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A method to measure the degree of control per se in the oral glucose tolerance test
(‘OMPIJ
I ERS AND
A Method
LIIOMEUI~AL
RESEARCH
22, 314-3’7
( 1%‘))
to Measure the Degree of Control Oral Glucose Tolerance Test HORACE
F. MARTIN,*,~ AND JOANNE
Keceivcd
RICHARD
per se in the
B. GOLDSIWN.$
HOI.OGGITAS”
April
IY.
IYW
Using the glucose and insulin values from a S-hr oral glucose tolerance test. nine quantitative measures have been developed to separate normal, “flat-curve.” and non-insulin-dependent diabetic (NIDD) patients. The purpose of these measures i\ to quantify per ye the degree of control operating in glucose homeoasis. A control index which i\ hased upon Swan’s minimizing principle and uses only the glucose values was cuccessful in asse>\ing the degree of control operating In glucose homeostasis. IWJ Ac.tJcmtc Prc\\. lnr
Physicians and biomathematicians have long suspected that there are a number of disease states defined by impairment of dynamic control (I 1. In homeostasis there exists an entire spectrum ranging from rigid overcontrol to total lack of control. The normals represent ideal control. This study presents alterations of glucose control with the “flat-curve” representing overcontrol while the lack of control is represented by non-insulindependent diabetes (NIDD). Review of the literature showed that Swan (2). Bellman (I). and Bellomo (3) have dealt with this problem with the practical objective of gauging the delivery of insulin by a mechanical pump in such a way that the fluctuation of glucose from some set point would always be a minimum. They were dealing with only two of the three states: normal and NIDD. If their goal of minimum fluctuation is applied it literally negates control. Biologically. the organism experiencing the phenomenon of life encounters many fluctuations and must deal with them successfully. For example, in the oral glucose tolerance test following the displacement of serum glucose the sugar concentration path seeks the original set point by oscillating toward it. It oscillates because the model is a coupled oscillator (4, 5) where the envelope is an 314
CONTROLINTHEGLUCOSETOLERANCETEST
315
exponential curve whose characteristics are well known-the rate of change being proportional to the displacement. The measure of control defined by Bellman and Swan is modified to include the overcontrol system which is the “flat-curve.” Several control measures were applied to observed oral glucose tolerance curves, and the best discriminator of the three clinical states was sought. The control index is a measure of the degree of control of glucose homeostasis.
POPULATIONS Normals. The criteria for normal is (1) a fasting blood sugar within the limits of 65-105 mg/dl and not a “flat-curve” (described below); (2) insulin less than 25 @/ml; (3) maxima of glucose and insulin in synchrony: (4) the absence of the use of the following drugs: steroids, nonsteroidal anti-inflammatory agents, hormones, diuretics, or recent or past use of diphenylhydantoin; (5) terminal insulin not significantly higher than fasting: (6) effective insulin as demonstrated by lowering of free fatty acids by 50% in 1 hr; (7) lowering of the growth hormone and cortisol by the glucose load; and (8) no glycosuria. The reference population consisted of 64 males and 83 females. The males were 32.5 ? 14.0 years of age, 69.6 2 2.6 inches tall, and weighed 188 ? 67 pounds. The females were 40.7 + 17.5 years of age, 63.4 -+ 3.8 inches tall, and weighed 179 ? 69 pounds. The males include two populations: 25 male patients were deemed to be normal by stringent external criteria which include history, physical examination, X ray, EKG, chemistry profile-20, hematology, etc., and the balance were deemed to be normal by less stringent criteria which include a complete history and internal laboratory data entirely within normal limits. There was no obvious difference between the two normal populations with respect to fasting glucose and maximum glucose and they were grouped together. Patients diagnosed us havingflat glucose tolerance curves. The “flat-curve” patients were selected from the patient population. The diagnosis is based upon a rise of less than 20 mg/dl in glucose from the fasting level throughout the period. The “flat-curve” population consisted of 16 males and 41 females. The flat glucose response is found more frequently among females. The males were 37.9 + 16.7 years of age, 69.1 + 4.0 inches tall, and weighed 169 + 42 pounds. The females were 36.6 2 11.3 years of age, 64.0 ? 2.2 inches tall, and weighed 158 2 44 pounds. Patients diagnosed as NIDD. The criteria for NIDD is a fasting glucose greater than 105 mg/dl and not a “flat-curve,” lack of synchrony of glucose and insulin maxima, no drug effects (see above), adequate insulin, and elevated glucose that did not return to fasting levels. When present, glycosuria was highly likely to be NIDD. The NIDD patients were selected from our patient population. The NIDD population consisted of 25 males and 17 females. The males were 58.4 +- 12.9 years of age, 68.4 + 3.5 inches tall, and weighed 194 -+
316
MAKTIN,GOLDSTEIN.ANDHOLOGGITAS
43 pounds. The females were 55.9 2 15.9 years of age. 62.9 2 2.6 inches tail, and weighed 179 t 51 pounds. METHODS (I) Mrthods usd irz tFw&h~k testing. Blood serum glucose and insulin are measured at the fasting time for a base value. The patient then receives 0.5 g of glucose orally per kilogram of body weight not to exceed 100 g. At regular intervals in the next 5 hr glucose and insulin levels are drawn giving approximately 10 points in all. Two methods of glucose analysis are given because of the span of time of the study. Blood serum glucose levels were determined by Autoanalyzer ferricyanide reduction technique (6, 7) and by glucose oxidase in a glucose analyzer (Beckman Instruments, Fullerton, CA). A study was done of Autoanalyzer ferricyanide and glucose oxidase complete with chart review and the mean was 90 and 89, respectively, for normals and I I8 and I IS. respectively, for abnormals showing no significant difference of the means between the two methods. The authors are aware that the literature documents S-10 as a positive bias. If this bias existed, it still would not affect our patient classification since data utilized for this diagnosis are the differences between successive glucose values and not their absolute values. Insulin was determined by a Rhode Island Hospital in-house modification adapted from Yalow and Berson (8) and Herbert et LII. (Y). (2) Nr~tFz~rit~~~//stclti,stic~u/. For each of the three groups (flat. normal. and NIDD) the arithmetic mean, standard deviation, and the coefficient of skewness were computed for the indices described below. These statistics were used in the Gram-Charlier equation (see Reference Interval section) to give a distribution shape. The prevalences were based upon approximately 500 patients at Rhode Island Hospital. The Mann-Whitney test (IO) was used to test differences between two populations because the distributions are often heavily skewed and as a nonparametric test it is less affected by extremes from the mean as in a parametric test. The reference range is determined by using Gram-Charlier Type A (three term) series to fit the distributions using a least-square iterative method based upon Marquardt’s compromise (f/-/3). This method was chosen because it will perform a dissection on mixed populations when the group classification is uncertain. The reference range is further compared to the reference range determined by the rank-order statistics (13. 15) of the population classihed as normal defined above. CONTROL PARAMETERS
Measures of the control of the system are now introduced. The first. called the control index (Cl), is based upon the Swan’s minimizing principle. It is the mean square deviation of glucose in the blood after absorption from the fasting level.
CONTROL
IN THE
GLUCOSE
TOLERANCE
TEST
317
Ga(t) = serum glucose at time t gB(f)
=
GB@)
-
GB(@
h(t) = insulin at time t
CI= {Lmax/‘max [GB(f) 0
- C&(O)]* dt]“*
Ii?
PI
The control index shown in Eqs. [I] and [2] is approximated using Eq. [3] which is based upon the Trapezoid rule for estimating integrals. Besides the control index four additional measures were determined from a two-dimensional time line spiral graph in glucose and insulin as shown in Fig. 1. The spiral path traces the variations of blood glucose on the x-axis and the corresponding insulin values on the y-axis starting at t = 0 and ending at t = 300 min. Usually there is a return to the fasting level after a few hours. The loop may be circular or elliptical or it might spiral around a few times near the end until it reaches the fasting conditions. The path length is measured by taking line segment distances and summing them over the 5-hr test. The line segment distances along the time line spiral graph shown in Fig. 1 are computed from the actual patient values by the formula disti = V[GB(ti) The glucose is measured microunits per milliliter.
- GB(ti-l)]’ in milligrams
SPIRAL
200-
TIME-LINE
+ [h(t;) - h(ti-,)I’. per deciliter
and the insulin
141 is in
GRAPH
-l E ; 150. 5
TIME=90 1
TI ME=300
-L TIME=0
‘0
4
50
100
GL”&E!
( r&?BDL
>
250
300
FIG. 1. The glucose-insulin time line spiral curve of a typical NIDD patient. The maximum distance is shown by the oblique dashed line along with the angle above the horizon. The time line traces from 0 to 300 min.
318
MARTIN.
GOLDSTEIN.
AND
HOLOCXITAS
The total distance (TD) around the time line graph is the sum of the segment distances. The measure of the percent in 90 min (P90) is found by taking the sum of the line segments in 90 min, dividing by the TD, and expressing the ratio as a percentage. The other two measurements are based upon the maximum geometric distance from the fasting point. Maximum distance (Max. Dist.) is this maximum geometric distance and (Angle) is the angle in degrees above the horizon. Note that in some patients this angle may be above 90” or below 0”. Besides the five parameters discussed (Cl, TD, P90, Max. Dist.. and Angle) there are an additional four measures based upon a linear regression in the changes in glucose versus the changes in insulin. These are found by taking differences in glucose and insulin values taken some 10 to I4 times over the 5-hr test. For example. the 10 time points may be at 0. 20. 40, 60, 90. 120. 180, 210. 140, and 300 min. Let f = 0, the fasting time and AGB(i)
= GH(~,) ~ G(fj-11,
for i = I. 2. .
. . II.
151
-
161
where A/Z(i) = l?(t,) - h(ti-1). Assuming
12 = no. of samples
I.
a linear relationship Ah(i)
= /& + ,GtAGa(i)
171
between the variables, the values of PO and /!I, are estimated as B,, and B, using elementary statistical methods. For example, see Fig. 2 as an illustration. The four measures based upon this regression line are sample correlation coefficient (Y), slope C/3, est. = II,), y-intercept (PO est. = II(,), and the Student t statistic (t-Stat) for either of the equivalent hypotheses: H,,: p = 0 H,: p + 0
or
H,,: p, = 0 H,: ,!3, =k 0.
,.....
.../
..:.
/...JJ
a -40 LINEAR
FIG.
2. Linear
regression
between
REGRESSION
the differences
in glucose
versus
the differences
in insulin.
CONTROL
IN THE
GLUCOSE
TOLERANCE
TABLE CONTROL
Group
Prevalence
Flat Normal NIDD Note. statistic test.
35% 25% 40% (I) Prevalences is given because
INDEX
(CI)
319
TEST
I
STATISTICS
FOR EACH GROUP
n
Mean
St. dev.
Skew.
57 147 42
14.39 30.01 78.98
4.33 9.81 26.71
0.92 0.99 0.51
Mann-Whitney z statistic for Group vs Normal
are based upon over 500 patients at Rhode these are large sample tests. (3) The significance
Significance
- 10.34
P < 0.0001
+9.25
P < 0.ooo1
Island Hospital. (2) The 7 is based upon a single-tail
RESULTS
As seen in Table I the control index (CI) proves to be very discriminatory. Table 1 illustrates that the NIDD patients have a much larger control index than their normal counterparts. This is an indication of much greater fluctuation of the glucose from the fasting state and the increased amount of time that it remains elevated. The “flat-curve” patients by comparison have a much lower control index illustrating their lack of fluctuation from fasting. This group is overcontrolled. Table 2 compares the total distance (TD) taken through the 5-hr spiral graph of the glucose-insulin time line. The “flat-curve” patients have a significantly shorter path than the normal patients. Although the NIDD patients have a slightly longer path than the normal its appearance is different in two ways: they are greatly shifted to the right (much higher glucose values) and they often do not come to a close at the end of the 5th hour. The control system is searching for the fasting conditions, but in comparison as seen in Table 3 it takes longer getting there.
TABLE TOTAL
Group Flat Normal NIDD
DISTANCE
Prevalence 35% 25% 40%
AROUND
SPIRAL
n
Mean
57 147 42
243. I 355.0 403.0
2
PATH (TD)
STATISTKS
FOR EACH
St. dev.
Skew.
Mann-Whitney ; statistic for Group vs Normal
136.5 178.2 194.5
2.52 1.66 2.04
-5.17 +I.98
GROUP
Significance P < 0.0003 P < 0.05
320
MARTIN.
GOLDSTEIN.
AND
TABL.E PERC-ENTAC;E
Group
OF WE
Prevalence
TOI AL PATH
TRAVELED FOR EACH
II
Mean
St. dev.
HOLOGGITAS
3 1~ .THE FIRS’I. GROIII’
Skew.
90 MINU-I.ES
(P90)
Mann-Whltne) .- statislic for Group vx Normal
Flat Normal
35% 25%
Sl 147
61.92 56.64
13.11 1 I.61
0. I9 0.41
17.56
NIDD
40%
42
46.41
7.62
I .33
5.44
Sl AT ISI I( s
Significance P
0.01 -
P c 0.oooI
Table 3 shows the significantly longer time that the NIDD takes to get back to near fasting levels of glucose and insulin. The measure shown in Table 4 is similar in nature to the control index but it is not nearly as discriminatory. The angle appears to be the second best discriminator (see Table 5). However, it is influenced by the weight of the patient. The obese patient produces significantly more insulin in response to the glucose challenge. Tables 6-9 show that the normal patient’s glucose tolerance curve has a strong linear correlation between the differences in glucose and insulin. The “flat-curve” and NIDD patients show some signs of breakdown of the system. But again, unfortunately. there is too much overlap among the three groups for effective discrimination. Besides the nonparametric Mann-Whitney test, two other comparisons were made: (I) the Student t test between two populations (flat vs normal and NIDD vs normal) where the samples have unequal variances and (2) a visual comparison based upon the Kolmogorov-Smirnov test of the cumulative distributions of the three groups for each of the nine measurements. The results of these comparisons were similar to the Mann-Whitney tests. For discrimination purposes alone only CI was capable of properly classifying the three groups. Where it is necessary to distinguish between NIDD and IDDM (insulin-depen-
TABL,E MAXIMUM
Group Flat Normal NIDD
Prevalence 35% 25% 40%
DISTANCE
(MAX.
4
D~sr .) S.TATISTICS
IF
Mean
St. dev.
57 147 47
104.5 155.7 195.3
68.2 92.7 97.9
Skew. 2.35 2.03 I .95
FOR EACH
GROUP
Mann-Whitney ; statistic for Group vs Normal
Significance
-4.77
P < 0.0001
t3.40
P -c 0.001
CONTROL
IN THE
GLUCOSE
TOLERANCE
TABLE ANGLE
Group Flat Normal NIDD
Prevalence 35% 25% 40%
STATISTICS
5 FOR EACH
n
Mean
St. dev.
Skew.
57 147 42
14.32 63.20 43.82
14.04 14.20 19.08
-0.14 -0.50 0.09
TABLE CORRELATION
Group Flat Normal NIDD
Prevalence 35% 25% 40%
COEFFICIENT
n
Mean
St. dev.
Skew.
57 147 42
0.659 0.790 0.721
0.211 0.133 0.144
-1.45 -1.19 -0.84
TABLE
Group
Prevalence
GROUP Mann-Whitney i statistic for Group.vs Normal +4.79
P < 0.0001
-5.63
P < 0.0001
FOR EACH GROUP Mann-Whitney ; statistic for Group vs Normal
P < 0.0001
-3.03
P < 0.01
7 FOR EACH GROUP
n
Mean
St. dev.
Skew.
Flat Normal
35% 25%
51 147
1.467 1.414
I.237 I.124
2.23 1.88
+0.37 -
NIDD
40%
42
0.643
0.558
1.68
-5.48
TABLE
Group Flat Normal NIDD
Prevalence 35% 25% 40%
Significance
-4.13
Mann-Whitney ; statistic for Group vs Normal
Y-INTERCEPT
Significance
6
(r) STATISTICS
SLOPE (B,) STATISTICS
321
TEST
(B,,) STATISTICS
Significance Not
significant -
P
8 FOR EACH
n
Mean
St. dev.
Skew.
51 I47 42
0.611 I.850 2.854
1.271 3.112 5.399
0.57 2.05 -2.80
GROUP Mann-Whitney i statistic for Group vs Normal
Significance
-2.28
P < 0.05
+3.95
P < 0.0001
322
MARTIN.
GOLDSTEIN.
AND
‘I’ABl,t
HOLOGGII’AS
Y
Mann-Whitnc) GKNlp
Prevalence
,I
Flat
?S’f
51
Normal NIDD
75”:.
117
40%.
43
Mean
St. dev.
: \latidic tier (irtwp VY Normal
Skew.
Sipnltiancc
2.949
I .93h
11.10
5.03
I'
o.000 I
4.13X
I .x55 I.212
0.x’) 0. I9
:.73
P
0.0001
3.068
dent diabetes mellitus) the best discriminators for this classification are slope. low correlation coefficient, or low t test. Also, an excessively high value of the angle in the spiral curve can be an indication of the degree to which the insulin production is being affected by obesity. In this study the best discriminatory measure was sought regardless of whether it was dependent or independent of other variables. REFERENCE
INTERVAL
FORTHE
CONTROL
INDEX
The primary method for finding a reference interval is the dissection of the cumulative data into three distributions: low, normal. and high. This is done by fitting the gathered data using nonlinear least-squares optimization on the frequencies. The authors have successfully used the &am-Charlier series on many reference interval/normal range calculations. The fit will be better if this asymptotic series is expanded for nonnormal distributions to include only the skewness term (1.5). This series is given as j’(x) = a(x) c 1 + ; pjHi
+ 7’j (p., ~ 3)H4 +
. ).
where (Y(X) = (liv(2-rr)) expt-x2/2), pi are central moments, Hermite polynomials. The first two terms are used and rewritten as .f’(.r; p. u. y, A) = A expt-:‘/2)
1
and H;(.v) are
I - z t: ~ ~‘13) . J
191
where ; = (.I- - p)/(r and y = coefficient of skewness. Basically. this form is a first-order perturbation of the normal frequency curve to give skewness. It is unimodal as long as l-y/ < I .648. This form is then written for a mixture of two distributions as jix)
= ,ftr; pi, UI. ye 1A,) + Jtx; puz.VT. yz. A?).
1101
Since the prevalence obtained of the “flat-curve” and NIDD patients were much smaller than their observed prevalences at the community at large would indicate. their Cl values were given weights of 3.6 and 5.6, respectively. to
CONTROL
FREQ
IN THE
I
CI
-
GLUCOSE
LOWER
END
TOLERANCE
323
TEST
DISSECTION
n
4 CONTROL
14 INDEX
(CI)
IN
24 MG/DL
FIG. 3. The control index is dissected at the lower end into two distributions: Each of the distributions is assumed to take the Gram-Charlier series form.
Flat and Normal.
adjust to the external prevalences of flat (35%), normal (25%), and NIDD (40%). The dissection process converged with good initial estimates in some 5 to 10 iterations. Figures 3 and 4 show the results of least-squares curve fitting of the mixed distributions. Figure 3 shows the dissection at the lower end for the control index and Fig. 4 shows the dissection at the upper end. The crossover point between the dissected populations with the assumed prevalences is used as the cutoff. The authors ignored societal values and chose to maximize the correct number of classifications. From the fit one can further produce tables of the probability of illness (or group membership), predictive values of a positive or negative test, estimates of the probabilities of false negatives and false positives, and the specificity, and sensitivity of the test. The predictive value of a positive test is the percentage of times that a positive test will detect a diseased individual. It is the ratio of the number or proportion of diseased persons with positive tests divided by the total number or proportion of persons with positive tests. Similarly, the predictive value of a negative test is the percentage of times that a negative test will detect nondiseased or healthy persons. It is the ratio of the number or proportion of nondiseased persons with CI
-
UPPER
END
DISSECTION
50 40 30 20 10 0
FIG.
4. The control
index
ie CONTROL
is dissected
k0 INDEX
at the upper
(CI)
IN
ii0 MGfDL
end into two distributions:
Normal
and NIDD
324
MAKTIN.GOLDSTEIN,
AND HOLOGGITAS
negative tests divided by the total number or proportion of persons with negative tests. Normal subjects with positive tests are known as false positives and diseased subjects with negative tests are known as false negatives. The specificity of a screening test is defined as the number of normal or nondiseased subjects negative to the test expressed as a percentage of all normal subjects tested. Sensitivity is defined as the number of diseased subjects with a positive test as a percentage of all disease subjects tested (I,‘). The cutoff values for the Cl were calculated as 19.77 and 48.67. At the lowel end (19.77) the probability of a false negative is X.66 and 3.15: for a false positive. The specificity and sensitivity at the lower end are, respectively, 91.4% and 96.95%.At the upper end (48.67) the probability of a false negative is 7. I% and for a Mse positive 3.5%. The specificity and sensitivity at the upper end are, respectively. 92.9% and 96.5%. The probabilities of illness and PV+ and PV- are shown in Tables IO and I I. Combining the two results from Tables IO and I I with respect to the weighted prevalences the totals give P(false positive) = 4.1%
sensitivity = 95.9%
P(false negative) = 9.8%
specificity = 90.256.
An alternate procedure for finding the reference interval is to use only the normal population values and to take a 2.5% cutoff of the top and botton end of the ordered statistics (14). This produces a normal range of 16.82 to 54.55 where these are ranked statistics X(4) and ,Y( 144). respectively. This range is fairly close as shown in Table I2 to that produced by the above procedure. In essense, as in many medical tests there is a gray area where one is in transition between the physiological and pathological states. Table I2 shows the distribution of the patients based upon the diagnosis criteria and the reference interval of the control index based upon each of the two methods. DISCUSSION
Perturbing the control system with glucose causes the blood serum glucose and insulin levels to fluctuate with an attempt by the system to return to fasting levels by seeking some set point. In the normal, the set point is closely approached in about 3 hr with continued dampened oscillations in the 4th and 5th hours due to minor perturbations induced by gluconeogenesis and glycogenolysis. Comparing the normal patients to those who have “flat-curves” the glucose in the latter returns to the fasting set point more quickly and with less amplitude displacement. The penalty for this increased response is that it takes very little perturbation to place the patient in hypoglycemia (less than 55 mgi dl). On the other hand, the NIDD patients frequently start at high levels of glucose and insulin at fasting, reaching still higher values of glucose and insulin. and while the system attempts to return to the fasting level within the 5 hr, it rarely succeeds.
CONTROL
IN THE
GLUCOSE TABLE
PROBABILITIES
CI 13 14 15 16 17 18 19 [19.77 20 21 22 23
TOLERANCE
325
10
ON THE
LOWER
P(flatlCI) 100.0% 99.5 98.5 96.4 92.1 83.5 67.7 50.0 44.1 20.0 4.5 0.0
TEST
END
pv+
PV-
100.0% 99.9 99.7 99.3 98.7 97.5 95.8 94.0 93.4 90.3 86.6 82.5
51.7% 56.0 61.8 69.0 77.1 85.0 91.7 95.51 96.3 98.9 99.9 100.0
The control index is different in the three states, normal, flat, and NIDD. The spiral path and its associated measures give some indication of the glucoseinsulin coupling. The flat glucose tolerance patients have significantly shorter paths but their percentage traveled in the first 90 min is similar to that of the normals (see Tables 2 and 3). The NIDD patients have similar path lengths as the normal patient population but the percentage traveled in the first 90 min is significantly less than in the normal group. The maximum geometric excursion from the fasting point is further in the NIDD along with its angle being more horizontal. In the flat glucose tolerance patient this maximum distance is less than normal as well as its angle being steeper than normal. The correlation coefficient of insulin differences on glucose differences gives the strength of a linear relationship between these two variables. The normal patients have a TABLE PROBABILITIES
Cl 30 35 40 [::.67 50 55 60 65 70 75
P(NIDD(CI) 0.0% 1.9 16.7 37.5 50.0 54.8 74.9 90.8 97.8 99.6 100.0
II
ON THE
UPPER
pv+ 78.5%~ 86.2 90.8 93.7 95.6 96.3 98.3 99.5 99.9 100.0 100.0
END
PV100.0% 99.9 99.1 96.9 94.31 93.2 88.1 81.8 74.9 68.0 61.8
326
MAKTIN.GOLDSTEIN.AND
DISTRIBUTION
HOLOGGITAS
OF POPULATIONS
Classification Actual Cl dissectton Cl
order
method statistic
method
stronger linear relationship than the flat or NIDD as shown in Tables 6 and 7. For the NIDD patient the slope is significantly less and the correlation significantly weaker. Several measures of control of glucose homeostasis have been proposed and statistically evaluated. Of these the control index was found to be the most discriminatory. The reference interval has been specified for the control index and associated probabilities are given. The time line spiral graph of glucoseinsulin coupling has given us a visual impression of the control and the ability to return to the origin (fasting values). The regression line showed the coupling between glucose and insulin, especially in the normal patient. The authors conclude that the flat glucose curves are not a disease involving issues of control since the time line curve returns to the fasting condition. However, the NIDD is a derangement of control as evidenced by the inability to return to the origin and diminished sensitivity of insulin to changes in glucose. This diminished sensitivity is shown by the more horizontal leaning of the NIDD spiral curves (angle). The authors believe that the control index introduced in this paper will be better as a single classifier than the current definitions used for flat and NIDD and that as a quantitative value it will better measure the degree or lack of control (disease). ACKNOWLEDGMENTS We and
thank
Ralph
Miech.
M.D.,
Ph.D.,
and
Clyde
B. Schechter.
M.D..
for
their
Press.
Princeton.
critical
reviews
clarifications.
REFERENCES I. 2.
BELLMAN, R. E. “Dynamic SWAN. G. W. “Applications 19x4.
.T. BELLOMO. G.. Optimal feedback 1. 5.
Princeton Programming.” of Optimal Control Theory
BRUNETTI. P.. CALABRESE, of glycemia regulation
Univ.
on Biomedicine.”
G.. hh%o’-l-1, in diabetics. Med.
Dekker.
I>.. SARI I, E.. ANU Biol. Eng. C’ornprrr.
NJ.
1957. New
York.
VINTENZI. A. 20, 329 (1982).
ACKERMAN. E.. ROSEVEAR, J. W.. AND MCGUCKLIN. W. F. A mathematical model of the glucose-tolerance test. P1lv.s. Med. Bid. 9, 203 (1964). CELESTE. R.. ACKERMAN. E., GATEWOOD, L. C.. REYNOLDS. C., AND MOLNAR, G. D. The role of glucagon in the regulation of blood glucose model studies. Bull. Mofh. Bid. 40. 59
(1978).
CONTROL
IN THE
GLUCOSE
TOLERANCE
TEST
327
6. HOFFMAN. W. S. A rapid photoelectric method for the determination of glucose in blood and urine. J. Biol. CI~ern. 120, 51 (1937). 7. Technicon Autoanalyzer method file N-2b, Technicon Instruments, Tarrytown, NY. R. YALOW, R. S., AND BERSON, S. A. Immunoassay of endogenous plasma in man. J. Clirz. Inuesr. 39, 1157 (1960). Y. HERBERT. V.. LAU, K. S., GOTTLIEB, AND BLEICHER. S. J. Coated charcoal immunoassay of insulin. J. C/in. Endocrine/. Mrtah. 25, 1375 (1965). 10. FREUND. J. E. “Modern Elementary Statistics.” 4th ed. Prentice-Hall, Englewood Cliffs. NJ, 1973. Il. MARQUARDT. D. W. An algorithm for least squares estimation of nonlinear parameters. SIAM J. Appl. Math. 11, 431 (1963). 12. MARTIN, H. F., GUDZINOWICZ, B. J., FANGER, H.. DRISCOLL, J. L., GOLDSTEIN, R., HOLOGCITAS. J., AND PEZZULLO. J. “Normal Values in Clinical Chemistry.” Chaps. 5 and 6, pp. 272786. Dekker, New York, 1975. 13. SCALES. L. E. “Introduction to Non-Linear Optimization.” Springer-Verlag. New York, 1985. 14. HERRERA. L. The precision of percentile in establishing normal limits in medicine. J. Lab. C/in. Med. 52, 34 (1958). 15. KENDALL, SIR M. G., AND STUART. A. “The Advanced Theory of Statistics,” 4th ed.. Vol. 1. pp. 159-192. Macmillan Co.. New York. 1977.
I ERS AND
A Method
LIIOMEUI~AL
RESEARCH
22, 314-3’7
( 1%‘))
to Measure the Degree of Control Oral Glucose Tolerance Test HORACE
F. MARTIN,*,~ AND JOANNE
Keceivcd
RICHARD
per se in the
B. GOLDSIWN.$
HOI.OGGITAS”
April
IY.
IYW
Using the glucose and insulin values from a S-hr oral glucose tolerance test. nine quantitative measures have been developed to separate normal, “flat-curve.” and non-insulin-dependent diabetic (NIDD) patients. The purpose of these measures i\ to quantify per ye the degree of control operating in glucose homeoasis. A control index which i\ hased upon Swan’s minimizing principle and uses only the glucose values was cuccessful in asse>\ing the degree of control operating In glucose homeostasis. IWJ Ac.tJcmtc Prc\\. lnr
Physicians and biomathematicians have long suspected that there are a number of disease states defined by impairment of dynamic control (I 1. In homeostasis there exists an entire spectrum ranging from rigid overcontrol to total lack of control. The normals represent ideal control. This study presents alterations of glucose control with the “flat-curve” representing overcontrol while the lack of control is represented by non-insulindependent diabetes (NIDD). Review of the literature showed that Swan (2). Bellman (I). and Bellomo (3) have dealt with this problem with the practical objective of gauging the delivery of insulin by a mechanical pump in such a way that the fluctuation of glucose from some set point would always be a minimum. They were dealing with only two of the three states: normal and NIDD. If their goal of minimum fluctuation is applied it literally negates control. Biologically. the organism experiencing the phenomenon of life encounters many fluctuations and must deal with them successfully. For example, in the oral glucose tolerance test following the displacement of serum glucose the sugar concentration path seeks the original set point by oscillating toward it. It oscillates because the model is a coupled oscillator (4, 5) where the envelope is an 314
CONTROLINTHEGLUCOSETOLERANCETEST
315
exponential curve whose characteristics are well known-the rate of change being proportional to the displacement. The measure of control defined by Bellman and Swan is modified to include the overcontrol system which is the “flat-curve.” Several control measures were applied to observed oral glucose tolerance curves, and the best discriminator of the three clinical states was sought. The control index is a measure of the degree of control of glucose homeostasis.
POPULATIONS Normals. The criteria for normal is (1) a fasting blood sugar within the limits of 65-105 mg/dl and not a “flat-curve” (described below); (2) insulin less than 25 @/ml; (3) maxima of glucose and insulin in synchrony: (4) the absence of the use of the following drugs: steroids, nonsteroidal anti-inflammatory agents, hormones, diuretics, or recent or past use of diphenylhydantoin; (5) terminal insulin not significantly higher than fasting: (6) effective insulin as demonstrated by lowering of free fatty acids by 50% in 1 hr; (7) lowering of the growth hormone and cortisol by the glucose load; and (8) no glycosuria. The reference population consisted of 64 males and 83 females. The males were 32.5 ? 14.0 years of age, 69.6 2 2.6 inches tall, and weighed 188 ? 67 pounds. The females were 40.7 + 17.5 years of age, 63.4 -+ 3.8 inches tall, and weighed 179 ? 69 pounds. The males include two populations: 25 male patients were deemed to be normal by stringent external criteria which include history, physical examination, X ray, EKG, chemistry profile-20, hematology, etc., and the balance were deemed to be normal by less stringent criteria which include a complete history and internal laboratory data entirely within normal limits. There was no obvious difference between the two normal populations with respect to fasting glucose and maximum glucose and they were grouped together. Patients diagnosed us havingflat glucose tolerance curves. The “flat-curve” patients were selected from the patient population. The diagnosis is based upon a rise of less than 20 mg/dl in glucose from the fasting level throughout the period. The “flat-curve” population consisted of 16 males and 41 females. The flat glucose response is found more frequently among females. The males were 37.9 + 16.7 years of age, 69.1 + 4.0 inches tall, and weighed 169 + 42 pounds. The females were 36.6 2 11.3 years of age, 64.0 ? 2.2 inches tall, and weighed 158 2 44 pounds. Patients diagnosed as NIDD. The criteria for NIDD is a fasting glucose greater than 105 mg/dl and not a “flat-curve,” lack of synchrony of glucose and insulin maxima, no drug effects (see above), adequate insulin, and elevated glucose that did not return to fasting levels. When present, glycosuria was highly likely to be NIDD. The NIDD patients were selected from our patient population. The NIDD population consisted of 25 males and 17 females. The males were 58.4 +- 12.9 years of age, 68.4 + 3.5 inches tall, and weighed 194 -+
316
MAKTIN,GOLDSTEIN.ANDHOLOGGITAS
43 pounds. The females were 55.9 2 15.9 years of age. 62.9 2 2.6 inches tail, and weighed 179 t 51 pounds. METHODS (I) Mrthods usd irz tFw&h~k testing. Blood serum glucose and insulin are measured at the fasting time for a base value. The patient then receives 0.5 g of glucose orally per kilogram of body weight not to exceed 100 g. At regular intervals in the next 5 hr glucose and insulin levels are drawn giving approximately 10 points in all. Two methods of glucose analysis are given because of the span of time of the study. Blood serum glucose levels were determined by Autoanalyzer ferricyanide reduction technique (6, 7) and by glucose oxidase in a glucose analyzer (Beckman Instruments, Fullerton, CA). A study was done of Autoanalyzer ferricyanide and glucose oxidase complete with chart review and the mean was 90 and 89, respectively, for normals and I I8 and I IS. respectively, for abnormals showing no significant difference of the means between the two methods. The authors are aware that the literature documents S-10 as a positive bias. If this bias existed, it still would not affect our patient classification since data utilized for this diagnosis are the differences between successive glucose values and not their absolute values. Insulin was determined by a Rhode Island Hospital in-house modification adapted from Yalow and Berson (8) and Herbert et LII. (Y). (2) Nr~tFz~rit~~~//stclti,stic~u/. For each of the three groups (flat. normal. and NIDD) the arithmetic mean, standard deviation, and the coefficient of skewness were computed for the indices described below. These statistics were used in the Gram-Charlier equation (see Reference Interval section) to give a distribution shape. The prevalences were based upon approximately 500 patients at Rhode Island Hospital. The Mann-Whitney test (IO) was used to test differences between two populations because the distributions are often heavily skewed and as a nonparametric test it is less affected by extremes from the mean as in a parametric test. The reference range is determined by using Gram-Charlier Type A (three term) series to fit the distributions using a least-square iterative method based upon Marquardt’s compromise (f/-/3). This method was chosen because it will perform a dissection on mixed populations when the group classification is uncertain. The reference range is further compared to the reference range determined by the rank-order statistics (13. 15) of the population classihed as normal defined above. CONTROL PARAMETERS
Measures of the control of the system are now introduced. The first. called the control index (Cl), is based upon the Swan’s minimizing principle. It is the mean square deviation of glucose in the blood after absorption from the fasting level.
CONTROL
IN THE
GLUCOSE
TOLERANCE
TEST
317
Ga(t) = serum glucose at time t gB(f)
=
GB@)
-
GB(@
h(t) = insulin at time t
CI= {Lmax/‘max [GB(f) 0
- C&(O)]* dt]“*
Ii?
PI
The control index shown in Eqs. [I] and [2] is approximated using Eq. [3] which is based upon the Trapezoid rule for estimating integrals. Besides the control index four additional measures were determined from a two-dimensional time line spiral graph in glucose and insulin as shown in Fig. 1. The spiral path traces the variations of blood glucose on the x-axis and the corresponding insulin values on the y-axis starting at t = 0 and ending at t = 300 min. Usually there is a return to the fasting level after a few hours. The loop may be circular or elliptical or it might spiral around a few times near the end until it reaches the fasting conditions. The path length is measured by taking line segment distances and summing them over the 5-hr test. The line segment distances along the time line spiral graph shown in Fig. 1 are computed from the actual patient values by the formula disti = V[GB(ti) The glucose is measured microunits per milliliter.
- GB(ti-l)]’ in milligrams
SPIRAL
200-
TIME-LINE
+ [h(t;) - h(ti-,)I’. per deciliter
and the insulin
141 is in
GRAPH
-l E ; 150. 5
TIME=90 1
TI ME=300
-L TIME=0
‘0
4
50
100
GL”&E!
( r&?BDL
>
250
300
FIG. 1. The glucose-insulin time line spiral curve of a typical NIDD patient. The maximum distance is shown by the oblique dashed line along with the angle above the horizon. The time line traces from 0 to 300 min.
318
MARTIN.
GOLDSTEIN.
AND
HOLOCXITAS
The total distance (TD) around the time line graph is the sum of the segment distances. The measure of the percent in 90 min (P90) is found by taking the sum of the line segments in 90 min, dividing by the TD, and expressing the ratio as a percentage. The other two measurements are based upon the maximum geometric distance from the fasting point. Maximum distance (Max. Dist.) is this maximum geometric distance and (Angle) is the angle in degrees above the horizon. Note that in some patients this angle may be above 90” or below 0”. Besides the five parameters discussed (Cl, TD, P90, Max. Dist.. and Angle) there are an additional four measures based upon a linear regression in the changes in glucose versus the changes in insulin. These are found by taking differences in glucose and insulin values taken some 10 to I4 times over the 5-hr test. For example. the 10 time points may be at 0. 20. 40, 60, 90. 120. 180, 210. 140, and 300 min. Let f = 0, the fasting time and AGB(i)
= GH(~,) ~ G(fj-11,
for i = I. 2. .
. . II.
151
-
161
where A/Z(i) = l?(t,) - h(ti-1). Assuming
12 = no. of samples
I.
a linear relationship Ah(i)
= /& + ,GtAGa(i)
171
between the variables, the values of PO and /!I, are estimated as B,, and B, using elementary statistical methods. For example, see Fig. 2 as an illustration. The four measures based upon this regression line are sample correlation coefficient (Y), slope C/3, est. = II,), y-intercept (PO est. = II(,), and the Student t statistic (t-Stat) for either of the equivalent hypotheses: H,,: p = 0 H,: p + 0
or
H,,: p, = 0 H,: ,!3, =k 0.
,.....
.../
..:.
/...JJ
a -40 LINEAR
FIG.
2. Linear
regression
between
REGRESSION
the differences
in glucose
versus
the differences
in insulin.
CONTROL
IN THE
GLUCOSE
TOLERANCE
TABLE CONTROL
Group
Prevalence
Flat Normal NIDD Note. statistic test.
35% 25% 40% (I) Prevalences is given because
INDEX
(CI)
319
TEST
I
STATISTICS
FOR EACH GROUP
n
Mean
St. dev.
Skew.
57 147 42
14.39 30.01 78.98
4.33 9.81 26.71
0.92 0.99 0.51
Mann-Whitney z statistic for Group vs Normal
are based upon over 500 patients at Rhode these are large sample tests. (3) The significance
Significance
- 10.34
P < 0.0001
+9.25
P < 0.ooo1
Island Hospital. (2) The 7 is based upon a single-tail
RESULTS
As seen in Table I the control index (CI) proves to be very discriminatory. Table 1 illustrates that the NIDD patients have a much larger control index than their normal counterparts. This is an indication of much greater fluctuation of the glucose from the fasting state and the increased amount of time that it remains elevated. The “flat-curve” patients by comparison have a much lower control index illustrating their lack of fluctuation from fasting. This group is overcontrolled. Table 2 compares the total distance (TD) taken through the 5-hr spiral graph of the glucose-insulin time line. The “flat-curve” patients have a significantly shorter path than the normal patients. Although the NIDD patients have a slightly longer path than the normal its appearance is different in two ways: they are greatly shifted to the right (much higher glucose values) and they often do not come to a close at the end of the 5th hour. The control system is searching for the fasting conditions, but in comparison as seen in Table 3 it takes longer getting there.
TABLE TOTAL
Group Flat Normal NIDD
DISTANCE
Prevalence 35% 25% 40%
AROUND
SPIRAL
n
Mean
57 147 42
243. I 355.0 403.0
2
PATH (TD)
STATISTKS
FOR EACH
St. dev.
Skew.
Mann-Whitney ; statistic for Group vs Normal
136.5 178.2 194.5
2.52 1.66 2.04
-5.17 +I.98
GROUP
Significance P < 0.0003 P < 0.05
320
MARTIN.
GOLDSTEIN.
AND
TABL.E PERC-ENTAC;E
Group
OF WE
Prevalence
TOI AL PATH
TRAVELED FOR EACH
II
Mean
St. dev.
HOLOGGITAS
3 1~ .THE FIRS’I. GROIII’
Skew.
90 MINU-I.ES
(P90)
Mann-Whltne) .- statislic for Group vx Normal
Flat Normal
35% 25%
Sl 147
61.92 56.64
13.11 1 I.61
0. I9 0.41
17.56
NIDD
40%
42
46.41
7.62
I .33
5.44
Sl AT ISI I( s
Significance P
0.01 -
P c 0.oooI
Table 3 shows the significantly longer time that the NIDD takes to get back to near fasting levels of glucose and insulin. The measure shown in Table 4 is similar in nature to the control index but it is not nearly as discriminatory. The angle appears to be the second best discriminator (see Table 5). However, it is influenced by the weight of the patient. The obese patient produces significantly more insulin in response to the glucose challenge. Tables 6-9 show that the normal patient’s glucose tolerance curve has a strong linear correlation between the differences in glucose and insulin. The “flat-curve” and NIDD patients show some signs of breakdown of the system. But again, unfortunately. there is too much overlap among the three groups for effective discrimination. Besides the nonparametric Mann-Whitney test, two other comparisons were made: (I) the Student t test between two populations (flat vs normal and NIDD vs normal) where the samples have unequal variances and (2) a visual comparison based upon the Kolmogorov-Smirnov test of the cumulative distributions of the three groups for each of the nine measurements. The results of these comparisons were similar to the Mann-Whitney tests. For discrimination purposes alone only CI was capable of properly classifying the three groups. Where it is necessary to distinguish between NIDD and IDDM (insulin-depen-
TABL,E MAXIMUM
Group Flat Normal NIDD
Prevalence 35% 25% 40%
DISTANCE
(MAX.
4
D~sr .) S.TATISTICS
IF
Mean
St. dev.
57 147 47
104.5 155.7 195.3
68.2 92.7 97.9
Skew. 2.35 2.03 I .95
FOR EACH
GROUP
Mann-Whitney ; statistic for Group vs Normal
Significance
-4.77
P < 0.0001
t3.40
P -c 0.001
CONTROL
IN THE
GLUCOSE
TOLERANCE
TABLE ANGLE
Group Flat Normal NIDD
Prevalence 35% 25% 40%
STATISTICS
5 FOR EACH
n
Mean
St. dev.
Skew.
57 147 42
14.32 63.20 43.82
14.04 14.20 19.08
-0.14 -0.50 0.09
TABLE CORRELATION
Group Flat Normal NIDD
Prevalence 35% 25% 40%
COEFFICIENT
n
Mean
St. dev.
Skew.
57 147 42
0.659 0.790 0.721
0.211 0.133 0.144
-1.45 -1.19 -0.84
TABLE
Group
Prevalence
GROUP Mann-Whitney i statistic for Group.vs Normal +4.79
P < 0.0001
-5.63
P < 0.0001
FOR EACH GROUP Mann-Whitney ; statistic for Group vs Normal
P < 0.0001
-3.03
P < 0.01
7 FOR EACH GROUP
n
Mean
St. dev.
Skew.
Flat Normal
35% 25%
51 147
1.467 1.414
I.237 I.124
2.23 1.88
+0.37 -
NIDD
40%
42
0.643
0.558
1.68
-5.48
TABLE
Group Flat Normal NIDD
Prevalence 35% 25% 40%
Significance
-4.13
Mann-Whitney ; statistic for Group vs Normal
Y-INTERCEPT
Significance
6
(r) STATISTICS
SLOPE (B,) STATISTICS
321
TEST
(B,,) STATISTICS
Significance Not
significant -
P
8 FOR EACH
n
Mean
St. dev.
Skew.
51 I47 42
0.611 I.850 2.854
1.271 3.112 5.399
0.57 2.05 -2.80
GROUP Mann-Whitney i statistic for Group vs Normal
Significance
-2.28
P < 0.05
+3.95
P < 0.0001
322
MARTIN.
GOLDSTEIN.
AND
‘I’ABl,t
HOLOGGII’AS
Y
Mann-Whitnc) GKNlp
Prevalence
,I
Flat
?S’f
51
Normal NIDD
75”:.
117
40%.
43
Mean
St. dev.
: \latidic tier (irtwp VY Normal
Skew.
Sipnltiancc
2.949
I .93h
11.10
5.03
I'
o.000 I
4.13X
I .x55 I.212
0.x’) 0. I9
:.73
P
0.0001
3.068
dent diabetes mellitus) the best discriminators for this classification are slope. low correlation coefficient, or low t test. Also, an excessively high value of the angle in the spiral curve can be an indication of the degree to which the insulin production is being affected by obesity. In this study the best discriminatory measure was sought regardless of whether it was dependent or independent of other variables. REFERENCE
INTERVAL
FORTHE
CONTROL
INDEX
The primary method for finding a reference interval is the dissection of the cumulative data into three distributions: low, normal. and high. This is done by fitting the gathered data using nonlinear least-squares optimization on the frequencies. The authors have successfully used the &am-Charlier series on many reference interval/normal range calculations. The fit will be better if this asymptotic series is expanded for nonnormal distributions to include only the skewness term (1.5). This series is given as j’(x) = a(x) c 1 + ; pjHi
+ 7’j (p., ~ 3)H4 +
. ).
where (Y(X) = (liv(2-rr)) expt-x2/2), pi are central moments, Hermite polynomials. The first two terms are used and rewritten as .f’(.r; p. u. y, A) = A expt-:‘/2)
1
and H;(.v) are
I - z t: ~ ~‘13) . J
191
where ; = (.I- - p)/(r and y = coefficient of skewness. Basically. this form is a first-order perturbation of the normal frequency curve to give skewness. It is unimodal as long as l-y/ < I .648. This form is then written for a mixture of two distributions as jix)
= ,ftr; pi, UI. ye 1A,) + Jtx; puz.VT. yz. A?).
1101
Since the prevalence obtained of the “flat-curve” and NIDD patients were much smaller than their observed prevalences at the community at large would indicate. their Cl values were given weights of 3.6 and 5.6, respectively. to
CONTROL
FREQ
IN THE
I
CI
-
GLUCOSE
LOWER
END
TOLERANCE
323
TEST
DISSECTION
n
4 CONTROL
14 INDEX
(CI)
IN
24 MG/DL
FIG. 3. The control index is dissected at the lower end into two distributions: Each of the distributions is assumed to take the Gram-Charlier series form.
Flat and Normal.
adjust to the external prevalences of flat (35%), normal (25%), and NIDD (40%). The dissection process converged with good initial estimates in some 5 to 10 iterations. Figures 3 and 4 show the results of least-squares curve fitting of the mixed distributions. Figure 3 shows the dissection at the lower end for the control index and Fig. 4 shows the dissection at the upper end. The crossover point between the dissected populations with the assumed prevalences is used as the cutoff. The authors ignored societal values and chose to maximize the correct number of classifications. From the fit one can further produce tables of the probability of illness (or group membership), predictive values of a positive or negative test, estimates of the probabilities of false negatives and false positives, and the specificity, and sensitivity of the test. The predictive value of a positive test is the percentage of times that a positive test will detect a diseased individual. It is the ratio of the number or proportion of diseased persons with positive tests divided by the total number or proportion of persons with positive tests. Similarly, the predictive value of a negative test is the percentage of times that a negative test will detect nondiseased or healthy persons. It is the ratio of the number or proportion of nondiseased persons with CI
-
UPPER
END
DISSECTION
50 40 30 20 10 0
FIG.
4. The control
index
ie CONTROL
is dissected
k0 INDEX
at the upper
(CI)
IN
ii0 MGfDL
end into two distributions:
Normal
and NIDD
324
MAKTIN.GOLDSTEIN,
AND HOLOGGITAS
negative tests divided by the total number or proportion of persons with negative tests. Normal subjects with positive tests are known as false positives and diseased subjects with negative tests are known as false negatives. The specificity of a screening test is defined as the number of normal or nondiseased subjects negative to the test expressed as a percentage of all normal subjects tested. Sensitivity is defined as the number of diseased subjects with a positive test as a percentage of all disease subjects tested (I,‘). The cutoff values for the Cl were calculated as 19.77 and 48.67. At the lowel end (19.77) the probability of a false negative is X.66 and 3.15: for a false positive. The specificity and sensitivity at the lower end are, respectively, 91.4% and 96.95%.At the upper end (48.67) the probability of a false negative is 7. I% and for a Mse positive 3.5%. The specificity and sensitivity at the upper end are, respectively. 92.9% and 96.5%. The probabilities of illness and PV+ and PV- are shown in Tables IO and I I. Combining the two results from Tables IO and I I with respect to the weighted prevalences the totals give P(false positive) = 4.1%
sensitivity = 95.9%
P(false negative) = 9.8%
specificity = 90.256.
An alternate procedure for finding the reference interval is to use only the normal population values and to take a 2.5% cutoff of the top and botton end of the ordered statistics (14). This produces a normal range of 16.82 to 54.55 where these are ranked statistics X(4) and ,Y( 144). respectively. This range is fairly close as shown in Table I2 to that produced by the above procedure. In essense, as in many medical tests there is a gray area where one is in transition between the physiological and pathological states. Table I2 shows the distribution of the patients based upon the diagnosis criteria and the reference interval of the control index based upon each of the two methods. DISCUSSION
Perturbing the control system with glucose causes the blood serum glucose and insulin levels to fluctuate with an attempt by the system to return to fasting levels by seeking some set point. In the normal, the set point is closely approached in about 3 hr with continued dampened oscillations in the 4th and 5th hours due to minor perturbations induced by gluconeogenesis and glycogenolysis. Comparing the normal patients to those who have “flat-curves” the glucose in the latter returns to the fasting set point more quickly and with less amplitude displacement. The penalty for this increased response is that it takes very little perturbation to place the patient in hypoglycemia (less than 55 mgi dl). On the other hand, the NIDD patients frequently start at high levels of glucose and insulin at fasting, reaching still higher values of glucose and insulin. and while the system attempts to return to the fasting level within the 5 hr, it rarely succeeds.
CONTROL
IN THE
GLUCOSE TABLE
PROBABILITIES
CI 13 14 15 16 17 18 19 [19.77 20 21 22 23
TOLERANCE
325
10
ON THE
LOWER
P(flatlCI) 100.0% 99.5 98.5 96.4 92.1 83.5 67.7 50.0 44.1 20.0 4.5 0.0
TEST
END
pv+
PV-
100.0% 99.9 99.7 99.3 98.7 97.5 95.8 94.0 93.4 90.3 86.6 82.5
51.7% 56.0 61.8 69.0 77.1 85.0 91.7 95.51 96.3 98.9 99.9 100.0
The control index is different in the three states, normal, flat, and NIDD. The spiral path and its associated measures give some indication of the glucoseinsulin coupling. The flat glucose tolerance patients have significantly shorter paths but their percentage traveled in the first 90 min is similar to that of the normals (see Tables 2 and 3). The NIDD patients have similar path lengths as the normal patient population but the percentage traveled in the first 90 min is significantly less than in the normal group. The maximum geometric excursion from the fasting point is further in the NIDD along with its angle being more horizontal. In the flat glucose tolerance patient this maximum distance is less than normal as well as its angle being steeper than normal. The correlation coefficient of insulin differences on glucose differences gives the strength of a linear relationship between these two variables. The normal patients have a TABLE PROBABILITIES
Cl 30 35 40 [::.67 50 55 60 65 70 75
P(NIDD(CI) 0.0% 1.9 16.7 37.5 50.0 54.8 74.9 90.8 97.8 99.6 100.0
II
ON THE
UPPER
pv+ 78.5%~ 86.2 90.8 93.7 95.6 96.3 98.3 99.5 99.9 100.0 100.0
END
PV100.0% 99.9 99.1 96.9 94.31 93.2 88.1 81.8 74.9 68.0 61.8
326
MAKTIN.GOLDSTEIN.AND
DISTRIBUTION
HOLOGGITAS
OF POPULATIONS
Classification Actual Cl dissectton Cl
order
method statistic
method
stronger linear relationship than the flat or NIDD as shown in Tables 6 and 7. For the NIDD patient the slope is significantly less and the correlation significantly weaker. Several measures of control of glucose homeostasis have been proposed and statistically evaluated. Of these the control index was found to be the most discriminatory. The reference interval has been specified for the control index and associated probabilities are given. The time line spiral graph of glucoseinsulin coupling has given us a visual impression of the control and the ability to return to the origin (fasting values). The regression line showed the coupling between glucose and insulin, especially in the normal patient. The authors conclude that the flat glucose curves are not a disease involving issues of control since the time line curve returns to the fasting condition. However, the NIDD is a derangement of control as evidenced by the inability to return to the origin and diminished sensitivity of insulin to changes in glucose. This diminished sensitivity is shown by the more horizontal leaning of the NIDD spiral curves (angle). The authors believe that the control index introduced in this paper will be better as a single classifier than the current definitions used for flat and NIDD and that as a quantitative value it will better measure the degree or lack of control (disease). ACKNOWLEDGMENTS We and
thank
Ralph
Miech.
M.D.,
Ph.D.,
and
Clyde
B. Schechter.
M.D..
for
their
Press.
Princeton.
critical
reviews
clarifications.
REFERENCES I. 2.
BELLMAN, R. E. “Dynamic SWAN. G. W. “Applications 19x4.
.T. BELLOMO. G.. Optimal feedback 1. 5.
Princeton Programming.” of Optimal Control Theory
BRUNETTI. P.. CALABRESE, of glycemia regulation
Univ.
on Biomedicine.”
G.. hh%o’-l-1, in diabetics. Med.
Dekker.
I>.. SARI I, E.. ANU Biol. Eng. C’ornprrr.
NJ.
1957. New
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