Allometric Scaling by the Length of the Circulatory Network


Background: Allometric scaling is a well-known research tool used for the metabolic rates of organisms. It measures the living systems with fractal physiology. The metabolic rate versus the mass of the living species has a definite scaling and behaves like a four-dimensional phenomenon. The extended investigations focus on the mass-dependence of the various physiological parameters. Objective: Proving the length of vascularization is the scaling parameter instead of mass in allometric relation. Method: The description of the energy balance of the ontogenic growth of the tumor is an extended cell-death parameter for studying the mass balance at the cellular level. Results: It is shown that when a malignant cellular cluster tries to maximize its metabolic rate, it changes its allometric scaling exponent. A growth description could follow the heterogenic development of the tumor. The mass in the allometric scaling could be replaced by the average length of the circulatory system in each case. Conclusion: According to this concept, the dependence of the mass in allometric scaling is replaced with a more fundamental parameter, the length character of the circulatory system. The introduced scaling parameter has primary importance in cancer development, where the elongation of the circulatory length by angiogenesis is in significant demand.

Share and Cite:

Szasz, A. (2021) Allometric Scaling by the Length of the Circulatory Network. Open Journal of Biophysics, 11, 359-370. doi: 10.4236/ojbiphy.2021.114013.

1. Introduction

The spatiotemporal organization of biosystems is complex. The complexity is driven by self-organization ( [1] [2] [3] ), and validated by new science: fractal physiology ( [4] [5] ), including the bioscaling processes ( [6] [7] ). Understanding the challenges of the complexity of human medicine requires the development of a new paradigm [8].

The Basal Metabolic Rate (BMR) shows allometric scaling of the mass of the organism [9], describes as the power function of the mass ( M ) [10]:

B M R M α (1)

The allometric relation connects the surface-controlled metabolic processes with the geometry of the given material, which uses the available energy. In the simple formulation, metabolic processes are surface-dependent, while the mass is proportional to the volume. Therefore, the exponent of their ratio mirrors the dimensionality, and consequently, the exponent is α = 2 / 3 . On the other hand, the complex living allometry shows the exponent as α = 3 / 4 instead of α = 2 / 3 [11], explaining the relationship between the three-dimensional surface and the four-dimensional volume. Metabolic scaling in solid tumors is significant, but its heterogeneity and its rapid development by intensive proliferation and the supporting vascularity [12] change the scaling behavior [13], and this is described as dynamic evolution [14].

Life in this context is “four-dimensional” based on its metabolic exchange processes [15]. The self-organized multicellular structure creates fractal arrangements, and their metabolic energy-exchange proceeds on fractal surfaces, maximizing the available energy-consumption, scaling the fluctuation of the metabolic power by the universal scaling law [16]. This optimization of energy consumption was rigorously tested in the context of the scaling idea and can be extended to broader mechanisms [17], such as the energy-consumption's subcellular level, including the mitochondria and respiratory complexes [17].

The scaling model has been shown to be valid in a broad category of living structures and processes. The primary physiological parameters exponentially depend on the mass of the body [18]. The allometry shows a structural, geometrical constraint for living organisms. Nevertheless, life is more complex than what can be determined by its geometrical structure. A self-similar spatial-temporal-fractal structure defines the self-organizing procedure both in space and time [19]. A particular noise (temporal fractal noise)—like a fingerprint of the self-organizing [20] —is a typical and general behavior of the living biomaterial [21]. The stochastic fluctuations have a characteristic effect on malignant development [22], acting in the apoptotic threshold of cancer [23], and is well observable in the growth process [24].

The measured structural patterns could be applied to evaluate the cancer development [25] [26], an example of this is the use of image analysis is done by a pathologist. The metabolic power not only depends on the size of the surface involved in active transport, but also on the flow-rate of the same active surface size. This dependence could modify the transport. In the case of Benthic invertebrates (n = 215), they have the lowest average scaling exponent because they metabolize in an anaerobic way. This can be written as: ( α m e a n = 0.63 , [near to 2/3], C I m e a n = 0.18 ), where α is the scaling exponent, and CI is the Confidence Interval [27]. However, the other studied animals (n = 496) have ( α m e a n = 0.74 , [near to α m e a n = 3 / 4 ], C I m e a n = 0.18 ) [28]. The scaling of the metabolic activity is also different in mitochondrial and non-mitochondrial processes [29]. Mitochondrial metabolism is always aerobic, and its scaling exponent is nearly α = 3 / 4 [30] [31]. When the oxygen supply is limited, the cell extends its ATP production to fermentation by non-mitochondrial respiration, where the allometric scaling exponent lowers to nearly α = 2 / 3 .

Based on the scaling theory, a general model for ontogenic growth has been proposed [32] [33] [34]. Allometry is a consequence of the evolution process [35]. The variation of the personal sizes of the organs and the whole body of the individuals can be addressed in the frame of the power-law. The high fractal dimension could be used as a significant prognostic factor in diseased tissues [36]. There is research on tumor growth evaluated from an ontogenic basis [14] [37] [38] in which the tumor is successfully described, despite the substantial heterogeneity of the blood-supply [39] and the cellular structures differing from their regular counterparts. If the whole tumor mass differs from the mass of the viable part of the tumor, and the viable part has a scaling by the complete tumor mass with a high confidence scaling exponent α = 0.78 then the inadequate metabolic supply causes an extension of the nectrotic tissue inside advanced tumors [40].

2. Method

The general model for ontogenic growth described tumor-cell growth needs to calculate the cell-production considering also the vanished cells in the energy balance [40]. We learned, however, how vital programmed cell-death (apoptosis) is in the development of the fetus of mammals [41], and we considered it as a basic biological phenomenon [42]. The concept of cell-death is crucial in cancer development, considering one of the hallmarks of the malignancy is its escape from apoptosis [43], and is instead more susceptible to a more drastic kind of death: necrosis [44]. Following the line and extensive discussion of numerous other authors [28] [30] [32] [33] [34], who adapted the death-free energy balance from the original [45] publication, we extended this view with the changes caused by the perished cells. This approach became even more relevant with the study of malignancies, where a large mass of the tumor could well involve non-living necrotic tissue, so the ontogenic calculations [37] [40] need modification based on their energy-balance.

The number of cancerous cells (Nc) is the difference between the newly produced cells (P), and the perished (due to apoptosis or necrosis) drop off cells (D) at the unit time, basically follow the method of [40]:

d N c d t = P D (2)

The value of the changing cells is zero, while production just equal to the perished cells ( P = D ). It is a realistic assumption that the perished cells are proportional to the complete cell number in unit time:

D = λ N c (3)

where λ is the cell death-rate in a tumor. Note, at the beginning of the tumor-growth the P is also proportional with N c , ( P = ξ N c ) and in this case, the tumor growth exponentially: N c ( 0 ) ( t ) = exp ( ξ λ ) t . When P is constant during the development, the balance of the cell number by the time:

d N c d t = P λ N c (4)

The P = c o n s t deviates from the assumption of [40]. Our consideration concentrates on the fact that the cellular production after the initial period of growth became constant due to the stabilized balance of the resources and the autonomic growth of cells in a supporting healthy host environment by resources. The situation in this phase is well similar to the in-vitro experiments of the monoculture system when the allometric exponent is zero [31]. The energy balance is determined by the transported energy-flux delivered by the bloodstream. The energy-transport current intensity, the metabolic rate (B), is divided into two parts: one produces new cells, while the other keeps the living set alive. Hence:

B = N c B c + E c P = N c B c + E c ( d N c d t + λ N c ) = N c B c + E c ( d N c d t + N c T c ) (5)

where B c is the metabolic rate of a cell, and E c is the necessary metabolic energy to create a new cell and λ 1 = T c is the average lifespan of a cell in the tumor. Consequently:

E c d N c d t = B N c ( B c + λ E c ) (6)

Metabolic energy can be scaled by exponent α,

B = B 0 N c α (7)

where B 0 is a normalizing factor that shows the metabolic rate in the unity of N c . Therefore, we obtain:

E c d N c d t = B 0 N c α N c ( B c + λ E c ) (8)


d N c d t = a c N c α b c N c

a c = B 0 E c ; b c = B c E c + λ (9)

By multiplying N c by the average mass of a single cell ( m c ) we now obtain the energy-balance for the full tumor-mass (m):

d m d t = a m α b m (10)


a = B 0 m c 1 α E c and b = B c E c + λ = B c E c + 1 T c (11)

This balance was previously similarly formulated [46]. The mass has a maximum limit M, asymptotic value, a saturation when no more real changes of the mass can be observed, so:

0 = d M d t = a M α b M (12)


M = ( a b ) 1 1 α = ( B 0 m c 1 α B c + λ E c ) 1 1 α (13)

3. Results

A death parameter of the single-cell characteristically appears in the energy-balance of the ontogenic growth of the tumor. The nutrients supply profoundly determines the death of cancer-cells. At least at larger tumor sizes, the cell growth never happens with optimal nutrition supply; the cells intensively compete for the available energy sources.

The exponent α is located in the interval 2 / 3 α 1 , and it is α = 3 / 4 at ideal basal conditions [15] [45]. The ideal nutrition supply supports ontogenic

growth. The “ideal” asymptotic mass ( M i d ) from (13) is: M i d = ( a b ) 4 , hence the BMR* in non-ideal conditions:

M = ( a b ) 1 1 α = ( M i d ) 1 4 ( 1 α ) B M R * = M α = ( M i d ) α 4 ( 1 α ) (14)

Substituting (14) into (10):

d m d t = a m α ( 1 ( m M ) 1 α ) (15)


d ( m M ) 1 α d t = a ( 1 α ) M 1 α ( 1 ( m M ) 1 α ) (16)

which has a sigmoidal solution:

( m M ) 1 α = 1 ( 1 ( m 0 M ) 1 ) e a ( 1 α ) t M 1 α = 1 exp ( a t ( 1 α ) M 1 α + ln ( 1 ( m 0 M ) 1 α ) ) = 1 e τ (17)


τ = a t ( 1 α ) M 1 α ln ( 1 ( m 0 M ) 1 α ) (18)

and m 0 is the mass at the start of a tumor (probably a few times m c ), the initial (just born) mass. The ratio (r) of the energy spent on keeping cells alive ( λ = 0 ) from (13) is:

r ( τ ) = N c B c B = B c m m c B 0 m α = b a m 1 α = ( m M ) 1 α = 1 e τ (19)

Using α = 3 / 4 for the ideal four-dimensional case, the solution is:

( m M ) 1 / 4 = 1 e τ , τ = a t 4 M 1 / 4 ln ( 1 ( m 0 M ) 1 / 4 ) (20)

This is formally the universal growth law [45], but has a difference in the values of b (see (11)) and M (see (13)), including the average life-time of the malignant cells (death rate λ) in ontology description. The M value became smaller by shortening the average life-time of the cells and elongating τ time approaching the saturatin of the mass.

4. Discussion

The four-dimensionality and the allometry with evolutional optimization require different approaches: as the evolutionary conditions have a higher than a four-dimensional allometric scaling. The tumor mass is a somewhat indefinite parameter because the whole environment of the tumor suffers from sub-optimal alimentation. Consequently, the mass does not describe the allometry well. A more fundamental parameter of the networking conditions is requested.

From the original “four-dimensional life” fractal concept, we get scaling of the characteristic volume (v) with a characteristic length (l) [15] [45]:

v = k l 4 (21)

where k is a constant.

When the mass density of the tumor is relatively homogeneous, we assume proportional relation between the mass and volume:

m v (22)

When l 0 is the average asymptotic length of the circulatory network of the organ, and M is the asymptotic mass, from (21) and (22) with other K constant:

M = K l 0 4 (23)

Consequently, from (23) and (21), we obtain:

r = ( m M ) 1 / 4 = ( K l 4 K l 0 4 ) 1 / 4 = l l 0 (24)

The fourths-root of the relative mass growth to the asymptotic value (the relative basal metabolic rate) corresponds to the relative ratio of the length of the circulatory network. The geometrical parameter of the vascularity offers a more evident intrinsic factor than the mass. The length looks essential in the allometric relations.

From (20) and (24) the geometric growth rate is obtained, where a universal law can describe the average relative length of the circulatory network:

r ( τ ) = 1 e τ , τ = a t 4 M 1 / 4 ln ( 1 r 0 ) (25)

where r 0 = ( m 0 M ) 1 / 4 . The ratio of the energy maintaining new cells is

R ( τ ) = ( 1 r ( τ ) ) = e τ . Assuming the average density of the cancerous cell colony in the experiments of Bru et al. [47], the scaling law could be determined by the characteristic lengths, which are (at α = 3 / 4 [45] ), m L 4 in ideal cases, consequently:

r ( τ ) = ( m M ) 1 α = ( L 4 L 0 4 ) 1 / 4 = L L 0 = 1 e τ (26)

where L 0 is the asymptotic size of the cancer-cell cluster. It is naturally assumed that L 0 L , then from equation (17) the Taylor expansion of τ could be truncated at its second term, so (26) will be the linear dependence as measured:

L ( t ) a 4 K t L 0 ln ( 1 L ( τ = 0 ) L 0 ) (27)

However, if the energy supply is not ideal (which is the case in almost all the developed tumors in-vivo), the results do not support the ideal scaling by α = 3 / 4 [38]. It is shown in a generalized model that the fractal surface and the covered volume are scaled rigorously [48].

In cases of sub-optimal alimentation (there is an energy-deficiency for optimal growth), the scaling-exponent changes, and it depends on the fractal dimension of the vascular network ( D v ) [48]. The shortage of energy for adequate alimentation is a usual condition for the rapidly proliferating structures. Two strategies can be followed to distribute the available (sub-optimal) energy resources: maximizing the metabolism on the surface of the cells. The elongation of the vascular network (angiogenesis) is the optimal strategy in this growing phase of

the tumor ( α 1 = 3 3 + D v 1 ) [49]. The optimizing strategy could change in the

advanced stages when the blood volume is limited despite the elongated vascular possibilities. In the advanced cases, the energy-distribution request a α 2 = 4 D v 2 4 exponent ( α 2 < α 1 , D v > 1 ) [49]. The growth of the mass would be in these cases (as described by (16)):

d m max ( 1 ) d t = a D v 1 ( 3 + D v 1 ) M D v 1 3 + D v 1 ( 1 ( m M ) D v 1 3 + D v 1 ) (28)


d m max ( 2 ) d t = a D v 2 4 M 4 D v 2 4 ( 1 ( m M ) 4 D v 2 4 ) (29)

The generalized form of relation (25) could be used in α 2 < α 1 , ( D v 1 , D v 2 > 1 ) exponents, when the allometry exponent is α . The phase 1 and phase 2 stages of tumors had been studied by various publications [49] [50] [51]. We choose two characteristic values to demonstrate the differences, D v 1 = 1.28 and D v 2 = 1.52 .

The time development well shows the different saturation time of the processes with various exponents (Figure 1 ).

The mass and the characteristic length are strictly connected:

m max ( 1 ) = K l 3 + D v D v and m max ( 2 ) = K l 4 D v 4 (30)

In general:

M = K l 0 1 1 α (31)

and therefore:

r = ( m max M ) 1 α = l l 0 (32)

For optimal distribution, we get the exact same result:

r = ( m o p t M ) 1 α = l l 0 (33)

The limited nutrition, the energetic control of the tumor could be considered as a part of the controlled therapy [52]. If the cell culture were to be placed on the tumour region, and the cell culture had the same or higher demands as the

Figure 1. Development of the relative length in time of vascular structure in a tumor at various allometric exponents: the normal, tumor-free tissue α = 3 / 4 (solid line) and in the Euclidean geometrical construction α = 2 / 3 (dotted line). The saturation time to reach the final length increases by the decreasing of the vascular fractal-dimension, ( α 1 = 0.701 , dashed line; and α 2 = 0.62 , dash-dotted line). The chosen sample parameters: m 0 = 1 a .u . and M = 1000 a .u . .

tumour tissue, then it could successfully compete against the cancer cells supplied from the same sources of energy. These in-silico results have not yet been verified experimentally, they are expected in the future.

5. Conclusions

The mass change to the more fundamental length of the vascular network in allometric scaling is generally proven in optimal metabolic conditions. We had shown the application in two basic kinds of non-optimal alimentation processes, too.

We proved that allometric scaling could eliminate the mass and an entirely intrinsic parameter, the average relative length of the circulatory network. The allometric model by the length directly connects the metabolic energy intake of the tumor with the length of the vascular system, as a supplier of energy. The derivation of the equations is rather general because the obtained fractal dimensions are model-independent. We regard the vascular length as more fundamental than the mass because the tumor volume is usually indefinitely smeared out, having a gradient formed by the mix of the precancerous and host cells. Hence, the fractal determination of the vascular network gives a more precise solution for allometric relations.


This work was supported by the Hungarian National Research Development and Innovation Office PIACI KFI grant: 2019-1.1.1-PIACI-KFI-2019-00011.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.


[1] Sornette, D. (2000) Chaos, Fractals, Self-Organization and Disorder: Concepts and Tools. Springer Verlag, Berlin.
[2] Walleczek, J. (2000) Self-Organized Biological Dynamics & Nonlinear Control. Cambridge University Press, Cambridge.
[3] Kauffman, S.A. (1993) The Origins of Order: Self-Organization and Selection in Evolution. Oxford University Press, New York.
[4] West, B.J. (1990) Fractal Physiology and Chaos in Medicine. World Scientific, Singapore.
[5] Bassingthwaighte, J.B., Leibovitch, L.S. and West, B.J. (1994) Fractal Physiology. Oxford University Press, New York.
[6] Brown, J.H. and West, G.B. (2000) Scaling in Biology. Oxford University Press, Oxford.
[7] Brown, J.H., West, G.B. and Enquis, B.J. (2005) Yes, West, Brown and Enquist’s Model of Allometric Scaling Is Both Mathematically Correct and Biologically Relevant. Functional Ecology, 19, 735-738.
[8] West, B.J. (2006) Where Medicine Went Wrong: Rediscovering the Path to Complexity. World Scientific, London.
[9] West, D. and West, B.J. (2012) On Allometry Relations. International Journal of Modern Physics B, 26, 116-171.
[10] Deering, W. and West, B.J. (1992) Fractal Physiology. IEEE Engineering in Medicine and Biology, 11, 40-46.
[11] Kleiber, M. (1961) The Fire of Life: An Introduction to Animal Energetics. Wiley, Hoboken.
[12] Tannock, I.F. (1968) The Relation Between Cell Proliferation and the Vascular System in a Transplanted Mouse Mammary Tumour. British Journal of Cancer, 22, 258-273.
[13] Milotti, E., Vyshemirsky, V., Sega, M. and Chignola, R. (2012) Interplay between Distribution of Live Cells and Growth Dynamics of Solid Tumours. Scientific Reports, 2, Article No. 990.
[14] Guiot, C., Delsanto, P.P.P., Carpinteri, A., et al. (2006) The Dynamic Evolution of the Power Exponent in a Universal Growth Model of Tumors. Journal of Theoretical Biology, 240, 459-463.
[15] West, G.B., Brown, J.H. and Enquist, B.J. (1990) The Four Dimension of Life: Fractal Geometry and Allometric Scaling of Organisms. Science, 284, 1677-1679.
[16] Labra, F.A., Marquet, P.A. and Bozinovic, F. (2007) Scaling Metabolic Rate Fluctuations. Proceedings of the National Academy of Sciences of the United States of America, 104, 10900-10903.
[17] West, G.B. and Brown, J.H. (2000) Scaling in Biology. Oxford University Press, Oxford.
[18] Calder, W.A. III (1996) Size, Function and Life History. Dover Publications Inc., Mineola.
[19] Schlesinger, M.S. (1987) Fractal Time and 1/f Noise in Complex Systems. Annals of the New York Academy of Sciences, 504, 214-228.
[20] Bak, P., Tang, C. and Wieserfeld, K. (1988) Self-Organized Criticality. Physical Review A, 38, 364-373.
[21] Musha, T. and Sawada, Y. (1994) Physics of the Living State. IOS Press, Amsterdam.
[22] Berryman, M.J., Spencer, S.L., Allison, A. and Abbott, D. (2004) Fluctuations and Noise in Cancer Development. Proceedings of SPIE—The International Society for Optical Engineering, 5471.
[23] Qui, B., Zhou, T. and Zhang, J. (2020) Stochastic Fluctuations in Apoptotic Threshold of Tumor Cells Can Enhance Apoptosis and Combat Fractional Killing. Royal Society Open Science, 7, Article ID: 190462.
[24] Fiasconaro, A., Ochab-Marcinek, A., Spagnolo, B. and Gudowska-Nowak, E. (2008) Monitoring Noise-Resonant Effects in Cancer Growth Influenced by External Fluctuations and Periodic Treatment. Physics of Condensed Matter, 65, 435-442.
[25] Chan, A. and Tuszynski, J.A. (2016) Automatic Prediction of Tumour Malignancy in Breast Cancer with Fractal Dimension. Royal Society Open Science, 3, Article ID: 160558.
[26] Tambasco, M. and Magliocco, A.M. (2008) Relation between Tumor-Grade and Computed Architectural Complexity in Breast Cancer Specimens. Human Pathology, 39, 740-746.
[27] Pamatmat, M.M. (1983) Measuring Aerobic and Anaerobic Metabolism of Benthic Infauna under Natural Conditions. Journal of Experimental Zoology, 228, 405-413.
[28] Moses, M.E., Hou, C., Woodruff, W.H., et al. (2008) Revisiting a Model of Ontogenic Growth: Estimating Model Parameters from Theory and Data. The American Naturalist, 171, 632-645.
[29] Voet, D., Voet, J.G. and Pratt, C.W. (2006) Fundamentals of Biochemistry. 2nd Edition, John Wiley and Sons, Inc., Hoboken.
[30] West, G.B. and Brown, J.H. (2005) The Origin of Allometric Scaling Laws in Biology from Genomes to Ecosystems: Towards a Quantitative Unifying Theory of Biological Structure and Organization. Journal of Experimental Biology, 208, 1575-1592.
[31] West, G.B., Woodruf, W.H. and Born, J.H. (2002) Allometric Scaling of Metabolic Rate from Molecules and Mitochondria to Cells and Mammals. Proceedings of the National Academy of Sciences of the United States of America, 99, 2473-2478.
[32] West, G.B., Brown, J.H. and Enquist, B.J. (2004) Growth Models Based on First Principles or Phenomenology? Functional Ecology, 18, 188-196.
[33] Banavar, J.R., Damuth, J., Maritan, A., et al. (2002) Modelling Universality and Scaling. Nature, 420, 626.
[34] West, G.B., Enquist, B.J. and Brown, J.H. (2002) Modelling Universality and Scaling—REPLY. Nature, 420, 626-627.
[35] Stevens, C.F. (2009) Darwin and Huxley Revisited: The Origin of Allometry. Journal of Biology, 8, Article No. 14.
[36] Delides, A., Panayiotides, I., Alegakis, A., et al. (2005) Fractal Dimension as a Prognostic Factor for Laryngeal Carcinoma. Anticancer Research, 25, 2141-2144.
[37] Guiot, C., Degiorgis, P.G., Delsanto, P.P., et al. (2003) Does Tumor Growth Follow a “Universal Law”? Journal of Theoretical Biology, 225, 147-151.
[38] Deisboeck, T.S., Guiot, C., Delsanto, P.P., et al. (2006) Does Cancer Growth Depend on Surface Extension? Medical Hypotheses, 67, 1338-1341.
[39] Nagy, J.A., Chang, S.-H., Shih, S.-C., Dvorak, A.M. and Dvorak, H.F. (2010) Heterogeneity of the Tumor Vasculature. Seminars in Thrombosis and Hemostasis, 36, 321-331.
[40] Herman, A.B., van Savage, M. and West, G.B. (2011) A Quantitative Theory of Solid Tumor Growth, Metabolic Rate and Vascularization. PLoS ONE, 6, e22973.
[41] Haanen, C. and Vermes, I. (1996) Apoptosis: Programmed Cell Death in Fetal Development. European Journal of Obstetrics & Gynecology and Reproductive Biology, 64, 129-133.
[42] Kerr, J.F.R., Wyllie, A.H. and Currie, A.R. (1972) Apoptosis: A Basic Biological Phenomenon with Wide-Ranging Implications in Tissue Kinetics. British Journal of Cancer, 26, 239-257.
[43] Lowe, S.W. and Lin, A.W. (2000) Apoptosis in Cancer. Carcinogenesis, 21, 485-495.
[44] Karsch-Bluman, A., Feiglin, A., Arbib, E., Stern, T., Shoval, H., Schwob, O., Berger, M. and Benny, O. (2018) Tissue Necrosis and Its Role in Cancer Progression. Oncogene, 38, 1920-1935.
[45] West, G.B., Brown, J.H. and Enquist, B.J. (2001) A General Model for Ontogenetic Growth. Nature, 413, 628-631.
[46] von Bertalanffy, L. (1957) Quantitative Laws of Metabolism and Growth. The Quarterly Review of Biology, 32, 217-231.
[47] Bru, A., Albertos, S., Subiza, J.L., Asenjo, J.L.G. and Bru, I. (2003) The Universal Dynamics of Tumor Growth. Biophysical Journal, 85, 2948-2961.
[48] Szasz, A. (2021) Vascular Fractality and Alimentation of Cancer. International Journal of Clinical Medicine, 12, 279-296.
[49] di Leva, A., Bruner, E., Widhalm, G., Michev, G., Tschabitscher, M. and Grizzi, F. (2012) Computer-Assisted and Fractal-Based Morphometric Assessment of Microvascularity in Histological Specimens of Gliomas. Scientific Reports, 2, Article No. 429.
[50] Grizzi, F., Russo, C., Colombo, P., Franceschini, B., Frezza, E.E., et al. (2005) Quantitative Evaluation and Modeling of Two-Dimensional Neovascular Network Complexity: The Surface Fractal Dimension. BMC Cancer, 5, 14.
[51] Ceelen, W., Boterberg, T., Smeets, P., Van Damme, N., et al. (2007) Recombinant Human Erythropoietin α Modulates the Effects of Radiotherapy on Colorectal Cancer Microvessels. British Journal of Cancer, 96, 692-700.
[52] Deisboeck, T.S. and Wang, Z. (2008) A New Concept for Cancer Therapy: Out-Competing the Aggressor. Cancer Cell International, 8, 19.

Copyright © 2024 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.