Problem 63 - Calculus for Biology and Medicine

Question

Hormone Implant You are studying an implanted contraceptive that releases hormone continuously into a patient's blood. The data in this question are from Nilsson et al. (1986). The device adds \(20 \mu \mathrm{g}\) of hormone to the blood each day. In the blood the hormone has first order elimination kinetics; \(4 \%\) of the hormone is eliminated each day. (a) Let the amount of hormone in the blood on day \(t\) be \(a_{t} .\) Write a word equation for the change in \(a_{t}\) over one day. (b) Put in mathematical formulas for each of the terms in your word equation from (a). (c) Assume that on day 0 no hormone is present in the patient's blood, in other words, \(a_{0}=0 .\) Use your equation from (b) to compute the amount of hormone in the blood on days \(1,2,3,4\), \(5,6 .\) (d) Over time the level of hormone in the blood converges to a limit. Find the value of this limit by looking for a fixed point of your recurrence relation in (b).

Step-by-Step Solution

Answer

(a) Hormone level change: added hormone + remaining hormone. (b) Equation: \( a_{t+1} = 0.96a_{t} + 20 \). (c) Hormone on days 1 to 6: 20, 39.2, 57.632, 75.32672, 92.3136512, 108.6211051. (d) Long-term level: 500.

Step 1: Establish the Word Equation

The hormone's daily change in the blood is the sum of the hormone added each day and the hormone remaining after elimination. Therefore, word equation is: \( \text{New hormone amount} = \text{Current hormone amount} + \text{Added hormone} - \text{Eliminated hormone}. \)

Step 2: Convert Word Equation to Mathematical Equation

Using the word equation, introduce mathematical expressions: \( a_{t+1} = a_{t} + 20 - 0.04a_{t} \). Simplify to get the equation for the next day's hormone amount: \( a_{t+1} = 0.96a_{t} + 20 \).

Step 3: Calculate Hormone Levels for Days 1 to 6

Start with \( a_0 = 0 \). Calculate each day:- Day 1: \( a_1 = 0.96(0) + 20 = 20 \)- Day 2: \( a_2 = 0.96(20) + 20 = 39.2 \)- Day 3: \( a_3 = 0.96(39.2) + 20 = 57.632 \)- Day 4: \( a_4 = 0.96(57.632) + 20 = 75.32672 \)- Day 5: \( a_5 = 0.96(75.32672) + 20 = 92.3136512 \)- Day 6: \( a_6 = 0.96(92.3136512) + 20 = 108.6211051 \).

Step 4: Determine the Long-term Hormone Level

Set \( a_t = a_{t+1} \) to find the limit \( L \) as \( t \to \infty \): \( L = 0.96L + 20 \). Solve for \( L \) to find: \( L = \frac{20}{1 - 0.96} = 500 \).

Key Concepts

Hormone Implant

A hormone implant is a small medical device inserted under the skin. Its purpose is to release a hormone steadily over a period of time.
This mechanism is particularly beneficial for treatments requiring consistent hormone levels, such as contraceptive methods. In this context, the hormone is released at a constant rate of 20 micrograms per day into the bloodstream.
When a hormone implant releases hormones into the blood, it's important to consider how the body processes and eliminates these hormones. With first-order elimination kinetics, the rate of elimination is proportional to the amount of hormone present. Here, 4% of the hormone in the blood is removed daily. This gradual reduction affects how long and how much of the hormone remains active in the body.
As the device continuously adds hormones each day, this ensures that there is always a steady amount present, even while some are being naturally eliminated.

Mathematical Modeling

Mathematical modeling plays a crucial role in understanding the behavior of hormones released by implants. It helps predict the concentration of hormones over time.
When constructing a model for this scenario, it integrates the addition of the hormone and its subsequent elimination. By using mathematical equations, one can provide a clear picture of how hormone levels change daily.

**The Basic Equation**
From the word equation that governs the hormonal change in blood, we derive a mathematical equation:

• New Hormone Amount = Current Hormone Amount + Added Hormone - Eliminated Hormone

This translates to:

• \( a_{t+1} = a_t + 20 - 0.04a_t \)

Simplifying, you get:

• \( a_{t+1} = 0.96a_t + 20 \)

By applying this model, predictions on the hormone levels for successive days can be made. This is vital for observing trends and understanding how the hormone stabilizes over time.

Recurrence Relation

Recurrence relations are a powerful tool in modeling dynamic processes, like the hormone release from an implant.
They are used to predict the next term in a sequence based on the current term and some constants.
For the hormone implant situation, the recurrence relation is derived from the equation:

• \( a_{t+1} = 0.96a_t + 20 \)

This formula demonstrates how the hormone amount at day \( t+1 \) can be calculated from the day \( t \).
The newly added hormone and the portion of hormone remaining after elimination influence each next step. Therefore, each day's calculation builds on the previous day's amount.
**Finding Long-term Stability**
To see the long-term behavior of this system, we make the assumption that the amount of hormone reaches a constant level or limit, \( L \).
Setting \( a_t = a_{t+1} \) in the recurrence relation helps to find this fixed point:

• \( L = 0.96L + 20 \)

Solving gives \( L = 500 \), indicating how the system stabilizes over time. This fixed point is key to predicting how effective the implant will be in maintaining the desired hormone level consistently.