# Entropy Stress and Scaling of Vital Organs over Life Span Based on Allometric Laws

^{*}

## Abstract

**:**

_{k}). Using this hypothesis, Kleiber’s law on metabolic rate of BS $({\dot{q}}_{\text{body}})$ for animals of different sizes was validated. In this work, a similar procedure is adopted in estimating total entropy generation rate of whole human body (${\dot{\mathsf{\sigma}}}_{\text{body}}$ , W/K) as a sum of product of specific entropy generation rate for each organ, ${\dot{\mathsf{\sigma}}}_{\mathrm{k},\mathrm{m}}$ (W/{K kg of organ k·}) and the organ mass at any given age (t). Further integrating over life span for each organ (t

_{life}), the lifetime specific entropy generated by organ k, ${\mathsf{\sigma}}_{\mathrm{k},\mathrm{m},\text{life}}$ (J of organ k/ {K kg organ k}) is calculated. Then lifetime entropy generation of unit body mass, ${\mathsf{\sigma}}_{\text{body},\mathrm{M},\text{life}}$ (J/{K kg body mass·}) is calculated as a sum of the corresponding values contributed by all vital organs to unit body mass and verified with previously published literature. The higher the ${\mathsf{\sigma}}_{\mathrm{k},\mathrm{m},\text{life}}$ , the higher the entropy stress level (which is a measure of energy released by unit organ mass of k as heat) and the irreversibility within the organ, resulting in faster degradation of organ and the consequent health problems for the whole BS. In order to estimate ${\dot{\mathsf{\sigma}}}_{\mathrm{k}}$ (W/K of organ k), data on energy release rate $(\dot{q})$ is needed over lifetime for each organ. While the Adequate Macronutrients Distribution Range (AMDR)/Adequate Intake (AI) publication can be used in estimating the energy intake of whole body vs. age for the human body, the energy expenditure data is not available at organ level. Hence the ${\mathsf{\sigma}}_{\mathrm{k},\mathrm{m},\text{life}}$ was computed using existing allometric laws developed for the metabolism of the organs, the relation between the m

_{k}of organ and body mass m

_{B}, and the body mass growth data m

_{B}(t) over the lifetime. Based on the values of ${\mathsf{\sigma}}_{\mathrm{k},\mathrm{m},\text{life}}$ the organs were ranked from highest to lowest entropy generation and the heart is found to be the most entropy-stressed organ. The entropy stress levels of the other organs are then normalized to the entropy stress level (NES

_{H}) of the heart. The NES

_{H}values for organs are as follows: Heart: 1.0, Kidney: 0.92, Brain: 0.46, Liver: 0.41, Rest of BS: 0.027. If normalized to rest of body (R), NES

_{R}, heart: 37, Kidney: 34, Brain: 17, Liver: 15, Rest of BS: 1.0; so heart will fail first followed by kidney and other organs in order. Supporting data is provided.

## 1. Introduction and Literature Review

**Figure 1.**Leading Causes of Natural Death based on 2001 data (adopted from [2] with permission).

_{2}, with SATP being 25 C and 1 atm, or 20.2 kJ/CSA L of O

_{2}; see Acronyms). Then, the oxygen used over life span must also be constant at about 45190 SATP-L of O

_{2}per kg body for all BS. Recently, the Rubner’s constant was revised by Spearman to be 590-1100 MJ/kg (excluding man) and 3025 MJ/kg (including man) over species spanning a body mass change of almost 50000 times [10]. Based on the belief that ROL is valid for all BS, the 25 year study on 30% calorie restriction (CR) in rhesus monkeys reported “improved survival” but does not increase the life span [11]. The study seems to suggest that the nutrient composition (e.g., higher % of sucrose, C

_{12}H

_{22}O

_{1}) and genetics affect longevity more than CR. The leading causes of death are heart disease and were shown to be same in the normally fed and CR monkeys.

_{2}required for combustion is released as ROS, a by-product of energy metabolism in the mitochondria by phosphorylation processes where electrons are transported. The ROS generated during metabolism may attack cells, result in cell copy error, and cause damage to proteins, lipids and DNA and probably leads to cell death [13]. The ROL theory and ROS model, however, are not necessarily equivalent since metabolic rate and ROS production rate are not always positively correlated; e.g. birds live longer with high metabolic rates [14].

_{B}) where E is the activation energy and $\overline{R}$ the universal gas constant, and hence ROS is body-temperature (T

_{B}) sensitive [12]. A fraction of local energy released is converted into thermal energy resulting in temperature rise and entropy generated while the remainder is used in production of ATP. Note that heat production is necessary to maintain the warm body temperature. The higher metabolic efficiency implies that lesser nutrients are metabolized for the delivery of same work, hence less ROS.

_{life}, σ

_{M,}body, life can be estimated.

## 2. Rationale and Objective

## 3. Analysis

- Lifetime energy expenditure (LSEE) and entropy generation (LSEG
_{k}) of each organ k, where k = B, H, K, L and the remainder R - Specific lifetime energy expenditure, LSEE
_{M}, (kJ/kg body mass) and specific entropy generation, LSEG_{M}, (kJ/{kg body mass K}) whole body - The % contribution by the each organ k to the overall energy consumption and % contribution by each organ to the total entropy generation of the whole body.

#### 3.1. Life Span Energy Expenditure of Body in Terms of Energy Expenditure of Vital Organs

_{k}(W/{K kg organ k}) at age (t) and ${\dot{\mathrm{q}}}_{\mathrm{M}}(\mathrm{t})$ is the specific metabolic rate of the whole body (W/kg body), then the life span energy expenditure LSEE

_{M}(J/kg body) is given as a sum of LSEE of each organ using the following expression:

_{k}as a function of age (t) are needed for each organ over the life span, in order to estimate LSEE

_{m}. Previous literature measured the difference between oxygen concentration at the inlet and exit of organs, estimated the energy release rate using heating value based on unit mass of oxygen (HHV

_{O2}) consumed, obtained ${\dot{q}}_{\mathrm{k}}$ the energy release rate) , and related the rate to mass m

_{k}through allometric laws. These laws have been used by Wang et al. to estimate metabolic rate of whole body. Following Wang et al. [24] ${\dot{\mathrm{q}}}_{\mathrm{k},\mathrm{m}}(\mathrm{t})$ can be expressed in terms of body mass using the allometric constants in Table 1 below:

_{k}and f

_{k}are allometric constants for organ k. Calder’s allometric laws for the organ mass to the body mass [25] is given as:

_{k}and d

_{k}being constants for organ k. Equation (3) can also be expressed in terms of organ mass k (Appendix A.1). Expressing m

_{B}as function of m

_{k}, replacing into Equation (3) and multiplying by organ mass m

_{k}, the energy release rate contributed by organ k is given as,

Organ, k | c_{k} | d_{k} | e_{k} | f_{k}^{+} | W ^{Contrib*} | |
---|---|---|---|---|---|---|

Brain | 0.01100 | 0.76 | 21.620 | −0.14 | 11.93 | 14.0 |

Heart | 0.00630 | 0.98 | 43.113 | −0.12 | 25.89 | 3.31 |

Kidneys | 0.00893 | 0.85 | 33.414 | −0.08 | 23.79 | 6.16 |

Liver | 0.03300 | 0.87 | 33.113 | −0.27 | 10.52 | 9.20 |

Rest, without BHKL | 0.93900 | 1.01 | 1.446 | −0.17 | 0.70 | 49.2 |

^{+}Recent literature suggests that oxygen accessibility by cells in the organ affects the values of f

_{k}. According to the proposed group combustion theory, smaller organs must have f

_{k}≈ 0 (isometric law) while the larger organs must have negative values with a limit of f

_{k}≈ −0.333 [43]. Also the % vital organ masses and SMR decrease with increasing body mass [45].

_{k}, d

_{k}, e

_{k}, and f

_{k}. Wang et al [25] used Calder’s data [26] to predict specific basal metabolic rate (SBMR

_{k}) of organ mass for a 70 kg person including the effect of fat free mass and showed that the computed metabolic rate of whole body with {Σ

_{k}SBMR

_{k}*m

_{k}} yields values similar to those of Kleiber’s allometric model. Elia et al. [27] presented the specific basal metabolic rate of vital organ k. Further Gallagher et al. [28] measured the mass of the vital organs (m

_{k}) and using Elia’s data, computed the summation {Σ

_{k}SBMR

_{k}* m

_{k}} and showed that the computed resting energy expenditure (REE) is same as measured REE. Finally, using Equations (3) and (4) in Equation (2):

_{k}, d

_{k}, e

_{k}, and f

_{k}are presented in Table 1. Since the heating values per unit stoichiometric oxygen are roughly constant (HHV

_{O2}) for any nutrient oxidized (see Table 2), the conventional method is to measure the blood flow rates to organs and the difference in oxygen concentration between the arterio-venous blood, and then multiply the oxygen consumption rate by HHV

_{O2}in order to obtain the metabolic rate.

Nutrients | Formulae | M, kg/kmol | St.O_{2}, kg/kg | RQ | HHV kJ/kg | HHV_{O2} kJ/kg O_{2} | ΔH_{C}° at 37°C MJ/kmol | h_{f} MJ/kmol | s°_{298} kJ/kmol K | ΔG_{c}° MJ/kmol | $\frac{\mathrm{\Delta}{\mathit{G}}_{\mathit{c}}^{\mathbf{0}}}{\mathrm{\Delta}{\mathit{H}}_{\mathit{c}}^{\mathbf{0}}}$ | ΔG_{M}° MJ/kmol | ΔS_{c}° kJ/kmol K | Metabol. eff. % |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Glucose | C_{6}H_{12}O_{6} | 180 | 1.066 | 1.0 | 15630 | 14665 | −2815 | −1260 | 212.0 | −2895 | 1.03 | −1790 | 259.5 | 38.2 |

Fat | C_{16}H_{32}O_{2} | 256 | 2.869 | 0.7 | 39125 | 13635 | −10035 | −835 | 452.4 | −9840 | 0.98 | −3125 | −630.1 | 32.2 |

Protein or Albumin | C_{72}H_{112}N_{2}O_{22}S (*1) | 1390 | 2.07 | 0.8 | 28893 (*3) | 13944 | −4480 | |||||||

Protein | C_{4.57}H_{9.03}N_{1.27} O_{2.25}S_{0.046} (*2) | 119 | 1.54 | 0.8 | 22790 | 14705 | −2720 | −384 | −2665 | 0.98 | −163.8 | 10.4 | ||

Protein [40] | C_{4.98} H_{9.8} N_{1.4}O_{2.5} | 117.3 | 1.413 | 0.83 | 19000 [41] | 13475 |

_{1.972}N

_{0.277}O

_{0.492}S

_{0.010}, with a molecular weight of 119.39 kg/kmol and a heating value of -2.721x10

^{6}kJ/kmol (5.5 kcal/g). kmol based on empirical molecular weight; (*3) Heating value from Boie equation [Chapter 4, ref.12].

#### 3.2. Life Span Entropy Generation (LSEG) in Terms of Entropy Generation of Vital Organs

_{k}(t) is known in terms of body mass (Equation 4). Using Equation 10 in Equation 1, the cumulative entropy generation of whole BS at age t can be computed.

#### 3.3. Availability Analysis

_{c})

_{n}is the change in Gibbs function of nutrient “n” during metabolism. Typically, macronutrient “n” contains chemical energy at low entropy level (high G

_{R}) (see arteries, inlet to organ k, in Figure 2) while the products (CO

_{2}and H

_{2}O) have high entropy (low G

_{P}) leaving through veins due to the release of a fraction of energy as heat $({\dot{Q}}_{k})$ and the remaining fraction as ATP. Due to release of heat, blood remains slightly hotter compared to normal body temperature. Hence typically (ΔG

_{c})

_{n}< 0. Section 3.3.3 will show that the life span entropy generation (LSEG) depends upon the metabolic efficiency and the life span energy expenditure (LSEE).

**Figure 2.**Distribution of Nutrients (fuel) to Cells of an organ k; Simplified Schematic (adopted from [31] and modified).

#### 3.3.1. Assumptions

- (a)
- The macronutrients or main nutrient groups CH, F and P are modeled using glucose, palmitic acid, and average amino acids composition respectively.
- (b)
- The η
_{n}, where n=CH, F and P are different for every nutrient but remain constant over time/age. - (c)
- The ATP, which is equivalent to work in thermodynamics, does not create irreversibility.
- (d)
- Energy requirements are related to body mass m
_{B}(t); statistical data on normal growth of body m_{B}(t) with age from the Summary Report 2007, US National Center for Environmental Assessment [33]. - (e)
- Life span of whole species could be defined since the birth and death are well defined; however it is difficult to define the life span of organs. Thus, only extent of degradation of organs is presented in terms of entropy generated during average life span.
- (f)
- The Gibbs free energy change of nutrients during metabolism, ΔG
_{c,n}is a function of temperature, pressure and mole fraction is approximately same as ΔG_{c,n}°, i.e., ΔG_{c,n}≈ ΔG_{c,n}° which implies that nutrients, oxidants, CO_{2}and H_{2}O exists as pure species in the reactants and products. - (g)
- In thermodynamic literature, the ratio of $|\mathrm{\Delta}{G}_{c,n}^{0}(T)|/|\mathrm{\Delta}{H}_{c,n}(T)|$ varies from 1.0 to 1.02 for most hydrocarbon fuels of general formulae C
_{x}H_{y}when lower heat value is used for the enthalpy of combustion of nutrient n. This is consistent with the findings of Brzustowski and Brena, who showed that the ratio of fuel availability to lower heat value ranges from 1.04 to 1.07 [32]. When higher heating values are used for the same fuels, the ratio varies from 0.9 to 0.96 for HC and from 0.98 to 1.03 for CH, and F. Hence,$$\left(\frac{|\mathrm{\Delta}{G}_{c,n}\xb0|}{HH{V}_{n}}\right)\approx 1$$ - (h)
- While general derivations assume that metabolic efficiency depends upon organ k, age (t) and type of nutrients (j) being oxidized, the quantitative results assume a weighted metabolic efficiency independent of organ k and age (t).

#### 3.3.2. Irreversibility of Organs and Heat Transfer from Organs

_{2}carriers enter at inlet (Figure 2), undergo metabolism, a part of the nutrients get oxidized and reduced O

_{2}and increased CO

_{2}along with unreacted nutrients exit the organ. Extending the availability analysis of Silva and Annamalai to each organ k, the general availability balance equation is written as [29,30]:

_{0}s and T

_{0}, ambient temperature. It is customary in USA to define ψ’ = ψ − ψ

_{0}as stream/flow availability where ψ

_{0}= (h

_{0}− T

_{0}s

_{0}) while in the European Union (EU) it is defined as stream exergy. The ψ’ = (h − T

_{0}s )-( h

_{0}− T

_{0}s

_{0}) represents the maximum work that could be delivered by the fluid at given state (T,P) when it is expanded to a dead state (T

_{0}, P

_{0}). The term ψ is called stream availability in EU but there is no parallel definition for ψ in USA. Under steady state conditions one may use either ψ’ or ψ in Equation (13) since the term ψ

_{0}disappears when calculating the difference between inlet and exit availabilities/exergies. For organs within biological systems, T

_{0}= T

_{B}, ψ =h − T

_{B}s = g

_{c,n}<0 for exergetic reactions. Using (14) and (15), quasi-steady state and no thermal energy reservoir and simplifying Equation (13) under steady state:

_{O2,j}is the stoichiometric oxygen mass per unit mass of nutrient j (e.g. for CH, ν

_{O2,CH}= 1.066 kg per kg of CH), ${\dot{\mathrm{m}}}_{\mathrm{O}2,\mathrm{k}}$ the total O2 consumed within organ k, ${\text{HHV}}_{\mathrm{O}2}$ is the heat value per unit stoichiometric oxygen and is approximately constant for most fuels and nutrients. Using Equation (17) entropy generated can be calculated. Table 2 shows properties for macronutrients CH, F and P.

_{2}consumed) and the term ${\mathsf{\eta}}_{\mathrm{N}}{\dot{\mathrm{m}}}_{\mathrm{O}2,\mathrm{n}}{\text{HHV}}_{\mathrm{O}2}$ represents the energy used for production of chemical work. Equation (22) suggests that lesser the work you obtain, more the energy available as metabolic heat which results in temperature raise and more thermal denaturation. It is apparent from Equation (22) that the heat transfer across the organ results in entropy generation of each organ k.

#### 3.3.3. Lifespan Energy Expenditure and Entropy Generation of Organs and Contribution by Organs to the Body

_{k}(t)} for organ k based on stoichiometric oxygen as:

_{k,m}of organ k per unit mass of organ k, ii) LSEG

_{k}contributed by organ k to the unit mass of the body. In order to compute LSEG one needs to express ${\dot{\mathrm{q}}}_{\mathrm{k},\mathrm{m}}$ in Equation (26) in terms of body mass m

_{B}(t) so that census data on average body weight vs. age (t) can be used and then integrated. As seen in Equation (3), most allometric laws for are expressed in terms of body mass (m

_{B}). Using Equation (3) in Equation (26), the specific entropy generation by each unit mass of organ is given as:

_{k}, and using Equation (4) for organ mass and then dividing by body mass, m

_{B}, one obtains specific entropy generation rate contributed by organ mass m

_{k}to each unit body mass. Thus:

_{k}= d

_{k}+f

_{k}and i

_{k}= e

_{k}c

_{k}. It is noted that the entropy contribution rate by organ k to each unit body mass can be obtained just by replacing the allometric constant e

_{k}and f

_{k}in Equation (27) by i

_{k}and j

_{k}-1 [i.e., Equation (28)]. Summing over all organs in thermal equilibrium within the body:

**Figure 3.**Curve Fitting Procedure for the Mass Growth and decrease. (Adopted and modified from [33]).

_{B,st}and finally, a relative short period of body mass decrease or negative growth. Following the conventional relations typically used in biology, the data was curve fitted with the following allometric form of equation:

_{B,st}and t

_{st,1}are the steady state body mass and the age at which stead body mass is reached respectively. The exponent d is defined as:

_{k}m

_{B,st}

^{fk}, one can obtain the specific entropy generated by organ k over lifespan t

_{life}as:

_{k},

_{I}= η

_{k},

_{II}= η

_{k},

_{III}are assumed to be same as “η” for all organs for the three periods and the short period III will be ignored. Thus t

_{st, 2}= t

_{life}. Modifying and simplifying Equation (33):

_{birth}*= t

_{birth}/t

_{life}, t

_{st,1}*= t

_{st,}/t

_{life}. Equation (34) is written as:

_{birth}, t*

_{st,1}, c, f

_{k})

_{k}representing Equation (36). If body mass is constant c=d=0 and t

_{birth}<< t

_{life}, then F (t*

_{birth}, t*

_{st,1}, c, f

_{k}) → 1 and hence from Equation (34), $\mathrm{T}{\mathsf{\sigma}}_{\mathrm{k},\mathrm{m},\text{life}}\approx {\mathrm{t}}_{\text{life}}\left(1-\eta \right){\dot{\mathrm{q}}}_{\mathrm{k},\mathrm{m},\text{st}},\text{and}{\dot{\mathrm{q}}}_{\mathrm{k},\mathrm{m},\text{st}}={\mathrm{e}}_{\mathrm{k}}{{\mathrm{m}}_{\mathrm{B},\text{st}}}^{{\mathrm{f}}_{\mathrm{k}}}$. Thus the F (t*

_{birth}, t*

_{st, 1}, c, f

_{k}) can be interpreted as a growth correction factor to the estimate based on the product of steady entropy generation rate and life span period.

Parameter | Y | F_{k} | Remarks |
---|---|---|---|

Life Time Specific Entropy Generation of organ k | $\frac{\mathrm{T}{\mathsf{\sigma}}_{\mathrm{k},\mathrm{m},\text{life}}}{{\mathrm{t}}_{\text{life}}\left(1-\eta \right){\dot{\mathrm{q}}}_{\mathrm{k},\mathrm{m},\text{st}}}$ | Equation (34) | ${\dot{\mathrm{q}}}_{\mathrm{k},\mathrm{m},\text{st}}\left(\frac{\mathrm{W}}{\text{kg}\text{organ}\mathrm{k}}\right)={\mathrm{e}}_{\mathrm{k}}{{\mathrm{m}}_{\mathrm{B},\text{st}}}^{{\mathrm{f}}_{\mathrm{k}}}$ |

Life Time Entropy Generation contribution by organ k to unit mass of body | $\frac{\mathrm{T}{\mathsf{\sigma}}_{\mathrm{k},\mathrm{M},\text{life}}}{{\mathrm{t}}_{\text{life}}\left(1-\eta \right){\dot{\mathrm{q}}}_{\mathrm{k},\mathrm{M},\text{st}}}$ | Equation (34) with f_{k} replaced by (f_{k}+d_{k}-1) | $\dot{q}$_{k,M,st} = c_{k}e_{k} m_{B,st} ^{(f}_{k}^{+d}_{k}^{−1)} |

Life Time Entropy Generation contribution by organ k to whole body | $\frac{\mathrm{T}{\mathsf{\sigma}}_{\mathrm{k},\text{life}}}{{\mathrm{t}}_{\text{life}}\left(1-\eta \right){\dot{\mathrm{q}}}_{\mathrm{k},\text{st}}}$ | Equation (34) with f_{k} replaced by (f_{k}+d_{k}) | $\dot{q}$_{k, st} = c_{k}e_{k} m_{B,st}^{(f}_{k}^{+d}_{k}^{)} |

Life Time Specific metabolic energy release by organ k | $\frac{{\mathrm{q}}_{\mathrm{k},\mathrm{m},\text{life}}}{{\mathrm{t}}_{\text{life}}{\dot{\mathrm{q}}}_{\mathrm{k},\mathrm{m},\text{st}}}$ | Equation (34) | ${\dot{\mathrm{q}}}_{\mathrm{k},\mathrm{m},\text{st}}\left(\frac{\mathrm{W}}{\text{kg}\text{organ}\mathrm{k}}\right)={\mathrm{e}}_{\mathrm{k}}{{\mathrm{m}}_{\mathrm{B},\text{st}}}^{{\mathrm{f}}_{\mathrm{k}}}$ |

Life Time metabolic energy contribution by organ k to unit mass of body | $\frac{{\mathrm{q}}_{\mathrm{k},\mathrm{M},\text{life}}}{{\mathrm{t}}_{\text{life}}{\dot{\mathrm{q}}}_{\mathrm{k},\mathrm{M},\text{st}}}$ | Equation (34) with f_{k} replaced by (f_{k}+d_{k}-1) | ${\dot{\mathrm{q}}}_{\mathrm{k},\mathrm{M},\text{st}}\left(\frac{\mathrm{W}}{\text{kg}\text{body}}\right)={\mathrm{c}}_{\mathrm{k}}{\mathrm{e}}_{\mathrm{k}}{{\mathrm{m}}_{\mathrm{B},\text{st}}}^{{\mathrm{f}}_{\mathrm{k}}+{\mathrm{d}}_{\mathrm{k}}-1}$ |

Life Time metabolic energy contribution by organ k to whole body | $\frac{{\mathrm{q}}_{\mathrm{k},\text{life}}}{{\mathrm{t}}_{\text{life}}{\dot{\mathrm{q}}}_{\mathrm{k},\mathrm{m},\text{st}}}$ | Equation (34) with f_{k} replaced by (f_{k}+d_{k}) | ${\dot{\mathrm{q}}}_{\mathrm{k},\text{st}}\left(\mathrm{W}\right)={\mathrm{c}}_{\mathrm{k}}{\mathrm{e}}_{\mathrm{k}}{{\mathrm{m}}_{\mathrm{B},\text{st}}}^{{\mathrm{f}}_{\mathrm{k}}+{\mathrm{d}}_{\mathrm{k}}}$ |

_{k}, d

_{k}, e

_{k}, and f

_{k;}lower case subscript “m” per unit mass of organ k, capital ”M” per unit mass of body. $\frac{{\mathrm{m}}_{\mathrm{B}}(\mathrm{t})}{{\mathrm{m}}_{\mathrm{B},\text{st}}}={\left\{\frac{\mathrm{t}}{{\mathrm{t}}_{\text{st},\text{ref}}}\right\}}^{\mathrm{d}},\mathrm{d}=\mathrm{c},{\mathrm{T}}_{\mathrm{B}}\dot{\mathsf{\sigma}}=\dot{\mathrm{Q}}\left(\mathrm{W}\text{as}\text{heat}\right)=\dot{\mathrm{q}}(\mathrm{W}\text{due}\text{to}\text{metabolism})(1-\eta ),{\dot{\mathrm{W}}}_{\text{ATP}}(\mathrm{W}\text{as}\text{work})=\dot{\mathrm{q}}\eta $.

_{k}and f

_{k}in Equation (34) by i

_{k}and j

_{k}-1. The LSEE

_{k}for each organ k can be estimated as well by setting η

_{k}= 0 (i.e., all metabolic energy released as heat) in Equation (34). Since metabolic heat contributed by organ k to unit mass of the body is given by ${\dot{\mathrm{Q}}}_{\mathrm{k},\mathrm{M}}$ (t) = ${\dot{\mathrm{q}}}_{\mathrm{k},\mathrm{M}}$ (t) *{1-η

_{k}(t)}, and since it was assumed that η

_{k}(t) remains constant at η

_{k}(assumption h) then multiplying LSEE

_{k}by (1-η

_{k}), the life span energy released as heat by organ k (LSEH

_{k}) can be estimated. Summing overall organs, the lifetime specific energy released as heat (LSEH) of the whole body is obtained. Table 3 summarizes Y for other lifetime parameters of organ k: specific entropy generation by k, contribution to entropy generation of unit mass of body by organ k, contribution to entropy generation of whole body by organ k, metabolic energy release by unit mass of organ k, metabolic energy contribution to unit mass of body by organ k and finally metabolic energy contribution by organ k to whole body. The methodology, presented in Appendix A.2, enables the metabolic and entropy stress level estimation over the life span by just measuring oxygen intake rate into organs and oxygen outflow rate from organs and masses of organs at ages t = t

_{st,1}(age at beyond which mass remains constant) and assuming the validity of allometric relations.

## 4. Results and Discussion

#### 4.1. Nutrient Data

_{O2}) is approximately constant for all three nutrients with an average of 14,335 kJ/kg of O

_{2}or 18.7 kJ/SATP L of O

_{2}consumed at standard atmospheric temperature (25 °C) and pressure (101 kPa) (SATP). Note that the medical and biological literature uses 0°C and pressure of 101 kPa (CSA). The entropy generated to energy release ratios for the three nutrients, ENER = {(1-η

_{n})/T

_{B}} are estimated as 0.00219 K

^{−1}, 0.00218 K

^{−1}and 0.00323 K

^{−1}for CH, F and P respectively. Since metabolic efficiencies are almost similar for glucose and fats, the ENER’s are similar. The proteins’ metabolic entropy is expected to be high due to the low efficiency of the acid cycle transforming proteins to ATP. Thus, thermal denaturation is severe for protein diet metabolism compared to glucose and fat.

#### 4.2. Growth Data

- Period I: t
_{birth}<t< t_{st, 1}, t_{st, 1}= 24 yrs, m_{B, st}= 84 kg, {m_{B}/m_{Bst}} = {t/t_{st, 1}}°^{.75} - Period II: 24 < t < 75; m
_{B}= m_{Bst}= 84 kg, t_{life}= 75 years - A short period of small weight loss (period III) after 70 yrs and prior to death was ignored.

#### 4.3. Results

#### 4.3.1. % Contribution by Vital Organs (BHKL) to Overall Metabolic Rates

_{B}

^{−}°

^{.133}

_{B}

^{−}°

^{.1}°

^{1}

**Figure 4.**Correlation of % of vital organ mass and contribution % by vital organs towards overall metabolism.

_{B}

^{−}

^{0.07}[24]. The % contribution to BMR by four vital organs ranges from 73% for BS of 0.1 kg, 38% for 70 kg human and 29% for a 1,000 kg animal while the % contribution to mass ranges from 8 for BS of 0.1 kg, 3.3% for 70 kg human and 2.3% for a 1,000 kg animal. From the correlation it is seen that the % contribution to BMR and % mass to total mass by vital organs BHKL decrease with age or increase in clock time. The fit is consistent with those of Snyder et al who found that BHKL and spleen organs of most mammals, constituting only about 5% of body weight spend about 60% of whole body REE [35]. It is seen that as the person growths, the vital organ mass % decreases and as such these metabolically active tissues (heart, lungs, brain, liver, and kidneys) contributes less to resting metabolic rate.

#### 4.3.2. Growth Correction Factor, F

_{birth}, t*

_{st, 1}, c, f

_{k}) for entropy generation with t

_{st, 1}

^{*}for various values of f

_{k}assuming c= 0.75. The f

_{k}values cover the range of values tabulated in Table 1. It is noted that SBMR of the body is higher at the time of birth due to high surface area to volume and hence entropy generation rate is higher. As the body weight increases, SBMR decreases and becomes lowest at time of death. If the steady mass is approached slowly (i.e., period I dominant), the period II becomes less and less, t

_{st, 1}is higher and, t

_{st, 1}* approaches unity and hence lifespan entropy generation will be higher. The organs which follow isometric law in metabolic rate must have f

_{k}=0; i.e., organ specific metabolic rate is constant over life period and hence entropy generation over life span can be simply given by a product of steady entropy generation rate and life span period.

**Figure 5.**The effect of variation of organ mass growth on correction factor F (t*birth, t*st,1 , c, f

_{k}) with tst,1 * , c= d= 0. 75.

#### 4.3.3. Specific Basal Metabolic Contribution (SBMR_{k}) by Organ k, t_{st}* = t_{st,}/t_{life}

**Figure 6.**Specific Metabolic rate of organs (SBMRk, W/kg organ mass), W contribution by respective organs (BMRk) and % contribution to overall metabolism for 70 kg human.

#### 4.3.4. Lifespan Specific Energy Expenditure (LSEE_{M})

_{M}is estimated to be 2832 MJ/kg body mass. Data collected by Speakman indicate LSEE

_{M}of 3025 MJ/kg [10] confirming the validity of current approach. The lifetime entropy generated per unit mass of body is calculated to be 6.3 MJ/ {kg body mass·K}.

% Nutrient consumed for metabolism, CH: F: P = | 55:30:15 |

Computed Fraction of O2 by nutrient “n”, f_{O2, n}. See Equation (24) = | 0.349:0.513: 0.138 |

Average metabolic Efficiency computed by Equation (24) = | 31.3% |

_{st, 1}= 24 years, t

_{st, 2}= 75 years, t

_{life}= 75 years, d=c= 0.75, Male, m

_{B, st}= 84 kg

**Table 5.**Life Span Entropy Contribution by each organ (MJ/kg body mass K) and metabolic contribution (GJ per kg body).

Organ | MJ/(kg body·K) | MJ/kg body |
---|---|---|

Sigma contrib. | Metabolic contrib. | |

Brain | 0.252 | 114 |

Heart | 0.784 | 352 |

Kidney | 0.589 | 266 |

Liver | 1.069 | 480 |

Remainder | 3.610 | 1620 |

Sum | 6.304 | 2832 |

#### 4.3.5. Life Span Organ Entropy Generation

**Figure 7.**Lifetime Specific entropy generation in MJ/{kg organ•K}, Metabolic Energy in GJ/ {kg organ} and % contribution to entropy by organs to overall entropy generation of BS.

_{H}) values are: Heart: 1.0, Kidney: 0.92, Brain: 0.46, Liver: 0.41, Rest of BS: 0.027. If reference organ is the rest of body (R), then NES

_{R}values are: Heart: 37, Kidney: 34, Brain: 17, Liver: 15, Rest of BS: 1.0; so heart will fail first followed by kidney and other organs in order.

#### 4.3.6. Life span Specific Entropy Generation of Whole Body

**Figure 8.**Lifespan entropy generation {MJ by organ k/ (K kg body mass} and metabolic heat contribution {GJ by organ k/kg body mass)} by respective organs.

#### 4.3.7. Effect of Nutrients

_{CH}moles of CH and (1-X

_{CH}) moles of fat, one can obtain X

_{CH}as [29]

_{CH}= 0.75, mass fraction of CH = 0.68, and ν

_{O2, stoich}= 1.64 kg per kg of nutrient mix; using Equation (24), η is estimated as 0.35 which is slightly higher than 0.31 (Table 4). The higher efficiency decreases the estimated lifetime entropy generation only by 6%.

#### 4.3.8. Relation to Life Span

- Looking at data on people living longer than 85 years, heart is cited as #1 cause agreeing with current entropy stress level [2]
- According to current hypothesis, the next organ must be kidney; however cancer is the statistical number 2 cause of death (Figure 2). Recently, Germaine Wong and her colleagues collected data from 3654 Australians within the age group 49–97 years over 10 year period and observed that decreased kidney function leads to an increased risk of developing cancer [38]. Chronic kidney disease is common in people with cardio-vascular disease. Kidney function is also related to progression to cardio-vascular disease; chronic kidney disease is a risk factor in other chronic diseases such as infections and cancer [38].
- Since ROS concentrations are generally higher with increased T
_{B}(i.e., metabolism which results in fraction of energy converted into heat) and hence, shorter life span, then decreased T_{B}must lead to prolonged lifespan. “On November 2006, a team of scientists from the Scripps Research Institute reported that transgenic mice which had body temperature 0.3–0.5 °C lower than normal mice indeed lived longer than normal mice.” [39]. Lifespan was 12% longer for males and 20% longer for females. Mice were allowed to eat as much as they wanted. However they had indicated that the effects of such a genetic change in body temperature on longevity are harder to study in humans. - The third cause happens to be brain as predicted by the MREG model.
- The effect of change in nutrient composition and metabolic efficiency on $\dot{\mathsf{\sigma}}$
_{n}(t) are apparent from Equation (17); when n= P, η_{n}is low (e.g., proteins), and hence $\dot{\mathsf{\sigma}}$_{n}(t) is higher indicating high protein diet leads to highest metabolic heat and irreversibility. It has been shown by Kapahi and his group that life spans of fruit flies are extended by using low protein diet [37].

## 5. Conclusions

- (1)
- The first and second laws of thermodynamics including availability analyses were applied to the vital organs of biological systems.
- (2)
- It is shown that that the sum of lifetime entropy generation contribution by all the vital organs to each unit body mass is $\frac{6.3MJ}{Kkgbodymass}$.
- (3)
- The lifetime specific entropy generation of vital organs for 84 kg person is estimated as follows (MJ/ {kg of organ·K}): Bran: 62.4, Heart = 135.4, Kidney: 124.1, Liver: 55.5, Rest of organs: 3.7. The vital organ under most severe stress was found to be heart in agreement with leading cause of natural death.
- (4)
- The total lifetime contribution by all the vital organs to each unit body mass is $\frac{6.3MJ}{Kkgbodymass}$.
- (5)
- The heart-normalized entropy stress (NES
_{H}) values are: Heart: 1.0, Kidney: 0.92, Brain: 0.46, Liver: 0.41, Rest of BS: 0.027. If normalized to rest of body (R), NES_{R}, heart: 37, Kidney: 34, Brain: 17, Liver: 15, Rest of BS: 1.0; so heart will fail first followed by kidney and other organs in order. Supporting data is provided. - (6)
- It is possible to estimate lifespan entropy stress just by measuring metabolic rate at the standard weight age (after which weight remains constant), and assuming that allometric laws are valid for organs.
- (7)
- Since ROS concentrations are generally higher with increased T
_{B}and hence, shorter life span, then decreased T_{B}must lead to prolonged lifespan.

_{k}) vs. body mass (m

_{B}) along with their metabolic rates and use these data for ranking the “entropy (or heat) stress” of organs. Future work must be conducted to compare organ-lifespan values of energy and entropy generated for an average individual with those of super-centenarians.

## Acknowledgements

## Acronyms

ADP | Adenosine di-phosphate |

AMDR/AI | Adequate macronutrient distribution range/Adequate Intake |

ATP | Adenosine tri-phosphate |

BHKL | Brain, heart, kidney, liver |

BMR | Basal Metabolic Rate |

BS | Biological system |

CCE | Cell copy error |

CDC | Center for Disease Control and Prevention |

CH | Carbohydrate |

CR | Calorie restriction diet |

CSA | Chemist Standard Atmosphere, 0 °C, 101 kPa |

DRI | Dietary reference intake |

EER | Energy expenditure requirements |

EER | Estimated energy requirements |

HHV | Higher or gross heating value, |

HHV_{O2, n} | higher heating value per unit mass of stoichiometric oxygen of nutrient n |

LSEG | Lifetime specific entropy generation (J/kg K) |

LSEE | Lifetime specific Energy Expenditure (J/kg) |

LSEH | Lifetime specific energy released as heat |

ME | Metabolic efficiency |

MREG | Modified Rate of Entropy Generation |

NES | Normalized Entropy Stress |

REG | Rate of Entropy Generation |

ROL | Rate of Living Theory |

ROS | Radical Oxygen Species |

SATP | Standard Atmospheric temperature and pressure (25 C, 1 atm) |

SBMR | Specific Basal Metabolic Rate, (W/kg K) |

US FNB | US Food and Nutrition Board |

VLSF | Vital life sustaining functions |

## Nomenclature

E | Energy, kJ |

F | Growth Correction factor |

G | Gibbs free energy, kJ |

h | Enthalpy, kJ/kg |

I | Irreversibility I, kJ |

$\dot{\mathrm{I}}$ | Irreversibility rate, kJ/s |

m | Mass, kg |

m_{B} | Body mass |

m_{k} | Mass of organ k |

${\dot{\mathrm{m}}}_{k}$ | Mass flow rate of nutrient n in organ k |

${\dot{\mathrm{m}}}_{O2,n,k}(t)$ | Consumption rate of oxygen by nutrient n in organ k |

P | Protein |

Q | Heat |

$\dot{\mathrm{Q}}$ | Heat transfer rate due to metabolic heat release
${\dot{q}}_{k}$ at organ k |

${\dot{\mathrm{q}}}_{k,m}$ | Specific metabolic energy release rate from organ k per unit mass of organ k |

${\dot{\mathrm{q}}}_{k,M}$ | Energy release rate of organ k contributed to the unit mass of body |

S | Entropy , kJ/ K |

s | Specific Entropy, kJ/kg K |

T_{B} | Body temperature, K |

t | Time or age |

t_{st} | Time to reach steady weight |

U | Internal energy |

W_{K} | Work delivered by metabolism at organ k |

$\mathrm{\Delta}\overline{\mathrm{G}}$ _{C}° | Gibbs free Energy for combustion |

$\mathrm{\Delta}\overline{\mathrm{G}}$ _{M}° | Gibbs free Energy for metabolism ( with ATP production) |

$\mathrm{\Delta}\overline{\mathrm{G}}$ °_{ATP} | Gibbs free energy |

## Greek Symbols

η | metabolic efficiency |

σ | Entropy generation, kJ/K |

σ _{M,k} | Entropy contribution to unit mass of body by whole organ k |

${\dot{\mathsf{\sigma}}}_{\mathrm{M}},$ | Entropy generation rate per unit body mass (W/kg body mass K) |

${\dot{\mathsf{\sigma}}}_{\mathrm{m},\mathrm{k}}$ | Specific entropy generation rate of organ k (W/{K kg of k}) |

Ψ | Stream availability, kJ/kg |

ν_{O2,n} | Stoichiometric oxygen mass per unit mass of nutrient n |

η_{n,k} | metabolic efficiency of nutrient n in organ k |

## Superscript

0 | Atmospheric conditions |

B,ref | Reference mass for body |

C | Combustion |

k | Organ k |

life | Life Span |

m | Specific referring to unit mass of organ |

M | Specific referring to unit mass of body |

n | Nutrient (n) |

P-R | Difference of value from products to reactants |

J | Nutrient j |

P | Products |

R | Reactants |

St | Steady |

## General Notes

## Appendix

## A1. Alternate Allometric Relations

_{k}. Solving for m

_{B}in terms of m

_{k}from Equation (4) as m

_{B}= {m

_{k}/c

_{k}}

^{1/dk}and using in Equation (3), one can obtain allometric laws in term of organ mass m

_{k}as

_{k}, and it is given as:

_{k}and using Equation(4), one can re-express metabolic energy release rate contributed by k to the hole body as:

## A2. Integration:

^{b},

^{d}

## References

- Olshansky, S.J.; Carnes, B.A. Quest for Immortality: Science at Frontiers of Aging; W.W. Nortons: New York, NY, USA, 2001. [Google Scholar]
- National Center for Health Statistics, 2001 Data Warehouse on Trends in Health and Aging. extensive data from http://www.cdc.gov/nchs/nvss/mortality_public_use_data.htm.
- Ebersole, P.; Hess, P.; Schmidt, L.A. Towards Healthy Aging-Human Needs and Nursing, 6th ed.; Mosby Inc.: London, UK, 2003. [Google Scholar]
- Schrodinger, E. What Is Life? Cambridge University Press: Cambridge, UK, 1944; p. 194. [Google Scholar]
- Azbel, M.Y. Universal Biological Scaling and Mortality. Proc. Nat. Acad. Sci. USA
**1994**, 91, 453–457. [Google Scholar] [CrossRef] - Kirkwood, T.B.L. Evolution of Ageing. Nature
**1977**, 270, 301–304. [Google Scholar] [CrossRef] [PubMed] - Hofman, M.A. Energy, Metabolism, Brain Size and Longevity in Mammals. Quart. Rev. Biol.
**1983**, 58, 495–512. [Google Scholar] [CrossRef] [PubMed] - Hofman, M.A. Evolution of brain size in neonatal and adult placental mammals: a theoretical approach. J. Theor. Biol.
**1983**, 105, 317–332. [Google Scholar] [CrossRef] - Pearl, R. The Biology of Death: Being a Series of Lectures Delivered at the Lowell Institute in Boston in December 1920; J. B. Lippincott & Co.: Philadelphia, PA, USA, 1922. [Google Scholar]
- Speakman, J.R. Body size, energy metabolism and lifespan-A Review. J. Exp. Biol.
**2005**, 208, 1717–1730. [Google Scholar] [CrossRef] [PubMed] - Julie, A.M.; George, S.R.; Beasley, T.M.; Tilmont, E.M.; Handy, A.M.; Herbert, R.L.; Longo, D.L.; Allison, D.B.; Young, J.E.; Bryant, M.; et al. Impact of caloric restriction on health and survival in rhesus monkeys from the NIA study. Nature
**2012**. [Google Scholar] [CrossRef] - Annamalai, K.; Puri, I. Combustion Science and Engineering; CRC Press: Boca Raton, FL, USA, 2007. [Google Scholar]
- DNA Repair at Encyclopedia of Aging. Available online: http://www.encyclopedia.com/doc/1G2-3402200112.html/ (accessed on 12 August 2012).
- Beckman, A. Free Radical Theory of Aging Matures. Phys. Rev.
**1998**, 78, 547–581. [Google Scholar] - Hershey, D.; Wang, H. A New Age-Scale for Humans; Lexington Books: New York, USA, 1980. [Google Scholar]
- Silva, C.A.; Annamalai, K. Entropy generation and human aging: lifespan entropy and effect of diet composition and caloric restriction diets. J. Thermodyn.
**2009**, 186723. [Google Scholar] [CrossRef] - Batato, M.; Deriaz, O.; Jequier, E.; Borel, L. Second Law Analysis of the Human Body. In Proceedings of Florence World Energy Research Symposium, Firenze, Italy, May 1990.
- Aoki, I. Entropy Flow and Entropy Production in the Human Body in Basal Conditions. J. Theor. Biol.
**1989**, 141, 11–21. [Google Scholar] [CrossRef] - Aoki, I. Effects of Exercise and Chills on Entropy Production in Human Body. J. Theor. Biol.
**1990**, 145, 421–428. [Google Scholar] [CrossRef] - Aoki, I. Entropy production in human lifespan: a thermodynamical measure for aging. Age
**1994**, 1, 29–31. [Google Scholar] [CrossRef] - Rahman, M.A. A novel Method for Estimating the Entropy Generation in a Human Body. Therm. Sci.
**2007**, 11, 75–92. [Google Scholar] [CrossRef] - Flyod, R.A.; West, M.; Hensley, K. Oxidative Biochemical Markers Cluse To Understanding Aging In Long Lived Species. Exp. Gerontol.
**2001**, 36, 619–640. [Google Scholar] [CrossRef] - Walford, R.L. Calorie Restriction, Eat Less, Eat Better, Live Longer. LE Magazine, Febuary 1998. [Google Scholar]
- Wang, Z.; Heshka, S.; Gallagher, D.; Boozer, C.; Kotler, D.P.; Heymsfield, S. Resting energy expenditure-fat-free mass relationship: new insights provided by body composition modeling. Am. J. Physiol. Endocrinol. Metab.
**2000**, 279, E539–E545. [Google Scholar] [PubMed] - Wang, Z.; Timothy, P.; O’Connor, S.H.; Steven, B.H. The Reconstruction of Kleiber’s Law at the Organ-Tissue Level. J. Am. Soc. Nutr. Sci.
**2001**, 131, 2967–2970. [Google Scholar] - Calder, W.A., III. Size, Function, and Life History; Dover: New York, NY, USA, 1996. [Google Scholar]
- Elia, M. Organ and tissue contribution to metabolic rate. In Energy Metabolism: Tissue Determinants and Cellular Corollaries; Kinney, J.M., Tucker, H.N., Eds.; Raven: New York, NY, USA, 1992; pp. 61–80. [Google Scholar]
- Gallagher, D.; Belmonte, D.; Deurenberg, P.; Wang, Z.; Krasnow, N.; Pi-Sunyer, F.X.; Heymsfield, S.B. Organ tissue mass measurement allows modeling of resting energy expenditure and metabolically active tissue mass. Am. J. Physiol.
**1998**, 275, E249–E258. [Google Scholar] [PubMed] - Annamalai, K.; Puri, I.; Jog, M. Advanced Thermodynamics Engineering, 2nd ed.; Taylor and Francis: Boca Raton, FL, UAS, 2011; Chapter 14; p. 1096. [Google Scholar]
- Silva, C.; Annamalai, K. Entropy generation and human aging: lifespan entropy and effect of physical activity level. Entropy
**2008**, 10, 100–123. [Google Scholar] [CrossRef] - Cardiovascular System at Wikibooks. Available online: http://en.wikibooks.org/wiki/Anatomy_and_Physiology_of_Animals/Cardiovascular_System/Blood_circulation (accessed on 23 December 2010).
- Brzustowski, T.A.; Brena, A. Second law analyses of energy processes. IV–The exergy of hydrocarbon fuels, Second Law analysis of Energy Processes. Trans. Canad. Soc. Mech. Eng.
**1986**, 10, 121–128. [Google Scholar] - National Center for Environmental Assessment, Office of Research and Development, U.S. Environmental Protection Agency. Summary Report of a Peer Involvement Workshop on the Development of an Exposure Factors Handbook for the Aging, Arlington, VA, USA, 14–15 February, 2007.
- Health Survey at the National Archives. Available online: http://www.dh.gov.uk/en/Publicationsandstatistics/PublishedSurvey/HealthSurveyForEngland/Healthsurveyresults/DH_4001334 (accessed on 15 December 2010).
- Snyder, W.S.; Cook, M.J. Report of the task group on reference men. International Commission on Radiological Protection, 1975; Pergamon Press: Oxford, UK.
- Dietary Reference Intakes for Energy, Carbohydrate, Fiber, Fat, Fatty Acids, Cholesterol, Protein, and Amino Acids (Macronutrients); National Academy Press: Washington, DC, USA, 2002.
- Edman, U.; Garcia, A.M.; Busuttil, R.; Sorensen, D.; Lundell, M.; Kapahi, P.; Vijg, J. Lifespan extension by dietary restriction is not linked to protection against somatic DNA damage in Drosophila melanogaster. Aging Cell
**2009**, 8, 331–338. [Google Scholar] [CrossRef] [PubMed] - Wong, G.; Hayen, A.; Chapman, J.; Webster, A.; Wang, J.; Mitchell, P.; Craig, J. Association of CKD and Cancer Risk in Older People. Clin. J. Am. Soc. Nephrol.
**2009**. [Google Scholar] [CrossRef] [PubMed] - Conti, B.; Sanchez-Alavez, M.; Winsky-Sommere, R.; Morale, M.C.; Lucero, J. Transgenic Mice with a Reduced Core Body Temperature Have an Increased Life Span. Science
**2006**, 314, 825–828. [Google Scholar] [CrossRef] [PubMed] - Madya, C.; Keutenedjian, E.; de Oliveira, S., Jr. Human body exergy metabolism. In Proceedings Of Ecos 2012–The 25th International Conference on Efficiency, Cost, Optimization, Simulation And Environmental Impact Of Energy Systems, Perugia, Italy, 26–29 June 2012.
- Hayne, D.T. Biological Thermodynamics. Cambridge University Press: London, UK, 2008. [Google Scholar]
- American Society of Heating, Refrigerating and Air-Conditioning Engineers. Physiological principles and thermal comfort. In Handbook of Fundamentals; ASHRAE: Atlanta, GA, USA, 1993; pp. 1–29. [Google Scholar]
- Annamalai, K. Group Combustion of Char Particles and Metabolism in Large Organs. In Spring Technical Meeting of the Central States Section of the Comb. Inst., Dayton, OH, USA, 22–24 April 2012. Paper No. 12S-130.
- Suarez, R.K.; Charles, A.D. Review: Multi-level regulation and metabolic scaling. J. Exp. Biol.
**2005**, 208, 1627–1634. [Google Scholar] [CrossRef] [PubMed]

© 2012 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Annamalai, K.; Silva, C.
Entropy Stress and Scaling of Vital Organs over Life Span Based on Allometric Laws. *Entropy* **2012**, *14*, 2550-2577.
https://doi.org/10.3390/e14122550

**AMA Style**

Annamalai K, Silva C.
Entropy Stress and Scaling of Vital Organs over Life Span Based on Allometric Laws. *Entropy*. 2012; 14(12):2550-2577.
https://doi.org/10.3390/e14122550

**Chicago/Turabian Style**

Annamalai, Kalyan, and Carlos Silva.
2012. "Entropy Stress and Scaling of Vital Organs over Life Span Based on Allometric Laws" *Entropy* 14, no. 12: 2550-2577.
https://doi.org/10.3390/e14122550