Taguchi method for partial differential equations with application in tumor growth.

UNLABELLED: The growth of tumors is a highly complex process. To describe this process, mathematical models are needed. A variety of partial differential mathematical models for tumor growth have been developed and studied. Most of those models are based on the reaction-diffusion equations and mass conservation law. A variety of modeling strategies have been developed, each focusing on tumor growth. MATERIAL AND METHODS: Systems of time-dependent partial differential equations occur in many branches of applied mathematics. The vast majority of mathematical models in tumor growth are formulated in terms of partial differential equations. We propose a mathematical model for the interactions between these three cancer cell populations. The Taguchi methods are widely used by quality engineering scientists to compare the effects of multiple variables, together with their interactions, with a simple and manageable experimental design. In Taguchi's design of experiments, variation is more interesting to study than the average. RESULTS: First, Taguchi methods are utilized to search for the significant factors and the optimal level combination of parameters. Except the three parameters levels, other factors levels other factors levels would not be considered. Second, cutting parameters namely, cutting speed, depth of cut, and feed rate are designed using the Taguchi method. Finally, the adequacy of the developed mathematical model is proved by ANOVA. According to the results of ANOVA, since the percentage contribution of the combined error is as small. CONCLUSIONS: Many mathematical models can be quantitatively characterized by partial differential equations. The use of MATLAB and Taguchi method in this article illustrates the important role of informatics in research in mathematical modeling. The study of tumor growth cells is an exciting and important topic in cancer research and will profit considerably from theoretical input. Interpret these results to be a permanent collaboration between math's and medical oncologists.

Differential equations with applications in cancer diseases.

UNLABELLED: Mathematical modeling is a process by which a real world problem is described by a mathematical formulation. The cancer modeling is a highly challenging problem at the frontier of applied mathematics. A variety of modeling strategies have been developed, each focusing on one or more aspects of cancer. MATERIAL AND METHODS: The vast majority of mathematical models in cancer diseases biology are formulated in terms of differential equations. We propose an original mathematical model with small parameter for the interactions between these two cancer cell sub-populations and the mathematical model of a vascular tumor. We work on the assumption that, the quiescent cells' nutrient consumption is long. One the equations system includes small parameter epsilon. The smallness of epsilon is relative to the size of the solution domain. RESULTS: MATLAB simulations obtained for transition rate from the quiescent cells' nutrient consumption is long, we show a similar asymptotic behavior for two solutions of the perturbed problem. In this system, the small parameter is an asymptotic variable, different from the independent variable. The graphical output for a mathematical model of a vascular tumor shows the differences in the evolution of the tumor populations of proliferating, quiescent and necrotic cells. The nutrient concentration decreases sharply through the viable rim and tends to a constant level in the core due to the nearly complete necrosis in this region. CONCLUSIONS: Many mathematical models can be quantitatively characterized by ordinary differential equations or partial differential equations. The use of MATLAB in this article illustrates the important role of informatics in research in mathematical modeling. The study of avascular tumor growth cells is an exciting and important topic in cancer research and will profit considerably from theoretical input. Interpret these results to be a permanent collaboration between math's and medical oncologists.

Ordinary differential equations with applications in molecular biology.

UNLABELLED: Differential equations are of basic importance in molecular biology mathematics because many biological laws and relations appear mathematically in the form of a differential equation. In this article we presented some applications of mathematical models represented by ordinary differential equations in molecular biology. MATERIAL AND METHODS: The vast majority of quantitative models in cell and molecular biology are formulated in terms of ordinary differential equations for the time evolution of concentrations of molecular species. Assuming that the diffusion in the cell is high enough to make the spatial distribution of molecules homogenous, these equations describe systems with many participating molecules of each kind. RESULTS: We propose an original mathematical model with small parameter for biological phospholipid pathway. All the equations system includes small parameter epsilon. The smallness of epsilon is relative to the size of the solution domain. If we reduce the size of the solution region the same small epsilon will result in a different condition number. It is clear that the solution for a smaller region is less difficult. We introduce the mathematical technique known as boundary function method for singular perturbation system. In this system, the small parameter is an asymptotic variable, different from the independent variable. In general, the solutions of such equations exhibit multiscale phenomena. Singularly perturbed problems form a special class of problems containing a small parameter which may tend to zero. CONCLUSIONS: Many molecular biology processes can be quantitatively characterized by ordinary differential equations. Mathematical cell biology is a very active and fast growing interdisciplinary area in which mathematical concepts, techniques, and models are applied to a variety of problems in developmental medicine and bioengineering. Among the different modeling approaches, ordinary differential equations (ODE) are particularly important and have led to significant advances. Ordinary differential equations are used to model biological processes on various levels ranging from DNA molecules or biosynthesis phospholipids on the cellular level.

Pancreatic exocrine function in patients with diabetes.

BACKGROUND: Decreased function of the exocrine pancreas is frequent in patients with diabetes. Our aim was to investigate clinical correlates of pancreatic exocrine failure in patients with diabetes. PATIENTS AND METHODS: We investigated exocrine function by assaying both elastase-1 concentration and chymotrypsin activity in 667 patients. We conducted separate analysis on patients with Type 1 diabetes and patients with Type 2 diabetes. Patients were separated into three groups according to whether both elastase-1 concentration and chymotrypsin activity were normal, or one or both were altered. RESULTS: A total of 667 consecutive patients were analysed, including 195 with Type 1 and 472 with Type 2 diabetes. Elastase-1 concentration was <200 µg/g in 23% of the patients. Chymotrypsin activity was <6 U/g in 26% of the patients. In 66% of the patients elastase-1 concentration was >200 ug/g and chymotrypsin activity >6 U/g. One test was below threshold in 19%, both in 15%. In patients with Type 1 diabetes, the three groups defined by results of elastase-1 concentration and chymotrypsin activity differed with regard to duration of diabetes and prevalence of glutamic acid decarboxylase antibodies, but not BMI or HbA(1c) , or prevalence of retinopathy, neuropathy, nephropathy or vascular disease. In patients with Type 2 diabetes, the three groups differed with regard to BMI, use of insulin and vascular disease, but not known duration. CONCLUSION: Factors associated with pancreatic exocrine failure differ in patients with Type 1 diabetes compared with patients with type 2 diabetes. In patients with Type 2 diabetes, association of decreased pancreatic exocrine function with BMI and vascular disease suggests a role of pancreatic arteriopathy.