Pharmaceutical Mathematics With Application To Pharmacy By Gowda Pdf

Index

Chapter 1

PARTIAL FRACTIONS

1.1 INTRODUCTION

We know the method of combining number of fractions into a single by taking LCM (least common multiple) of denominators of the fractions. Now, here we have to learn, converse problem, that is splitting up of a given fraction into a number of simpler fractions called Partial Fractions.

In this Chapter, we study the method of resolving a given fraction into Partial fractions.

1.1.1 Polynomial

An algebraic expression of the form f(x) = a0xn + aixn_1 + ... + an is called a polynomial in x, where a0 ^ 0, a1, a2, ... an are real constants and x is an unknown variable. The highest power of x that exists in the expression is called the degree of the polynomial.

1.1.2 Rational Fraction

of two polynomials P(x) and Q(x) ^ 0 is called a rational fraction.

1.1.3 Proper and Improper Fractions

is a proper fraction, if the degree of the numerator

P(x)is less than the degree of the denominator Q(x).

is a improper fraction, if the degree of numerator P(x) is greater than or equal to the degree of the denominator Q(x).

1.2 PARTIAL FRACTION

Consider

1.2.1 Rules to Resolve into Partial Fraction

1. Denominator containing linear distinct factors

A and B are constants to be determined

Take LCM for RHS – side equation

Cancel the denominator of LHS and RHS

Substitute x = -1 in equation (2)

Substitute x = -2 in equation

Take equation (1), substitute A and B

2. Denominator containing linear repeated factors:

The fraction should be resolve as

Then the values of A, B, C are to be determined

Take terms LCM and add the RHS side

Substitute x = -2 in equation (2)

3. Denominator containing non-repeated non-factorizable factor. To each non-repeated, non-factorizable quadratic factor of the type (ax² + bx + c) of the denominator,

Take L.C.M

Substitute A = 1, B = −1, C = 0 in equation (1)

1.3 APPLICATION OF PARTIAL FRACTION IN CHEMICAL KINETICS AND PHARMACOKINETICS

1. In a second-order kinetics, the differential rate expression is given by the equation

To solve this equation. We have to obtain the partial fraction of 1

As per Partial fraction rule

Take LCM

But

Substitute A and B in equation (1)

2. First order Absorption (Infusion Method)

For a drug that enters the body by an apparent first-order absorption process, is eliminated by a first – order process and distributes in the body according to a one – compartment model, is given by equation.

by applying Laplace transform

To find the value of X, we have to find the partial fraction of the equation

Take LCM in RHS

Substitute the values A and B

3. The rapid intravenous injection of a drug that distribute in the body according to two compartment model system with elimination occurring from the central compartment is as shown in model and the amount of drug in the central compartment asc is given by equation (function)

has to be resolved into partial fraction.

After resolution into partial fraction, we will arrive into the equation

4. The disposition function for the central compartment in three compartmental model is given by the equation.

Amount of drug in central compartment asc, which is product of input and disposition is given by the fraction.

After resolving into partial fraction, the equation will be reduced into the form

5. When the drug is administered intravenously at a constant rate, rate of change of drug in the body with respect time can be obtained by the Linear differential equation.

After simplification by applying Laplace Transform

has to be resolved into partial fraction form

Take LCM

Exercise 1.1

Resolve into Partial Fraction

Answers

Chapter 2

LOGARITHMS

2.1 INTRODUCTION

In the absence of calculators and computers, logarithms play an important role in calculations. The operations of multiplication and division are replaced by addition and subtraction and those of raising to a power and extracting a root by multiplication John Napier (1550-1617) introduced the concept of logarithms and Henry Briggs (1556-1631) modified them to suit calculations.

2.1.1 Definition of Logarithm

If a is a positive real number other than 1 such that ax = m, then x is called the logarithm of m to the base a. We write x = logam (a > 0, a ^ 1)

Thus, if ax = m, then x = logam and

If x = loga(m), then ax = m

For example

The above examples indicate that indices lead to logarithms and vice-versa. We define the logarithm as follows:

The logarithm of a number to a given base is the index or the power to which the base should be raised in order to get the number

for example, logiol = 0, logel = 0, logee = 1, logiolO = 1

The logarithms to base 10 are called common logarithms. The logarithms to the base e are called Napierian Logarithms or Natural.

[e is called the exponential constant. It is an irrational number lying between 2 and 3].

The graph of y = log10x when x is a positive real number.

when x = 1, y = log101 = 0

when x = 10, y = log1010 = 1

when x = 100, y = log10100 = 2 and so on

Thus, if 0 < x < 1, y is negative and if x > 1, y is positive. The graph is as shown in Fig. 2.1.

Fig. 2.1

Fig. 2.2

The graph of y = loge x when x is a positive real number.

When 0 < x < 1, y is negative and when x > 1, y is positive. The graph is as shown Fig. 2.2

2.2 THEOREMS ON LOGARITHMS

Theorem 1

Note: loga = loga m + loga n expect in the following cases

Theorem 2

Substitute m = ay in (1), we get

Theorem 3: loga(mn) = n loga m

Theorem 4:

2.3 COMMON LOGARITHMS

Logarithms to the base 10 are called common logarithms. These were first introduced by Henry Briggs in 1615. If 'n' is the given number and x denote the logarithm of 'n' w.r.t. base 10, then we have 10x = n. This equation shows that the common logarithm of a number consists of an integral part and a fractional part and sometimes it can be negative also.

For example, 254 > 10² and < 10³

. log10254 = 2 +