^{1}

^{*}

^{2}

^{2}

In biological systems, self-time differs from the physical time scale. Biological self-time is the characteristic of individual biosystems and invariantly describes the various dynamic processes in organisms. This biological invariance is introduced by using 2 basic theorems: Allometric scaling and the Weibull psychometric function. Our objective was to precisely describe the timing phenomenon in biosystems and provide a framework to further develop this analogy for other fields.

The lifetime of biological objects is measured by their dynamical development. Studies have a surprising universality by the self-organizing [^{−8} oxygen molecules per heartbeat [

T l t = n h b / l t / f h ∝ M 1 / 4 (1)

Self-time can be introduced on a thermodynamic optimizing basis when entropy production is constant over time [

Our objective is to study the self-time of biological objects in comparison to the coordinate time (clock time) measured by the observervations outside the studied bio-system, to develop an analogy for brittle materials by introducing the Weibull function (cumulative form of two-parametric Weibull distribution) for survival times. We transform time-scales and show the accuracy of the fit of Weibull based calculation to standard allometric scaling.

The generalized cumulative Weibull distribution [

W ( x ) = exp ( − ( x x 0 ) n ) , (2)

where n is the shape factor, or form parameter, and x_{0} is the scale parameter. This function is widely used in physiology and psychometry [

Weibull’s statistics was originally developed to describe the fracture of brittle materials [_{S}) and the survival rate probability define the percentage of sample breaks when multiple probes are tested. When processing the statistical data, the cumulative probability of survival at a homogeneous σ stress on a sample with V volume is:

P S ( σ , V ) = exp ( − V V 0 ( σ σ 0 ) n ) , (3)

where V_{0} is the reference volume, σ_{0} is the reference stress, and n is the Weibull’s form factor of the function. When experimenting with N samples, NP_{S} pieces of this count will not fail even under the maximum stress of σ. The reference stress and volume should be introduced into the exponent for dimensional reasons. When we have a sample where V = V_{0} and σ = σ_{0}, and these are both independent from the form factor n, by solving the Equation (3) we obtain:

P S 0 = exp ( − V 0 V 0 ( σ 0 σ 0 ) n ) ≅ 0.3678. (4)

Therefore, the survival probability of such a sample is 36.78%.

The above Weibull function (3) can be rewritten by using the familiar 2-parametric shape:

P S ( σ , V ) = exp ( − V 1 / n ( V 0 1 / n σ 0 ) ( σ x 0 ) n ) . (5)

The relationship between the individual scale parameter characteristics can be readily associated with the Weibull statistics (2) used in the strength theory:

x = σ V 1 / n and x 0 = σ 0 V 0 1 / n . (6)

It is possible to determine the survival probability of the same stress and volume relationship, i.e., the scaling law from the Weibull distribution in (3), when we have the same survival probability:

σ = c o n s t V 1 / n , (7)

which means a smaller sample is stronger and stabler than a larger one.

The Weibull function is based on 2 primary properties: volume dependence and self-similarity. To formulate the above equations in another way, imagine, that a sample is composed of 2 parts (

By applying the Weibull Function (3) to the scenario shown in _{1} + V_{2} sample volume is:

P S ( σ , V ) = exp ( − V 1 + V 2 V 0 ( σ σ 0 ) n ) = P S ( σ , V 1 ) P S ( σ , V 2 ) . (8)

From (7), sample failure occurs at the weakest location (the structural fault location). In the Weibull theory, the structural failure probability is proportional to the sample volume. The effect of the form factor is shown in

Step-function occurs at large form factors. In other words, if σ ≤ σ_{0}, each sample component survives, however, when σ > σ_{0}, the entire sample fails. Materials composed of links with equal strength, that is, homogeneous chains like metals, show such behavior.

A recent study attempted to describe the connection between fractal geometry and the circulatory system with Weibull’s survival function by using the age of the patient and body weight as its parameters [

(ontogenetic growth [

The above assumption is not far from strength theory, where repetitive stress would likely be the best parameter to statistically investigate and characterize failure. This characterization is called the fatigue test. When each stress cycle takes a nearly identical course, the measurement can be reduced to a single parameter, and the load is characterized by the spending time. Statistical functions can describe both inanimate objects and organisms; although living organisms are far more complex than objects described with mechanical rheology. Indeed, we found many similarities between machine parts and the organs of a living body by using lifespan approximations with the Weibull function [

Malignant tumors are the result of altered biological activity in an organ. Tumor dynamism overloads normal systemic functions by their aggressive demands for nutrients and thus alters organ function. Hence, in addition to the tumor’s metastasis, there are 3 direct reasons why organ failure may occur due to this malignant process:

• the cancerous metabolism overwhelms normal physiological consumption;

• tumor metabolism becomes comparable to the host organ metabolism;

• the geometric scale of the tumor makes normal organ function impossible.

Consequently, tumor size influences the individual’s lifespan. Tumor volume is related to the structural failure probability. This parameter is constant over time for healthy individuals; however, the tumor size is time dependent. Therefore, there is a qualitative difference between the volumetric parameters of the Weibull law and the so-called “faulty volume” of the disease. Naturally, the growing cancerous tumor reduces survival probability. The increased, faulty tissue volume increases the probability of structural incompatibility problems and stress caused by the growing tumor increases the probability of failure. Thus, the analogy between mechanical failure and survival is as follows:

• mechanical stress, σ, is analogous to survival time t (or observation time during treatment);

• the reference stress, σ_{0}, is analogous to the scale parameter t_{0} (the reference time in the therapy, when the survival probability is 36.8%);

• the volume, V, is analogous to the actual size (volume) or the actual metabolic activity (energy consumption rate) of the tumor, G(t). This function is additive and could be the volume or the metabolic rate. Both parameters are additive in the destructive process;

• the reference volume, V_{0}, is analogous with the reference size (volume) or the reference metabolic activity (energy consumption rate) of the tumor (G_{0}).

Hence, the Weibull law (as in (3) and (2)), which corresponds to survival by time (t) distribution is:

W ( t ) = exp ( − ( t t 0 ) n ) ; (9)

alternatively, when considered in the context of an administered therapy, it is analogous with (3):

P S ( t ) = exp ( − G ( t ) G 0 ( t t 0 ) n ) . (10)

As discussed above, tumor size could be geometric (volume) or metabolic (energy consumption). Extra energy demands are due to the high proliferative and metabolic processes of cancer cells. Indeed, tumor energy consumption may be several times higher than the metabolism of healthy cells, and energy is supplied by intensive glucose production from non-oxidative glycolysis [

The size of the host organism and its net metabolic rate (basal metabolic rate, R_{0}) can be expressed by the actual body mass (m) of the healthy adult individual, as introduced by the allometric considerations (allometric law [

R 0 ∝ m α . (11)

In case of satisfactory nourishment, α = α 0 = 3 / 4 [

R 0 M ∝ M α − 1 . (12)

These allometric considerations could be applied for organs, where M is the final, stable organ mass, and R_{0} is the organ’s basal metabolic rate.

Tumor growth satisfies the allometric metabolic rule; that exhibits universal growth dynamics [_{c}) depends on production (proliferation, P) and cell death (annihilation, A); A is proportional to n_{c} as shown below:

d n c d t = P − A and A = λ n c , (13)

where λ^{−1} = T is the average life of the cell.

Energy balance has 3 components: energy to support current cells (the metabolic rate of a single cell, R_{c}), energy to produce new cells (E_{c}) and external work (W_{e}) on the system. All of these factors originate from the energy flux of blood flow (I) through capillary terminals:

I = R c n c + P E c + W e = R c n c + ( d n c d t + λ n c ) E c + W e . (14)

Due to allometric considerations [

I = R 0 n c α and W e = C 0 n c α , (15)

where α ≤ 1 [

E c d n c d t = ( R 0 − C 0 ) n c α − n c ( R c + λ E c ) . (16)

Multiply by the mass of an individual cell (m_{c}):

d m d t = a m α − b m (17)

a = ( R 0 − C 0 ) m c ( 1 − α ) E c , b = R c E c + λ .

Due to energy flux changes by the α power of the mass during metabolism to maintain homeostasis, (dm/dt) is positive, and thus the maximum mass (M) is limited and can be expressed by the mass of a cell (m_{c}):

a M α − b M = 0 → M = ( a b ) 1 ( 1 − α ) = ( R 0 m c ( 1 − α ) R C + λ E C ) 1 ( 1 − α ) (18)

When substituting b from (18) into (17), we get the Verhulst-Pearl differential equation [

d m d t = a m α ( 1 − ( m M ) ( 1 − α ) ) . (19)

The solution of this differential equation describes a sigmoid curve:

( m M ) 1 − α = 1 − e − τ , (20)

where

τ = a ( 1 − α ) t M ( 1 − α ) − ln ( 1 − ( m 0 M ) ( 1 − α ) ) ; (21)

moreover, m_{0} is the initial (birth) mass.

The ratio of energy flux for stationery stabilization and metabolism is:

r ( t ) = n c R c I = m R c m c R 0 m α = b a m 1 − α . (22)

Using (18) and (20), the relative metabolic rate for maintaining stationary equilibrium is:

r m ( t ) = ( m M ) 1 − α = 1 − e − τ ; (23)

moreover, the universal energy-function used for growth is:

R m ( t ) = 1 − r ( t ) = e − τ . (24)

r(t) and R(t) are the same functions of τ for all organs with m mass; the time-scale τ may be regarded as a biological self-time. τ is invariant for the organ or organism; it is a time scale that is determined wholly by the biological system and not by any outside processes. The existence of physiologic time, which is different from clock time, was previously hypothesized [

Equation (20) is a universal function, and so the above considerations are well suited to living organisms [

d 2 m d t 2 = 0. (25)

Here, mass and growth rate are:

m = ( 3 4 ) 4 M ≅ M 3 ; (26)

d m d t = 27 a 256 M 2 / 3 ≅ 0.1 a M 2 / 3 ,

when α = α 0 = 3 / 4 . Thus, relative metabolic rate (r_{m}) and the part of metabolism related to growth ( R m = 1 − r m ) are:

r m = 3 4 and R m = 1 4 . (27)

Several investigations have estimated that 2/3 ≤ α ≤ 1 depends on the fractal geometry of the nutrient supply of the tumor [

Identical tumor size (G(τ) = G_{0}) and biological self-time (τ = τ_{0}) could differ from the physical time (t) periods, and the survival probability (P_{S}) in this situation would be approximately 36.8%. However, the connection between the two-time scales, according to (20), is:

t = M ( 1 − α ) a ( 1 − α ) [ τ + ln ( 1 − ( m 0 M ) ( 1 − α ) ) ] . (28)

The difference depends on the metabolic rate per unit volume, the initial size of the tumor and actual vascularization. Generally, scaling biological self-time shows the same survival probability of lifespan from (10) to any actual probability:

G ( τ ) G 0 ( τ τ 0 ) n ≅ c o n s t . (29)

Hence,

τ ≅ c o n s t τ 0 ( G ( τ ) G 0 ) 1 n . (30)

Thus, survival time decreases as tumor size increases. The higher the value of m, the more it reduces the dependency rate. The physical timescale using (30) is:

t ≅ c o n s t M ( 1 − α ) a ( 1 − α ) [ τ 0 ( G ( τ ) G 0 ) 1 n + ln ( 1 − ( m 0 M ) ( 1 − α ) ) ] . (31)

If we substitute (18) into (31) we get:

t ≅ c o n s t 1 b ( 1 − α ) [ τ 0 ( G ( τ ) G 0 ) 1 n + ln ( 1 − ( m 0 M ) ( 1 − α ) ) ] . (32)

Consequently, if the enlarged initial mass decreases, the surface supply increases the physical lifetime. It is also clear from (32) that individuals with higher G(τ) would exhibit a reduced survival, especially when n is relatively small.

From above, the Weibull law is applicable to biological self-time and also correlates with physical time. Lifespans connected to the same survival probability have a dependence as shown in (10) and similarly to (29):

G ( τ ) G 0 ( t t 0 ) n ≅ c o n s t , (33)

from which we conclude, that survival time decreases as tumor size increases:

t ≅ c o n s t t 0 ( G ( t ) G 0 ) 1 n . (34)

The greater the value of n, the more independent t is of G.

We know from [

Φ = c m α , (35)

where c is a constant with a peculiar dimension, that is, the nutrient flux per unit weight of the system. This is not a good measure since it is not additive. The additive rate is:

G ( t ) : = Φ 1 α = c 1 α m ( t ) and G ( τ ) : = Φ 1 α = c 1 α m ( τ ) ; (36)

alternatively, any of its homogeneous linear function.

From a mathematical point of view, 2 cases will produce a biparametric Weibull function:

G ( τ ) ≅ c o n s t = K or G ( τ ) ≅ τ δ . (37)

The G(τ) ≈ const case occurs when the tumor has almost reached its final size. This scenario may occur if tumor vascularization cannot sufficiently supply the required nutritional demand and/or due to interventional therapy. However, here α ≈ 1, denoting conformity with surrounding surface supply. This case is termed as a stagnant cancer. The second case is equivalent to the long physical time related to short biological time. This phenomenon could occur with a large M (matured tumor mass) or when the tumor supply is volumetric. τ could be m_{0} ≈ 1 and α ≈ 1; hence, the observed real-time survival would be accompanied with a very short biological self-time. This case represents a rapidly growing tumor.

When τ is small, the Equation (20) is well approximated by the linear function 1 − e − τ ≅ τ . Comparing this biological self-time scale set to the allometric law, the rate of weight is:

G ( τ ) = m ( τ ) = M ( ( m M ) 1 − α ) 1 1 − α ≅ M τ 1 1 − α . (38)

Substituting (37) for the Weibull function using the analogy of biological self-time, we get:

P S = exp ( − M M 0 ( τ τ 0 n / q ) q ) = exp ( − ( τ τ 0 n / q ( M M 0 ) 1 / q ) q ) (39)

q = n + 1 1 − α ,

where the selected reference is the metabolic flux of the tumor with the power α, and M_{0} corresponds to the healthy reference G_{0}. If the tumor is near to the entirely surface-determined supply (α ≈ 1), the exponent will be large, the phenomenon will become digital (1 or 0, e.g., all or none) and the deviation of survival times will be low. The scaling law in this case would be:

τ ≅ c o n s t ⋅ τ 0 1 − α ( M 0 M ) 1 q . (40)

This equation again shows that in the case of tumor surface feeding, survival time becomes independent of the parameters.

When the first condition of (37) is valid, then:

τ ≅ c o n s t ⋅ τ 0 ( M 0 M ) 1 n . (41)

When the second condition of (37) is valid (metabolic rate is high), then:

G ( τ ) = a 1 / α m ( τ ) = a 1 / α M ( ( m M ) 1 − α ) 1 1 − α ≅ a 1 α M τ 1 1 − α . (42)

When we substitute (41) into the Weibull function obtained by using the biological self-time analogy, we get:

P S = exp ( − a 1 / α M a 0 4 / 3 M 0 ( τ τ 0 n / q ) q ) = exp ( − ( τ τ 0 n / q ( a 0 4 / 3 M 0 a 1 / α M ) 1 / q ) q ) . (43)

Concerning the metabolic flux of the reference tumor with allometric exponent α, we used α_{0} = 3/4 for the healthy (τ_{0}, reference) exponent.

It again appears that at a near surface supply condition, the exponent will be large and a digital behavior will occur. From this point of view, the scaling law is:

τ ≅ c o n s t ⋅ τ 0 1 − α ( a 0 4 3 M 0 a 1 α M ) 1 q (44)

Again, the above equation describes universal behavior, namely, that survival time increasingly becomes independent in the case of surface supply. According to the second equation of (42), a simple deduction to the real physical time scale from the above equation to describe biological self-time leads to:

τ ≅ a ( 1 − α ) t M 1 − α = b ( 1 − α ) t . (45)

Substituting (44) into (43), we get the Weibull law associated with the physical time scale:

P S = exp ( − a 1 / α M a 0 4 / 3 M 0 ( τ τ 0 1 − α ) q ) = exp ( − ( t τ 0 1 − α b ( 1 − α ) ( a 0 4 / 3 M 0 a 1 / α M ) 1 / q ) q ) . (46)

Consequently:

t ≅ c o n s t ⋅ τ 0 1 − α b ( 1 − α ) ( a 0 4 3 M 0 a 1 α M ) 1 q . (47)

We again see that the supply tends to become localized at the surface when α → 1, and so the growth of the tumor becomes independent of the survival parameters.

Several theories describe the growth of cell populations (including tumors). The time dependence of tumor size has several competing theories, but recently hyperbolic growth models have been proposed [

d ( m M ) d t = ( 1 − m M ) λ β λ t λ − 1 , (48)

where M is the final tumor size. This equation is actually the Weibull equation. The solution is:

m M = 1 − ( 1 − m M ) exp ( − ( t β ) λ ) ; (49)

or slightly modified:

m M = 1 − e − τ ; (50)

τ : = ( t β ) λ − ln ( 1 − m 0 M ) .

If m 0 ≪ M , then:

m M = 1 − e − ( t β ) λ , (51)

and the equation is returned to the well-known Weibull law. Considering (20), the allometric law could be formulated as:

m M = ( 1 − e − τ ) 1 / ( 1 − α ) . (52)

A simple comparison shows the validity of the approximation of the allometric law using the Weibull’s function (

The Shannon entropy of the Weibull functions for fits to α = 2 3 and α = 3 4 are s 2 / 3 ≅ 1.403 , and s 2 / 3 ≅ 1.452 ; respectively. This ≅ 3.5% difference shows a more certain death (less lifetime) in the α = 2 3 case than in α = 3 4 , because the growing entropy shows a growing uncertainty.

This observation reveals a more profound relationship and can be proven by rigorous mathematical calculation. If α and β are large values in (50), then the following linear approximation is valid:

m M ≈ ( t β ) λ . (53)

Thus, rapidly growing tumors could also be described by the Weibull evolutive equation. Note, that the probability of cumulative survival covariates with the normalized tumor size, i.e.:

m M = 1 − e − ( t β ) λ ≅ F S ( t ) . (54)

The same can also be formulated in a more approximate version. The cumulative survival probability is the self-similar function of (1 − m/M). In other words, the slopes of Equations (55) and (56) are equal, and the straight lines are parallel.

ln ( t ) ↦ ln ( ln ( F S ( t ) ) ) ; 55)

ln ( t ) ↦ − ln ( ln ( 1 − m M ) ) . (56)

This formulation allows the fitting of the β scale parameter of survival to the final mass of the tumor (or to fit other tumor characteristics). In other words, ln ( F S ( t ) ) is a self-similar function of ln(1 − m/M), so:

ln ( ln ( F S ( t ) ) ) = K 1 ln ( K 2 ) ln ( 1 − m M ) , (57)

where K_{1} and K_{2} are constants.

We proposed an intrinsic time model that differs from clock time. We analyzed the self-time for tumor growth and showed its scaling based on allometric scaling as well as compared it to the Weibull physiologic function. The allometric function could be well approached by the Weibull function which highlights the intrinsic values of both the bioscaling and the Weibull physiologic function for living objects.

This research was supported by a Hungarian Competitiveness and Excellence Program grant (NVKP_16-1-2016-0042).

The authors declare no conflicts of interest regarding the publication of this paper.

Szasz, O., Szigeti, P.G. and Szasz, A. (2019) The Intrinsic Self-Time of Biosystems. Open Journal of Biophysics, 9, 131-145. https://doi.org/10.4236/ojbiphy.2019.92010