Cholesterol Level in Humans - Differential Equations

S. F. Ellermeyer

Contents

1  Introduction

High levels of cholesterol in the blood are known to be a risk factor for heart disease. Cholesterol is produced biosynthetically in the liver for use in the construction of cell walls and is absorbed from foods containing saturated fatty acids. In the average American adult, the total amount of cholesterol circulating in the blood is about 200 mg/dl. (Source: Grolier's Multimedia Encyclopedia)

In this project, you will study a mathematical model for the cholesterol level of an individual. The model predicts cholesterol level as a function of the individual's ``natural'' cholesterol level, cholesterol intake, and metabolism of cholesterol. This module is an expansion of an exercise that appears in the book Differential Equations by Blanchard, Devaney, and Hall (1997, Brooks/Cole Publishing Company).

2  The Model

In the book Differential Equations (Blanchard, et. al., 1997), the authors propose a mathematical model for the cholesterol level of an individual. The proposed model is the differential equation

• t is time measured in days.

• C( t) is the individual's cholesterol level at time t measured in mg/dl.

• L is the individual's ``natural'' cholesterol level that would result from a diet excluding all fatty foods.

• E is the individual's cholesterol intake measured in mg/day.

• k₁ is a parameter that measures how rapidly the individual's body responds to deviations in cholesterol level from the natural cholesterol level.

• k₂ is a parameter that measures the rate at which the individual's body produces cholesterol from foods that have been ingested.

2.1  Questions to Answer

1. What are the units of measurement of [dC/dt]?

2. What are the units of measurement of k₁? (Hint: Refer to the model (1). The units of measurement of the right hand side of this differential equation must be the same as for the left hand side.)

3. What are the units of measurement of k₂?

3  Analyzing the Model

Let us consider the case of two (fictitious) twin brothers - Bubba and Biff. Because they are identical twins, these brothers both have the same ``natural'' cholesterol level of L = 140 mg/dl and the same production and absorption parameters of k₁ = 0.1 and k₂ = 0.05. When we first encounter Bubba and Biff, they are 22 years old and, because they have been living at home together and eating the same low-fat meals (cooked by their mother), they both have the same daily cholesterol intake of E = 80 mg/day. One day, Bubba decides that it is time to move away from home and get a place of his own. He finds a rather nice one bedroom apartment with affordable rent and moves in. Unfortunately, Bubba's new apartment complex is located right next door to Junior's Fried Chicken and Catfish Emporium (which has daily all-you-can-eat specials for $5.95).

3.1  Bubba

For a young man out on his own for the first time, the formation of healthy eating habits is often not a high priority. Such is the case with Bubba. Returning home to his apartment each night after a grueling morning at work at his part time job followed by an even more grueling afternoon of classes at the local university, Bubba is tired and hungry. The blinking neon sign of Junior's beckons. As Bubba becomes settled in his routine of nightly meals at Junior's, his daily cholesterol intake becomes E = 250 mg/day.

According to the model (1), Bubba's cholesterol level is modeled by

3.1.1  Exercises

1. Sketch the phase line for the differential equation (2) and use the phase line to sketch several typical solutions of this differential equation (by hand).

2. Find the general solution of the differential equation (2). (Show your computations.)

3. Find the particular solution of the initial value problem ( 3).

4. If Bubba were to maintain this high cholesterol diet for a very long time (say, a year or more), what would his (approximate) cholesterol level be? Explain how you arrive at your conclusion.

3.2  Biff

Biff also works part time and is a student at the local university. However, he chooses to live at home where he continues to eat his mother's cooking.

3.2.1  Exercises

Assuming that Biff's cholesterol level is C₀ = 180 mg/dl, at time t₀ = 0 :

1. Write down the initial value problem that models Biff's cholesterol level.

2. Solve the initial value problem for Biff's cholesterol level.

3. Does the model's prediction about Biff's cholesterol level seem reasonable to you? Explain.

3.3  General Analysis of the Model

One of the intended benefits of the cholesterol model (1) is that we hope to be able to use the model to try to understand how a person's cholesterol level is determined by the various parameters that are involved in formulating the model. Given all of the parameters of the model (which, as we have seen, are determined by a particular person's physiology and eating habits), we would like to answer quantitative questions such as

• If this person continues on his present diet, what will his cholesterol level be one month from now?

and qualitative questions such as

• Under the current conditions (parameter values), will this person's cholesterol level rise or fall?

Another question of interest that is both qualitative and quantitative in nature is

• What is this person's ``long-term'' predicted cholesterol level?

This last question is obviously very important. A person whose long-term predicted cholesterol level is very high will want to take measures (diet, exercise, etc.) which will lower the long-term predicted level.

As you have discovered in carrying out the preceding exercises, the general form of the initial value problem that models a person's cholesterol level is

3.3.1  Exercises

1. Write the parameter M (of the differential equation in ( 4)) in terms of L, E, k₁, and k₂.

   M = _________________.

2. Because of the biological meanings of the parameters L, E, k₁, and k₂, all of these parameters are assumed to be positive numbers (with the exception that E could be zero). Explain why we can conclude from this that M must also be a positive number. Write down the value of M for Bubba and for Biff.

   Bubba¢s M = _________________

   Biff¢s M = _____________________.

3. Find the solution of the initial value problem (4).

   C( t) = _________________________________.

4. For the solution you computed above, determine lim_t®¥C( t) . What is the meaning of this limit in terms of cholesterol level?

5. Draw the phase line for the differential equation

   dC dt = k1( M-C)

   (assuming that k₁ and M are positive parameters).

4  Varying the Model Parameters

The cholesterol model (1) predicts the behavior of a person's cholesterol level (over some time period) as a function of the parameters L, E, k₁, and k₂. Thus far, we have assumed that these parameters are all constants. However, for most people, these parameters are probably not really constant. They change as the individual changes eating habits, exercise habits, etc. In what follows, you will study the effects of changing the parameter E.

4.0.2  Exercises

1. Assuming that the parameters L, E, k₁, and k₂ are allowed only to be positive numbers (with the exception that E could be zero), explain why it must always be the case that M ³ L.

2. Taking fixed values of k₁, k₂, and L (you can make up some values but they should be positive of course!), treat E as the only parameter and describe how the phase portrait of

   dC dt = k1( M-C)

   changes as E is decreased slowly toward zero. What does this mean in terms of how long term predicted cholesterol level changes in response to changes in eating habits?

3. Do any bifurcations occur as E is decreased slowly toward zero?