Growth in solid heterogeneous human colon adenocarcinomas: comparison of simple logistical models

Three models of simple logistical growth were used to describe volumetric growth in heterogeneous tumours. Two clonal subpopulations (designated as clone A and clone D) originally obtained from a human colon adenocarcinoma were used to produce solid xenograft tumours in nude mice. Volumetric growth of tumours produced from pure cells alone was compared to that produced from 50% A:50% D, 88% A:12% D, and 9% A:91% D admixtures. Gompertzian analysis of the in vivo growth data indicated significant differences in both the initial growth rates and final asymptotic limiting volumes of the pure versus the admixed tumours. Verhulstian and modified Verhulstian models were also used to derive regression curves from the same data. The fit of the curves was compared with each other using standard (Akaike, 1974; Schwartz, 1978) information criteria. In four of the five tumour populations the Gompertz equation fitted best. Only in the 88% A:12% D tumours did the modified Verhulst model fit best. The deviations from the regression curves, the residuals, for all three models were systematically distributed. These systematic errors are likely to be the result of using simplified logistical models to describe the growth kinetics of interacting populations in heterogeneous tumours.     

Growth in solid heterogeneous human colon adenocarcinomas: comparison of simple logistical models

Abstract. Three models of simple logistical growth were used to describe volumetric growth in heterogeneous tumours. Two clonal subpopulations (designated as clone A and clone D) originally obtained from a human colon adenocarcinoma were used to produce solid xenograft tumours in nude mice. Volumetric growth of tumours produced from pure cells alone was compared to that produced from 50% A:50% D, 88% A:12% D, and 9% A:91% D admixtures. Gompertzian analysis of the in vivo growth data indicated significant differences in both the initial growth rates and final asymptotic limiting volumes of the pure versus the admixed tumours. Verhulstian and modified Verhulstian models were also used to derive regression curves from the same data. The fit of the curves was compared with each other using standard (Akaike, 1974; Schwartz, 1978) information criteria. In four of the five tumour populations the Gompertz equation fitted best. Only in the 88% A:12% D tumours did the modified Verhulst model fit best. The deviations from the regression curves, the residuals, for all three models were systematically distributed. These systematic errors are likely to be the result of using simplified logistical models to describe the growth kinetics of interacting populations in heterogeneous tumours.     

Analysis of free growth and post-treatment regrowth of C3H mammary carcinoma in individual mice

Free growth and post-Doxorubicin treatment regrowth of the C3H mammary carcinoma were analysed in individual mice. In both cases, the Gompertzian function provided a better fit than the exponential function, and the difference was statistically significant (P less than 0.001, chi 2 test). No comprehensive Gompertzian function was found, and each individual tumour growth or regrowth was described by a specific curve. Nevertheless, although both individually measured alpha 0 and beta, Gompertzian parameters varied from one animal to another, in both free-growing and post-treatment regrowing tumours a strong linear correlation between alpha 0 and beta was found. A parallelism test was performed to verify if there exists any treatment-induced alteration. The two regression lines appeared to be identical, however.     

Gompertzian growth pattern correlated with phenotypic organization of colon carcinoma,malignant glioma and non-small cell lung carcinoma cell lines

Abstract. In the current study we present a Gompertzian model for cell growth as a function of cell phenotype using six human tumour cell lines (A-549, NCI-H596, NCI-H520, HT-29, SW-620 and U-251). Monolayer cells in exponential growth at various densities were quantified over a week by sulforhodamine B staining assay to produce cell-growth curves. A Gompertz equation was fitted to experimental data to obtain, for each cell line, three empirical growth parameters (initial cell density, cell-growth rate and carrying capacity – the maximal cell density). A cell-shape parameter named deformation coefficient D (a morphological relationship among spreading and confluent cells) was established and compared by regression analysis with the relative growth rate parameter K described by the Gompertz equation. We have found that coefficient D is directly proportional to the growth parameter K . The fit curve significantly matches the empirical data ( P  < 0.05), with a correlation coefficient of 0.9152. Therefore, a transformed Gompertzian growth function was obtained accordingly to D . The degree of correlation between the Gompertzian growth parameter and the coefficient D allows a new interpretation of the growth parameter K on the basis of morphological measurements of a set of tumour cell types, supporting the idea that cell-growth kinetics can be modulated by phenotypic organization of attached cells.     

Verhulstian analysis of the growth of transplantable mammary tumours in sialoadenectomized mice

In this report we have analysed data published in 1989 by Inui et al. (Incidence of precancerous foci of mammary glands and growth rate of transplantable mammary cancers in sialoadenectomized mice. J. Natl Cancer Inst. 81, 1660) involving the effects of perturbation of the epidermal growth factor (EGF) status of mammary tumour-bearing mice on subsequent volumetric responses. Removal of an endogenous EGF stimulus by surgical ablation of the submaxillary glands, the major EGF-producing organ in mice, produced significantly slower growth of rodent mammary neoplasms, decreased success rate of transplantation, and an increase in the latent period before growth occurred. Administration of i.p. EGF (5 micrograms/mouse/day) to sialadectomized tumour-bearing mice would however, increase tumour growth rate. Data were analysed using the Verhulst equation which indicated that the observed effects on tumour volumetrics by either sialoadenectomy or EGF administration could be interpreted as being produced through paracrine pathways. The use of the Verhulstian analysis indicates that it is possible to analyse neoplastic responses and infer whether paracrine or autocrine pathways are involved.     

Characteristic Species Dependent Growth Patterns of Mammalian Neoplasms

Evidence is presented, arising from an analysis of published data on tumour growth in three species of laboratory animals and in human multiple myeloma supporting a species specific relation between two supposedly independent parameters in the Gompertz equations frequently used to quantify tumour growth curves. This evidence supports the conjecture of Norton et al. (1976), based on their observations of the growth kinetics of a murine melanoma and a rat mammary carcinoma, that such a relation may be a general feature of tumour growth. Published data on the growth of xenografts of human colorectal tumours in immune-deprived mice suggests that the observed growth relation reflects the ability of a particular species to support a tumour of a certain maximum size. the existence of this relation greatly simplifies the task of predicting complete patterns of undisturbed neoplastic growth in these species.     

Submandibular salivary glands: influence on growth rate and life span in mice

Submandibular glands accumulate a variety of growth factors, especially in male mice. Surgical excision of these glands (sialoadenectomy) results in alterations in several organs and systems including the liver, skin and reproductive system. We studied the life-long consequences of sialoadenectomy in male mice. Animals were operated at the age of 10 weeks. Thereafter, body weight and food and water intake were controlled until death. Few weeks after surgery, body weight was lower in sialoadenectomized than in control mice. The difference remained stable until the age of 80 weeks. In spite of the lower body weight, food intake was higher in sialoadenectomized mice than in controls. The first death of sialoadenectomized mice occurred 10 weeks earlier than that of the first control, and the initial death rate in sialoadenectomized mice was almost twice the rate in controls. After 100 weeks of life, the death rate increased in control mice, but suddenly decreased in sialoadenectomized mice. The consequence was that the mean life span of the last 25% surviving animals was 10 weeks longer in sialoadenectomized than in control mice. Autopsy examination suggests that the effect of sialoadenectomy on death rate may be the consequence of a contrasting effect on tumour growth. Our results indicate that submandibular glands, or rather the factors derived from these glands, have contrasting roles in tumour growth. At early ages they may be survival factors and protect tissues, whereas at later ages they may stimulate the growth of transformed cells.     

A principle of fractal-stochastic dualism and Gompertzian dynamics of growth and self-organization

The emergence of Gompertzian dynamics at the macroscopic, tissue level during growth and self-organization is determined by the existence of fractal-stochastic dualism at the microscopic level of supramolecular, cellular system. On one hand, Gompertzian dynamics results from the complex coupling of at least two antagonistic, stochastic processes at the molecular cellular level. It is shown that the Gompertz function is a probability function, its derivative is a probability density function, and the Gompertzian distribution of probability is of non-Gaussian type. On the other hand, the Gompertz function is a contraction mapping and defines fractal dynamics in time-space; a prerequisite condition for the coupling of processes. Furthermore, the Gompertz function is a solution of the operator differential equation with the Morse-like anharmonic potential. This relationship indicates that distribution of intrasystemic forces is both non-linear and asymmetric. The anharmonic potential is a measure of the intrasystemic interactions. It attains a point of the minimum (U(0), t(0)) along with a change of both complexity and connectivity during growth and self-organization. It can also be modified by certain factors, such as retinoids.     

Mathematical models for optimizing tumour radiotherapy. I: A simple two compartment cell-kinetic model for the unperturbed growth of transplantable tumours

Tumour growth kinetics has been analysed on the basis of interactions between two compartments comprising the proliferating and non-proliferating cells. Starting from the differential equations of growth of the cell-populations in the two compartments and assuming the process of intercompartmental cell transfers to be linear, an analytic expression on the variation of growth-fraction with the age of the tumour is obtained. The restricted conditions on the cell-cycle time and cell-loss-rate, under which these differential equations lead to a Gompertzian growth of the tumour, are critically analysed. The formalism permits the estimation of some important cell-kinetic parameters, like growth-fraction or cell-loss-factor, from a knowledge of the tumour-growth curve, cell-cycle-time and a single measurement of the cell-loss-rate of the matured tumour, provided the tumour follows a Gompertzian growth. The validity of the model has been verified with the experimental data on 4 different transplantable murine tumour systems. Usefulness of the model has been demonstrated by making some interesting predictions regarding the radiation response of the tumours from the cell-kinetic parameters.     

On the Gompertzian growth in the fractal space-time

An analytical approach to determination of time-dependent temporal fractal dimension b(t)(t) and scaling factor a(t)(t) for the Gompertzian growth in the fractal space-time is presented. The derived formulae take into account the proper boundary conditions and permit a calculation of the mean values b(t)(t) and a(t)(t) at any period of time. The formulae derived have been tested on experimental data obtained by Schrek for the Brown-Pearce rabbit's tumor growth. The results obtained confirm a possibility of successful mapping of the experimental Gompertz curve onto the fractal power-law scaling function y(t)=a(t)tb(t) and support a thesis that Gompertzian growth is a self-similar and allometric process of a holistic nature.     

On time-space of nonlinear phenomena with Gompertzian dynamics

This paper describes a universal relationship between time and space for a nonlinear process with Gompertzian dynamics, such as growth. Gompertzian dynamics implicates a coupling between time and space. Those two categories are related to each other through a linear function of their logarithms. Moreover, we demonstrate that the spatial fractal dimension is a function of both scalar time and the temporal fractal dimension. The Gompertz function reflects the equilibrium of regular states, that is, states with dynamics that are predictable for any time-point (e.g., sinusoidal glycolytic oscillations) and chaotic states, that is, states with dynamics that are unpredictable in time, but are characterized by certain regularities (e.g., the existence of strange attractor for any biochemical reaction). We conclude that both this equilibrium and volume of the available complementary Euclidean space determine temporal and spatial expansion of a process with Gompertzian dynamics.     

PV1 down‐regulation via shRNA inhibits the growth of pancreatic adenocarcinoma xenografts

PV1 is an endothelial‐specific protein with structural roles in the formation of diaphragms in endothelial cells of normal vessels. PV1 is also highly expressed on endothelial cells of many solid tumours. On the basis of in vitro data, PV1 is thought to actively participate in angiogenesis. To test whether or not PV1 has a function in tumour angiogenesis and in tumour growth in vivo, we have treated pancreatic tumour‐bearing mice by single‐dose intratumoural delivery of lentiviruses encoding for two different shRNAs targeting murine PV1. We find that PV1 down‐regulation by shRNAs inhibits the growth of established tumours derived from two different human pancreatic adenocarcinoma cell lines (AsPC‐1 and BxPC‐3). The effect observed is because of down‐regulation of PV1 in the tumour endothelial cells of host origin, PV1 being specifically expressed in tumour vascular endothelial cells and not in cancer or other stromal cells. There are no differences in vascular density of tumours treated or not with PV1 shRNA, and gain and loss of function of PV1 in endothelial cells does not modify either their proliferation or migration, suggesting that tumour angiogenesis is not impaired. Together, our data argue that down‐regulation of PV1 in tumour endothelial cells results in the inhibition of tumour growth via a mechanism different from inhibiting angiogenesis.     

Effect of monoclonal antibodies to early pregnancy factor (EPF) on the in vivo growth of transplantable murine tumours

Summary Neutralisation studies with monoclonal antibodies (mAbs) specific for early pregnancy factor (EPF) have shown it to be essential for the continuation of pregnancy in mice and the growth of some tumour cells in vitro. These studies report that the mAbs are also able to limit the growth of two murine tumour lines transplanted s. c. The development of MCA-2 tumours in CBA mice was unaffected by the injection of 1 mg anti-EPF IgM at the time of tumour cell inoculation. However, four doses of 500 µg anti-EPF, injected one dose per day for 4 days after tumour cell inoculation, significantly retarded tumour development such that no tumours were palpable on day 13. A similar dose regimen of control IgM had no effect on tumour size. Dose/response studies revealed that lower doses of anti-EPF administered after tumour cell inoculation were effective in retarding the growth of the MCA-2 tumours. The effect of anti-EPF mAb administration on the growth rate of palpable B16 tumours established s. c. in C57BL/6 mice was also determined. Tumours injected with 6 mg anti-EPF 5/341 or anti-EPF 5/333 mAbs showed significant decrease in the uptake of [³H]thymidine into tumour tissue, measured 16 h after injection. Furthermore, titration of sera for active EPF showed that a significant reduction in the EPF titre was associated with a significant inhibition of tumour DNA synthesis. Thus it appears that neutralisation of EPF retards tumour growth both in vitro and in vivo. In vitro the effects must be due to anti-EPF mAb interfering with a direct mechanism that contributes to the maintenance of cells in the active growing phase. However, in vivo host immunological mechanism that are modified to allow tumour survival may also be affected. The presence of EPF-induced suppressor factors curculating in the serum of tumour-bearing mice has been confirmed and the contribution of such factors to tumour progression must now be investigated.     

The multi-site tumour transplantation model for human endometrial carcinoma: a statistical evaluation

We studied the effect of multi-site tumour transplantation on tumour growth by implanting varying numbers of EnCa 101 human endometrial tumours in athymic mice. One treatment group received a single tumour per mouse, another group received two tumours per mouse and a third group received four tumours per mouse. Tumour growth was sustained in all animals by implantation of oestradiol-17 beta pellets. We observed positive correlation between tumours within the same mouse, which implies that individual tumours are not statistically independent. The correlation is sufficiently large that failure to account for it in statistical design and analysis could result in studies with insufficient power and in spurious assertions of significance. Regression modelling of tumour growth curves showed that mean tumour volume per animal is not affected by the number of tumours growing on the animal; that is, the data are consistent with the null hypothesis that mean tumour volume is the same regardless of the number of tumours present. Our results therefore suggest that the use of multiple tumours per animal can increase the precision of experiments without loss of validity and at relatively little cost. However, correct and efficient analysis of the data so obtained requires more sophisticated techniques than those--such as fixed-effects analysis of variance and the two sample t-test--that assume independence of tumours.     

Spatio-temporal pattern formation on spherical surfaces: numerical simulation and application to solid tumour growth

In this paper we examine spatio-temporal pattern formation in reaction-diffusion systems on the surface of the unit sphere in 3D. We first generalise the usual linear stability analysis for a two-chemical system to this geometrical context. Noting the limitations of this approach (in terms of rigorous prediction of spatially heterogeneous steady-states) leads us to develop, as an alternative, a novel numerical method which can be applied to systems of any dimension with any reaction kinetics. This numerical method is based on the method of lines with spherical harmonics and uses fast Fourier transforms to expedite the computation of the reaction kinetics. Numerical experiments show that this method efficiently computes the evolution of spatial patterns and yields numerical results which coincide with those predicted by linear stability analysis when the latter is known. Using these tools, we then investigate the r?le that pre-pattern (Turing) theory may play in the growth and development of solid tumours. The theoretical steady-state distributions of two chemicals (one a growth activating factor, the other a growth inhibitory factor) are compared with the experimentally and clinically observed spatial heterogeneity of cancer cells in small, solid spherical tumours such as multicell spheroids and carcinomas. Moreover, we suggest a number of chemicals which are known to be produced by tumour cells (autocrine growth factors), and are also known to interact with one another, as possible growth promoting and growth inhibiting factors respectively. In order to connect more concretely the numerical method to this application, we compute spatially heterogeneous patterns on the surface of a growing spherical tumour, modelled as a moving-boundary problem. The numerical results strongly support the theoretical expectations in this case. Finally in an appendix we give a brief analysis of the numerical method. Received: 27 July 2000 / Revised version: 15 August 2000 / Published online: 16 February 2001     

CYTOKINETICS OF TUMOUR AND ENDOTHELIAL CELLS AND VASCULARIZATION OF LUNG METASTASES IN C3H/HE MICE

A model of lung metastases was developed using intravenous injection of tumour cell aggregates of spontaneous C3H/He mammary tumours in syngeneic mice. the growth rate of lung tumours decreased with increasing tumour volume, with mean host survival of 46 days. the cytokinetics of individual tumours ranging between 0.004 and 4.2 mm³ in volume were studied. the labelling index (LI) ranged between 12 and 17%, the DNA synthesis time (T_s) being 9–10 hr. the growth fraction (GF) ranged between 26 and 38%. the cell cycle time (T_c) was found to be 18–19 hr. the LI and the GF decreased with increasing tumour volume doubling time (T_d). No correlation was found between the tumour volume and T_c. the LI of endothelial cells within these tumours, ranging between 0.004 and 4.2 mm³ in volume was 14–15% and endothelial cell proliferation was not affected by tumour growth. Vascular parameters were also determined for these tumours as a function of tumour volume. Vascular volume increased with increasing tumour size while the percentage of capillary vessels decreased. the cellular volume to capillary volume ratio increased with increasing tumour volume. Necrosis was observed in 0.27 mm³ tumours and increased with increasing tumour size. The results from these studies suggested that the age-dependent decrease in proliferative activity of tumour cells growing in the lung is related to change in effective vascularity.     

Alteration of properties of cell contacts during progression of hepatomas

Mechanical, electrical and physicochemical properties of the cell contacts in normal, regenerating liver and nine hepatoma strains were examined. Three patterns of contact alteration were found in tumours. They coincide with definite steps in tumour progression. Only some contact destabilization is found in hepatomas with minimal deviations. Contact stability can be restored to an almost normal degree by addition of macromolecular adhesive factor washed out from the normal liver by Ca²⁺-free solution. This factor does not act on the cell contacts of the hepatomas during the further steps of progression. The decrease in mutual adhesiveness and the non-divergence of the daughter cells play an important role during the next steps of tumour progression. Possible mechanisms of contact alteration and the significance of these alterations for tumour progression are discussed.     

Effect of local interleukin-2 treatment on spontaneous tumours of different immunogenic strength

Transplantable tumour lines established from spontaneous tumours of BALB/c, CBA, and DBA/2 mice displayed different immunogenic strength. This report describes tumour susceptibility to interleukin-2 (IL-2) therapy in relation to tumour immunogenicity. The following tumour lines were used: X5, X6, and X9 mammary tumours of DBA/2, BALB/c, and CBA origin respectively, X7 carcinoma of BALB/c and X18 papilloma of CBA mice. Two spontaneous tumours of long transplantation history, SL2 lymphoma (SL2) of DBA/2 and Madison lung carcinoma M109 (M109) of BALB/c origin, were used as control systems. Experimental mice were transplanted with different inocula of tumour cells at day 0; treatment with IL-2 was initiated on days 1–3 or delayed until day 10 and consisted of daily injections of low doses of 5000 or 20 000 U/mouse given five times a week for a period of 3 weeks. Treatment of SL2 (2 × 10⁴ cells injected i.p.) consisted of i.p. injections of 5000 or 20 000 U IL-2/mouse given on days 10–14 after tumour transplantation. IL-2 therapy of SL2-bearing DBA/2JIco mice resulted in a significant proportion of cures; however, no response to IL-2 treatment was achieved in SL2-bearing DBA/2CrIiw mice. BALB/c mice with the i.p. transplant of M109 responded to IL-2 treatment with 40% increase in lifespan. The low-dose IL-2 therapy of the five spontaneous tumours resulted, in general, in transient growth inhibition of the i.m. transplants of lines X5, X6, and X7 provided that IL-2 was administered locally. The therapeutic effect depended on the number of transplanted tumour cells, the best results being achieved at cell numbers close to the dose-inducing tumour growth in 50% of animals. We found that the spontaneously arising tumours responding to IL-2 treatment were all slowly growing and immunogenic (X6 and X7) or might have viral association (X5) and, as such, might express foreign antigens. The data suggest a correlation between tumour immunogenicity and the therapeutic effect. However, IL-2 can still exert some effect against tumours with negligible immunogenicity. Received: 16 July 1998 / Accepted: 5 October 1998