Letters

Address correspondence to Randy L.Anderson, MD, Department of Public HealthSciences, Bowman Gray School of Medicine,Medical Center Boulevard, Winston-Salem,NC 27157.

Acknowledgments—The authors arefunded in part by National Institutes of HealthNHLB1 Grant NIH 2-194-811-9443; N1NDSGrant NIH NS 20618-08; and N1NDS GrantNIH 1PO1-NS-27500-02.

References1. Tai MM: A mathematical model for the de-

termination of total area under glucose tol-erance and other metabolic curves. Diabe-tes Care 17:152-154, 1994

2. Swokowski EW: Calculus with Analytic Ge-ometry. 3rd ed. Boston, MA, Prindle, We-ber and Schmidt, 1983, p. 260-261

3. FairesJD, Faires BT: Calculus. 2nd ed. NewYork, NY, Random House, 1988, p. 497-498

Modeling MetabolicCurves

While not denigrating the Taimodel (1) for the total area undermetabolic curves, nor denying

its validity, we believe the T variate func-tion also deserves to be better known. Itcannot only be used to approximate thetotal area under the curve (AUC) but alsoto describe these curves functionally withestimates of their characteristic parame-ters and to separate their secretion andclearance phases. The formula for thisfunction is

(1) y = y0

where y e {insulin, C-peptide, glucoseconcentrations}, y0 is a basal value, t istime, and A, a, and b art found by simplyfitting the data to the straight line form of(1):

(2) ln(y - .yO) = lnA + a\nt - bt

in which ln(x) represents the natural log-arithm. By differentiating}' with respect tot, one can find the secretion and clearancerates as the first and second terms respec-tively of

(3)dy/dt = Aaf - Abtae~hl

The total above basal area under the rele-vant response curve, AUC, is then foundby integrating (1) with respect to t be-tween the limits of 0 and o° to obtain

(4) AUC = AY(a + l)/ba + !

in which F(x) represents the T function,values for which are in standard tables (2)(hence the name of the function).

An example of the use of the Fvariate function in this context may befound in Shannon et al. (3). The T variatemodel does not require seeding with ini-tial values, nor does it require many sam-pling times to increase its accuracy.

A. G. SHANNON,PHD

D. R. OWENS, MD

From the School of Mathematical Science,University of Technology, Sydney, Australia.

Address correspondence to A. G. Shannon,PhD, University of Technology, Sydney, P.O.Box 123, Broadway, New South Wales 2007,Australia.

References

1. Tai MM: A mathematical model for the de-termination of total area under glucose tol-erance and other metabolic curves. Diabe-tes Care 17:152-154, 1994

2. Abramowitz M, Stegun IA: Handbook ojMathematical Functions. New York, Dover,1965

3. Shannon AG, Ollerton RL, Owens DR, Lu-zio S, Wong CK, Colagiuri S: Mathematicalmodel for estimating pre-hepatic insulinsecretion from plasma C-peptide (Ab-stract). Diabetes 40 (Suppl. 1):13A, 1991

Reply From Mary Tai

I would like to thank all of the readerswho have reviewed and responded tothe publication of Tai's model. 1 am

particularly grateful to those who hawconfidence in my intention of publishingand have considered Tai's model an effec-tive tool in calculating total area under ametabolic curve. However, three of thereaders expressed their concerns on thefollowing issues. My replies are presentedas follows.

To Dr. BenderThe originality of Tai's model. While adoctoral candidate working on my disser-tation at Columbia University in H)81, Ineeded to calculate total area under acurve. During a session with my statisticaladvisor, and after examining several alter-native methods, I worked out the modelin front of him. The concept behind it isobviously common sense, and one doesnot have to consult the trapezoid rule tofigure it out. The trapezoid rule is reallynot Nobel Prize material, such as the dou-ble helix or jumping genes. I also used theformulas to calculate the areas of a squareor a triangle without knowing whoserules were being followed. Fortunately, Ido not have to answer that for you.Why I call it Tai's model. 1 neverthought of publishing the model as a greatdiscover)' or accomplishment; it was notpublished until 14 years later, in l(-)(-)4.Because of its accuracy and easy applica-tion, many colleagues at the Obesity Re-search Center of St Luke's-Roosevelt I los-pital Center and Columbia Universitybegan using it and addressed it as "Tai'sformula" to distinguish it from others.Later, because the investigators were un-able to cite an unpublished work, 1 sub-mitted it for publication at their requests.Therefore, my name was rubber-stampedon the model before its publication.

According to Mcrriam Webster'sDictionary, a model can be defined as "a

DIABETES CARE, VOLUME 17, NUMBER 10, OCTOBER 1994 1225

Letters

type of design of product;" "a descriptionused to visualize something that cannotbe directly observed;" or "a system of pos-tulates, data, and inferences presented asa mathematical description of an enti-ty. . . ." Even if Tai's model were based onthe trapezoid rule concept, according tothe definition of a model, I have workedout a "design" (mathematical expression)for the "structure units" (individual areas)on my own. In other words, I have pre-sented the original concept into a func-tioning mathematical description that canbe easily observed and applied. Followingthe above definition, I therefore carefullynamed the mathematical description asTai's "model" rather than "formula" to in-dicate that 1 have used existing formulasfor small area calculations.

My intention in publishing themodel is therefore to share, rather than togain honor or glory with its publication,because there is none. Many other inves-tigators probably thought about the samething, but maybe they did not bother tofollow up or produce a model (or thesame model). You indicated that I proba-bly did work this out on my own and I amgrateful for your "probability," because Idid indeed do so with a witness present.Maybe I can address the model as my cre-ation based on fact rather than yourdoubtful "probability." Besides, if 1 do notaddress the model as "Tai's," other inves-tigators who wish to cite it will.The precision of Tai's model. BecauseTai's model is based on the calculations ofindividual squares and triangles, its pre-cision is obviously absolute. You are cor-rect in saying that I have verified the va-lidity of the formula by comparison withits approximation, meaning countingsquares.The size of n. Following the statisticalprinciple that you consider elementary, itis correct that n does represent numbersof data sets. However, in this case, ele-mental principle simply does not apply.The hypothesis here is the validity of theformula. The acceptance or rejection ofthe hypothesis is not based on the find-ings of each individual data set, as is a

general rule in an experimental study. Itshould not be difficult to see that eachdata set here represents the findings fromits respective research protocol and an-swers its individual research questionsrather than answering the validity of Tai'smodel. Furthermore, because the sameformula was used for each data set, thedegree of accuracy on the resultant totalarea obtained will be exactly the same foreach set. Therefore, increasing n of thedata set does not increase statistical poweras you suggested.

I introduced other formulas sim-ply for the purpose of comparison. Be-cause the formula cannot be comparedwith its approximation and there are lim-ited formulas available, I decided tocount, because every published curve hasbeen based on counting squares. I alsobelieve, if one increases the N I am talkingabout, meaning the number of methods,one can better verify the validity of Tai'smodel.

To Dr. WoleverAfter receiving your recent graphic repre-sentation of your formula, I began to re-alize that I have indeed misunderstoodyour formula as some other readers did.Your incremental area is the area abovethe baseline rather than the total area un-der the curve including the baseline area.I apologize for the misapplication of yourunique formula, which I do fully support.I also acknowledge that you, too, haveindeed used the concept of adding trian-gles and rectangles in your mathematicalmodel for the total increment. I also ap-preciate your idea of weighing, because Idid weigh the total area under an arc and,as you know, that might be the only way.

To Dr. Anderson and Ms. MonacoTai's model is designed to calculate totalarea under a metabolic curve that is plot-ted by connecting experimental points xv

yi with straight lines as shown in Fig. 1 inmy article. Because the metabolic curve isnot an arc, the exact area can be calcu-

lated without assumption and approxi-mation. If a smooth arc represents thetrue curve, it is obtainable only whenXj —> oo and Ax —» 0, as presented in thetrapezoid rule or calculus, and it is virtu-ally impossible in an experimental condi-tion.

Finally, I would like to correctsome typographical errors in my article:

On p. 153, the correct formulashould be

1 "Area = - ^ Xi^](yi_1 + _yf)

i = 1

and in example 1:

1Area = - 30 [(95 + 147) + (147

+ 124) + . . .]

MARY M. TAI, MS, EDD

From the Obesity Research Center, St Luke's-Roosevelt Hospital Center, New York; and theDepartment of Nutrition, School of Educa-tion, New York University, New York, NewYork.

Address correspondence to Mary M. Tai,MS, EdD, Obesity Research Center, Women'sHospital, 10th Floor, St. Luke's-RooseveltHospital Center, 114 Street at Amsterdam Av-enue, New York, NY 10025.

Addendum toMonaco#s andAnderson's Letter

Tai responds that her formula isbased on the sum of the areas ofsmall triangles and rectangles and is

not based on the sum of the areas of trap-ezoids (the trapezoidal rule). As is evidentin the following figure and algebra, thesmall triangle and the contiguous rectan-gle form a trapezoid. The sum of the area

1226 DIABETES CARE, VOLUME 17, NUMBER 10, OCTOBER 1994