The Background for the Diabetes Detection Model

Transcription

1 The Background for the Diabetes Detection Model James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University November 23, 2014

2 Outline The Background for the Diabetes Model

3 Abstract This Lecture discusses the background for the diabetes detection model.

4 We are now going to discuss a very nice model of diabetes detection which was presented in the classic text on applied mathematical modeling by Braun, Differential Equations and Their Applications. In diabetes there is too much sugar in the blood and the urine. This is a metabolic disease and if a person has it, they are not able to use up all the sugars, starches and various carbohydrates because they don t have enough insulin. Diabetes can be diagnosed by a glucose tolerance test (GTT). If you are given this test, you do an overnight fast and then you are given a large dose of sugar in a form that appears in the bloodstream. This sugar is called glucose. Measurements are made over about five hours or so of the concentration of glucose in the blood. These measurements are then used in the diagnosis of diabetes. It has always been difficult to interpret these results as a means of diagnosing whether a person has diabetes or not. Hence, different physicians interpreting the same data can come up with a different diagnosis, which is a pretty unacceptable state of affairs!

5 We are now going to discuss a criterion developed in the 1960 s by doctors at the Mayo Clinic and the University of Minnesota that was fairly reliable. It showcases a lot of our modeling in this course and will give you another example of how we use our tools. We start with a simple model of the blood glucose regulatory system. Glucose plays an important role in vertebrate metabolism because it is a source of energy. For each person, there is an optimal blood glucose concentration and large deviations from this leads to severe problems including death. Blood glucose levels are autoregulated via standard forward and backward interactions like we see in many biological systems. An example is the signal that is used to activate the creation of a protein which we discussed earlier.

6 The signaling molecules are typically either bound to another molecule in the cell or are free. The equilibrium concentration of free signal is due to the fact that the rate at which signaling molecules bind equals the rate at which they split apart from their binding substrate. When an external message comes into the cell called a trigger, it induces a change in this careful balance which temporarily upgrades or degrades the equilibrium signal concentration. This then influence the protein concentration rate. Blood glucose concentrations work like this too, although the details differ. The blood glucose concentration is influenced by a variety of signaling molecules just like the protein creation rates can be. The hormone that decreases blood glucose concentration is insulin.

7 Insulin is a hormone secreted by the β cells of the pancreas. After we eat carbohydrates, our gastrointestinal tract sends a signal to the pancreas to secrete insulin. Also, the glucose in our blood directly stimulates the β cells to secrete insulin. We think insulin helps cells pull in the glucose needed for metabolic activity by attaching itself to membrane walls that are normally impenetrable. This attachment increases the ability of glucose to pass through to the inside of the cell where it can be used as fuel. So, if there is not enough enough insulin, cells don t have enough energy for their needs. The other hormones we will focus on all tend to change blood glucose concentrations also.

8 Glucagon is a hormone secreted by the α cells of the pancreas. Excess glucose is stored in the liver in the form of Glycogen. There is the usual equilibrium amount of storage caused by the rate of glycogen formation being equal to the rate of the reverse reaction that moves glycogen back to glucose. Hence the glycogen serves as a reservoir for glucose and when the body needs glucose, the rate balance is tipped towards conversion back to glucose to release needed glucose to the cells. The hormone glucagon increases the rate of the reaction that converts glycogen back to glucose and so serves an important regulatory function. Hypoglycemia (low blood sugar) and fasting tend to increase the secretion of the hormone glucagon. On the other hand, if the blood glucose levels increase, this tends to suppress glucagon secretion; i.e. we have another back and forth regulatory tool.

9 Epinephrine also called adrenalin is a hormone secreted by the adrenal medulla. It is part of an emergency mechanism to quickly increase the blood glucose concentration in times of extremely low blood sugar levels. Hence, epinephrine also increases the rate at which glycogen converts to glucose. It also directly inhibits how much glucose is able to be pulled into muscle tissue because muscles use a lot of energy and this energy is needed elsewhere more urgently. It also acts on the pancreas directly to inhibit insulin production which keeps glucose in the blood. There is also another way to increase glucose by converting lactate into glucose in the liver. Epinephrine increases this rate also so the liver can pump this extra glucose back into the blood stream.

10 Glucocorticoids are hormones like cortisol which are secreted by the adrenal cortex which influence how carbohydrates are metabolized which is turn increase glucose if the the metabolic rate goes up. Thyroxin is a hormone secreted by the thyroid gland and it helps the liver form glucose from sources which are not carbohydrates such as glycerol, lactate and amino acids. So another way to up glucose! Somatotrophin is called the growth hormone and it is secreted by the anterior pituitary gland. This hormone directly affect blood glucose levels (i.e. an increase in Somatotrophin increases blood glucose levels and vice versa) but it also inhibits the effect of insulin on muscle and fat cell s permeability which diminishes insulin s ability to help those cells pull glucose out of the blood stream. These actions can therefore increase blood glucose levels.

11 Now net hormone concentration is the sum of insulin plus the others. Let H denote this net hormone concentration. At normal conditions, call this concentration H 0. There have been studies performed that show that under close to normal conditions, the interaction of the one hormone insulin with blood glucose completely dominates the net hormonal activity. That is normal blood sugar levels primarily depend on insulin-glucose interactions. So if insulin increases from normal levels, it increases net hormonal concentration to H 0 + H and decreases glucose blood concentration.

12 On the other hand, if other hormones such as cortisol increased from base levels, this will make blood glucose levels go up. Since insulin dominates all activity at normal conditions, we can think of this increase in cortisol as a decrease in insulin with a resulting drop in blood glucose levels. A decrease in insulin from normal levels corresponds to a drop in net hormone concentration to H 0 H. Now let G denote blood glucose level. Hence, in our model an increase in H means a drop in G and a decrease in H means an increase in G! Note our lumping of all the hormone activity into a single net activity is very much like how we modeled food fish and predator fish in the predator prey model.

13 The idea of our model for diagnosing diabetes from the GTT is to find a simple dynamical model of this complicated blood glucose regulatory system in which the values of two parameters would give a nice criterion for distinguishing normal individuals from those with mild diabetes or those who are pre diabetic. Here is what we will do. We describe the model as G (t) = F 1 (G, H) + J(t) H (t) = F 2 (G, H) where the function J is the external rate at which blood glucose concentration is being increased. There are two nonlinear interaction functions F 1 and F 2 because we know G and H have complicated interactions.

14 Let s assume G and H have achieved optimal values G 0 and H 0 by the time the fasting patient has arrived at the hospital. Hence, we don t expect to have any contribution to G (0) and H (0); i.e. F 1 (G 0, H 0 ) = 0 and F 2 (G 0, H 0 ) = 0. We are interested in the deviation of G and H from their optimal values G 0 and H 0, so let g = G G 0 and h = H H 0. We can then write G = G 0 + g and H = H 0 + h.

15 The model can then be rewritten as (G 0 + g) (t) = F 1 (G 0 + g, H 0 + h) + J(t) (H 0 + h) (t) = F 2 (G 0 + g, H 0 + h) or g (t) = F 1 (G 0 + g, H 0 + h) + J(t) h (t) = F 2 (G 0 + g, H 0 + h)

16 Now as we explained in the Calculus One for Biologist course, Starting Calculus for Biologists, for a function of two variables like this, there is an analogue of the idea of approximating a function by its tangent line which is called a tangent plane. We need to introduce some new ideas just to get to what we want to in this model. Please see the other text for more details. The partial derivative of a function F (x, y) with respect to its first variable x is found by treating the second variable as a constant and taking the derivative normally. It is denoted by F x or just F x. The partial derivative of a function F (x, y) with respect to its second variable y is found by treating the first variable as a constant and taking the derivative normally. It is denoted by F y or just F y. For example, if F (x, y) = x 2 y 3, then F x = 2xy 3 and F y = 3x 2 y 2.

17 Now it turns out the tangent plane to F (x, y) at the point (x 0, y 0 ) is given T (x, y) = F (x 0, y 0 ) + F x (x 0, y 0 )(x x 0 ) + F y (x 0, y 0 )(y y 0 ) and if you replace F (x, y) by its tangent plane at the point (x 0, y 0 ), you naturally make some error usually. Call this error E F. Then we have F (x, y) = F (x 0, y 0 ) + F x (x 0, y 0 )(x x 0 ) + F y (x 0, y 0 )(y y 0 ) + E F. We want to use this idea on our functions F 1 and F 2 at the optimal values G 0 and H 0.

18 We have F 1 (G 0 + g, H 0 + h) = F 1 (G 0, H 0 ) + F 1 g (G 0, H 0 ) g + F 1 h (G 0, H 0 ) h + E F1 F 2 (G 0 + g, H 0 + h) = F 2 (G 0, H 0 ) + F 2 g (G 0, H 0 ) g + F 2 h (G 0, H 0 ) h + E F2 but the terms F 1 (G 0, H 0 ) = 0 and F 1 (G 0, H 0 ) = 0, so we can simplify to F 1 (G 0 + g, H 0 + h) = F 1 g (G 0, H 0 ) g + F 1 h (G 0, H 0 ) h + E F1 F 2 (G 0 + g, H 0 + h) = F 2 g (G 0, H 0 ) g + F 2 h (G 0, H 0 ) h + E F2

19 It seems reasonable to assume that since we are so close to ordinary operating conditions, the errors E F1 and E F2 will be negligible. Thus our model approximation is g (t) = F 1 g (G 0, H 0 ) g + F 1 h (G 0, H 0 ) h + J(t) h (t) = F 2 g (G 0, H 0 ) g + F 2 h (G 0, H 0 ) h Next, we can reason out the algebraic signs of the four partial derivatives. This will be what we will do in the next lecture.