Low-intensity combination chemotherapy maximizes host survival time for tumors containing drug-resistant cells

Recent clinical trials have shown that for some cancers, high-intensity alternating chemotherapy does not significantly improve either survival times or response rates compared with nonalternating therapy. The current study uses optimal control to determine the best way to treat a tumor that contains drug-resistant cells that cannot be destroyed. The delivery of two non-cross-resistant chemotherapeutic agents is limited by bounds on the drug concentration and the dose intensity. This ensures that the drug toxicity stays within a tolerable range. The aim of the therapy is to maximize the host survival time, defined as the time over which the tumor burden can be kept below a fixed bound. The model is posed as a free terminal time, optimal parameter selection problem in which the constraints are continuously parametrized by time and the number of courses of therapy is free to vary. New theory is developed so that the optimal parameter selection problem can be solved as a sequence of fixed terminal time problems using existing optimal control software. Numerical simulations of Gompertz tumor growth showed that a treatment maintaining a high tumor burden doubled and sometimes tripled with survival time under aggressive therapy. When these simulations were repeated using exponential and logistic tumor growth models, the tumor burden during treatment had little influence upon survival time. In all simulations, survival time was not extended by delivering the anticancer drugs concurrently instead of staggering the treatment arms.     

Comparison of definitions of the lag phase and the exponential phase in bacterial growth.

Different definitions for the lag time and of the duration of the exponential phase can be used to calculate these quantities from growth models. The conventional definitions were compared with newly proposed definitions. It appeared to be possible to derive values for the lag time and the duration of the exponential phase from the growth models and differences between the various definitions could be quantified. All the different values can be calculated from the growth parameters microm, lambda and alpha. Therefore, it appeared to be unnecessary to use complicated mathematical equations; simple equations were adequate. For the Gompertz model the conventional definition of the lag time did not differ appreciably from the newly proposed definition. The end-point of the exponential phase and thus the duration of the exponential phase differed considerably for the two definitions. For the logistic model the two definitions lead to considerable differences for all quantities. It is recommended that the conventional definition is used for calculating the lag time. For the duration of the exponential phase it is recommended that the new definition is used. The value can be calculated, however, directly from the conventional growth parameters.     

Comparison of definitions of the lag phase and the exponential phase in bacterial growth

M.H. ZWIETERING, F.M. ROMBOUTS AND K. VAN 'T RIET. 1992. Different definitions of the lag time and of the duration of the exponential phase can be used to calculate these quantities from growth models. The conventional definitions were compared with newly proposed definitions. It appeared to be possible to derive values for the lag time and the duration of the exponential phase from the growth models, and differences between the various definitions could be quantified. All the different values can be calculated from the growth parameters μ _m , and a. Therefore, it appeared to be unnecessary to use complicated mathematical equations: simple equations were adequate. For the Gompertz model the conventional definition of the lag time did not differ appreciably from the newly proposed definition. The end-point of the exponential phase and thus the duration of the exponential phase differed considerably for the two definitions. For the logistic model the two definitions lead to considerable differences for all quantities. It is recommended that the conventional definition is used for calculating the lag time. For the duration of the exponential phase it is recommended that the new definition is used. The value can be calculated, however, directly from the conventional growth parameters.     

A Stochastic Model of Cancer Growth Subject to an Intermittent Treatment with Combined Effects: Reduction in Tumor Size and Rise in Growth Rate

A model of cancer growth based on the Gompertz stochastic process with jumps is proposed to analyze the effect of a therapeutic program that provides intermittent suppression of cancer cells. In this context, a jump represents an application of the therapy that shifts the cancer mass to a return state and it produces an increase in the growth rate of the cancer cells. For the resulting process, consisting in a combination of different Gompertz processes characterized by different growth parameters, the first passage time problem is considered. A strategy to select the inter-jump intervals is given so that the first passage time of the process through a constant boundary is as large as possible and the cancer size remains under this control threshold during the treatment. A computational analysis is performed for different choices of involved parameters. Finally, an estimation of parameters based on the maximum likelihood method is provided and some simulations are performed to illustrate the validity of the proposed procedure.     

A Comparison and Catalog of Intrinsic Tumor Growth Models

Determining the mathematical dynamics and associated parameter values that should be used to accurately reflect tumor growth continues to be of interest to mathematical modelers, experimentalists and practitioners. However, while there are several competing canonical tumor growth models that are often implemented, how to determine which of the models should be used for which tumor types remains an open question. In this work, we determine the best fit growth dynamics and associated parameter ranges for ten different tumor types by fitting growth functions to at least five sets of published experimental growth data per type of tumor. These time-series tumor growth data are used to determine which of the five most common tumor growth models (exponential, power law, logistic, Gompertz, or von Bertalanffy) provides the best fit for each type of tumor.     

A novel approach using transcomplementing adenoviral vectors for gene therapy of adrenocortical cancer.

Current therapies for adrenocortical carcinomas do not improve the life expectancy of patients. In this study, we tested whether a gene-transfer therapy based upon a suicide gene/prodrug system would be effective in an animal model of the disease. We employed E4- and E1A/B-depleted, herpes simplex virus-thymidine kinase-expressing adenoviral mutants that transcomplement each other within tumor cells, hereby improving transgene delivery and efficacy by viral replication in situ. Transcomplementation of vectors increased the fraction of transduced of tumor cells. This increase was accompanied by greater tumor volume reduction compared to non-transcomplementing approaches. Survival time improved with non-replicating vectors plus GCV compared to controls. However, transcomplementation/replication of vectors led to a further significant increment in anti-tumor activity and survival time (p < 0.02). In treated animals, we observed a high number of apoptotic nuclei both adjacent to and distant from injection sites and sites of viral oncolysis. Ultrastructural analyses exhibited nuclear inclusion bodies characteristic of virus production in situ, and provided further evidence that this therapy induced apoptotic cell death within tumor cells. We conclude that the efficacy of suicide gene therapy is significantly amplified by viral replication and, in combination with GCV, significantly reduces tumor burden and increases survival time.     

A general bilinear model to describe growth or decline time profiles

Linear models are widely used because of their unrivaled simplicity, but they cannot be applied for data that have a turning-or rate-change-point, even if the data show good linearity sufficiently far from this point. To describe such bilinear-type data, a completely generalized version of a linearized biexponential model (LinBiExp) is proposed here to make possible smooth and fully parametrizable transitions between two linear segments while still maintaining a clear connection with the linear models. Applications and brief conclusions are presented for various time profiles of biological and medical interest including growth profiles, such as those of human stature, agricultural crops and fruits, multicellular tumor spheroids, single fission yeast cells, or even labor productivity, and decline profiles, such as age-effects on cognition in patients who develop dementia and lactation yields in dairy cattle. In all these cases, quantitative model selection criteria such as the Akaike and the Schwartz Bayesian information criteria indicated the superiority of the bilinear model compared to adequate less parametrized alternatives such as linear, parabolic, exponential, or classical growth (e.g., logistic, Gompertz, Weibull, and Richards) models. LinBiExp provides a versatile and useful five-parameter bilinear functional form that is convenient to implement, is suitable for full optimization, and uses intuitive and easily interpretable parameters.     

Simultaneous identification of growth law and estimation of its rate parameter for biological growth data: a new approach

Scientific formalizations of the notion of growth and measurement of the rate of growth in living organisms are age-old problems. The most frequently used metric, “Average Relative Growth Rate” is invariant under the choice of the underlying growth model. Theoretically, the estimated rate parameter and relative growth rate remain constant for all mutually exclusive and exhaustive time intervals if the underlying law is exponential but not for other common growth laws (e.g., logistic, Gompertz, power, general logistic). We propose a new growth metric specific to a particular growth law and show that it is capable of identifying the underlying growth model. The metric remains constant over different time intervals if the underlying law is true, while the extent of its variation reflects the departure of the assumed model from the true one. We propose a new estimator of the relative growth rate, which is more sensitive to the true underlying model than the existing one. The advantage of using this is that it can detect crucial intervals where the growth process is erratic and unusual. It may help experimental scientists to study more closely the effect of the parameters responsible for the growth of the organism/population under study.     

Model-Based Tumor Growth Dynamics and Therapy Response in a Mouse Model of De Novo Carcinogenesis

Tumorigenesis is a complex, multistep process that depends on numerous alterations within the cell and contribution from the surrounding stroma. The ability to model macroscopic tumor evolution with high fidelity may contribute to better predictive tools for designing tumor therapy in the clinic. However, attempts to model tumor growth have mainly been developed and validated using data from xenograft mouse models, which fail to capture important aspects of tumorigenesis including tumor-initiating events and interactions with the immune system. In the present study, we investigate tumor growth and therapy dynamics in a mouse model of de novo carcinogenesis that closely recapitulates tumor initiation, progression and maintenance in vivo. We show that the rate of tumor growth and the effects of therapy are highly variable and mouse specific using a Gompertz model to describe tumor growth and a two-compartment pharmacokinetic/ pharmacodynamic model to describe the effects of therapy in mice treated with 5-FU. We show that inter-mouse growth variability is considerably larger than intra-mouse variability and that there is a correlation between tumor growth and drug kill rates. Our results show that in vivo tumor growth and regression in a double transgenic mouse model are highly variable both within and between subjects and that mathematical models can be used to capture the overall characteristics of this variability. In order for these models to become useful tools in the design of optimal therapy strategies and ultimately in clinical practice, a subject-specific modelling strategy is necessary, rather than approaches that are based on the average behavior of a given subject population which could provide erroneous results.     

Secretome compartment is a valuable source of biomarkers for cancer-relevant pathways

In principle, targeted therapies have optimal activity against a specific subset of tumors that depend upon the targeted molecule or pathway for growth, survival, or metastasis. Consequently, it is important in drug development and clinical practice to have predictive biomarkers that can reliably identify patients who will benefit from a given therapy. We analyzed tumor cell-line secretomes (conditioned cell media) to look for predictive biomarkers; secretomes represent a potential source for potential biomarkers that are expressed in intracellular signaling and therefore may reflect changes induced by targeted therapy. Using Gene Ontology, we classified by function the secretome proteins of 12 tumor cell lines of different histotypes. Representations and hierarchical relationships among the functional groups differed among the cell lines. Using bioinformatics tools, we identified proteins involved in intracellular signaling pathways. For example, we found that secretome proteins related to TGF-beta signaling in thyroid cancer cells, such as vasorin, CD109, and βIG-H3 (TGFBI), were sensitive to RPI-1 and dasatinib treatments, which have been previously demonstrated to be effective in blocking cell proliferation. The secretome may be a valuable source of potential biomarkers for detecting cancer and measuring the effectiveness of cancer therapies.     

A flexible sigmoid function of determinate growth

A new empirical equation for the sigmoid pattern of determinate growth, 'the beta growth function', is presented. It calculates weight (w) in dependence of time, using the following three parameters: t(m), the time at which the maximum growth rate is obtained; t(e), the time at the end of growth; and w(max), the maximal value for w, which is achieved at t(e). The beta growth function was compared with four classical (logistic, Richards, Gompertz and Weibull) growth equations, and two expolinear equations. All equations described successfully the sigmoid dynamics of seed filling, plant growth and crop biomass production. However, differences were found in estimating w(max). Features of the beta function are: (1) like the Richards equation it is flexible in describing various asymmetrical sigmoid patterns (its symmetrical form is a cubic polynomial); (2) like the logistic and the Gompertz equations its parameters are numerically stable in statistical estimation; (3) like the Weibull function it predicts zero mass at time zero, but its extension to deal with various initial conditions can be easily obtained; (4) relative to the truncated expolinear equation it provides more reasonable estimates of final quantity and duration of a growth process. In addition, the new function predicts a zero growth rate at both the start and end of a precisely defined growth period. Therefore, it is unique for dealing with determinate growth, and is more suitable than other functions for embedding in process-based crop simulation models to describe the dynamics of organs as sinks to absorb assimilates. Because its parameters correspond to growth traits of interest to crop scientists, the beta growth function is suitable for characterization of environmental and genotypic influences on growth processes. However, it is not suitable for estimating maximum relative growth rate to characterize early growth that is expected to be close to exponential.     

Lifetime growth in wild meerkats: incorporating life history and environmental factors into a standard growth model

Lifetime records of changes in individual size or mass in wild animals are scarce and, as such, few studies have attempted to model variation in these traits across the lifespan or to assess the factors that affect them. However, quantifying lifetime growth is essential for understanding trade-offs between growth and other life history parameters, such as reproductive performance or survival. Here, we used model selection based on information theory to measure changes in body mass over the lifespan of wild meerkats, and compared the relative fits of several standard growth models (monomolecular, von Bertalanffy, Gompertz, logistic and Richards). We found that meerkats exhibit monomolecular growth, with the best model incorporating separate growth rates before and after nutritional independence, as well as effects of season and total rainfall in the previous nine months. Our study demonstrates how simple growth curves may be improved by considering life history and environmental factors, which may be particularly relevant when quantifying growth patterns in wild populations.     

Modeling the growth of Enterococcus faecium in bologna sausage.

A study to set up mathematical models which allow the prediction of Enterococcus faecium growth in bologna sausage (mortadella) was carried out. Growth curves were obtained at different temperatures (5, 6, 12, 15, 25, 32, 35, 37, 42, 46, 50, 52, and 55 degrees C). The Gompertz and logistic models, modified by Zwietering, were found to fit with the representation of experimental curves. The variations of the parameters A (i.e., the asymptotic value reached by the relative population during the stationary growth phase), mu m (i.e., the maximum specific growth rate during the exponential growth phase), and lambda (i.e., the lag time) with temperature were then modeled. The variation of A with temperature can be described by an empirical polynomial model, whereas the variation of mu m and lambda can be described by the Ratkowsky model modified by Zwietering and the Adair model, respectively. Data processing of these models has shown that the minimum growth temperature for E. faecium is 0.1 degrees C, the maximum growth temperature is 53.4 degrees C, and the optimal growth temperature is 42 to 45 degrees C.     

Stochastic Gompertz model of tumour cell growth

In this communication, based upon the deterministic Gompertz law of cell growth, a stochastic model in tumour growth is proposed. This model takes account of both cell fission and mortality too. The corresponding density function of the size of the tumour cells obeys a functional Fokker--Planck equation which can be solved analytically. It is found that the density function exhibits an interesting "multi-peak" structure generated by cell fission as time evolves. Within this framework the action of therapy is also examined by simply incorporating a therapy term into the deterministic cell growth term.     

A stochastic model in tumor growth

A stochastic model of solid tumor growth based on deterministic Gompertz law is presented. Tumor cells evolution is described by a one-dimensional diffusion process limited by two absorbing boundaries representing healing threshold and patient death (carrying capacity), respectively. Via a numerical approach the first exit time problem is analysed for the process inside the region restricted by the boundaries. The proposed model is also implemented to simulate the effects of a time-dependent therapy. Finally, some numerical results are obtained for the specific case of a parathyroid tumor.     

Growth and annual survival estimates to examine the ecology of larval lamprey and the implications of ageing error in fitting models

This study used existing western brook lamprey Lampetra richardsoni age information to fit three different growth models (i.e. von Bertalanffy, Gompertz and logistic) with and without error in age estimates. Among these growth models, there was greater support for the logistic and Gompertz models than the von Bertalanffy model, regardless of ageing error assumptions. The von Bertalanffy model, however, appeared to fit the data well enough to permit survival estimates; using length‐based estimators, annual survival varied between 0·64 (95% credibility interval: 0·44–0·79) and 0·81 (0·79–0·83) depending on ageing and growth process error structure. These estimates are applicable to conservation and management of L. richardsoni and other western lampreys (e.g. Pacific lamprey Entosphenus tridentatus) and can potentially be used in the development of life‐cycle models for these species. These results also suggest that estimators derived from von Bertalanffy growth models should be interpreted with caution if there is high uncertainty in age estimates.     

A cell kinetics model for prostate cancer and its application to clinical data and individual patients

A cell kinetics model is developed to describe the evolution of prostate cancer (PC) from diagnosis to PC specific death. Such a model can be used to estimate an individual's eventual outcome and thus to inform decisions about therapy. To describe the observed clinical progression, the model must postulate three PC cell populations that are (1) local to the prostate and sensitive to hormones, (2) regional and hormone sensitive, and (3) systemic and hormone resistant. A set of coupled first-order differential equations describes the exponential growth of a PC tumor as well as its transformation from a local to systemic disease. The time dependence of the solutions is scaled to the doubling time of the prostate specific antigen (PSADT) because it characterizes the tumor growth for the individual. The conversion from local to systemic cell populations is described with a parameter α that can be associated with the Gleason score. The model also has three critical cell populations that describe (1) the initiation of the non-local populations, (2) the saturation level of the local tumor, and (3) the cell count likely to cause PC specific death. These parameters are calibrated by reproducing published PC clinical data and survival tables. The model is then applied to individuals with complete PC diagnostic data in order to calculate the progression to PC specific death. One man has early stage PC as described in the ‘vignette’ patient of Walsh et al. (2007. N. Engl. J. Med. 357, 2696-2705). The second man has a more serious condition and has undergone both local and systemic treatments. Unfortunately, I am that patient.     

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

Abstract. Growth data on 60 multicellular spheroids of MCF-7 human breast cancer cells were fitted, on an individual basis, by the Gompertz, Bertalanffy and logistic equations. MCF-7 spheroids, initiated and grown in medium containing oestrogens, exhibited a growth rate that decreased continuously as spheroid size increased. Plots of spheroid volume v. time generated sigmoid curves that showed an early portion with an approximately exponential volume increase; a middle region or retardation phase characterized by a continuously decreasing growth rate; and, finally, a late segment or plateau phase approaching zero growth rate, that permitted an estimate of the maximum spheroid size (V_max). Growth curves generated by MCF-7 spheroids under different experimental conditions (hormones, drugs and radiation exposures) can be compared after normalization. Linearized forms of the fitted Gompertz curves provided a convenient way to express differences in growth rate.