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.

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Background: **Mathematical** modeling of biological processes is widely used to enhance quantitative understanding of bio-medical phenomena. This quantitative knowledge can be applied in both clinical and experimental settings. Recently, many investigators began studying **mathematical** models of **tumor** response to radiation therapy. We developed a simple **mathematical** **model** to simulate the **growth** of **tumor** volume and its response to a single fraction of high dose irradiation. The modelling study may provide clinicians important insights on radiation therapy strategies through identification of biological factors significantly influencing the treatment effectiveness. Methods: We made several key assumptions of the **model**. **Tumor** volume is composed of proliferating (or dividing) cancer cells and non-dividing (or dead) cells. **Tumor** **growth** rate (or **tumor** volume doubling time) is proportional to the ratio of the volumes of **tumor** vasculature and the **tumor**. The vascular volume grows slower than the **tumor** by introducing the vascular **growth** retardation factor, θ . Upon irradiation, the proliferating cells gradually die over a fixed time period after irradiation. Dead cells are cleared away with cell clearance time. The **model** was applied to simulate pre-treatment **growth** and post-treatment radiation response of rat rhabdomyosarcoma tumors and metastatic brain tumors of five patients who were treated with Gamma Knife stereotactic radiosurgery (GKSRS).

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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).

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The work presented here is the first step in an iterative modelling process [7], 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.

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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 [9] and [10]. 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 [11] and [12] or an implicit **model** by using the Neural Network [13]. 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

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by Misra and Pandey [3] in order to explore different aspects of the oesophageal swallowing of food bolus. A **mathematical** **model** was developed [4] to explore the enzymatic action of DNA Knots and links. Another **mathematical** **model** was formulated and analyzed [5] with aim to identify transcription factor binding sites of genes. In [6] 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 [7]. In another study carried out by developing a **mathematical** **model** an attempt was made by Misra et al. [8] 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 [9], while non-Newtonian characteristics of blood flow in contracting/ expanding arteries were explained in [10] by developing different **mathematical** models. Recent theoretical studies reported in [11] and [12] on nanofluid flow, temperature distribution and entropy generation in porous microchannel/microfluidic tubes bear promises of important applications in human microcirculatory systems.

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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** ().

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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.

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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.

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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

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.

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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.

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).

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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.

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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 [8] 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 [16] discussed a method for estimating the sticking eciency from experimental data and a modication of the **model** of Jackson [14]. 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 [4] an inverse methodology based on least squares, was used together with an aggregation **model** developed in [3], in which **growth** occur only in single cells and not aggregates, to t data obtained from [13]. 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

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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 [3], the relative cheapness and ease of in vitro experiments in comparison to animal experiments has made **tumor** spheroid assays very popular.

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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.

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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.

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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 [1] [2] [3], biology [4] [5] [6] [7], power engineering [8] [9] [10] and so on. Very recently, considerable research results have been reported on the **parameter** **estimation** based on discrete observation. To be special, [11] used the least squares method to estimate the parameters, also obtained the point estimators and confidence intervals as well as joint confidence regions. [12] 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. [13] studied the asymptotic behaviour of parametric estimator for nonstationary reflected Ornstein-Uhlenbeck process by applying maximum likelihood **estimation**. For the stochastic logistic diffusion **model**, [14] 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. [15] 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. [16] 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. [17] presented some basic aspects of adequate numerical analysis for the random extensions such as numerical regularity and mean square convergence. [18] 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.

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In earlier papers [1,2], we developed **mathematical** models and a computational scheme for investigating the non-intuitive trade-offs between incurred test-costs and the resulting product quality (field performance, expected return rates, etc.) along the entire assembly, test and repair chain. The **model** **parameter** **estimation** described here is for various different test scenarios that are combinations of different property types being measured (continuous or discrete) and different data recorded (the measurement itself or simply whether the product passed or failed).

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