Nutrient supply, cell spatial correlation and Gompertzian tumor growth.

Gompertzian tumor growth can be reproduced by mitosis, related to nutrient supply, with local spatial cell correlations. The global energy constraint alone does not reproduce in vivo data by the observed values of the nutrient expenditure for the cell activities. The depletion of the exponential growth, described by the Gompertz law, is obtained by mean field spatial correlations or by a small word network among cells. The well-known interdependence between the two parameters of the Gompertz growth naturally emerges and depends on the cell volume and on the tumor density.

Introduction

Tumor growth is a complex phenomenon which includes sustaining proliferative signaling, evading growth suppressors, resisting cell death, inducing angiogenesis and activating invasion and metastasis (Hanahan and Weinberg 2011).

The tumor evolution (size, chemotherapy and radiotherapy effects) can be described by macroscopic growth laws, coarse-grain approximations of complex cell dynamics at microscopic level.

In particular, the Gompertz law (GL) has been extensively applied after the seminal paper by Norton (1988) on breast cancer and confirmed by a recent analysis (Vaghi 2020) as a very useful tool for a quantitative understanding of the tumor growth.

The GL, originally formulated as an actuarial curve for the population of England almost two centuries ago (Gompertz 1825), describes the dynamics of a variety of natural phenomena : magnetic hysteresis (Stauffer and Stanley 2017), kinetics of enzymatic reactions (Murray 1989), oxygenation of hemoglobin (Murray 1989), intensity of photosynthesis as a function of concentration (Murray 1989), drug dose–response curve, dynamics of growth (e.g., bacteria, normal eukaryotic organisms and cancer) (Laird et al. 1965a, b; Calderon and Kwembe 1991; Waliszewski et al. 1998), spread of COVID-19 (Castorina et al. 2020).

For human mortality, a derivation of the GL can be obtained by the reliability theory designed for man-made machines (Gavrilov and Gavrilova 2001) and by a different model (Shklovskii 2005) which relates the human survival probability with an exponentially rare escape of abnormal cells from immunological response.

Possible theoretical bases of the Gompertz growth for biological systems have been addressed in the literature, since long time and from different points of view (Wheldon 1988; Bajzer and Vuk-Pavlovic 1997; Savageau 1979; Witten 1985; Kendal 1985; Frenzen and Murray 1986; Gyllenberg and Webb 1989; Makany 1991; Ling and He 1993; Qi et al. 1993; Bajzer 1999; Afenya and Calderon 2000; Bajzer and Vuk-Pavlovic 2000; Mombach et al. 2002; Waliszewski and Konarski 2003). More recently, the GL has been discussed by a biochemical approach (Anguelov et al. 2017) and by statistical mechanics methods (Castorina and Zappala 2006), where the energetic balance is considered as the key dynamical ingredient (for a recent review of tumor growth laws, see ref. Jarrett (2018)). Indeed, the idea that the GL dynamics can be considered as an optimization problem, as, for example, an “energy” budget problem (where “energy” can have many different meanings) could explain why it emerges at macroscopic level for so different systems.

In the general classification of growth laws (Castorina et al. 2006), the GL is in the U2 universality class, i.e., it depends on two parameters, with a strong reduction with respect to the large number of dynamical variables of the microscopic models.

A crucial feature of the macroscopic evolution laws is that the feedback effect is global: It depends on the total population N at time t, whereas the microscopic dynamics is local, due, for example, to competition for resource. This aspect becomes more tricky for the GL where the global dependence has the typical nonlinear ln[N] behavior.

In this letter, we study cancer growth by considering mitosis with specific cell local spatial correlations which globally reproduce the GL. The purpose of the analysis is twofold. One shows the crucial role of cell spatial correlations to reproduce in vivo data by GL with realistic values of the growth parameters. Indeed, the model based on the nutrient supply balance, without the local dynamical competition for resources, is not able to describe the tumor progression with the observed cell metabolic activities. Moreover, it is well known that the two GL growth parameters are linearly correlated and this property naturally arises by the link between the spatial correlation effects and cell properties.

Although the language of a biological system is used, the results are more general and can be applied to any local bifurcation process with spatial correlation among participants.

In the next section, we recall the GL, and in Sect. 2, the models are proposed. Section 3 contains our comments and conclusions. “Appendices A and B” are devoted to mathematical details and to the time evolution based on the global nutrient supply constraint only.

Gompertz growth laws

General macroscopic growth laws for a population N(t) are solutions of the differential equation (for a classification see Castorina et al. (2006))where f(N) is the specific growth rate and its N dependence describes the feedback effects during the time evolution. If constant, the growth follows an exponential pattern, with no limit for . On the other hand, a saturation is obtained by the Gompertz equation, i.e.,where and are constants and is the initial value. By definingone getswhere is the carrying capacity, i.e., the steady state is reached for , when N is equal to .

A different, equivalent, formulation of the Gompertz equation isIndeed, the solution of the previous equation turns out to bewhich givesBy Eqs. (6, 7), one easily obtains Eq. (2) from Eq. (5) and identifies with and .

In the next sections, n(t) refers to the number of cells at time t, since we study the tumor mass progression.

Nutrient supply and cell spatial correlations

A common feature of cancer cell metabolism is the ability to acquire necessary nutrients from a frequently nutrient-poor environment and utilize these nutrients to both maintain viability and build new biomass (Pavlova and Thompson 2016; Fernandez-de-Cossio-Diaz and Vazquez 2017; DeBerardinis et al. 2008).

After the initial phase, the total amount of nutrients is not able to sustain the exponential trend of the complete population of mitotic cells and angiogenic stimulator controls are crucial ingredients to increase the energetic balance. The extremely large nutrient consumption of tumor cells originates a competition for resources which is local and spatially correlated. Different spatial correlations give macroscopic laws with various global and nonlinear feedback effects.

Let us assume, without loss of generality, that is the end of the exponential phase with cells and let us call the average duplication time of the total number of mitotic cells (which is different from the duplication times of a single cell and of the whole tumor). Let us call with the time intervals and n(k) the cell population at the beginning of interval k.

After the exponential phase, only a number of cells smaller than n(k) can replicate and, according to the previous definition, is the time when this condition starts. Apoptosis is initially neglected, and its role in increasing the available nutrients for cell mitosis will be discussed later (see “Appendices A and B”).

Since a limited number of mitosis are possible, let us call the number of quiescent cells and the number of duplicating cells during the interval k, that isThe average number of cells at the beginning of interval is therefore:The local competition for growth implies a spatial correlation, which decreases with the distance between pairs of cells and plays a crucial role. In fact, in Appendices A and B one shows that a model of the tumor evolution, based on the nutrient budget without spatial correlations, is not able to reproduce by GL the, in vivo, breast cancer data with realistic values of the growth parameters. In the next subsections, two different spatial correlation models are discussed from which the GL emerges at global level.

Mean field approximation

According to previous discussion, let us write the number of mitotic cells as a fraction of the total number and a depletion due to spatial correlation effects:where describes the exponential phase (see “Appendix A”), is a function of the distance between cell pairs (i, j), , is the Kronecker delta, is a dimensionfull constant related to cell size, and is the cell fraction competing for nutrient supply.

A model with competition depending on a power law of the cell distance has been proposed in Ref. Mombach et al. (2002), where different growth laws are determined by the relation between the power of spatial correlation, , and the, assumed, fractal dimension of the cellular structure. Here, we explicitly show the dimensionfull constants in Eq. (10), interpreted in terms of the typical cell size and exponential rate, with a clear understanding of the interrelation among growth parameters that is a crucial aspect of the tumor Gompertzian progression (Vaghi 2020). Moreover, no fractal dimension has been introduced since the GL is obtained, in the mean field approximation, without geometrical self-similar pattern in the branching process.

For a spherical and homogeneous system in 3D, let us assumeand let us identify with the cell volume.

For large n, it turns out (Mombach et al. 2002)where is the constant density, R is the time-dependent maximum size of the system, and is a minimum characteristic length (of order of cell size, ).

For the considered system, one hasand, by Eqs. (9–12), in the mean field approximation (Mombach et al. 2002), one getswithandin time unit.

The comparison with the GL immediately identifies (taking into account the correct time units, ) , . Moreover, by Eq. (14) one getsThe consistent identification of the parameters can be checked by in vivo data. Indeed, for breast cancer (Norton 1988) the parameter turns out to follow a log-normal distribution function with average and variance 0.71 (with time in month unit) and then (month . Moreover, cells (in 1 liter), corresponding to the density cells and the typical average cell radius is . By Eqs. (14–17), one gets ():

a) linear correlation between and This is a crucial aspect of the application of GL to cancer growth: The two parameters turn out to be linearly correlated for different tumor phenotypes (Wheldon 1988; Vaghi 2020);

(b) (month), ranging from for day to for days. For the initial value (Norton 1988), the number of mitosis in is about .

Finally, let us notice that the dependence of leads to a generalized logistic growth law.

A small world network correlation

The function and behavior of any given tumor cell are affected by interactions with its neighboring cells which send and receive messages in the form of direct contacts and secreted signaling molecules. This dynamics is limited to small distances from the cell position, and in this sense, a solid tumor can be considered a system of groups of interacting cells with global effects transmitted by cell to cell. According to this point of view, an advanced solid cancer can be qualitatively seen as a cell system with clustering and small correlation length.

In network analysis (Latora 2017), those properties identify the, so called, small world (SM) network (Watts and Strogatz 1998), where the clustering coefficient and the correlation length, L, have a precise quantitative definition in terms of number of nodes and links. In this respect, the suggestion is that, analogously to other biological systems, the cell spatial correlation can be described in terms of small world network, i.e., local connections and a small number of steps to reach distant cells. The average spatial correlation among cells is, accordingly, related to the average geodesic (i.e., shortest path) length L in the SM network, defined bywhere is the shortest geodesic distance between nodes (cells) i and j. In SM, for , one has (Watts and Strogatz 1998; Latora 2017)By taking into account the maximum volume V for a finite, homogeneous, system, let us write the number of mitotic cells as a fraction of n(t) with a decreasing contribution due to SM network:where is a dimensionfull constant. By Eqs. (19, 20) and , for large n, one getsand the specific rate turns out to beThe comparison with the Gompertz equation givesandcompletely consistent with the condition for , (Norton 1988) and the previous value of .

Apoptosis

Cell death, in the form of apoptosis, and cell proliferation are linked by biochemical effects. Moreover, in the energetic balance, apoptosis permits to transfer some nutrient from metabolic activity to mitosis as quantitatively expressed in “Appendices A and B.”

However, without spatial correlations, the effect of apoptosis on the global nutrient balance does not solve the problem of the consistency of the GL growth parameters to fit in vivo data with the observed values of the metabolic tumor cell activities (see “Appendices A and B” ).

Comments and Conclusions

The success of macroscopic growth laws in describing the time evolution of dynamical systems in many different sectors is astonishing. Indeed, with a small number of parameters one can fit large sets of data. Even more surprising is that they contain global and nonlinear feedback effects, although the microscopic dynamics is, in general, local. In other terms, the coarse-grain patterns catch the most important aspects of the underlying phenomena, strongly reducing the large number of parameters typical of the microscopic models.

The proposed results suggest that a branching process with global constraint on the energy budget is not able to reproduce breast cancer in vivo data by GL if the competition for resources is not included: The local spatial correlation among cells is a crucial ingredient to obtain the observed progression.

The linear interdependence of the two GL growth parameters emerges by comparison between the average spatial correlation distances and the typical cell size.

The results are based on a spherical and homogeneous system. On the other hand, heterogeneity is a well-known feature of tumor growth, in particular under radiochemotherapy. This important aspect can be included in the proposed framework by considering different subpopulations with various GL specific rates (Castorina et al. 2009). On the other hand, without spatial correlations, the unique parameter which drives the tumor progression is related to nutrient expenditures for mitosis and for the other metabolic activities (i.e., the parameter in Eq. (32), “Appendix A”). Therefore, also if different strains characterize the tumor evolution, it is difficult to obtain a GL description of in vivo data with the observed values of .