Abstract
Complex systems, in many different scientific sectors, show coarsegrain properties with simple growth laws with respect to fundamental microscopic algorithms. We propose a classification scheme of growth laws which includes human aging, tumor (and/or tissue) growth, logistic and generalized logistic growth and the aging of technical devices. The proposed classification permits to evaluate the aging/failure of combined new biotechnical “manufactured products”, where part of the system evolves in time according to biologicalmortality laws and part according to technical device behaviors. Moreover it suggests a direct relation between the mortality levelingoff for humans and technical devices and the observed small cure probability for large tumors.
Keywords:
Growth laws; Aging; Tumor growth; 89.75.kBackground
Complex systems with millions of interacting elementary parts are often considered computationally irreducible Wolfram ( [1984]); Wolfram ( [2002]) which means that the only way to decide about their evolution is to let them evolve in time.
On the other hand, there is an impressive number of experimental verifications, in many different scientific sectors, that coarsegrain properties of systems, with simple laws with respect to fundamental microscopic alghoritms, emerge at different levels of magnification providing important tools for explaining and predicting new phenomena.
In this respect, a priori unrelated systems show similar emergent properties and if an unexpected effect is found experimentally in a field, a similar effect, “mutatis mutandis”, should also be sought in similar experiments in other fields. Therefore a useful tool to greatly facilitate the cross fertilization among different fields of research is a general classification of growth laws Castorina et al. ( [2006]).
A very important example is the Gompertz law (GL) Gompertz ( [1825]) which applies to human mortality tables (i.e. aging) and tumor growth Steel ( [1977]); Wheldon ( [1988]); Norton ( [1988]).
In general, a growth problem is characterized by a function f (t), which describes the time evolution of some macroscopic quantity, and by the specific rate, α, defined as (1/f)(df/dt)=α(t). In the GL αhas an exponential dependence on time:
where a and b are constants. In aging f (t) indicates the survival probability; while with regards to tumor growth it corresponds to the number of cells N(t) (depending on the specific case a and b can be positive or negative).
For technical devices the specific rate of the survival probability has a powerlaw time behavior
with n>1, called Weibull law (WL) Barlow and Proschan ( [1975]); Rigdon and Basu ( [2000]). The analogy with the biological systems is intriguing (for clarity, as necessary, one defines the specific rate α_{h}(t) for the human mortality, α_{f}(t) for the technical systems and α_{c}(t) for tumor growth) and deeper than the similarity between eq. (1) and eq. (2).
Indeed, many independent analyses of experimental data on humans and animals suggest that at advanced ages (more than 8590 years for humans) there is a deceleration in mortality Gavrilov and Gavrilova ( [1991]); Vaupel et al. ( [1998]); Olshansky ( [1998]): in the large range 20  85 years for humans the mortality rate is well described by the Gompertz law and then there is a latelife mortality (although a definite conclusion has yet to be reached Gavrilov and Gavrilova ( [2011])). A similar trend is observed for technical devices Economos ( [1979]), confirming the analogy between biological and technical systems.
The understanding of aging and of latelife mortality is still an open problem and many interesting models have been proposed to explain the similar behavior in metabolic systems and in technical devices Gavrilov and Gavrilova ( [2001]). Moreover, a unifying language for the description of performance of metabolic and technical production and distribution has been recently suggested Becker et al. ( [2011]) to implement the idea that the robustness of metabolic systems with respect to enviromental changes could represent a useful model for technical systems.
In this letter, rather than focusing on specific models, we shall address the generalization of the classification scheme of growth laws to include human aging, tumor (and/or tissue) growth, logistic and generalized logistic growth and the aging of technical devices. We shall consider two applications of the proposed approach: a) a method to evaluate the aging/failure of combined new biotechnical “manufactured product”, where part of the system evolves in time according to biologicalmortality laws and part is a technical device; b) an interpretation of the “tumor size effect”, i.e. the small cure probability for large tumor Stanley et al. ( [1977]); Bentzen and Thomas ( [1996]); Huchet et al. ( [2003]), in analogy with the latelife mortality in aging.
Results
Let us start with the general classification scheme. It turns out that a classification of the growth laws according to the simple equation (1/f)(df/dt)=α(t) is obtained by considering the power expansion in αof the function ( see ref. Castorina et al. ( [2006]) for details)
which for b_{0}=0 and b_{i}=0 for i>1 gives a time independent specific rate α_{0} and therefore an exponential growth; for b_{0}≠0 and b_{i}=0 for i>1 describes a linear time dependent specific rate and again an exponential growth; at the first order in α, for b_{0}=0, b_{1}≠0 and b_{i}=0 for i>1, reproduces an exponential time behavior of the specific growth and therefore the GL; the second order term , O(α^{2}), for b_{0}=0, b_{1}b_{2}≠0 and b_{i}=0 for i>2 generates the logistic and generalized logistic growth.
The feedback effect, that is the dependence of the specific growth rate α on the function f (t), can be easily derived by the temporal behaviour of the specific rate. For the GL for a growing number of cells, N(t), one has the well known logarithmic non linearity,
and for the (generalized) logistic law one gets the typical powerlaw behavior
where a,b,c,γ are constants and the carrying capacity,
In order to describe technical devices, the previous classification scheme has to
be generalized since the specific growth rate of Weibull law has a power law dependence
on time which is not reproduced by eq. (3). The behavior
Note that: a) 0<(n−1)/n<1 and the nth term in the power series in α^{(n−1)/n}tends for large n to α, i.e. to the Gompertz law; b) the term b_{0}≠0, i.e. the exponential growth, has been neglected because one considers the GL, the generalized logistic or more complex growth laws for the biological systems (there is no problem to include this term in the expansion) ; 3) the first sum in the expansion has fractional powers that recall a Puiseux expansion.
As a byproduct of the proposed classification scheme one can easily evaluate the aging/failure of combined new biotechnical “manufactured products” by taking explicitely into account the mutual “interference” between the aging behavior of the biological part and the failure of the technical one. The “interference” effect strongly depends on the typical time scales in the coefficients c_{n} and b_{n} in the previous expansion: if the lifetime of the technical device is much larger than the lifetime of the biological part ( or viceversa) there is essentially no effect Muller et al. ( [1988]).
Let us first consider aging/failure of a combined biotechnological “manufactured product”, where part of the system evolves in time according to GL, i.e. the term O(α), and the behavior of technical part is described by a single term O(^{αn−1/n}),i.e.
By introducing dimensionless variables in time unit 1/b_{1}, i.e. τ=b_{1}t,
where
which describes the combined effect of the two growth laws. The quantitative effect
is depicted in Figures (
1,
2) where the previous function is plotted for different values of n at fixed
Figure 1. Comparison of the GL, the WL and the combined effect for a biotechnical device for
Figure 2. Comparison of the GL, the WL and the combined effect for a biotechnical device for
The next step is to include the term
By repeating analogous calculations it turns out that
In Figure
3 is shown that the term
Figure 3. Comparison for
Therefore the general expansion of Φ(α) in eq. (6) can describe the aging/failure of any biological and technical system including the levelingoff at late mortality which is obtained by taking into account the term O(α^{2}) in Φ(α), i.e. by the transition from the GL or WL to a logistic type law Horiuchi and Wilmoth ( [1998]).
The proposed unification scheme suggests a practical method to understand growth patterns. Given a set of data on some growth process, the first step of the analysis is a fit in power of α of the derivative of the specific growth rate, i.e. of the function Φ(α). Therefore : a) if the best fit is linear, the growth is a Gompertzian one; b) if the best fit is quadratic, look at the sign of the coefficients of the expansion. For b_{1}>0 and b_{2}<0 the growth is logistic (or generalized logistic) corresponding to a competitive dynamics; c) if the best fit indicates a fractional power the growth follows the WL. Of course, it is always possible to obtain a better agreement with data by increasing the number of coefficients. However, should increasing the number of parameters indicate only a marginal improvement in the description of data one concludes that the added terms in the expansion are irrelevant.
Discussion and conclusions
Let us now consider the crossfertilization among different sectors.
As previously discussed, there is a deceleration of mortality in aging at late time which is described as a “transition” from a Gompertz law to a generalized logistic behavior. On the other hand, tumors evolve in time according to the GL. The obvious indications is to verify if a phenomenon corresponding to the deceleration of mortality, i.e. a transition from the GL to a power law, exists for cancer growth at a later time. As we shall see, this aspect has strong consequences on the therapy.
For tumor growth the b_{1}α term gives the GL in eq. (4) and the introduction of the O(α^{2}) term corresponds to the power law nonlinear feedback in eq. (5). Therefore one has to investigate if at latelife of a tumor growth there is such a modification in the dependence of the specific growth rate on the cell number N(t). Since direct informations “in vivo” are almost impossible, the question has to be addressed in an indirect way by considering radiotherapy.
The radiotherapic tumor treatment consists in series of radiation doses at fixed time intervals. However tumors start to regrow in the interval between two treatments : the regrowth during radiotherapy is therefore an important clinical parameter Kim and Tannock ( [2005]) and the probability of treatment benefit critically depends on the tumor regrowth pattern.
The so called “tumor size effect” is a reduction of radiotherapeutic results for large tumors ( which , presumably, has grown since long time). The dependence of the surviving fraction on the tumor volume was already observed by Stanley et al. in 1977 in lung tumors Stanley et al. ( [1977]) and reemphasized by Bentzen et al. and Huchet et al. in Bentzen and Thomas ( [1996]); Huchet et al. ( [2003]).
The effect of regrowth rate on radiotherapy has been quantitatively investigated in ref. Castorina et al. ( [2007]) and the results clearly indicate that to understand the tumor size effect the regrowth rate for large tumor has to follow a power law Guiot et al. ( [2003]) rather than the GL.
From this point of view the “tumor size effect” is a phenomenon which indicates that in late time tumor growth there is a change from a GL specific rate to a power law behavior, corresponding to the deceleration in mortality at advanced age.
One should conclude that such a common feature in aging and in failure in biological and/or technical systems should be considered as a “bifurcation” or a “phase transition” in the specific growth rate at large time from GL or WL to a logistic or generalized logistic behavior.
In closing, the general expansion of Φ(α) in eq. (6) can describe the growth/aging/failure of biological and technical systems and the transition to a different (“phase”) specific growth rate at latelife could be a common feature of those systems independently on the microscopic dynamics.
Competing interests
The authors declare that they have no competing interests.
Author’s contributions
PB and PC carried out the entire study in collaboration. Both authors read and approved the final manuscript.
Acknowledgements
It is a pleasure to thank H. Satz for helpful comments and discussions. PC thanks for the hospitality the Department of Physics of Bielefeld University.
References

Barlow RE, Proschan F (1975) Statistical theory of reliability and testing. Probability models. New York: Wiley.

Becker T, et al. (2011) Flow control by periodic devices: a unifying language for the description of traffic, production and metabolic systems. J Stat Mech. P05004

Bentzen SM, Thomas HD (1996) Tumor volume and local control probability: clinical data and radiobiological interpretations. Int Radiat Oncol Biol Phys 36:247251

Castorina P, Deisboeck TS, Gabriele P, Guiot C (2007) Growth Laws in Cancer: Implications for Radiotherapy. Rad Res 169:349

Castorina P, Delsanto PP, Guiot C (2006) Classification Scheme for Phenomenological Universalities in Growth Problems in Physics and Other Sciences. Phys Rev Lett 188701:96

Economos AC (1979) A nongompertzian paradigm for mortality kinetics of metazoan animals and failure kinetics of manufactured products. AGE 2:7476 Publisher Full Text

Gavrilov LA, Gavrilova NS (1991) The Biology of life span: a quantitative approach. N.Y: Harwood Academic Publisher.

Gavrilov LA, Gavrilova NS (2001) The reliability theory of aging and longevity. J Theor Biol 213(4):527545 PubMed Abstract  Publisher Full Text

Gavrilova NS, Gavrilov LA (2011) Mortality measurement and modeling beyond age 100, Symposium Orlando, Living to 100 monograph (published on line),. The Society of Actuaries, 2011.

Gompertz B (1825) On the nature of the function expressive of the law of human mortality and a new mode of determining life contingencies. Phil Trans R Soc 115:513 Publisher Full Text

Guiot C, Degiorgis PG, Delsanto PP, Gabriele P, Deisboeck TS (2003) Does tumor growth follow a ’universal law. J Theor Biol 225:147151 PubMed Abstract  Publisher Full Text

Horiuchi S, Wilmoth JR (1998) Deceleration in tha age pattern of mortality at older ages. Demography 35:391 PubMed Abstract  Publisher Full Text

Huchet A, Candry H, Belkaceni Y (2003) L’effet volume en radiotherapie. Premiere parie: effect volume et tumeur. Cancer Radiotherapie 7:7989 PubMed Abstract  Publisher Full Text

Kim JJ, Tannock IF (2005) Repopultaion of cancer cells during therapy: an important cause of treatment failure. Nat Rev Cancer 5:516525 PubMed Abstract  Publisher Full Text

Muller CH, et al. (1988) The survival rate of patient with pacemaker is essentially the same of the normal population. Eur Heart J 9:1003 PubMed Abstract  Publisher Full Text

Norton LA (1988) Gompertzian model of human breast cancer growth. Cancer Res 48:70677071 PubMed Abstract  Publisher Full Text

Olshansky SJ (1998) On the biodemography of aging: a review essay. Population and Development Review 24:381393 Publisher Full Text

Rigdon SE, Basu AP (2000) Statistical methods for the reliability of repairable systems. New York: Wiley.

Stanley JA, Shipley WU, Steel GG (1977) Influence of tumor size on the hypoxic fraction and therapeutic sensitivity of Lewis lung tumour. Br J Cancer 36:10513 PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Steel GG (1977) Growth kinetics of tumours. Oxford: Clarendon Press.

Vaupel JW, Carey JR, Christensen K, Johnson T, Yashin AI, Holm NV, Iachine IA, Kannisto V, Khazaeli AA, Liedo P, Longo VD, Zeng Y, Manton K, Curtsinger JW (1998) Biodemographic trajectories of longevity. Science 280:855860 PubMed Abstract  Publisher Full Text

Wheldon TE (1988) Mathematical models in cancer research. Bristol: Adam Hilger Publisher.

Wolfram S (2002) A New Kind of Science. Wolfram media, Champaign, USA.