A Mathematical Model of Chemotherapeutic Drug for Tumor Treatment

This work considers a model that represents the procedure for tumor treatment, which consists tumor cell energy and specific dose of selenium with the combination of Adriamycin. The tumor cell energy depends upon tumor cell density and the selenium with the combination of Adriamycin is a specific drug that works against tumor cell energy. The problem in the form of partial differential equations for the tumor cells density, tumor cell energy and the effect of selenium with the combination of Adriamycin. The result of this work suggests that the tumor cell energy decrease by the effect of selenium with the combination of selenium with the combination of Adriamycin, if tumor cell energy decreases then the size of tumor reduces. Introduction: Main focus on the work because combination of selenium enhanced chemotherapeutic effect of Adriamycin in tumor cells beyond that seen with the Adriamycin(Doxorubicin) used alone. Selenium is a natural health product widely used in the treatment and prevention of cancers, but not good for large chemotherapeutic treatment of tumor. The energy in all cells originates from the energy of the sun’s light quanta, which is convicted by photosynthetic plants into cellular molecules. These plants are used as food sources by various microorganisms and animals. Life is an energy process. It takes energy to operate muscles, extract wastes, make a new cell, heal wounds, even to think. It’s in an organism’s cells where all this energy is spent. In some cells, as much as a half of a cells energy output is used to transfer molecules across the cell membrane. Cell movements require energy and thousands of energy-hungry chemical reactions go on in every living cell, every second, every day. The casual agents are certain chemicals, radiation, and viruses that behave as mutagens by acting at the level of DNA. However, it has also been proved that cancer is a genetic disease caused by multiple mutations within the DNA of cells. The researchers [3] have discovered that an apparently nontoxic cellular “energy blockers” can eradicate large liver tumors grown, in rats. Liver cancer usually isn’t detected in people until it’s difficult or impossible to treat, and many other aggressive cancers spread to the liver, so we need more treatment options. Martin et. al. [1] a quadruple drug combination consisting of a triple-drug combination of N-(phosphonacetyl)-L-aspartate (PALA) + 6-methylmercaptopurine riboside (MMPR) + 6-amino-nicotinamide (6-AN), designed to primarily deplete cellular energy in tumor cells, + selenium with the combination of Adriamycin (Adria) yielded significantly enhanced anticancer activity (i.e., tumor regressions) over that produced by either Adria alone at maximum tolerated dose (MTD) or by the triple-drug combination, against large, spontaneous, autochthonous murine breast tumors. selenium with the combination of Adriamycin is a type of antibiotic used specifically in the treatment of cancer. It interferes with the multiplication of cancer cells and slows or stops their growth and spread in the body. Stolfi et. al. [2] this report describes a highly active chemotherapeutic drug combination. This quadruple drug combination, administered on a 10-11-day schedule, produced an impressive partial tumor regression rate of 67% of large, spontaneous, autochthonous, murine breast tumors and a tumor regression rate of 74% of first-passage transplants of the spontaneous breast tumors. Selenium with the combination of Adriamycin, which is administrated intravenously only. It is most commonly used in treatment of all cancers (Breast, Stomach, Lymphomas etc.). Selenium with the combination of Adriamycin is chemotherapy drug that interrupts the cell cycle, effectively cell growth. Selenium with the combination of Adriamycin degrades rapidly in solution; a fluorometric method was developed to determine the precise dose use in treatments. But selenium with the combination of Adriamycin also has more serious side effects that limit the amount you can safely take. At a certain level, selenium with the combination of Adriamycin increases the risk of heart damage. The selenium with the combination of Adriamycin works by impairing DNA synthesis, a crucial feature of cell division, and this is able to target rapidly dividing cells. Selenium with the combination of Adriamycin is a very serious anticancer medication with definite potential to do great harm as well as great good. It used alone or in combination with other chemotherapy drugs. Ward and King [10] consider the effect of cellular material as well, mainly because it is very simple to construct an analogous model to that of Ward and King [9] for monolayer cultures, so that a direct comparison of the effect of drugs on the two types culture can be made the approach used in this model of Ward & King [12], which assumes source of cellular material outside a 3-dimensional monolayer case. Anderson et. al. [13] works to examine fluid flow through the theoretical network structures. 2. Our model: Mathematically, for an untreated tumor the work equation may be reasonably quantified by a single partial differential equation . c J c t λ ∂ = ∇ + ∂ (1) in which c(x,t) designates the tumor cell density and λ denotes the cell proliferation rate. Where ( ) J D c μ = ∇ , the gradient of the potential μ produces a flux J is proportional to μ ∇ . The D is Fickian diffusion coefficient represent the active motility of tumor cells, which in this derivation may depend on x, t and c. The tumor spread is assumed to be spherically symmetric in this model, and x measures the distance from the center. .[ ( )] c D c c t μ λ ∂ = ∇ ∇ + ∂ (2) Associate with a spatial distribution of cells, an energy den-


Introduction:
Main focus on the work because combination of selenium enhanced chemotherapeutic effect of Adriamycin in tumor cells beyond that seen with the Adriamycin(Doxorubicin) used alone. Selenium is a natural health product widely used in the treatment and prevention of cancers, but not good for large chemotherapeutic treatment of tumor. The energy in all cells originates from the energy of the sun's light quanta, which is convicted by photosynthetic plants into cellular molecules. These plants are used as food sources by various microorganisms and animals. Life is an energy process. It takes energy to operate muscles, extract wastes, make a new cell, heal wounds, even to think. It's in an organism's cells where all this energy is spent. In some cells, as much as a half of a cells energy output is used to transfer molecules across the cell membrane. Cell movements require energy and thousands of energy-hungry chemical reactions go on in every living cell, every second, every day. The casual agents are certain chemicals, radiation, and viruses that behave as mutagens by acting at the level of DNA. However, it has also been proved that cancer is a genetic disease caused by multiple mutations within the DNA of cells. The researchers [3] have discovered that an apparently nontoxic cellular "energy blockers" can eradicate large liver tumors grown, in rats. Liver cancer usually isn't detected in people until it's difficult or impossible to treat, and many other aggressive cancers spread to the liver, so we need more treatment options.
Martin et. al.
[1] a quadruple drug combination consisting of a triple-drug combination of N-(phosphonacetyl)-L-aspartate (PALA) + 6-methylmercaptopurine riboside (MMPR) + 6-amino-nicotinamide (6-AN), designed to primarily deplete cellular energy in tumor cells, + selenium with the combination of Adriamycin (Adria) yielded significantly enhanced anticancer activity (i.e., tumor regressions) over that produced by either Adria alone at maximum tolerated dose (MTD) or by the triple-drug combination, against large, spontaneous, autochthonous murine breast tumors. selenium with the combination of Adriamycin is a type of antibiotic used specifically in the treatment of cancer. It interferes with the multiplication of cancer cells and slows or stops their growth and spread in the body. Stolfi et. al.
[2] this report describes a highly active chemotherapeutic drug combination. This quadruple drug combination, administered on a 10-11-day schedule, produced an impressive partial tumor regression rate of 67% of large, spontaneous, autochthonous, murine breast tumors and a tumor regression rate of 74% of first-passage transplants of the spontaneous breast tumors.
Selenium with the combination of Adriamycin, which is ad-ministrated intravenously only. It is most commonly used in treatment of all cancers (Breast, Stomach, Lymphomas etc.). Selenium with the combination of Adriamycin is chemotherapy drug that interrupts the cell cycle, effectively cell growth. Selenium with the combination of Adriamycin degrades rapidly in solution; a fluorometric method was developed to determine the precise dose use in treatments. But selenium with the combination of Adriamycin also has more serious side effects that limit the amount you can safely take. At a certain level, selenium with the combination of Adriamycin increases the risk of heart damage. The selenium with the combination of Adriamycin works by impairing DNA synthesis, a crucial feature of cell division, and this is able to target rapidly dividing cells. Selenium with the combination of Adriamycin is a very serious anticancer medication with definite potential to do great harm as well as great good. It used alone or in combination with other chemotherapy drugs. Ward and King [10] consider the effect of cellular material as well, mainly because it is very simple to construct an analogous model to that of Ward and King [9] for monolayer cultures, so that a direct comparison of the effect of drugs on the two types culture can be made the approach used in this model of Ward & King [12], which assumes source of cellular material outside a 3-dimensional monolayer case. Anderson et. al. [13] works to examine fluid flow through the theoretical network structures.

Our model:
Mathematically, for an untreated tumor the work equation may be reasonably quantified by a single partial differential equation in which c(x,t) designates the tumor cell density and λ denotes the cell proliferation rate. Where .
Associate with a spatial distribution of cells, an energy den-RESEARCH PAPER sity n(c), which is an internal energy per unit volume of an evolving spatial pattern so that the total energy N(c) in a volume is given by

Numerical result: The parameters
To solve the partial differential equation, I use the Matlab 6.0, a partial differential equation solver. In this result, I see that the tumor cell density is going to decrease as well as the effect of selenium with the combination of Adriamycin is increases. It shows that if tumor cell density decreases with the effect of selenium with the combination of Adriamycin then tumor cell energy decreases with the dose of selenium with the combination of Adriamycin because tumor cell energy is depend upon the tumor cell density.  Figure 1 shows the 3D and 2D representation in the above diagrams.
The figure 1 shows the surface plot tumor cell density with respect to time T and radius x. I consider the time 0 to 100 h and radius 0 to 1 mm. I analyzed the growth of tumor cell density in above figure shows the initially, tumor cell density is cells at center of tumor and cells at the boundary of the tumor. After the 100h the tumor cell density is almost same at the center and boundary of the tumor, without the effect of selenium with the combination of Adriamycin. The figure shows at initially max. at the center of tumor but it is less at the boundary.     Result and discussion. An enzyme (Adenosine Triphoshate ATP) provides chemical energy for the cell, ATP release energy by releasing a phosphoric acid radical. Then, energy derived from the cellular nutrient causes the acceptor molecule and phosphoric acid to recombine to form new ATP. The entire process continues over and over again. Without that energy, blood vessels cannot grow to the site of a tumor, and without the nutrient supply in blood, tumors cannot grow larger than a pinhead.
The aim to develop a model for studying avascular tumor growth with the effect of dose of the selenium with the combination of Adriamycin. The equations are solved numerically in this case. Throughout the paper, our philosophy when RESEARCH PAPER modeling has been to use the simplest functional forms that capture the physical phenomena. The numerical simulation mainly involved the study of the effects on tumor cell survival of the dimensionless parameter α , which encapsulates the extent of penetration of the chemotherapeutic drug. The simulation emphasize that chemotherapeutic drug penetration is a crucial factor in determining drug effectiveness. The growth of tumor in a spherical shape has been examined in order to describe the initial stages. We have introduced a tumor cell energy model and used tumor cell density with the effect of selenium with the combination of Adriamycin to describe the movement of tumor cells. In section 3 the numerical results suggest that the tumor cell density decreases with the effect of selenium with the combination of Adriamycin. It has been observed for different values of effect of selenium with the combination of Adriamycin. The tumor cell energy decreases with the effect of selenium with the combination of Adriamycin because tumor cell energy fully depends on the tumor cell density.