Draft:Traditional and modified surface rule

            Traditional and modified surface rule

                                  Introduction

  In 1839, professor of mathematics Sarrus and doctor of medicine Rameaux theoretically substantiated the traditional surface rule , according to which the metabolic rate of warm-blooded living beings is proportional to the surface area of their body or the mass (associated with it through volume and density) to the ⅔ power:

                         P = ksS         (1s)

                         P = km2/3,	 (1m)

where S is the body surface area of a warm-blooded living being, m²;

m is its mass, kg;

P is the metabolic rate (intensity) corresponding to the power of energy consumption for the notations accepted in physics, W = J/s;

k_s is the proportionality coefficient when expressing the surface rule in terms of area, W/m²;

k is the proportionality coefficient when expressing the surface rule in terms of mass, W/kg^2/3.

The units of measurement in Formulas (1_s) and (1_m) are given in the SI; traditionally, physiologists often use the initially accepted units of measurement: P = [kcal/h], m = [g], S = [cm²]. Clearly, the values of the same proportionality coefficients will also be different when using different measurement units but can easily be converted into each other.

The surface rule follows from the law of conservation of energy, according to which, in order to maintain a constant body temperature, the amount of heat released from its surface to the environment per time unit must be exactly equal to the amount of heat released inside the body during the same time and proportional to the power of energy consumption or metabolism intensity – in terms accepted in physiology.

Max Rubner in 1883 was the first to experimentally prove the validity of the surface rule in meticulous experiments on dogs of different sizes , and apparently, the rule is deservedly associated with his name, frequently referred to as Rubner’s surface rule, although its real authors are Sarrus and Rameaux.

At the beginning of the 20th century, researchers tried to place all mammal species on a single from-mouse-to-elephant curve, with a uniform coefficient of proportionality, ignoring the physical-geometric meaning of this coefficient and of the exponent in Formula (1m). This approach inevitably led to exponents different from the ⅔ power and to confusion with pseudoscientific justifications for the results obtained , which are, in fact, "statistical artifacts" . For instance, the famous physiologist Max Kleiber approximated experimental data for various mammals, from rats (M = 0.15 kg) to oxen (M = 679 kg), by the dependence P = kM^b (both coefficients k and b varied), and in 1932 published Equation :

                               P = 73.3M0.74      (2),

where P = [kcal/day], M = [kg].

Two years after Kleiber’s publication, Brody with colleagues , having performed the same manipulations with an even larger number of animals («from mice to elephants»), practically confirmed the value of the exponent for mass in Formula (2) – 0.734. In 1961, Max Kleiber , to simplify calculations and without compromising their accuracy, proposed "rounding" the coefficients in Equation (2) as follows:

                              P = 70M0.75         (3)

The units and notations in Formula (3) are the same as in Formula (2).

Despite the fact that other studies on individual groups of mammals gave exponents that were very different from 0.75 (for instance, for animals weighing 2.4–100 grams, b equaled 0.23, and with the range expanded to 260 grams, the exponent for mass became 0.46 ), the scientific community established the opinion that the surface rule (1m) is not always satisfied, and the exponent in the metabolic equation is closer to 0.75 = ¾ than to 0.67 ≈ ⅔, and this is based only on statistical processing of experimental data and serves the convenience of calculations, so that it is possible to describe the resting metabolism of all mammals with a single equation. Clearly, with this approach, all attempts to physically substantiate Equation (3) look unconvincing , which, however, does not prevent even modern researchers from still referring to it and considering it a full-fledged «Kleiber's law» ; the European Federation of Animal Science (EFAS) in 1965 even adopted body weight to the ¾ power as a reference basis for determining metabolic rate .

In 1982, Heusner began to rehabilitate the surface rule (1m) – he showed for seven species of mammals (from a 16-gram mouse to a 922-kg bull) that the difference in energy expenditure for basal metabolism between them can be taken into account only by changing the proportionality coefficient k, and the power with mass is to always remain unchanged and equal to ⅔ ≈ 0.67, as required by the theory. Interestingly, with an increase in the weight of animals by more than 57,000 times (922/0.016), the proportionality coefficient for them changed by only three times: 1.91 in the deer mouse (Peromyscus) and 6.06 in the bull (Bos taurus) (at P = [W], M = [kg]) . Below is the contribution to the value of the proportionality coefficient k, made by geometric differences in body structure and by physical ones: variations in the normal body temperatures of different animal species, the temperatures of their environment, and the heat-protective properties of fur/wool.

Heusner’s empirical evidence could not fully convince some scientists of the inviolability and binding nature of the surface rule: "... there is no doubt that the regression line of metabolic rate depending on body size will have a slope very close to 0.75, and not to 0.67". At the same time, however, it was added: "Until recently, attempts to explain deviations from the exponent determined by the surface were unsatisfactory" and "somewhat metaphysical" , and "unfortunately, we still do not truly understand the principles that determine the slope of the regression line we have obtained for mammals of all known sizes" .

Thus, on an uncertain natural scientific foundation, Kleiber’s simple statistical approximation established itself as an immutable physical law that still exists. The explanation here is completely unrelated to scientific arguments: "Kleiber outlived both Brody and Benedict by ~20 years, and with his strong personality he greatly encouraged use of the ¾ exponent. In later publications, he sometimes uses 'mass ¾' instead of 'metabolic rate'. … His strong selling of the simple ¾ power of body mass for most things metabolic likely contributed to the relationship becoming a 'rule"' or 'law'" (, pp. 191, 192). In addition, «Kleiber's law» allowed conveniently calculating the metabolic rate of farm animals based on their weight using a slide rule, which was necessary for determining the doses of food additives or drugs. This circumstance remained relevant until electronic calculators appeared in the late 1960s .

     1. Modern techniques and models confirming the surface rule

At the end of the 20th – beginning of the 21st century, such arguments exhausted themselves, and more and more researchers, following Heusner, considered «Kleiber’s law» to be only a "statistical artifact" [4] or simply a convenient approximation of experimental data . Glazier in 2005 summarized the matter as follows: "For more than 70 years, the ¾-power rule has had a dominant influence on the field of metabolic scaling. Some useful and insightful work has been produced within this paradigm, but further progress requires that we look beyond the ¾-power rule to consider the great diversity of metabolic scaling relationships that exist in the living world" (, p. 331). However, by "metabolic diversity" most researchers understood only different values of the proportionality coefficient k and the exponent b in the allometric equation, while its expression remained unshakable:

                         P = kMb       (4)

The experimental data were approximated precisely by this formula with independent variation of the desired coefficients k and b. That is, although «Kleiber’s law» itself was overthrown, its work continues to live: different values of the exponent b of the allometric equation (4) are calculated for different "sets" of animals only using an approximation of experimental data that has no physical meaning, and additional parameters are introduced into this equation in the form of new terms or coefficients, also in a completely arbitrary manner.

Information about other physical factors influencing the metabolic rate forced researchers to think about expanding the number of parameters of the allometric equation. For instance, juvenile sow bugs (Porcellio scaber) exhibit a lower metabolic rate and, accordingly, a lower exponent of body weight in the allometric equation (b = 0.687) at 30 °C than at 20 °C: b = 0.917 . Moreover, "all other things being equal, a polar mammal living at −10 °C has a body temperature ∼2.7 °C warmer and a BMR higher by ∼40% than a tropical mammal of similar size living at 25 °C" . Thus, the need to take into account both body and ambient temperature in formulas for calculating metabolic rate became obvious. For instance, an attempt to take this into account within the framework of Kleiber’s statistical paradigm was made in : the authors simply multiplied the right side of allometric Equation (4) by body and ambient temperatures and, taking the logarithm, obtained the following approximating expression for the general case:

                  ${\textstyle ln(BMR)=a_{0}+bln(Bm)+a_{1}T_{b}+a_{2}T_{a},}$         (5)

where BMR is the basal metabolic rate, W;

B_m is the mass of the animal, g;

T_b, T_a is the body and ambient temperature, respectively, ℃;

b is the exponent in allometric Equation (4), calculated by approximating the experimental data BMR_i (B_mi,T_bi,T_ai);

a₀, a₁, a₂ are the constant coefficients calculated with the same approximation.

Approximating the experimental data BMR_i (B_mi, T_bi, T_ai) for 462 mammals, the following coefficients were obtained in Equation (5) (, Appendix S2):

            $\ln(BMR)=-7.4344+0.684\ln(B_{m})+0.105T_{b}+0.0097T_{a}$        (5*)

Without taking the ambient temperature into account (the member  in Equation (5) is missing) and with a set of experimental data for 505 mammals, the approximation results were as follows (, Appendix S2):

              $\ln(BMR)=-8.06+0.683\ln(B_{m})+0.118T_{b}$           (5**)

Finally, approximating a set of experimental data for the same 505 mammals with Equation (4), the exponent was b = 0.695, and with the experimental array expanded to 634 animal species, it was b = 0.709 .

Thus, including body and ambient temperatures even in a linear statistical (rather than physical) metabolic model removes the exponent of mass from the value of «Kleiber’s law» ¾ = 0.75, and it approaches the value in the surface rule – ⅔ ≈ 0.67. In accordance with the results obtained, the authors of summarize: "A simple allometric power rule is thus not an appropriate statistical model for describing the scaling of BMR with Bm in mammals. The variation of scaling with size, together with the variation of body temperature and metabolic rate with ambient temperature all point to an important role for heat flow in the metabolic physiology of mammals. Heat flow was central to early discussions of mammalian metabolism, but has been largely ignored in more recent models based on vascular architecture. We suggest that any complete physical model of metabolic scaling in endotherms must include explicit consideration of heat flow".

Afterward, "explicit consideration of heat flow" appeared in the experimental work by a Canadian scientist and in the theoretical model of Spanish and British researchers .

The breakthrough work by Jacopo P. Mortola refuses to determine the body surface temperature, as was done previously , and measures the temperature distribution over the body cover surface (wool, hair, etc.), using the obtained data to calculate the heat released from the animal. The results of turned out to be much more significant than the modest stated goal of the work: to find out whether the average body cover surface temperature changes systematically depending on the size of animals located at the same ambient temperature. The answer to this question was negative, but, in addition to achieving the goal, two more important problems of energy exchange of warm-blooded organisms were solved: the method for determining the heat lost by an animal based on the measured temperatures of the body cover surface T_bs and the environment T_a was formed, and indicators of heat loss scaling depending on the mass of organisms, which practically coincided with the predicted surface rule b ≈ 2/3 for all three ranges of ambient temperatures – T_a ϵ [20÷22], [22÷25] and [25 ÷27] ℃, were calculated ([20], Fig. 7). Heat loss in this case was not determined by classical calorimetric methods but calculated by the measured T_bs, T_a and body cover surface area A :

            $W_{\Sigma }=W_{r}+W_{c}=A\sigma F\epsilon (T_{bs}^{4}-T_{a}^{4})+1.4A(T_{bs}-T_{a})^{1.25}/L^{0.25}$      (6)

where W_Σ, W_r, W_c are total, radiation (first term) and convective (second term) heat losses from the animal’s body cover surface, respectively, W;

A is the body cover surface area, m²;

σ is the Stefan-Boltzmann constant, σ ≈ 5.67∙10^-8 (W/m²/K⁴);

F_ε is the coefficient that takes into account the ratio of the areas of the radiating surface and the enclosing structures surrounding it; when the ratio is less than ¼ (which was always observed, since the animal enclosures were quite spacious), one can approximately assume that Fε ≈ ε, where ε is the degree of blackness of the radiating body;

T_bs, T_a is the area-average surface temperature of the animal’s body cover and ambient temperature, K;

L is the characteristic size of the animal, corresponding to the diameter of the horizontal cylinder around it which is calculated by the formula: L = 0.0064∙M^0.3506 = [m], where M is the mass of the animal in grams.

Whenever possible, the mass of the animal was measured by weighing; otherwise, it was determined with standard bibliographic references by sex and age , resulting in calculation error and scattered results. Body surface area A = [m²] was calculated using the approximate formula A = 0.1∙M^2/3, where M is the mass of the animal in kg. This also led to the accumulation of errors, because the numerical coefficient in this formula, frequently referred to as the Meeh factor, generally speaking, differs in various animals – for cats, small dogs (up to 4 kg), and horses it equals 0.1, and for rats, mice, guinea pigs, and swine it is 0.09, while for humans it is 0.116 (, in SI units measurements).

In addition to the above, the author cites a whole series of assumptions and simplifications in the algorithm for determining heat loss, thereby warning colleagues against using the results obtained in strict quantitative models and positioning them only as rough estimates of the required parameters. However, in , the experimental results from were supplemented with data from other sources and used to quantify the metabolic rate of five animals: mouse (Mus musculus), chinchilla (Chinchilla lanigera), guinea pig (Cavia porcellus), «pig (Sus scrofa)», and camel (Camelus bactrianus). These animals were chosen from all those studied in because there were reliable experimental data on the intensity of their metabolism. For example, for the mouse, chinchilla, guinea pig, pig and camel, Heusner and McNab provide similar data on metabolic rate, differing by a few percent if they are converted into specific values (see small print), because the masses of animals in the sources may be different.

Recalculating the usual metabolic rate P1 = [W] of an animal with mass m₁ = [kg] from the specific metabolic rate p_s2 = [W/kg] of an animal of the same species with mass m₂, it is necessary to keep in mind that the specific metabolic rate, generally speaking, depends on the mass, i.e. refers only to the mass for which it was calculated. For instance, knowing the specific metabolic rate p_s2 ≈ 0.772 (W/kg) of Sus scrofa with mass m₂ = 135 (kg) (, p. 34), one cannot use the p_s2 value directly to calculate the metabolic rate of Sus scrofa of another mass m₁ = 295 (kg), given in (, p. 120): P₁ ≠ 0.772∙295 ≈ 228 (W), even if both animals are absolutely isometric, since p_s = f(m). Indeed, the specific metabolic rate is obtained by dividing both sides of Equation (1_m) by mass: p_s = P/m = km^2/3/m = k/m^1/3, that is, p_s is a function of mass even if the animals are isometric – k = const. Thus, the formula for calculating the metabolic rate P₁ of an animal with mass m₁ from the specific metabolic rate p_s2 of an isometric animal of another mass m₂ of the same species is as follows: k = p_s2m₂^1/3 = p_s1m₁^1/3, from which p_s1 = p_s2(m_2/m₁)^1/3 or P₁ = p_s1m₁ = p_s2m₂^1/3m₁^2/3. That is, the specific intensity of Sus scrofa with mass m₁ = 295 (kg) from the above example will differ significantly from the same parameter of Sus scrofa with mass m₂ = 135 (kg): p_s1 = 0.772(135/295)^1/3 ≈ 0.595 (W/kg), and p_s2 ≈ 0.772 (W/kg), respectively, P₁ = 0.595∙295 ≈ 0.772∙135^1/3∙295^2/3 ≈ 175.5 (W) ≠ 228 (W). All the specific metabolic values of the animals listed below were recalculated in accordance with the described algorithm, except for the mouse (Mus domesticus) because the mass values of mice in and are close: 26 and 25 grams.

The metabolic rates of chinchilla, guinea pig, pig and camel, calculated from thermograms of surface integument from [20], differed from those obtained experimentally in by only 7-13% . This confirms the possibility of creating a new method for quantitative rapid assessment of the metabolic rate of warm-blooded living beings, based on optical measurement of the temperature field of the surface of their body, which can compete with existing methods of direct and indirect calorimetry.

However, the calculated values of the mouse metabolic rate differed from the experimental ones by approximately 40%, even after changing the calculation formula (6) to a more adequate one for this animal species . This circumstance requires a more thorough approach both to the creation of calculation methods and the measurement of the parameters included in them, and to the direct experimental determination of the metabolic rate of small animals.

Despite a number of unresolved methodological issues listed above, Jacopo P. Mortola rehabilitated the surface rule for mammals through calculations and experiments and laid the foundation for quantitative thermography, which has enormous development potential.

                  2. Modified surface rule

“Kleiber’s law” is just a convenient statistical approximation of experimental data, with undeniable advantages – it is easy to record and universal to apply yet devoid of any physical meaning.

In shows that the traditional surface rule (1_m) is a special case of the law of conservation of energy, applied to a specific living organism in stable temperature and humidity conditions and in a stationary heat and energy state, which ensures the constancy of the proportionality coefficient between metabolic rate and mass to the ⅔ power. Changes in the physical and geometric properties of a living creature, the conditions of its body heat release or heat removal inevitably lead to variations in the proportionality coefficient in the traditional surface rule, which remains constant only for animals of the same species, isometric in body shape and performing the same activity, or at rest, while the posture also matters.

Applying the law of conservation of energy in general to metabolic processes allowed formulating a modified surface rule (7) – it is identical to the traditional (1_m) one, yet the proportionality coefficient kvar in it is not a constant but determined by the physical and geometric characteristics of living beings, the conditions of heat release and removal in their organisms, including: body shape coefficient, its average density, body efficiency for a specific type of activity or at rest, average values of the integral degree of blackness and convective heat transfer coefficient from the body cover surface and open areas, values of heat transfer by evaporation from body surface and during respiration, normal body temperature (if the animal is warm-blooded), average temperature of the body cover surface and open areas, thermophysical properties of supporting surfaces and their temperature, ambient temperature and humidity, in the general case – dimensions of the room in which the animal is located, and the physical properties of its enclosing structures.

                              P = kvarm2/3      (7)

In a quantitative expression of the proportionality coefficient kvar in the modified surface rule (7) was obtained through the listed parameters, and an example of its calculation for a person standing at rest based on thermal imaging was given, while the calculated value fell approximately in the middle of the known experimental data range.

The proportionality coefficient kvar in the modified surface rule (7) can be expressed both using the densities of heat fluxes removed by evaporation and during respiration (8), and by expressing them in terms of shares of the total heat removal (9) :

          $k_{var}=k_{m}(q_{be}+k_{sl}(q_{le}+q_{lb})+k_{c}q_{c}+k_{r}q_{r}+k_{tc}q_{tc})/(1-f_{c})$       (8)

          $k_{var}=k_{m}(k_{c}q_{c}+k_{r}q_{r}+k_{tc}q_{tc})/(1-f_{c})/(1-k_{e}-k_{b})$       (9)

where q_be is the average heat flow density (thermal power) removed from a unit of body surface area by evaporation, including imperceptible, W/m²;

k_sl = S_l/S_b is the dimensionless coefficient taking into account the proportion of the respiratory tract surface area S_l in the body surface area S_b;

q_le is the average heat flow density (thermal power) removed from a unit of respiratory tract area by evaporation, W/m²;

q_lb is the average heat flow density (thermal power) removed from a unit of respiratory tract area by heating of inhaled/exhaled air, W/m²;

k_c = S_c/S_b is the dimensionless coefficient taking into account the share of the surface area S_c, from which heat is removed by convection, in the body surface area S_b;

q_c is the average heat flow density (thermal power) removed by convection from a unit of surface area S_c, W/m²;

k_r = S_r/S_b is the dimensionless coefficient taking into account the share of the surface area S_r, from which heat is removed by radiation, in the body surface area S_b;

q_r is the average heat flow density (thermal power) removed by radiation from a unit of surface area S_r, W/m²;

k_tc = S_tc/S_b is the dimensionless coefficient taking into account the share of the surface area S_tc, from which heat is removed by thermal conductivity, in the body surface area Sb;

q_tc is the average heat flow density (thermal power) removed by thermal conductivity from a unit of surface area S_tc, W/m²;

k_е=Q_e/Q_∑ – dimensionless coefficient taking into account the share of heat removed by evaporation from the surface of the body (Q_e) from the total heat removal from the animal’s body (Q_∑);

k_b=Q_b/Q_∑ – dimensionless coefficient that takes into account the share of heat removed during respiration from the total heat removal from the animal’s body (Q_∑).

When expressing the mass coefficient k_m through the shape coefficient k_v and density ρ:

       $k_{var}=k_{v}(q_{be}+k_{sl}(q_{le}+q_{lb})+k_{c}q_{c}+k_{r}q_{r}+k_{tc}q_{tc})/(1-f_{s})/\rho ^{2/3}$       (8*)

       $k_{var}=k_{v}(k_{c}q_{c}+k_{r}q_{r}+k_{tc}q_{tc})/(1-f_{c})/(1-k_{e}-k_{b})/\rho ^{2/3}$             (9*)

where k_v is the shape coefficient which is the coefficient of proportionality between the body surface area and its volume to the ⅔ power, a dimensionless quantity:  k_v=S_b/V^2/3=S_b/(m/ρ)^2/3= ρ^2/3S_b/m^2/3= ρ^2/3k_m, from where k_m= k_v/ρ^2/3;

V is the body volume, m³;

m is the body mass, kg;

ρ is the average body density, kg/m³;

k_m = k_v/ρ^2/3 is the mass coefficient, often called the Meeh factor, which is a coefficient of proportionality between the surface area of a mammal’s body and its mass to the ⅔ power, m²/kg^2/3.

The shape coefficient k_v=Sb/V^2/3 for simple bodies, although being constant, has different values, for example, k_v ≈ 4.84 for a sphere and k_v = 6 for a cube. For complex animal bodies, its values not only differ among species but, as will be shown in , can vary by 10–15% among specimens of different physiques of the same species.

Substituting into Formulas (9) and (9*) the expressions of convective qc, radiative qr and conductive qtc heat flows known from thermophysics, we obtain expressions for the metabolic rate in accordance with the modified surface rule (7):

$P=k_{m}(k_{c}\alpha (T_{bs}-T_{a})+k_{r}\epsilon \sigma (T_{bs}^{4}-T_{a}^{4})+k_{tc}\lambda _{eq}(T_{br}-T_{r})/\delta _{\Sigma })m^{2/3}/(1-f_{c})/(1-k_{e}-k_{b})$  (7*)

$P=k_{v}(k_{c}\alpha (T_{bs}-T_{a})+k_{r}\epsilon \sigma (T_{bs}^{4}-T_{a}^{4})+k_{tc}\lambda _{eq}(T_{br}-T_{r})/\delta _{\Sigma })m^{2/3}/(1-f_{c})/(1-k_{e}-k_{b})/\rho ^{2/3}$    (7**)

where α is the average convective heat transfer coefficient, W/m²/K;

T_bs is the average temperature of the body cover and open areas surface, К;

T_a is the ambient temperature, K;

ε is the average integral degree of blackness of the body cover and open areas surfaces, dimensionless value; if the room in which measurements are made is small-sized, namely, when the ratio of the radiating surface area to the enclosing structures area is more than ¼, the degree of emissivity will be calculated using more complex formulas ;

σ = 5.67∙10^-8 is Stefan-Boltzmann’s constant, W/m²/K⁴;

λ_eq is the equivalent thermal conductivity coefficient of the supporting structure and surface coverings on which the subject sits or lies, for example, a chair or couch when measuring RMR or BMR, respectively, W/m/K; the formula for calculating the equivalent thermal conductivity of a multilayer structure is not given here, yet it can be found in thermal reference books or in ;

δ_Σ is the total thickness of the supporting structure, including surface covers, m;

T_br is the average temperature of the body over the supporting structure, K;

T_r is the surface temperature of the back side of the supporting structure, K.

If the subject lies, sits, or stands on the ground or on the massive floor, the heat from the body surface in contact with the specified mass is dissipated in a semi-infinite space. For this case, conductive heat removal into a semi-infinite array will be calculated using a more complex algorithm and formulas which must be substituted into the numerators of Equations (7*–9*) instead of the conductive component of heat transfer through the support structure q_tc. The final formula for calculating the conductive heat flow density q_tcf from the soles of a person standing on a massive base is given in :

             $q_{tcf}=\pi \lambda _{f}\lambda _{g}(T_{bf}-T_{ag})/(b\lambda _{f}\ln(4a/b)+\pi \delta _{f}\lambda _{g})$        (10)

where a,b are the length and width of the shoe sole, assuming its rectangular shape, m;

δ_f is the thickness of the shoe sole including sock and insole, m;

λ_f is the equivalent thermal conductivity coefficient of the shoe sole including sock and insole, taking into account the thermal resistance between them, W/m/K;

λ_g is the thermal conductivity coefficient of the base (floor) array, W/m/K;

T_bf is the surface temperature of the foot sole, ℃;

T_ag is the temperature in the base array, ℃; for the floors of multi-story buildings, T_ag can be considered equal to the indoor air temperature; for the ground or floors on the ground floors of buildings, the T_ag value is to be clarified;

q_tcf is the density of conductive heat flow from rectangular soles of Size axb of a person standing on a massive base, W/m².

Two important details are noteworthy. First, in case of heat removal from a body part surface of a mammal into the floor or soil, a stationary thermal regime (in which the corresponding heat flow and all the determining parameters will be constant values) will occur only after some time, which must be taken into account when planning the experiment. Second, as seen from Formula (10), the value of the heat flow density depends on the shoe size, for instance, from the sole of a child q_tcf will be higher than from that of an adult; this does not allow using the value once calculated by Formula (10) in all cases; for each specific area of contact with the base array, an individual calculation is necessary, and this applies not only to shoes but also to cases of animals lying on the floor during measurement.

Thus, with a sufficiently long monotonous activity or inactivity of a living creature in constant ambient temperature, the right-hand sides of Equations (8–9*) will be constants, because all the parameters included in them are constants. Accordingly, the metabolic rate P, both measured by indirect calorimetry and calculated by Formulas (7*–7**), will also be a constant value. For the basal metabolic rate (BMR) to be a stable, reproducible value, strict conditions for its determination are to be followed, one of which is the recommended period before measurements, when a stationary thermal regime is established between the body and the environment.

The coefficients k_var, calculated using any of Formulas (8-9*), theoretically are to have the same values; besides, they are to coincide with the experimentally determined proportionality coefficient k in the traditional surface rule (1_m) under the same conditions of measurements. The task now is to determine to what extent the present theoretical research corresponds to reality.

One option here is given in based on data from thermal imaging images of a young man at rest in a spacious room with T_a = 20 ℃, standing on a tiled floor in Size 43 sneakers with porous rubber soles 15 mm thick, dressed in a cotton T-shirt, a long-sleeved sweatshirt, and loosely fit trousers. In this case, the average temperature over the clothing surface was 25 ℃, the temperature of open body areas was 33 ℃, respectively, averaged over the entire radiating surface T_bs ≈ 25.6 ℃; the temperature of the feet soles was 34 ℃; the degree of blackness of cotton fabric in accordance with reference data was equal to ε = 0.77; the average value of the mass coefficient (Meeh factor) for a person was taken from (, p. 90, table 7.3) – k_m = 0.116 (m²/kg^2/3) (measurement units converted to SI); the value of the efficiency of the human body at rest was f_c = 0.15 ; the average convective heat transfer coefficient over the clothing surface was calculated in specifically for this case: α = 2.74 (W/m²/K); the coefficient of increase in heat transfer surface due to clothing was k_cl = 1.078; the coefficient of radiating surface area as a share of the total cover surface area k_r ≈ 0.977; the coefficient taking into account the share of the conductive heat removal area from the body through two shoe soles to the floor was k_tcf ≈ 0.03; the heat removed from the skin surface during breathing and evaporation as a share of total heat transfer was (k_e+k_b) ≈ 0.22; the sole dimensions were b = 10 centimeters wide and a = 30 centimeters long; the thermal conductivity of the porous rubber sole was λ_f = 0.06 (W/m/K); the thermal conductivity of the tile floor array was λ_g = 1.3 (W/m/K) .

Substituting into (9*) specific expressions for the heat flow densities of a person standing at rest yields a general formula for calculating the proportionality coefficient in the modified surface rule (7):

     $k_{vars}=k_{m}(k_{cl}k_{r}(\alpha (T_{bs}-T_{a})+\epsilon \sigma (T_{bs}^{4}-T_{a}^{4}))+k_{tcf}q_{tcf})/(1-f_{c})/(1-(k_{e}+k_{b}))$       (11)

Substituting the above values of the input parameters into (10) gives the value of the heat flow density from the soles to the floor:

   $q_{tcf}=3.14\cdot 0.06\cdot 1.3\cdot (34-20)/(0.1\cdot 0.06\cdot \ln(4\cdot 0.3/0.1)+3.14\cdot 0.015\cdot 1.3)\thickapprox 45(W/m^{2})$

Substituting the calculated value of q_tcf and the values of other parameters into (11) yields the following for a standing person at rest:

$k_{vars}=0.116(1.078\cdot 0.977(2.74(25.6-20)+0.77\cdot 5.67\cdot 10^{-8}(298.6^{4}-293^{4}))+0.03\cdot 45)/$

$/(1-0.15)/(1-0.22)\thickapprox 7.73(W/kg^{2/3})$

Taking into account the difference in energy consumption of a standing and lying person at rest by 1.7 times ([12], p. 335, Table 6.31), for the latter k_varl = k_vars/1.7 ≈ 4.55 (W/kg^2/3).

Comparing the calculated proportionality coefficient with its experimental values obtained from the traditional surface rule (1_m) according to various sources is illustrative. According to measurements taken at the beginning of the 20th century for a person weighing 70 kg, k ≈ 4.87 (W/kg^2/3) (, p. 40); various classical textbooks and even modern compendiums (, p. S498) give the average value of specific BMR ≈ 1 (kcal/kg/hour) ≈ 1.163 (W/kg). Multiplying it by the average weight of 70 kg gives P_c = 70∙1.163 ≈ 81.4 (W), from which k ≈ 81.4/70^2/3 ≈ 4.79 (W/kg^2/3); in , for a man of the end of the 20th – beginning of the 21st century, the average BMR ≈ 0.892 (kcal/kg/hour) ≈ 1.0374 (W/kg) is indicated; then for an individual weighing 70 kg, P_c ≈ 1.0374∙70 ≈ 72.38 (W), whence k ≈ 72.38/70^2/3 ≈ 4.26 (W/kg^2/3).

Thus, the value of the proportionality coefficient k_varl ≈ 4.55 (W/kg^2/3), calculated in accordance with the modified surface rule, falls approximately in the middle of the range of empirical coefficients k ϵ [4.26÷4.87] (W/kg^2/3)given in the literature and experimentally measured within the framework of the traditional surface rule (1_m), which indicates the adequacy of the modified surface rule and the possibility of using it both to quantitatively determine the metabolic rate of living organisms and to theoretically explain variations in the proportionality coefficients and in their BMR levels.

In particular, the modified surface rule can help resolve well-known controversies about the reasons specific BMR varies in people depending on body mass index, gender, and even age. For instance, a decrease in BMR with increasing weight is often explained by a relative increase in metabolically inert adipose tissue and a decrease in metabolically active muscle mass , which is argued against by other authors disputing the “imaginary” inertia of adipose tissue and, conversely, proving the passivity of muscle mass at rest: the mass of skeletal muscles increases from birth to adulthood by about 40 times while BMR, on the contrary, decreases by almost 2.5 times, which is precisely a consequence of the “metabolic passivity” of muscle tissue at rest, as other metabolically active organs also increase in size with age, maintaining a high resting metabolism. Lower values of specific metabolic rate in average women compared to average men are also unlikely to be a consequence of gender differences only, because the metabolic rate in women with normal weight was significantly higher (by 17%) than RMR in obese men – 0.926 and 0.791 kcal/kg/hour, according to . Even the aging factor is not decisive in reducing the metabolic rate, because it was not found among individuals whose weight remained virtually unchanged over the 17-year observation period .

The above experimental phenomena can be explained by purely geometric reasons affecting heat transfer from the body to the environment; these reasons are determined by the shape coefficient k_v in the modified surface rule – Formulas (7**, 8*, 9*). The fact is that the said coefficient k_v, the ratio of the body surface area to its volume to in the ⅔ power, is a constant only for simple geometric shapes such as a sphere or a cube, while for cylinders, its values vary significantly depending on the ratio of the height-to-base radius n = h/r: S_z = k_vz(n)V_z^2/3. In (, p. 28) the dependence of the cylinder shape coefficient on the ratio of its height to base radius is derived:

                              $k_{vz}(n)=2\cdot {\sqrt[{3}]{\pi }}(n+1)/n^{2/3}$         (12)

Figure 1. Dependence of the cylinder shape coefficient k_vz on the ratio of its height to base radius n = h/r (Formula (12)).

Figure 2. Rough model of a human figure made of cylinders of various diameters and heights.

Since the bodies of most animals can be schematically represented as cylinders of various heights and base radii, Figure 1 clarifies the reasons and limits for changes in their shape coefficients and, consequently, BMR in accordance with the modified surface rule (7**) become clear. For instance, based on a rough “cylindrical” model of the human body (Figure 2) in , it was calculated that with the same mass m ≈ 83 (kg) but different physique (height 155.6 and 193.5 cm), the shape coefficient for a tall and thin person would be greater than for a short and fat one by approximately 12%, which would lead, taking into account density (see (7**, 8*, 9*)), to the 8.6% difference in their metabolic levels. These values are sufficient to explain most of the BMR variations in the phenomena described above, and, accordingly, remove the far-fetched reasons for their manifestation. Notably, Formula (12) is not directly applicable to calculating the overall shape coefficient of a cylindrical model of a person, since all the cylinders that make up the body are either completely devoid of bases (arms and legs in Figure 1) or the bases are only partially present (the torso) because at the bases, they are adjacent to other parts of the body – hands, feet, and neck, respectively. For these cases, based on Formula (12) in , special equations were derived, and the dependence of the general shape coefficient of a cylindrical human model on its geometric parameters was formalized.

Generally speaking, the cylindrical model of a person was implemented only to clearly show the mechanism of geometric causes influencing metabolic rate. The exact values of the shape coefficients of living organisms are to be determined immediately as the ratio of the body surface area to the volume to the ⅔ power, which can be measured by modern methods, for example, using a 3D scanner. With the same 3D scanner and scales, the mass coefficient k_m (the Meeh factor), included in the invariants of the modified surface rule Formulas (7*, 8, 9), can be immediately determined, which will also allow excluding density from calculations.

                                     Conclusions:

1.   “Kleiber’s law” is just a convenient statistical approximation of experimental data, with undeniable advantages – it is easy to record and universal to apply yet devoid of any physical meaning. The traditional surface rule, on the contrary, can be considered a special case of the law of conservation of energy manifested in living organisms; however, as for any special case, its application is limited by strict conditions, in particular, the isometry of animal bodies and stable ambient temperature. If these conditions are not met, the value of the proportionality coefficient in the traditional surface rule (1_m) changes, which is inconvenient for describing the metabolic rate both in different animal species and its variations within one species.

2.   Applying the law of conservation of energy in general form to connect the power of energy consumption of living beings in a state of monotonous activity or at rest with their mass to the ⅔ power allowed deriving the modified surface rule (invariant Equations (7, 7*, 7**)) , the proportionality coefficient in which, calculated, for example, by Formula (11) for a standing individual, is not a constant value but depends on the ambient temperature and humidity and the physical and geometric parameters of a particular organism, including: body shape features through the shape coefficient k_v, equal to the ratio of the body surface area to its volume to the ⅔ power; density ρ; gender through body efficiency f and features of body geometry; age through body surface temperature; thermophysical properties of body cover – fur in animals and clothing in humans; etc.

3.   The traditional surface rule does not help explain the significant differences in proportionality coefficients k in different animal species “from 1.91 for the deer mouse (Peromyscus) to 6.06 for the bull (Bos) (units – Watts and kilograms)” (, p. 68); large differences in energy consumption of warm-blooded animals of approximately the same mass but different species, for example, Chinchilla lanigera (m_c = 485 g) and Sciurus carolinensis (m_c = 440 g): P_c = 1.28 (W) and P_s = 2.08 (W) (, pp. 40, 44); lower specific power consumption of lighter females compared to heavier males, etc. Applying the modified surface rule allows not only explaining these differences but also calculating the corresponding energy parameters.

4.   New express methods for determining basal metabolic rate (BMR) or resting metabolic rate (RMR) using the modified surface rule can be implemented with the following equipment: scales of the corresponding measurement interval; a 3D scanner capable of determining the surface area and volume of bodies of arbitrary shapes; a thermal imager with the ability to average temperature within specified zones and the entire body as a whole, preferably with an error not exceeding ±0.1 °C, because the results of determining the required characteristics are quite sensitive to variations in surface temperature. In addition, all participants in the experiments – if they are people – are to be dressed in the same way and in equally fitted clothing, for example, thermal underwear, which will eliminate the amount of error associated with different clothing. In fact, different clothes have the same effect on the measurement results both when using direct and indirect calorimetry; with thermal imaging, the reason for the difference will be clearly visible immediately, even before the final results are obtained.

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