or larger body and adjust metabolic rates of the body (${\dot{q}}_B$) accordingly. These puzzles were answered in the author’s earlier work by linking the field of oxygen-deficient combustion (ODC) of fuel particle clouds (FC) in engineering to the field of oxygen-deficient metabolism (ODM) of cell clouds (CC) in biology. The current work extends the ODM hypothesis to predict the whole-body metabolic rates of 114 BS and demonstrates Kleiber’s power law {${\dot{q}}_B = a {M_B}^b$}. The methodology is based on the postulate of Lindstedt and Schaeffer that “150 ton blue whale. and the 2 g Etruscan shrew.. share the same.. biochemical pathways” and involve the following steps: (i) extension of the effectiveness factor relation, expressed in terms of the dimensionless group number G (=Thiele Modulus²), from engineering to the organs of BS, (ii) modification of G as G_OD for the biology literature as a measure of oxygen deficiency (OD), (iii) collection of data on organ and body masses of 116 species and prediction of SOrMR_k of organ k of 114 BS (from 0.0076 kg Shrew to 6650 kg elephant) using only the SOrMR_k and organ masses of two reference species (Shrew, 0.0076 kg: RS-1; Rat Wistar, 0.390 kg: RS-2), (iv) estimation of ${\dot{q}}_B$ for 114 species versus M_B and demonstration of Kleiber’s law with a = 2.962, b = 0.747, and (v) extension of ODM to predict the allometric law for maximal metabolic rate (under exercise, {${\dot{q}}_{B,MMR} = a_{MMR} {M_B}^{b_{MMR}}$}) and validate the approach for MMR by comparing b_MMR with the literature data. A method of detecting hypoxic condition of an organ as a precursor to cancer is suggested for use by medical personnel

1. Introduction, Literature Review and Objectives

Recently, efforts are underway to link the fields of thermodynamics and combustion with biology in order to understand the virus evolution, develop empirical formulae for viruses, and analyze biological processes through Gibbs function and energy release within biological systems (BS) [1,2,3]. The current study extends previous work linking the field of oxygen-deficient combustion (ODC) with oxygen-deficient metabolism (ODM) [3] to: (i) predict the specific organ metabolic rates (SOrMR_k) of vital organ k of mass m_k (SOrMR_k in W/kg of k, where k = Kidneys (Kids), Heart, Brain and Liver) for 114 BS ranging in mass from 0.0076 to 6650 kg, using organ mass (m_k) data of all species and the SOrMR_k of two reference species (RS-1 with lowest body mass, M_B: Shrew, 0.0076 kg; RS-2 with M_B,RS-2 > M_B,RS-1 Rat Wistar, 0.390 kg; M_B of RS-2 much higher than M_B of RS-1), (ii) obtain the whole-body metabolic rates (BMR) of 116 species as a function of body mass (M_B) by summing the organ metabolic rates (OrMR_k) of individual organs (OrMR_k = SOrMR_k × m_k) with known organ masses (m_k) and (iii) demonstrate (a) Kleiber’s law for BS at rest and (b) the allometric law for maximal metabolic rate (MMR) during exercise. More importantly, the ODM framework introduces a dimensionless group (G_OD)_k for organ k for the biology literature to indicate the extent of OD within an organ.

1.1. Kleiber’s Law and Organ Metabolic Rates

Allometry refers to how the characteristics of biological species (BS), including morphological traits (e.g., brain size) and physiological traits (e.g., metabolic rate (MR), life span), change with body mass. The allometric relation for MR of BS is given as:

$${\dot{q}}_B = a {M_B}^b$$

(1)

where M_B is body mass (kg) and (${\dot{q}}_B$), MR is in watts. In 1932, Kleiber obtained a = 3.4, b = 0.74 for M_B ranging from 0.15 to 679 kg, known as Kleiber’s law [4,5], which persisted for over 70 years [6]. The specific basal metabolic rate (SBMR) is written as SBMR = ${\dot{q}}_{B,M} = {\displaystyle \frac{{\dot{q}}_B}{M_B }}$, W/kg of body mass)

$${\displaystyle \frac{{\dot{q}}_B}{M_B}} = {\dot{q}}_{B,M} = a M_B b^{\prime }$$

(2)

where b′ = b − 1 = −0.26. Nutrients consumed through the mouth are essential for energy release ${\dot{q}}_B$ through oxidation. The review in Ref. [7] suggests that O₂, which enters through nasal intake, must also be considered a “nutrient” since it is essential for energy release. The energy release rate (ERR) is related to the oxygen consumption rate $\left\{{\dot{m}}_{O2,B} \right\}$ within whole body since ERR_B = $\left\{{\dot{m}}_{O2,B} \right\}HH{V}_{O2}$, where HHV_O2 is the higher heating or gross heating value per unit mass of oxygen consumed and it is almost constant at about 14,335 J/g of O₂ for most fuels and nutrients [3]. Thus, the scaling function for metabolic rate with body size M_B is explained with two existing theories:

(I)

If O₂ supply from blood to the cells of tissues falls below a critical level $\left\{{\dot{m}}_{O2,crit} \right\}$, then its uptake $\left\{{\dot{m}}_{O2} \right\}$ is limited by the supply from blood vessels, which is a common assumption used in the classical WBE (West, Brown and Enquist) hypothesis [8] for demonstrating Kleiber’s law. West, a theoretical physicist, and ecologists Jim Brown and Brian Enquist proposed a fractal or “nutrient (including oxygen) distribution network” (i.e., the circulatory system of animal or vascular system of plants) hypothesis (also referred to as the “upstream” or supply side [9], or “outward-directed vascular network” [10]) and illustrated Kleiber’s law by minimizing the heart’s work required to pump a unit amount of blood; i.e., a network that minimizes the pressure difference (P_aorta − P_cap). The scaling function is explained with O₂ delivery to cells as the limiting factor.

(II)

Thermodynamics relates pressure losses (P_aorta − P_cap) to entropy generation. Thus, Bejan [11,12], proposed that architectures and organs must develop in such a way that resistance to flow current (e.g., water flow in trees) must be minimized, or equivalently, that entropy generation is minimized, resulting in lower energy consumption and food requirements.

The biologists have raised the following issues as unknown: “The allometric size relationship is somehow ‘programmed’ into cells, although the factors that let them know whether they are in a small or large organism are still unknown” [13]. That is, existing biological data indicate that organs increase their metabolic rates per unit mass when within a smaller body and vice versa. In earlier work, the author proposed the “Oxygen-Deficient Metabolism (ODM)” or “downstream” hypothesis to explain these unknowns [3] and introduced a dimensionless group (G_OD,k) for organs in the biology literature. In the current work, the same ODM hypothesis is extended to predict the whole-body basal metabolic rate ${\dot{q}}_B$ as a function of M_B for 116 BS and present a method of estimating (G_OD,k) of organ k for use by the medical community.

1.2. Literature Review

While combustion is a rapid oxidation process that typically occurs at high oxygen mass fraction, Y_O2,air = 0.23 (mole fraction = 0.21 or 210,000 ppm), resulting in a significant temperature rise, the metabolism is a slow oxidation process that typically occurs at a low oxygen mass fraction, Y_O2 ($Y_{O_2} = 0.0415 x p_{O_2}$ [3]) with a small rise in temperature rise for the body. The term p_O2 is the partial pressure of O₂ in mm of Hg. While the p_O2 in the alveolar is 106 mm of Hg, the p_O2 in tissues is about 40–50 mm Hg, and dissolved O₂ is on the order of 2–5 ppm, thus resulting in a lower temperature rise in the body containing almost 50–75% water. Sometimes, the biology literature calls the energy release (${\dot{q}}_B$) as “heat produced” or “power produced” [14], while the engineering literature defines ${\dot{q}}_B$ as the energy release rate (ERR) by all the cells within the body. The ${\dot{q}}_B$ is a sum of the work delivery rate, ${\dot{W}}_B$, (i.e., ATP delivery rate, approximately 25% of ${\dot{q}}_B$) and the heat transfer rate (${\dot{Q}}_B$, approximately 75% of ${\dot{q}}_B$, [15]) due to the temperature difference (ΔT) across the cells and the rest of the body.

The required O₂ uptake $\left\{{\dot{m}}_{O2} \right\}$ (biology uses volume units ${\dot{V}}_{O2}$ in mL/min, ${\dot{m}}_{O2}$ in mg/min = 1.42 × ${\dot{V}}_{O2}$) by mitochondria in the cells is supplied by blood vessels via capillaries, followed by diffusion from capillaries to the cells immersed in the interstitial fluid (IF) and then from cells to mitochondria.

The hypotheses used for demonstrating Kleiber’s law {${\dot{q}}_B$ vs. M_B} fall under two broad groups: (I) Homogeneous and (II) Heterogeneous.

(I) 

Homogeneous hypothesis:

This hypothesis considers the whole body as a system; the hypotheses include: (i) the law of surface area to volume (S/V) ratio of the body, which balances the energy released over volume V with heat loss via skin to the ambience, yielding b = 2/3, b′ = −1/3, known as Rubner’s law (1883), (ii) WBE’s fractal geometry [8] (geometry of circulatory system: macro and microcirculation), which relies on minimization of dissipative energy in the vascular system supplying oxygen and nutrients. Savage et al. showed that the WBE model is applicable only for BS of infinite body size (or network) [16] and when finite size is included, it yields scaling exponents as a function of body size. Further, Weibel and Hoppeler [17] question the universal models based on “the fractal design of the vasculature and the fractal nature of the total effective surface of mitochondria and capillaries” since they predict b = ¾ for both basal and maximal metabolic rates. Silva et al. [18] suggest that there are mathematical and conceptual errors in network models, weakening the proposed theoretical arguments. The same review suggests that the power law exponent b should vary between 2/3 and 1 based on “metabolic level” (activity level of the organism or metabolic intensity). Painter et al. [19] agreed with the assumption of blood volume ∝ M_B but questioned the assumptions of uptake nutrient consumption rate (called total current in the network) proportional to blood volume, (iii) Assuming the number of cells per kg body mass {n_cell,m} in mammals is mass invariant, Ahluwalia [20] used the “downstream” hypothesis and Kleiber’s law {Equation(2)} to obtain a relation for the average cell metabolic rate (CMR, Watts/cell} as

$${CMR}_{avg}= a_{CMR} M_B b^{\prime }, a_{CMR} = {\displaystyle \frac{a }{ n_{cell,m}}}, b^{\prime }= -{\displaystyle \frac{1}{4}}$$

and the reaction–diffusion equation with a source term was solved for two simplified cases: (a) Desorption controlled metabolism (zero-order kinetics): ${\dot{w}}_{O2,cell} ={\dot{w}}_{O2,cell,\max }, Y_{O2}>>k_{MM}$ (Michaelis Menten constant) as in many in vitro samples {See Equation (16) in Ref. [3]}, and (b) adsorption controlled metabolism (first-order): ${\dot{w}}_{O2,cell} = {\displaystyle \frac{{\dot{w}}_{O2,cell,\max }}{k_{MM}}}{Y}_{O2}(r), Y_{O2}(r)<<k_{MM}$. Ref. [20] presents results on MR in terms of Thiele Thiele modulus, Ψ_T. It was shown that b′ for the whole body satisfies the following bounds: −1/3 {high Ψ_T} < b′ < 0 {low Ψ_T} under first-order kinetics. Note that these bounds on b′ are given for the whole body while the ODM hypothesis presented in Ref. [3] presents F_k at the organ level {−1/3 < F_k < 0}; cf. Equation (7). Other models include (iv) network structures [10], (v) quantum mechanics [21], and (vi) topology [11].

(II) 

Heterogeneous Hypothesis:

A.

Empirical Allometric Relations (EAR): The heterogeneous hypothesis considers the whole-body metabolic rate ${\dot{q}}_B$ as a sum of the OrMR_k with k = kids, H, Br, L and RM where RM represents all the remaining weakly metabolizing tissues. The mass of RM is given as

$$m_{RM}=M_B-m_{vit},m_{vit}= {\displaystyle \sum m_k}, k=Kids,H,Br,L$$

• (i) 

  Body Mass Based Allometry: Wang et al. used a heterogeneous or reductionist approach for estimating the whole-body metabolic rate {${\dot{q}}_B$} [22,23,24]:

  $${\dot{q}}_B=a {M_B}^b={\displaystyle \sum _k {\dot{q}}_{k,m} m_k} , M_B= {\displaystyle \sum _k m_k }, k= Kids,H,Br,L,RM$$

  (3)

  where ${\dot{q}}_{k,m}$ is the SOrMR_k of kth organ {W/kg of k} given by the body mass based allometry (BMA):

  $${\dot{q}}_{k,m}=e_{k,6}{M_B}^{f_{k,6}}, f_{k,6} < 0 , k= Kids,H,Br,L,RM$$

  (4)

Here afterwards, this method of computing SOrMR_k using the empirical allometric relations {EAR, Equation (4)} will be termed as EAR method. Wang et al. presented allometric relations for organ masses [23]:

$$m_k=c_{k,6}{M_B}^{d_{k,6}}, d_{k,6}>0, k= Kids,H,Br,L,RM$$

(5)

Using Equations (4) and (5) in Equation (3), the ${\dot{q}}_B$ is obtained as

$${\dot{q}}_B={\displaystyle \sum _k {\dot{q}}_{k,m} m_k}= \left({\displaystyle \sum _k c_{k,6} e_{k,6} {M_B}^{d_{k,6}+f_{k,6}}}\right) , k= Kids,H,Br,L,RM,$$

(6)

Based on data on the organ mass and ${\dot{q}}_{k,m}$ of six species (ranging from 0.48 kg of rat to 70 kg of human), Ref. [23] tabulates the constants c_k,6, d_k,6, e_k,6 and f_k,6.

Table 1. Allometric Constants for Organ Mass, Energy Release (Metabolic) Rate. Note: Values are based on six species: Rat (M_B = 0.45 kg), Rabbit (2.5 kg), Cat (3 kg), Dog (10 kg), Human-1 (65 kg), Human-2 (70 kg). Constants c_k,6, d_k,6, e_k,6, and f_k,6 are from Ref. [23]; density from Ref. [24]). (Adopted and modified from Wang–5 organ model, Table 4 of Ref. [27]). ${\dot{q}}_{k,m }\left({\displaystyle \frac{W}{kg of organ k}}\right)=e_{k,6}{M}_B^{f_{k,6}}= E_{k,6} {m_k}^{F_{k,6}}, m_k = c_{k,6} {M_B}^{d_{k,6} }; F_{k,6}= \left({\displaystyle \frac{f_{k,6}}{d_{k,6}}}\right), E_{k,6}=\left\{{\displaystyle \frac{e_{k,6}}{ {c_{k,6}}^{\frac{f_{k,6}}{d_{k,6}}}}}\right\}; m_k,M_B in kg$. HHV_O2 = 14,335 kJ/kg of O₂ or 18.7 kJ/SATP L of O₂ or 20.5 J/CST mL of O₂ or 18.1 J/mL of O₂ at 36.2 °C. HHVO₂, kJ/L O₂ for M_B from 71 to 92 kg = 15.818 + 5.17 × RQ [32].

Organ	ρk, g/cc	ck,6 1 kg	dk,6 2	ek,6 3	fk,6 4	mk (85 kg Human)	Ek,6	Fk,6	$$\mathbf{(}\mathbf{85} \mathbf{kg} \mathbf{Human}\mathbf{)} {\dot{\mathit{q}}}_{\mathit{k},\mathit{m}}$$	ck,116 [3]	dk,116 [3]	OEFk (5 kg Human)
Kidneys (Kids) 5	1.05	0.007	0.85	33.41	−0.08	0.31	20.94	−0.094	0.11	0.00631	0.728	0.085
Heart (H)	1.06	0.006	0.98	43.11	−0.12	0.47	23.04	−0.122	0.15	0.00580	0.932	0.49
Brain (Br)	1.036	0.011	0.76	21.62	−0.14	0.32	9.42	−0.184	0.044	0.0108	0.886	0.376
Liver (L)	1.06	0.033	0.87	33.11	−0.27	1.57	11.49	−0.310	0.19	0.0286	0.872	0.50
RM 6		0.939	1.01	1.45	−0.17	83.44	1.44	−0.168	0.19	0.940	1.007

¹ 50–70 kg Human brains indicate jump in mass from 1.2 kg to 1.4 kg compared to sheep of comparable body mass of 52 kg with m_Br = 0.11 kg. Human. ² Same as footnote (1). ³ Elia values for “e_k” are [33]: Kids, H, Br, L, SM,AT, RM-ex 2: 21.3, 21.3, 11.62, 9.7, 0.63, 0.22, 0.58 W/kg [24] and f_k = 0; m_RM-ex2 = M_B-m_vit-m_SM-m_AT. Krebs reports that the SOrMRk of organs decreases with an increase in body mass, and the order of decrease is the same as the decrease in SBMR of the body [26]. The constants ck,₆, dk,₆, etc., are based on data from six species [23], and ck,₁₁₆, dk,₁₁₆, etc., are based on 116 species [3]. Elia constant SOrMRk (W/kg) for Kids, H, Br, L, and RM: i.e., ek, 21.3, 21.3, 11.62, 9.7, and 0.58 W/kg and f_k for Elia = 0. Later et al. [27] for species M_B: 70–80 kg, e_RM: 0.463 W/kg, f_RM = 0, q_RM,m = constant, AT mass isometric with body mass [28]. ⁴ Ref. [29] cites Hepatocytes: fk = −0.17 to 0.21; kidney cortex: −0.11 to −0.07, brain: −0.07, spleen: −0.14 and lung: −0.10. For SM based on 49 species, ck_,49 = 0.061, dk_,49 = 1.09, MB from 0.006 to 6600 kg [30]. ⁵ Gutierrez: kidneys m_{K ∝ mB} ^0.85; for liver m_{L ∝ mB} ^{0.87 to 0.89} [31]. ⁶ Allometric relation for mass of RM yields different values compared to m_RM = M_B – m_bital where m_bital is based on allometric constants.

tabulates the allometric constants c_k,6, d_k,6, e_k,6 and f_k,6 [23,24,25]. Since d_k,6 > 0, organ sizes are positively related to M_B, while the SOrMR (Equation (4)) are negatively correlated with body mass (f_k,6 < 0). The additional subscript “6” indicates that the empirical constants are based on data from six species. Several other supporting data from references [26,27,28,29,30,31,32] are provided in footnotes of Table 1.

If one uses the data on c_k,6 and d_k,6 tabulated in Table 1, the resulting mass percentages (m_k × 100/M_B) for a 70 kg human are: 0.37% (Kids), 0.58% (Heart), 0.40% (Brains) and 1.90% (Liver), with the vital organ mass percentage at 3.25%. The corresponding energy percentages are 7.36% (Kids), 12.81% (Heart), 3.97% (Brain), and 17.10% (Liver), with the vital organ energy percentage at 41%. The RM for the 5-organ model is 66.2 kg, while the masses of Kids, H, Br, and L are 0.31, 0.33, 1.4 and 1.8 kg, respectively; this results in 94.5% of body mass being RM, while vital organs account for 5.5%. As opposed to a majority of BS, human brain masses are relatively larger, and the allometric relation underpredicts m_Br for humans. Thus, human brain mass is estimated using the encephalization quotient (EQ), which is the ratio of measured brain mass to the mass predicted with allometry. Gallagher et al. report that for a reference human of 70 kg [33], brain mass is 1.4 kg while the brain mass is predicted as 0.28 kg from allometry indicating a high EQ. Vital organs contribute {mass % in parenthesis) 8.7% (0.4, Kids), 8.2% (0.5%, Heart), 21.6% (2%, Brain), and 20.2% (2.6%, Liver) of the total BMR [28]. The underprediction of human brain mass and energy percentage is due to the EQ factor, as humans have the highest EQ (i.e., a larger brain size compared to animals of similar mass). This additional brain mass enhances cognitive abilities beyond general brain mass versus body mass scaling laws.

• (ii) 

  Organ Mass Based Allometry (OMA) Exponents for SOrMR_k {or ${\dot{q}}_{k,m}$}: It is noted that “f_k,6” in BMA for the vital organs of the six species in EAR by Wang et al. [23] are all negative. To explain the negative values of f_k,6 in BMA for organs, and answer the puzzle on “how the organs inside the body know that they are in smaller or larger body and adjust the metabolic rates”, the BMA is replaced by organ mass-based allometry (OMA) using the relation between organ mass and body mass {Equation (5)} [3]. Thus,

  $$SOrM{R}_k= {\dot{q}}_{k,m}=e_{k,6}{M_B}^{f_{k,6}}=E_{k,6}{m_k}^{F_{k,6}}, f_{k,6} < 0, F_{k,6} < 0, k=Br,H,K,L,R$$

  (7)

  $$F_{k,6}=\left({\displaystyle \frac{f_{k,6}}{d_{k,6}}}\right), E_{k,6}=\left({\displaystyle \frac{e_{k,6}}{{c_{k,6}}^{(f_{k,6}/d_{k,6})}}}\right)$$

See Table 1 for the listing of E_k,6 and F_k,6. The F_k,6 becomes more and more negative for increasing organ mass. Ref. [3] explains the rationale for F_k,6 vs. m_k using the ODM hypothesis. It is apparent from the constant e_k,6 and f_k,6, or E_k,6 and F_k,6 that different organs consume oxygen at different rates, thus indicating different O₂ profiles within IF. Since the ODM hypothesis is used to predict SOrMR_k and demonstrate Kleiber’s law in the current work, a brief outline of Ref. [3] on ODC and ODM is presented below for the convenience of readers.

B.

Group or Oxygen-Deficient Combustion (GC or ODC) in Engineering: The engineering literature models the combustion of dense fuel particle suspension (e.g., coal suspensions fired into a boiler) using a spherical fuel cloud (FC) of radius R_FC, mass m_FC and number density of fuel particles n_FC, with its surface exposed to a known oxygen mass fraction at the surface Y_O2,FC,s. Each particle located at “r” from the center of the spherical cloud is subjected to Y_O2(r) and consumes oxygen at the rate of ${\dot{w}}_{O2,p}(r)$:

$${\dot{w}}_{O2,p}(r) =C_{ch,p} Y_{O2}(r),$$

(8)

where the characteristic oxygen consumption rate by the particle, C_ch,p, and the relation for C_ch,p changes depending on kinetics control (C_Ch,p = C_Ch,p,kin) with a first-order reaction, or diffusion control (C_Ch,p = C_Ch,p,dif). The basic relations for C_ch,p are given in Ref. [3]. A detailed description, along with engineering results for (i) O₂ consumption rate of the whole cloud and (ii) effectiveness factor {η_eff,FC} for a spherical FC, are presented in Refs. [3,34] in terms of dimensionless group G; they are briefly presented here.

The solutions for ${\dot{w}}_{O2,FC}$, are presented in terms of the effectiveness factor (η_eff,FC) of the FC, which is defined as a ratio of the O₂ consumption rate by all particles within the cloud to the rate of consumption of O₂ in the case that each particle within the FC is subjected to Y_O2,FC,s:

$${\eta }_{eff,FC} ={\displaystyle \frac{{\dot{w}}_{O2,FC}}{{\dot{w}}_{O2,FC}\left(Y_{O2,FC,s}\right)}} or {\displaystyle \frac{ERR}{ERR with Y_{O2,FC,s}}} or {\displaystyle \frac{{\dot{w}}_{O2,FC,m}}{{\dot{w}}_{O2,FC,m}\left(Y_{O2,FC,s}\right)}} = {\displaystyle \frac{SERR}{SERR with Y_{O2,FC,s}}},$$

(9)

where the energy release rate $ERR= {\dot{w}}_{O2,FC} HH{V}_{O2}$, and the specific energy release rate $SERR= {\displaystyle \frac{ERR}{m_{FC}}}$. Using oxygen profiles Y_O2(r) within the cloud, the relation for η_eff,FC is given as:

$${\eta }_{eff,FC} = {\displaystyle \frac{Rate of O2 consumption rate of FC }{Rate of O2 consumption rate of FC if all particles \exp osed to Y_{O2,FC,s}}} = {\displaystyle \frac{3}{\sqrt{G}}} \left\{ {\displaystyle \frac{1}{\tanh \left(\sqrt{G}\right)}}-{\displaystyle \frac{1}{\sqrt{G}}}\right\}$$

(10)

The dimensionless group G for FC is defined as:

$$G={\displaystyle \frac{C_{ch,p} n_{FC} {R_{FC}}^2}{\rho D}}={{\Psi }_T}^2$$

(11)

and the G number for FC is shown to be related to Thiele Modulus, Ψ_T (G = Ψ_T²) in the porous char combustion literature [28,29]. Using Equation (10), the effectiveness factor can be plotted against G as shown in Figure 1. It is noted that G ∝ R_FC², and since the mass of the fuel cloud, m_FC ∝ R_FC³ and hence G ∝ m_FC^(2/3), assuming a constant number density of fuel particles (n_FC). The caption of Figure 1 describes the results for η_eff,FC vs. G for the fuel clouds.

A.

Oxygen-Deficient Metabolism (ODM) in Biology: The oxygen diffuses from capillaries towards the metabolic cells contained within interstitial fluid (IF). Even though the biology literature suggests a radial diffusion distance of the order of 100 μm (where pO₂ ≈ 0) from the capillaries, the actual path may be longer, leading to a decreased O₂ transport rate (or decreased effective diffusivity) to the mitochondria. The diffusive O₂ transport rate is affected due to the following [35,36]: (i) closely packed cells (number density of cells, n, or crowding effect) [13] (ii) tortuous oxygen path, (iii) amount of aqueous fluid, (iv) extracellular structures or cell barriers, and (v) presence of cytoplasm (which alone reduces D by 30 times the normal level). As a result, cells far away from the capillaries cannot maintain the required O₂ flow for ATP production [37] leading to oxygen deficiency (OD).

B.

ODM in Organs and Significance: Hypoxic conditions (low pO2 in cells) decrease the oxygen consumption rate ${\left\{{\dot{w}}_{O2} \right\}}_{cell}$ by cells, while anemic conditions or a reduction in blood flow [38] or reduced Hb contents cause a decrease in O₂ supply to the cells from capillaries. Hypoxic conditions cause a reduction in ATP production rate, leading to “bioenergetic collapse” [39]. These conditions lead to sleeper cells. The OD in cells results in the production of protein called HIF-1α (hypoxia-induced factor), a dimeric protein complex that causes a metabolic shift from oxidative phosphorylation (OXPHOS) to glycolysis. HIF enables the activity of genes to switch from oxidative phosphorylation to glycolytic pathways [40] for energy and ATP release. ODM promotes a switch to glycolysis, where only two ATP are obtained per CH compared to 32 ATP via oxidative phosphorylation, resulting in a decrease in overall energy release [41] and rapid cell division due to more intermediaries provided by glycolysis which serves as a source of energy for cancer cells [42]. It is well known that oxygen deficiency (OD) or hypoxia contributes to several diseases, including cancer, stroke, anemia, and heart disease. There appears to be a positive correlation between the mass of an organ and the number of cancer cases [43], which is attributed to the link between excess fat in organs and obesity.

C.

Modeling ODM in Cell Clouds:

(i)

Phenomenological type of ODM Model: Singer et al. studied the role of OD or the “crowding effect” on the metabolic rates of in vitro organ samples and developed a phenomenological type of model [44]. Just like particles in FC, the cells near the aerobic surface undergo high SOrMR_k while those cells near the anerobic core may undergo only glycolysis. Recently, Botte et al. [45] dealt with the effects of cell packing on O₂ consumption within densely packed cells of 3D cultures. They introduced proximity index (PI, a measure of cell–cell interaction), an uptake coefficient (ϕ, a measure of area around the cell through which O₂ passes), and dimensionless group called LSU (local supply to uptake ratio = ϕ/D), which is similar to G_OD. They studied the effects of cell packing medium on CMR and used LSU to correlate CMR. In the author’s opinion, LSU may be called LUS (local uptake to supply via diffusion), which also aligns with engineering interpretations of G_OD,k {cf. Equation (14)}.

(ii)

Detailed ODM Model following ODC Literature in Engineering: More detailed ODM models were developed by Annamalai by adapting the ODC literature from engineering to biology [3]. Unlike the Krogh cylinder model, where the capillary is placed on the axis (COA) of a cylinder containing metabolic cells within IF, the ODM model uses a spherical cloud of cells (CC) of radius R_CC having n_CC, cells per unit volume with capillaries on the surface (COS) of CC with mass of CC, m_cc {Figure 2a}. The COS model is also known as the “solid cylinder” model in biology when geometry for modeling uses cylindrical geometry [46]. A detailed comparison between ODC and ODM models, and relations for several variables of interest, are presented in Ref. [3]. In ODM, the fuel cloud is replaced by a cell cloud (CC), particles are replaced by cells of the organ k in BS, and Y_O2,FC,s becomes Y_O2,CC,s. The radius R_FC is replaced by R_CC, the ERR is replaced by the organ metabolic rate (OrMR_k) and SERR is replaced by SOrMR_k in cell clouds, defined as SOrMR_k = (OrMR_k)/m_CC}k. The G number in engineering [47]) is also replaced by G_OD,k for organ k. The oxidation rate for each particle is replaced by the cell metabolic rate {Figure 2d}. These relations will be summarized in the Section 2.2.

Just like FC, there exists three zones of operation of CC: (I) Dilute {G_OD,k < 1, η_eff,k ≈ 1, F_k based on OMA ≈ 0, η_eff,k ≈ 1}, (II) Dense Cloud {1 < G_OD,k < 100, η_eff,k < 1}, F_k < 0 and (III) Very dense Cloud {G_OD,k > 100, {e.g., Liver}, ${\eta }_{eff,k} \to {\displaystyle \frac{3}{\sqrt{G_{OD,k}}}}$ ∝ m_k^(−1/3) or F_k → (−1/3) as G_OD,k becomes large}. More details for CC model are provided in the caption of Figure 2. Those cells near the surface are aerobic with a higher rate of oxidation of nutrients, while those cells near the core of the cloud are anaerobic with very little oxygen. Cells near the core may undergo only glycolysis {Figure 2b} leading to the production of protein HIF-1α.

While resting, limited capillaries are perfused with blood (Figure 2b, surface at Y_O22,cc,s), whereas under exercise, almost all the capillaries are perfused (Figure 2c) increasing Y_O22,CC,s at the surface of CC {e.g., Maximum Metabolic Rate (MMR) when there is increased blood flow through selected organs during exercise}.

The literature review suggests that despite several hypotheses outlined in Section 0 for the 3/4 law, “the hunt for an explanation of the 3/4 law continues” [48]. The current ODM hypothesis focusses on the metabolic rate he controlled by “downstream demand-side” oxygen consumption of all the cells within an organ and provides another “hunt” for an explanation of the 3/4 law.

Figure 2. (a) Capillary system: Cells can be placed within a cylinder with capillary on axis (COA, Figure 2a) or cells can be enclosed by capillaries on surface (COS). The (COS) model is known as the solid “cylinder” model in biology for a cylindrical geometry (Figure 2a) [49] (b) COS model for spherical cloud of cells subjected to fixed oxygen mass fraction at the surface Y_O2,CC,s with cell located at “r” consuming O₂ at the rate of ${\dot{w}}_{O2,cell,k}(r)$; capillaries are partly perfused. The effective area for O₂ exchange is limited since only 25% to 35% of available capillaries are perfused under BMR conditions [50] {e.g., resting conditions resulting in lower Y_O2,CC,s}. (c) COS model with mostly perfused capillaries {e.g., exercise, resulting in higher Y_O2,CC,s} for a spherical cloud of cells. (d) Oxidation at cell level with local Y_O2(r) supplying oxygen to the cell and nutrients oxidized to CO₂ and H₂O. COS model is close to ODC model in the engineering literature. The O₂ diffuses towards the center of cell cloud (CC) of radius R_CC and mass m_CC. Cells adjacent to the surface containing capillaries are aerobic, while those farther surfaces become hypoxic, and the farthest cells are necrotic cells. Detailed results for cell clouds are given in Column 4 of Table 2 in Ref. [3] Certain regions within organs may undergo only glycolysis due to a lack of O₂. note that for the partly perused capillaries as shown in (b) Y_O2,CC,s. (d) Single cell within IF with diffusion of O₂, oxidation and release of CO₂ and H₂O. During exercise, capillaries are mostly perfused (Figure 2c).

Figure 2. (a) Capillary system: Cells can be placed within a cylinder with capillary on axis (COA, Figure 2a) or cells can be enclosed by capillaries on surface (COS). The (COS) model is known as the solid “cylinder” model in biology for a cylindrical geometry (Figure 2a) [49] (b) COS model for spherical cloud of cells subjected to fixed oxygen mass fraction at the surface Y_O2,CC,s with cell located at “r” consuming O₂ at the rate of ${\dot{w}}_{O2,cell,k}(r)$; capillaries are partly perfused. The effective area for O₂ exchange is limited since only 25% to 35% of available capillaries are perfused under BMR conditions [50] {e.g., resting conditions resulting in lower Y_O2,CC,s}. (c) COS model with mostly perfused capillaries {e.g., exercise, resulting in higher Y_O2,CC,s} for a spherical cloud of cells. (d) Oxidation at cell level with local Y_O2(r) supplying oxygen to the cell and nutrients oxidized to CO₂ and H₂O. COS model is close to ODC model in the engineering literature. The O₂ diffuses towards the center of cell cloud (CC) of radius R_CC and mass m_CC. Cells adjacent to the surface containing capillaries are aerobic, while those farther surfaces become hypoxic, and the farthest cells are necrotic cells. Detailed results for cell clouds are given in Column 4 of Table 2 in Ref. [3] Certain regions within organs may undergo only glycolysis due to a lack of O₂. note that for the partly perused capillaries as shown in (b) Y_O2,CC,s. (d) Single cell within IF with diffusion of O₂, oxidation and release of CO₂ and H₂O. During exercise, capillaries are mostly perfused (Figure 2c).

1.3. Objectives

The objectives of the current work are to (i) link the field of group or oxygen-deficient combustion (ODC) in engineering with the field of oxygen-deficient metabolism (ODM) in biology, (ii) adopt the relations developed for energy release rate per unit mass (SERR, W/kg fuel cloud) to predict SOrMRk (W/kg of organ k) of 116 BS ranging in body mass from 0.0075 to 6650 kg, using known organ mass data (m_k) and SOrMRk of two BS, referred to as Reference Species (RS), where RS-1 has the lowest body mass, M_B,RS-1 (Shrew, 0.0075 kg, η_eff,k ≈ 1), and RS-2 has a much higher body mass, M_B,RS—2 >> M_B,RS—1 (Rat Wistar of 0.390 kg, η_eff,k < 1), (iii) estimate the whole-body metabolic rate versus M_B by summing the predicted organ metabolic rates, and validate the ODM method by demonstrating Kleiber’s law for BS at rest with an exponent close to 0.75, (iv) predict an hypothetical upper metabolic rate (UMR) for organs, and, consequently, for the whole body, assuming all cells within each organ metabolize without oxygen concentration gradients; (v) obtain the maximal metabolic rate (MMR) during exercise, when most of the capillaries at the cell cloud surface are perfused, and show that the whole-body allometric law under exercise yields an exponent close to 0.87, as reported in the literature; (vi) suggest methods of estimating G_OD,k and hence extent of OD in organs for use by medical personnel.

2. Materials and Methods

2.1. ODM Hypothesis

The ODM hypothesis assumes that each organ k consists of multiple cell clouds (CC), with each cell cloud having a mass m_cc,k and radius R_CC,k, which is related to the organ mass m_k by R_CC,k α m_k^1/3. It is also possible that capillaries do not fully cover the entire spherical surface enclosing the cells (Figure 2b). Typically, 25–35% of an organ’s capillaries are perfused at rest, with perfusion increasing during exercise since there is increased metabolic demand; further a higher perfusion percentage results in a higher Y_O2,cc,s. The effective area for O₂ exchange is limited, since only a fraction of capillaries on surface are perfused under BMR conditions [50], which affects the O₂ diffusion distance and, hence, the metabolic rate within the organ. Using the Krogh–Erlang equation, Ostergaad demonstrated that only 10% of SM capillaries are perfused at rest, but more capillaries are recruited during exercise [51]. Further details can be found in Section 2.2 and Section 3.4.

Smaller species have smaller organs (e.g., shrew of mass 0.0075 kg), while larger species have larger organs, as indicated by d_k > 0 in the allometric exponents for organ sizes (Table 1). Thus, the smaller organs of smaller species may have shorter distances for O₂ diffusion from capillaries to cells, allowing metabolism to occur as if each cell within the CC is exposed to the same O₂ concentration as Y_O2,CC,s (η_eff ≈ 1), resulting in SOrMR_k of RS-1 ≈ SOrMR_k,iso, of same organ k of any BS. However, isolated cell metabolic rates can still vary from organ to organ, even in smaller species, due to differences in functional requirements, cell reactivity, and O₂ transport rate (e.g., heart tissues containing M_b can deliver O₂ at a faster rate resulting in increased effective transport coefficients for O₂ from capillaries to mitochondria). As organ mass increases (e.g., in the liver), η_eff,k < 1, and SOrMR_k ≈ η_eff,k × SOrMR_k,iso, where it is assumed that for any given organ k, SOrMR_k,iso remains the same for all BS regardless of body size.

2.2. Methodology

Detailed comparisons of the governing conservation equations, and several relations for various parameters in the fields of ODC in engineering and ODM in the biology literature are presented in Ref. [3]. These include: (i) conservation equations for both fields, (ii) the ERR of a single particle in a FC and a single cell in a CC in terms of Y_O2, (iii) oxygen profiles in FC versus CC, (iv) the dimensionless number G for FC and G_OD,k for the CC of organ k and (v) the SERR of FC (W/kg of cloud) in terms of the effectiveness factor and G and specific organ metabolic rate (SOrMR_k) of CC (W/kg of cell cloud of organ k) in terms of the effectiveness factor η_eff,k and G_OD,k. The methodology and relevant relations for the current work are briefly summarized below.

(i) 

Metabolic Rate of single cell located at r in CC {Figure 2}

$${\dot{w}}_{O2,cell,k}(r) ={\left(C_{ch,cell}\right)}_k Y_{O2}(r)$$

(12)

where C_Ch,cell, characteristic cell O₂ consumption rate {W/kg cell} when Y_O2 = 1; the relations for C_Ch,cell under first-order kinetic control or diffusion controlled O₂ consumption rates are given in Ref. [3] in terms of k_MM constants, transport properties and cell diameter.

(ii) 

Oxygen Profiles within CC 

Adopting the engineering literature [34], the profile for oxygen mass fraction within the cell cloud of organ k, is given as {See Table 3, items 10 and 11 in Ref. [3]}.

$${\displaystyle \frac{Y_{O2,k}(\xi )}{{\left(Y_{O2,k}\right)}_{\xi =1}}} = {\displaystyle \frac{Y_{O2,k}(\xi )}{\left(Y_{O2,CC,s}\right)}} =\left({\displaystyle \frac{1}{\xi }}\right){\displaystyle \frac{\mathit{Sinh} \left({G_{OD,k}}^{1/2}\xi \right)}{\mathit{Sinh} \left({G_{OD,k}}^{1/2}\right)}}, \xi = \left({\displaystyle \frac{r}{R_{CC,k}}}\right)$$

(13)

where for organ k

$$G_{OD,k}={\displaystyle \frac{{\left(C_{ch,cell}\right)}_k n_{CC,k} {R_{CC,k}}^2}{\rho D}} = {\displaystyle \frac{{\left(C_{ch,cell}\right)}_k n_{CC,k} {R^3}_{CC,k}}{\left\{\rho \left(\frac{D}{R_{CC,k}}\right)\right\}{R^2}_{CC,k}}}={\displaystyle \frac{Charactristic O_{2}consumption rate by cell cloud}{Charactristic O_{2}transport rate to cells from cell cloud surface }}$$

(14)

where R_CC,k ∝ m_k^1/3, C_ch,cell, characteristic O₂ consumption rate of cell. Note that G_OD,k = Ψ_T,k² where Ψ_T,k is called Thiele Modulus in porous char oxidation literature [28]. The larger the radius R_CC,k or organ mass m_k, the higher the G_OD,k or Ψ_T,k, and the characteristic O₂ consumption rate by cell cloud. The term ${\left(C_{ch,cell}\right)}_k n_{CC,k} R_{CC,k}^3$ is the characteristic consumption (or uptake) rate by CC, and $\left\{\rho \left({\displaystyle \frac{D}{R_{CC,k}}}\right)\right\}{R}_{CC,k}^2$ characteristic diffusion rate from the CC surface to the CC. The G_OD,k is almost similar to LUS (defined as local uptake to supply ratio) presented in Ref. [45]. It is noted that lowest mass fraction of O₂ is at the center of the CC with ${\displaystyle \frac{Y_{O2,k}(0)}{Y_{O2,CC,s}}} ={\displaystyle \frac{1}{\mathit{Sinh} \left({G_{OD,k}}^{1/2}\right)}}$. The higher the consumption rate of O₂ compared to supply via diffusion, the higher the G_OD,k, the lower the Y_O2(0), and the lower the η_eff,_k and vice versa..

(iii) 

Effectiveness Factor of Spherical CC and Specific Organ Metabolic Rate {SOrMR_k} 

Adopting the same procedure as in engineering,

$${\eta }_{eff,k} ={\displaystyle \frac{{\dot{w}}_{O2,m,k}}{{\dot{w}}_{O2,m,k}\left(Y_{O2,CC,s}\right)}}={\displaystyle \frac{{\left\{SOrMR\right\}}_k}{{\left\{SOrMR\left(Y_{O2,CC,s}\right)\right\}}_k }}= 3 {\displaystyle \underset{0}{\overset{1}{\int }}} \left\{{\displaystyle \frac{Y_{O2,k}(r)}{{\left(Y_{O2,CC,s}\right)}_k}}\right\}\xi d\xi , \xi ={\displaystyle \frac{r}{R_{CC,k}}}, Spherical CC$$

(15)

With Y_O2 profile from Equation (13), the η_eff,k is derived as:

$${\eta }_{eff,k} ={\displaystyle \frac{{\left\{SOrMR\right\}}_k}{{\left\{SOrMR\left(Y_{O2,CC,s}\right)\right\}}_k }}={\displaystyle \frac{{\left\{SOrMR\right\}}_k}{{\left\{SOrMR\right\}}_{k,iso} }}= {\displaystyle \frac{3}{\sqrt{G_{OD,k}}}} \left\{ {\displaystyle \frac{1}{\mathit{tanh}\left(\sqrt{G_{OD,k}}\right)}}-{\displaystyle \frac{1}{\sqrt{G_{OD,k}}}}\right\}$$

(16)

where k = Kids, H, Br, and L Using the definition of η_eff,k, the SOrMR_k is given as

$${\left\{SOrMR\right\}}_k = {\eta }_{eff,k} {\left\{SOrM{R}_{k,iso}\right\}}_{BS} = {\eta }_{eff,k} {\left\{ {\dot{q}}_{\mathrm{k},\mathrm{m},\mathrm{iso} }\right\}}_{BS}$$

(17)

where ${\left\{{\dot{q}}_{\mathrm{k},\mathrm{m},\mathrm{iso} }\right\}}_{BS} ={\left\{SOrMR\left(Y_{O2,CC,s}\right)\right\}}_{k,BS}={\left\{{\dot{q}}_{\mathrm{k},\mathrm{m},\mathrm{iso} }\right\}}_{RS-1}={\left\{{\dot{q}}_{\mathrm{k},\mathrm{m} }\right\}}_{RS-1}$. Equation (17) reveals that SOrMR_k is a function of G_OD,k in biology. One may classify the CC as dilute, dense and very dense clouds: (I) Dilute CC: G_OD,k < 1, η_eff,k ≈ 1 (Figure 2, η_eff, replaced by η_eff,k, G replaced by G_OD,k). For Zone I, Equation (16) leads to η_eff,k =1 and hence ${\dot{q}}_{m,k} \propto {m_k}^0$. Thus, in OMA, F_k → 0 for smaller organs, e.g., F_k for Kidney is −0.094, (II) Dense CC: 1 < G_OD,k < 100. As the organ size increases, mass increases, G_OD,k increases, η_eff,k the effectiveness factor of organ k decreases. (III) Very dense CC, or Sheath CC: G_OD,k > 100 (e.g., Liver, a large organ), η_eff,k is very low and ${\eta }_{eff,k} \to {\displaystyle \frac{3}{\sqrt{G_{OD,k}}}}$ [34]. Since G_OD,k ∝ R_CC,k² ∝ m_k^(2/3); Thus ${\eta }_{eff,k} \to {\displaystyle \frac{3}{\sqrt{G_{OD,k}}}} \propto {m_k}^{-(1/3)}, G_{OD,k}>100$. F_k in the organ mass based allometry (OMA, Equation (7)) becomes −1/3 for large organs (e.g., Liver). The ODM hypothesis leads to the bounds: −1/3 < F_k < 0 for vital organs. The data for liver, a large organ shows that F_k = −0.31 while F_k = −0.094 for Kidney, a smaller organ (Table 1).

iv 

Metabolic Rate of Vital organs {${\dot{q}}_{vit }$}: Using Equation (17) for the vital organs, the metabolic rates of vital organs of any BS:

$$\begin{array}{l}{\dot{q}}_{vit } = {\eta }_{eff,Kid} {\dot{q}}_{Kid,m,iso} m_{Kids}+ {\eta }_{eff,H} {\dot{q}}_{H,m,iso} m_H + \\ {\eta }_{eff,Br} {\dot{q}}_{Br,m,,iso}{m}_{Br} +{\eta }_{eff,L }{\dot{q}}_{L,m,,iso} m_L \end{array}$$

(18)

where ${\dot{q}}_{k,m,iso} = {\dot{q}}_{k,m,RS-1}$, k = kid, H, Br and L.

v 

Metabolic Rate of Remaining Mass (RM) of Tissues {${\dot{q}}_{RM}$} for any BS

The remaining mass of organs (RM) represents a sum of all “minor”(i.e., weakly metabolizing) organs (e.g., SM, skin, Adipose Tissue, etc.) within the body, and the specific metabolic rate of RM (W/kg of RM) is needed. There are several possible approaches: (a) Select data for each of minor organ if available, estimate the effectiveness factor for all minor organs, and adopt a similar procedure outlined for vital organs, (b) Use the EAR for RM: ${\dot{q}}_{RM,m}= e_{RM,6} {M_B}^{f_{RM,6}}$; from Table 1, e_RM,6 = 1.45, f_RM,6 = −0.17, ${\dot{q}}_{RM,m}$ in W/kg, (c) Assume Elia’s constant values for ${\dot{q}}_{RM,m}$ as 0.581 W/kg for all BS. In the current work, methods (b) and (c) are adopted for estimating ${\dot{q}}_{RM,m}$.

$${\dot{q}}_{RM} = {\dot{q}}_{RM,m} m_{RM} , m_{RM} = M_B -m_{vit}$$

(19)

vi 

Whole Body Metabolic Rate (${\dot{q}}_B$) under Rest

Adopting the heterogeneous method, the whole body metabolic rate under rest, {${\dot{q}}_B$} is given as

$$\begin{array}{l}{\dot{q}}_B = {\dot{q}}_{vit} + {\dot{q}}_{RM} ={\eta }_{eff,Kids} {\dot{q}}_{Kid,m,iso} m_{Kids}+ {\eta }_{eff,H} {\dot{q}}_{H,m,iso} m_H + \\ {\eta }_{eff,H} {\dot{q}}_{Br,m,,iso}{m}_{Br} +{\eta }_{eff,L }{\dot{q}}_{L,m,,iso} m_L+ {\dot{q}}_{RM,m} m_{RM} \end{array}$$

(20)

vii 

Metabolic Rate of RM {${\dot{q}}_{RM,Ex}$} and Whole-Body Metabolic Rate {${\dot{q}}_{B,Ex}$} under Exercise

During exercise, blood flow to the skeletal muscle (SM) is increased to supply the required oxygen and nutrients. Thus, SM becomes metabolically more active compared to other tissues within the RM due to increased capillary perfusion, increasing from approximately 25% at rest to nearly 100% during exercise.

$${\dot{q}}_{SM-Ex} = {\dot{q}}_{SM,m} m_{SM}$$

(21)

Relations for ${\dot{q}}_{SM,m}$ depends upon the Y_O2,CC,s and hence the percentage of capillaries perfused. The remaining mass of tissues under exercise is given as

$${\dot{q}}_{B,Ex} = {\dot{q}}_{vit,Ex} + {\dot{q}}_{SM,Ex}+{\dot{q}}_{RM,Ex}$$

(22)

where ${\dot{q}}_{vit,Ex}$ is different from ${\dot{q}}_{vit}$ under rest due to the percentage of capillaries perfused under exercise are different from those at rest. The ${\dot{q}}_{RM,Ex}$ during exercise is given as,

$${\dot{q}}_{RM,Ex} = {\dot{q}}_{RM,Ex,m} m_{RM,Ex} , m_{RM,Ex} =M_B -m_{vit}-m_{SM}$$

(23)

viii 

Upper Metabolic Rate (UMR, ${\dot{q}}_{B,UPR}$) and Maximum Metabolic Rate (MMR, ${\dot{q}}_{B,MMR}$) of Whole Body

(a)

UMR: The Upper metabolic MR of organ k (not the maximum MR) and hence, the whole-body MR can be estimated by setting η_eff,k = 1 for all organs {i.e., no oxygen gradients within CC}, including RM.

(b)

MMR: When the CC surface is covered with more perfused capillaries, the Y_O2,CC,s increases, and most of the cells are aerobic, resulting in the MMR, and leading to a whole-body allometric law with an exponent higher than 0.75. The MMR is obtained for two cases: (a) Lower limit on MMR by keeping O₂ gradients but accounting for more perfusion on CC surface; (b) upper limit on MMR by setting η_eff,k = 1 (i.e., no O₂ gradients, setting η_eff,k = 1 for all organs, including SM, and RM,Ex) during exercise. The percentage of capillaries perfused under rest and exercise conditions is shown in Table 2. The percentage of perfusion affects Y_O2,CC,s. The change in Y_O2,CC,s is given by the following relation:

$${\displaystyle \frac{{\left(Y_{O2,CC,s}\right)}_{\mathrm{k},\mathrm{MMR}}}{{\left(Y_{O2,CC,s}\right)}_{\mathrm{k},\mathrm{rest}}}} = {\displaystyle \frac{Blood flow rate to organ k under MMR}{Blood flow rate to organ k under \mathrm{Re}st}}$$

(24)

The increased Y_O2,CC,s causes isolated rates to increase (i.e., higher, thereby increasing the whole-body metabolic rate. Note that the blood flow rate to organ k is given by the product of % blood flow to organ k* circulation rate of blood/100. The pumping rate of blood by the heart changes depending on the resting or exercise conditions, leading to a whole-body allometric law with an exponent higher than 0.75.

Table 2. Capillary Perfusion Under Rest and Exercise. Note: Capillary perfusion percentage does not affect SOrMRk and BMR estimations; they affect only Y_O2,CC,s or MMR under exercise. In ODM method [49]. Cardiac Output = 5.8 (rest), 9.5 (m ild exercise), 17.5 LPM (maximal exercise). Vital Organ Blood flow: 2.7 (Rest), 2.4 (Mild Exercise), 2.5 LPM (Maximal Exercise). Vital Blood Flow %: 47 (Rest), 25 (Mild Exercise), 14 (Maximal Exercise). (Vital Blood flow)_Ex/Vital Blood flow)_Rest = 1 (Rest), 0.89 (Mild Exercise), 0.93 (Maximal Exercise).

Organ	Rest (mL/min)	Mild Exer (mL/min)	Maximal (mL/min)	Rest %	Max. Exercise %	Max Ex/Rest Ratios
Kidney	1100	900	600	19.0	3.4	0.55
Heart	250	350	750	4.3	4.3	3
Brain	750	750	750	12.9	4.3	1
Others (i.e., liver, spleen)	600	400	400	10.3	2.3	0.67
Skeletal muscle	1200	4500	12,500	20.7	71.4	10.42
RM, Ex (GI + skin + others)	2500	3000	2900	32.8	14.3	1.32

Table 2. Capillary Perfusion Under Rest and Exercise. Note: Capillary perfusion percentage does not affect SOrMRk and BMR estimations; they affect only Y_O2,CC,s or MMR under exercise. In ODM method [49]. Cardiac Output = 5.8 (rest), 9.5 (m ild exercise), 17.5 LPM (maximal exercise). Vital Organ Blood flow: 2.7 (Rest), 2.4 (Mild Exercise), 2.5 LPM (Maximal Exercise). Vital Blood Flow %: 47 (Rest), 25 (Mild Exercise), 14 (Maximal Exercise). (Vital Blood flow)_Ex/Vital Blood flow)_Rest = 1 (Rest), 0.89 (Mild Exercise), 0.93 (Maximal Exercise).

Organ	Rest (mL/min)	Mild Exer (mL/min)	Maximal (mL/min)	Rest %	Max. Exercise %	Max Ex/Rest Ratios
Kidney	1100	900	600	19.0	3.4	0.55
Heart	250	350	750	4.3	4.3	3
Brain	750	750	750	12.9	4.3	1
Others (i.e., liver, spleen)	600	400	400	10.3	2.3	0.67
Skeletal muscle	1200	4500	12,500	20.7	71.4	10.42
RM, Ex (GI + skin + others)	2500	3000	2900	32.8	14.3	1.32

2.3. Estimation of OD Number (G_OD,k) and Specific Organ Metabolic Rate of Organ k {SOrMR_k} of Any BS

Equation (16) presents η_eff,k in terms of the dimensionless number G_OD,k. If G_OD,k of an organ is known, then η_eff,k can be estimated; or, if the actual MR of an organ under interactive and isolated modes is known, so that η_eff,k can be estimated, then G_OD,k can be solved. The predictions on SOrMR_k of organs require the estimation of G_OD,k of each organ of 116 species; difficulty arises in the estimation of G_OD,k {See Equation (14)} since it requires a knowledge of (i) the reactivity of cells within organ k undergoing metabolism (C_Ch,cell) either under diffusion or kinetic control, (ii) the overall transport coefficient D of oxygen from capillaries to mitochondria, (iii) capillary number density, cell cloud radius which may change within an organ k and (iv) percentage of capillaries perfused for each organ in the 116 BS, which affects (Y_O2,CC,s)_k. Two methods are proposed for estimating G_OD,k and SOrMR_k of organs:

(A) 

Basic Method: This approach requires basic data for C_Ch,cell, D, % of capillaries perfused for each organ k. These properties may vary by an order of magnitude. Consequently, greater uncertainty exists in the estimation of G_OD,k and SOrMR_k due to variations in these parameters across the organs of 116 species.

(B) 

Ratio Method or Reference Species (RS) Method: To circumvent the data collections, minimize the wide variations in properties and reduce the uncertainties, the current work proposes references species (RS) method, also known as the ratio method. Two reference species are selected: RS-1 is selected as the BS with the lowest body mass (e.g., Shrew, M_B = 7.6 g with η_eff,k ≈ 1), and RS-2 is selected as a BS with significantly higher body mass (e.g., Rat Wistar, M_B = 390 g, almost 50 times heavier than that of RS-1; so η_eff,k < 1). The method is based on the premise that the metabolic behavior of organ k of BS with the lowest body mass is close to “isolated” rate with almost no oxygen gradient because of very few cells within the organ k; i.e., all cells are at Y_O2,CC,s, and hence η_eff,k ≈ 1. Then the isolated metabolic rate of organ k of any other BS ${\dot{q}}_{\mathrm{k},\mathrm{m},\mathrm{iso} }\approx {\dot{q}}_{\mathrm{k},\mathrm{m},\mathrm{RS}-1 }$ since basic architecture of cell is same for almost all mammalian species. Justification is as follows. Makarieva et al. [52] demonstrated that SBMR varied from 0.3 W/kg to 9 W/kg (a 25-fold variation), despite a 10²⁰-fold difference in body mass for “bacteria to elephants and algae to trees”. Then it follows that the cell metabolic rate (CMR) does not vary significantly across BS, since the number of cells per unit mass is similar across BS. This view is confirmed by Lindstedt and Hoppeler [53], who stated that the “150-ton blue whale”, which is 75 million times the mass of the 2 g Etruscan shrew, “shares the same architecture… organ systems, biochemical pathways”. For RS-2, the G_OD,k falls within the dense zone (i.e., the steeper part of η_eff,k vs. G_OD,k, Zone II in Figure 2); hence (η_eff,k)_RS-2 < 1. With the known SOrMR_k data for RS-1 and RS-2, (η_eff,k)_RS-2 is estimated as

$${\left\{{\eta }_{eff,k}\right\}}_{RS-2} ={\displaystyle \frac{\left(SOrM{R}_{k,RS-2}\right)}{\left(SOrM{R}_{k,iso,RS-2}\right)}} \approx {\displaystyle \frac{\left(SOrM{R}_{k,RS-2}\right)}{\left(SOrM{R}_{k,RS-1}\right)}}$$

(25)

The corresponding (G_OD,k)_RS-2 is evaluated using Equation (16). This RS method reduces uncertainty in the results by relying on ratios, thus canceling out the effects of several biological properties. With the assumption of a constant cell diameter for a given organ k across BS [16], constant number density of cells and using the definition of G_OD,k (Equation (14)), G_OD,k ∝ R_CC,k² and since the mass of the cell cloud, m_CC,k ∝ R_CC,k³, then G_OD,k ∝ m_CC,k^2/3 ∝ m_k^2/3. Knowing the organ masses for any BS (other than RS-1 and RS-2), the G_OD,k values can be estimated using the following relation:

$${\displaystyle \frac{{\left(G_{OD,k}\right)}_{BS}}{{\left(G_{OD,k}\right)}_{RS-2}}}={\left({\displaystyle \frac{m_{k,BS}}{m_{k,RS-2}}}\right)}^{(2/3)}$$

(26)

And the corresponding η_eff,k is estimated using Equation (16). Thus, the SOrMR_k for k = Kids, H, Br and L, ${\dot{q}}_{vit}$, and ${\dot{q}}_{RM}$, are determined from Equation (17), Equation (18), and Equation (19), respectively. Note that the RM consists of several organs with metabolically weak cells. Hence ${\dot{q}}_{RM,m}$ is estimated by two methods: (I): Body mass based EAR (Table 1, [23]); (II) Elia’s constant for SOrMR_RM (Table 1). Finally, ${\dot{q}}_B$ can be estimated from Equation (20), with the known organ masses of 116 species. Results are presented in the next section. A step-by-step procedure is presented in the Supplementary Materials A placed on MDPI website.

3. Results and Discussion

The results are presented in the following order: 1. (A) EAR for all organs and (B) EAR for vital organs with Elia’s constant used for RM, 2. (A) ODM for vital organs with EAR for RM, and comparison with results from the EAR for all organs [23], (B) ODM for vital organs with Elia’s constant for RM (C) effects of changing the selection of RS-2 on ODM results 3. Comparison of the energy contribution percentage by vital organs using the ODM method versus the EAR method, 4. (A) Allometric constants for the upper metabolic rate (UMR), assuming no oxygen gradients within organs (B) Allometric constants for the maximum metabolic rate (MMR), based on known perfusion percentages at rest and during exercise (Table 2), with comparisons to literature data.

3.1. Whole Body Metabolic Rate Using EAR for All Organs and the Effect of Elia’s Constant for ${\dot{q}}_{RM,m}$ on Whole-Body Allometry

A.

Empirical Allometric Relations (EAR) for all Organs: Wang et al. [22] used the EAR constants (Table 1) for all organs of 116 species in estimating the SOrMRk, then summed up the OrMR_k {=SOrMRk * mk} with known organ masses to obtain the whole-body metabolic rate (${\dot{q}}_B$) and plotted ${\dot{q}}_B$ vs. M_B {Figure 3}, which yielded Kleiber’s law with a = 3.216 and b = 0.756.

Table A1. Data on body and organ masses (kg), Specific Organ Metabolic rates (W/k organ), and comparison of Whole-body metabolic rates from ODM and EAR. Empirical Allometric Rule (EAR), ${\dot{q}}_{k,m} = e_{k,6} {M_B}^{f_{k,6}}, k=Kid, H,Br,L$. Original data on organ and body masses are from ref. [22].

BS #	Species	MB, kg	${\dot{\mathit{q}}}_{\mathit{k}\mathit{i}\mathit{d},\mathit{m}}$ W/kg	${\dot{\mathit{q}}}_{\mathit{H},\mathit{m}}$ W/kg	${\dot{\mathit{q}}}_{\mathit{B}\mathit{r},\mathit{m}}$ W/kg	${\dot{\mathit{q}}}_{\mathit{L},\mathit{m}}$ W/kg	${\dot{\mathit{q}}}_{\mathit{R}\mathit{M},\mathit{m}}$ W/kg	100 × mkidskg	100 × mH, kg	100 × mBr, kg	100 × mL kg	100 × mvit kg	Vit ERR % ODM	Vit ERR % EAR	${\dot{\mathit{q}}}_{\mathit{B}}$ ODM W	${\dot{\mathit{q}}}_{\mathit{B}}$ EAR, W
1	Shrew/Sorex araneus	0.00755	50.2	76.8	43.3	122.5	3.3	0.011	0.011	0.015	0.038	0.68	71.8	71.8	0.09	0.09
2	Crocidura russula	0.00953	49.2	74.7	41.9	115.1	3.2	0.013	0.008	0.017	0.055	0.86	75.7	75.7	0.09	0.11
3	Lasiurus borealis	0.01377	47.7	71.5	39.8	104.3	3	0.011	0.014	0.017	0.035	1.3	55.4	55.4	0.09	0.1
4	Lasionycteris noctivagans	0.01478	47.5	70.9	39.4	102.3	2.9	0.013	0.016	0.016	0.033	1.4	53.2	53.2	0.1	0.1
5	Mus musculus	0.01539	47.3	70.6	39.2	101.2	2.9	0.028	0.007	0.036	0.068	1.4	77.1	77.1	0.13	0.14
6	Myodes glareolus	0.01536	47.3	70.6	39.2	101.3	2.9	0.024	0.01	0.035	0.067	1.4	74.1	74.1	0.13	0.14
7	Microtus agrestis	0.01531	47.3	70.6	39.2	101.4	2.9	0.017	0.012	0.039	0.063	1.4	69.8	69.8	0.12	0.14
8	Neomys fodiens	0.01616	47.1	70.2	38.9	99.9	2.9	0.022	0.014	0.025	0.055	1.5	66.6	66.6	0.12	0.13
9	Blarina brevicauda	0.01764	46.8	69.5	38.4	97.6	2.8	0.021	0.018	0.032	0.093	1.6	71.8	71.8	0.15	0.17
10	Apodemus sylvaticus	0.01807	46.7	69.3	38.3	97	2.8	0.026	0.014	0.057	0.11	1.6	78.1	78.1	0.17	0.2
11	Microtus	0.02119	46.1	68	37.4	92.9	2.8	0.036	0.015	0.058	0.11	1.9	77.3	77.3	0.18	0.2
12	Peromyscus leucopus	0.02239	45.9	67.5	37.1	91.6	2.7	0.03	0.015	0.074	0.12	2	76.4	76.4	0.19	0.22
13	Apodemus flavicollis	0.02513	45.4	66.6	36.5	88.8	2.7	0.034	0.018	0.061	0.1	2.3	70.9	70.9	0.19	0.2
14	Nyctalus noctula	0.02532	45.4	66.6	36.5	88.6	2.7	0.013	0.037	0.032	0.05	2.4	45	45	0.15	0.15
15	Microtus arvalis	0.02703	45.1	66	36.1	87.1	2.6	0.055	0.019	0.039	0.19	2.4	81.7	81.7	0.25	0.28
16	Mouse	0.02797	45	65.8	36	86.3	2.6	0.051	0.016	0.05	0.18	2.5	80.4	80.4	0.24	0.27
17	Gerbillus perpallidus	0.02998	44.8	65.2	35.6	84.7	2.6	0.027	0.013	0.058	0.1	2.8	65.1	65.1	0.19	0.2
18	Mustela nivalis	0.03219	44.5	64.7	35.3	83.1	2.6	0.043	0.036	0.18	0.16	2.8	75.5	75.5	0.27	0.31
19	Acomys minous	0.0423	43.5	62.6	33.9	77.2	2.5	0.032	0.018	0.09	0.09	4	57.2	57.2	0.22	0.22
20	Jaculus jaculus	0.04804	43	61.7	33.3	74.6	2.4	0.029	0.045	0.12	0.11	4.5	54.3	54.3	0.27	0.27
21	Rhabdomys pumilio	0.05002	42.9	61.4	33.1	73.8	2.4	0.041	0.021	0.06	0.18	4.7	63.6	63.6	0.29	0.3
22	Talpa europaea	0.05117	42.8	61.2	33	73.4	2.4	0.036	0.031	0.1	0.15	4.8	59.6	59.6	0.29	0.29
23	Glaucomys volans	0.05495	42.5	60.7	32.7	72	2.4	0.059	0.056	0.19	0.29	4.9	72.1	72.1	0.4	0.45
24	Arvicola terrestris	0.06168	42.1	59.9	32.1	69.8	2.3	0.07	0.028	0.11	0.26	5.7	69.9	69.9	0.38	0.39
25	Glis glis	0.08386	41.1	57.8	30.8	64.3	2.2	0.068	0.048	0.15	0.32	7.8	64.3	64.3	0.47	0.48
26	Tamias striatus	0.10377	40.4	56.3	29.8	60.7	2.1	0.081	0.066	0.24	0.29	9.7	59.9	59.9	0.52	0.52
27	Octodon degus	0.12921	39.6	54.9	28.9	57.3	2	0.11	0.041	0.19	0.48	12.1	64.9	64.9	0.64	0.64
28	Tupaia glis	0.14107	39.3	54.3	28.6	55.9	2	0.11	0.117	0.34	0.34	13.2	56.7	56.7	0.65	0.66
29	Rat	0.1496	39.1	54	28.3	55.1	2	0.14	0.07	0.23	0.92	13.6	72.9	72.9	0.86	0.94
30	Cebuella cebuella	0.16266	38.9	53.4	28	53.8	2	0.19	0.086	0.44	1.35	14.2	79.9	79.9	1.06	1.25
31	Rattus norvegicus	0.20987	38.1	51.8	27	50.3	1.9	0.15	0.087	0.23	0.92	19.6	64.2	64.2	0.97	1
32	Cheirogaleus medius	0.23103	37.8	51.3	26.6	49	1.9	0.1	0.093	0.28	0.63	22	52.4	52.4	0.89	0.88
33	Rat	0.25004	37.5	50.8	26.3	48	1.8	0.21	0.094	0.2	1.2	23.3	66.6	66.6	1.13	1.18
34	Mustela erminea	0.2585	37.4	50.6	26.2	47.6	1.8	0.23	0.25	0.57	1	23.8	62.8	62.8	1.19	1.27
35	Helogale parvula	0.2603	37.4	50.5	26.2	47.5	1.8	0.25	0.15	0.52	1.11	24	67	67	1.2	1.27
36	Sciurus vulgaris	0.2742	37.2	50.2	26	46.8	1.8	0.17	0.17	0.63	0.55	25.9	52.9	52.9	1.02	1.04
37	Callithrix jacchus	0.3118	36.8	49.5	25.5	45.2	1.8	0.29	0.28	0.73	1.78	28.1	69.6	69.6	1.55	1.73
38	Saguinus fuscicollis	0.3304	36.6	49.1	25.3	44.5	1.7	0.19	0.33	0.78	1.44	30.3	61.2	61.2	1.47	1.6
39	Rat	0.3372	36.6	49	25.2	44.3	1.7	0.23	0.1	0.19	0.8	32.4	51.9	51.9	1.14	1.1
40	Rat (Wistar)	0.3901	36.1	48.2	24.7	42.6	1.7	0.28	0.11	0.19	1.43	37	59.7	59.7	1.43	1.44
41	Sciurus niger	0.4127	36	47.9	24.5	42	1.7	0.3	0.25	0.75	1.07	38.9	56	56	1.48	1.51
42	Sciurus carolinensis	0.5959	34.9	45.8	23.3	38	1.6	0.32	0.28	0.75	1.64	56.6	52.8	52.8	1.92	1.93
43	Saguinus oedipus	0.6237	34.8	45.6	23.1	37.6	1.6	0.31	0.37	1	2.09	58.6	55.7	55.7	2.12	2.21
44	Mustela putorius	0.64	34.7	45.4	23	37.3	1.6	0.4	0.48	1.04	2.88	59.2	61.3	61.3	2.39	2.6
45	Leontopithecus chrysomelas	0.642	34.7	45.4	23	37.3	1.6	0.41	0.38	1.32	1.89	60.2	57.1	57.1	2.15	2.26
46	Guinea pig	0.7996	34	44.3	22.3	35.2	1.5	0.56	0.23	0.47	2.7	76	57.6	57.6	2.46	2.49
47	Potorous tridactylu	0.8091	34	44.2	22.3	35	1.5	0.62	0.48	1.14	2.37	76.3	56.7	56.7	2.55	2.65
48	Erinaceus europaeus	0.9493	33.6	43.4	21.8	33.6	1.5	0.89	0.55	0.43	4.96	88.1	65.7	65.7	3.27	3.59
49	Sylvilagus floridanus	0.972	33.5	43.3	21.7	33.4	1.5	0.63	0.48	0.79	3.2	92.1	55.4	55.4	2.91	3
50	Ondatra zibethicus	0.9915	33.4	43.2	21.6	33.2	1.5	0.58	0.3	0.47	2.6	95.2	50.6	50.6	2.7	2.67
51	Saimiri boliviensis	1.0026	33.4	43.1	21.6	33.1	1.5	0.67	0.65	2.9	1.94	94.1	54.7	54.7	2.87	3.14
52	Martes foina	1.406	32.5	41.4	20.6	30.2	1.4	0.73	0.98	1.9	3.49	133.5	49	49	3.72	3.92
53	Mephitis mephitis	1.4488	32.4	41.3	20.5	30	1.4	0.66	0.6	0.98	1.74	140.9	37	37	3.17	3.11
54	Trichosurus vulpecula	1.5504	32.2	40.9	20.3	29.4	1.3	1.35	0.9	1.27	3.32	148.2	52.1	52.1	3.91	4.04
55	Martes martes	1.603	32.1	40.8	20.2	29.2	1.3	0.88	1.08	2.05	3.79	152.5	48.6	48.6	4.08	4.29
56	Cebus apella	1.7499	31.9	40.4	20	28.5	1.3	1.04	1.34	5.08	4.93	162.6	56.7	56.7	4.75	5.44
57	Eulemur macaco macaco	1.8753	31.7	40	19.8	28	1.3	1.42	0.91	2.42	7.78	175	61.8	61.8	5.22	5.76
58	Chrotagale owstoni	1.9598	31.6	39.8	19.6	27.7	1.3	1.28	1.16	2.33	4.41	186.8	50.1	50.1	4.72	4.97
59	Vulpes corsac	2.0752	31.4	39.6	19.5	27.2	1.3	0.88	2.17	3.41	3.56	197.5	41.2	41.2	4.82	5.31
60	Lemur catta	2.0746	31.4	39.6	19.5	27.2	1.3	1.12	1.17	2.28	7.29	195.6	54.4	54.4	5.33	5.76
61	Eulemur fulvus fulvus	2.5002	31	38.7	19	25.9	1.2	0.95	1.18	2.25	4.34	241.3	40.3	40.3	5.21	5.31
62	Felis silvestris	2.573	30.9	38.6	18.9	25.7	1.2	1.54	1.03	3.81	5.02	245.9	49.9	49.9	5.62	5.93
63	Didelphis virginiana	2.6336	30.8	38.5	18.8	25.6	1.2	2.29	1.21	0.83	15.73	243.3	66.9	66.9	7.24	8.35
64	Aonyx cinerea	2.675	30.8	38.4	18.8	25.4	1.2	3.06	1.51	3.59	10.64	248.7	66.1	66.1	6.97	7.97
65	Leopardus geoffroyi	3.1002	30.4	37.7	18.4	24.5	1.2	3.07	1.6	3.21	5.84	296.3	54.6	54.6	6.61	7.12
66	Lepus europaeus	3.3386	30.2	37.4	18.2	24	1.2	1.85	2.89	1.48	9.04	318.6	45.2	45.2	7.26	7.86
67	Dasyprocta punctata	3.4002	30.2	37.3	18.2	23.9	1.2	2.13	3.63	2.28	10.88	321.1	48.8	48.8	7.81	8.81
68	Potos flavus	3.9203	29.8	36.7	17.8	23	1.2	1.44	2.11	3.11	16.57	368.8	53.1	53.1	8.84	9.82
69	Dasyprocta azarae	4.1004	29.7	36.5	17.7	22.7	1.1	2.27	3.04	2.38	9.35	393	44.1	44.1	8.22	8.83
70	Varecia rubra	4.2004	29.6	36.4	17.6	22.5	1.1	2.24	1.81	3.57	7.22	405.2	43.7	43.7	7.87	8.21
71	Alouatta sara	4.3996	29.5	36.2	17.5	22.3	1.1	0.99	2.4	5.65	8.12	422.8	38.7	38.7	8.2	8.75
72	Monkey	4.5	29.5	36.1	17.5	22.1	1.1	2.1	2.3	4.2	11	430.4	46.5	46.5	8.85	9.48
73	Martes pennanti	4.7907	29.3	35.8	17.3	21.8	1.1	2.11	2.74	4.12	11.3	458.8	44.5	44.5	9.22	9.9
74	Trachypithecus vetulus	4.9996	29.2	35.7	17.2	21.5	1.1	1.54	1.92	7.2	9	480.3	42.3	42.3	9.03	9.64
75	Lutrogale perspicillata	5.1002	29.2	35.6	17.1	21.4	1.1	4.85	4.85	6.22	15.2	478.9	56.1	56.1	10.83	12.76
76	Chlorocebus pygerythrus	5.3005	29.1	35.4	17.1	21.2	1.1	1.21	4.26	8.08	8.9	507.6	37.1	37.1	9.58	10.7
77	Lutra lutra	5.3253	29.1	35.4	17	21.2	1.1	6.11	5.14	4.78	25.5	491	64.2	64.2	12.38	15.2
78	Proteles cristata	5.3998	29	35.3	17	21.1	1.1	2.43	9.06	3.99	18.2	506.3	42.4	42.4	11.44	13.97
79	Agouti paca	5.4599	29	35.3	17	21	1.1	2.22	1.76	3.21	14	524.8	45.5	45.5	10.04	10.49
80	Macaca nigra	5.5997	28.9	35.2	16.9	20.9	1.1	1.86	2.39	10.52	9.5	535.7	44.1	44.1	9.95	10.98
81	Puma yagouaroundi	5.9007	28.8	35	16.8	20.6	1.1	3.91	2.96	4.3	11.6	567.3	47.1	47.1	10.6	11.4
82	Hylobates concolor	6.5502	28.6	34.5	16.5	20	1.1	3.52	5.82	13.78	29.3	602.6	57.9	57.9	14.08	17.55
83	Prionailurus viverrinus	7.3003	28.3	34.1	16.3	19.4	1	5.59	3.35	5.29	16	699.8	51	51	12.78	13.99
84	Macropus agilis	7.7003	28.2	33.9	16.2	19.2	1	4.63	6.02	3.08	20.3	736	45.7	45.7	13.71	15.33
85	Lontra canadensis	7.9003	28.1	33.8	16.1	19	1	7.47	5.41	4.25	25.5	747.4	56.8	56.8	14.83	17.15
86	Dolichotis patagonum	8.4296	28	33.5	16	18.7	1	3.6	6.51	3.65	15.8	813.4	37	37	13.72	15
87	Symphalangus syndactylus	8.5002	28	33.5	15.9	18.7	1	4.37	5.15	14.3	29.4	796.8	54.3	54.3	15.87	18.81
88	Colobus guereza	9.7498	27.6	32.9	15.6	18	1	2.33	3.7	8.65	17.1	943.2	36.5	36.5	14.79	15.66
89	Felis chaus	9.7999	27.6	32.9	15.6	18	1	8.19	4.83	4.97	15.3	946.7	48	48	15.26	16.77
90	Lynx canadensis	10.0003	27.6	32.8	15.6	17.9	1	5.49	3.88	8.26	15.8	966.6	43.4	43.4	15.26	16.45
91	Dog	10	27.6	32.8	15.6	17.9	1	7	8.5	7.5	42	935	55.4	55.4	18.73	22.64
92	Hystrix indica	11.2543	27.3	32.4	15.3	17.3	1	5.24	5.62	4.07	25.5	1085	42	42	17.39	18.81
93	Theropithecus gelada	11.4021	27.3	32.3	15.3	17.3	1	3.8	7.72	14.09	23.6	1091	40.9	40.9	17.83	20.31
94	Pudu puda	12.898	27	31.9	15	16.7	0.9	1.99	5.05	6.16	20.6	1256	29.5	29.5	17.75	18.41
95	Gazella gazella	14.9969	26.7	31.3	14.7	16	0.9	4.06	12	7.93	32.7	1443	34.9	34.9	21.79	24.58
96	Castor fiber	15.5662	26.6	31.2	14.6	15.9	0.9	7.83	4.4	4.89	34.5	1505	44.1	44.1	21.97	23.47
97	Macaca arctoides	15.87	26.5	31.1	14.6	15.8	0.9	5	6.1	11.8	24.1	1540	35.8	35.8	21.3	22.85
98	Lynx lynx	17.5008	26.3	30.7	14.4	15.4	0.9	7.95	9.3	9.43	26.4	1697	37.4	37.4	23.35	25.65
99	Capreolus capreolus	20	26	30.3	14.1	14.8	0.9	8	16	10	48	1918	39.3	39.3	27.93	32.35
100	Cuon alpinus	19.9964	26	30.3	14.1	14.9	0.9	7.64	15.8	11.6	34.6	1930	35.2	35.2	26.68	30.54
101	Dog	20.388	26	30.2	14.1	14.8	0.9	9.2	15.3	9.6	44.7	1960	39.6	39.6	27.98	32.17
102	Mandrillus sphinx	23.0249	25.7	29.8	13.8	14.3	0.9	4.99	7.6	16.8	33.1	2240	32.2	32.2	27.95	29.87
103	Papio hamadryas	23.2493	25.7	29.7	13.8	14.3	0.9	8.03	10.3	17.4	39.2	2250	37.4	37.4	29.35	32.45
104	Zalophus californianus	33.9579	24.9	28.4	13.1	12.9	0.8	20.59	16.8	31	127.4	3200	54.7	54.7	45.67	56.18
105	Hydrochaeris hydrochaeris	33.9875	24.9	28.4	13.1	12.9	0.8	10.35	10.4	8.4	69.6	3300	36.1	36.1	39.32	42.2
106	Canis lupus chanco	38.0209	24.7	28.1	12.9	12.5	0.8	20.69	30.3	14	97.1	3640	42.9	42.9	46.66	56.35
107	Sheep	52.006	24	27	12.3	11.5	0.8	16	28	10.6	96	5050	32.5	32.5	54.93	61.68
108	Reference women	58.015	23.8	26.7	12.1	11.2	0.7	27.5	24	120	140	5490	51.8	51.8	65.07	83.63
109	Human	59.97	23.8	26.6	12.1	11.1	0.7	25	32	130	170	5640	51.4	51.4	68.58	90.32
110	Reference man	70.04	23.5	26.1	11.8	10.6	0.7	31	33	140	180	6620	50.8	50.8	75.95	98.83
111	Panthera tigris altaica	74.9716	23.3	25.9	11.7	10.4	0.7	42.46	30.5	34.2	110	7280	41.6	41.6	72.84	84.7
112	Hog	125.33	22.3	24.4	10.9	9.1	0.6	26	35	12	160	12,300	25	25	102.8	109.95
113	Dairy cow	487.9	20	20.8	9	6.3	0.5	116	188	40	646	47,800	25.6	25.6	308.5	353.68
114	Horse	600.28	19.6	20.3	8.7	6	0.5	166	425	67	670	58,700	24.2	24.2	366.4	457.67
115	Steer	699.8	19.4	19.9	8.5	5.7	0.5	100	230	50	500	69,100	16.5	16.5	392.43	434.45
116	Elephant	6650.4	16.1	15.2	6.2	3.1	0.3	120	220	570	630	7.00 × 105	4	4	2292.2	2327.2

in Appendix A tabulates the body mass, organ masses, and ${\dot{q}}_B$ with EAR method for al organs (last column, Table A1) for 116 species.

B.

EAR for Vital Organs and Elia Constant for RM: In order to study the effect of using EAR for RM, the author replaced the EAR for RM with Elia’s constant for RM {${\dot{q}}_{RM,m}$ of 0.581 W/kg} and computed the whole-body metabolic rate. The constants in Kleiber’s law become a = 2.486 and b = 0.781 (Figure 3). It is seen from that that the slope b increased from 0.76 to 0.78, representing a 3.31% increase in the exponent b indicating that the slope “b” is dominated by vital organs.

Figure 3. Comparison of Kleiber’s Law {${\dot{q}}_B = a {M_B}^b$} using EAR for all organs with the Kleiber’s law with EAR for vital organs and Elia constant for RM. (a) The BMA allometric constants in SOrMR_k relations were derived from organ metabolic rate data of six species ranging in mass from 0.48 kg to 70 kg [21]. By extending these same organ allometric constants to 116 species and using available organ mass data for these species (ranging from 0.0075 kg to 6650 [20]), the resulting allometric constants using EAR for all organs were a = 3.216 and b = 0.756. (b) With EAR for vital organs and Elia’s constant for RM, {${\dot{q}}_{RM,m}$ = 0.581 W/kg}, the estimated allometric constants were a = 2.489 and b = 0.781. A minority view holds that the b for BMR is approximately 0.67 [54], rather than the widely accepted value of 0.75.

Figure 3. Comparison of Kleiber’s Law {${\dot{q}}_B = a {M_B}^b$} using EAR for all organs with the Kleiber’s law with EAR for vital organs and Elia constant for RM. (a) The BMA allometric constants in SOrMR_k relations were derived from organ metabolic rate data of six species ranging in mass from 0.48 kg to 70 kg [21]. By extending these same organ allometric constants to 116 species and using available organ mass data for these species (ranging from 0.0075 kg to 6650 [20]), the resulting allometric constants using EAR for all organs were a = 3.216 and b = 0.756. (b) With EAR for vital organs and Elia’s constant for RM, {${\dot{q}}_{RM,m}$ = 0.581 W/kg}, the estimated allometric constants were a = 2.489 and b = 0.781. A minority view holds that the b for BMR is approximately 0.67 [54], rather than the widely accepted value of 0.75.

3.2. Whole Body Metabolic Rate Using ODM Hypothesis and Comparison with Results from EAR Method

(A) 

ODM and EAR for SOrMR_k of RM: The ODM model uses the relation for effectiveness factor of four vital organs to predict SOrMR_k and using the allometric law for ${\dot{q}}_{RM,m}$. Figure 4 shows the results for the whole-body metabolic rate {${\dot{q}}_B$} vs. M_B obtained.

$${\dot{q}}_B \left(W\right)= a {M_B}^b, a=2.963, b=0.747, M_B=0.0075 to 6650 kg$$

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The same figure provides a comparison with Wang’s results using EAR. Note that the ODM method relies only on data from two reference species, RS-1 and RS-2, to predict SOrMR_k and whole-body metabolic rates for the remaining 114 species. The current ODM method for estimating SOrMRk is validated, as it supports Kleiber’s law.

Table A1 compares the ODM-based metabolic rate, ${\dot{q}}_B$ for 116 species with those ${\dot{q}}_B$ using the EAR method (last two columns). If the error percentage is defined as {MR with ODM–MR with EAR) × 100/MR with EAR}, then the highest error occurs for a 60 kg human at 24.07%. The average error across all 116 species is 8.14%.

(B) 

ODM for Vital Organs and Elia’s Constant SOrMRk for RM: When the SOrMR_k of vital organs from the ODM method and Elia’s constant for RM are used, a = 2.242 and b = 0.772. The results show that the allometric constant b increases from 0.747 to 0.781 (Figure 5), indicating 3.35% increase in the slope of b. This increase in b is nearly the same as in the EAR method (Section 3.1). The residual mass (non-vital mass) seems to play a minor role in determining the exponent b, since the vital organs are more metabolically active. It is apparent from Figure 4 and Figure 5 that the current ODM method for the estimation of SOrMR_k is validated, as it supports Kleiber’s law.

(C) 

Effects of Change in Selection of RS-2 species in ODM method: When BS of masses of 337 g {45 times M_B of shrew, 0.86 times M_B of Rat Wistar} and 84 g {11 times M_B of shrew, 0.21 times M_B of Rat Wistar} were selected for RS-2, the “b” changes by 0.1% and −0.12%, respectively, indicating a negligible change in “b”.

Figure 5. Effects of Energy release from RM on Kleiber’s law using ODM method. Based on 116 BS with MB from 0.0075 to 6650 kg: ${\dot{q}}_B = a {M_B}^b$ (i) Vital Organs under ODM and RM following EAR: a = 2.963 b = 0.747, (ii) Vital Organs under ODM and RM with Elia’s constant of 0.581 W/kg: a = 2.242, b = 0.772. The percentage increase in b for the ODM method is the same as the percentage increase in the EAR method.

Figure 5. Effects of Energy release from RM on Kleiber’s law using ODM method. Based on 116 BS with MB from 0.0075 to 6650 kg: ${\dot{q}}_B = a {M_B}^b$ (i) Vital Organs under ODM and RM following EAR: a = 2.963 b = 0.747, (ii) Vital Organs under ODM and RM with Elia’s constant of 0.581 W/kg: a = 2.242, b = 0.772. The percentage increase in b for the ODM method is the same as the percentage increase in the EAR method.

3.3. Vital Organ Contribution Percentage with ODM Method and Comparison of Results with Empirical Allometric Relations (EAR) Method

As a further validation of the ODM method, the predicted percentage contribution of vital organs is compared with literature data. Figure 6 shows the computed ERR from the four vital organs vs. m_vit, while Figure 7 compares the percentage energy contribution of vital organs estimated using ODM with those obtained using EAR. If the allometric fit for ${\dot{q}}_{vit}$ is expressed as ${\dot{q}}_{vit} = {\alpha }_{vit} {m_{vit}}^{\beta vit}$ with ${\dot{q}}_{vit}$ in watts, m_vit in kg, and vital energy contribution as $\% vit = {\gamma }_{vit} {M_B}^{{\nu }_{vit}}$, the fits yield α_vital = 12.69, β_vital = 0.74, γ_vit = 51.85, ν_vil = −0.115, while the previous literature with the EAR method for all vital organs suggests α_vit = 15.67, β_vit = 0.77. γ_vit = 49.67 and ν_vil = −0.101 [23]. The energy contribution estimated from ODM appears to agree with data from the EAR method for M_B up to 500 kg. The OD in organs results in a slope of ${\dot{q}}_{vit}$ vs. m_vit that is less than 1, indicating the increase in ${\dot{q}}_{vit}$ is less than proportional to the increase in the vital organ masses.

3.4. The Upper Metabolic of Whole-Body {UMR_B} and Maximum Metabolic Rate of Whole-Body {MMR_B}

The η_eff,k is a function of oxygen gradients within cell clouds; the steeper the gradients lower is η_eff,k. The ${\dot{q}}_B$ of any BS is a strong function of SOrMR_k {=η_eff,k ${\dot{q}}_{k,m,iso}$}, and ${\dot{q}}_{k,m,iso}$ is affected by Y_O2,CC,s, which depends on the percentage of capillaries perfused at the CC surface. Equation (20) states that whole-body metabolic rate {${\dot{q}}_B$} increases with the increase in η_eff,k {i.e., ${\dot{q}}_{k,m,} \to {\dot{q}}_{k,m,iso}$} for metabolically dominant vital organs and the remaining tissue masses.

(A) 

Hypothetical Upper Metabolic Rates of Organs and whole body: The resting or basal metabolic rate (BMR) is based on oxygen consumption, typically with partial perfusion from capillaries. What if there is no oxygen concentration gradient? What is the effect of oxygen gradients on “b”? Would this result in an isometric scaling law (b = 1 or b′ = 0), despite differences in organ masses? Mathematically, it can be shown that b ≠ 1 or b′ ≠ 0 due to differing SOrMR_k of organs, rather than differences in organ masses. By setting η_eff,k = 1 for all organs, a hypothetical UMR_B for the whole body can be obtained. Two cases were studied:

(i)

Using EAR for RM, the whole-body allometric relation is given as ${\dot{q}}_{B,UMR} = a_{UMR} {M_B}^{ b_{UMR}}$ where a_UMR = 6.28, b_UMR = 0.864 {Figure 8}; it is apparent that “b” increases from 0.747 to 0.864 in the absence of O₂ gradients within vital organs. As such, the difference between ${\dot{q}}_{B,UMR}$ and ${\dot{q}}_B$ is due to the effects of O₂ gradients within vital organs.

(ii)

Assuming the absence of O₂ gradients in all organs, the allometric constants become a_UMR = 8.160, b_UMR = 0.933 {Figure 8}. Thus, the absence of O₂ gradients in RM increases “b_UMR” from 0.864 to 0.933. It appears that BMR follows almost isometric law.

Figure 8. Effects of oxygen gradients on Kleiber’s law. The ODM-based BMR with finite η_eff,k yields Kleiber’s law: ${\dot{q}}_B = 2.963 {M_B}^{ 0.747}$, all ${\dot{q}}_B$ in Watts. When oxygen gradients disappear, Kleiber’s law is modified as: ${\dot{q}}_{B,UMR} = a_{UMR} {M_B}^{ b_{UMR}}$. Two cases: (i) (⬤): Absence of O₂ gradients in vital organs and ${\dot{q}}_{RM}$ following EAR, a_UMR = 6.282, b_UMR = 0.864 (ii) (■) Absence of O₂ gradients in all organs, including RM, a_UMR = 8.160, b_UMR = 0.933. Slope “b” generally increases when there are no oxygen gradients.

Figure 8. Effects of oxygen gradients on Kleiber’s law. The ODM-based BMR with finite η_eff,k yields Kleiber’s law: ${\dot{q}}_B = 2.963 {M_B}^{ 0.747}$, all ${\dot{q}}_B$ in Watts. When oxygen gradients disappear, Kleiber’s law is modified as: ${\dot{q}}_{B,UMR} = a_{UMR} {M_B}^{ b_{UMR}}$. Two cases: (i) (⬤): Absence of O₂ gradients in vital organs and ${\dot{q}}_{RM}$ following EAR, a_UMR = 6.282, b_UMR = 0.864 (ii) (■) Absence of O₂ gradients in all organs, including RM, a_UMR = 8.160, b_UMR = 0.933. Slope “b” generally increases when there are no oxygen gradients.

(B) 

Maximum Metabolic Rates of Organs: The cardiac output at rest is approximately 5–6 LPM, with capillaries partially perfused on CC surface {about 25–30% of capillaries in vital organs [50] and 15–25% in SM; also see Figure 2b showing perfused capillaries on surface of cell cloud}. At rest, the Table 2 indicates that about 50% of blood pumped by the heart flows is directed to four vital organs, Ref. [55] states that it could be as high as 80%.

Under exercise, cardiac output increases to approximately 25–35 LPM. For maximal O₂ consumption (${\dot{V}}_{O2max}$), increased blood flow rates lead to a higher capillary perfusion percentage, thereby increasing Y_O2,CC,s. Since ${\dot{q}}_B= \left|{\dot{Q}}_B\right|+{\dot{W}}_B$ increases during exercise, both ${\dot{W}}_B$ {ATP work} and |${\dot{Q}}_B$| {heat loss from skin} must also increase, leading to a rise in internal temperature. Thus, the rise in |${\dot{Q}}_B$| is accompanied by increased blood flow through the outer skin to facilitate heat dissipation, resulting in blood flow redistribution. The primary organ contributing to MMR is SM. Further blood flow is diverted from various organs (e.g., stomach, kidneys) and SM is almost 100% perfused during exercise [56]. It is noted from Table 2 that blood flows to organs are re-distributed and vital organ blood flow % are 47, 25, 14 respectively under rest, mild exercise and maximal exercise conditions. For example, kidney perfusion accounts only for 20–30% of resting blood flow [55,56,57] i.e., the flow through kidneys decreases under exercise {see Table 2), indicating decreased Y_O2,CC,s and decreased SOrMR_kid. Table 2 summarizes the blood flow distribution to vital organs: kid, H, Br and L and total blood flow redistribution to vital organs. Even though the total blood flow to vital organs is only 92% of resting flow, blood flow is diverted from kidney and liver to heart since it requires increased work output to increase the cardiac output by almost 3 times (=17,500/5800). It is remarkable that ratio of blood flow to heart muscles under exercise to resting flow is 3 (=750/250, last column Table 2 for heart). The SM receives almost 10 times resting blood flow (last column Table 2 for SM) under maximal exercise.

The increased ERR (${\dot{q}}_{B,MMR}$) is driven by a higher percentage of perfused capillaries (SM, which comprises about 40% of the body mass almost perfused to 100% [56]) and decreased vascular resistance due to an increased diameter of small arteries (100–300 μm) [58] thus affecting the scaling law for ${\dot{q}}_{B,MMR}$. The increased O₂ delivery during exercise is also due to a lower pH (i.e., increased CO₂, or increased acidity), reduced oxy-Hb affinity, and hence, an increased release of O₂ from Hb, which promotes higher Y_O2,CC,s. The OEF increases from 0.25 to 0.33 [20,55] at rest to almost 0.75 [55] for MMR.

Since SM plays a major role in metabolism during exercise, allometric laws for mass of SM vs. M_B and increased blood flow are used in the ODM model: (i) Prange’s relation for mass of SM in kg: m_SM = 0.061 M_B^1.09 [30] (M_B from 0.01 to 10,000 kg); (ii) Kayser: m_SM = 0.093 M_B^1.142 [59]; (iii) Painter: m_SM = 0.0961 M_B^1.06 [60]; (iv) White: m_SM = 0.0645 M_B^1.02 [61]; hence, mass of SM is 5.1 kg for a 58 kg person according to the Prange law {used in current work} but 9.6 kg according to the Kayser law, but the literature suggest SM is 24.4 kg for a 58 kg person [62].

For the ODM model during exercise, m_RM,ex = M_B − m_vit − m_SM. According to the ODM hypothesis, the increase in MMR is due to an increased O₂ supply to organs {particularly to SM and Heart}, with a higher perfusion percentage on the CC surface, thus increasing Y_O2,CC,s and possibly due to an increased η_eff,CC (See Section 4.8 in Ref. [63])} originating from an increased pO₂. With the following relations,

$${\left({\displaystyle \frac{Y_{O2,cc,s,Ex}}{Y_{O2,cc,s,\mathrm{Re}st}}}\right)}_k= \left\{{\displaystyle \frac{Blood Flow to organ k Under Exercise}{Blood Flow to k under Rest}}\right\} , k= Kids, H, Br,L, SM$$

(28)

$${\dot{q}}_{k,m,Ex} = \left\{{\dot{q}}_{k,m,\mathrm{Re}st} {\left( {\displaystyle \frac{Y_{O2,cc,s,Ex}}{Y_{O2,cc,s,\mathrm{Re}st}}}\right)}_k\right\}, {\dot{q}}_{SM,m,Ex} = \left\{{\dot{q}}_{SM,m,\mathrm{Re}st} {\left( {\displaystyle \frac{Y_{O2,cc,s,Ex}}{Y_{O2,cc,s,\mathrm{Re}st}}}\right)}_{SM}\right\},\left\{{\dot{q}}_{RM,Ex,m}\right\} = \left\{{\dot{q}}_{RM,m} \right\}{\left( {\displaystyle \frac{Y_{O2,cc,s,Ex}}{Y_{O2,cc,s,\mathrm{Re}st}}}\right)}_{RM}$$

(29)

$${\dot{q}}_{SM,m,\mathrm{Re}st} = {\dot{q}}_{RM,m,\mathrm{Re}st} = e_{RM,6} {M_B}^{ f_{RM,6}}, e_{RM,6}=1.45, f_{RM,6}= -0.17$$

(30)

For predicting MMR using the ODM model, Equations (28)–(30) are used, along with data on blood flow ratios presented in Table 2 (last column). The myoglobin (Mb), which aids in transport of O₂ in H and SM, increases during exercise indicating an increase in effective diffusivity “D” {i.e., lower G_OD,k, k = H, SM} and the core cells may also obtain O₂ thus decreasing OD and increasing {η_eff,CC}_k. The highest possible values {η_eff,CC}_k are 1 for SM and H. The blood flow is reduced for kidneys (flow under exercise/flow under rest = 0.55) and liver (0.67), but increased for H (3), SM (10.42), and RM-ex (1.32) (Table 2). Changing perfusion % alters Y_O2,CC, according to ODM hypothesis.

If ${\dot{q}}_{B,MMR} = a_{MMR} {M_B}^{ b_{MMR}}$, the constants a _MMR and b_MMR are estimated for the following cases:

(i)

Vital organs are under a finite oxygen gradient, but Perfusion Ratio Change during exercise: a_MMR = 4.221 and b_MMR = 0.8. The slope is higher compared to the resting case due to increased SOrMR_k for SM, H, RM-ex but decreased for kidneys and liver. Further, the slope is higher compared to the slope for UMR vs. MB, primarily due to the increased perfusion ratios of blood to SM. For RM-ex, the SOrMR_k is given by the product of allometric laws of RM as at rest and perfusion ratio. Figure 9 compares the results for ${\dot{q}}_{B,MMR}$ under ODM with the literature data for ${\dot{q}}_{B,MMR}$.

(ii)

Vital organs in absence of oxygen gradient but Perfusion Ratio Change during exercise: Oxygen gradient may become zero due to increase in Mb in the SM and Heart, which results in faster diffusion of O₂. Thus, b increases to 0.942. When η_eff,CC is set to 1 for all vital organs and SM {i.e., no O₂ gradient during exercise}, but RM-ex still follows the allometric law with a correction for the perfusion ratio of 1.32, then a_MMR = 8.651 and b_MMR = 0.939, indicating an almost isometric law. It is observed that even if all cells are irrigated with the same O₂ concentration, the exponent b_MMR is not equal to 1 due to varying organ masses.

Figure 9. Comparison of MMR (${\dot{q}}_{B,MMR }$, ♦ and ⬤) computed using ODM model with the literature data (▲) for M_B from 0.008 to 6650 kg. ${\dot{q}}_{B,MMR} = a_{MMR} {M_B}^{ b_{MMR}}$. The ODM-based BMR with finite η_eff,k at rest yields Kleiber’s law: ${\dot{q}}_B = 2.963 {M_B}^{ 0.747}$, all in Watts. For estimation under exercise, SM mass in kg: m_SM = 0.061 M_B^1.09 [30]. m _RM-Ex = M_B − m _vit − m_SM, ${\dot{q}}_{RM-ex} = a_{RM,ex} {M_B}^{ b_{RM-ex}}$. Cases studied: (i) ♦ Vital organ with finite η_eff (finite O₂ gradients) ${\dot{q}}_{SM,m}$ and ${\dot{q}}_{RM,Ex,m}$ follow EAR, Perf. Ratio of (Table 2), a_SM = a_RM, b_SM = b_RM, a_MMR = 4.221, b_MMR = 0.8, ”b” increases to 0.8 during exercise from 0.747 at rest due to change in capillary perfusion from rest to exercise mode (ii) ⬤ No oxygen gradients and hence η_eff,k = 1 for all vital organs and SM; but RM-ex follow EAR corrected for perfusion ratio, a_RM-ex = a_RM, b_RM-ex = b_RM (Table 1), a_MMR. = 8.651 b_MMR = 0.939, (iii) literature data-Painter: (▲), V_O2,max converted into Watts using ${\dot{\mathrm{q}}}_{\mathrm{B},\mathrm{MMR}} (\mathrm{watts}) = 0.342 {\mathrm{V}}_{\mathrm{O}2\max } (\mathrm{mL}/\min )$ assuming 1 mL of O₂ releases 20.5 J; a_MMR. = 40.46, b_MMR = 0.872; Ref. [17] cites b_MMR = 0.942.

Figure 9. Comparison of MMR (${\dot{q}}_{B,MMR }$, ♦ and ⬤) computed using ODM model with the literature data (▲) for M_B from 0.008 to 6650 kg. ${\dot{q}}_{B,MMR} = a_{MMR} {M_B}^{ b_{MMR}}$. The ODM-based BMR with finite η_eff,k at rest yields Kleiber’s law: ${\dot{q}}_B = 2.963 {M_B}^{ 0.747}$, all in Watts. For estimation under exercise, SM mass in kg: m_SM = 0.061 M_B^1.09 [30]. m _RM-Ex = M_B − m _vit − m_SM, ${\dot{q}}_{RM-ex} = a_{RM,ex} {M_B}^{ b_{RM-ex}}$. Cases studied: (i) ♦ Vital organ with finite η_eff (finite O₂ gradients) ${\dot{q}}_{SM,m}$ and ${\dot{q}}_{RM,Ex,m}$ follow EAR, Perf. Ratio of (Table 2), a_SM = a_RM, b_SM = b_RM, a_MMR = 4.221, b_MMR = 0.8, ”b” increases to 0.8 during exercise from 0.747 at rest due to change in capillary perfusion from rest to exercise mode (ii) ⬤ No oxygen gradients and hence η_eff,k = 1 for all vital organs and SM; but RM-ex follow EAR corrected for perfusion ratio, a_RM-ex = a_RM, b_RM-ex = b_RM (Table 1), a_MMR. = 8.651 b_MMR = 0.939, (iii) literature data-Painter: (▲), V_O2,max converted into Watts using ${\dot{\mathrm{q}}}_{\mathrm{B},\mathrm{MMR}} (\mathrm{watts}) = 0.342 {\mathrm{V}}_{\mathrm{O}2\max } (\mathrm{mL}/\min )$ assuming 1 mL of O₂ releases 20.5 J; a_MMR. = 40.46, b_MMR = 0.872; Ref. [17] cites b_MMR = 0.942.

Validation of ODM-Based MMR Allometry:

The predicted values for b_MMR range from 0.798 to 0.942, with an average of 0.87. The upper value of b_MMR indicates almost an isometric law. It is believed that MMR must follow an isometric law since the “cost” of transportation (e.g., treadmill, jogging) must be proportional to body mass, meaning SMMR {specific maximum metabolic rate, W/kg} must not differ between smaller and larger species during exercise. Ref. [9] states that when a 20 g mouse and a 500 kg racehorse run at their maximum capacity, their specific maximal metabolic rate (W/g) is nearly the same. This finding agrees with the ODM model, indicating all cells within an organ are subjected to oxygen concentrations close to their highest possible values.

The literature data mostly reports ${\dot{V}}_{O2max}$ {mL of O₂ per min} vs. M_B under exercise. It is converted into watts using HHV_O2 of 20.5 J/mL of O₂. ${\dot{q}}_{B,MMR} = 0.342 {\dot{V}}_{O2max}$ where ${\dot{q}}_{B,MMR}$ in Watts and ${\dot{V}}_{O2max}$ in mL/min. Painter collected data on MMR for 32 mammalian BS ranging from 0.007 kg (pygmy mice) to 575 kg (cattle), found that b_MMR = 0.872 (95% CI: b_MMR = 0.812–0.931) and attributes the increase from 0.75 at rest to 0.872 under exercise to the increased O₂ transport to cells (with the heart pumping rate as the limiting step [60]). Weibel et al. [17] conducted treadmill experiments in animals to measure ${\dot{V}}_{O2max}$ (highest rate for 5 min) and reported a_MMR = 118 mL of O₂/{min kg^0.872} or 40.4 W/kg^0.872, with b_MMR = 0.872 for 34 mammalian species, including both athletic and non-athletic groups (0.007 to 500 kg). The upper bound for b_MMR yields b_MMR = 0.942 when η_eff = 1(zero O₂ gradients) for vital organs and SM, and the lower bound is 0.798 {Figure 9}. Ref. [17] reports b_MMR = 0.942 for BS of M_B from 0.02 to 500 kg and 0.872 for the non-athletic group. The MMR per kg body mass is a weak function of M_B, which agrees with the data on {${\dot{V}}_{O2max}$}/M_B in ref. [9]. It appears that the athletic group may have a higher concentration of Mb, promoting zero oxygen gradients {Figure 9}. The predicted upper bound on b_MMR of 0.942 is higher than most reported values of 0.87 since η_eff,CC is set at 1 for all vital organs, even though the % perfused for blood flow through kidneys seems to decrease during exercise, indicating η_eff,CC < 1 for Kidneys.

Other literature data are as follows: Talyor et al. [64] and Ref. [9]: b_MMR = 0.87 − 0.88 for homeotherms. Refs. [6,60] (see Figure 6 in Ref. [6]) b_MMR = 0.872; Agutter b_MMR = 0.86 [54]. b_MMR = 0.81 for M_B = 0.3 to 300 kg but increases to 0.86 for M_B = 0.3 to 500 kg [62]. Single Flow Network model b_MMR = 6/7=0.86 [65].

While the predicted slopes (b_MMR) for the ODM method appear to be closer to literature data, the predicted a_MMR (8.436) is low compared to literature data of a_MMR = 40.46. The MMR is largely driven by the high MR of SM, and the predicted low values of a_MMR originate from the allometric relation of SM and body mass used in the current ODM model. According to Weibel and Hoppeler [17], S, SM is about 42% of body mass in the athletic wood mouse (small animal), 45% in the pronghorn, and 25% in the goat, with an average of 36% of body mass. Further, skeleton mass varies significantly, with the shrew at 5% and the elephant at 25% [53]. These findings indicate a wide variation in SM mass across body sizes, which alters the values of a_MMR.

The current results for MMR are validated further with the data reported by Midorikawa et al. [62]. The VO₂max (during maximal exercise) of Sumo wrestlers is about 30 mL/min/kg or 10.25 W/kg, attributed to SM, liver and kidneys [62]. For a 58 kg individual, reported data show ${\dot{q}}_{B,MMR}$ = 1320 W, while the predicted value is ${\dot{q}}_{B,MMR} = 446 W$. Why do measured values exceed predictions from the ODM model? The allometry for SM predicts a mass of 5.1 kg for 58 kg human, whereas the measured value is 24 kg for a 58 kg person! When the author used the actual SM mass of 24 kg (without using allometric SM mass) and m_RM-EX = M_B − m_SM − m_vit = 58 − 24 − 5.4 = 28.6 kg, the predicted ${\dot{q}}_{B,MMR}$ increased to 1045 W (${\dot{q}}_{RM,ex}$ = 296 W, EAR) while the reported data for ${\dot{q}}_{B,MMR}$ is 1320 W.

3.5. A Method of Tracking G_OD,k for Organs During Growth of Humans or Any Other BS by Medical Personnel

If O₂ diffusion follows an increasingly tortuous path for certain populations as humans grow, the effective diffusion coefficient decreases, causing G_OD,k variation to become much steeper than normal. This indicates higher ODM and a greater likelihood of energy release adaptation by the body via glycolysis at a specific (G_OD,k)_gly particularly near the core of CC. Thus, estimating (G_OD,k) as M_B(t) during the growth is of interest. How can medical personnel determine G_OD,k?

(I)

Direct Method: Measure Organ Mass (m_k) and use known SOrMR_k of same organ k of RS-1: Measure blood flow rate and the change in O₂ concentration between the arterial and venous ends of the organ to estimate OrMR_k. Directly measure organ masses using CT scan or MRI, then estimate SOrMR_k (=OrMR_k/m_k) and compare with SOrMR_k of the shrew (RS-1, i.e., isolated). Estimate η_eff,k and determine G_OD,k of organ k using Equation (16).

(II)

Ratio method for Same BS: Assume that (G_OD,k at any age/G_OD,k at birth) = ${({\mathrm{m}}_{\mathrm{k}}/{\mathrm{m}}_{\mathrm{k},\mathrm{birth}})}^{\mathcal{l}\mathrm{k}}$ if G_OD,k at birth and m_k,birth are known. Typically, ${\mathcal{l}}_{\mathrm{k}}$ ≈ 2/3.

(III)

G_OD,k in terms of Body Mass data M_B(t): The ODM method presents SOrMRk in terms of a powerful dimensionless parameter G_OD,k, which is proportional to ${{\mathrm{m}}_{\mathrm{k}}}^{\mathcal{l}\mathrm{k}}$. Using the allometric law for organ masses (Equation (5)), $G_{OD k} \propto M_B{\left(t\right)}^{{\mathcal{l}}_k d_k}, t in years$ where ${\mathcal{l}}_{\mathrm{k}}$ = 2/3 and d_k values are tabulated in Table 1. This relation seems to indicate that G_OD,k grows abnormally if body mass M_B grows abnormally.

(IV)

Ratio Method, G_OD,k in terms of measured Organ Masses and Reference Species RS-2: Assuming Rat Wistar as RS-2 and knowing G_OD,k of RS-2, one can determine G_OD,k if organ mass data are available.

$${\displaystyle \frac{ G_{OD,k}\left(t\right)}{G_{OD,k,RS-2}}} = {\left\{ {\displaystyle \frac{ m_k\left(t\right)}{m_{k,RS-2}}}\right\}}^{{\mathcal{l}}_k}, {\mathcal{l}}_k = {\displaystyle \frac{2}{3}}, t in years$$

(31)

For example, selecting organ masses for a 10 kg (or 1 year old) infant from Ref. [66], the estimated G_OD,k values and corresponding η_eff,k (in parentheses) are: 65 (0.33) for the kidneys, 148 (0.23) for the heart, 1007 (0.092) for the brain, and 522 (0.13) for the liver. For a dog of similar 10 kg body mass, the brain is much smaller, resulting in a significantly lower G_OD,Br (179), and a higher η_eff,Br (0.21), leading to a higher metabolic rate per unit mass of the dog’s brain compared to a human’s. As a human grows to 70 kg, the G_OD,k increases while η_eff,k decreases. Using the same reference, the values become: 160 (0.22) for the kidneys, 520 (0.13) for the heart, 1260 (0.082) for the brain, and 1422 (0.077) for the liver. Note the rapid growth of the liver and the slow growth of the brain mass for a healthy human. If organ mass m_k(t) is measured as a function of age in years, then Equation (31) can be used to estimate G_OD,k.

Organ G_OD,k# and Cancer: While the literature data indicates positive correlation between the mass of an organ and the number of cancer cases [43] which is attributed to link between excess fat in organs and obesity, the author speculates that ODM results in increased production, accumulation and concentration of HIF-1α,resulting in cellular adaptation to glycolysis for energy release. The glycolysis path provides many intermediaries for the creation of new cells and hence may contribute to the onset of cancer. ODM indicates that increased organ size may promote glycolysis within the core of the cell cloud, thus promoting cancer; however, HIF may behave like Y_O2,avg, indicating lower average HIF concentration within CC with increase in m_k. Thus, the rate of increase in the number of cancer cases with the size of BS may not be significant. This speculation is confirmed by data as confirmed by the data in the literature [67]; it is also known as Peto’s paradox. Thus, the ODM hypothesis may be of interest to oncologists, since the “elevated” G_OD,k number may be used to “detect” possible onset of cancer in the future, just like elevated Prostate-Specific Antigen (PSA) levels are used as an indicator of prostate cancer (see Section 5).

4. Summary and Conclusions

The earlier literature: (i) adopted the Krogh cylinder or COA model for prediction of metabolic rate while the ODM hypothesis uses COS model to explain the variation of allometric exponents with the size of organs, (ii) formulated the classical WBE’s “upstream” blood flow network model to explain Kleiber’s law $\left( {\dot{q}}_B = a {M_B}^b \right)$, while the current approach uses a “downstream” cell-level hypothesis to obtain Kleiber’s law; (iii) used a heterogeneous approach to obtain the empirical allometric relations for organs, which were then used for estimating the BMR of the whole body and to validate the hypothesis by demonstrating the Kleiber’s law with a = 3.216 and b = 0.756 for 116 BS [23]; since cell multiplication concept is used for larger BS, and if oxygen gradients were absent b tends towards 1 agreeing with Glazier’s hypothesis [68].

Since biologists raised several puzzles, such as (i) why F_k values are negative in OMA {Equation (7)} for k = Kids, H, Br and (ii) how organs “know” they are in a smaller or larger body mass and adjust their metabolic rates accordingly. The author’s previous work answered these puzzles by linking the field of ODC literature in engineering to ODM in biology [3]. The same reference showed that the allometric exponent {F_k} becomes more negative with the increasing mass of the organ.

The ODM hypothesis with COS model is now extended in demonstrating Kleiber’s law by adopting the literature on effectiveness factor in engineering to biology as η_eff,CC for cell clouds within an organ. Data on two reference species {RS-1: Shrew of 0.0076 kg, RRS-2: at Wistar of 0.390 kg} are used to (i) predict SOrMR_k of organ _k for 114 BS; (ii) demonstration of Kleiber’s power law (${\dot{q}}_B = a {M_B}^b$) with a = 2.962, b = 0.747 for M_B = 0.0075 to 6650 kg; (iii) obtain allometric law for maximal metabolic rate {MMR}, and validation with literature data; (iv) show that the MMR occurs due to increased blood perfusion % which increases O₂ concentration.

A method for estimating the magnitude of dimensionless (G_OD,k) for organs is “suggested” for use by medical personnel to assess whether the abnormal increase of G_OD,k of organs with growth {e.g., human growth from 2 kg to 70 kg} indicates a progression toward oxygen deficiency, glycolysis and the onset of organ cancer.

5. Future Work

Whether all the secrets of organ, Kleiber’s, and MMR allometries in biology can be revealed from oxygen-deficient combustion engineering remains an open question. Additional supporting data are needed either to confirm or question the ODM hypothesis. Further studies are needed for statistical data to determine whether cancer development correlates with abnormal increases in G_OD,k, and to assess its relationship with cancer occurrence. A more precise allometric relationship is needed for SM mass relative to body mass M_B, since it directly affects the predicted MMR for the ODM hypothesis. While current study focusses on Capillary on Surface (COS) for cell cloud model, future study is needed for (i) Krogh-type capillary on axis (COA) model incorporating the ODM method, (ii) develop G_OD,k number for COA and (iii) check whether Kleiber’s law still holds good. Further studies are needed in estimating the fraction of “lethal” volume containing sleeper cells.