################################################################# # Programs Related to Nonlinear Least Squares in FORTRAN77 # # Version 3.5 # ################################################################# Kiyoshi Yamaoka Faculty of Graduate School of Pharmaceutical Sciences Kyoto University, Kyoto 606, Japan [1] Introduction We have developed several programs for curve fitting by non-linear least squares. ALL programs are written in Microsoft BASIC(1-7). Unfortunately, the syntax of the BASIC on personal computers is different from those on UNIX machines and mainframe computers. Therefore, these programs were translated from BASIC to FORTRAN. It was confirmed all programs run on IBM compatible machines (i.e. on Windows). [2] Comments on Program Files This diskette (3.5') for the standard IBM-PC (720K) contains one document file and six programs written in FORTRAN. MULTI.TXT: this file SULTIF.FOR:simulation program using direct definition of model equations MULTIF.FOR:curve fitting program using direct definition of model equations SRUNGEF.FOR:simulation program using ordinary differential equations MRUNGEF.FOR:curve fitting program using ordinary differential equations SFILTF.FOR:simulation program using fast inverse Laplace transform (FILT) MFILTF.FOR:curve fitting program fast inverse Laplace transform The first letters S and M in file name mean simulation and curve fitting, respectively. The files, MULTIF.FOR, MRUNGEF.FOR and MFILTF.FOR correspond to the programs, MULTI, MULTI(RUNGE), and MULTI(FILT), respectively. [3] Definitions of Pharmacokinetic Models Let's consider the following pharmacokinetic models as a example. Cp=D/Vd exp(-ket) (1) CP=FDka/Vd/(ka-ke) (exp(-ket)-exp(-kat))/(ka-ke) (2) where Cp:plasma concentration, D:dose, Vd:volume of distribution, ke:elimination rate constant, F:absorption ratio, ka:absorption, rate constant, t:time. Eq.(1) is one-compartment model following rapid intravenous administration, and Eq.(2) is one-compartment model with first-order absorption. The parameters Vd and ke are assumed to be common in Eqs.(1) and (2). Eqs.(1) and (2) mean that two time courses of the same drug are measured following rapid intravenous and oral (or intra-muscular) administration. These models can be defined in the subroutine FFUNC in SULTIF.FOR or MULTIF.FOR as follows. SUBROUTINE FFUNC(J,P,T,CP) DIMENSION P(20) GOTO(100,200,300,400,500)J 100 CP=P(1)/R(1)*EXP(-P(2)*T) RETURN 200 CP=P(1)*P(3)*P(4)/(P(3)-P(2))*(EXP(-P(2)*T)-EXP(-P(3)*T)) RETURN 300 CONTINUE RETURN 400 CONTINUE RETURN 500 CONTINUE RETURN END where P(1):Vd, P(2):ke, P(3):ka, P(4):F and R(1):D. It should be noted that P(1), P(2).. are parameters to estimate, R(1),R(2).. are constans. Lines 100 and 200 are the equations defined by the user. We construct the differential equations corresponding to the models represented by Eqs.(1) and (2). dC1/dt = -ke C1 dC2/dt = -ke C2 + DFka/Vd exp(-kat) and the initial conditions are C1(0)=D/Vd C2(0)=0 These differential equation can commonly be defined in the subroutine DIFF of SRUNGEF.FOR and MRUNGEF.FOR. SUBROUTINE DIFF(DC,C,T,P) DIMENSION C0(15),DC(15),C(15),P(20) COMMON NN,ND,DT,TT(2000),CC(2000,15),R(20) DC(1)=-P(2)*C(1) DC(2)=-P(2)*C(2)+R(1)P(3)*P(4)/P(1)*EXP(-P(4)*T) END Initial conditions can be defined in the subroutine INI of SRUNGE.FOR and MRUNGE.FOR. SUBROUTINE INI(C0,P) DIMENSION C0(15),P(20) COMMON NN,ND,DT,TT(2000),CC(2000,15),R(20) C0(1)=R(1)/P(1) C0(2)=0. END where DC(i) means the differential of C(i) with respect to t and C0(i) means the initial value. P(1):Vd, P(2):ke, P(3):F, P(4):ka, R(1):D. Transforming Eqs.(1) and (2) into the Laplace domain, we get the following equations. CC1(s)=D/Vd/(s+ke) (3) CC2(s)=FDka/Vd/(s+ke)/(s+ka) (4) where CC1(s) and CC2(s) are the Laplace transforms of C1(t) and C2(t), respectively, s is the Laplace variable with respect to t. It should be noted that the parameters to estimate are real, whereas CC1(s), CC2(s) and s are complex. These equations can be defined in the subroutine FFUNC of SFILTF.FOR and MFILTF.FOR as follows. SUBROUTINE FFUNC(J,T,P,S,CCP) DIMENSION P(20) COMMON R(20) COMPLEX S,CCP GOTO(100,200,300,400,500)J 100 CCP=P(1)/(S+P(2)) RETURN 200 CCP= P(3)*P(4)*R(1)/P(1)/(S+P(2))/(S+P(4)) RETURN 300 CONTINUE RETURN 400 CONTINUE RETURN 500 CONTINUE RETURN END The new version of MFILTF.FOR becomes more compact and several times faster than the old version. [4] Example Runs (SULTI and MULTI) The followings are example runs of SULTIF(or SRUNGEF and SFILTF). SULTIF ########################### # SIMULATION BY EQUATIONS # ########################### NUMBER OF EQUATIONS ? 2 NUMBER OF PARAMETERS ? 4 P( 1) ? 100 P( 2) ? 0.5 P( 3) ? 2.0 P( 4) ? 1 INITIAL, AND FINAL TIME ? 0,8 TITLE OF T-AXIS ? TIME(HR) TITLE OF CP-AXIS ? CP(UG/ML) CP(UG/ML) 100.000+1 I I 1 I 75.000+ I 1 I 2 2 I 2 2 50.000+ 1 I 2 1 2 I 1 2 2 I 1 2 25.000+ 1 1 2 I 1 2 2 I 1 2 2 2 I 1 2 2 2 2 2 2 2 .000+2 1 1 2 2 ++---------+---------+---------+---------+---------+ .00 1.60 3.20 4.80 6.40 8.00 TIME(HR) DO YOU CONTINUE(Y/N)? N Stop - Program terminated. The followings are example runs of MULTIF(or MRUNGEF and MFILTF). Table 1 presents two time courses of cefotiam, an antibiotics, following intravenous dose(50mg/kg) and intra-muscular dose(50mg/kg) Table 1 ----------------------------------------- time(min) Cp^iv(ug/ml) Cp^im(ug/ml) ----------------------------------------- 5 102 6.03 10 56.2 9.37 15 24.8 13.4 20 22.0 11.8 40 8.04 10.8 60 3.10 8.97 90 -- 6.31 --------------------------------------- MULTIF ####################################### # MULTI LINES FITTING # ####################################### E)XIT, I)NPUT, R)EAD, W)RITE, S)TOP I NUMBER OF LINES(1-5)? 2 NUMBER OF POINTS(1)? 6 T 1( 1) , CP 1( 1) ? 5,102 T 1( 2) , CP 1( 2) ? 10,56 T 1( 3) , CP 1( 3) ? 15,24.8 T 1( 4) , CP 1( 4) ? 20,22 T 1( 5) , CP 1( 5) ? 40,8.04 T 1( 6) , CP 1( 6) ? 60,3.1 NUMBER OF POINTS ( 2) ? 7 T 2( 1) , CP 2( 1) ? 5,6.03 T 2( 2) , CP 2( 2) ? 10,9.73 T 2( 3) , CP 2( 3) ? 15,13.4 T 2( 4) , CP 2( 4) ? 20,11.8 T 2( 5) , CP 2( 5) ? 40,10.8 T 2( 6) , CP 2( 6) ? 60,8.97 T 2( 7) , CP 2( 7) ? 90,6.31 E)XIT, I)NPUT, R)EAD, W)RITE, S)TOP W FILE NAME TO WRITE ? CEFO.IN E)XIT, I)NPUT, R)EAD, W)RITE, S)TOP E (0)GAUSS-NEWTON METHOD (1)DAMPING GAUSS NEWTON METHOD (2)MODIFIED MARQUARDT METHOD (3)SIMPLEX METHOD WHICH ALGORITHM DO YOU SELECT ? 1 SUBJECT NAME ? TEST NUMBER OF LINES (1-5) ? 2 NUMBER OF PARAMETERS ? 4 NUMBER OF POINTS ( 1) ? 6 WEIGHT OF DATA(0,1,2) ? 0 --- CONSTRAINT ON P(I) --- (1) NO CONSTRAINT (2) P(I)=Q(I)*Q(I) (3) P(I)=B+(A-B)*SIN^2(Q(I)) (4) P(I)=B+(A-B)*(EXP(Q(I))/(1+EXP(Q(I))) WHICH CONSTRAINT DO YOU SELECT(1,2,3,4) ? 1 NUMBER OF CONSTANTS ? 1 R(1) ? 50 NUMBER OF PARAMETERS ? 4 INITIAL P( 1) ? 200.0 INITIAL P( 2) ? 0.1 INITIAL P( 3) ? 0.01 INITIAL P( 4) ? 1.0 INITIAL SS= .11770E+04 LOOP= 1 DAMP= 1 P( 1)= 179.761400 P( 2)= .113317 P( 3)= .010926 P( 4)= .929711 SS= .12938E+03 LOOP= 2 DAMP= 1 P( 1)= 183.394800 P( 2)= .118916 P( 3)= .011166 P( 4)= .915666 SS= .11615E+03 LOOP= 3 DAMP= 1 P( 1)= 184.405700 P( 2)= .119696 P( 3)= .011126 P( 4)= .917513 SS= .11603E+03 **** CEFO.IN **** BY DAMPING GAUSS NEWTON METHOD WEIGHT =1/CP^(0) CONSTRAINT ON P(I): NO CONSTRAINT LOOP = 4 AIC = 69.800030 FILAL P( 1)= 184.466400 S.D.= 12.125880 FILAL P( 2)= .119741 S.D.= .008441 FILAL P( 3)= .011121 S.D.= .006667 FILAL P( 4)= .917819 S.D.= .433933 SS= .11603E+03 DO YOU CONTINUE(Y/N) ? Y ## TIME COURSE 1 ## T 1= 5.000000 ,CP 1 = 101.368300( 102.000000) T 1= 10.000000 ,CP 1 = 55.704090( 56.000000) T 1= 15.000000 ,CP 1 = 30.610620( 24.800000) T 1= 20.000000 ,CP 1 = 16.821200( 22.000000) T 1= 40.000000 ,CP 1 = 1.533900( 8.040000) T 1= 60.000000 ,CP 1 = .139874( 3.100000) MINIMUM,MAXIMUM OF T-AXIS ? 0,60 NAME OF T-AXIS ? TIME NAME OF CP-AXIS ? CP(IV) CP(IV) 184.466+* I I I 138.350+ * I I * I O 92.233+ * I * I * I O 46.117+ ** I ** I O***O I ******* O .000+ * ************************O ++---------+---------+---------+---------+---------+ .00 12.00 24.00 36.00 48.00 60.00 TIME SAVE(Y/N) ? Y FILE NAME ? CEFO1.OUT ## TIME COURSE 2 ## T 2= 5.000000 ,CP 2 = 6.871439( 6.030000) T 2= 10.000000 ,CP 2 = 10.275770( 9.730000) T 2= 15.000000 ,CP 2 = 11.794960( 13.400000) T 2= 20.000000 ,CP 2 = 12.297230( 11.800000) T 2= 40.000000 ,CP 2 = 10.966180( 10.800000) T 2= 60.000000 ,CP 2 = 8.881471( 8.970000) T 2= 90.000000 ,CP 2 = 6.370930( 6.310000) MINIMUM,MAXIMUM OF T-AXIS ? 0,90 NAME OF T-AXIS ? TIME NAME OF CP-AXIS ? CP(IM) CP(IM) 13.400+ O I ***** I *** O ****** I * *O** 10.050+ *O **** I ***O* I * ***** I * ***** 6.700+ *****O I *O I I 3.350+ * I I I .000+* ++---------+---------+---------+---------+---------+ .00 18.00 36.00 54.00 72.00 90.00 TIME SAVE(Y/N) ? Y FILE NAME ? CEFO2.OUT (0) GAUSS-NEWTON METHOD (1) DAMPING GAUSS NEWTON METHOD METHOD (2) MODIFIED MARQUART METHOD (3) SIMPLEX METHOD (-1)END WHICH ALGORITHM DO YOU SELECT (0-3, OR -1) ? -1 Stop - Program terminated. The content of CEFO.IN is ----------------------------- 2 6 5.00000, 102.00000 10.00000, 56.00000 15.00000, 24.80000 20.00000, 22.00000 40.00000, 8.04000 60.00000, 3.10000 7 5.00000, 6.03000 10.00000, 9.73000 15.00000, 13.40000 20.00000, 11.80000 40.00000, 10.80000 60.00000, 8.97000 90.00000, 6.31000 90.00000, 6.31000 ----------------------------- The content of cefo1.out is ---------------------------- 6 5.00000, 102.00000 10.00000, 56.00000 15.00000, 24.80000 20.00000, 22.00000 40.00000, 8.04000 60.00000, 3.10000 50 .00000, 219.97080 1.22449, 184.18580 2.44898, 154.22230 3.67347, 129.13330 4.89796, 108.12580 6.12245, 90.53586 7.34694, 75.80742 8.57143, 63.47502 9.79592, 53.14886 11.02041, 44.50257 12.24490, 37.26286 13.46939, 31.20092 14.69388, 26.12513 15.91837, 21.87507 17.14286, 18.31643 18.36735, 15.33670 19.59184, 12.84171 20.81633, 10.75261 22.04082, 9.00337 23.26531, 7.53870 24.48980, 6.31230 25.71429, 5.28541 26.93878, 4.42557 28.16327, 3.70562 29.38776, 3.10279 30.61225, 2.59802 31.83673, 2.17537 33.06123, 1.82148 34.28571, 1.52516 35.51020, 1.27705 36.73470, 1.06930 37.95918, .89534 39.18367, .74969 40.40816, .62773 41.63265, .52561 42.85714, .44010 44.08163, .36851 45.30612, .30856 46.53061, .25836 47.75510, .21633 48.97959, .18114 50.20408, .15167 51.42857, .12700 52.65306, .10634 53.87755, .08904 55.10204, .07455 56.32653, .06242 57.55102, .05227 58.77551, .04377 60.00000, .03665 P( 1)=, .22730 P( 2)=, .14500 P( 3)=, .48726 P( 4)=, .00470 R( 1)=, 50.00000 AIC= 90.56375 --------------------------------- The content of cefo2.out ---------------------------- 7 5.00000, 6.03000 10.00000, 9.73000 15.00000, 13.40000 20.00000, 11.80000 40.00000, 10.80000 60.00000, 8.97000 90.00000, 6.31000 50 .00000, .00000 1.83673, 2.96389 3.67347, 5.29116 5.51020, 7.10872 7.34694, 8.51832 9.18367, 9.60155 11.02041, 10.42385 12.85714, 11.03766 14.69388, 11.48503 16.53061, 11.79968 18.36735, 12.00863 20.20408, 12.13353 22.04082, 12.19175 23.87755, 12.19720 25.71428, 12.16101 27.55102, 12.09211 29.38775, 11.99766 31.22449, 11.88335 33.06122, 11.75378 34.89796, 11.61260 36.73469, 11.46274 38.57143, 11.30652 40.40816, 11.14582 42.24490, 10.98210 44.08163, 10.81657 45.91837, 10.65013 47.75510, 10.48355 49.59184, 10.31740 51.42857, 10.15214 53.26530, 9.98814 55.10204, 9.82567 56.93877, 9.66495 58.77551, 9.50614 60.61224, 9.34936 62.44898, 9.19471 64.28571, 9.04224 66.12244, 8.89201 67.95918, 8.74404 69.79591, 8.59834 71.63265, 8.45491 73.46938, 8.31375 75.30612, 8.17485 77.14285, 8.03819 78.97959, 7.90376 80.81632, 7.77152 82.65306, 7.64146 84.48979, 7.51353 86.32653, 7.38773 88.16326, 7.26400 90.00000, 7.14234 P( 1)=, .27026 P( 2)=, .12015 P( 3)=, 1.06559 P( 4)=, .00920 R( 1)=, 50.00000 AIC= 69.92988 ---------------------------------- ------------------------------------------------------------------ [5] Fast Inverse Laplace Transform (FILT) In the pharmacokinetic field, the Laplace transform has been frequently used for the linear models. The compartment models described by simultaneous ordinary differential equations are usually solved by the Laplace transform method excepting the most simple models. The dispersion model described by partial differential equations are also solved by the Laplace transform. Though the transformed equation is ordinarily obtained in the routine work, the inverse Laplace transform from the Laplace domain to the time domain is usually difficult. Even if the inverse transform is possible, the solution in the time domain is often given in the form of a complicated integral or an infinite series which is difficult to handle. The fast inverse Laplace transform (FILT) is an algorithm which numerically generates the time course curve from the Laplace-transformed model equation. FILT was first proposed by Hosono in the field of the wave optics [T.Hosono, Radio Science, 16,1015 (1981)]. MULTI(FILT) is a nonlinear regression analysis program where FILT is combined with the least squares program MULTI [K.Yamaoka et al., J.Pharm.Dyn., 4, 879(1981)]. [6] INSTALLING AND RUNNING MULTI(FILT) MULTI(FILT) is written in FORTRAN77 that supports the arithmetic of complex numbers. Therefore, FORTRAN80 on CP/M or the old version of MS-FORTRAN which is limited to the arithmetic of real numbers is improper to use MULTI(FILT). The minimum system requirements for MULTI(FILT) are 1. 320 K bytes random access memories at least. 2. Preferably two disk drives. 3. MS-DOS(PC-DOS) version 2.0 or later. SFILT is a program for computer simulation using FILT algorithm. The details about SFILT is given in the following section. MULTI(FILT) and SFILT include a subroutine for the definition of model equations. When using these programs, the user is supposed to use FORTRAN system diskette in A drive and the program diskette in B drive. [7] EXAMPLE RUN OF MULTI(FILT) Initially, the user must define the Laplace-transformed pharmacokinetic model in the subroutine FFUNC by means of an editor such as WordStar or WordMaster etc. For example, the one-compartment model is given by the following equation. Cp(t) = D/Vd exp(-ket) (5) where Cp(t) is plasma concentration, D is dose, Vd is volume of distribution, ke is elimination rate constant and t is time. The Laplace-transformed equation corresponding to Eq.(1) is given by CCp(s) = D/Vd 1/(s+ke) (6) where CCp(s) is transformed plasma concentration and s is Laplace variable corresponding to t. The defined equation in MEQ1.FOR is as follows. C ===== FUNCTION FOR MULTI(FILT) ====== SUBROUTINE FFUNC(J,T,P,S,CCP) DIMENSION P(20) COMPLEX S,CCP GO TO (100,200,300,400,500) J 100 CCP=P(1)/(S+P(2)) RETURN 200 CONTINUE RETURN 300 CONTINUE RETURN 400 CONTINUE RETURN 500 CONTINUE RETURN END where CCP is CCp(s), P(1) is D/Vd, P(2) is ke and S is Laplace variable. It should be noted that CCP and S are complex variables and P(1) and P(2) are real variables. When another variable is adopted in the definition, the declaration of COMPLEX or REAL at the beginning of the subroutine is necessary. The following is an example to transform MEQ1.FOR to MEQ1.OBJ. FOR1 B:MFILT,B:MFILT; PAS2 The next link of MULTI(FILT) and the execution are done by the following commands, assuming that MFILT.FOR is previously compiled to MFILT.OBJ in the same way as mentioned above. LINK B:MFILT,B:MFILT; B:MFILT The example RUN is as follows. ======================================= MULTI LINES FITTING BY FAST INVERSE LAPLACE TRANSFORM ======================================= (0)GAUSS NEWTON METHOD (1)DAMPING GAUSS NEWTON METHOD (2)MODIFIED MARQUARDT METHOD (3)SIMPLEX METHOD WHICH ALGORITHM DO YOU SELECT ? 1 -- selection of algorithm SUBJECT NAME ? ONE -- data name NUMBER OF LINES (1-5) ? 1 -- number of time courses WEIGHT OF DATA (0,1,2) ? 0 -- WT=CP^0 which consequently means 1 NUMBER OF PARAMETERS ? 2 -- P(1) and P(2) NUMBER OF POINTS ( 1) ? 4 -- in this case, four points T 1( 1) , CP 1( 1) ? 1,10.1 T 1( 2) , CP 1( 2) ? -- time course data 2,5.2 T 1( 3) , CP 1( 3) ? 3,2.3 T 1( 4) , CP 1( 4) ? 4,1.2 --- CONSTRAINT ON P(I) --- (1) NO CONSTRAINT (2) P(I)=Q(I)*Q(I) (3) P(I)=B+(A-B)*SIN^2(Q(I)) (4) P(I)=B+(A-B)*(EXP(Q(I))/(1+EXP(Q(I))) WHICH CONSTRAINT DO YOU SELECT (1,2,3,4) ? 1 -- no constraint is selected INITIAL P( 1) = ? 20 INITIAL P( 2) = ? 0.7 INITIAL SS= .12271E+00 LOOP= 1 DAMP= 1 P( 1)= 20.747670 P( 2)= .710940 SS= .71331E-01 LOOP= 2 DAMP= 1 P( 1)= 20.377990 P( 2)= .701354 SS= .70923E-01 LOOP= 3 DAMP= 1 P( 1)= 20.572140 P( 2)= .706740 SS= .69009E-01 LOOP= 4 DAMP= 1 P( 1)= 20.532370 P( 2)= .705109 SS= .68991E-01 LOOP= 5 DAMP= 1 P( 1)= 20.510900 P( 2)= .704429 SS= .68951E-01 LOOP= 6 DAMP= 3 P( 1)= 20.551640 P( 2)= .705669 SS= .68909E-01 **** ONE **** BY DAMPING GAUSS NEWTON METHOD WEIGHT =1/CP (0) - CONSTRAINT ON P(I) - NO CONSTRAINT LOOP = 7 AIC = -6.700156 -- Akaike's information criterion DP FOR JACOBIAN= .001000 FINAL P( 1)= 20.553290 S.D.= .667971 FINAL P( 2)= .704723 S.D.= .022700 SS= .68904E-01 DO YOU CONTINUE (Y/N) ? Y T 1= 1.000000 ,CP 1 = 10.148260( 10.100000) T 1= 2.000000 ,CP 1 = 5.010694( 5.200000) T 1= 3.000000 ,CP 1 = 2.473990( 2.300000) T 1= 4.000000 ,CP 1 = 1.221586( 1.200000) MINIMUM, AND MAXIMUM OF T-AXIS ? 0,4 MINIMUM, AND MAXIMUM OF CP-AXIS ? 0,20 NAME OF T-AXIS ? TIME NAME OF CP-AXIS ? CONC CONC 20.000+** I * I * I * 15.000+ ** I * I ** I ** 10.000+ O* I *** I *** I *** 5.000+ **O* I ****** I ****O**** I *******O .000+ ++---------+---------+---------+---------+---------+ .00 .80 1.60 2.40 3.20 4.00 TIME DO YOU CONTINUE (Y/N) ? N Stop - Program terminated. The O and * in the plots are the observed and theoretical values, respectively. In this example, the single time course is assumed. When several time courses are available, the model equations may be defined at lines 100, 110, 120, ... in the subroutine FFUNC. [8] EXAMPLE RUN OF SFILT SFILT is a simulation program using FILT algorithm. To RUN SFILT, the Laplce-transformed equations must be defined in the subroutine FFUNC. Let's pick up the following model. Cp = Co exp(-ket) (7) Cp = CokaF/(ka-ke) (exp(-ket) - exp(-kat)) (8) where Co(=D/Vd) is initial concentration, ke is elimination constant, F is extent of availability, ka is absorption rate constant. Eqs(7) and (8) are one-compartment models with intravenous(IV) and oral(PO) administrations, respectively. The definition for this model in subroutine FFUNC is as follows. SUBROUTINE FFUNC(J,T,P,S,CCP) DIMENSION P(20) COMPLEX S,CCP GO TO (100,200,300,400,500) J 100 CCP=P(1)/(S+P(2)) RETURN 200 CONTINUE RETURN 300 CONTINUE RETURN 400 CONTINUE RETURN 500 CONTINUE RETURN END where P(1) is Co, P(2) is F, P(3) is ke and P(4) is ka. The coefficient 10 at line 200 is the factor to adjust the PO concentration to IV concentration. COMPILE, LINK and RUN of SFILT are done in the same way as MULTI(FILT) FOR1 SFILT,SFILT; PAS2 LINK SFILT,SFILT; SFILT The following is the example execution of SFILT. ===== FUNCTION SIMULATION PROGRAM ===== ===== BY FILT ===== NUMBER OF EQUATION ? 2 NUMBER OF PARAMETERS ? 4 P( 1)= 180 P( 2)= 0.918 P( 3)= 7.18 P( 4)= 0.667 MINMUM, AND MAXIMUM OF T-AXIS ? 0,1.5 MINIMUM, AND MAXIMUM OF Y-AXIS ? 0,179 NAME OF X-AXIS ? TIME NAME OF Y-AXIS ? CP CP 179.987+1 I I I 1 134.990+ I 222222 I 1 222 222222 I 22 222222 89.993+ 12 2222222 I 1 2222222 I 2 22222222 I 2 1 44.997+ 1 I 2 11 I 11 I 11111 .000+ 1111111111111111111111111111111111 ++---------+---------+---------+---------+---------+ .00 .30 .60 .90 1.20 1.50 TIME DO YOU CONTINUE (Y/N) ? N Stop - Program terminated. The numbers 1 and 2 in plots correspond to the IV and PO concentrations, respectively. [REFERENCES] 1. Y.Yano, K.Yamaoka and H.Tanaka: A Nonlinear Least Squares Program, MULTI(FILT), Based on Fast Inverse Laplace Transform for Micro- computers, Chem.Pharm.Bull. 37, 1035-1038 (1989). 2. K.Yamaoka, T.Nakagawa and T.Uno: Application of Akaike's Information Criterion (AIC) in the Evaluation of Linar Pharmacokinetic Equations, J.Pharmacok.Biopharm., 6, 165-175 (1978). 3. K.Yamaoka et al.: An Analysis Program MULTI(ELS) Based on Extended Nonlinear Least Squares Method for Microcomputers, J.Pharmacobio- Dyn., 9, 161-173(1986). 4. K.Yamaoka and H.Tanaka: A New Version of MULTI(ELS) for Extended Nonlinear Least Squares, J.Pharmacobio-Dyn., 10, 26-34 (1987). 5. K.Yamaoka and T.Nakagawa: A Nonlinear Least Squares Program Based on Differential Equations, MULTI(RUNGE), for Microcomputers, J.Pharm.Dyn., 6, 595-606 (1983). 6. K.Yamaoka, Y.Tanigawara, T.Nakagawa and T.Uno: A Pharmacokinetic Analysis Program (MULTI) for Microcomputer, J.Pharm.Dyn., 4, 879-885(1981). 7. K.Yamaoka, et al.: A Nonlinear Multiple Regression Program, MULTI2(BAYES), Based on Bayesian Algorithm for Microcomputers, J.Pharmacobio-Dyn., 8, 246-256(1985).