1

HUMAN BODY AND HUMAN TUMOR

GROWTH KINETICS: A MODEL

Antonio Casolari

Introduction

A model describing microbial inactivation and growth kinetics as affected by

chemical and physical agents (Casolari, 1981), suggested some years ago, led to some

understanding of a series of phenomena otherwise unexplained. The expected

evolution of such a modeling activity would be to try the application of the same

basic principle to the growth kinetics of eukaryotic organisms and mostly of metazoa.

The human embryo and human tumor growth kinetics were quite attractive as

immediate targets. Some difficulties arisen very early in borrowing the model

describing prokaryotic organisms. Seemingly, a quite peculiar mechanisms

supporting the growth of the eukaryon and entirely unknown to the prokaryotic cells,

it’s working. In fact, the prokaryotic and the metazoan cell although fundamentally

alike they differ in several respects and mostly in the surface / volume ratio. I am

inclined to believe indeed, that the quite prominent characteristic of the metazoan cell

resides in the low value of this ratio and that further studies could show substantial

connections with this character and the ultimate outcome represented by the

metazoan organisms and very likely with their evolution into metazoa.

Approach to the model

As previously shown (Casolari, 1981), a definite prominence of the water in

supporting the molecular interactions leading to such a relevant phenomena of

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prokaryons as are the growing and dying mechanisms, can be backed up on the basis

of bringing together the kinetic concern with the known physiologic evidence. The

hypothesis was that kinetic behavior of eukaryons could be expected to be held on the

same basis. If it would be the case, the growth curve of metazoan organisms would be

described, in the first approximation, by the same basic equation:

Co = Ct(1 – u*t)

[1]

where “u” is the number of water molecules carrying the energy useful for the

metabolic activities to be accomplished with the expected rate, i.e. the water

molecules carrying an energy value Eu greater than the expected one from the

Maxwellian distribution in the simplified form:

u = No . e - Eu / R*T

[2]

being No the number of total available water molecules, and “t” is the time (hours)

after which the Ct cell number is reached starting from the Co initial cells level.

According to equation (1):

Log Ct = Log Co / (1 - u * t) [3]

after which:

u = (1 - Log Co / Log Ct ) / t [4]

and being:

Eu = R*T*(No - Ln u) [5]

it follows that knowing the Ct value for an organism held ‘t’ hours in a defined

medium at the temperature T = 273.15 + °C, the entire growth curve could be

described, since the parameter ‘u’ is constant, temperature independent. Nevertheless,

trying to describe the growth curve of the mouse ascite tumor (Ciaranfi et al., 1971) it

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was evident that the ‘u’ value could not be constant, by applying the above functions,

while it decreased steadily increasing the tumor cell number. Such a situation could

be produced by a water decrease in the environment, occurring as the cell number

increases, owing to both the capture action of water required to build new cells, and

also by the water capture by the hydration shell of by-products delivered into the

environment by metabolizing cells.

Both mechanisms are active in the growing prokaryons, while in a less dramatic

measure. The application of the previously reported relationship for prokaryons was:

µ = Ln 2 / [u * Ln Ct - 60 * Ln Co * (M+ (L*Ct )2 )]

[6]

However, describing the bacterial growth curves, the ‘M’ and ‘L’ parameters, and

mostly the ‘L’ one, affected the fitting of the curves only at cell concentrations

greater than about 1/10-1/50 of the maximum attainable cell value (ca. 109.5

), while at

lower cell concentrations the ‘u’ value was very constant.

The water content of a bacterial cell of average size [~1 µm3] amounts to about 10

10

molecules; while an eukaryotic cell of average size will hold more than 1013

water

molecules. Hence, before attempting to work out an ad hoc solution to the above

unexpected problem, it was tried to ascertain if an alike behavior would be peculiar to

the growing animal cell.

To this end, the growth curve of the human embryo up to the birth and then to the full

growth of human body was taken into consideration. Studying the human body

growth kinetics, the mammalian cell was regarded as a spherical cell having 12.4 µm

mean diameter, which leads to an eukaryon volume of 103 µm

3 , having an average

80% water content, what leads to about CW = 2.676867 * 1013

water molecules.

Since the water content of the growing body will decreases during the entire life

period, it was attempted to mediate the quite different values of the body water

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content as the embryonic and post-natal development is concerned (Kelly et al., 1951;

Tables Geigy, 1963), attaining the following relationship:

FRw = 3.046876 - 0.182782472 * Log Ct [7]

- where FRw means the free-water content -

able to meet the requirements of the model. Fig. 1 shows some experimental body

water content values compared with the expected ones from equation (7) .

Since the model required definition of the total available water molecules necessary

for description of the human embryo development kinetics, two possibilities were

considered: (a) the water content of the pregnant woman together with the developing

embryo, and (b) water content required by the embryo alone. The latter hypothesis

proved to be inconsistent with an adequate description of the growing embryo

kinetics, because in this case the embryo would not overcome the weight of about

one kg. Consequently, the following relationship was used:

Taw = Pw + (Ct / 109 ) * W * FRw [8]

where Taw is the total available water, Pw is the number of water molecules held in a

30 kg of water (i.e. taking 30 kg water as being the 60% of the body weight of a

pregnant woman of mean weight), that is Pw = 1.26588233 * 1027

; W is the number

of water molecules contained in one g of water, i.e.: W = 3.34608334 * 1022

; Ct is

the embryo-cell / body-cell number and Ct /109 is the weight (g) of Ct cells.

The requirement that the number the water molecules ‘u’ carrying suitable energy

Eu would be a quantity continuously decreasing as embryo and body weight increase,

was tentatively approached assuming that each growing cell would produce NX

metabolite molecules able to lead to a decrease of the growing rate of the metazoan

cell itself.

Accordingly, the following relationship was used:

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AvW = Taw - Ct * (Cw * FRw + NX * WL) [9]

where AvW was the actual available water and WL was the number of water

molecules bound by each NX molecule. Equation (5) becomes:

Ln u = Ln AvW – Eu / R*T [10]

where Eu was in cal / mole , R = 1.987 and the absolute temperature T = 273.15 +

37°C.

The experimental values of the starting weight of human embryo available from the

literature (Tables Geigy, 1963; Hamilton et al., 1978) were not lower than 0.02 g,

which corresponds to a Ct value of 2*107 cells.

The series of ‘u’ values able to describe with the required accuracy the entire growth

curve of the human body were related to the number of cells by three relationship: the

first one describing the growth up to the seventh month of pregnancy:

Log u7 = - 1.48421632 - 0.182782472 * Log Ct

[11]

the second one, from the seventh month to the birth:

Log ub = - 0.33504552 – 0.27845682 * Log Ct

[12]

and the last equation, after the birth:

Log u = 12.99743970 - 1.32609003 * Log Ct

[13]

The above reported three relationships could be certainly represented by a single

function of higher degree, but the breakage into three functions intersecting at the

seventh month of prenatal development and at the birth, seems to suggest that some

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event of such a paramount importance does occur in the final period of the prenatal

development, and at the birth, greatly affecting the growth kinetics.

At the birth and afterwards, the model requires that the equation (8) would be

changed into the following:

Taw = (Bw + (Ct /109)*W)*FRw [14]

to neglect the water content of the mother, while considering the water content of the

new-born only, and actually the water content of the new individual, where Bw is the

water fraction of the new-born, that is 0.74 * (1012.6

/ 109 ) * 3.34608334 * 10

22

leading to Bw = 9.908294 * 1025

.

Replacing the equation (8) with the equation (14) does agree very well with the

sudden change in the relationship between Log u and Ct, occurring in the growing

rate at the starting of the post-natal development.

The events occurring at the seventh month of the prenatal development, determining

the visible change of growth kinetics, are not obvious. Quite likely, it could be linked

to an increased cell weight disjoined from an increase in the cell number (Hamilton et

al., 1978), as the growth curve [Fig. 3] seems to suggest.

This event can be justified by the model?

The last parameter, and a quite important one, related to the ‘u’ value, and whose

magnitude requires to be defined is the NX value (see rela-tionship 9). NX is the

number of cell molecules really affecting the growth of the mammalian cell. It was

found, by trial and trial, that the best results (i.e., the growth curve fitting and the

physiological concern) could be obtained by a value of NX increasing with the

increasing too of the cell number of the growing embryo; the reverse was true for

‘AL’ molecules. In spite of the paramount relevance of NX values as regards the

ability of the model to fit the experimental data, the best first order equation linking

NX to Ct [ i.e. Log NX = f(Ct )] had a general correlation coefficient not higher than

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0.9999 and it was then useless without decreasing significantly the accuracy of the

fitting, mostly at cell concentrations lower than one thousand. Anyhow, since the

utilized computer was unable to handle numbers greater than 1038

, it was impossible

to me to obtain a better fitting. Different computer facilities and or mathematician

ability would meet the requirement. [For the present description of the human body

growth curve, the used computing algorithm allowed to overcome the problem].

Finally, being defined the magnitude of ‘u’ and Aw parameters, the useful energy

value for the metabolic activity of the mammalian cell was easy [eq. 5] found to be

Eu = 40,564.6442 cal/mole, i.e. 160, 738.318 J / mole.

Human embryo and human body growth curve

Fig. 2 shows the human body growth curve, from the early fetal stage up to the age of

20 years. Fig. 3 shows the human growth curve during the first 88 weeks. As can be

seen, the degree of fitting can be regarded as being quite satisfactory. It was claimed

(Lewis and Hartman, 1933), on the basis of data obtained from different cleavage

stages of the eggs of Macacus rhesus, that three days after fertilization the number of

cells would be close to 16. As can be seen from Fig. 3b, this event is exactly expected

from the model. The first duplication requires, according to the model, 34.6 hours,

and the second one, 19.6 hours. Afterwards, the duplication time decreases down to a

minimum value of 8 hours, after which it increases steadily. The growth of the human

embryo during the first ten days is described by a typical sigmoid curve, as

represented in Fig. 3c.

Fig. 4 shows the duplication time behavior as a function of the number of embryo

cells. As can be seen, the shape of the represented curve is entirely different from

that of a prokaryon, in what is very close to an asymmetric parabola. It recalls the

curve describing the growth rate of a procaryon as a function of the temperature,

though reversed as regards the magnitude of the abscissa. This reversion seems to

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suggest that fertilized egg is very prone to grow, being the environmental conditions

very suitable for cell multiplication, while after only few duplications (five) some

factors suddenly affect the natural cell trend to the development.

As it is well known (Forfar and Arneil, 1978) and as expected from the model, the

birth occurs after 38 weeks and at this time the number of cells of the embryo is about

1012.5

.

It might be asked why the delivery occurs after about 9.5 months. I believe that the

pregnant woman has two possibility, at this time: to eject the embryo or to die.

Really, according to the model, the early increasing of the duplication time of the

mammalian cells (i.e. the early decreasing of the doubling rate) as compared with the

behavior of a prokaryotic cell, comes from the peculiar ability of the metazoan cell to

establish characteristically multiple inter-cells linkages. As a matter of fact it can be

envisioned that the procaryotic cell differs substantially from the eukaryotic one,

because the latter cannot avoid the linking of itself with neighboring eukaryons,

owing to the presence at the surface of the cell membrane a kind of molecules that

unavoidably bind to molecules of the same type located at the surface of the

neighboring cells.

This linkage among eukaryons is the prerequisite for tissues building. While losing

the ability of freely swimming in a watery environment, the metazoan cell organized

as tissue, seems to acquire unavoidably the characteristic to proceed towards a series

of phenomena leading in the end to liberty eprivation, that is leading to senescence

and death of the entire organism.

Really, according to the model, the growth curve of the human body can be described

only taking into account a defined type of molecules synthesized by the metazoan cell

itself and located at the outer surface of the membrane, i.e. the above called NX

metabolites, and afterwards called Gp molecules.

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This means that suggested model cannot envisage a human body growth curve, while

unaffected by molecules leading unavoidably to cell death.

A hypothesis anything but unlikely!

On the other hand, prokaryotic cells can reproduce at will, without any restriction,

transfer by transfer, without advises of biosenescent process, unless leaved to crowd

together in an intoxicated environment.

But prokaryotic cells do not link together; actually, they are free living units.

Nevertheless, while living in an environment densely populated of cells is not always

an enjoying condition, as procaryons prove, it is the unavoidable linkage with the

other members of the community that unavoidably leads to the death, as the metazoan

cells prove, according to the model.

So, I am inclined to believe that the observed decrease of the growing rate of the

mammalian cells after five duplications only, i.e. at cell concentration greater than 32

units, would mean that some difficulties are encountered very early, by the

mammalian cell, as regards the probability for a free development; and since an

increase of the duplication time cannot be biologically regarded otherwise than a

senescent symptom, the human life must be regarded in practice as a steadily

approach to the death.

In this view, the metazoan organism arose from a damaged free living cell,

synthesizing molecules designed to the outer surface of the cell membrane and forced

by their own structure to link together with molecules of the same type carried out by

the neighboring cells of the same type. The above linkage, quite extraneous to

prokaryons, while not unknown, restricts the individual motility of the metazoan cells

and above all decreases the probability of the cell to be struck by the water molecules

carrying useful energy, and able to trigger and to keep the metabolic activities at the

right rhythm. The above restriction is mostly a physical one, since the cell surface

facing the watery environment decreases further on by the cell joining together. It is

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for the above reported reason, perhaps, that the value of the useful energy obtained

for the growing human body, already expected to be in the mesophilic range, is on the

contrary higher than the value obtained for thermophilic bacteria (Casolari,1981).

Really, only water molecules having an energy value high enough to overcome the

wall represented by the crowding eukaryons in a tissue, can be regarded as water

molecules carrying useful energy.

As expected from the model, the number of Gp molecules synthesized by the

mammalian cell increases about 1,000 times from the beginning of the embryo

development (about 108 Gp /cell at the stage of’ two cells) up to the body age of about

20 years old (about 1011

Gp molecules/cell at a cell level of 1013.8

). At the same time,

the level of the AL molecules, i.e. the water molecules available per each Gp

molecule, it decreases of about 17 Log cycles. From the above values it follows that

at the human age of about 20 years, the concentra-tion of the Gp molecules is very

close to the osmolality value, that is about 0.3 moles / kg water. It can be expected

that both the increasing of Gp concentration and the decreasing availability of water

molecules, do work together in the biosenescence progression of the human body. As

a matter of fact, the increasing number of Gp molecules would extend the linkages

among mammalian cells, and the decreasing water content would support the

reinforcement of the same linkages in consequence of the increased proximity of the

Gp molecule’s binding sites quite deprived of their own hydration shell.

Nevertheless, there is perhaps no absolute system.

Sometimes, few eukaryons lose the ability of linking together, becoming unable to

synthesize properly the Gp molecules, as a result of a modified DNA information. As

a result, they separate from the neighboring cells as much as the synthetic property is

modified and they resume the duplication activity with comparatively more or less

extent. The event occurs owing to the increased probability for water molecules

carrying useful energy to hit such type of cells, for the greater exposition of the cell’s

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having lessened links with neighboring ones, following the mutated equipment of

surface molecules.

These cells will become able to go away from the site of origin, in a more or less

degree, following the magnitude of the acquired inability to supply “normal” Gp

molecules to the outer surface of the membrane.

These cells become insensitive to the “contact inhibition”, to a more or less degree, in

proportion of the extent of the surface molecules structure change.

Accordingly, these ”rejuvenated” cells [RC] may lose also the ability to bind the

synthesized abnormal Gp molecules (AGp) to the cell surface, thus delivering them in

the environment. Such AGp molecules affect seriously the metabolic activity of

normal cells. The network of AGp molecules delivered in the body fluids, by

diffusion and through vessels, increases further as the inability of the normal cells to

fix such AGp molecules to the surface membrane increases too, as it decreases the

differentiation of RC . The most prominent effect due to the increase of AGp

molecules in neighboring tissues and in the end in the entire organism, can be

recognized in a some kind of cytotoxicity, leading in the end to the cell death.

The above reported one would be the process of carcinogenesis, as can be outlined

according to the model.

On the other hand, if the ability to synthesize normal Gp molecules is modified to a

degree not high enough, or of a kind not different enough, the delivering of AGp

molecules into the neighboring tissues and entire organism does nor occur. Such cells

keep on the outer surface a high enough number of AGp molecules (a phenomenon

that would occur more frequently) and by a process very close to the one of the

normal cell they form strong enough linkages - and/or in a number high enough -

with neighboring cells (to an extent related to the actual difference between GP and

AGp molecules) to build a tumor-like structure. Following the type and/or extent

[the nature] of such inter-cells linkages, as it come from the kind of the modification

undergone by Gp molecules, the growing activity may be self-limiting and or

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anyhow not affecting the body cells down to the host death. To a process of the latter

kind would be ascribed the one of the benign tumor development.

Human tumor growth kinetics.

The principal object of studies about the tumor cell population kinetics is to explain

how a tumor attains its particular growth rate, and thus to reach a more informed

judgement on methods of growth control (Steel, Adams and Barret, 1966).

As at is well known, the size and weight of solid tumors could be directly measurable

parameters. Estimates of the size can be obtained by external measurements using

vernier caliper (tumors arising in the skin or in soft tissues), the measures including

the greatest and/or mean diameter, tumor area or volume (Lala, 1971). Most data can

be obtained from sequential measurements on plain films as in chest X-rays for

pulmonary metastases and primary bronchial carcinomas, or from mammography for

breast cancers, or on films where a contrast medium has been used (Tubiana and

Malaise, 1976). It has been shown, as for breast cancers and pulmonary metastases,

that there is a close agreement between the size of the excised tumor measured

directly and that calculated from radiographs showing the tumor in situ [Contesso et

al., 1971; Weiss, 1971]. But only lesions from untreated patients can be considered

suitable for studies of the growth rata of tumors, while the precision of the measures

is affected by both the time intervals between observations and the growth rate itself

of the tumor. Conflicting situations arise very often and mostly concerning ethical

considerations, obviously, since to obtain a high number of measurements, the

growing tumor would not be subjected to a treatment of any kind, leaving the illness

to worsen. It follows that the majority of observations have been made on the basis of

only a few (2-4) radiographs.

When, as it is the case with experimental tumors [animal solid neoplasms,

spontaneous or transplanted, animal ascitic tumors, leukemia] it is valid to calculate

an average growth curve, the result is usually a smooth curve which is convex

13

upwards in semi-Log plot [Tubiana, 1971; Sommers, 1973; Steel, 1982). The curve

may be fitted by the common Gompertz equation [McCredie et al., 1965; Laird,

1969):

St = So * e (A/a*(1 – exp (-a*t)))

[15]

where St and So are the tumor size at the time t and the initial size, respectively; ‘A’

is the specific growth rate in the observation period and ‘a’ is the rate of exponential

decay of A.

Clearly, the equation [15], able to describe a wide variety of biological, physical,

technological, economic events (Sommers, 1973) does add very little or nothing to

the knowledge of the cancerous phenomenon (Steel, 1982).

Tumor growth is rarely exponential, but it is often convenient to regard it of such a

kind over short periods of time (Steel, 1982); if tumor increases linearly with time,

the specific growth rate K would be:

k = (dV/V) *dt [16]

leading to:

V = Vo * e k*t

[17]

and

Ln Vt = Ln Vo + K*t [18]

so that:

Dt = 0.693 / K [19]

where Dt is the tumor volume doubling time.

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Really, it is impossible to distinguish an exponential growth curve from a

Gompertzian one over such a short period of observation that is achieved in the

practice (Steel and Lamerton, 1966; Tubiana and Malaise, 1976).

Different solutions to the problem were approached [Mendelshon, 1963; Hoffman,

l953; Till et al., 1964; Dethelfsen et al., 1968; Summers, 1966], while their relative

merits stand not yet established.

Very likely, the situation reflects a lack of hypotheses about tumor kinetics, defined

enough to be tested with the required accuracy.

Really, the published models, more or less mathematically complex, provide one or

more kinetic parameter while devoid of defined, although hypothetical, physiological

meaning. It must be recognized moreover, the complete lack [to my knowledge] of

human tumor growth curves over which one could exercise his aim of modeling.

The main experimental data abundance concerns the distribution of the tumor

doubling times. It is commonly accepted that the distribution of the time required to

double the diameter, the volume or the mass of a tumor, can be expected to be of the

log-normal type (Sommers, 1973; Tubiana and Malaise, 1976; Steel, 1982).

The median Dt value is expected to be 50 days (Steel, 1977), 58 days (Charbit et al.,

1971); only 0.1% of the Dt having a value equal or lover than one week and equal or

higher than one year [Tubiana and Malaise, 1976; Steel, 1982).

Tumor size and Dt may vary independently [Steele and Buell, 1973; Kusama et al.,

1972; Tubiana and Malaise, 1976).

A short Dt is believed to indicate a bad prognosis, whether applied to small or to

large tumors. Nevertheless, a large tumor size is an adverse prognostic factor

irrespective of Dt, since it is very often associated with metastases to regional nodes

(Steele and Buell, 1973). Some authors have claimed that cancers which grow more

rapidly are more likely to metastasize [Gershon-Cohen et al., 1963; Slack et al., 1969;

Weiss, 1971; Chahinian, 1972].

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A relationship could be expected between the Dt value and the survival time (St).

Kusama et al. (1972) have shown that breast cancers having 17 days Dt 15

months lead to 2.3 St 9.3 years. Steele and Buell (1973) have shown that about

20% of patients carrying a primary bronchial carcinoma with a Dt lower than 120

days can survive more than 5 years, while up to 39% of those having the same type of

cancer but with a Dt greater than 135 days might survive more than 5 years.

According to Weiss (1971), at Dt values lower than six months will correspond a St

lower than 4.3 years, while at Dt greater than six months, the St would be greater.

Collecting data of about 180 cases of tumors of five different types [embrional,

haematosarcomas, mesenchimal sarcomas, squamous cell carcinomas and

adenocarcinomas] Tubiana and Malaise (1976) showed that a rough relationship does

exist between Dt and St. From their data, this relationship can be obtained:

St = 7.6172 + 0.1479 * Dt [20]

while the correlation coefficient is quite low (r = 0.93).

According to Steel and Lamerton (1966) the problem of measuring cell production

rate in a tumor is essentially the same as in normal tissues.

The growth kinetics of human tumors, as expected from the proposed model, are

obtained just assuming that the tumor mass grows like an human embryo, the rate of

growth being affected by the number of GP molecules till bound to the surface of the

clonogenic cell. It follows that cancerogenic cells having a GP level not very lover

than the one pertaining to normal cells are expected to grow at a rate not very greater

than the rate of the cells of the host. While the cancerous cells having much lowered

GP molecules level, they will grow very much faster. Anyhow, the reference rate of a

normal mammalian cell is the one expected by the model for a 20 years old person,

that is a cell having a Dt equal or greater than about 1,000 days.

Fig. 5 shows the growth curves of human tumors, as expected by the suggested

model. As can be seen, the curves are represented on the basis of the clonogenic cell

16

number changing as a function of the time, while in parallel ordinates the tumor

weight and the tumor diameter are reported too, for the sake of comparison with more

usual parameters [the latter case is based on the hypothesis of tumors having a

spherical mass].

On the same figure, the Dt values expected at the cell concentration of 109 /tumor

mass [i.e. at a tumor diameter of 1.24 cm on a spherical basis and at the weight of one

g) are also reported.

As can be seen from Fig. 5, the human tumor growth curves are concave upwards, as

expected, and cannot be recognized as exponential unless, according to Tubiana and

Malaise [1976] and Steel (1982) among others, only a limited portion of the curve

[i.e. perhaps one to three Log cycles, as a function of the growing rate of the tumor]

is considered. As previously said, the decreasing rate of the tumors – going from the

curves in the left side of the Fig. 5 towards those on the right side - is obtained

considering that the number of GP molecules still bound to the clonogenic cells

would increase, just coming from the curves in the left side of the Fig. 5, towards

those in the right side.

The number of GP molecules linked to the clonogenic cells changes very little,

although increasing, coming from tumors growing rapidly to the tumors growing

more slowly. At a cell number of 109 the number of GP molecules per clonogenic

unit ranges between 104.8136

and 104.8168

while the Dt ranges between 6.6 and 132

days, respectively. At the tumor cell number ranging from 109

[one g tumor, 1.24 cm

diameter) to 1011

[about 100 g tumor, 5.76 cm diameter], the GP molecules level

ranges between 6 * 104 and 1.5 * 10

7. The latter range is very close to the one

obtained by Salmon (1973) for multiple myeloma.

In a mammalian cell of a 20 years old person the GP molecules amount to 2-6 * 1010

/

single cell. Nevertheless, in order to describe the tumor growth kinetics according to

the suggested model, it is required that the cancerous cell had quite completely lost

the GP molecules linked to the membrane surface at the starting of the cancerogenic

17

process (101 –10

2 clonogenic cells), while increasingly producing GP molecules as

the tumor mass increases. A decrease of molecular equipment of such an extent must

inevitably follows some kind of genetic mutation, whichever the physico-chemical

agent would be primary responsible. Coming back to the above outlined mechanisms

of growth inhibition by cell-to-cell linking through GP molecules, the cancerous cells

had not completely lost the ability to synthesize this type of molecules, but produce

AGp molecules, quite unrecognizable by the cell surfaces of the host. Moreover,

since the number of AGp molecules does not differ significantly, though different

tumors are growing at very different rate, it follows that it can be expected that it

would be the chemical structure of the AGp molecules, the biochemical one basis of

the observed wide range of the clonogenic cell malignancy level, i.e. the wide range

of both Dt and St values, and mostly the metastatic power of the tumor.

The nature of the GP molecules and a therapy outline.

The suggested model cannot go deeper into the carcinogenic process. Nevertheless,

making an attempt to identify the nature of the above called GP molecules, I am very

inclined to believe, on several basis [Rosen et al., l973; Bramwell and Harris, l978,

1979; Ray et al., 1979; Omary et al., 1980; Gahmberg, 1981), that they are

glycoproteins. It can be expected that AGp molecules differ mainly in the

carbohydrate moiety from normal GP molecules, and that the extent of the difference

would be log-normally distributed,

on the analogy of the Dt distribution.

If this would be the case, whether contiguous cells have altered affinity for Gp and/or

AGp molecules, a suitable therapy could bring clonogenic cells to restore some kind

of bonds with neighboring ones, so that a tissue-like structure must be restored,

together with a cell metabolism though differing from the normal one to some degree,

so that the cancerogenic process would be slowed down to be stopped. The new-

18

linking process would be achieved by promoting modification (oxidation, reduction,

methylation?) of suitable AGp molecules site.

The human tumor doubling time.

According to the suggested model, tumors having the same characteristics, as it is

expected for single metastases ( Spratt and Spratt, 1964; Tubiana and Malaise, 1976),

it must have the same Dt, while at the same level of the clonogenic cells number.

Otherwise, also tumors having the same characteristics must have a Dt value

decreasing with the increasing tumor mass.

Nevertheless, according to the literature (Kusama et al. 1972; Steele and Buell, 1973;

Tubiana and Malaise, 1976) there is no correlation between tumor size and Dt,

because at the same size the Dt value is a property of physiological characteristics of

the tumor considered. Such a situation is well described with reference to the Fig. 5.

The metastatic condition.

According to the model, the probability of finding tumor metastases is expected to be

higher in the hosts carrying a tumor having a low Dt (i. e. a high doubling rate), since

high doubling rates are expected to be coupled with highly modified GP molecules,

as said above, and thus with clonogenic cells having a greater motility owing to the

lower contact-inhibition. Several authors besides (Gersom et al, 1963; Slak et al.,

1969; Weiss, 1971; Chahinian, 1972) proved that tumors growing more rapidly are

more likely to metastasize.

The survival time.

Of quite relevant interest is the relationship always shown by the majority of the

clinicians, between the doubling time and the probability of survival, often reported

19

as survival time, St. It is quite useless to make a comparison between the clinical data

and those expected from the model, since the physical, surgical and pharmacological

treatments applied to the patients, inevitably affect the survival time, even though at a

limited extent. Nevertheless, from a pure theoretical standpoint, the two parameters

are expected by the model to be linked by a linear relationship. Tubiana and Malaise

(1976) have outlined a linear relationship between Dt and St, although with a low

correlation coefficient, as seen above. As a matter of fact, the St value is highly

affected by actual tumor mass at the time at which the tumor itself reaches the clinical

level, besides by the Dt.

Furthermore, the mass reached by the tumor at the time of death is not unequivocally

defined, although several authors do agree with a tumor mass very close to the one

gained by an embryo at the birth, saying at about 1012.5

clonogenic cells level. For

computing convenience, I fixed the tumor cells number leading to the death of the

host at 1012.64

cancerous units and I considered four values of the tumor mass at

which the cancer gains the clinical level [another value questioned by the clinicians,

and really linked to the malignancy level of the tumor], i.e. 107.82

, 108.73

, 109.93

and 1010.84

cancerous cells. The above clonogenic cells levels correspond to tumor

mass, assumed to be spherical, having respectively a diameter of 0.5, 1.0, 2.54 and

5.08 cm. The relationships expected from the model between Dt and St are the

following:

a t = 0 . 5 cm:

St = 0.0131 + 1.887 * Dt

at = 1 . 0 cm:

St = - 0.0003 + 1.2001 *Dt

at = 2 . 5 4 cm:

St = 0.0103 + 0.6103 * Dt

20

at = 5.08 cm:

St = 0.0122 + 0.3283 * Dt

being always the St value expressed in months and the Dt value in days.

Graphical representation of the above relationships is reported in Figs. 6 and 7.

As it results from the above equations and from the Figs. 6 -7, the most relevant

parameter affecting the host survival is the Dt, which means the extent of the ability

acquired by the tumor to escape the contact inhibition, and ultimately the extent of

the difference between GP and AGp molecules.

As may be seen from Figs. 6-7 too, the earliest the cancer could be evidenced, as

besides it is well known, the longer will be the time available for the cure.

The silent growth

It is a widespread practice in finding a tumor growth curve [or very often a sketch of

curve] to extrapolate to the initial size of the first small clone of cancerous cells,

aiming to define the time at which the tumor induction has been completed.

Really, as stated by Steel and Lamerton (1966) “.. the total portion of a growth curve

which can normally be obtained from a human tumor, is insufficient for more than

the first differential (slope) and the second differential (curve) as wells to be

established in a semi-Log plot”. From the growth data of an experimental multiple

myeloma Salmon (1973) obtained by backward titration a “silent time” [the time

during which the tumor grows at sub-clinical level] of about 20 years on exponential

basis, and only 1-2 years on Gompertzian basis. The above values show very well the

magnitude of the range over which does crowd the actual uncertainty.

According to the suggested model, the carcinogenic process had begun, as shown in

Fig. 8, not more than few years before it gained the clinical level, quite irrespective of

the magnitude of the tumor at the time at which the Dt is recorded.

21

Being the average Dt value of the human tumors about two months (Steel, 1982),

backward extrapolation yields a mean silent time of 1.5 years.

Radiotherapy and chemotherapy

Experimental inactivation curves of animal and human tumor cells have typical

shapes of the inactivation curves already discussed for prokaryons (Casolari, 1981),

i.e. they are sigmoid, exponential or concave in semi-log plot, according to the

physico-chemical agent applied and the cycle phase of the growing cells (Breur,

1966; Van Peperseel, l972; Malaise et al., 1972; Van Peperseel and Breur, 1974;

Skipper and Shabel, 1982).

The aim of the radiotherapy is to kill the greatest number of the tumor cells while

sparing the normal tissue as much as possible (Tubiana, 1971). Taking into account

the very moderate inactivation of the cancerous cells brought about by radiation and

currently used chemicals, very often restricted to only 2-5 Log cycles (Denekamp and

Fowler, 1977; Fowler and Denekamp, 1977; Hall, 1977; Kallman and

Rockwell,1977; Skipper and Shabel, 1982), the aim to cure the cancer by available

physical and/or chemical means alone can be regarded as quite hopeless. Combined

treatments would be expected of greater effectiveness, above all if coupled, when

possible, with the surgical practice

[Corbett et al., 1975; Fugman et al., l970; Griswold and Corbett, l976; Skipper and

Shabel, 1982]. It is well known that few surviving clonogenic cells would be the

starting points for the quite unavoidable recurrence phenomena (Skipper et al., 1964;

Wilcox, 1966; Brockman, 1974; Shabel, 1975). So, the applied therapy must kill the

highest number of cancerogenic cells. As it is extensively discussed otherwise for

prokaryions [Casolari, 1981], the destruction of the highest number of cells – up to

the highest attainable – can be obtained only by increasing at will – when feasible –

the killing power of the applied agents. The recurrence phenomena can be avoided

22

only by decreasing the survival probability of clonogenic cells to the lowest possible

level. And this target can be obtained with the highest probability by therapies

employing multiple physical and/or chemical agents, simultaneously.

Only simultaneous application of lethal - radiative/thermal/chemical - agents could be

effective in killing tumor cells.

Carcinogens and carcinogenesis

The known carcinogens are expected to bind to the DNA. If the above suggested

model is not very far from the reality, the cancerogenic substances are such

chemicals able to link to DNA sites coding for the chemical characteristics of the

GP molecules and mostly their carbohydrate moiety.

Conclusions

According to the model, the growth rate of the human cell decreases continuously,

beginning from the embryonic level of 32 cells. There is only one requirement that

must be met in order to describe the human body and the human tumor development

curves: the continuously decreasing growth rate of the mammalian cell must be due

to the extent of the cell-to-cell adhesion, causing an increasing inhibition of the

metabolic activity. The inter-cells linkages would be brought about by molecules

linked to the outer surface of the cell membrane. The above molecules are expected

to be glycoproteins.

The carcinogenic substances interact with the DNA fraction coding for the surface

glycoproteins and probably the carbohydrate moiety. The cells carrying modified

glycoproteins at the surface become clonogenic to a more or less degree, according,

to the extent of the modification occurred in the glycoproteins at the cell surface.

Meeting this requirement, all relevant observations concerning human growth curve,

23

(senescence), death and the human tumor growth behavior can be described. From

the above single basis, a therapeutic approach to tumor diseases can be envisaged.

REFERENCES

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