- Email: [email protected]
A novel model for allometric scaling laws for different organs
Chaos, Solitons and Fractals 27 (2006) 1108–1114 www.elsevier.com/locate/chaos
A novel model for allometric scaling laws for different organs Ji-Huan He a
a,*
, Zhende Huang
b
College of Science, Donghua University, 1882 Yan’an Xilu Road, Shanghai 200051, China b Bioinformatics Research Centre, Donghua University, Shanghai 200051, China Accepted 12 April 2005
Communicated by Prof. Helal
Abstract ðDþN =6Þ=ðDþ1Þ The power function of metabolic rate scaling for an organ is established as Borgan T organ , where Borgan is the metabolic rate of the organ, Torgan is its mass, D is its fractal dimension of the total cell boundary, and N is its cellÕs degree of freedom of motion. This prediction agrees quite well with the experimental data for the brain, liver, heart, and kidneys, and it explains very well the reason why the maximal metabolic rate induced by exercise scales with M0.86 rather then M0.75. Ó 2005 Elsevier Ltd. All rights reserved.
1. Introduction Allometric scaling laws in biology have attracted considerable attention after the insightful works by West et al. [1,2], Darveau et al. [3], and others [4–10]. Allometric analysis is also a powerful mathematical tool in engineering [11–18]. Wang et al. [19] reconstructed KleiberÕs law [20] at the organ–tissue level, and they found that four metabolically active organs, brain, liver, kidneys and heart, have high specific resting metabolic rates when compared with the remaining less-active tissues, such as skeletal muscle, adipose tissue, bone and skin. Brain, liver, kidneys and heart together account for 60% of resting energy expenditure in humans, even though the four organs represent <6% of body mass. At present and despite decades of concentrated effort, we do not have a rational theory which could explain different scaling laws for different organs. Though there exist dueling theories [21–30] to explain allometric scaling laws in biology, it is intriguing that, as far as the present authors know, no general model in a cell level exists to mechanistically explain scaling laws. A fresh approach, therefore, is still much-needed, and we conclude that cell fractal is the key, and we name the new branch as cytofractalogy, which is different from that suggested by West et al. [1,2].
*
Corresponding author. Tel.: +86 21 62379917; fax: +86 21 62378926. E-mail address: [email protected] (J.-H. He).
0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.04.082
J.-H. He, Z. Huang / Chaos, Solitons and Fractals 27 (2006) 1108–1114
1109
2. Cytofractalogy: fractal approach to allometrical scaling laws in biology Now we consider a multicellular organ, and assume that there are n basal cells with characteristic (or typical) radius r. The metabolic rate B of the organ scales linearly with respect to its total surface: B nr2 ¼ A;
ð1Þ
where A is the total surface of all cells in the organ. The total surface of cell boundaries is of fractal construction; we, therefore, have A r D;
ð2Þ
where D is the fractal dimension of the total cell surface. Due to the smallness of the cells, we can assume that cells are space-filling (thus, for example, the nutrition can reach each cell in the organism). Consequently, the fractal dimension, D, tends to 3, i.e., D = 3 for most three-dimensional organs [8]. Combination of (1) and (2) leads to the following scaling relationship: n r.
ð3Þ
The mass M of the organ scales linearly with respect to its total volume of cells: M nr3 .
ð4Þ
Note that (4) is not valid for bone tissue. From the scaling relations (1), (3), and (4), we obtain the well-known KleiberÕs 3/4 allometric scaling law [8], which reads B M 3=4 .
ð5Þ
In 1977, Blum [31] suggested that the 3/4-law can be understood by a four-dimensional approach. In D-dimension space, the ‘‘area’’ A of the hypersurface enclosing an D-dimensional hypervolume scales like A V (D1)/D. When D = 4, we have A V3/4, a four-dimensional construction: B M ðD1Þ=D .
ð6Þ
In view of E1 NaschieÕs E-infinity theory [32–38], we modify the scaling law (6) in the form B M ðDH 1Þ=DH ¼ M 0.764 ;
ð7Þ
(1)
where DH is the expectation value of the Hausdorff dimension of e [32–38]: 3 X 1 1 n/n ¼ ¼ 4.23606 ; ð8Þ DH ¼ Dim eð1Þ ¼ H / n¼0 pffiffiffi where / ¼ ð 5 1Þ=2 is the golden mean, which is the building stone of E1 NaschieÕs e(1) network [32–38]. Note that D = 4 is the dimension of classical spacetime upon which EinsteinÕs theory is based. This is only an approximation of the true geometry of the universe in the large. Similarly D = 4 is used in four-dimensional life [2], and we argue that E1 NaschieÕs theory might lead to an accurate prediction of metabolic rate. The main application of the E-infinity theory shows miraculous scientific exactness, especially in determining theoretically coupling constants and the mass spectrum of the standard model of elementary particles [32–38]. Now we consider a leaf of a plant or a hepatic cellular plate (see Fig. 1). For an approximate two-dimensional (2-D) organ, the fractal dimension, D, in (2) tends to 2, i.e., D = 2 [8]; as a result, we have nr2 = A r2, leading to the result: n r0 .
ð9Þ
We, therefore, obtain a 2/3-law for 2-D organs [8]: B M 2=3 .
ð10Þ
It is obvious that the Rubner 2/3-law [39] is valid for 2-D lives, and KleiberÕs law for 3-D lives as predicted by He and Chen [8]. If the exponent in (10) is linked to the gold mean, then Rubner 2/3-law can be modified as B M / ¼ M 0.618 ð11Þ for two-dimensional organs. It is interesting to note the scaling relationship n r (see (3)), which is valid for 3-D organs. Remember that r is a space dimension, so the number of cells in an organ endows another life dimension [9]. If a cell is isolated from a heart of
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Fig. 1. Diagram showing the structure of liver lobule, which is composed of many slice-like hepatic cellular plates, each hepatic plate being one- to two-cell thick; almost no relative motion exists, so the motionÕs degree of freedom is N = 0. If we use a box with characteristic length, L, to enclose cells and there is one cell, the total surface area is A, where A is the surface area of one cell; the total surface area becomes 4A if we use a box with length of 2L; consequently, the fractal dimension of the total cell surfaces is D = ln(4A/A)/ ln2 = 2.
a rat, or heart cells are cultivated on a plane (D = 2), no life dimension is endowed. Therefore, these cells have no life functional characteristic. However, if sufficient number of heart cells are cultivated and accumulated together in three dimensions, the isolated cardiac cell begins to beat [40], and the life dimension is endowed. We have also illustrated that n r0 (see (9)) for 2-D lives. It is really very interesting and gives much insight into explanation of transplantation of 2-D tissues, for example, most plants, liver and cornea. The life characteristic of the 2-D organs does not seriously depend upon the number of cells involved. This is the reason why a destroyed leaf can still make photosynthesis, and can replace the dead cells, whereas the adult human brain cannot replace lost neurons [41]. In adult rodents, however, one region of the brain, the subventricular zone, generates thousands of neurons every day [42]. There could be important implications for future developments in neuroregenerative therapy by constraining brain cells in a niche of unique organization in the adult human brain to be of 2-D tissue.
3. Maximal metabolic rate Cytofractalogy elucidates the otherwise intractable problem of changes in the exponent between states of rest and of exercise. When an animal exercises intensely, each cell in the body needs more nutrition and oxygen to provide the requisite energy output. This can be achieved by increase in both lung and the area available for exchange of resources with the environment. At rest, body cells are approximately spherical. Now the metabolic rate induced by exercise scales as follows: Bmotion nr2 E;
ð12Þ
where E is the kinetic energy of a cell, which can be expressed as E ¼ 12qu21 þ 12qu22 þ 12qu23 þ 12I 1 x21 þ 12I 2 x22 þ 12I 3 x23 ;
ð13Þ
where q is the density of the cell, u1, u2, and u3 are velocities in the x-, y- and z-directions, respectively; I1, I2 and I3 are inertia coefficients, and x1, x2 and x3 are the angular velocities.
J.-H. He, Z. Huang / Chaos, Solitons and Fractals 27 (2006) 1108–1114
1111
The potential energy and kinetic energy of a cell will change during motion, but the total energy is assumed to be conserved in the procedure: U + E = constant, where U is the potential energy. The potential energy scales like U rd, or E rd, where d is an unknown scaling exponent. Eq. (12), therefore, can be re-written in the form: Bmotion nr2þd .
ð14Þ
The value, 2 + d, is actually the fractal dimension of a cell in motion. By a similar manipulation as illustrated above, we can easily obtain the following relationship: Bmotion M ðDþdÞ=ðDþ1Þ .
ð15Þ
If we consider an extreme condition when the body is in an extreme exercise, each cell tends to enlarge its surface in order to obtain as much energy from its environment. As a result, there exists no space between cells, which are accumulated to form an integral construction. The maximal metabolic rate, under such an extreme case, scales linearly with blood volume or lung volume: Bmax V B
ð16Þ
Bmax V L ;
ð17Þ
or
where VB is the total blood volume and VL is the total lung volume. The observed data [1] show that VB M1.00 and VL M1.05, according to which we have Bmax M.
ð18Þ
Comparison of (15) and (18) yields the maximal value of d: dmax ¼ 1.
ð19Þ
The constant, d, depends upon its motion pattern. Generally, a motion can be decomposed into six components: three velocities in the x-, y- and z-directions, and three angular velocities, so we set d ¼ N=6;
ð20Þ
where N is the degree of freedom of motion. A cell is at rest when N = 0; and when N = 6, a cell moves in the x-, y- and z-directions and spins around the x-, y- and z-axes. Finally, we have Bmotion M ðDþN =6Þ=ðDþ1Þ .
ð21Þ
When an animal exercises seriously, each rest cell begins to move in all directions. Consequently, the average degree of freedom of motion is 3, or N = 3. From (21) we have Bmotion M0.875, which agrees very well with TaylorÕs data [43]: M 0.86 . B motion
It is well known that cells in, for example, heart are in motion when the body is at rest. We can obtain the following scaling relationship for an organ by similar derivations as illustrated above, which reads: ðDþN =6Þ=ðDþ1Þ Borgan T organ ;
ð22Þ
where Borgan is the metabolic rate of an organ and Torgan is its mass. Although the model (22) makes few simplifying assumptions, its power rests on the fundamental physical and biological principles as well as a realistic feature of biomechanics. It accurately predicts scaling exponents (see Table 1).
Table 1 Organ–tissue level metabolic rate–mass relationship and our prediction 0.60 Bliver M body
B–M relation [19]
T–M relation [19] T M 0.87
B–T relation observed B M 0.68
0.62 Bbrain M body
0.76 T brain M body
0..81 Bbrain T brain
Bkidneys Bheart
0.77 M body
0.86 M body
Bmotion
0.86 M body
liver
body
T kidneys T heart
0.85 M body
0.98 M body
liver
liver
Bkidneys Bheart
0.90 T kidneys
0.87 T heart
Predicted by (22) D = 2, N = 0; 0.67 D = 3, N = 3; 0.87 D = 3, N = 4; 0.91 D = 3, N = 3; 0.87 D = 3, N = 3; 0.87
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It is a known fact that animals at rest or undergoing starvation or at hibernation display low or basal metabolic rates, while displaying higher levels under conditions of high nutrient availability or in motion [3]. This phenomenon is consistent with our prediction (22) where the degree of freedom of motion, N, is taken into account. For a vegetative state patient, or a person in the condition of slow wave sleep, the cellÕs degree of freedom of motion in human brain reduces up to zero, i.e., N ! 0, and we predict 0.87 0.75 .87 ¼ 39.9% for man with 70 kg. ðT brain T brain Þ=T 0brain
A 40% decrease in the global cerebral metabolism, whereas the observed decrease is about 45% for comatose patients [44,45] and 44% for slow wave sleep [44,46,47].
4. Discussion It is mysterious that a liver in mammals acts like a leaf of plant which scales like B M2/3. The liver contains 50,000– 100,000 individual lobules. The liver lobule is constructed around a central vein. The lobule itself is composed of many slice-like hepatic cellular plates like many leaves piled up, each hepatic plate being one- or two-cell thick. Therefore, the liver obeys the same scaling law as a leaf of plant (see Fig. 1), and it is of two-dimensional construction (D = 2). The pressure in portal vein leading into the liver averages about 9 mm Hg, and the pressure in the hepatic vein leading from the liver into the vena cava averages almost exactly 0 mm Hg. This small pressure difference shows that the resistance to blood flow through the liver sinusoids is normally low, resulting in almost no relative motion of the liver lobules (N = 0). 0.90 Our model predicts, for example, accurately how kidneys must obey the law: Bkidneys T kidneys . The kidney contains about 1,000,000 nephrons. Each nephron is composed of a glomerulus through which fluid is filtered from the blood. Blood enters the glomerulus through the afferent arteriole and then leaves through the efferent arteriole [48,49]. Pulsing blood pressure causes each nephron to move in the x-, y-, and z-directions and spin around one direction periodically (see Fig. 2); accordingly, the total degree of freedom of motion for each nephron is 4, i.e., N = 4.
Fig. 2. SEM micrograph of glomerulus in the kidney. Obviously, it is of 3-D construction (D = 3). The blood pressure acts on the glomerulus in all directions, the force system acting on the glomerulus can be equivalently reduced to a system of three forces (F1, F2, and F3) in the x-, y-, and z-directions and one couple (M ). So the motionÕs degree of freedom is N = 3 + 1 = 4.
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1113
5. Conclusion We conclude that our cytofractalogy approach can successfully explain the different scaling laws for different organs. Of course, if El NaschieÕs E-infinity theory is considered, the scaling relationships (21) and (22) can be modified as follows: Bmotion M ðDH þN =6ÞðDH þ1Þ ; Borgan
H þN =6Þ=ðDH þ1Þ T ðD . organ
ð23Þ ð24Þ
Application of the present theory to biology will be discussed in future papers.
Acknowledgments The authors thank Professor Hari M. Srivastava of University of Victoria, Canada for his kind help in improving English, the authors also thank Jinsong Yao of Shanghai Jiaotong University for providing Figs. 1 and 2, and Refs. [48,49], and for taking part in the discussion of the explanation of liver and kidney. In this work J.-H. He proposed the original idea and made all theoretic deduction; Z.D. Huang took part in discussion of explanation of liver and kidneys. J.-H. He also thanks Shi-Huang Shao, Juan Zhang, Hong-Mei Liu, Yu-Qin Wan, Yan Yang, and Qian Guo for their helpful discussions.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]
West GB, Brown JH, Enquist BJ. Science 1997;276:122–6. West GB, Brown JH, Enquist BJ. Science 1999;284:1677–9. Darveau CA, Suarez RK, Andrews RD, Hochachka PW. Nature 2002;417:166–70. Banavar JR, Damuth J, Maritan A, Rinaldo A. Proc Nat Acad Sci USA 2002;99(16):10506–9. Kuikka JT. Int J Nonlin Sci Numer Simul 2003;4:317–28. Kuikka JT. Int J Nonlin Sci Numer Simul 2002;3:81–8. Kuikka JT. Int J Nonlin Sci Numer Simul 2005;6:107–12. He JH, Chen H. Int J Nonlin Sci Numer Simul 2003;4:429–32. He JH, Zhang J. Fifth dimension of life and the 4/5 allometric scaling law for human brain. Cell Biol Int 2004;28(11):809–15. He JH. A brief review on allometric scaling in biology. Lect Notes Comput Sci 2004;3314:652–8. Xiao X, Yao JS, Shao SH, et al. Particular symmetry in RNA sequence of SARS and the origin of SARS coronavirus. Int J Nonlin Sci Numer Simul 2005;6(2):181–6. Zhang J. Int J Nonlin Sci Numer Simul 2004;5(1):37–44. Zhang J, Qu JP. Int J Nonlin Sci Numer Simul 2004;5(1):97–8. He JH. Allometric scaling law in conductive polymer. Polymer 2004;45(26):9067–70. He JH, Wan YQ, Yu MY. Allometric scaling and instability in electrospinning. Int J Nonlin Sci Numer Simul 2004;5(3):243–52. He JH, Wan YQ, Yu JY. Application of vibration technology to polymer electrospinning. Int J Nonlin Sci Numer Simul 2004;5(3):253–62. He JH. Mysterious pi and a possible link to DNA sequencing. Int J Nonlin Sci Numer Simul 2004;5(3):263–74. He JH, Wan YQ. Allometric scaling for voltage and current in electrospinning. Polymer 2004;45(19):6731–4. Wang ZM, OÕConnort TP, Heshka S, Heymsfield SB. J Nutr 2001;131:2967–70. Kleiber M. Hilgardia 1932;6:315–53. Dodds PS, Rothman DH, Weitz JS. J Theor Biol 2001;209(1):9–27. Riisgard HU. Ecol Lett 1998;1(2):71–3. White CR, Seymour RS. Proc Nat Acad Sci USA 2003;100(7):4046–9. West G, Woodruff WH, Brown JH. Proc Nat Acad Sci USA 2002;99:2473–8. Damuth J. Proc Nat Acad Sci USA 2001;98(5):2113–4. Weibel ER. Nature 2002;417:131–2. Beuchat CA. Science 1997;27(8):369–73. Dawson TH. Science 1998;281:751. Banavar JR, Damuth J, Maritan A, Rinaldo A. Nature 2003;421(6924):713–4. West GB, Savage VM, Gillooly J, Enquist BJ, Woodruff WH, Brown JH. Nature 2003;421:713. Blum JJ. On the geometry of four-dimensions and the relationship between metabolism and body mass. J Theor Biol 1997;64:599–601. El Naschie MS. Int J Nonlin Sci Numer Simul 2005;6(2):94–7.
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[41] [42] [43] [44] [45] [46] [47] [48] [49]
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El Naschie MS. Int J Nonlin Sci Numer Simul 2004;5(3):191–8. El Naschie MS. Chaos, Solitons & Fractals 2004;19:209–36. El Naschie MS. Chaos, Solitons & Fractals 2004;22:495–511. El Naschie MS. Chaos, Solitons & Fractals 2005;24:653–7. El Naschie MS. Chaos, Solitons & Fractals 2004;20:649–54. He JH. Int J Nonlin Sci Numer Simul 2005;6(2):93–4. Rubner M. Zeitschr Biol 1883;19:536–62. NASA microgravity research highlights, Advancing Heart Research, Reprinted from the Spring 2000 issue of Microgravity News. Detailed information is published in Amer J Physiol 277 (Heart Circ Physiol 46), H433–H444 (1999) and Biotech Bioengrg 64, 580–589 (1999). Rakic P. Nature 2004;427:685–6. Sanal N, Tramontin AD, Qulnones-Hinojosa A, Barbaro NM, et al. Nature 2004;427:740–4. Taylor CR et al. Resp Physiol 1981;44:25–37. Laureys S, Antoine S, Boly M, et al. Brain function in the vegetative state. Acta Neurol Belg 2002;102:177–85. Levy DE, Sidtis JJ, Rottenberg DA, et al. Differences in cerebral blood flow and glucose utilization in vegetative versus locked-in patients. Ann Neurol 1987;22:673–82. Buchsbaum MS, Gillin JC, Wu J, et al. Regional cerebral glucose metabolic rate in human sleep assessed by positron emission tomography. Life Sci 1989;45:1349–56. Maque P, Peters J, Aerts J, et al. Functional neuroanatomy of human rapid-eye-movement sleep and dreaming. Nature 1996;383:163–6. Arthur C, Guyton MD. Human physiology and mechanism of disease. Philadelphia: WB Saunders Company; 1992 [p. 204–6]. Arthur C, Guyton MD. Textbook of medical physiology. Philadelphia: W.B. Saunders Company; 1991 [p. 771–2].
A novel model for allometric scaling laws for different organs Ji-Huan He a
a,*
, Zhende Huang
b
College of Science, Donghua University, 1882 Yan’an Xilu Road, Shanghai 200051, China b Bioinformatics Research Centre, Donghua University, Shanghai 200051, China Accepted 12 April 2005
Communicated by Prof. Helal
Abstract ðDþN =6Þ=ðDþ1Þ The power function of metabolic rate scaling for an organ is established as Borgan T organ , where Borgan is the metabolic rate of the organ, Torgan is its mass, D is its fractal dimension of the total cell boundary, and N is its cellÕs degree of freedom of motion. This prediction agrees quite well with the experimental data for the brain, liver, heart, and kidneys, and it explains very well the reason why the maximal metabolic rate induced by exercise scales with M0.86 rather then M0.75. Ó 2005 Elsevier Ltd. All rights reserved.
1. Introduction Allometric scaling laws in biology have attracted considerable attention after the insightful works by West et al. [1,2], Darveau et al. [3], and others [4–10]. Allometric analysis is also a powerful mathematical tool in engineering [11–18]. Wang et al. [19] reconstructed KleiberÕs law [20] at the organ–tissue level, and they found that four metabolically active organs, brain, liver, kidneys and heart, have high specific resting metabolic rates when compared with the remaining less-active tissues, such as skeletal muscle, adipose tissue, bone and skin. Brain, liver, kidneys and heart together account for 60% of resting energy expenditure in humans, even though the four organs represent <6% of body mass. At present and despite decades of concentrated effort, we do not have a rational theory which could explain different scaling laws for different organs. Though there exist dueling theories [21–30] to explain allometric scaling laws in biology, it is intriguing that, as far as the present authors know, no general model in a cell level exists to mechanistically explain scaling laws. A fresh approach, therefore, is still much-needed, and we conclude that cell fractal is the key, and we name the new branch as cytofractalogy, which is different from that suggested by West et al. [1,2].
*
Corresponding author. Tel.: +86 21 62379917; fax: +86 21 62378926. E-mail address: [email protected] (J.-H. He).
0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.04.082
J.-H. He, Z. Huang / Chaos, Solitons and Fractals 27 (2006) 1108–1114
1109
2. Cytofractalogy: fractal approach to allometrical scaling laws in biology Now we consider a multicellular organ, and assume that there are n basal cells with characteristic (or typical) radius r. The metabolic rate B of the organ scales linearly with respect to its total surface: B nr2 ¼ A;
ð1Þ
where A is the total surface of all cells in the organ. The total surface of cell boundaries is of fractal construction; we, therefore, have A r D;
ð2Þ
where D is the fractal dimension of the total cell surface. Due to the smallness of the cells, we can assume that cells are space-filling (thus, for example, the nutrition can reach each cell in the organism). Consequently, the fractal dimension, D, tends to 3, i.e., D = 3 for most three-dimensional organs [8]. Combination of (1) and (2) leads to the following scaling relationship: n r.
ð3Þ
The mass M of the organ scales linearly with respect to its total volume of cells: M nr3 .
ð4Þ
Note that (4) is not valid for bone tissue. From the scaling relations (1), (3), and (4), we obtain the well-known KleiberÕs 3/4 allometric scaling law [8], which reads B M 3=4 .
ð5Þ
In 1977, Blum [31] suggested that the 3/4-law can be understood by a four-dimensional approach. In D-dimension space, the ‘‘area’’ A of the hypersurface enclosing an D-dimensional hypervolume scales like A V (D1)/D. When D = 4, we have A V3/4, a four-dimensional construction: B M ðD1Þ=D .
ð6Þ
In view of E1 NaschieÕs E-infinity theory [32–38], we modify the scaling law (6) in the form B M ðDH 1Þ=DH ¼ M 0.764 ;
ð7Þ
(1)
where DH is the expectation value of the Hausdorff dimension of e [32–38]: 3 X 1 1 n/n ¼ ¼ 4.23606 ; ð8Þ DH ¼ Dim eð1Þ ¼ H / n¼0 pffiffiffi where / ¼ ð 5 1Þ=2 is the golden mean, which is the building stone of E1 NaschieÕs e(1) network [32–38]. Note that D = 4 is the dimension of classical spacetime upon which EinsteinÕs theory is based. This is only an approximation of the true geometry of the universe in the large. Similarly D = 4 is used in four-dimensional life [2], and we argue that E1 NaschieÕs theory might lead to an accurate prediction of metabolic rate. The main application of the E-infinity theory shows miraculous scientific exactness, especially in determining theoretically coupling constants and the mass spectrum of the standard model of elementary particles [32–38]. Now we consider a leaf of a plant or a hepatic cellular plate (see Fig. 1). For an approximate two-dimensional (2-D) organ, the fractal dimension, D, in (2) tends to 2, i.e., D = 2 [8]; as a result, we have nr2 = A r2, leading to the result: n r0 .
ð9Þ
We, therefore, obtain a 2/3-law for 2-D organs [8]: B M 2=3 .
ð10Þ
It is obvious that the Rubner 2/3-law [39] is valid for 2-D lives, and KleiberÕs law for 3-D lives as predicted by He and Chen [8]. If the exponent in (10) is linked to the gold mean, then Rubner 2/3-law can be modified as B M / ¼ M 0.618 ð11Þ for two-dimensional organs. It is interesting to note the scaling relationship n r (see (3)), which is valid for 3-D organs. Remember that r is a space dimension, so the number of cells in an organ endows another life dimension [9]. If a cell is isolated from a heart of
1110
J.-H. He, Z. Huang / Chaos, Solitons and Fractals 27 (2006) 1108–1114
Fig. 1. Diagram showing the structure of liver lobule, which is composed of many slice-like hepatic cellular plates, each hepatic plate being one- to two-cell thick; almost no relative motion exists, so the motionÕs degree of freedom is N = 0. If we use a box with characteristic length, L, to enclose cells and there is one cell, the total surface area is A, where A is the surface area of one cell; the total surface area becomes 4A if we use a box with length of 2L; consequently, the fractal dimension of the total cell surfaces is D = ln(4A/A)/ ln2 = 2.
a rat, or heart cells are cultivated on a plane (D = 2), no life dimension is endowed. Therefore, these cells have no life functional characteristic. However, if sufficient number of heart cells are cultivated and accumulated together in three dimensions, the isolated cardiac cell begins to beat [40], and the life dimension is endowed. We have also illustrated that n r0 (see (9)) for 2-D lives. It is really very interesting and gives much insight into explanation of transplantation of 2-D tissues, for example, most plants, liver and cornea. The life characteristic of the 2-D organs does not seriously depend upon the number of cells involved. This is the reason why a destroyed leaf can still make photosynthesis, and can replace the dead cells, whereas the adult human brain cannot replace lost neurons [41]. In adult rodents, however, one region of the brain, the subventricular zone, generates thousands of neurons every day [42]. There could be important implications for future developments in neuroregenerative therapy by constraining brain cells in a niche of unique organization in the adult human brain to be of 2-D tissue.
3. Maximal metabolic rate Cytofractalogy elucidates the otherwise intractable problem of changes in the exponent between states of rest and of exercise. When an animal exercises intensely, each cell in the body needs more nutrition and oxygen to provide the requisite energy output. This can be achieved by increase in both lung and the area available for exchange of resources with the environment. At rest, body cells are approximately spherical. Now the metabolic rate induced by exercise scales as follows: Bmotion nr2 E;
ð12Þ
where E is the kinetic energy of a cell, which can be expressed as E ¼ 12qu21 þ 12qu22 þ 12qu23 þ 12I 1 x21 þ 12I 2 x22 þ 12I 3 x23 ;
ð13Þ
where q is the density of the cell, u1, u2, and u3 are velocities in the x-, y- and z-directions, respectively; I1, I2 and I3 are inertia coefficients, and x1, x2 and x3 are the angular velocities.
J.-H. He, Z. Huang / Chaos, Solitons and Fractals 27 (2006) 1108–1114
1111
The potential energy and kinetic energy of a cell will change during motion, but the total energy is assumed to be conserved in the procedure: U + E = constant, where U is the potential energy. The potential energy scales like U rd, or E rd, where d is an unknown scaling exponent. Eq. (12), therefore, can be re-written in the form: Bmotion nr2þd .
ð14Þ
The value, 2 + d, is actually the fractal dimension of a cell in motion. By a similar manipulation as illustrated above, we can easily obtain the following relationship: Bmotion M ðDþdÞ=ðDþ1Þ .
ð15Þ
If we consider an extreme condition when the body is in an extreme exercise, each cell tends to enlarge its surface in order to obtain as much energy from its environment. As a result, there exists no space between cells, which are accumulated to form an integral construction. The maximal metabolic rate, under such an extreme case, scales linearly with blood volume or lung volume: Bmax V B
ð16Þ
Bmax V L ;
ð17Þ
or
where VB is the total blood volume and VL is the total lung volume. The observed data [1] show that VB M1.00 and VL M1.05, according to which we have Bmax M.
ð18Þ
Comparison of (15) and (18) yields the maximal value of d: dmax ¼ 1.
ð19Þ
The constant, d, depends upon its motion pattern. Generally, a motion can be decomposed into six components: three velocities in the x-, y- and z-directions, and three angular velocities, so we set d ¼ N=6;
ð20Þ
where N is the degree of freedom of motion. A cell is at rest when N = 0; and when N = 6, a cell moves in the x-, y- and z-directions and spins around the x-, y- and z-axes. Finally, we have Bmotion M ðDþN =6Þ=ðDþ1Þ .
ð21Þ
When an animal exercises seriously, each rest cell begins to move in all directions. Consequently, the average degree of freedom of motion is 3, or N = 3. From (21) we have Bmotion M0.875, which agrees very well with TaylorÕs data [43]: M 0.86 . B motion
It is well known that cells in, for example, heart are in motion when the body is at rest. We can obtain the following scaling relationship for an organ by similar derivations as illustrated above, which reads: ðDþN =6Þ=ðDþ1Þ Borgan T organ ;
ð22Þ
where Borgan is the metabolic rate of an organ and Torgan is its mass. Although the model (22) makes few simplifying assumptions, its power rests on the fundamental physical and biological principles as well as a realistic feature of biomechanics. It accurately predicts scaling exponents (see Table 1).
Table 1 Organ–tissue level metabolic rate–mass relationship and our prediction 0.60 Bliver M body
B–M relation [19]
T–M relation [19] T M 0.87
B–T relation observed B M 0.68
0.62 Bbrain M body
0.76 T brain M body
0..81 Bbrain T brain
Bkidneys Bheart
0.77 M body
0.86 M body
Bmotion
0.86 M body
liver
body
T kidneys T heart
0.85 M body
0.98 M body
liver
liver
Bkidneys Bheart
0.90 T kidneys
0.87 T heart
Predicted by (22) D = 2, N = 0; 0.67 D = 3, N = 3; 0.87 D = 3, N = 4; 0.91 D = 3, N = 3; 0.87 D = 3, N = 3; 0.87
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It is a known fact that animals at rest or undergoing starvation or at hibernation display low or basal metabolic rates, while displaying higher levels under conditions of high nutrient availability or in motion [3]. This phenomenon is consistent with our prediction (22) where the degree of freedom of motion, N, is taken into account. For a vegetative state patient, or a person in the condition of slow wave sleep, the cellÕs degree of freedom of motion in human brain reduces up to zero, i.e., N ! 0, and we predict 0.87 0.75 .87 ¼ 39.9% for man with 70 kg. ðT brain T brain Þ=T 0brain
A 40% decrease in the global cerebral metabolism, whereas the observed decrease is about 45% for comatose patients [44,45] and 44% for slow wave sleep [44,46,47].
4. Discussion It is mysterious that a liver in mammals acts like a leaf of plant which scales like B M2/3. The liver contains 50,000– 100,000 individual lobules. The liver lobule is constructed around a central vein. The lobule itself is composed of many slice-like hepatic cellular plates like many leaves piled up, each hepatic plate being one- or two-cell thick. Therefore, the liver obeys the same scaling law as a leaf of plant (see Fig. 1), and it is of two-dimensional construction (D = 2). The pressure in portal vein leading into the liver averages about 9 mm Hg, and the pressure in the hepatic vein leading from the liver into the vena cava averages almost exactly 0 mm Hg. This small pressure difference shows that the resistance to blood flow through the liver sinusoids is normally low, resulting in almost no relative motion of the liver lobules (N = 0). 0.90 Our model predicts, for example, accurately how kidneys must obey the law: Bkidneys T kidneys . The kidney contains about 1,000,000 nephrons. Each nephron is composed of a glomerulus through which fluid is filtered from the blood. Blood enters the glomerulus through the afferent arteriole and then leaves through the efferent arteriole [48,49]. Pulsing blood pressure causes each nephron to move in the x-, y-, and z-directions and spin around one direction periodically (see Fig. 2); accordingly, the total degree of freedom of motion for each nephron is 4, i.e., N = 4.
Fig. 2. SEM micrograph of glomerulus in the kidney. Obviously, it is of 3-D construction (D = 3). The blood pressure acts on the glomerulus in all directions, the force system acting on the glomerulus can be equivalently reduced to a system of three forces (F1, F2, and F3) in the x-, y-, and z-directions and one couple (M ). So the motionÕs degree of freedom is N = 3 + 1 = 4.
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5. Conclusion We conclude that our cytofractalogy approach can successfully explain the different scaling laws for different organs. Of course, if El NaschieÕs E-infinity theory is considered, the scaling relationships (21) and (22) can be modified as follows: Bmotion M ðDH þN =6ÞðDH þ1Þ ; Borgan
H þN =6Þ=ðDH þ1Þ T ðD . organ
ð23Þ ð24Þ
Application of the present theory to biology will be discussed in future papers.
Acknowledgments The authors thank Professor Hari M. Srivastava of University of Victoria, Canada for his kind help in improving English, the authors also thank Jinsong Yao of Shanghai Jiaotong University for providing Figs. 1 and 2, and Refs. [48,49], and for taking part in the discussion of the explanation of liver and kidney. In this work J.-H. He proposed the original idea and made all theoretic deduction; Z.D. Huang took part in discussion of explanation of liver and kidneys. J.-H. He also thanks Shi-Huang Shao, Juan Zhang, Hong-Mei Liu, Yu-Qin Wan, Yan Yang, and Qian Guo for their helpful discussions.
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