# Entropy Generation and Human Aging: Lifespan Entropy and Effect of Physical Activity Level

^{*}

## Abstract

**:**

## 1. Introduction

_{2}, exhaled CO

_{2}, and the knowledge of the chemical composition of food intake. The metabolic energy input varies depending upon resting conditions, the basal metabolic rate (BMR) and the extent of physical activity level (PAL). If two experiments are performed on the same organism, the first one in resting condition (BMR) while the second one is performing some measured work (PAL), the difference in the amount of food metabolized ${\dot{\mathrm{Q}}}_{\text{Chem},\text{PAL}}$ and ${\dot{\mathrm{Q}}}_{\text{Chem},\text{BMR}}$ (second and first cases) can be expressed as energy input for conversion to work and compared to the work output ($\dot{\mathrm{W}}$) in order to obtain the metabolic work efficiency (MWE). These experiments have confirmed that the efficiency of muscular work is fairly consistent 25-30 percent [4].

_{life}, σ

_{m,life}can be estimated. If LSEGm= σ

_{m,life}(kJ/kg-K) and the average ${\dot{\sigma}}_{\mathrm{m}}$ is a constant throughout a life span as assumed by Annamalai and Puri [6,7] then such a model seems to yields a relation qualitatively similar to that of Rubner. More details are provided in later sections.

## 2. Literature Review

#### 2.1 Allometry

^{b}

- a)
- Metabolic rates: Metabolism involves oxidation of glucose, fats and proteins in the nutrients, which require oxygen for oxidation. The O
_{2}consumption rate is given by the above general law [8] with:- y= O
_{2}consumption rate in mL/hr, x= mass of body in g - a= 4 - 4.2, b = 0.68

- a = 0.064, b = 0.734 with y = mL O
_{2}/min and x = body mass in g

- y= specific metabolic rate (W/kg), x= mass of body in kg
- a= 3.55, b = -0.26

- b)
- Mass of brain: Same equation (1) with:
- y = brain mass in g, x = body mass in g
- a = 0.043, b = 2/3 to 0.73

- c)
- Body surface area (BSA) [http://ajpendo.physiology.org/cgi/content/full/281/3/E586]
- y, BSA in m
^{2}, x, body mass in kg - a= 0.1173, b = 0.6466, R
^{2}= 0.9914

^{c}z

^{β}

- y = lifespan, days
- x = body mass in g
- z = brain mass in g

#### 2.2 Lifespan

_{m,max}as 10,000 kJ/kg_K. Although this calculation is simplistic, its values seem reasonable when compared to statistical data, and most importantly, yielded a scaling law for life span with mass for all BS.

## 3. Rationale and Objective

- Accounting for the metabolic efficiency (production of ATP) in the BS. We will start with the basic chemical reactions typical of metabolism to determine the efficiency of ATP production, or MWE, for each of the three main nutrient groups. In the BS, energy not converted to ATP is disposed of as heat, thus, increasing entropy.
- Accounting for the effects of age, body mass, physical activity level and diet composition on entropy generation.
- Using the maximum allowable entropy generation per unit mass as a criteria in defining life span. As long as the organism performs its life functions, entropy continues to be generated. If we calculate the maximum lifespan entropy, then we can study the effect of the different factors on the time needed to reach lifespan entropy.

_{m,max}concept, then compare with empirical data available for the BS.

#### 3.1 Hypothesis:

**within**the BS. This is the approach taken in the current paper, in which the estimated (daily) energy requirements (EER) of the BS were used to calculate the metabolic rates, and then an availability analysis was applied to the metabolic reactions of the main nutrient groups to obtain the entropy generated. The hypothesis presumes that nutrients not metabolized leave the BS without significant change in their entropy, and that differences between the entropy inflow/outflow of water and air are small; such an assumption is supported by the work of Hershey and Wang in which the entropy exchange due to breathing (considering the change in both temperature and chemical composition of breathing air) was less than 2.5% of the total entropy generated by the human body.

## 4. Analysis

#### 4.1 Phenomenological Analysis

_{0}). Assuming the surface temperature is the same as T, the minimum specific metabolic rate (${\dot{q}}_{\mathrm{m}}$) required to overcome heat loss is given as [6]:

_{H}, the heat transfer coefficient in kW/m

^{2}K and $\dot{\mathrm{Q}}$, heat loss rate in kW. If the body surface area of BS, A ∝ R

^{2}

_{Char}where R

_{char}is the characteristic size of BS and mass of BS, m ∝ R

_{Char}

^{3}, then A ∝ m

^{2/3}(empirical relation A ∝ m

^{0.664})

_{Char}∝ 1/m

^{1/3}

_{m}is constant. The empirical fit of experimental data indicates:

_{m}= 3.552 (W/kg

^{0.74}) and ${\dot{q}}_{\mathrm{m}}$ is given in (W/kg). While simple theory predicts the exponent to be -0.333, empirical experimental fit yields the exponent to be -0.26. If ATP production is accounted for, the heat production rate must be more than the value given by Eq. (6).

_{life}) with the same specific metabolic rate:

_{life}=2,840 MJ/kg. This is clearly an overestimate since specific metabolic rate keeps decreasing with increase in the mass. If Rubner’s hypothesis of specified q

_{m,life}(J/kg) is correct, then Eq. (7) yields:

_{life}∝ m

^{1/3}

_{b}is the body temperature. If BS is maintained at constant m, T and P, a first approximation yields dS/dt =0; thus,

_{b}= 310 K , Eq.(11) yields the lifetime entropy generated as 9753 kJ/kg K. Integrating with life span, it is apparent that:

_{life}∝ m

^{n}

#### 4.2 Rigorous Analysis

#### a) Overview on present methodology

#### b) Assumptions

- ○
- The essential nutrients are Carbohydrates (CH), fats (F) and proteins (P).
- ○
- CH, F and P will be modeled using glucose, palmitic acid, and average amino acids composition respectively.
- ○
- The time span of interest is much longer compared to metabolic reaction time scales; thus, quasi-steady approach is valid.
- ○
- The MWE for conversion of CH, F and P energy into ATP is different for every nutrient but remains constant over the whole life span. Thus, a child who is physically active requires more metabolic rate to produce necessary ATP rather than increased MWE.
- ○
- Metabolic rate and MWE are the same throughout the body including all body organs. It is known that oxidation near heart cells require more ATP, requiring increased MWE or increased metabolic rate.
- ○
- Actual energy requirements are used instead of basal metabolism in order to include the entropy generated by physical activity.
- ○
- Functioning normally, human beings and other large organisms cannot tolerate changes of body temperature of more than a few degrees
^{16}, and can be considered isothermal. One may assume that biochemical reactions in humans proceed at constant temperatures and pressures. - ○
- It is presumed that entropy generated for the metabolic part of the body determines the life span of BS since they produce ATP which enables the body to perform life sustaining functions and the metabolic mass of the body is proportional to body total mass.
- ○
- The nutrient consumption data are based on normal gravity environment.
- ○
- It will be assumed that a low-active physical activity level is the best representation of the healthy average individual activity.
- ○
- In order to keep the physics simple, we assume all of ATP is converted into work and there is no dissipative heat produced by ATP.
- ○
- All the processes within BS occur isothermally.
- ○
- For warm blood invertebrate, only 10% of the feed is converted into body substance called efficiency of conversion of ingested food (ECI). Note that for cold blooded vertebrate the ECI could be as high as 44 % (e.g. German Cockroaches). Thus the analysis is essentially for warm blooded vertebrate only; hence it is assumed that the feed ration is essentially used for conversion to ATP and dissipative heat.

^{th}percentile height and weight.

#### 4.3 Governing Equations

#### a) Energy Conservation:

_{T,k}, kJ/kg of k) includes the sum of internal energy (u in kJ/kg), flow work to cross the boundaries, and kinetic and potential energies of the advection terms for each species k. The term E ( in kJ) includes internal (U in kJ), kinetic, potential, and chemical energies, among others.

#### b) Availability Balance:

_{CV}) and (

_{R}) in Eq. (14) correspond to the control volume and thermal reservoirs respectively. For any ideal gas component k, the specific entropy (s

_{k}) of component k is given as

_{k}is the mole fraction of the component k, p

_{k}, partial pressure of component k in bars, R

_{k}is gas constant of species k {kJ/(kg K)} and $\overline{\mathrm{R}}$ is universal gas constant {8.314 kJ/(kmole K)}. The p

_{ref}is typically 1 bar. A statement of Eq. (14) is that the availability accumulation rate of any system is equal to the availability input due to heat transfer from external thermal reservoirs (e.g. radiant heaters used near incubators), less the availability transfer through work, and change due to advection and availability loss through irreversibility (=To ${\dot{\sigma}}_{\text{cv}}$). The entropy generation within selected CV is due to internal irreversibility caused by chemical reactions within all the reacting cells of BS, external irreversibility due to temperature gradient near the skin and irreversibility due to inhaled air being at T

_{0}, which is different compared to T. When temperature T

_{0}is set equal to T, body temperature of BS (e.g. CV boundary just below skin), then the irreversibility estimation or entropy generation rate ${\dot{\sigma}}_{\text{cv}}$ is due to all irreversible processes occurring within the BS.

#### 4.4 Simplifications

#### a) Gibbs free energy and Isothermal Chemical Reactions:

_{0}in Eq. (14) is set as body temperature T and neglecting kinetic and potential energies and external heat inputs, then the stream availability in Eq. (15) is equal to the Gibbs free energy and Eq. (14) becomes

_{k}= h

_{k}-T s

_{k}where h

_{k}is enthalpy and s

_{k}is entropy of the constituent; the lower is h

_{k}and the higher is s

_{k}, lower is “g

_{k}”. Like temperature T, pressure P, and enthalpy h, the “s” and g are also properties and value for any given pure component k, the s and g are fixed once T and P are specified. Eq. (18) is equally applicable to each organ of BS across which reactants enter and products leave. Adding over all metabolic cells within the body, the ${\dot{\mathrm{m}}}_{\mathrm{k}}$ represents total mass of nutrient species k entering the body. Macro-nutrients (reactants) are constantly injected at T, the required oxygen is supplied with air inhaled at T, and products exhausted through exhaling at T.

_{p1}] and [X

_{R1}] stands for products and reactants molar fractions.

#### b) Metabolism, Work in Biological Systems and Bioenergetics:

_{6}H

_{12}O

_{6}+ 6O

_{2}➔ 6CO

_{2}+ 6H

_{2}O

_{c}º is Gibb’s free energy for a conventional combustion process. The macro-nutrients contain chemical energy at low entropy level (high G

_{R}) while the products (CO2 and H2O) have high entropy (low G

_{P}) due to release of thermal energy. Hence typically ΔG

_{c}º <0. Now consider the metabolic oxidation of glucose along with ATP production, which can be expressed as:

_{6}H

_{12}O

_{6}+ 6O

_{2}➔ 6CO

_{2}+ 6H

_{2}O + 36 ATP

_{M}º is the Gibb’s free energy change for a metabolic reaction. Then the work potential of ATP for running the BS is:

_{j,P}, enthalpy of products per kg of j and S

_{j,P}, entropy of products per kg of j . As η

_{j}→0 for j = CH, F, and P, all the chemical energy is converted into dissipative heat.

_{4.57}H

_{9.03}N

_{1.27}O

_{2.25}S

_{0.046}with an empirical molecular weight of 119.39 kg/kmol and a heating value of -385 MJ/kmol (5.5 kcal/g) calculated by the Boie equation which is widely used in combustion literature [6,7]. It is interesting to note that efficiency of protein metabolism is about 1/3 of the efficiency for carbohydrates and fats. This may explain why energy is obtained from proteins in very small quantities and usually when other sources of energy are not available. Thus, bodily functions of a system BS with protein intake must be highly impaired compared to BS consuming glucose. Cells being made mostly of protein molecules may get consumed in the event other sources of energy intake are not available leading to cell death and impairing body functions.

#### 4.5 Entropy Generation

_{,j}is the kmoles of j reacted per unit time ( kmole/s), M

_{j}, molecular weight of j and $\mathrm{\Delta}{\overline{\mathrm{G}}}_{\mathrm{c},\mathrm{j}}$ change in Gibbs function per kmole (kJ/kmole of j). Summing up over the three reactions involving CH, F and P,

_{c,j}= ΔH°

_{c,j}-TΔS°

_{c,j}≈ ΔH°

_{c,j}. Thus ${\dot{\mathrm{m}}}_{\mathrm{j}}(\mathrm{t})\mathrm{\Delta}{{\mathrm{G}}^{\circ}}_{\mathrm{c},\mathrm{j}}(\mathrm{T},\mathrm{P})$ ≈ ${\dot{\mathrm{m}}}_{\mathrm{j}}(\mathrm{t})$ ΔH°

_{c,j}, metabolic rate [see Ref 6]. Thus the numerator in Eq (27) could be approximately interpreted as wasted heat for the whole BS. Eq. (27) can be equally extended to individual organs (heart, liver, kidney etc) if system is elected around each organ and $\dot{\sigma}$, η

_{j}and {${\dot{\mathrm{m}}}_{\mathrm{j}}(\mathrm{t})$ ΔH°

_{c,j}} are interpreted as entropy generation rate, metabolic efficiency and metabolic rate of the organ.

_{j}is low (e.g. proteins), then $\dot{\sigma}$ is higher. Note that $\dot{\sigma}$ reaches a maximum when η → 0 (all energy is dissipated as heat) and $\dot{\sigma}$ → 0 as η → 1. The latter result is due to the assumption that ATP simply serves as the “work” equivalent in thermodynamics. However, as ATP is used for pumping blood, the pressure loss through blood vessels is converted into heat. Also it is known that about 75% of energy released from ATP in muscles goes to mechanical work of a contraction, while 25% is released as heat. Thus, an equivalent metabolic efficiency (typically lower than true values) could be used to account for conversion of a part of ATP into heat. The specific entropy generation per unit mass of the BS with time (aging) is given as:

## 5. Input data and procedure

#### 5.1 Fuel Data

_{2}. Based on the above numbers and using 14,335 kJ/kg of O2 (or 18.7 kJ/L of O2), the ROL theory suggests that the lifetime oxygen consumed is 41.2- 77.4 kg of O

_{2}/kg body mass (31,510-59,280 liter/kg body mass, without man) and 211 kg of O

_{2}/kg body mass (161,560 liter/kg body mass) including man. With Rubner’s data of 836 MJ/kg, the O2 consumption is 58.3 kg of O

_{2}/kg body mass (44,650 Liters/kg). Compare this with the metabolic potential (total O

_{2}consumed during lifespan per kg bodyweight) for a 2 g shrew or a 100,000 kg blue whale is approximately 38,000 L of O

_{2}consumed or 8.5 kmol ATP/kg body mass per lifetime [20].

#### 5.2 Statistical Databases

Glucose | C_{6}H_{12}O_{6} | 180 | 15630 | 14665 | -1260 | 212.0 | 38.2 |

Protein | C_{4.57}H_{9.03}N_{1.27}O_{2.25}S_{0.046} | 119 | 22790 | 14705 | -385 | 10.4 | |

Fat | C_{16}H_{32}O_{2} | 256 | 39125 | 13635 | -835 | 452.4 | 32.2 |

**Table 2.**PAL walking equivalences [28].

PAL category | PAL value | (m/d at 2-4 mph) (*) | |

Sedentary | 1.0 – 1.39 | ||

Low Active | 1.4 – 1.59 | 1.5, 2.2, 2.9 for PAL=1.5 | |

Active | 1.6 – 1.89 | 3.0, 4.4, 5.8 for PAL=1.6 | |

5.3, 7.3, 9.9 for PAL=1.75 | |||

Very Active | 1.9 – 2.5 | 7.5, 10.3, 14.0 for PAL=1.9 | |

12.3, 16.7, 22.5 for PAL=2.2 | |||

17.0, 23.0, 31.0 for PAL=2.5 |

#### 5.3 Adequate Macronutrients Distribution Range (AMDR) / Adequate Intake (AI)

#### 5.4 Calculation Procedure

- ○
- Use of 0.25-year increments (trimesters, 91.5 days) within the age range from 0 to 90 years.
- ○
- Use of growth data from CDC to obtain body size and weight as function of age for both genders and 50th percentile.
- ○
- ○
- ○
- Apply Eq. (27) to obtain entropy generated.
- ○
- Generate tables and charts for entropy generation per trimester and calculate cumulative entropy generated.
- ○
- Express all data as per unit total body mass.
- ○
- Though calculations are made for different activity levels, it is assumed that the Low Active (LA) PAL represents the best average of the healthy individual.

^{th}percentile (height and weight) population, low active physical activity level (LA PAL) and the average value of the recommended AMDR/AI range is considered the base case. The average lifespan for the U.S. population is obtained from the CIA World Fact Book [29], and is found to be 74.63 years for males and 80.36 years for females (2004).

**Table 3.**EER (kcal/day) Correlation for average male individuals [28].

0-3 months (89 x weight of infant[kg] -100) + 175 | ||||||

4-6 months (89 x weight of infant[kg] -100) + 56 | ||||||

7-12 months (89 x weight of infant[kg] -100) + 22 | ||||||

13-35 months (89 x weight of child[kg] -100) + 20 | ||||||

EER for Boys 3 through 8 years, EER = TEE + Energy Deposition | ||||||

EER = 88.5 - 61.9 x Age[y] + PA x (26.7 x Weight[kg] + 903 x Height[m]) + 20 | ||||||

Where PA is the physical activity coefficient: | ||||||

PA = 1.00 if PAL is estimated to be > 1.0 < 1.4 (Sedentary) | ||||||

PA = 1.13 if PAL is estimated to be > 1.4 < 1.6 (Low Active) | ||||||

PA = 1.26 if PAL is estimated to be > 1.6 < 1.9 (Active) | ||||||

PA = 1.42 if PAL is estimated to be > 1.9 < 2.5 (Very Active) | ||||||

EER for Boys 9 through 18 Years, EER = TEE + Energy Deposition | ||||||

EER = 88.5 - 61.9 x Age[y] + PA x (26.7 x Weight[kg] + 903 x Height[m]) + 25 | ||||||

Where PA is the physical activity coefficient, equal to the 3-8 years range. | ||||||

EER for Total Energy Expenditure (TEE) for Men 19 years and older | ||||||

EER = 662 - 9.53 x Age [y] + PA x (15.91 x Weight [kg] + 539.6 x Height [m]) | ||||||

Where PA is the physical activity coefficient: | ||||||

PA = 1.00 if PAL is estimated to be > 1.0 < 1.4 (Sedentary) | ||||||

PA = 1.11 if PAL is estimated to be > 1.4 < 1.6 (Low Active) | ||||||

PA = 1.25 if PAL is estimated to be > 1.6 < 1.9 (Active) | ||||||

PA = 1.48 if PAL is estimated to be > 1.9 < 2.5 (Very Active) |

**Table 4.**AMDR/AI Data [28].

Carbohydrates | Total Fats | Protein | |||||

RDA/AI | AMDR | RDA/AI | AMDR | RDA/AI | AMDR | ||

Years | (g/d) | (%) | (g/d) | (%) | (g/d) | (%) | |

Infants | |||||||

0 | 0.5 | 60 | 31 | 9.1 | |||

0.5 | 1 | 95 | 30 | 13.5 | |||

Children | |||||||

1 | 3 | 130 | 45-65 | 30-40 | 13 | 5-20 | |

4 | 8 | 130 | 45-65 | 25-35 | 19 | 10-30 | |

Male | |||||||

9 | 13 | 130 | 45-65 | 25-35 | 34 | 10-30 | |

14 | 18 | 130 | 45-65 | 25-35 | 52 | 10-30 | |

19 | 30 | 130 | 45-65 | 20-35 | 56 | 10-35 | |

31 | 50 | 130 | 45-65 | 20-35 | 56 | 10-35 | |

50 | 70 | 130 | 45-65 | 20-35 | 56 | 10-35 | |

>70 | 130 | 45-65 | 20-35 | 56 | 10-35 | ||

Female | |||||||

9 | 13 | 130 | 45-65 | 25-35 | 34 | 10-30 | |

14 | 18 | 130 | 45-65 | 25-35 | 46 | 10-30 | |

19 | 30 | 130 | 45-65 | 20-35 | 46 | 10-35 | |

31 | 50 | 130 | 45-65 | 20-35 | 46 | 10-35 | |

50 | 70 | 130 | 45-65 | 20-35 | 46 | 10-35 | |

>70 | 130 | 45-65 | 20-35 | 46 | 10-35 | ||

Pregnancy | |||||||

≤18 | 175 | 45-65 | 20-35 | 71 | 10-35 | ||

19 | 30 | 175 | 45-65 | 20-35 | 71 | 10-35 | |

31 | 50 | 45-65 | 20-35 | 71 | 10-35 | ||

Lactation | |||||||

≤18 | 210 | 45-65 | 20-35 | 71 | 10-35 | ||

19 | 30 | 210 | 45-65 | 20-35 | 71 | 10-35 | |

31 | 50 | 210 | 45-65 | 20-35 | 71 | 10-35 |

## 6. Results

#### 6.1 Base case

^{th}percentile (weight and height) and low active PAL.

^{3}/min which corresponds to a heat release rate of 8.4 W (or 2.4 W/kg body mass) [32] which is higher compared to an adult ( ≈ 0.8 W/kg).

Av. Lifespan [5] | 74.63 | 80.36 |

_{m,life}at 74.63 years age is 11,508 kJ/kg-K, while females at 80.36 years age have σ

_{m,life}of 11,299 kJ/-K. These values are close together (less than 2% difference), enforcing the idea of a fixed amount of entropy generation per unit body mass during a lifetime. We will assume σ

_{m,life}= 11,404 kJ/kg-K for subsequent parametric studies on a human lifespan.

_{m,life}for Beagle bitches to be 3,874 kJ/kg-K. We were expecting the lifetime specific entropy generation to be similar among species, but for a dog to reach the same level of entropy generation as a human would require a life span of 43 years, which is 3 times the average of 13.5 years typical of Beagles. However, some similarities with the human data were observed as puppies generate 3 times more entropy than adult dogs. More research in this area is necessary to fully understand how the concept translates between different biological species.

#### 6.2 Effect of physical activity level

Case | Male | Female | ||

Sedentary | 85.05 | 95.75 | ||

LA - | Low Active (base) | 73.78 | 81.61 | |

A - | Active | 63.78 | 69.53 |

#### 6.3 Gravitational Effects

## 7. Conclusions

- ○
- The basic laws of thermodynamics were applied to biological systems, using a combination of laws of thermodynamics and available information from biochemistry literature and updated CDC databases. Entropy generated was determined for metabolism of the typical components of the human diet, and total entropy generation was estimated through numerical integration for the average population.
- ○
- Data on average lifespan was used to obtain lifetime limit entropy, which was found equal to 11,404 kJ/kg-K. This value of entropy predicts life span within 1.5% of the life span from literature (predicted: 73.78 and 81.61 years; Literature: 74.63 and 80.36 years; males and females respectively).
- ○
- ○
- The higher the specific metabolic rate (kW/kg), the higher the specific entropy generation rate (kW/kg), and the faster we approach the specific entropy generation limit over a lifetime (kJ/kg K).
- ○
- The entropy generation rate depends upon the type of ration fed to BS.
- ○
- When a non-zero gravity environment is approximated as a change from low active to sedentary PAL, it is possible to predict changes on the lifespan of astronauts based on the exposition time to the weightless condition. It was found that a male astronaut will extend his life span 1 year for every 9.5 years he spent in space. For a female astronaut, this time was estimated as 7.5 years.
- ○
- The present approach presumes that the ATP acts as work currency, generating no entropy.
- ○
- While the present analysis for entropy generation is conducted considering the human as a whole system, the analysis can be extended to determine the entropy generation for each organ of the system (heart, kidney, liver etc) and to determine which degenerates rapidly as long as metabolic rates and metabolic efficiencies of the organ are known.

## Acknowledgments

## Appendix A - Terminology

#### Basal Metabolic Rate (BMR)

#### Total Energy Expenditure (TEE)

#### Physical Activity Level (PAL)

#### Estimated Energy Requirements (EER)

#### Databases

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**MDPI and ACS Style**

Silva, C.; Annamalai, K. Entropy Generation and Human Aging: Lifespan Entropy and Effect of Physical Activity Level. *Entropy* **2008**, *10*, 100-123.
https://doi.org/10.3390/entropy-e10020100

**AMA Style**

Silva C, Annamalai K. Entropy Generation and Human Aging: Lifespan Entropy and Effect of Physical Activity Level. *Entropy*. 2008; 10(2):100-123.
https://doi.org/10.3390/entropy-e10020100

**Chicago/Turabian Style**

Silva, Carlos, and Kalyan Annamalai. 2008. "Entropy Generation and Human Aging: Lifespan Entropy and Effect of Physical Activity Level" *Entropy* 10, no. 2: 100-123.
https://doi.org/10.3390/entropy-e10020100