# Top PDF Mathematical Model and Parameter Estimation for Tumor Growth ### Mathematical Model and Parameter Estimation for Tumor Growth

Numerous models in applied mathematics are expressed as a system of partial differential equations involving certain coefficients. In this work, we consider a tumor growth model originally proposed by Ward and King in 1997. Our main goal is to find an efficient and accurate numerical method for identification of parameters in the model (an inverse problem) from measurements of the evolving tumor over time. The so-called direct problem, in this case, is to solve a system of coupled nonlinear partial differential equations for given fixed values of the unknown parameters. We compare several derivative free and gradient based methods for the solution of the inverse problem which is formulated as an optimization problem with a constraint that is a system of partial differential equations (PDEs). Finally, we modify the original model to include a random parameter and solve the new optimization problem using the Monte Carlo method. The thesis is organized as follows. In the first two intro- ductory chapters, we discuss the original model and the non-dimensionalized version of the model equations. The next chapter is devoted to the optimization formulation of the inverse problem. In the following chapters, we compare performances of the optimization methods. In the final chapter, we discuss the performance comparison of the optimization methods for the cases where the random parameter in the model follows either uniform or truncated normal distributions. ### A mathematical model of tumor growth and its response to single irradiation ### A Parameter Estimation Model of G CSF: Mathematical Model of Cyclical Neutropenia

e fit of t e untre ted an reate data f r Dogs 0, 118 and 127. This confirms that the new model, with the G-CSF coupled to the cell population dynamics, is capable of reproducing the data. The least squares dif- ferences between the FFT analysis and the data were not significantly less than the reported values. Figure 1 shows the data and analysis for the other four dogs (Dogs 101, 113, 117 and 128), again with daily treatment. Re- call that these were the estimated, not fitted, values for the treated parameters and note the quality of the fits. Thus, we are able to match observed data without auto- mated parameter fitting based simply on an examination of the treated data and the parameter changes for Dogs 100, 118 and 127. For each dog, we performed simula- tions comparing daily treatment, treatment every other day, and every three days. We find that particularly for Dogs 100, 101, 118 and 127, changing the period of the treatment can significantly affect the nature of the oscil- lations. It shows the results of treating Dog 118 every other day, rather than every day. We have also explored the effects of changing the time at which the treatment is initiated. In most cases, this did not significantly change the long-term behavior. However, for Dog 127 the am- plitude of the oscillations was significantly reduced when the treatment was initiated in the latter half of the cycle. More specifically, measured from day 1, we find that smaller oscillations occur if treatment is initiated on day 8 or afterwards, or on days 2 or 5 (see Figure 1). When treatment was initiated on other days, larger oscillations in the model resulted. It should also be noted that in- creasing the G-CSF dosage in the model sometimes helped to stabilize oscillations (Dog 127), but in several cases (Dogs 100, 128 and 101) a dosage increase from 5 μg/kg to a dosage in the range 15 - 25 μg/kg caused some FFT analysis to fail. In that analysis, the differentiation rate out of the stem cells was so high, and the apoptosis rate in the stem cells was so high, that the stem cell population was no longer able to maintain itself. For the other dogs, there was always a dosage that was suffi- ciently high to terminate the FFT analyze, but it was sometimes a factor of 10 higher than the actual dosage given (see Table 1). ### Parameter estimation for an allometric food web model

The work presented here is the first step in an iterative modelling process , in which our first attempts with the inverse problems inform our future efforts in describ- ing the system of interest. These inverse problems are simply an attempt to minimize the distance between real observations and the proposed dynamics that model our system. In carrying out the inverse problems, we arrive at a set of parameters for a mathemat- ical model (in this case, the ATN model) that allow us to best-fit the observed data. However, the first attempt to solve an inverse problem naturally invites us to question the accuracy of the data, including the properties its observation errors might possess, as well as how much information about our system could be possibly represented in a given data set. In evaluating the performance of the inverse problems, we often find that the mathematical and/or statistical models require reformulation. Thus we usually must statistically address the nuances of inherent errors in data collection. Arriving at these statistical and mathematical model reformulations yields new inverse problems to solve, and so we repeat the modelling process until a sufficient theoretical framework has been developed to compare with experimental data. In this way, the inverse problems represent an experiment in themselves; we have hypothesized a mathematical and a sta- tistical model and seek to investigate their validity and shortcomings within the context of our study system. ### Study of Adaptive Model Parameter Estimation for Milling Tool Wear

source signals, it is not easy to extract the feature information of tool wear from complex signals in time-domain, frequency-domain. In addition, many past methods were developed to monitor tool wear by measuring spindle and feed motor power (current) and proved that the tool wear is very sensitive to the change of the cutting power  and . In the cutting process, techniques for tool wear monitoring are being used widely using the spindle and feed motor power. It does not interfere with cutting process by measurement equipment and the machine tool was not formed by a reworking process. However, generation mechanisms of the milling tool wear are more complex and in the view of various factors that affect tool wear, it is difficult to build the exact practical analysis model. Therefore, it is necessary to use experimental data to ensure the analysis and model. In some general methods, an explicit model is built by using Multivariate Linear Regression analysis method  and  or an implicit model by using the Neural Network . MLR method for monitoring tool wear by measuring spindle and feed motor power is to establish a mathematical model between milling cutting parameters and the classification by fuzzy pattern using MLR analysis. Then, tool wear model for spindle and feed motor power is established. Tool wear value is predicted by tool wear model. Tool wear model is adjusted using cutting parameters to give it better dynamic, fuzzy and real-time characteristics. Therefore, it will be effective to be used in the nonlinear predictive control systems. The NN method for monitoring tool wear by measuring spindle and feed motor power is to establish a Neural Network model which contains milling cutting parameters and cutting power. Then, tool wear network model is trained by using several experimental data of tool wear in different cutting process. Tool wear value is predicted by the Network model. Several problems exist with these methods; (1) It is diffcult to establish an exact practical analysis model between milling cutting parameters and tool wear. (2) The model based on spindle and feed motor power is used to recognize tool wear and can also cause larger error in a different cutting process by using the MLR method because tool wear model coefficients are fixed, that is, of low-precision and limiting applications. (3) The results of prediction are ### Mathematical Modelling of the Dynamics of Tumor Growth and its Optimal Control

by Misra and Pandey  in order to explore different aspects of the oesophageal swallowing of food bolus. A mathematical model was developed  to explore the enzymatic action of DNA Knots and links. Another mathematical model was formulated and analyzed  with aim to identify transcription factor binding sites of genes. In  performed a mathematical analysis for investigating in single-species and host-parasite system, with special reference to period-doubling bifurcations and chaos. A mathematical model was formulated and analyzed to investigate synchronization among tumor-like cell aggregations coupled by quorum sensing . In another study carried out by developing a mathematical model an attempt was made by Misra et al.  to find theoretical estimate of arterial blood flow during cancer treatment by the method of electromagnetic hyperthermia. The peristaltic flow of bile in a patheological state was explored in , while non-Newtonian characteristics of blood flow in contracting/ expanding arteries were explained in  by developing different mathematical models. Recent theoretical studies reported in  and  on nanofluid flow, temperature distribution and entropy generation in porous microchannel/microfluidic tubes bear promises of important applications in human microcirculatory systems. ### Ergodicity of stochastic smoking model and parameter estimation

Recently, Lahrouz et al. [] studied that a stochastic mathematical model of smoking has stability under certain conditions. And many scholars have studied the eﬀects of stochas- tic noises on the biological model: Gard [] pointed out that permanence in the corre- sponding deterministic model is preserved in the stochastic model if the intensities of the random ﬂuctuations are not too large; Gray et al. [] discussed the impacts of stochastic noises on one-dimensional stochastic SIS model; Zhang and Chen [] presented new suf- ﬁcient conditions for the existence and uniqueness of a stationary distribution of general diﬀusion processes, which is eﬃcient for the stochastic smoking model (). ### The effect of statistical error model formulation on the fit and selection of mathematical models of tumor growth for small sample sizes

As neither of our proposed methods of choosing a statistical error model a priori yielded results for our small longitudinal data sets, we examine the effects of the choice of the statistical error model (γ values) on the performance of each mathematical model using four criteria: visual fit, standard errors (SEs), mean square errors (MSEs), and consistency of parameter estimates across γ values. When comparing visual fits, we look for discrepancies between the model fit and the data graphically. If there is little to no discrepancy, we conclude that it is a reasonable visual fit for the data. Standard errors of parameter estimates describe the uncertainty of the estimate. If the SEs of a particular parameter are on the same order of magnitude as the parameter estimate, there is a great deal of uncertainty in that parameter estimate. If the SEs are larger than the parameter estimate, the level of uncertainty in the parameter estimation is too high to draw reasonable conclusions. Mean squared error is a measure of how well the model fits the data. If the MSEs are consistent across statistical models and SEs for certain parameters are relatively large, we conclude that the model fits are not sensitive to those parameters. If MSEs vary across statistical models, we conclude that certain statistical models capture the dynamics of the data set more accurately than others. We examine the consistency of parameter estimates among the different statistical models to see if changing γ has an effect on the parameter values. ### MATHEMATICAL MODEL ON DNA MUTATION AND TUMOR FORMATION

The fact that tumor occurs from uncontrolled cell growth basically due to DNA mutation is to a large extent beyond human control but effort can be made to reduce mutation. Supported by the stability analysis in this work, the study showed that the influence of environmental agents on the DNA awakens mutation, hence, disorganizing the already stated programme of the system. It is important that we stay away from cigarette smoking and the likes and alcohol intake as these sends very fast signals to the system, damaging normal protein synthesis, transfer, etc. ### Mathematical Model Solid Tumor At The Stage Of Angiogenesis With Immune Response

During its solid stage a tumor is small enough to take in nutrients and to expel waste by diffusion. However diffusion is not sufficient to support any continued growth of the tumor. This is because the tumor consumes nutrients at a rate proportional to its volume whereas the supply of ### Online State Space Model Parameter Estimation in Synchronous Machines

Abstract: In this paper a new approach based on the Least Squares Error method for estimating the unknown parameters of the 3 rd order nonlinear model of synchronous generators is presented. The proposed approach uses the mathematical relationships between the machine parameters and on-line input/output measurements to estimate the parameters of the nonlinear state space model. The field voltage is considered as the input and the rotor angle and the active power are considered as the generator outputs. In fact, the third order nonlinear state space model is converted to only two linear regression equations. Then, easy-implemented regression equations are used to estimate the unknown parameters of the nonlinear model. The suggested approach is evaluated for a sample synchronous machine model. Estimated parameters are tested for different inputs at different operating conditions. The effect of noise is also considered in this study. Simulation results declare that the efficiency of the proposed approach. ### Mathematical Modeling For The Tumor Growth In The Bladder

The early detection of bladder cancer is a challenging problem, due to structure of cancer cells. The aim of this paper is to presents mathematical modeling describe the tumor growth in the bladder. This model is used for detecting the cancer in its early stages. The segmentation results will be used as a base for a computer aided diagnosis (CAD) system for early detection of the cancer which will improve the chances of survival of patient. ### Mathematical analysis for tumor growth model of ordinary differential equations

In 1906, Hamer has developed one of the earliest epidemic model. He try to apply post- germ-theory thinking towards the solution of two specific quantitative problems whereas the relationship between numbers mosquitoes regarding incidence of malaria; and the regular recurrence of measles epidemic (Anderson & May, 1992).From the work of Hamer, then in 1916-1917, the classical SIR model originated from the papers of Ross and Ross and Hudson. (Magal & Ruan, 2014). After year of 1926, SIR model become famous since fundamental contribution of Kermack and Mckendrick which describes the transmission of infectious disease between susceptible and infective individuals and provides the basic framework for almost all later epidemic models they also include stochastic epidemic models using Monte Carlo simulations or known as individual-based models (IBM) in their paper. (Magal & Ruan, 2014). The model categorizes population into Susceptible, Infected and Recovered. In this model, susceptible individuals in S-stage have chance to be infected and progress to infection I-stage until recovery to R-stage. The flow is shown in figure 1 (Chong & Zee, 1927). ### Threshold Parameter for the Control of Unemployment in the Society: Mathematical Model and Analysis

DOI: 10.4236/jamp.2018.612214 2566 Journal of Applied Mathematics and Physics fects leading to communal clashes and the rise of such groups such as Boko Ha- ram, Niger Delta Militant, Armed Robbery, Prostitution and child trafficking constituting hiccups to lifes and properties. Also, there research revealed that unemployment in Nigeria increases from 21.1% in 2010 to 23.9% in 2011 with youth unemployment at over 50% from 2011 to 2013, there is about increament of 16% unemployment growth rate in Nigeria. The impact of their paper also revealed that Government should invest on education heavily so that graduate will be self reliance instead of job seekers. ### Parameter estimation in a structured algal coagulation-fragmentation model

Obtaining experimental values for parameters that govern the aggregation pro- cess, such as, sticking eciency contact eciency and growth rate of aggregates has proven to be dicult. Alldredge and Mcgillivary  observed that occulated diatoms in the size range from 0.2 to 7.6 mm had probability up to 0.88 of stick- ing to each other. The paper  discussed a method for estimating the sticking eciency from experimental data and a modication of the model of Jackson . Their conclusion was that for some species of phytoplankton stickiness increases when nutrients are depleted and for other species stickiness remained almost con- stant. In  an inverse methodology based on least squares, was used together with an aggregation model developed in , in which growth occur only in single cells and not aggregates, to t data obtained from . Their conclusions were that the modeled contact eciencies are small, in comparison with the actual ones, giving rise to low densities for large aggregates (> 0:6 mm 3 ), a corroboration of ### Mathematical Models of Avascular Tumor Growth

Because there are three distinct stages (avascular, vascular, and metastatic) to cancer development, researchers often concentrate their eﬀorts on answering speciﬁc questions on each of these stages. This review aims to describe the current state of mathematical modeling of avascular tumor growth, i.e., tumors without blood vessels. This is not to say that this is the most important aspect of tumor growth—on the contrary, from a clinical point of view angiogenesis and vascular tumor growth together with metastasis are what cause the patient to die, and modeling and understanding these is crucial for cancer therapy. Nevertheless, when attempting to model any complex system it is wise to try and understand each of the components as well as possible before they are all put together. Avascular tumor growth is much simpler to model mathematically, and yet contains many of the phenomena which we will need to address in a general model of vascular tumor growth. Moreover, the ease and reproducibility of experiments with avascular tumors means that the quality and quantity of experimental evidence exceeds that for vascular tumors, for which it is often diﬃcult to isolate individual eﬀects. In particular, because some (but certainly not all) tumor cell lines grown in vitro form spherical aggregates , the relative cheapness and ease of in vitro experiments in comparison to animal experiments has made tumor spheroid assays very popular. ### Parameter Identifiability and Parameter Estimation of a Diesel Engine Combustion Model

The paper is organized as follows. In the second section the phenomenological model is presented as well as the resulting mathematical system. The third section is devoted to identifiability: some definitions are recalled, piecewise identifiability tests and results are given. The fourth section treats the transformation of the relations obtained from identifiability previous study. Section 5 is devoted to numerical procedure: the numerical estima- tion of derivatives and the resulting estimation procedure and numerical results are given. Finally, conclusions are drawn in the last section. ### AN ENHANCED METHOD OF LMS PARAMETER ESTIMATION FOR SOFTWARE REALIABILITY MODEL

Software reliability growth models are designed to make predictions. Predictions of actual reliability or failure rate time needed to reach a given reliability target and things like that. In practice, software reliability growth models encounter major challenges. First of all, software testers seldom follow the operational profile to test the software, so what is observed during software testing may not be directly extensible for operational use. Secondly, when the number of failures collected in a project is limited, it is hard to make statistically meaningful reliability predictions. Thirdly, some of the assumptions of Software reliability Growth Model (SRGM) are not realistic e.g. the assumption that the faults are independent of each other, that each fault has the same chance to be detected in one class; and that corrections of a fault never introduces new fault (Deepak Pengoria and Saurabh 2009). Vladimir Zeljkovic et al (2011), the authors have made a study on reliability and show that the software reliability cannot be calculated during the design phase. If adequate data on system failure is collected throughout the project during testing phase, the models could apply on the parameter to predict the reliability. They have also observed that the importance of reliability estimation during the testing phase. ### Parameter Estimation for the Continuous Time Stochastic Logistic Diffusion Model

DOI: 10.4236/ojs.2017.76072 1040 Open Journal of Statistics into random variables. Therefore, it is more reasonable to use the stochastic differential equation with to describe the real systems disturbed by random noises. For example, the stochastic logistic diffusion model has been widely used in the field of social life, application of stochastic logistic diffusion model has been used in the field of applied economics   , biology    , power engineering    and so on. Very recently, considerable research results have been reported on the parameter estimation based on discrete observation. To be special,  used the least squares method to estimate the parameters, also obtained the point estimators and confidence intervals as well as joint confidence regions.  used conditional least squares and weighted conditional least squares method to study the parameter estimation of two-type continuous-state branching processes with immigration based on low frequency observations at equidistant time points.  studied the asymptotic behaviour of parametric estimator for nonstationary reflected Ornstein-Uhlenbeck process by applying maximum likelihood estimation. For the stochastic logistic diffusion model,  worked out the optimization problem with respect to stationary probability density and provide a new equivalent, an ergodic method is used to show the almost surely equivalency between the time averaging yield and sustainable yield.  considered a stochastic logistic growth model involving both birth and death rates in the drift and diffusion coefficients and the associated complete Fokker-Planck equation is also established to describe the law of the process.  focused on stochastic dynamics involve continuous states as well as discrete events, and obtain weak convergence of the underlying system, and utilized the structure of limit system as a bridge to invest stochastic permanence of original system driving by a singular Markov chain with a large number of states.  presented some basic aspects of adequate numerical analysis for the random extensions such as numerical regularity and mean square convergence.  improved two mathematically tractable cases: at the limit of the number of individuals and at the limit of basic reproduction ratio. In the discrete observations, we let the time interval tends to 0 to get more accurate result. Therefore, this means that parametric inference based on continuous time observation is much more accurate in dealing with parameter estimation problem. During the estimation processing, two important theories have been used to estimate parameters based on continuous observation in the existing literature. One is denoting Radon-Nikodym derivative with likelihood ratio and the other one is by using stationary distribution of solution. 