Journal of Pharmacokinetics and Biopharmaceutics, Vol. 6, No. 1, 1978
`
`SCIENTIFIC COMMENTARY
`
`A Comparison of Numerical Integrating Algorithms
`by Trapezoidal, Lagrange, and Spline Approximation
`
`K. C. Yeh I and K. C. Kwan 1
`
`Received June 9, 1977--Final August 23, 1977
`
`In the trapezoidal method, linear interpolation between data points tends to overestimate or
`underestimate the area, depending on the concavity o[ the curve. In some instances, area
`estimates can be obtained by linear interpolation of logarithmically tranfformed data. Two
`alternative algorithms based on known interpolating [unctions have been implemented for area
`calculations. In the Lagrange method, the linear interpolations are replaced by cubic polyno/nial
`interpolations. In the spline method, the cubic [unctions are further modified so that the fitted
`curves are completely smooth. This report describes their computing procedures with numerical
`examples.
`
`KEY WORDS: numerical integrating algorithms; trapezoidal approximation; Lagrange
`method; spline method.
`
`I N T R O D U C T I O N
`
`It is customary in biopharmaceutics to use a trapezoidal method to
`calculate areas under the concentration-time curve. The popularity of this
`method may be attributed to its simplicity both in concept and in execution
`(1,2). However, in cases where changes in curvature between data points
`are excessive or there are long intervals between data points, large
`algorithm errors are known to occur.
`To circumvent the curvature problem, two alternative algorithms
`based on interpolating polynomials have been devised and implemented
`for area calculations in these laboratories. These polynomials are known as
`spline functions (3,4) and Lagrange
`interpolating functions (5). The
`purpose of this report is to describe computational procedures of these two
`methods, to compare their solutions along with those obtained by the
`
`1 Merck Sharp & Dohme Research Laboratories, West Point, Pennsylvania 19486.
`
`79
`oo9o-466x/78/o2o0-oo79505.oo/o @ 1978 Plenum Publishing Corporation
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`80
`
`Yeh and Kwan
`
`linear and log trapezoidal methods, and to discuss the relative merit of the
`methods.
`
`NUMERICAL METHODS
`The purpose of a numerical method is to obtain practical solutions
`which otherwise would have been difficult or impossible to achieve.
`Because of two contributing factors, the solutions are seldom error free.
`First, experimental errors in the data are inevitable. These are called input
`errors. Second, additional errors are incurred when data are processed to
`produce numerical solutions. These are called algorithm errors. There are
`two types of algorithm errors (6). The truncation error is the difference
`between the true functional value and that calculated by numerical
`approximation. The round off error results from the fact that only a finite
`number of digits can actually be retained after each computational step,
`and any excess digits are lost.
`In biopharmaceutics, experimental data such as plasma concentrations
`are usually recorded at discrete time points. The purpose of using an
`approximating function in the present case is to connect all data points so
`that reliable values of areas can be calculated by integration. Although the
`selection of a particular procedure is somewhat subjective, two basic
`factors are usually considered: speed and accuracy. When calculations are
`to be performed manually, easy and simple methods are clearly preferable.
`However, with the advent of high-speed electronic computers, accuracy
`becomes the major consideration since computational steps can be pro-
`grammed and executed swiftly. Thus a method that increases the accuracy
`of solutions by minimizing algorithm errors should be attempted whenever
`the procedure is compatible with the limitations of available facilities.
`Because of their convenient mathematical properties, polynomials are
`the most widely used among various curve-fitting approaches. The four
`procedures described below represent the application of polynomials to
`area calculations.
`
`Linear Trapezoidal Method
`The linear trapezoidal method is the best known numerical integrating
`method. The functional value y between two adjacent points (ti-1, Yi-1)
`and (ti, Yi) is approximated by a straight line:
`y = a + bt
`
`(1)
`
`where
`
`a =
`(tiYi-1
`-- yiti-1)/hi
`b--(yi-yi-1)/hi
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`A Comparison of Numerical Integrating Algorithms
`
`81
`
`and
`
`hi = ti -
`
`ti-1
`
`Integrating equation 1 from t;-1 to ti gives the incremental area in that
`interval:
`
`[AUC]~ 1 =
`
`I t i
`
`i - - 1
`
`y dt = ı89
`
`+ yi-1)
`
`(2)
`
`To obtain the cumulative area over the interval [h, In], where n is the
`number of data points, the above procedure is repeated for each i, where
`i = 2, 3,..., n. The cumulative area then becomes
`
`[AUClt~ - [AUC]t 1 + [AUC]t2+''' + [AUClt:_ 1
`
`t
`
`(3)
`
`t
`
`- -
`
`t 2
`
`t 3
`
`It is apparent that linear interpolation between data points will tend to
`underestimate the area when data form a convex curve and to overestimate
`when the curve is concave. Further, the greater the hl, the greater would be
`the error. The magnitude of error would also depend on the oscillatory
`nature of the curve, or the lack thereof, between data points.
`
`Log Trapezoidal Method
`A direct modification of the linear trapezoidal method is the so-called
`log trapezoidal method. In this modified version, the y values are assumed
`to vary linearly within each sampling interval on a semilogarithmic scale:
`
`In (y) = In (Yi-1)+ (t- ti 1)" In (yi/yi-1)/hi
`
`On integration, one obtains
`
`[AUC]ttI_I = hi(yi- yi-1)/In (yi/yi-1)
`
`(4)
`
`(5)
`
`In pharmacokinetics, equation 5 is most appropriate when applied to
`data which appear to decline exponentially. Under such condition, the
`error produced is independent of hi. However, the method may produce
`large errors when used in an ascending curve, near a peak, or in a steeply
`descending polyexponential curve. Furthermore, the method cannot be
`used if either concentration is zero or if the two values are equal. When
`they occur, equation 2 can be used as an alternative approach. Despite
`these limitations, the log trapezoidal method can be used advantageously
`in specific situations or in combination with a second method to yield
`optimal solutions.
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`82
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`
`Lagrange Method
`In the Lagrange method, the linear function of equation 1 is replaced
`by a cubic polynomial:
`
`(6)
`y = ai + bit + ci te q- dlt 3
`To interpolate between t~_x - t -< t~, the equation is fitted to the nearest
`four data points (ti-2, y~-e), (t~_~, Yi-1), (ti, y~), and (t~+l, yi+a). The function
`is thus forced to pass through all four points. The shape of the fitted curve
`in the middle interval may not always be linear, but may be parabolic, or
`sigmoidal with one inflection point. The four coefficients, a~, bi, c~, and d~,
`can be obtained by using the Lagrange multiplier formula (5), or by solving
`the following system of equations:
`
`(7)
`
`4
`ti
`
`)
`
`(8)
`
`ti3+1_1 di
`t2+1
`Yi+
`t,+1
`Once the coefficients are obtained, the incremental area on the inter-
`val [ti-1, tl] is calculated by integrating equation 6:
`+ sbi(ti
`1
`t.
`1
`e
`3
`[AUC],',_ 1 =aih~
`t~-l)+~ci(t~
`ti_a)+zdi(t~
`3
`1
`2
`-
`-
`As an example, Fig. 1 shows the cubic polynomial
`(9)
`y = 1 +7.5t- 5.5t2+ t 3
`passing through (0,1), (1,4), (2,2), and (3,1). The area over the interval
`[1,2] by equation 8 is 3.17, whereas that by the trapezoidal method is 3.0
`
`4
`
`-
`
`6.
`
`5
`
`4
`
`>_ 3
`
`2
`
`I
`
`:
`."
`~
`
`0
`
`~
`.-
`
`"
`
`~
`
`o~
`
`*%~ ~176
`
`o
`"%,~
`
`".,~
`
`,J
`~
`oo ,~
`.~176
`
`I
`
`i
`2
`
`I
`5
`
`Fig. 1. A cubic polynomial fitted to four data points.
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`A Comparison ot Numerical Integrating Algorithms
`
`83
`
`In order to use equation 6, the data must be theoretically smooth over
`the interval [ti-2, t~+l]. Under this condition, the cubic polynomial will
`usually give a better approximation than a single straight line in the middle
`interval [ti-1, ti] because the two contiguous points also contribute to
`defining the functional behavior over this subinterval. Functional approx-
`imations over the intervals [ti-2, ti-1] and [ti, ti+l] tend to be less reliable in
`that shaping constraints are one-sided.
`Equation 7 can be applied serially for each i, where i = 3, 4 . . . . . n - 1,
`but not for the two end intervals, It1, t2] and [tn-1, tn]. To fit these two
`intervals, the nearest three points are used and fitted with a parabola:
`y = ai q- bit + cit 2
`(10)
`The three coefficients are calculated by solving a system of three
`simultaneous linear equations, analogous to equation 7, and the cor-
`responding areas are obtained by integrating equation 10.
`The cumulative area over the entire interval [q, t,] is computed by
`summation, using equation 3.
`Areas calculated by the integration of parabolic equations (equation
`10) are subject to greater error than those calculated by the integration of
`cubic equations (equation 6). In part, this is because approximations are
`usually better near the middle segment than at the two ends. Since equa-
`tion 10 is quadratic, it may yield a minimum or a maximum that "should"
`not have been there. Unwanted oscillations may also occur with cubic
`equations. Therefore, it is desirable to monitor the interpolated curve over
`the entire interval. The monitoring can be accomplished by sampling a set
`of interpolated values within each interval [t~-~, t~]. In so doing, the suit-
`ability of the fitted curve can be evaluated intuitively in relation to the
`prevailing understanding or assumptions concerning the underlying kinetic
`mechanism,
`Thus in the Lagrange method the n experimental points are linked by
`n-1 smooth curves, with each data point forming the knot of the chain(The
`fitted curve is piecewise smooth; i.e., it is differentiable within each interval
`[t~_~, t~], but not at the data points. In other words, each knot forms a
`singular point and has no definitive first derivative because the two tangent
`lines of adjacent cubic functions at t~ may not coincide. This is similar to the
`trapezoidal method wherein the fitted curve is piecewise linear. However,
`the curve connected by serial cubic polynomials will look more curvilinear
`and natural than the polygonal curve formed by the linear trapezoidal
`method.
`The use of serial low-degree (cubic) polynomials, each of which is
`fitted to a local region, is preferable to the use of a single high-degree
`polynomial. While a polynomial of degree n-1 or less can be expected to
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`84
`
`Yeh and Kwan
`
`pass through n points, excessive oscillations on the fitted curve would be
`inevitable. On the other hand, the linear interpolations of either of the
`trapezoidal methods are overly simplistic, even though the problem of
`spurious oscillations is completely avoided. In a practical sense, the
`Lagrange method can be considered a compromise between the two
`extremes.
`
`Spline Method
`General spline functions are defined as piecewise polynomials of
`degree k. The pieces are connected at each of the several knots such that
`the fitted curve and its first k-1 derivatives are continuously differentiable
`over the interval [h, tn] (3).
`In this method, the knots are taken to be the data points themselves
`and k is defined to be 3. Thus the procedure of interpolation by cubic
`splines is similar to the Lagrange method except for the additional con-
`straint of differentiability at each data point.
`Description of spline functions can be found in many sources (3,4,7).
`The derivation presented below follows closely that of Dunfield and Read
`(8).
`
`Consider equation 6, which is a cubic function. Differentiating it twice
`gives the following linear expression:
`= 2Ci + 6dlt
`(11)
`Equation 11 shows that )" is linear over each interval It;-1, ti]. Rewri-
`ting it in the following form
`
`Yi-1 (t,- t)+~ (t- ti-D
`
`the equation can be integrated to yield
`
`.
`-2 _]_ Yi
`--Yi-l (t i -t)
`~i{t-ti-1)
`2hi
`
`-2 _~_
`$1
`
`s
`
`(12)
`
`(13)
`
`t]3+ ~i (t-tg-1)3+slt+s2
`)ii-l"
`,
`6h~
`y = ~
`{ti -
`where sl and s2 are integration constants. Evaluating equation 14 at t~_~
`and b, respectively, gives
`
`(14)
`
`Yi-1 (hi)3 + Site-1 + s2
`Yi-1 = "~i
`
`Ji
`Yi =~i
`
`(hi)3 + Slti +$2
`
`(15)
`
`(16)
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`A Comparison of Numerical Integrating Algorithms
`
`from which the two constants can be solved:
`
`h;
`,
`1
`81 = hi (y -- yi-1)--6
`
`(jJi-)Ji-1)
`
`1
`
`s2 =
`
`(t,yi-1
`
`-
`
`hi
`
`..
`(tiyi-
`
`
`
`-
`
`fiti-1)
`
`85
`
`(17)
`
`(18)
`
`All quantities in equation 14 are known except ~i-1 and J~i, the second
`derivatives at each data point. These unknown values are determined as
`follows.
`Evaluating equation 13 at ti-1 from the right interval [ti-~, ti] and from
`the left interval [ti-2, ti-1] gives the following two equations, respectively:
`
`3)i-1
`
`.9i-~
`
`- Yi-lh~
`2
`iJi-lhi-1
`- - +
`2
`
`(yi - yi-1)
`~ -
`-
`
`hi
`(Yi-1- Yl-2)
`hi-1
`
`(~i -
`h i - -
`
`:91-1)
`6
`(Yi-1- Yi-2)
`hi-1 - -
`6
`
`(19)
`
`(20)
`
`where hi-1 = ti-1 - ti-2.
`Combining Equations 19 and 20 gives a single expression after rear-
`rangement:
`
`hi
`1
`hi-1
`-6 Yi-2+3(hi-1- hi-1)~i-~ +-~Yi
`
`(yi - yi-1)
`hi
`
`(yi-l- yl-2)
`hi-1
`
`(21)
`
`Thus n-2 equations can be generated from equation 21 where i =
`3, 4 . . . . . n. Two extra equations are required to solve for n unknowns.
`They can be obtained by specifying two additional conditions. In the
`present case, these are f2 = ~)'3, and ~'~-1 = ~'n. These third derivatives are
`obtained by differentiating equation 11 at i = 2, 3, n-l, and n:
`Y'2 = (Y2 -- fi)/h2
`Y'3 =- (Y3 -- y2)/h3
`})'n-1 = (j~n-1 -- yn-2)/hn-1
`(24)
`f. = (f. - ~-O/h.
`(25)
`Combining equations 22 and 23, and equations 24 and 25, respec-
`tively, gives
`
`(22)
`(23)
`
`1
`
`+L
`(__i
`i
`-hn_ ~n_2-\hn_ a h.]~._ 1 h
`j~.=0
`
`(27)
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`86
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`
`With the addition of the last two equations, the n unknown :fi values
`are determined by solving equations 21, 26 and 27 simultaneously. The
`incremental area over each interval [ti-a, ti] is calculated by integrating
`equation 14:
`
`3 hi
`[AUf]t'i_l ~-- ~
`
`..
`(Yi "nt- ffi-1) "q- hi[1Sl(tl 4- t,-1) + s2]
`
`(28)
`
`As before, the cumulative area over the interval [h, t,] is obtained by
`summation, using equation 3.
`The functional behaviors of the fitted curve in each interval [t--a, t~]
`are determined not only by the nearest two or four data points but also by
`all others. Evidently, data in the immediate neighborhood are the most
`influential.
`Because of these additional constraints, the fitted curve and its first
`derivatives are completely smooth. This is in contrast to the other methods
`wherein the fitted curve is only piecewise smooth. Since the spline function
`is composed of serial cubic polynomials, the actual values of the inter-
`polated curve should also be sampled to monitor the functional behavior of
`the fitted curve. This is especially critical when data are scattered or
`sampling intervals are large. Under these conditions, the cubic equations
`may produce extraneous inflection points and cause serious under- or
`overestimations of areas.
`In practice, equations 26 and 27 are applicable to general types of
`kinetic data. In specific situations where additional information is avail-
`able, there may be other functions (e.g., exponential) that are more
`representative of the data. When used appropriately, such modifications
`should increase the accuracy of the solutions.
`
`SIMULATIONS AND RESULTS
`
`A series of computations were performed using each of the above
`procedures. These exercises were designed to examine the validity of the
`algorithms as well as to compare the reliability of the solutions under
`simulated conditions. The following examples may serve to provide some
`perspective on the relative merits of the four methods in specific circum-
`stances.
`Example 1. The data were assumed to be linear over the interval
`[h, tn]. The results, shown in Table I, indicate that all three methods gave
`exact and identical solutions. In this example, the linear trapezoidal
`method should yield the correct answer since assumption of linearity
`between points is identical to the nature of data. Moreover, the obser-
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`A Comparison of Numerical Integrating Algorithms
`
`87
`
`Table I. Estimation of Areas Under Three Straight Lines
`
`i
`
`1
`2
`3
`4
`5
`6
`7
`
`t
`
`0
`0.2
`1.0
`3.0
`4.0
`5.5
`8.0
`
`[AUC]o 8 by
`Linear trapezoidal method
`Lagrange method
`Spline method
`
`Simulated Yl
`
`Case 1
`
`Case 2
`
`Case 3
`
`0
`0.2
`1.0
`3.0
`4.0
`5.5
`8.0
`
`32.0
`32.0
`32.0
`
`8.0
`7.8
`7.0
`5.0
`4.0
`2.5
`0
`
`32,0
`32.0
`32.0
`
`4.0
`4.0
`4.0
`4.0
`4.0
`4.0
`4.0
`
`32.0
`32.0
`32.0
`
`vation that both the Lagrange and the spline methods also produced
`correct solutions indicates that all polynomials had degenerated to linear
`functions and that undue oscillations had not been produced.
`
`Example 2. Data displayed in Table II represent sampled values on a
`monoexponentially decaying curve:
`
`y = 16 exp ( - OAt)
`
`(29)
`
`On a semilogarithmic plot, equation 29 forms a straight line. The
`theoretical area values were obtained by using the log trapezoidal method
`(equation 5).
`
`Table II, Incremental Areas and the Corresponding Algorithm Errors for Example 2
`
`Area on subinterval [t~-l, ti] a'b
`
`ti
`
`Y/
`
`Theoretical
`
`0
`0.2
`0.5
`1,0
`2.0
`3.0
`4.0
`6.0
`9.0
`
`16.0000
`14.7699
`13.0997
`10.7251
`7.1893
`4.8191
`3.2303
`1 . 4 5 1 5
`0.4372
`
`.
`3.0753
`4.1754
`5.9364
`8.8396
`5.9254
`3.9719
`4.4471
`2.5358
`
`Linear
`trapezoidal
`
`.
`
`.
`3.0770(0.05)
`4.1804 (0.12)
`5.9562(0.33)
`8.9572 (1.33)
`6.0042 (1.33)
`4.0247 (1.33)
`4.6818 (5.28)
`2.8330 (11.72)
`
`Values shown are rounded off for display.
`b Values in parentheses are percent algorithm errors.
`
`Lagrange
`
`Spline
`
`
`.
`(0.00)
`3.0754
`4.1754 (-0.00)
`5.9363 (-0.00)
`8.8374 (-0.02)
`5.9231 (-0.04)
`3.9695 (-0.06)
`4.4256 (-0.48)
`2.3368 (-7.85)
`
`(0.00)
`3.0754
`4.1754 (-0.00)
`5.9364
`(0.00)
`8.8392 (-0.01)
`5.9250 (-0.01)
`3.9725
`(0.02)
`4.4340 (-0.30)
`2,6346
`(3.90)
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`88
`
`Yeh and Kwan
`
`It is evident from Table II that the linear trapezoidal method consis-
`tently produced large, positive errors, while the Lagrange method yielded
`relatively small negative ones. In contrast, output errors produced by the
`spline method were generally smaller and were more randomly distributed.
`However, both the Lagrange and the spline methods produced pro-
`portionately large error for the last increment because the sampling inter-
`val is large and there are no later data to tie the cubic function. In this
`example, the log trapezoidal method is obviously the best since its solutions
`are identical to the theoretical values.
`Example 3. Simulated plasma concentrations were generated based
`on an oral two-compartment open model with elimination occurring from
`the central compartment (Scheme I):
`ka
`
`G
`
`>B-'~E
`
`T
`Scheme I
`
`Differential equations corresponding to the model are as follows:
`0 = -kaG
`
`G=l koG-k~2G +l k21T-Ic~oG
`
`(30)
`
`(31)
`
`(32)
`~" = k12 VCp - k2l T
`= klo VCp
`(33)
`where G, B, T, and E are drug amounts in the absorption compartment,
`the central compartment (including blood), the peripheral compartment,
`and
`the elimination compartment (sum of biotransformation and
`excretion), respectively; Cp is the plasma concentrations; V is the appar-
`ent volume of distribution of
`the
`central compartment; and
`k12, k21, and kl0 are first-order rate constants for the designated pro-
`cesses. To simulate the possibility that absorption efficiency may be related
`to transient location of absorption sites, the absorption parameter ka was
`assumed to be time dependent. After several experimentations, the
`following empirical equation was employed:
`
`ka = Aa exp [- (A2t-A3) 2] +A4x/}
`where An, A2, A3, and A4 are constants.
`
`(34)
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`A Comparison of Numerical Integrating Algoritimas
`
`89
`
`Table III. Time-Dependent Variables Generated by the Runge-Kutta
`Procedure a'b
`
`i
`
`1
`2
`3
`4
`5
`6
`7
`8
`9
`10
`11
`12
`
`tl (hr)
`
`G (mg)
`
`ka (hr -1)
`
`C o (ixg/ml)
`
`0
`0.25
`0.5
`1.0
`1.5
`2.0
`3.0
`4.0
`6.0
`9.0
`12.0
`16.0
`
`75,0
`64.6909
`48.8621
`15,2885
`2.1823
`0,2677
`0.0308
`0,0153
`0.0039
`0.0003
`0.0
`0.0
`
`0.3092
`0,8288
`1.4496
`3.1992
`4.3524
`3.7487
`0.9961
`0.6140
`0.7425
`0.9093
`1,0473
`1,2093
`
`0
`1.1616
`2.7103
`5,1988
`4,9576
`3,8694
`2.3532
`1.4732
`0.5847
`0.1464
`0.0366
`0.0058
`
`~Parameter used: dose = 75 mg; V = 8 liters, klz = 0.4 hr -1, k z l =
`1.Shr -~, k_13o=0.6hr -1, Al=4.0hr -l, Aa=l.0hr -1, A3=1.6,
`A 4 = 0.3 hr- /z.
`program
`integration
`numerical
`~The G.E.
`time-sharing
`RKPBI/KKPB2 (10) was employed. Step size was set to vary with
`sampling interval and equaled 0.0625 h~.
`
`IO
`
`i ~ 0.05
`\
`
`0ol
`
`r
`
`\
`
`,
`
`,
`
`2
`
`4
`
`I0
`8
`6
`TiME, HOURS
`
`12
`
`14
`
`16
`
`o ~
`
`3
`
`uJ
`
`o
`
`I -
`
`O-
`
`Fig, 2. Time course of simulated plasma levels.
`Circles represent sampled values.
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`90
`
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`
`In the present context, such an assumption also defies a closed-form
`solution to the rate equations. Therefore, solutions of differential equa-
`tions were obtained by the fourth-order Runge-Kutta procedure (9).
`Actual values of model parameters and the generated time-dependent
`quantities are shown in Table III. The time course of the plasma concen-
`trations is illustrated in Fig. 2.
`Using the 12 values of Cp "sampled" at times indicated in the table,
`areas under the curve were calculated by each of the four methods. Table
`IV lists the incremental and the total areas. In order to eliminate the large
`end error discussed in the previous example, plasma levels during the last
`interval were assumed to decline monoexponentially, and equation 5 was
`used to calculate the last increment in all computations. This was adapted
`as a standard procedure since, in actual practice, sampling of plasma
`concentrations usually terminates in the log linear phase. Thus, in the
`spline method, the following equation was used in place of equation 27,
`which fulfills the continuity and smoothness of requirements at the (n-1)th
`point:
`
`Yn--1.
`hn-1
`hn-1
`
`Yn-2 +""2"-- jJn--l.~ = "--~n In (yn/y.-x)
`6
`
`(y,-1 - Y,-2)
`hn-1
`
`(35)
`
`Comparisons of the algorithm errors are given in Fig. 3. These values
`were obtained by comparing the numerically calculated incremental areas
`with the corresponding theoretical areas. It is apparent that the largest
`errors were produced by the linear trapezoidal method. In particular,
`there were large negative deviations around the peak and large positive
`deviations elsewhere. While significant error reductions were ob-
`served with both the Lagrange and the spline methods, the smallest
`absolute errors were produced by the latter. It should be noted that
`the log trapezoidal method produced the best area estimates after the
`fourth hour. Figure 2 shows that the curve practically declines log-
`linearly after 4 hr. Since the log trapezoidal method can best approxi-
`mate the nature of the data, it should be superior to any of the other
`three empirical methods. One could use a combination procedure of
`applying the spline method for t - 4 hr and the log trapezoidal method
`for t >4 hr. The last column of Table IV shows the excellent result by
`such a combination.
`Example 4. Twelve sets of data were generated by introducing simu-
`lated random experimental errors, corresponding to a coefficient of varia-
`tion of 10%, into the sampled Cp values shown in Table III. These values
`were then rounded off to retain only two decimal places, as shown in Table
`
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`!
`
`o
`
`bLast increment calculated from equation 5,
`aValues in parentheses are percent errors.
`
`15.5991 (-0.1)
`
`15.5896 (-0,1)
`
`15.5165 (-0.6)
`
`15.3512 (- 1.7)
`
`(1.7)
`
`15.8725
`
`15.6125
`
`0.0668
`(0.0)
`0.2376
`(0.04)
`0.9496
`(0.01)
`1.9230 (0.04)
`1.8781
`(0.06)
`3.0314 (-0.16)
`2.2071
`(0.12)
`2.6293
`(0.13)
`2.0631 (-0.77)
`0.4804
`(0.20)
`0.1327 (-0.96)
`
`trapezoidal
`Spline and log
`
`0.0668
`(0.0)
`0.2378
`(0.08)
`0.9450
`(0.48)
`1.9179 (-0.23)
`1.8781
`(0.06)
`3.0314 (-0.16)
`2,2071
`(0.12)
`2.6293
`(0.13)
`2.0631 (-0.77)
`0.4804
`(0.20)
`0.1327 (-0.96)
`
`0.0668
`(0.0)
`0.2210 (-7.02)
`0.9097 (-4.19)
`1.9023 (-1.04)
`1.8710 (-0.33)
`3.0461
`(0.33)
`2.2254
`(0.95)
`2.6136 (-0.47)
`2.0431 (-1.74)
`(0.24)
`0.4805
`(2.35)
`0.1371
`
`Spline
`
`Lagrange
`
`0.0668
`(0.0)
`0.2376
`(0.04)
`0.9496
`(0.01)
`1.9230
`(0.04)
`1.8790
`(0.10)
`3.0487
`(0.42)
`2.1955 (-0.41)
`2.5386 (-3.13)
`1.9102 (-8.13)
`0,4570 (-4.70)
`(8.38)
`0.1452
`
`trapezoidal
`
`Log
`
`0,0668
`(0.0)
`0.2745 (15.50)
`1.0967 (15.49)
`2.0579
`(7.05)
`1.9132
`(1.92)
`3.1113
`(2.48)
`2.2068
`(0.10)
`2.5391 (-3.31)
`1.9773 (-4.90)
`(0.96)
`0.4840
`(8.38)
`0.1452
`
`trapezoidal
`
`Linear
`
`0.0668
`0.2377
`0.9495
`1.9223
`1.8771
`3.0361
`2.2045
`2.6259
`2.0792
`0.4794
`0.1340
`
`Theoretical
`
`Total
`
`16.0
`12.0
`9.0
`6.0
`4.0
`3.0
`2.0
`1.5
`1.0
`0.50
`0.25
`
`t~
`
`Area on subinterval [ti, ti-1] ~'~
`
`Table IV. Incremental Areas and the Corresponding Algorithm Errors for Example 3
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`
`SPLINE METHOD
`
`+5 l
`
`/ 0 m
`
`+5 ~
`
`~ m
`
`--
`
`--
`
`II
`
`LAGRANGE METHOD
`_ ~-'~z +5 ]IIi LOG TRAPEZOIDAL METHOD
`
`ae ~ +i01 I+15-+5 LINEARIIm TRAPEZOIDALIII IIMETHODI I I
`
`
`
`~
`Iw
`
`-5
`
`I
`I
`I
`I
`I
`I
`1
`I
`I
`/
`I
`h I h2 h~ h 4 h 5 h 6 h7 h 8 h 9 hlo hll
`At
`
`h i
`Fig. 3. Algorithm errors produced by the
`four numerical methods. Errors for the last
`increment (hll) were obtained by using
`equation 5.
`
`Table V. Twelve Sets of Simulated Plasma Concentrations (~g/ml) Containing
`Random Errors
`
`Sampling time (hr)
`
`0 0.25
`
`0.5
`
`1
`
`1.5
`
`2
`
`3
`
`4
`
`6
`
`9
`
`12
`
`16
`
`0.04 0.01
`0 1.09 3.19 5.34 5.33 3.96 2.32 1.47 0.51 0.15
`0 1.17 3.20 5.13
`4.38 3.19 2.31
`1.18 0.57 0.17 0.04 0.01
`0 1.07 2.87 5.49 5.44 4.06 1.97
`1.31 0.44 0.15 0.04 0.01
`0 1.08 2.27 6.08 5.35 4.56 2.53
`1.26 0.57 0.12 0.04 0.01
`0 1.20 2.94 5.00 4.15 3.93 2.77
`1.79 0.59 0.16 0.04 0.01
`0 1.02 2.45 5.17 4.41
`3.70 2.38 1.36 0.62 0.15 0.03 0.01
`0 1.17 2.08 4.63 4.27 3.83 2.63 1.40 0.52 0.15 0.04 0.01
`0 1.30 2.99 4.58 5.02 3.39 2.18 1.63 0.63 0.16 0.03 0.01
`0 1.21 3.05 5.67 5.74 4.06 2.01
`1.29 0.54 0.16 0.04 0.01
`0 1.30 2.82 5.13 4.77 3.87 2.20 1.54 0.63 0.16 0.04 0.01
`0 1.31 2.82 5.78 5.31
`4.46 2.69 1.65 0.55 0.15 0.04 0.01
`0 1.27 2.91
`5.13 4.98 4.51
`2.52
`1.52 0.54 0.13 0.04 0.01
`
`Test
`set
`
`1
`2
`3
`4
`5
`6
`7
`8
`9
`10
`11
`12
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`

`M
`
`i
`
`0
`
`0
`
`0.46
`
`4.34
`9.69
`0.59
`1.25
`- 2.72
`- 5.28
`- 3.46
`4.04
`5.15
`-2.67
`- 6.79
`1.38
`
`-- 0.07
`
`3.74
`8.97
`0.28
`1.03
`- 2.27
`- 6.29
`- 4.35
`3.46
`3.68
`-2.88
`- 7.49
`1.24
`
`2.18
`
`5.90
`11.52
`2.14
`3.24
`- 0.17
`- 3.79
`- 2.15
`5.57
`6.38
`-0.82
`- 5.19
`3.57
`
`Spline
`
`Lagrange
`
`trapezoidal
`
`Linear
`
`15.684 • 0.755
`
`15.601 • 0.745
`
`15.954 i 0.754
`
`Mean =k SD
`
`16.291
`17.125
`15.704
`15.807
`15.187
`14.788
`15.073
`16.243
`16.417
`15.196
`14.553
`15.829
`
`Spline
`
`16.196
`17.013
`15.657
`15.774
`15.259
`14.631
`14.933
`16.152
`16.187
`15.162
`14.443
`15.806
`
`Lagrange
`
`16.533
`17.412
`15.947
`16.118
`15.587
`15.022
`15.277
`16.482
`16.608
`15.485
`14.802
`16.170
`
`trapezoidal
`
`Linear
`
`12
`11
`10
`9
`8
`7
`6
`5
`4
`3
`2
`1
`
`set
`Test
`
`Percent output error
`
`Cumulative area (~g-hr/ml)
`
`Table VI. Cumulative Areas Calculated by the Three Numerical Methods
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`
`V. Cumulative areas and the corresponding output errors were calculated
`using the three empirical procedures identical to those of Example 3. 2
`Two important observations are found in the results given in Table VI.
`First, almost without exception, areas calculated by the spline method were
`higher than those of the Lagrange method and lower than those of the
`trapezoidal method. Second, the output errors of each data set were similar
`in magnitude and in sign regardless of the algorithm, and the average
`cumulative areas were comparable to those calculated from error-free Cp
`values (Table IV).
`With regard to the first observation, an inspection of the results shows
`that, for a given set of data, output errors by each of the three methods are
`either all positive or all negative. Among those with negative errors,
`invariably the smallest deviations were produced by the trapezoidal
`method. Since the loci of points are predominantly concave upward, posi-
`tive biases associated with the trapezoid method (+1.7%) tend to
`compensate best the negative effect of input errors. The reason that
`Lagrange interpolations consistently produced the smallest deviations
`among those with positive errors may be in part related to the negative bias
`associated with the Lagrange method (-0.6%) in the present example.
`With regard to the second observation, simulated errors in each data
`set were not completely balanced. As a result, a given set would be
`composed of either more positively deviated values and fewer negatively
`deviated values or vice versa. These errors would inevitably contribute to
`the output errors, regardless of the algorithm used. It is also obvious that
`the exact allocation of the random errors would also affect the actual
`output values. However, with increasing number of data set, the mean
`areas and output errors should converge, in a stochastic manner, to cor-
`responding limiting values that are characteristic of the algorithm. The
`magnitude of standard deviation, on the other hand, should reflect the
`scattering of data only. Comparing Tables IV and VI, these appear to be
`the case,
`Monitoring of the interpolated curves revealed no unusual oscillations
`in any of the 12 data sets. An examination of such behavior is given in the
`next example.
`Example 5. Table VII is based on the data tabulated in Table III with
`three modifications. First, the data point at 9 hr is deleted as might occur
`when a sample is missed. Second, the data are rounded off to two decimal
`places only. Third, the data point at 4 hr is given a positive deviation (1.59
`instead of 1.47 ~g/ml). Columns 3-5 include the linear trapezoidal, log
`
`2Two Fortran subroutines implementing the Lagrange and spline methods are available on
`request.
`
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`

`A Comparison of Numerical Integrating Algorithms
`
`95
`
`Table VII. Summary of Example 5 a
`
`t (hr) Cp (~xg/ml)
`
`Linear
`trapezoidal
`
`Log
`trapezoidal
`
`Spline
`
`Combined
`spline
`and log
`trapezoidal
`
`0.1324(-1.2)
`0.1324(-1.2)
`(8.2)
`0.1450
`(8.2)
`0.25 1.16(-0.1) 0.1450
`0.4801
`(0.1)
`0.4801
`(0.1)
`0.4557(-4.9)
`(0.6)
`0.5 2.71(-0.0)
`0.4825
`2.0634 (-0.8)
`2.0634 (-0.8)
`1.9072 (-8.3)
`1
`5.20
`(0.0)
`1.9750 (-5.0)
`2.6302
`(0.2)
`2.6302
`(0.2)
`2.5395(-3.3)
`1.5
`4.96
`(0,1)2.5400(-3.3)
`2.2083
`(0.2)
`2.2083
`(0.2)
`2.1962 (-0.4)
`2
`3.87
`(0.0) 2,2075
`(0.1)
`3.0227 (-0.4)
`3.0227 (-0.4)
`3.0471
`(0.4)
`3
`2.35 (-0.1) 3.1100
`(2.4)
`1.9339
`(3.0)
`1.9339
`(3.0)
`1.9453
`(3.6)
`4
`1.59
`(7.9)
`1.9700
`(4.9)
`2.0031
`(4.2)
`2.0930
`(8.9)
`2.0031
`(4.2)
`6
`0.58(-0.8) 2.1700 (12.9)
`
`.
`.
`.
`.
`9
`N.S."
`1.2116
`12
`0.04 (9.3) 1.8600 (56.7) 1.2116
`(2.1) 0.7812(-34.2)
`(2.1)
`0.01 (72.4) 0.0866 (29.6) 0.0866 (29.6) 0.0866 (29.6) 0.0866 (29.6)
`16
`Total
`16.5466
`(6.0) 15.5373 (-0.5) 15.4317 (-1.2) 15.7723
`(1.0)
`
`aValues in parentheses are percent errors.
`bNo sample.
`
`o
`
`0.01
`
`,
`
`,
`0
`
`,
`
`,
`2
`
`,
`4
`
`,
`
`,
`IO
`8
`6
`TIME, HOURS
`Fig. 4. Example showing the production of
`spurious oscillation between