Conclusion:

1. Mathematical Modeling of Central Nervous System and Algorithms for The Prediction of Cerebrospinal Fluid Flow AAA, BBB, CCC, MSc Evening (Semester 4), CASPAM, BZU What happens in extreme cases? Introduction to Microvasculature Microvasculature refers to smallest system of blood vessels in a body, including those responsible for microcirculation (like in brain). • Algorithms to Construct Microvasculature: • Vessel and Boundary Avoidance • Constrained Constructive Optimization Cerebrospinal fluid flow (CSF): CSF stands for Cerebrospinal fluid : a clear and colorless fluid that is continuously produced and absorbed. CSF flows in the ventricles within the brain, around the surface of the brain and spinal cord. Fig 3. Simple and Complex Domain Constrained Constructive Optimization algorithm: By Hagen-Poiseuille Law Pressure drop = Resistance x Flow P which implies that min V= such that new (x,y) Vessel and Boundary Avoidance Algorithm: ; = Fig 1.Basic Parameters in Brain Physics Problem statement: Development of a model of Hydrocephalus: Hydrocephalus is a condition characterized by excessive accumulation of fluid in the brain. Communicating hydrocephalus occurs when cerebrospinal fluid (CSF) can still flow among the ventricles. Noncommunicating hydrocephalus, also called "obstructive" hydrocephalus, occurs when the flow of CSF is blocked. Fig 4. vessel and boundary avoidance scheme Fig 5. Optimize structure to minimize volume. Conclusion: The mathematical model & the algorithms provide for improving vasculature changes with hydrocephalusand may allow for CSF drainage through the microvasculature.Thelateral ventricle volume is almost 18 times larger in the hydrocephalic case.Development of algorithm to create microvasculature is 80% accurate. Mathematical illustration: We take the following variables: Distance from to boundary No. of boundary points Unit vector of Unit vector from boundary Length of new segment ; avoidance exponent =New vessel =+ References: 1. D. Beard, J. Bassingthwaighte. The Fractal Nature of Myocardial Blood Flow Emerges from a Whole-Organ Model of Arterial Network. J Vasc Res 2000; 37:282–296. 2. Nolte J, Sundsten JW. The human brain: an introduction to its functional anatomy. 5. Mosby; St. Louis: 2002. Fig 2. Mathematical Models for Improved Understanding Treatment

2. MATH IN COSMOLOGY ABC (The Women University of Multan) MATHEOTIC MIND STUDY OF BLACK HOLE Observing a black hole directly isn't something you can just roll out of bed and do. You also can't just fire a probe in there and read some data. They're black holes and that's not how this works. To prove an idea about black holes, cosmologists have to do some major number-crunching. If the numbers check out, that means that the theory makes sense and would be true. It remains a theory since there's (usually) no way to empirically prove it. This math is not the polynomial factoring you remember from 10th grade algebra. This is math that would melt regular people brains.different math model used for this like river model MATHEMATICAL COSMOLOGY Cosmology, the study of the universe as a whole At the heart of cosmology, there's mathematics, and to comprehend the universe, it is not only necessary to observe and measure it, but also to understand its inner workings;.it takegood understanding of the equations that govern the evolution of the cosmos. HUBBLE LAW FOR SPEED OF GLAXY MATHEMATICAL MODEL FOR STAR FORNMATION A simple mathematical model of the formation of the stars is established and put in computational algorithm. This algorithm enables us to know more about the formation of the star. The model contains three active components: cool atomic clouds, cool molecular clouds, and active young stars. Each of these components may interact with the other components or with the rest of the Galaxy. The model is therefore an open system connected with two mass reservoirs outside the system T=A+M+S. . FIRST MATHEMATICAL MODEL The "Friedmann model" is a model of the Universe governed by the Friedmann equation, which describes how the Universe expands from singularity. Sometimes written as two independent equations, it's based on Einstein's general theory of relativity for gravity, and with two very important assumptions it forms the basis for our understanding of the evolution and structure of our Universe. These assumptions, together called "the cosmological principle", are that the Universe is homogeneous, and that it's isotropic CURRENT TASK One of the greatest challenges we face today is attempting to create a mathematical grand "theory of everything" Key Image 2 Acknowledgments ALLAH ALMIGHTY .MY PARENTS AND TEACHERS References ARTICLE REVEALING NATURE OF COSMOS MATHEMATICS OF GLOBAL warming FRIEDMAN EQUATION

3. Basic drug combination theory : Loewe additivity ABC (The Women University of Multan) Matheotic Mind Key Image 3 Application of Loewe additivity for estimating rational drug combinations and optimal doses : The Loewe additivity with mathematical modeling of hepatitis C virus (HCV) replication The aim is to optimize the antiviraldrug combination. Additivity: is regarded as a primary criterion for evaluating drug combination effects. To conceptualize Loewe additivity, let us consider the simplest situation, in which a combination of drugs A and B has a synergistic, additive or antagonistic effect. If the effect is additive, the Loewe additivity is defined as Key Image 5 + = 1(1) To evaluate a drug combination effect by the CI, we require dose–response curves of drugs A and B, from which we can determine and . where and are the concentrations of drugs A and B respectively in the combined dose, and and are the respective concentrations of drugs A and B thatproduce the same effect as the drug combination. Evaluation of the Effect of Single Drug: The effect of a single drug E is modeled by the Hill function as follows:. Combination Index (CI)If two drugs do not mutually interact, they can be related through the combination index (CI) based on the mass action law derived by Chou and Talalay. Conceptual diagram showing the effects of two anti-HCV drug effects on HCV replication The alternative drug combination theory: Bliss independence Bliss independence defines the expectation of a combined drug effect, calculated by multiplying the probabilities of the individual drugs+ (5) CI = + (2) where is the maximum effect, c is the drug concentration, h is the Hill coefficient that determines the steepness of the dose–response curve, and IC50 is the concentration at which E exerts 50% of its maximum effect Substituting Eq. (3) into Eq. (1) and rearranging in the case of = 1, we obtain: When CI < 1, the relationship is synergetic, when CI = 1, it is additive, and when CI > 1, it is antagonistic + = 1 (04) Acknowledgments: I am thankful to Allah Almighty and then my worthy Teachers who provided insight and expertise that greatly assisted me in preparing this poster. References Ho DD, Neumann AU, Perelson AS, Chen W, Leonard JM, Markowitz M: Rapid turnover of plasma virions and CD4 lymphocytes in HIV-1 infection. Nature 1995, 373:123–126. Perelson AS: Modelling viral and immune system dynamics. Nat Rev Immunol 2002, 2:28–36. Numerically solving Eq. (4) for E, we can predict the additive effect at any drug concentrations and

4. Mathematics in Metabolism How a Medicine is Absorbed in the body ABC (M.ScMathematics Semester II), CASPAM, BZU Formation of Differential Equation We need to subtract the elimination part from absorption part for our model. Our differential equation is as follows If we substitute V=15 , A=0.5 , F=2 , D= 800 , E=0.4 where C is the variable dependent on time, then Solving this equation, we get the concentration at t Pharmacokinetics Pharmacokinetics is a process wherely substances like food or drugs are ingested into the body via mouth or needles and then processed. Here we will concentrate on drugs. This process takes place in 5 steps: (1) Liberation (2) Absorption (3) Distribution (4) Metabolism (5) Excretion Various drugs having different levels of concentration and known levels of distribution, metabolism and excretion in the body Illustration by Differential Equation We model such situation with a differential equation having (1) absorption part and (2) elimination part. We take the following variables: D = drug dose given , V= volume distributed in the body C= concentration of drug at time t F=fraction of dose that is absorbed , t= time A=absorption rate constant , E=elimination rate constant C(t)= 533.3(e-0.4t – e-0.5t) Different Steps of Pharmacokinetics Role of MATHEMATICS When the drug is injected into the body, the concentration of the drug is zero. As the drug moves around in the body and is metabolised, its concentration increases. There comes a point when the concentration no longer increases and begins to decrease. This is the stage when the drug is fully distributed and metabolism is taking place. As time goes on, the concentration gets lesser and lesser and finally falls below a certain effective amount. Absorption and Elimination Part Absorption part depends on the amount of the drug given, the fraction that is absorbed and the absorption rate constant that decreases with time. The expression for absorption is given by, (A) (F) (D) (e-At) Elimination part depends on elimination constant, volume distributed in the body and the concentration left of the drug. The expression for this part is given by (E) (V) (C) The concentration increases upto around 2 and then levels off , finally decreasing to almost zero at t=24 Conclusion Too much concentration of a drug may result in death, as suicide attempters this way. Consequently, only that concentration of medicine should be taken that is prescribed by the doctor such as if he has said to take 1 pill a day then it means only one pill will be absorbed in the body in 24 hours. References Murray Bourne (articles about Math education) www.IntMath.com Preparing a Syringe Absorption rate constant decreases with time

5. Mathematical Analysis Of A Model Of Nuclear Reactor Dynamics AAA, BBB, CCC , CASPAM, BZU