Inferring the effect of therapy on tumors showing stochastic Gompertzian growth

The Distance Between: An Algorithmic Approach to Comparing Stochastic Models to Time-Series Data

While mean-field models of cellular operations have identified dominant processes at the macroscopic scale, stochastic models may provide further insight into mechanisms at the molecular scale. In order to identify plausible stochastic models, quantitative comparisons between the models and the experimental data are required. The data for these systems have small sample sizes and time-evolving distributions. The aim of this study is to identify appropriate distance metrics for the quantitative comparison of stochastic model outputs and time-evolving stochastic measurements of a system. We identify distance metrics with features suitable for driving parameter inference, model comparison, and model validation, constrained by data from multiple experimental protocols. In this study, stochastic model outputs are compared to synthetic data across three scales: that of the data at the points the system is sampled during the time course of each type of experiment; a combined distance across the time course of each experiment; and a combined distance across all the experiments. Two broad categories of comparators at each point were considered, based on the empirical cumulative distribution function (ECDF) of the data and of the model outputs: discrete based measures such as the Kolmogorov-Smirnov distance, and integrated measures such as the Wasserstein-1 distance between the ECDFs. It was found that the discrete based measures were highly sensitive to parameter changes near the synthetic data parameters, but were largely insensitive otherwise, whereas the integrated distances had smoother transitions as the parameters approached the true values. The integrated measures were also found to be robust to noise added to the synthetic data, replicating experimental error. The characteristics of the identified distances provides the basis for the design of an algorithm suitable for fitting stochastic models to real world stochastic data.

Age at cancer diagnosis by breed, weight, sex, and cancer type in a cohort of more than 3,000 dogs: Determining the optimal age to initiate cancer screening in canine patients

The goal of cancer screening is to detect disease at an early stage when treatment may be more effective. Cancer screening in dogs has relied upon annual physical examinations and routine laboratory tests, which are largely inadequate for detecting preclinical disease. With the introduction of non-invasive liquid biopsy cancer detection methods, the discussion is shifting from how to screen dogs for cancer to when to screen dogs for cancer. To address this question, we analyzed data from 3,452 cancer-diagnosed dogs to determine the age at which dogs of certain breeds and weights are typically diagnosed with cancer. In our study population, the median age at cancer diagnosis was 8.8 years, with males diagnosed at younger ages than females, and neutered dogs diagnosed at significantly later ages than intact dogs. Overall, weight was inversely correlated with age at cancer diagnosis, and purebred dogs were diagnosed at significantly younger ages than mixed-breed dogs. For breeds represented by ≥10 dogs, a breed-based median age at diagnosis was calculated. A weight-based linear regression model was developed to predict the median age at diagnosis for breeds represented by ≤10 dogs and for mixed-breed dogs. Our findings, combined with findings from previous studies which established a long duration of the preclinical phase of cancer development in dogs, suggest that it might be reasonable to consider annual cancer screening starting 2 years prior to the median age at cancer diagnosis for dogs of similar breed or weight. This logic would support a general recommendation to start cancer screening for all dogs at the age of 7, and as early as age 4 for breeds with a lower median age at cancer diagnosis, in order to increase the likelihood of early detection and treatment.

Modified leaky competing accumulator model of decision making with multiple alternatives: the Lie-algebraic approach

In this communication, based upon the stochastic Gompertz law of population growth, we have reformulated the Leaky Competing Accumulator (LCA) model with multiple alternatives such that the positive-definiteness of evidence accumulation is automatically satisfied. By exploiting the Lie symmetry of the backward Kolmogorov equation (or Fokker–Planck equation) assoicated with the modified model and applying the Wei–Norman theorem, we have succeeded in deriving the N-dimensional joint probability density function (p.d.f.) and marginal p.d.f. for each alternative in closed form. With this joint p.d.f., a likelihood function can be constructed and thus model-fitting procedures become feasible. We have also demonstrated that the calibration of model parameters based upon the Monte Carlo simulated time series is indeed both efficient and accurate. Moreover, it should be noted that the proposed Lie-algebraic approach can also be applied to tackle the modified LCA model with time-varying parameters.

Optimal minimum variance-entropy control of tumour growth processes based on the Fokker-Planck equation

The authors demonstrated an optimal stochastic control algorithm to obtain desirable cancer treatment based on the Gompertz model. Two external forces as two time-dependent functions are presented to manipulate the growth and death rates in the drift term of the Gompertz model. These input signals represent the effect of external treatment agents to decrease tumour growth rate and increase tumour death rate, respectively. Entropy and variance of cancerous cells are simultaneously controlled based on the Gompertz model. They have introduced a constrained optimisation problem whose cost function is the variance of a cancerous cells population. The defined entropy is based on the probability density function of affected cells was used as a constraint for the cost function. Analysing growth and death rates of cancerous cells, it is found that the logarithmic control signal reduces the growth rate, while the hyperbolic tangent-like control function increases the death rate of tumour growth. The two optimal control signals were calculated by converting the constrained optimisation problem into an unconstrained optimisation problem and by using the real-coded genetic algorithm. Mathematical justifications are implemented to elucidate the existence and uniqueness of the solution for the optimal control problem.

Inference on an heteroscedastic Gompertz tumor growth model

We consider a non homogeneous Gompertz diffusion process whose parameters are modified by generally time-dependent exogenous factors included in the infinitesimal moments. The proposed model is able to describe tumor dynamics under the effect of anti-proliferative and/or cell death-induced therapies. We assume that such therapies can modify also the infinitesimal variance of the diffusion process. An estimation procedure, based on a control group and two treated groups, is proposed to infer the model by estimating the constant parameters and the time-dependent terms. Moreover, several concatenated hypothesis tests are considered in order to confirm or reject the need to include time-dependent functions in the infinitesimal moments. Simulations are provided to evaluate the efficiency of the suggested procedures and to validate the testing hypothesis. Finally, an application to real data is considered.

Two-Parameter Stochastic Weibull Diffusion Model: Statistical Inference and Application to Real Modeling Example

This paper describes the use of the non-homogeneous stochastic Weibull diffusion process, based on the two-parameter Weibull density function (the trend of which is proportional to the two-parameter Weibull probability density function). The trend function (conditioned and non-conditioned) is analyzed to obtain fits and forecasts for a real data set, taking into account the mean value of the process, the maximum likelihood estimators of the parameters of the model and the computational problems that may arise. To carry out the task, we employ the simulated annealing method for finding the estimators values and achieve the study. Finally, to evaluate the capacity of the model, the study is applied to real modeling data where we discuss the accuracy according to error measures.

A Note on Estimation of Multi-Sigmoidal Gompertz Functions with Random Noise

The behaviour of many dynamic real phenomena shows different phases, with each one following a sigmoidal type pattern. This requires studying sigmoidal curves with more than one inflection point. In this work, a diffusion process is introduced whose mean function is a curve of this type, concretely a transformation of the well-known Gompertz model after introducing in its expression a polynomial term. The maximum likelihood estimation of the parameters of the model is studied, and various criteria are provided for the selection of the degree of the polynomial when real situations are addressed. Finally, some simulated examples are presented.

Logistic Growth Described by Birth-Death and Diffusion Processes

We consider the logistic growth model and analyze its relevant properties, such as the limits, the monotony, the concavity, the inflection point, the maximum specific growth rate, the lag time, and the threshold crossing time problem. We also perform a comparison with other growth models, such as the Gompertz, Korf, and modified Korf models. Moreover, we focus on some stochastic counterparts of the logistic model. First, we study a time-inhomogeneous linear birth-death process whose conditional mean satisfies an equation of the same form of the logistic one. We also find a sufficient and necessary condition in order to have a logistic mean even in the presence of an absorbing endpoint. Then, we obtain and analyze similar properties for a simple birth process, too. Then, we investigate useful strategies to obtain two time-homogeneous diffusion processes as the limit of discrete processes governed by stochastic difference equations that approximate the logistic one. We also discuss an interpretation of such processes as diffusion in a suitable potential. In addition, we study also a diffusion process whose conditional mean is a logistic curve. In more detail, for the considered processes we study the conditional moments, certain indices of dispersion, the first-passage-time problem, and some comparisons among the processes.

Adaptive non-linear control for cancer therapy through a Fokker-Planck observer

In recent years, many efforts have been made to present optimal strategies for cancer therapy through the mathematical modelling of tumour-cell population dynamics and optimal control theory. In many cases, therapy effect is included in the drift term of the stochastic Gompertz model. By fitting the model with empirical data, the parameters of therapy function are estimated. The reported research works have not presented any algorithm to determine the optimal parameters of therapy function. In this study, a logarithmic therapy function is entered in the drift term of the Gompertz model. Using the proposed control algorithm, the therapy function parameters are predicted and adaptively adjusted. To control the growth of tumour-cell population, its moments must be manipulated. This study employs the probability density function (PDF) control approach because of its ability to control all the process moments. A Fokker-Planck-based non-linear stochastic observer will be used to determine the PDF of the process. A cost function based on the difference between a predefined desired PDF and PDF of tumour-cell population is defined. Using the proposed algorithm, the therapy function parameters are adjusted in such a manner that the cost function is minimised. The existence of an optimal therapy function is also proved. The numerical results are finally given to demonstrate the effectiveness of the proposed method.

A hyperbolastic type-I diffusion process: Parameter estimation by means of the firefly algorithm

A stochastic diffusion process, whose mean function is a hyperbolastic curve of type I, is presented. The main characteristics of the process are studied and the problem of maximum likelihood estimation for the parameters of the process is considered. To this end, the firefly metaheuristic optimization algorithm is applied after bounding the parametric space by a stagewise procedure. Some examples based on simulated sample paths and real data illustrate this development.

A branching process model of heterogeneous DNA damages caused by radiotherapy in in vitro cell cultures

This paper deals with the dynamic modeling and simulation of cell damage heterogeneity and associated mutant cell phenotypes in the therapeutic responses of cancer cell populations submitted to a radiotherapy session during in vitro assays. Each cell is described by a finite number of phenotypic states with possible transitions between them. The population dynamics is then given by an age-dependent multi-type branching process. From this representation, we obtain formulas for the average size of the global survival population as well as the one of subpopulations associated with 10 mutation phenotypes. The proposed model has been implemented into Matlab^© and the numerical results corroborate the ability of the model to reproduce four major types of cell responses: delayed growth, anti-proliferative, cytostatic and cytotoxic.

Kinetic Models for Predicting Cervical Cancer Response to Radiation Therapy on Individual Basis Using Tumor Regression Measured In Vivo With Volumetric Imaging

This article describes a macroscopic mathematical modeling approach to capture the interplay between solid tumor evolution and cell damage during radiotherapy. Volume regression profiles of 15 patients with uterine cervical cancer were reconstructed from serial cone-beam computed tomography data sets, acquired for image-guided radiotherapy, and used for model parameter learning by means of a genetic-based optimization. Patients, diagnosed with either squamous cell carcinoma or adenocarcinoma, underwent different treatment modalities (image-guided radiotherapy and image-guided chemo-radiotherapy). The mean volume at the beginning of radiotherapy and the end of radiotherapy was on average 23.7 cm(3) (range: 12.7-44.4 cm(3)) and 8.6 cm(3) (range: 3.6-17.1 cm(3)), respectively. Two different tumor dynamics were taken into account in the model: the viable (active) and the necrotic cancer cells. However, according to the results of a preliminary volume regression analysis, we assumed a short dead cell resolving time and the model was simplified to the active tumor volume. Model learning was performed both on the complete patient cohort (cohort-based model learning) and on each single patient (patient-specific model learning). The fitting results (mean error: ∼ 16% and ∼ 6% for the cohort-based model and patient-specific model, respectively) highlighted the model ability to quantitatively reproduce tumor regression. Volume prediction errors of about 18% on average were obtained using cohort-based model computed on all but 1 patient at a time (leave-one-out technique). Finally, a sensitivity analysis was performed and the data uncertainty effects evaluated by simulating an average volume perturbation of about 1.5 cm(3) obtaining an error increase within 0.2%. In conclusion, we showed that simple time-continuous models can represent tumor regression curves both on a patient cohort and patient-specific basis; this discloses the opportunity in the future to exploit such models to predict how changes in the treatment schedule (number of fractions, doses, intervals among fractions) might affect the tumor regression on an individual basis.

Modeling the Interplay Between Tumor Volume Regression and Oxygenation in Uterine Cervical Cancer During Radiotherapy Treatment

This paper describes a patient-specific mathematical model to predict the evolution of uterine cervical tumors at a macroscopic scale, during fractionated external radiotherapy. The model provides estimates of tumor regrowth and dead-cell reabsorption, incorporating the interplay between tumor regression rate and radiosensitivity, as a function of the tumor oxygenation level. Model parameters were estimated by minimizing the difference between predicted and measured tumor volumes, these latter being obtained from a set of 154 serial cone-beam computed tomography scans acquired on 16 patients along the course of the therapy. The model stratified patients according to two different estimated dynamics of dead-cell removal and to the predicted initial value of the tumor oxygenation. The comparison with a simpler model demonstrated an improvement in fitting properties of this approach (fitting error average value <5%, p < 0.01), especially in case of tumor late responses, which can hardly be handled by models entailing a constant radiosensitivity, failing to model changes from initial severe hypoxia to aerobic conditions during the treatment course. The model predictive capabilities suggest the need of clustering patients accounting for cancer cell line, tumor staging, as well as microenvironment conditions (e.g., oxygenation level).

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.

Inferring the effect of therapies on tumor growth by using diffusion processes

Modeling the effect of therapies in cancer animal models remains a challenge. This point may be addressed by considering a diffusion process that models the tumor growth and a modified process that includes, in its infinitesimal mean, a time function modeling the effect of the therapy. In the case of a Gompertz diffusion process, where a control group and one or more treated groups are examined, a methodology to estimate this function has been proposed by Albano et al. (2011). This method has been applied to infer the effect of cisplatin and doxorubicin+cyclophosphamide on breast cancer xenografts. Although this methodology can be extended to other diffusion processes, it has an important restriction: it is necessary that a known diffusion process adequately fits the control group. Here, we propose the use of a stochastic process for a hypothetical control group, in such a way that both the control and the treated groups can be modeled by modified processes of the former. Thus, the comparison between models would allow estimating the real effect of the therapy. The new methodology has been validated by inferring the effects in breast cancer models, and we have checked the robustness of the procedure against the choice of stochastic model for the hypothetical control group. Finally, we have also applied the methodology to infer the effect of a therapeutic peptide and ovariectomy on the growth of a breast cancer xenograft, and its efficiency in modeling the effect of different treatments in the absence of control group data is shown.

Nonlinear stochastic Markov processes and modeling uncertainty in populations

We consider an alternative approach to the use of nonlinear stochastic Markov processes (which have a Fokker-Planck or Forward Kolmogorov representation for density) in modeling uncertainty in populations. These alternate formulations, which involve imposing probabilistic structures on a family of deterministic dynamical systems, are shown to yield pointwise equivalent population densities. Moreover, these alternate formulations lead to fast efficient calculations in inverse problems as well as in forward simulations. Here we derive a class of stochastic formulations for which such an alternate representation is readily found.