Diabetes is a disease of metabolism which is characterized by too much sugar in the these measurements are used in diagnosis of diabetes. …
Project for Math 224
DETECTION OF DIABETES Diabetes is a disease of metabolism which is characterized by too much sugar in the blood and urine Because of the lack of insulin a hormone, the patients body is unable to burn off all its sugar, starches and carbohydrates Diabetes is usually diagnosed by a glucose tolerance test GTT In GTT, the patient comes to the hospital after an overnight fast and is given a large dose of glucose the kind of sugar in the bloodstream In the next few hours, several measurements of the concentration of blood glucose are made and these measurements are used in diagnosis of diabetes This project concerns one of the criteria for interpreting the results of a GTT The model is based on the following: Basic Biological Facts: Glucose is a source of energy for all organs and tissues Each
individual has an optimal blood glucose concentration, from which any large deviation would cause a serious pathological condition The blood glucose concentration is influenced and controlled by various kinds of hormones, among which the predominant factor is insulin In this project, for the sake of simplicity we shall ignore the effect of other hormones Insulin is secreted
by the pancreas After we eat any carbohydrates, the pancreas is signaled to secrete more insulin Also the glucose in the bloodstream directly stimulates the pancreas to secrete insulin The insulin in return facilitates tissue uptake of glucose by attaching itself to the impermeable membrane walls, opening the door for glucose to pass through the membrane to the center of cells, where glucose is consumed Let Gt and Ht be the concentrations of blood glucose and insulin at time t, respectively Then G and H satisfy 1 dG f1 G, H Et , dt dH f2 G, H , dt
2 where Et represents external rate of change for G, and f1 and f2 represent internal rate of change for G and H , respectively a It is known that after an overnight fast, the concentrations of glucose and insulin in the patients blood stabilize at their optimal values, ie, Gt constant G0 , Ht constant H0 Using this fact, show that f1 G0 , H0 0 f2 G0 , H0 Let g G - G0 , h H - H0 , then by 1, we have dg f1 g G0 , h H0 Et , dt 2 dh f2 g G0 , h H0 , dt This system is often hard to solve In case that g and h are small, 2 can be approximated by a linear system as follows: By the tangent plane approximation
taught in Calculus III, f1 g G0 , h H0 f1 G0 , H0 f2 g G0 , h H0 f2 G0 , H0 f1 f1 G0 , H0 g G0 , H0 h , G H
f2 f2 G0 , H0 g G0 , H0 h , G H if g and h are small Thus 2 can be approximated by dg f1 G0 , H0 g f1 G0 , H0 h Et dt G H 3 f2 f2 dh G0 , H0 g G0 , H0 h dt G H Recall f1 G0 , H0 0 f2 G0 , H0 This approximation is good if g and h are small This procedure is called the linearization of 2 at point G0 , H0 In system 3, f1 f1 f2 f2 G0 , H0 , G0 , H0 , G0 , H0 , G0 , H0 G H G H are unknown because functions f1 and f2 are unknown However, it is possible to determine their signs b By using the Basic Biological Facts, explain why
f2 H G0 , H0 f1 f1 G G0 , H0 , H G0 , H0
and
are negative and
f2 G G0 , H0
is positive
3 [Hint: When discussing the sign of
f1 G G0 , H0 ,
assume in 3 that Et 0, and
h0 0, g0 0 what does this assumption mean biologically?, then argue that g 0 0 ] Now 3 can be written as dg -a1 g - a2 h Et , dt dh -a3 h a4 g , dt
4
where a1 , a2 , a3 and a4 are positive constants c By eliminating h and
dh dt
from 4, show that gt satisfies
d2 g dg dE a1 a3 a1 a3 a2 a4 g a3 Et 2 dt dt dt Except for the very short
time interval in which glucose load is being ingested after the arrival of the patient at the hospital, Et and hence
dE dt t
are identically zero Set
t 0 at the time of the completion of the ingestion Then for t 0, gt satisfies 5 where
a1 a3 2
g 2g 2 g 0 ,
, 2 a1 a3 a2 a4 is called the natural frequency of equation 5
This is exactly the equation for the springmass system d Using 5, show in detail that every solution gt of 5 is of the form e-t coswt - , if 2 - 2 0 underdamped case gt C1 e-t C2 te-t , if 2 - 2 0 critically damped case C1 e1 t C2 e2 t , if 2 - 2 0 overdamped case where w 2 - 2 , 1 - 2 - 2 , 2 - - 2 - 2 , and , , C1
and C2 are constants This part is a repeat of what you learned in the class e Show that lim Gt G0
t
4 Now we have for t 0, G0 e-t coswt - , if 2 - 2 0 , Gt G0 C1 e-t C2 te-t , if 2 - 2 0 , G0 C1 e1 t C2 e2 t , if 2 - 2 0 It should be reminded that this formula holds only approximately because we derive it by using 4 which is an approximation of the precise model 1 The constants G0 , , , , , C1 and C2 are unknown constants They can be determined by taking several measurements of the patients blood
glucose f How can G0 be found experimentally; ie, what measurements, under what conditions, should be taken to estimate G0 ? g In the overdamped case 2 - 2 0, how would you find , , C1 and C2 ? In numerous experiments, it was observed that is insensitive to experimental errors in measuring G Therefore we choose as the basic descriptor of the response to a glucose tolerance test Let T 2/ T is called the natural period of the system 5 The data from many doctors and hospitals lead to the following Criterion for Diabetes: diabetes Remark: The model we discussed above can only be used to diagnose mild diabetes, T 4 hours implies normalcy, T 4 hours indicates mild
since the linearized system 3 is a good approximation of 2 only if g and h are small Very large deviations g of G from its optimal value G0 imply severe diabetes Before using data to determine T 2/ of a patient as discussed in g, we have to know if we are in the underdamped, critically damped, or overdamped case What distinguishes the underdamped case from the other two cases is that in the former, any nontrivial solution gt of 5 changes the sign infinitely many times, while in the latter cases, gt can change its
sign at most once By a nontrivial solution of 5, we mean a solution of 5 which is not identically zero
5 h Show that in the critically damped or overdamped case, any nontrivial solution gt of 5 can change its sign at most once Thus, if data indicate that gt changes its sign more than once, we know we are in the underdamped case Now lets look at two examples Example 1 After an overnight fast, this patients blood glucose concentration is 75
mg glucose/100 ml blood His blood glucose concentration 1 hour, 2 hours and 3 hours after fully absorbing a large amount of glucose is 90, 62, 81 mg glucose/100 ml blood, respectively i Argue by using h that this is the underdamped case j Show that in the underdamped case, the time interval between any consecutive zeros of any nontrivial solution gt of 5 is greater than T /2 k Now show that the patient in Example 1 is normal Example 2 A patients blood glucose concentration is 70 mg glucose/100 ml blood after
an overnight fast His blood glucose concentration 1 hour, 2 hours, 3 hours and 4 hours after fully absorbing a large amount of glucose is 95, 70, 65, 65 mg glucose/100 ml blood l Using f and g, determine if this patient is a diabetic Assume
2 - 2 0 Solving the system you wrote for C1 , C2 , 1 , 2 in g is pretty difficult The reasonable thing to do is to introduce two new variables instead of 1 , 2 , namely, x1 e1 , x2 e2 after you find x1 , x2 , you just take i ln xi Write the system for the unknowns C1 , C2 , x1 , x2 The new system is simpler, but still difficult to solve You may try to eliminate C1 , C2 by dividing all equations by C1 Alternatively, you can use Matlab to solve the algebraic equationssee its manual
Source:ahrp.org
