The Calculus Behind Generic Drug Equivalence Stanley R. Huddy and Michael A. Jones Stanley R. Huddy srh@fdu.edu, MRID183534, ORCID -3-33-231) isanassistantprofessorat Fairleigh Dickinson University in Teaneck, New Jersey. His research focuses on dynamic behaviors and synchronization patterns on networks of nonlinear systems as well as on their applications. In his free time, Huddy enjoys mountain biking, snowboarding, playing the drums, and watching Netflix with his wife. Downloaded by [68.193.163.2] at 9:9 22 December 217 Michael A. Jones maj@ams.org, MRID64157,ORCID -2-321-342) is an associate editor at the American Mathematical Society s Mathematical Reviews in Ann Arbor and editor of Mathematics Magazine.Thisishis second article in The College Mathematics Journal that originated from listening to the radio while driving in his car; the first was about the National Football League s overtime rules. When the second author was listening to Jeremy Greene being interviewed on National Public Radio s Science Friday about his book on the history and science of generic drugs [3], he thought that the requirements for a drug to be considered a generic equivalent to a brand name drug involved calculus. This was solidified when he saw the words areas under the curve in quotes) while reading about the history of testing for generic equivalence [4,p.111]. It is not enough that two drugs contain the same amount of the active ingredients to be bioequivalent. Other factors, like the coating of a pill, may affect how an individual is able to absorb and to eliminate the active ingredient. For this reason, the U.S. Food and Drug Administration FDA) requires a statistical comparison of three values from clinical trials when checking for bioequivalence between a generic drug and a brand name drug: the maximum concentration C max of the active ingredient, the peak concentration time t max, and the total amount AUC of drug that enters the system for area under the curve). All three values involve fundamental ideas from calculus. Rather than consider a statistical analysis, we flip this around by assuming that the concentration function for an orally taken drug under a single-compartment model is known, at least in form, with two key model parameters unknown. We differentiate the concentration function to find C max and t max and integrate the concentration function to find AUC. There is good reason why the FDA focuses on these values: We show that knowing any two of the three of t max, C max,andauc is enough to reconstruct Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/ucmj. doi.org/1.18/7468342.217.139152 MSC: 92B5, 97M6 2 C THE MATHEMATICAL ASSOCIATION OF AMERICA
the concentration function. The analyses hinge on an application of the Lambert W function a function that cannot be expressed in terms of elementary functions. Maximum concentration and area under the curve The concentration-time curve describes how much of a drug is present in an individual s blood plasma as a function of time. For an oral dose of a drug under a singlecompartment model which treats the body as a single uniform compartment), there are different but equivalent representations of this concentration function. In the pharmacokinetics literature [5, 7 ], this concentration function is given by Ct) = λ afc V λ a ) e t e λ at ) 1) Downloaded by [68.193.163.2] at 9:9 22 December 217 where C, F,andV are constants, λ a > istheabsorptionrate,and > istheelimination rate. Specifically,C is the amount of drug administered at t =, F is the fraction of C that is absorbed, and V is the apparent volume of distribution the volume into which a given mass of drug would need to be diluted in order to give an observed concentration). While C is known at the time of administration, F and V can be estimated for a particular drug if they are not already available in the literature. The parameters λ a and are patient specificand,generally,λ a for orally administered drugs. In Figure 1, the increasing portion of the graph represents the absorption phase and the decreasing portion represents the elimination phase. If the concentration function is known for a specific patient,then differentiation and integration can be used to find the maximum concentration and the area under the curve, respectively. To find the maximum concentration, we find the critical points of the concentration function by differentiating the concentration function: C t) = λ afc V λ a ) e t + λ a e λ at ). The only critical point occurs at t max = lnλ a / )/λ a ), which is referred to as the peak concentration time. It is easy to check that this critical point provides a maximum by using the first or second derivative test.) Substituting t max into 1) yieldsthe 3 C max Ct ) 2 1 AUC t max 2 4 t 6 8 Figure 1. Concentration-time curve generated by 1) withparameters =.693, λ a =.247, F = 1, C = 5, and V = 1.37; from [5,p.11]. VOL. 49, NO. 1, JANUARY 218 THE COLLEGE MATHEMATICS JOURNAL 3
maximum concentration of the drug, C max = λ [ ) )] afc λe ln λ a / ) λa ln λ a / ) exp exp V λ a ) λ a λ a 2) where expx) = e x represents the exponential function. By multiplying the right side of the above equation by an appropriate representation of 1, namely, ) / ) λa ln λ a / ) λa ln λ a / ) 1 = exp exp, λ a λ a the maximum concentration may be rewritten and simplified as Downloaded by [68.193.163.2] at 9:9 22 December 217 C max = λ [ afc λa V λ a ) = λ a FC V ] ) λa /λ a ) λa ) λa /λ a ) λa = FC V λa ) λe /λ a ). 3) Recall that the area under the curve AUC)givesthetotalamountofdrugthatenters the system, which can be found by integration. For our single-compartment model, the area under the curve is the improper integral from zero to infinity of the concentration function: λ a FC V λ a ) e t e λ at ) dt = lim b = λ afc V λ a ) lim b b [ e t λ a FC V λ a ) e t e λat ) dt + e λat λ a Recovering the absorption and elimination rates ] b = FC V. 4) Because the concentration function is determined by λ a and, if we can write λ a and in terms of C max and AUC, thenwewillbeabletowritetheconcentrationfunction in terms of C max and AUC. It is easy to write in terms of AUC from 4): = FC VAUC. It is a bit more difficult to determine λ a.first,werewrite2) as VC max FC = λa ) λe /λ a ). To simplify notation, define K by taking logarithms of both sides so that K = ln VCmax FC ) ) λe = lnλ a ) ln )) <. 5) λ a 4 C THE MATHEMATICAL ASSOCIATION OF AMERICA
tangent line ln x ) 1 secant line 2 ln x ) λ e x + h Figure 2. Comparing the slopes of the tangent line of y = ln x at x = and the secant line between, ln )and + h, ln + h)). Downloaded by [68.193.163.2] at 9:9 22 December 217 We use the relationship between the slopes of secant lines and tangent lines of the natural logarithm function to show that K > 1. Since λ a >,writeλ a = + h for some h >. Then K may be written as times the slope of a secant line where ) ) ln λa ln lnλe + h) ln =. λ a + h) Because the natural logarithm function is concave down, the slope of the secant line through, ln )and + h, ln + h)) is less than the slope of the tangent line for y = ln x at x = see Figure 2) sothat Therefore, ln + h) ln h ) ln λa ln K = λ a < d dx ln x x=λe = 1. > 1 = 1 as claimed. Multiplying 5) byλ a )/ )andaddingln )tobothsidesgives Kλ a + K + ln ) = lnλ a ). By exponentiating both sides, we obtain e Kλ a/ e K = λ a and finally ) Ke K Kλa = e Kλa λe. With f x) = xe x,thetermsk, λ a,and are related by ) Kλa f K) = f. 6) VOL. 49, NO. 1, JANUARY 218 THE COLLEGE MATHEMATICS JOURNAL 5
Figure 3. K and Kλ a / as related by y = xe x. Downloaded by [68.193.163.2] at 9:9 22 December 217 If f were one-to-one, then we could take the inverse of f on both sides to equate the two arguments. But 1 < K < andλ a > give Kλ a / < 1 < K <, thus 6) implies that f is not one-to-one. Indeed, Figure 3 shows that f is two-to-one over the domain of interest, ), which contains K and Kλ a /. Using the Lambert W function At this point, there is still a fair amount of work needed to be done to isolate λ a.for y 1/e, ), there are two real solutions x to the equation y = xe x ;wedenotethe solution in which x 1asx = W 1 y) andthesolutioninwhichx 1asx = W y). These W 1 and W are the two real branches of the Lambert W function, the inverse of f z) = ze z for z C.ThegraphsofW 1 andw with their respective domains 1/e, ) and [ 1/e, ) appearinfigure 4 left and right, respectively. Now that we have introduced the Lambert W function, we can isolate λ a.applying W 1 to 6) yields )) W 1 Ke K Kλa ) = W 1 f K)) = W 1 f = Kλ a W 1 y) 2 1, 1 e W y) 2 1, 1 e 4. 5.5 y 1 4. 5.5 y 1 Figure 4. Left, the real branch W 1 of the Lambert W function with domain 1/e, ). Right, the real branch W of the Lambert W function with domain [ 1/e, ). 6 C THE MATHEMATICAL ASSOCIATION OF AMERICA
where the last equality follows because Kλ a / < 1. Hence, λ a = W 1Ke K ) K FC VAUC. The Lambert W function, also called the omega function, cannot be expressed in terms of elementary functions, but W 1 Ke K )canbecalculatednumerically.thelambert W function is often a built-in function in different numerical packages. See [1, 6 ] for more on the Lambert W function and its applications.) Because the concentration function for the single-compartment model depends on λ a and,andλ a and can be written as functions of C max and AUC,theconcentration function may be written as a function of C max and AUC. Downloaded by [68.193.163.2] at 9:9 22 December 217 Recovering the rates from any two of the parameters In this section, we show that not only is the concentration function for a single compartment model determined by C max and AUC,butitisalsodeterminedfromtheother two pairs from {C max, t max, AUC}. First, we show that λ a and can be written in terms of AUC and t max.rearranging t max = lnλ a / )/λ a )leadsto t max e t max = t max λ a e t maxλ a which is precisely f t max ) = f λ a t max ). Because λ a t max < t max < and f is two-to-one in the region of interest, we have λ a t max < 1 < t max <. Again a graph analogous to Figure 3 is key. Hence, applying the W 1 branch of the Lambert W function gives W 1 λe t max e t max ) = W 1 f t max )) = W 1 f λ a t max )) = λ a t max. From this and 4), we can write and λ a in terms of AUC and t max : = FC VAUC, λ a = W 1 t max e tmaxλe ) t max. 7) Now we show that λ a and can be written in terms of C max and t max.takingthe natural logarithm of 3) gives which implies that ) FC lnλ a / ) lnc max ) = ln = ln V λ a = lnfc /V ) lnc max ) t max. FC V ) t max VOL. 49, NO. 1, JANUARY 218 THE COLLEGE MATHEMATICS JOURNAL 7
This also gives us λ a as a function of C max and t max because λ a was written as a function of and t max in 7). For the single compartment model of an orally taken single-dose drug, any pair from {C max, t max, AUC} uniquely determines λ a and and, therefore, uniquely determines the concentration function. This supports the FDA requirement for collecting data on C max, t max,andauc in their testing of whether a generic drug is bioequivalent to a name brand drug. Downloaded by [68.193.163.2] at 9:9 22 December 217 Area under the curve in practice Because the concentration function is not known in practice, t max, C max,andauc are approximated from observed concentration values. As the names imply, the greatest observed concentration value is used to approximate C max and the time at which this observation occurs is used for t max. Numerical integration is used to approximate AUC from the observed concentration values, viewed as points on the concentration curve. In the pharmacology literature, the trapezoidal rule is a common method for estimating a definite integral. The trapezoidal rule steps through the observed concentration values and requires the calculation of averages between consecutive values. As AUC is defined by an improper integral, the trapezoid rule is used on a finite interval and a different method is used to approximate the tail. Thus, AUC = Ct) dt = t Ct) dt + t Ct) dt where t is the time of the last observed concentration value. The trapezoidal rule applied to any interval [t k 1, t k ] for k = 1,...,n + 1 with t n+1 = t )is T k = tk t k 1 Ct) dt Ck 1 + C k 2 ) t k t k 1 ) wherec k 1 andc k are the k 1)st and kth observed concentration values, respectively. This procedure is repeated for each pair of observed concentration values up until t and these averages T 1,...,T n are summed to estimate AUC. Figure 5 illustrates this process using data points from the curve in Figure 1. The second piece of the AUC 3 C max Ct ) 2 C k 1 1 C k T k C* t max t k 1 t k t* 8 t Figure 5. Concentration-time sample data of the concentration curve from Figure 1.The sample with the largest measured concentration is C max and the last recorded sample is C. 8 C THE MATHEMATICAL ASSOCIATION OF AMERICA
estimate is t Ct) dt C where C is the last observed concentration. This approximation is based on the assumption that no more of the drug is being absorbed into the blood stream. Mathematically, this means that e λ at max is assumed to be zero when evaluating the improper integral t Ct) dt. See [5, pp. 131 134] for information about when this approximation is used in practice. Downloaded by [68.193.163.2] at 9:9 22 December 217 Concluding remarks For a single compartment model of an oral-dose drug, we were able to show that the concentration function can be determined by any two of the three parameters t max, C max, and AUC. This justifies why the FDA requires patient data on these three parameters when testing if a generic drug is bioequivalent to a name brand drug. There is more work to be done. It would be interesting to relate t max, C max,and AUC to the concentration function if the drug is delivered intravenously, if an orally taken drug is time released, or if a multi-compartment model better reflects how the body interacts with the drug. Because the single compartment intravenous model is simpler than the oral-dose model [5, p.9],we expect the analysis in that case to be straightforward and suitable for a student project. We are interested in the calculus of bioequivalence, but bioequivalence is more of a measure of the statistical equivalence of key pharmacokinetic parameters. The statistics of bioequivalence is beyond this article, but see [2, Ch.22]forafirst step. Acknowledgments. We thank two anonymous referees for their thoughtful comments and suggestions on how to improve the exposition of this article, including the use of Figure 3. Summary. To show bioequivalence of generic and brand name drugs, the Food and Drug Administration FDA) requires a statistical comparison of three pharmacokinetic values that measure aspects of the drugs concentrations. These three values are related to calculus. We show that there is good reason why the FDA considers these values, as any two of the three is enough to recover the concentration of the drug over time for an orally taken drug using a single-compartment model. The results hinge on applications of the Lambert W function. References [1] Corless, R. M., Gonnet, G. H., Hare, D. E. G., Jeffrey, D. J., Knuth, D. E. 1996). On the Lambert W function. Adv. Comput. Math. 5: 329 359. [2] De Muth, J. E. 214).Basic Statistics and Pharmaceutical Statistical Applications, 3rd ed. Boca Raton, FL: CRC. [3] Flatow, I. 212). The Science of Sameness: Developing Generic Medications. Science Friday, National Public Radio. sciencefriday.com/segments/the-science-of-sameness-developing-generic-medications/. [4] Greene, J. A. 214). Generic: The Unbranding of Modern Medicine. Baltimore: Johns Hopkins Univ. Press. [5] Jambhekar, S. S., Breen, P. J. 219). Basic Pharmacokinetics. London:PharmaceuticalPress. [6] Rathie, P. N., Swamee, P. K., Ozelim, L. C. de S. M. 212). Lambert W-function revisited: Applications in science. In: Agarwal, A. K., ed. Proceedings of the 1th Annual Conference and the 11th Annual Conference of the Society for Special Functions and their Applications. Chennai: Soc. Spec. Funct. Appl., pp. 93 15. [7] Shargel, L., Yu, A. B. C. 212). Applied Biopharmaceutics & Pharmacokinetics,7thed.NewYork:McGraw- Hill. VOL. 49, NO. 1, JANUARY 218 THE COLLEGE MATHEMATICS JOURNAL 9