A Classification Scheme for Phenomenological Universalities in Growth Problems

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by  P. Castorina

A Classification Scheme for Phenomenological Universalities in Growth Problems
P.Castorina(a,b), P.P.Delsanto(c,d,e), C.Guiot(d,f)1
1(a) Department of Physics, University of Catania, Italy
(b) INFN-Catania, Italy
(c) Department of Physics, Politecnico di Torino, Italy
(d) CNISM, Sezioni di Torino Universita’ e Politecnico, Italy
(e) Bioindustry Park of Canavese, Ivrea, Italy
(f) Department of Neuroscience, Universita’ di Torino, Italy
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A classification in universality classes of broad categories of phenomenologies, belonging to dif-
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ferent disciplines, may be very useful for a crossfertilization among them and for the purpose of
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pattern recognition. We present here a simple scheme for the classification of nonlinear growth

n problems. The success of the scheme in predicting and characterizing the well known Gompertz,
a Westandlogisticmodels,suggeststousthestudyofahithertounexploredclassofnonlineargrowth
J problems.

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Present efforts towards the understanding of complex A reliable macroscopic analysis of a complex system

] systems in physics, biology,economics and social science requires two fundamental ingredients: non linearity and
h
p require complementary microscopic and macroscopic de- stochasticity. Nonlinearityismorefundamentalbecause
- scriptions. In fact, due to the complexity ofthe underly- the stochastic behaviour requires a non linear dynamics.
o
ingdynamicsandtheunboundedvarietyofexternalcon- Therefore non linearity must be considered as the fun-
i
b ditions, a fundamental approach is missing. Microscopic damental feature of these systems and in this letter we
.
s models depend on such a large number of parameters consider general growth problems based on this crucial
c that they often lose almost any predictive power, even aspect. We shall show that different “degrees of nonlin-
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s when the calculations do not become forbiddenly diffi- earity”(asspecifiedbelow)correspondtovariousgrowth
y
cultortimeconsuming. Ontheotherhand,macroscopic patterns, which can be systematically classified.
h
descriptionsareofteninadequateanddonottakeadvan- For this purpose, let us consider the very broad class
p
[ tage of the enormous progress that has been achieved at of growth phenomena, which may be described by the

the microscopic level in recent years. An intermediate simple law:
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(mesoscopic) approach[1, 2, 3] may be very fruitful, but
v dY(t)
9 abridgingamongthevariouslevels[4]isnotalwayseasy =α(t)Y(t) (1)
dt
5 to accomplish.
0 Adifferentapproachhasconsequentlyemergedforthe where α(t) represents the specific growth rate, which
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treatment of problems , which do not directly require may vary with time, of a given variable Y(t). By intro-
0
6 a detailed description of the system to be investigated. ducing the nondimensional variables τ = α(0)t, y(t) =
0 The idea is to exploit the spectacular advancement of Y(t)/Y(0) and a(τ)=α(t)/α(0), Eq.(1) becomes:
/ interdisciplinary research, which has taken place in the
s
dy(τ)
c last two decades or so, involving e.g. the relevance of =a(τ)y(τ) (2)
i scale laws, complexity and nonlinearity in virtually all dτ
s
y disciplines.
with y(0) = a(0) = 1. By defining the time variation of
h
In this context many patterns have been discovered, a(τ) through a function Φ(a):
p
which are remarkably similar, although they concern
:
v completelydifferentphenomenologies. Thisishardlysur- da(τ)
Φ(a)= (3)
i
X prising,sinceoftenthe”background”mathematicsisthe − dτ
same. We shall call them “phenomenological universal-
r we obtain a system of two differential equations, which
a ities” [5], in the sense that they refer to a ”transversal”
may generate a variety of growth patterns, according to
generality (not to a uniformly general behaviour within
the explicitformofΦ(a),andisusuallyanalyzedbythe
a given class of phenomena).
standard fixed points and characteristic curves methods
As examples of universality we can quote the “life’s
[9].
universal scaling laws” [6], which will be discussed later,
In this contribution we are not directly interested in
and the “universality of nonclassical nonlinearity” [7].
this aspect, but we wish to show, instead, how the non-
Thelattersuggeststhatunexpectedeffects,suchasthose
linear terms in Φ(a) affect the growthdynamics process.
recently discovered by P. Johnson and collaborators [8]
We assume that a(τ) <1 and expand Φ(a) in power
and called by them “Fast Dynamics”, may be found as | |
series
well, although possibly with quite different manifesta-
tions in other fields of research. Φ(a)=Σ∞ b an (4)
n=0 n
2
in which we retain only a limited number of N + 1 i.e. all Gompertz curves are (under the mentioned pro-
terms. Borrowingfromthe languageofphasetransitions viso) identical.
[10] , we define, as belonging to the phenomenological LetusnowturnourattentiontotheclassU2,i.e. N=2.
universality class of order N (which we shall call UN, From Eqs. (6) and (3) and Φ(a)=a+ba2, where b=b
2
N=1,2,...),theensembleofallthephenomenology,which , it follows
maybesuitablydescribedbytruncatingtheseriesatthe
dy
power n=N. In the following we shall analyse in detail =α yp β y (9)
2 2
dτ −
the classes U1, U2 and U3 and provide a description of
their nonlinear properties. where α = (1+b)/b, p = 1 b and β = 1/b with the
2 2
−
The “linear”behaviourofthe systemcorrespondsto a solution
constantspecificgrowthrate,i.e. Φ(a)=0(orb =0for
n
y =[1+b bexp( τ)]1/b (10)
anyn). Theny(τ)followsapurelyexponentiallaw. Also
− −
the case b = 0 with all b = 0 for n 1, can be easily
0 6 n ≥ Byidentifyingy withthemassofabiologicalsystem,y =
shown to lead to an exponential growth. Since we are
m, and defining the asymptotic mass (m =y =1)
0 0
interested only in the nonlinear effects, we shall assume
b0 = 0. This does not cause any loss of generality, since M =limτ→∞m(τ)=(1+b)1/p (11)
onecanalwaysexpandΦinthevariableβ =a c,where
c is a solution of Σ∞ b cn = 0. In the β expa−nsion the it is easy to show that Eqs. (9) and (10) correspond
n=0 n
coefficient of β0 vanishes. Likewise, again without any to the well known allometric West equation for the case
loss of generality , we can set b =1, as one would have p=3/4 [15]. In their ontogenetic growth model, m rep-
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from an expansion in the variable γ =a/b . resents the mass of any living organism, from protozoa
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In order to study the various classes of universality to mammalians (including plants as well). By redefin-
and obtain the corresponding differential equations and ing their mass and time variables z = 1 (y/m)b and
−
solutions, we write from Eqs. (2) and (3): θ = τ +lnb blnM they obtain the very elegant pa-
− −
rameterless universal law
dy
Φ(a) =ay (5)
− da z =exp( θ) (12)
−
from which it follows: which fits well the data for a variety of different species,
ada rangingfromshrimpstohenstocows. Itisinterestingto
lny = +const (6)
−Z Φ(a) note that, in a subsequent work [16], West and collabo-
ratorsgiveaninterpretationofθ asthe “biologicaltime”
Bysolvingthepreviousequationwithrespecttothevari- , based on the organism’s internal temperature.
able a(τ) and then substituting into Eq. (2), one obtain An extension of West’s law to neoplastic growths has
thedifferentialequationcharacterizingtheclass. Thein- beenrecently suggestedby C.Guiot, P.P.Delsanto,T.S.
tegrationconstantcanbeeasilyobtainedfromtheinitial Deisboeck and collaborators [17, 18]. Although an un-
conditions. ambigous fitting of experimental data is much harder in
Letus thenstartbyconsideringthe classU1,i.e. with tumors(exceptforculturesinvitroofmulticellulartumor
N=1. From Eq. (6) and Φ(a) = a , it immediately spheroids),the extensionseemsto workwell. Ofcourse,
follows: particularlyinvivo,othermechanismsmustbetakeninto
dy account,suchasthepressurefromthesurroundingtissue
=y ylny (7)
[19]. Another important issue is the actual value of the
dτ −
exponentp,whichhasbeentheobjectofastrongdebate
with the solution
[20]. Recently C. Guiot et al. [21] have proposed that p
mayvarydinamicallywiththefractalnatureoftheinput
y =exp[1 exp( τ)] (8)
− − channels (e.g. at the onset of angiogenesis).
Eq.(7) represents the “canonical”form of U1 differential Althoughit is notobviousfroma comparisonbetween
equations and corresponds to the Gompertz law, orig- Eq. (7) andEq. (9), U1 representsa specialcase (b=0)
inally introduced [11] in actuarial mathematics to eva- ofU2,asitobviouslyfollowfromthe powerexpansionof
lute the mortality tables and, nowdays, largely applied Φ (which has b=0 in U1). This can be verified directly
todescribeeconomicalandbiologicalgrowthphenomena. by carefully performing the limit b 0 in Eq. (10) . In
→
For example, the Gompertz law gives a very good phe- fact it is interesting to plot y vs. τ in a sort of phase
nomenological description of the tumor growth pattern diagram ( see Fig. 1) .
[12], [13] and it can be related to the energetic cellular This leads to a very suggestive interpretation of Eq.
balance[14]. Itisremarkablethatitdoesnotcontainany (9). Having added a term to the Φ(a) expansion, we
free parameter ( except for the scale and linear parame- gain, in U2, the possibility of adding a “new” ingredi-
ters which have not been included, as discussed before), ent, which turns out to be a different dimensionality of
3
To conclude, we have developed a simple scheme,
which allows the classification in nonlinear phenomeno-
logical universality classes of all the growth problems,
which can be described by Eqs. (2) and (3). We have
foundthatthefirstclassU1correspondstotheGompertz
curve, which has no free parameters (apart from scale
and linear ones). The second class U2 includes all the
Westlike and logistic curves and has a free parameter b:
whenb=0wefallbackintoU1 (Gompertz). The success
of the scheme in obtaining the classes U1 and U2 when
one or two terms are retained in the expansion of Φ(a)
has suggested to us to investigate the class U3, which is
generatedby simply adding one more term (see Eq. 13).
FIG. 1: - Growth curves belonging to the class U2. From To our knowledge, this class has never been investigated
the top to the bottom the values of the parameter b are before. Aremarkableresultisthateachnewclassaddsa
−0.25,−0.1,0.1,0.25,0.5 respectively. The solid curve (b =
new“ingredient”(orgrowthmechanism). E.g. U2allows
0,p = 1) corresponds to the Gompertzian (U1), while the
for the possible presence of two dimensionalities in the
dashed one refers to the value proposed in [6] p = 3/4
energy flux. U3 extends such a possibility to the growth
(b=1/4).
term ( the time derivative). In addition to its intrinsic
elegance [24] the concept of universality classes may be
the “energy flux” i.e. input, output and consumption usefulforseveralreasonsofapplicativerelevance. Infact
(metabolism). E.g. the first term on the RHS of Eq. itgreatlyfacilitatesthecrossfertilizationamongdifferent
(9) may be related[22] to the premise that the tendency fields of research by implicitly suggesting that a method
of natural selection to optimise energy transport has led ofanalysis,whichisprovenadvantageousinonestudy,be
totheevolutionoffractal-likedistributionnetworkswith tried andeventually adopted in others. Also,if anunex-
anexponentp for their terminalunits vs. anexponent1 pectedeffectisfoundexperimentallyinafield,similaref-
for flux mechanisms related to the total number of cells. fects“mutatismutandis”shouldalsobesoughtinsimilar,
When b=0, p=1 and we lose the new ingredient, thus although unrelated, experiments in other fields. Finally,
falling back into U1. ifadetailedstudy isperformedto recognizethe patterns
Thisisconfirmedalsobyconsideringthelogisticequa- that are characteristics of the most relevant classes (and
tions,correspondingtoeq.(9)withnegativeb. Theusual subclasses),thiscouldgreatlyhelpinclassifyingandfit-
logistic equation is obtained for p=2. As well known in ting new sets of experimental data independently of the
populationdynamics[23],inthiscasethenewingredient field of application.
is the competition for resources. Acknowledgements
Finally we consider the class U3. Writing We wish to thank Drs. M. Griffa and F. Bosia for
their help and useful discussions. This work has been
Φ(a)=a(1+ba+ca2) (13) partly supported by CviT (Centre for the development
of a Virtual Tumor).
from Eq. (6) it follows
da
=K lny (14)
Z 1+ba+ca2 −
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