Math 121 - Calculus for Biology I Spring Semester, 2004 Derivative of ex and ln(x)
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1. Prozac
2. Half-life of a Drug
3. Norfluoxetine Kinetic Model
4. Derivative of the Exponential
5. Application of the Derivative to the Prozac Kinetic Model
6. Height vs Weight Relationship
7. Derivative of ln(x)
8. Worked Examples
9. References

Prozac

• Fluoxetine, commonly known by its trade name Prozac, is a selective serotonin reuptake inhibitor (SSRI)
• This drug is used to treat depression, obsessive compulsive disorder, and a number of other neurological disorders
• It works by preventing serotonin from being reabsorbed too rapidly from the synapses between nerve cells, prolonging its availablity, which improves the patient's mood
• Fluoxetine is metabolized in the liver and transformed into a slightly less potent SSRI, norfluoxetine
• Both compounds bind to plasma protein, then become concentrated in the brain (up to 50 times more concentrated)
• Fluoxetine and norfluoxetine are eliminated from the brain with characteristic half-lives of 1-4 days and 7-15 days, respectively

• It is very important to understand the kinetics of the drug in the body
• Drugs metabolized into another active form make modeling more complex
• Models below examine first order kinetic models for the concentrations of fluoxetine (F(t)) and norfluoxetine (N(t)) in the blood

Half-Life of a Drug

• Half-life of a drug is similar to the half-life of radioactive material
• A subject taking a 40 mg oral dose of fluoxetine rapidly exhibits a blood stream concentration of 21 ng/ml
• One study of healthy volunteers showed the half-life of fluoxetine was 1.5 days
• Assume instantaneous uptake of the drug, then the initial blood concentration of fluoxetine is

F(0) = 21 ng/ml.

• When a drug is either filtered out by the kidneys or metabolized by some organ such as the liver proportional to its concentration, then the drug is said to exhibit first-order kinetics
• The drug decays exponentially with a characteristic half-life
• Fluoxetine is metabolized in both the brain and liver, so satisfies the kinetic equation

F(t) = 21e^-kt

• With a half-life of 1.5 days, we have

F(1.5) = 10.5 = 21e^-1.5k

• Solving this equation for k,

e^1.5k = 2

^{k = ln(2)/1.5 = 0.462}

• A good model for blood plasma concentration of fluoxetine is

F(t) = 21e^-0.462t

Norfluoxetine Kinetic Model

• Fluoxetine is metabolized in the liver and through a hepatic biotransformation becomes norfluoxetine (through a demethylation)
• Norfluoxetine continues to act as potent and specific serotonin reuptake inhibitor
• The half-life is taken to be 9 days for norfluoxetine
• A reasonable model using linear kinetics for the blood plasma concentration of norfluoxetine is

N(t) = 27.5(e^-0.077t - e^-0.462t).

• Below is a graph of the fluoxetine and norfluoxetine concentrations from the models

• Determine the rate of change of fluoxetine and norfluoxetine
• Find the time of maximum blood plasma concentration of norfluoxetine and what that concentration is
• To solve these problems, we need to learn the formula for the derivative of the exponential function.

Derivative of e^x

• The exponential function e^x is a special function
• It's the only function (up to a scalar multiple) that is the derivative of itself

• From the definition of the derivative and using the properties of exponentials

.

• One definition of the number e is the number that makes

• Geometrically, the function e^x is a number raised to the power x, whose slope of the tangent line at x = 0 is 1.

General rule for the derivative of e^kx

• The derivative of e^kx is

Example 1: Find the derivative of

Solution: From our rule of differentiation and the formula above, we have

Application of the Derivative to the Prozac Kinetic Model

• Find the rate of change of fluoxetine and norfluoxetine
• The function modeling blood plasma concentration of fluoxetine is

F(t) = 21e^-0.462t

• So the derivative is

F '(t) = (-0.462)21e^-0.462t = -9.702e^-0.462t

• The model for the blood plasma concentration of norfluoxetine is

N(t) = 27.5e^-0.077t - 27.5 e^-0.462t

• So its derivative is

N '(t) = (-0.077)27.5e^-0.077t - (-0.462)27.5 e^-0.462t

= 12.705e^-0.462t - 2.1175 e^-0.077t

• The rate of change of blood plasma concentration of fluoxetine at times t = 2 and 10 is

F '(2) = -9.702e^-0.462(2) = -3.85 ng/ml/day

F '(10) = -9.702e^-0.462(10) = -0.0956 ng/ml/day

• The rate of change of norfluoxetine blood plasma concentration at times t = 2 and 10 is

N '(2) = 12.705e^-0.462(2) - 2.1175 e^-0.077(2). = 3.23 ng/ml/day

N '(10) = 12.705e^-0.462(10) - 2.1175 e^-0.077(10). = -0.855 ng/ml/day

• These calculations show that at t = 2 the blood plasma concentration of fluoxetine is dropping quite rapidly, while blood plasm concentration of norfluoxetine is rising at a similar rate
• The calculations at t = 10 show that the blood plasma concentration of both compounds are falling at fairly slow rates

Maximum Concentration of Norfluoxetine

• The maximum concentration of norfluoxetine is found by determining when the derivative is equal to zero

N '(t) = 12.705e^-0.462t - 2.1175 e^-0.077t = 0

2.1175 e^-0.077t = 12.705e^-0.462t

• This gives

• The maximum occurs at

0.385t = ln(6)

t_max = 4.654 days

• The maximum blood plasma concentration of norfluoxetine is

N(t_max) =16.01 ng/ml

Height and Weight Relationship for Children

The average height and weight of girls in the U. S.

• Below is a graph of the data, showing the height as a function of weight of girls ages 5 through 13 using data on the average height and weight of girls in the U. S.

• Ehrenberg noted that there was a logarithmic relationship between the height and the weight of children
• The formula for the height, H, as a function of weight, w, is given by

• We would like to find the find the rate of change of height with respect to weight for the average girl.

Derivative of ln(x)

• The derivative of the natural logarithm, ln(x), is given by the formula

• This relationship is most easily demonstrated after learning the Fundamental Theorem of Calculus in Math 122, which centers about the integral.

Derivative of the Height and Weight Relationship for Children

• The relationship for height as a function of weight was given by

H(w) = 49.5ln(w) - 34.14

• This is easily differentiated with respect to w, using the derivative for the natural logarithm

• From this relationship, it is clear that as the weight increases, the rate of change in height decreases

• For example, at a weight of 20 kg.

  H '(20) = 49.5/20 = 2.475 cm/kg

  while at a weight of 49.5 kg

  H '(49.5) = 49.5/49.5 = 1 cm/kg

• Note that this is not the rate of change of the height as a function of the age, which we saw to be nearly linear

Example 3: Find the derivative of

Solution: From our properties of logarithms and the formula above,

f(x) = ln(x²) = 2ln(x)

Worked Examples