Gompertzian Growth Research Articles

Cancer Modeling: A Personal Perspective

C ancer modeling comes in a wide variety of styles. Indeed, it can involve almost any type of applied mathematics. My personal favorite approach is the use of probability models to understand how genetic mutations lead to cancer progression, metastasis, and resistance to therapy. Ordinary differential equations can be used to study the growth of tumor cell populations, often leading to a conclusion of Gompertzian growth [21]. PDE models using cell densities and nutrient concentrations as state variables can be used to analyze various spatiotemporal phenomena; see [13]. Individual and agent-based models that treat cells as discrete objects with predefined rules of interaction can offer an improvement over PDE methods in some situations, such as the study of angiogenesis, the development of new blood vessels to bring nutrients to a growing tumor [1]. For a comparison of individual-based and continuum approaches in one particular example, see [4]. Agent-based systems are one of many computationally intensive methods [24] and are often components of multiscale models (see [16], [6], and [8]). Rather than spend the entire article in the land of generalities with random pointers to the literature, I will next give a description of a useful, simple, and flexible model: multitype branching processes. The types represent stages in the cancer progression. For example, in colon cancer, type 1 cells have one copy of the gene APC inactivated, type 2 cells have both copies inactivated, type 3

Evolutionary Kinetics of Graphene Formation on Copper

It has been claimed that graphene growth on copper by chemical vapor deposition is dominated by crystallization from the surface initially supersaturated with carbon adatoms, which implies that the growth is independent of hydrocarbon addition after the nucleation phase. Here, we present an alternative growth model based on our observations that oppose this claim. Our Gompertzian sigmoidal growth kinetics and secondary nucleation behavior support the postulate that the growth can be controlled by adsorption-desorption dynamics and the dispersive kinetic processes of catalytic dissociation and dehydrogenation of carbon precursors on copper.

An extension of Gompertzian growth dynamics: Weibull and Fréchet models

In this work a new probabilistic and dynamical approach to an extension of the Gompertz law is proposed. A generalized family of probability density functions, designated by Beta • (p,q), which is proportional to the right hand side of the Tsoularis-Wallace model, is studied. In particular, for p=2, the investigation is extended to the extreme value models of Weibull and Frechet type. These models, described by differential equations, are proportional to the hyper-Gompertz growth model. It is proved that the Beta• (2,q) densities are a power of betas mixture, and that its dynamics are determined by a non-linear coupling of probabilities. The dynamical analysis is performed using techniques of symbolic dynamics and the system complexity is measured using topological entropy. Generally, the natural history of a malignant tumour is reflected through bifurcation diagrams, in which are identified regions of regression, stability, bifurcation, chaos and terminus.

A dynamic model for tumour growth and metastasis formation

A simple and fast computational model to describe the dynamics of tumour growth and metastasis formation is presented. The model is based on the calculation of successive generations of tumour cells and enables one to describe biologically important entities like tumour volume, time point of 1st metastatic growth or number of metastatic colonies at a given time. The model entirely relies on the chronology of these successive events of the metastatic cascade. The simulation calculations were performed for two embedded growth models to describe the Gompertzian like growth behaviour of tumours. The initial training of the models was carried out using an analytical solution for the size distribution of metastases of a hepatocellular carcinoma. We then show the applicability of our models to clinical data from the Munich Cancer Registry. Growth and dissemination characteristics of metastatic cells originating from cells in the primary breast cancer can be modelled thus showing its ability to perform systematic analyses relevant for clinical breast cancer research and treatment. In particular, our calculations show that generally metastases formation has already been initiated before the primary can be detected clinically.

Optimal response to chemotherapy for a mathematical model of tumor–immune dynamics

An optimal control problem for cancer chemotherapy is considered that includes immunological activity. In the objective a weighted average of several quantities that describe the effectiveness of treatment is minimized. These terms include (i) the number of cancer cells at the terminal time, (ii) a measure for the immunocompetent cell densities at the terminal point (included as a negative term), (iii) the overall amount of cytotoxic agents given as a measure for the side effects of treatment and (iv) a small penalty on the terminal time that limits the overall therapy horizon which is assumed to be free. This last term is essential in obtaining a well-posed problem formulation. Employing a Gompertzian growth model for the cancer cells, for various scenarios optimal controls and corresponding responses of the system are calculated. Solutions initially follow a full dose treatment, but then at one point switch to a singular regimen that only applies partial dosages. This structure is consistent with protocols that apply an initial burst to reduce the tumor volume and then maintain a small volume through lower dosages. Optimal controls end with either a prolonged period of no dose treatment or, in a small number of scenarios, this no dose interval is still followed by one more short burst of full dose treatment.

Abstract 5079: A mathematical approach to earlier cancer detection: Using blood biomarker assays to monitor growth of a tumor from a single cell

Abstract Early cancer detection may eventually be enabled by a quick, highly sensitive assay to monitor abnormal cancer biomarker levels in blood, followed by a localized imaging scan to visualize tumor burden. To improve screening of high-risk patients prior to imaging, blood assay sensitivity/specificity is being increased and biomarker secretion processes are being studied, but it is not clear which strategies will provide better detection of smaller tumors. To address this, we developed a linear 1-compartment model with 9 parameters to describe plasma biomarker kinetics in relation to the growth of cancer, beginning with a single tumor cell. In this model, plasma biomarker levels are dependent on the number of biomarker-secreting cells (cancerous and healthy), the biomarker secretion rate, the percent of biomarker entering plasma from the interstitium, and the elimination rate of plasma biomarker. To exemplify a realistic case, we based parameter values on ovarian cancer and CA125 secretion because CA125 secretion rates and assay sensitivities are readily available in published literature. Gompertzian tumor growth was modeled using published human ovarian carcinoma growth rates. The model predicted that tumor burden must grow to a volume of (20.4 mm)3 to reach plasma biomarker levels that are greater than in 99.9% of healthy patients; this may require >10 years after the birth of the initial tumor cell. These predictions are consistent with published cancer progression timelines recently uncovered from sequencing of metastatic genome data. Furthermore, by perturbing parameter values over log-orders of magnitude, we identified alternative strategies that could allow detection of <1 mm diameter tumors (Table). Applying this model to other cancers and associated biomarkers may pinpoint characteristics of ideal biomarkers and the assays used to detect them, thereby helping to identify cancers at earlier stages, when they can be more effectively treated. Citation Format: {Authors}. {Abstract title} [abstract]. In: Proceedings of the 102nd Annual Meeting of the American Association for Cancer Research; 2011 Apr 2-6; Orlando, FL. Philadelphia (PA): AACR; Cancer Res 2011;71(8 Suppl):Abstract nr 5079. doi:10.1158/1538-7445.AM2011-5079

Inferring the effect of therapy on tumors showing stochastic Gompertzian growth

The present work deals with a Gompertz-type diffusion process, which includes in the drift term a time-dependent function C( t) representing the effect of a therapy able to modify the dynamics of the underlying process. However, in experimental studies is not immediate to deduce the functional form of C( t) from a treatment protocol. So a statistical approach is proposed in order to estimate this function when a control group and one or more treated groups are observed. In order to validate the proposed strategy a simulation study for several interesting functional forms of C( t) has been carried out. Finally, an application to infer the net effect of cisplatin and doxorubicin+cyclophosphamide in actual murine models is presented.

An optimal control approach to cancer treatment under immunological activity

Mathematical models for cancer treatment that include immunological activity are considered as an optimal control problem with an objective that is motivated by a separatrix of the uncontrolled system. For various growth models on the cancer cells the existence and optimality of singular controls is investigated. For a Gompertzian growth function a synthesis of controls that move the state into the region of attraction of a benign equilibrium point is developed.

Growth curve analysis of asymptomatic and symptomatic meningiomas

Meningiomas sometimes appear to stop growing after attaining a large size. Commonly used exponential growth models do not reflect this phenomenon. We attempted to find the best curve to simulate their growth. Fifty-two patients with meningioma were followed up for 3.1-21.7years (mean 7.5years) with four or more imaging studies each. Thirty-one patients had asymptomatic tumors. The other 21 patients with residual or recurrent tumor were followed up after surgery. Time-volume curves for each tumor were plotted. Nonlinear regression analyses were performed against power, exponential, logistic, and Gompertzian curves. Time-volume curves corresponded to the Gompertzian and logistic growth curves better than to power or exponential curves. When simulating time-volume curves with Gompertzian curves, the majority of benign meningiomas began to slow their growth before patient age of 80years. Twenty-three of 31 asymptomatic meningiomas had already passed the inflection point before diagnosis. In contrast, this happened less frequently in symptomatic tumors. Especially, all six atypical meningiomas continued to grow quasi-exponentially. Sigmoid curves that approach a plateau were better descriptors of the growth of benign meningiomas than were curves of unlimited growth. However, atypical meningiomas were unlikely to slow their growth.

Clinical evidence for overcoming capecitabine resistance in a woman with breast cancer terminating in radiologically occult micronodular pseudo-cirrhosis with portal hypertension: a case report

IntroductionWe report a case of stage IV breast cancer terminating in an unusual picture of radiologically occult micronodular pseudo-cirrhosis. Contrast-enhanced computed tomography showed no evidence of metastatic breast cancer within the liver. Unlike the few previously reported cases of intrasinusoidal spread of breast cancer, our patient was palliated with a transjugular intrahepatic portosystemic shunt along with salvage chemohormonal therapy. In addition, our patient demonstrated proof of the principle of the dependence of capecitabine (Xeloda) efficacy on dose scheduling as predicted by laboratory studies based on Gompertzian tumor growth and the Norton-Simon hypothesis.Case presentationWe report the case of a 52-year-old Caucasian woman who developed radiological signs of portal hypertension without radiological evidence of hepatic metastasis five years after being diagnosed with metastatic breast cancer. She was receiving chemotherapy for stage IV breast cancer initially thought to be metastatic only to the bones. During salvage therapy with high-dose estradiol (Estradiol valerate), vinorelbine (Navelbine) and bevacizumab (Avastin), she suddenly developed signs of portal hypertension confirmed on computed tomography and by portal and systemic venous pressure measurements. Drug toxicity due to bevacizumab (Avastin) was initially and incorrectly entertained as a cause. The patient underwent palliative transjugular intrahepatic portosystemic shunt and transhepatic venous liver biopsy, which revealed the presence of rapidly progressive and uncontrolled metastatic breast cancer. The new discovery of radiologically occult intrasinusodal hepatic metastases with secondary micronodular cirrhosis was found to be the cause of her sudden onset portal hypertension. The patient's resistance to capecitabine (Xeloda) was reversed by changing the schedule of medication to biweekly 7/7 (7 days ingesting drug alternating with 7 days off drug) from the 14/7 (14 days ingesting drug alternating with a 7 day rest period) day schedule approved by the US Food and Drug Administration.ConclusionThis case report demonstrates an unusual presentation of radiographically occult hepatic metastasis from breast cancer palliated with transjugular intrahepatic portosystemic shunt. All patients with advanced breast cancer developing unexpected portal hypertension should be considered candidates for liver biopsy despite normal computed tomography of the liver imaging results. This is the first report of a reversal of clinical resistance to capecitabine (Xeloda) by changing from the schedule of 14/7 day to a biweekly 7/7 day schedule. This suggests that a biweekly schedule may be best for some patients.

Dose-Time-Effect Modeling for Cancer Cell Growth in In-vitro Experiments

In this article, we propose several dose-time-effect models, which combine the commonly used median effect model with the commonly used growth curve models, such as the exponential growth curve or Gompertzian growth curve. We apply the nonlinear least squares regression and the transformation on both the response variable and the regression function to estimate the parameters in the models. The Akaike information criterion (AIC) is used to select a parsimonious model. We illustrate the proposed procedures by fitting the dose-time-effect models to two case studies from in-vitro experiments. For each case study, the selected model fits the experimental data reasonably well. By comparing different dose-time-effect models with the selected parsimonious model for each case study, we found that the cells killed or injured by treatment still contributed to the total cell count. By accounting for these killed or injured cells in our models, the relationship between dose and the fraction of cells surviving treatment can be more accurately elucidated.

Dynamics of Gompertzian tumour growth under environmental fluctuations

We investigate the effect of correlated additive and multiplicative Gaussian white noise on the Gompertzian growth of tumours. Our results are obtained by solving numerically the time-dependent Fokker–Planck equation (FPE) associated with the stochastic dynamics. In our numerical approach we have adopted B-spline functions as a truncated basis to expand the approximated eigenfunctions. The eigenfunctions and eigenvalues obtained using this method are used to derive approximate solutions of the dynamics under study. We perform simulations to analyze various aspects, of the probability distribution, of the tumour cell populations in the transient- and steady-state regimes. More precisely, we are concerned mainly with the behaviour of the relaxation time ( τ ) to the steady-state distribution as a function of (i) of the correlation strength ( λ ) between the additive noise and the multiplicative noise and (ii) as a function of the multiplicative noise intensity ( D ) and additive noise intensity ( α ). It is observed that both the correlation strength and the intensities of additive and multiplicative noise, affect the relaxation time.

Scheduling of angiogenic inhibitors for Gompertzian and logistic tumor growth models

The problem of scheduling a given amount of angiogenic inhibitors is consideredas an optimal control problem with the objective of maximizing the achievable tumor reduction.For a dynamical model for the evolution of the carrying capacity of the vasculature formulatedin [15] optimal controls are computed for both a Gompertzian and logistic model of tumor growth.While optimal controls for the Gompertzian model typically contain a segment along whichthe control is singular, for the logistic model optimal controls are bang-bang with atmost two switchings.

The dynamics of tumor growth and cells pattern morphology

The mathematical modeling of tumor growth is an approach to explain the complex nature of these systems. A model that describes tumor growth was obtained by using a mesoscopic formalism and fractal dimension. This model theoretically predicts the relation between the morphology of the cell pattern and the mitosis/apoptosis quotient that helps to predict tumor growth from tumoral cells fractal dimension. The relation between the tumor macroscopic morphology and the cell pattern morphology is also determined. This could explain why the interface fractal dimension decreases with the increase of the cell pattern fractal dimension and consequently with the increase of the mitosis/apoptosis relation. Indexes to characterize tumoral cell proliferation and invasion capacities are proposed and used to predict the growth of different types of tumors. These indexes also show that the proliferation capacity is directly proportional to the invasion capacity. The proposed model assumes: i) only interface cells proliferate and invade the host, and ii) the fractal dimension of tumoral cell patterns, can reproduce the Gompertzian growth law.

A cellular automata model for avascular solid tumor growth under the effect of therapy

Tumor growth has long been a target of investigation within the context of mathematical and computer modeling. The objective of this study is to propose and analyze a two-dimensional stochastic cellular automata model to describe avascular solid tumor growth, taking into account both the competition between cancer cells and normal cells for nutrients and/or space and a time-dependent proliferation of cancer cells. Gompertzian growth, characteristic of some tumors, is described and some of the features of the time-spatial pattern of solid tumors, such as compact morphology with irregular borders, are captured. The parameter space is studied in order to analyze the occurrence of necrosis and the response to therapy. Our findings suggest that transitions exist between necrotic and non-necrotic phases (no-therapy cases), and between the states of cure and non-cure (therapy cases). To analyze cure, the control and order parameters are, respectively, the highest probability of cancer cell proliferation and the probability of the therapeutic effect on cancer cells. With respect to patterns, it is possible to observe the inner necrotic core and the effect of the therapy destroying the tumor from its outer borders inwards.

Poster - Thurs Eve-04: Mathematical modeling of liver metastases tumour growth and control with radiotherapy.

Generating an optimized radiation treatment plan requires understanding the factors affecting tumour control. Mathematical models of tumour dynamics may help in future studies of factors predicting tumor sensitivity to radiotherapy. In this study, a time-dependent differential model, incorporating biological cancer markers, is presented to describe pre-treatment tumour growth, response to radiation, and recurrence. The model uses Gompertzian growth to model pre-treatment tumour growth. The effect of radiotherapy is handled by a realistic cell-kill term that includes a volume-dependent change in tumour sensitivity. Post-treatment, a Gompertzian, accelerated, delayed repopulation is employed. As proof-of-concept, we examined the fit of the model's prediction using various liver enzyme levels as markers of metastatic liver tumour growth in a liver cancer patient. Enzyme data were derived from repeated serum blood tests, and radiation treatment-related data were retrieved from the patient's chart. A tumour clonogen population model was formulated. Each enzyme was coupled to the same tumour population, and served as surrogates of the tumour. This dynamical model was solved numerically and compared to the measured enzyme levels. By minimizing the mean-squared error of the model enzyme predictions, we determined the following tumour model parameters: growth rate prior to treatment was 0.7% per day; the fractional radiation cell kill for the prescribed dose was 60% per day; and tumour repopulation rate was 3.5% per day. These preliminary results provided the basis to test the model in a larger series of patients, to apply biological markers for improving the efficacy of radiotherapy by determining the underlying tumour dynamics.

A computational model for cancer growth by using complex networks

In this work we propose a computational model to investigate the proliferation of cancerous cell by using complex networks. In our model the network represents the structure of available space in the cancer propagation. The computational scheme considers a cancerous cell randomly included in the complex network. When the system evolves the cells can assume three states: proliferative, non-proliferative, and necrotic. Our results were compared with experimental data obtained from three human lung carcinoma cell lines. The computational simulations show that the cancerous cells have a Gompertzian growth. Also, our model simulates the formation of necrosis, increase of density, and resources diffusion to regions of lower nutrient concentration. We obtain that the cancer growth is very similar in random and small-world networks. On the other hand, the topological structure of the small-world network is more affected. The scale-free network has the largest rates of cancer growth due to hub formation. Finally, our results indicate that for different average degrees the rate of cancer growth is related to the available space in the network.

Defining the Optimal Schedule of Drug Administration: Art or Science?

Defining the Optimal Schedule of Drug Administration: Art or Science?

Optimal Schedule of Paclitaxel: Weekly Is Better

Optimal Schedule of Paclitaxel: Weekly Is Better

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 t b t and support a thesis that Gompertzian growth is a self-similar and allometric process of a holistic nature.