Evaluation of the growth rate of MCF-7 breast cancer multicellular spheroids using three mathematical models

An on-lattice agent-based Monte Carlo model simulating the growth kinetics of multicellular tumor spheroids

PURPOSE

To develop an on-lattice agent-based model describing the growth of multicellular tumor spheroids using simple Monte Carlo tools.

METHODS

Cells are situated on the vertices of a cubic grid. Different cell states (proliferative, hypoxic or dead) and cell evolution rules, driven by 10 parameters, and the effects of the culture medium are included. About twenty spheroids of MCF-7 human breast cancer were cultivated and the experimental data were used for tuning the model parameters.

RESULTS

Simulated spheroids showed adequate sizes of the necrotic nuclei and of the hypoxic and proliferative cell phases as a function of the growth time, mimicking the overall characteristics of the experimental spheroids. The relation between the radii of the necrotic nucleus and the whole spheroid obtained in the simulations was similar to the experimental one and the number of cells, as a function of the spheroid volume, was well reproduced. The statistical variability of the Monte Carlo model described the whole volume range observed for the experimental spheroids. Assuming that the model parameters vary within Gaussian distributions it was obtained a sample of spheroids that reproduced much better the experimental findings.

CONCLUSIONS

The model developed allows describing the growth of in vitro multicellular spheroids and the experimental variability can be well reproduced. Its flexibility permits to vary both the agents involved and the rules that govern the spheroid growth. More general situations, such as, e. g., tumor vascularization, radiotherapy effects on solid tumors, or the validity of the tumor growth mathematical models can be studied.

Classical mathematical models for description and prediction of experimental tumor growth

Despite internal complexity, tumor growth kinetics follow relatively simple laws that can be expressed as mathematical models. To explore this further, quantitative analysis of the most classical of these were performed. The models were assessed against data from two in vivo experimental systems: an ectopic syngeneic tumor (Lewis lung carcinoma) and an orthotopically xenografted human breast carcinoma. The goals were threefold: 1) to determine a statistical model for description of the measurement error, 2) to establish the descriptive power of each model, using several goodness-of-fit metrics and a study of parametric identifiability, and 3) to assess the models' ability to forecast future tumor growth. The models included in the study comprised the exponential, exponential-linear, power law, Gompertz, logistic, generalized logistic, von Bertalanffy and a model with dynamic carrying capacity. For the breast data, the dynamics were best captured by the Gompertz and exponential-linear models. The latter also exhibited the highest predictive power, with excellent prediction scores (≥80%) extending out as far as 12 days in the future. For the lung data, the Gompertz and power law models provided the most parsimonious and parametrically identifiable description. However, not one of the models was able to achieve a substantial prediction rate (≥70%) beyond the next day data point. In this context, adjunction of a priori information on the parameter distribution led to considerable improvement. For instance, forecast success rates went from 14.9% to 62.7% when using the power law model to predict the full future tumor growth curves, using just three data points. These results not only have important implications for biological theories of tumor growth and the use of mathematical modeling in preclinical anti-cancer drug investigations, but also may assist in defining how mathematical models could serve as potential prognostic tools in the clinic.

Explicit separation of growth and motility in a new tumor cord model

We investigate a new model of tumor growth in which cell motility is considered an explicitly separate process from growth. Bulk tumor expansion is modeled by individual cell motility in a density-dependent diffusion process. This model is implemented in the context of an in vivo system, the tumor cord. We investigate numerically microscale density distributions of different cell classes and macroscale whole tumor growth rates as functions of the strength of transitions between classes. Our results indicate that the total tumor growth follows a classical von Bertalanffy growth profile, as many in vivo tumors are observed to do. This provides a quick validation for the model hypotheses. The microscale and macroscale properties are both sensitive to fluctuations in the transition parameters, and grossly adopt one of two phenotypic profiles based on their parameter regime. We analyze these profiles and use the observations to classify parameter regimes by their phenotypes. This classification yields a novel hypothesis for the early evolutionary selection of the metastatic phenotype by selecting against less motile cells which grow to higher densities and may therefore induce local collapse of the vascular network.

Hyperbolastic growth models: theory and application

Background

Mathematical models describing growth kinetics are very important for predicting many biological phenomena such as tumor volume, speed of disease progression, and determination of an optimal radiation and/or chemotherapy schedule. Growth models such as logistic, Gompertz, Richards, and Weibull have been extensively studied and applied to a wide range of medical and biological studies. We introduce a class of three and four parameter models called "hyperbolastic models" for accurately predicting and analyzing self-limited growth behavior that occurs e.g. in tumors. To illustrate the application and utility of these models and to gain a more complete understanding of them, we apply them to two sets of data considered in previously published literature.

Results

The results indicate that volumetric tumor growth follows the principle of hyperbolastic growth model type III, and in both applications at least one of the newly proposed models provides a better fit to the data than the classical models used for comparison.

Conclusion

We have developed a new family of growth models that predict the volumetric growth behavior of multicellular tumor spheroids with a high degree of accuracy. We strongly believe that the family of hyperbolastic models can be a valuable predictive tool in many areas of biomedical and epidemiological research such as cancer or stem cell growth and infectious disease outbreaks.

Dose dependence of the growth rate of multicellular tumour spheroids after irradiation

The present study investigated differences in the growth rate of multicellular tumour spheroids of the MCF-7 line of human breast cancer before and after their irradiation. Growth of the spheroids was analysed according to a model based on a Gompertz function. In this model, normalization to a common initial volume is achieved in a way that enables meaningful comparisons to be made between the results obtained for each spheroid. For irradiated spheroids the model includes an additional term to take account of sterilized cells. We found that the growth rate observed before irradiation is not fully recovered by irradiated spheroids and that growth recovery reduces with higher irradiation doses. Surviving fractions obtained at doses below 3 Gy are comparable with those found in clonogenic assays on spheroids of the same cellular line. At larger doses, discrepancies between the different studies are considerable.

In vitro preclinical models for a rational design of chemotherapy combinations in human tumors

Today, drug combinations are frequently used in the treatment of cancer to increase therapeutic efficacy. Currently used clinical protocols for cancer combination therapies are mainly obtained empirically or on the basis of results from previous clinical trials. Information obtained from clinical protocols is invaluable, but it is time-consuming, expensive and does not provide data on the biochemical and molecular mechanisms of interaction of the drugs used in combination treatments at cellular level. Therefore, in vitro drug combination studies on established cell lines or primary cell cultures play an important role in designing and optimising combination protocols. A variety of in vitro assays and different mathematics models have been developed to investigate cytotoxic effects and to analyse the type of drug interactions. Increased knowledge of the cellular targets of traditional and new drugs and the development of new technologies have resulted in a new role for the in vitro tests which are no longer used only to evaluate the cytotoxic effects of drugs, but also to investigate the interference on cell cycle, induction of apoptosis and molecular or biochemical interactions. A review on in vitro preclinical tests used to evaluate the effects of drug combinations and to design the rationale of combined chemotherapy protocols is presented.

Use of the recursion formula of the Gompertz survival function to evaluate life-table data

The recursion formula of the Gompertz function is an established method for the analysis of growth processes. In the present study the recursion formula of the Gompertz survival function 1n S(t + s) = a + b x ln S(t) is introduced for the analysis of survival data, where S(t) is the survival fraction at age 1, s is the constant age increment between two consecutive measurements of the survival fraction and a and b are parameters. With the help of this method--and provided stroboscopial measurements of rates of survival are available--the Gompertz survival function, instead of the corresponding mortality function, can be determined directly using linear regression analysis. The application of the present algorithm is demonstrated by analysing two sets of data taken from the literature (survival of Drosophila imagoes and of female centenarians) using linear regression analysis to fit survival or mortality rates to the corresponding models. In both cases the quality of fit was superior by using the algorithm presently introduced. Moreover, survival functions calculated from the fits to the mortality law only poorly predict the survival data. On the contrary, the results of the present method not only fit to the measurements, but, for both sets of data the mortality parameters calculated by the present method are essentially identical to those obtained by a corresponding application of a non-linear Marquardt-Levenberg algorithm to fit the same sets of data to the explicit form of the Gompertz survival function. Taking into consideration the advantages of using a linear fit (goodness-of-fit test and efficient statistical comparison of survival patterns) the method of the recursion formula of the Gompertz survival function is the most preferable method to fit survival data to the Gompertz function.

Interaction between ionizing radiation, estrogens and antiestrogens in the modification of tumor microenvironment in estrogen dependent multicellular spheroids

MCF7 human breast cancer cells growing as multicellular spheroids were examined as a model of three-dimensional cellular organization. Estrogen-free medium inhibited spheroid formation. In medium containing estrogens, the antiestrogen hydroxytamoxifen decreased the spheroid growth rate. Analyses with the recursion formula after Gompertz fitting showed that the rate of exponential decrease in growth rate (alpha) was alpha 0.099 +/- 0.013 d-1, and the decrease in alpha' was 0.061 +/- 0.015 d-1 for 0.1 microM hydroxytamoxifen and control spheroids respectively. MCF7 cells which had been growth arrested in an estrogen-free medium showed a significant decrease in radiosensitivity (surviving fraction at 2 Gy, SF2 = 63%) when compared with 0.1 nM 17 beta-estradiol-treated cells (SF2 = 38%). No differences in radiosensitivity were seen in MCF7 spheroids in estrogen-supplemented medium (radiation dose necessary to control 50% of spheroids (SCD50) was 5.51 Gy; derived alpha, beta and SF2 were 0.301 +/- 0.110 Gy-1, 0.018 +/- 0.005 Gy-2, and 51% respectively) when compared with monolayer cultures in the same medium (alpha = 0.316 +/- 0.059 Gy-1, beta = 0.023 +/- 0.006 Gy-2 and SF2 = 50%). In the spheroid model, manipulating the cellular environment, i.e., with estrogen treatment, modulates sensitivity to ionizing radiation.