Experimental Designs for Mixtures of Chemicals along Fixed Ratio Rays

Transcription

1 Chemial Mixture Experimental Deign for Mixture of Chemial along Fixed Ratio Ray Stephanie L Meadow, Chri Genning, W an Carter Jr, and Dong-Soon Bae Pfizer In, Groton, Connetiut, USA; Department of Biotatiti, Virginia Commonwealth Univerity, Medial College of Virginia, Rihmond, Virginia, USA; Center for Environmental Toxiology and Tehnology, Department of Environmental and Radiologial Siene, Colorado State Univerity, Fort Collin, Colorado, USA Experimental deign i important when tudying mixture/ombination of hemial The traditional approah for tudying mixture/ombination of multiple hemial involve repone urfae methodology, often upported by fatorial deign Although uh an approah permit the invetigation of both the effet of individual hemial and their interation, the number of deign point needed to tudy the hemial mixture beome prohibitive when the number of ompound inreae Fixed ratio ray deign have been developed to redue the amount of experimental effort when interet an be retrited to a peifi ray We fou on the deign and analyi iue involved in tudying mixture/ombination of ompound along fixed ratio ray of the ompound To obtain the inferene regarding the interation among the ompound, we how that the only data required are thoe along the fixed ratio ray Key word: additivity, interation index, iobologram Environ ealth Perpet (uppl 6): () Determining and haraterizing the nature of interation among omponent of a ombination of drug or hemial i a problem of urrent interet ( i the number of drug/hemial in a mixture) Although aement baed on ingledrug/hemial expoure enable u to aquire fundamental knowledge about individual drug or hemial under arefully ontrolled ondition, they do not reflet realworld expoure Thu, it i often of interet to tudy the effet of expoure to multiple drug/hemial Of ultimate interet in uh tudie i the determination and haraterization of interation among the omponent in a mixture For example, Genning et al () report on a tudy of the nature of the interation involving the mixture of four metal The four metal hoen for the tudy were areni (A), admium (Cd), hromium (Cr), and lead (Pb), whih are among the top ontaminant in ite frequeny ount by the Ageny for Toxi Subtane and Dieae Regitry (ATSDR) Completed Expoure Pathway Site Count Report () In addition, human health rik aement aoiated with expoure to diinfetion byprodut in drinking water i of onern beaue of the widepread expoure of peron who reeive diinfeted water Other example of human expoure to ombination of agent an be found in the treatment of numerou dieae inluding aner, AIDS, diabete, and athma Thee example illutrate the importane of tudying mixture/ombination of drug or hemial Determining departure from additivity for a ombination of drug or hemial i a problem that ha been onidered by many author ( 7) Clai Methodology for Deteting and Charaterizing Departure from Additivity Iobologram The lai method for deteting and haraterizing departure from additivity between ombination of drug or hemial i the iobologram The iobologram, introdued a a graphial tool by Fraer (8,9), i a plot of a ontour of ontant repone of the doe repone urfae aoiated with the ombination uperimpoed on a plot of the ame ontour under the aumption of additivity It ue wa extended by Loewe and Muihnek (), Loewe (), and Berenbaum (), and reviewed by Gener (), Weinger (4), and Berenbaum (5) For a two-omponent mixture, the analyi of an iobologram ompare the oberved iobol (eg, ombination doe/onentration aoiated with 5% repone [ 5 ]) to the line of additivity The line of additivity i formed by oining the 5 aoiated with eah of the individual omponent alulated from the doe repone data for the individual omponent Figure preent illutration of poible iobologram for a ombination of two drug/hemial A indiated, if the iobol i below the line of additivity, a ynergim i laimed On the other hand, if the iobol i above the line of additivity, an antagonim i laimed owever, there are hortoming aoiated with the ue of iobologram For intane, the method ued in the ontrution of an iobologram typially doe not take data variability into aount Additionally, beaue it i a graphial method, iobologram effetively are limited to the tudy of ombination of two or three drug or hemial Interation index The interation index, introdued by Berenbaum (), provide a onvenient method to determine and haraterize departure from additivity for a ombination of > or omponent The interation index, II, i defined by X II = CEM X + K+ CEM, X CEM + µ µ µ [] i the number of omponent, X, X,,X are the doe in ombination aoiated with a deired effet, and µ (CEM i ), i =,, i the doe of the ith omponent that, when adminitered alone, produe the ame effet When the interation index, defined in Equation, i equal to, the omponent interat additively When II i greater than, the omponent interat antagonitially; when II i le than, the omponent interat ynergitially Again, it hould be noted that the individual omponent doe repone information i required to alulate the interation index A deribed by Berenbaum (), the interation index i diretly related to the iobologram, ie, when II =, the iobol i oinident with the line of additivity; when II >, the iobol bow above the line of additivity; and when II <, the iobol bow below the line of additivity An advantage of uing the interation index over the iobologram i that the interation index i not limited to ombination/mixture of ut two or three omponent owever, a developed by Berenbaum (), the biologial variability aoiated with the data i not taken into aount by the interation index Statitial model Statitiian frequently ue model of the form Thi artile i part of the monograph Appliation of Tehnology to Chemial Mixture Reearh Addre orrepondene to C Genning, Dept of Biotatiti, Virginia Commonwealth Univerity, Eat Marhall St, B-9-A, Rihmond, VA 98- USA Telephone: (84) Fax: (84) genning@hvuedu Thi reearh wa upported by the US Environmental Protetion Ageny National Center for Environmental Aement (Cininnati, O) ooperative agreement CR-878 Reeived 8 Deember ; aepted September Environmental ealth Perpetive VOLUME SUPPLEMENT 6 DECEMBER 979

2 Chemial Mixture Meadow et al Y = + x + x x + xx x + K+ xx Kx to approximate the relationhip between a repone of interet, Y, and onentration of hemial (x,x,,x ) Carter et al (4) howed that a relationhip exit between the interation index propoed by Berenbaum () and the parameter in a tatitial model aoiated with the interation of the omponent of the ombination Without lo of generality, onider that the ombination/mixture of interet involve two hemial and that the repone i ontinuou Therefore, following the logi of Carter et al (4), for the linear model ae, the relationhip between the repone and the doe or onentration of the omponent in ombination an be expreed a [] µ i the mean repone, E(Y), i the unknown interept, i the unknown lope parameter aoiated with the firt omponent, i the unknown lope parameter aoiated with the eond omponent, i the unknown parameter aoiated with the interation of the two omponent, and x and x are the doe of the repetive hemial From the model defined in Equation, the µ (CEM i ) for the repetive omponent an be derived to be Thu, after algebrai manipulation, the model defined in Equation beome or µ = + x + x + x x x x + + = µ µ µ x x x = + µ ( µ ) µ, ( CEM )= µ µ ( CEM )= µ µ i i i i i = i = = ik i k K i = = k= <k Therefore, it follow that when =, the ombination of omponent and i additive, ie, the iobologram i oinident with the line of additivity and the interation index equal Similarly, when >, the ombination of omponent and i ynergiti, ie, the iobologram bow below the line of additivity and the interation index i < ; and when <, an antagonim i preent, ie, the iobologram bow above the line of additivity and the interation index i > Thi demontrate the algebrai equivalene between the tatitial model and the interation index Genning et al (6) demontrated the experimental onvergene of the tatitial modeling approah and the interation index The number of omponent that an be onidered in the tatitial model an be generalized to, and data variability i appropriately aounted for in the reulting inferene The variou method ued to determine and laify departure from additivity deribed above ue both ingle-ompound data a well a ombination data Conider the ituation in whih the ingle-ompound doe repone data are not available The approah diued in the following etion of thi artile allow one to tet the null hypothei of additivity uing only ombination/mixture data olleted along a fixed ratio ray Thu, ingle-ompound doe repone data are not needed New Methodology Problem with tatitial modeling are aoiated with the ize of the experiment required to generate data to upport the model Fatorial experiment, eg, or, are often onidered When i large uh experiment may not be feaible, o in a ene, thi approah i limited to ombination of relatively few drug or hemial An alternative to the traditional fatorial deign for tudying interation, and the deign to be onidered here, i the ray deign Ray deign, deribed by Martin (7), Mantel (8), Finney (9), Bruden and Vidmar (), and other, are ued to tudy mixture of drug or hemial at a fixed mixing ratio, [a :a : :a ], i = with the total doe, t, varying The fration of the total doe repreented by the ith drug or hemial i a i, and the amount of the ith drug or hemial in the mixture i a i t Thi approah i appealing beaue the dimenionality of the tudy i redued along eah ray, ie, eah ray an be onidered a an individual drug or hemial, with only the total doe varying For example, in a tudy involving drug or hemial, the fitted model baed on a repone urfae approah i a ( + )- dimenional urfae In ontrat, the fitted model baed on a ray deign define a et of two-dimenional doe repone urve What an be tated about departure from additivity in the mixture? Meadow () howed that when mixture data are olleted along a fixed ratio ray, the additivity model redue to a imple linear regreion model In addition, the interation model redue to a higher-order polynomial model Thu, the tet for additivity i equivalent to the tet of the adequay of the imple linear regreion model Conider that the ombination/mixture of interet involve drug or hemial and that the repone of interet i ontinuou The underlying additivity model, ie, the model with no ro-produt term i defined by Y = + x + + x, [] Y i the oberved repone, x i i the doe of the ith drug or hemial, i an unknown parameter aoiated with the interept, and i i an unknown parameter aoiated with the lope of the ith drug or hemial When the mixing ratio are invoked, the doe of the ith drug or hemial i x i = a i t, a i i the mixture fration for the ith drug or hemial and t i the total doe A a reult, the additivity model beome Y= + at+ at+ K+ at = + a + a + K+ a t, = + t = a i i i = [4] For onveniene, we aume the experimental region along the fixed ratio ray in term of total doe i tranformed to the region t Thu, under additivity, the doe repone relationhip along the ray an be deribed with a imple linear regreion on total doe It alo follow that when the lope of the regreion line for total doe i a + a + + a, the interation index, defined in Equation equal Thi would ugget that ingle-omponent doe repone data would Doe hemial Doe hemial Figure Illutration of iobologram for a ombination of two drug/hemial The dahed line i the line of additivity When the iobol bow below the line of additivity, a ynergim i laimed; when the iobol bow above the line of additivity, an antagonim i demontrated 98 VOLUME SUPPLEMENT 6 DECEMBER Environmental ealth Perpetive

3 Chemial Mixture Mixture of hemial along fixed ratio ray be required In the abene of ingle-hemial data, the lope i for eah drug/hemial alone i unknown, o that the hypothei : = i a i annot be diretly teted i = owever, onider the following model: Y = + x + x x Notie that thi model i the model that would be upported by a fatorial experiment, and the i, ik, term are oeffiient aoiated with the variou two-fator, threefator, and higher-order interation The ray deign will not upport thi model; however, invoking the mixing ratio aoiated with the ray deign reult in Y = + a i i t aia i t + + aa a i = = k = <k [5] It follow that interation among pair of hemial omponent are aoiated with eond-degree term, interation among three hemial are aoiated with third degree term, and o on Of ultimate interet i the determination of departure from additivity among a partiular ombination/mixture of drug or hemial When omparing the model under additivity to the interation model, defined in Equation 4 and 5, repetively, evidene of urvature indiate departure from additivity, ie, there i interation among the ompound if at leat one i, i =,, Thu, any polynomial lak of fit aoiated with the additive model Y = + t would be aoiated with a lak of additivity Meadow () howed that the tet tatiti for the null hypothei of additivity, : =, i given by M i = i = = i k ik t K aa Ka K t = + t + t + t + + i = + i = = k= <k + K+ i i i i i = = xx x K xx Kx R F = ( θ ) θ ik i k + K+ t [6] Experimental Deign Experimental deign impliation for tudying a omponent mixture inlude the following: Plae a minimum of + point on the ray of interet to maximize the power of the tet for lak of fit of the additivity model, ie, Y = + t Repliate the experiment at thee point to make the lak of fit tet poible When the repone variable i ontinuou and the method of leat quare ha been ued to etimate the model parameter, Meadow () howed that we an inorporate the tatitial reult of Jone and Mithell () to determine value of total doe that maximize the deign ability to detet lak of fit or departure from additivity The overall lak of fit anwer the quetion of whether there i a departure from additivity Reetion of hi hypothei that imultaneouly tet that the interation parameter are equal to zero, ie, : = = = =, implie that interation i preent among the hemial globally Thu, if the overall tet for additivity i reeted, tet of the form : = : uing ohberg () orretion for multiple teting, an be ued to anwer the quetion of whether a -fator interation exit If uh an interation i deteted, reall that i i, i = = i < i k ik i = = k= = aa = aa a, et ere, interet will be foued on whih of the -fator interation are preent Thi an be determined by performing additional ray experiment at the ame ratio a were preent on the original ray Illutration The methodology introdued in thi artile i illutrated with ytotoxiity data obtained from aeing interation among A, Cd, Cr, and Pb in human keratinoyte The experimental data were obtained from R Yang and olleague at Colorado State Univerity (Fort Collin, Colorado) The end point of interet i the perent viability of treated NEK (normal human epidermal keratinoyte) ell uing the MTT (-[4,5-dimethylthiazol--yl]-,5-diphenyltetrazolium bromide) aay The mixture point of interet for A, Cr, Cd, and Pb ontained the etimated doe/onentration aoiated with 5% lethality (LD 5 ) of 77 µm, 49 µm, 6 µm, and µm, repetively Thi X olution wa erially diluted at a : ratio to get,, 7,, 4, and 4 dilution group Double deionized water wa ued a the vehile ontrol in all ae After expoure to individual metal or metal expoure, ell were re-fed with freh metal-free KGM medium and inubated for day prior to viability analyi by the MTT aay Detail of the experimental protool and method are deribed ele (,4) and are not inluded here The ummary tatiti for the LD 5 mixture data preented in Table are linearized ytotoxiity repone data from Genning et al () The nonlinear additivity model eleted for fitting the ingle-ompound data by Genning et al () wa baed on a Gompertz funtion the mean viability wa modeled a µ= α + γ 4 exp exp + ixi i = From thi model, α i the parameter aoiated with the minimum mean repone, and γ i the range of mean repone value Therefore, for thi example, it i reaonable to aume that a tranformation on the repone, onditioning on the value of α = 876 and γ = 9 obtained by Genning et al (), will indue linearity in the additivity model A a reult, the additivity model beome µ α log log = + x + x + x + x 4 4 Table Summary tatiti for the LD 5 mixture data baed on linearized ytotoxiity repone data from Genning et al a NEK ell Mixture dilution Total doe b (µm) Sample mean Sample variane Sample ize a Data from Genning et al () b LD 5 mixing ratio (77 µm, 49 µm, 6 µm, and µm) for A, Cr, Cd, and Pb, repetively Overall, four data value are miing beaue of tranformation on the repone Environmental ealth Perpetive VOLUME SUPPLEMENT 6 DECEMBER 98

4 Chemial Mixture Meadow et al Table Etimated model parameter for the additivity model given in Equation 7 and the interation model given in Equation 8 Parameter Etimate SE p-value Additivity model 6 < (t) 4 < SS RES = 84, df RES = 66 Interation model 96 < (t) 6 9 < (t ) (t ) (t 4 ) SS RES = 49, df RES = 6 Abbreviation: df RES, degree of freedom aoiated with SS RES ; SS RES, reidual um of quare A hown in New Methodology, beaue the mixture data were olleted along a fixed ratio ray, the additivity model an be rewritten a [7] Additionally, the interation model along the ame fixed ratio ray beome µ µ m, add m, interation y α = log log = + t y α = log log 4 4 = + t + t + t + t [8] Therefore, onditioning on the value of α = 876 and γ = 9, the tranformation y 876 log log 9 on the oberved repone for the mixture data wa performed The additivity model given in Equation 7 and the interation model given in Equation 8 were fit to the mixture data uing the method of leat quare The GLM (general linear model) proedure of SAS (5) wa ued to etimate the unknown parameter in Equation 7 and 8 Parameter etimate and their p value are provided in Table Figure preent the fitted onentration effet urve under additivity for total onentration of the four metal along the ray aoiated with the LD 5 mixing ratio Aterik () indiate the oberved tranformed repone at the even dilution point From thi figure, there i ome quetion a to whether the data fall along the line of additivity In omparion, Figure preent the oberved mixture data and the fitted interation (higher-order polynomial) model Dot ( ) indiate the deign loation of the total doe value eleted by the Λ -optimal deign, whih are preented by total doe of, 6, 594, 7, and 87 µm Notie that the value eleted a the Λ -optimal deign are ymmetrially pread throughout the total doe region, a the maority of the point ued in the urrent tudy are direted toward the lower total doe region An enlarged verion of the lower total doe region of the plot of the fitted interation model i preented in Figure 4 The tet tatiti for the null hypothei of additivity, i given by R F = ( θ ) θ ( SSRe,redued SS ) Re, full = Table preent the overall tet for departure from additivity given in Equation 6 Baed on thi tet, we reet the null hypothei of additivity (p value < ) and onlude that at leat one of the -fator interation exit, =, 4 Beaue the overall tet for additivity i reeted, it i of interet to determine whether two-, three-, or four-fator interation are preent Therefore, we want to tet the following hypothee uing ohberg orretion () for multiple teting: 4 = ( ) = 7 99 :, = A A : = v : B B : = v : C 4 C 4 : = v : Tranformed viability Figure Oberved repone and fitted urve under the additivity model for the fit of the mixture data uing the NEK ell Tranformed viability Figure Oberved and predited repone for the higher-order polynomial model for the fit of the mixture data uing the NEK ell Dot ( ) indiate the deign loation of the total doe value eleted by the Λ -optimal deign Tranformed viability Figure 4 Oberved and predited repone for the higher-order polynomial model for the fit of the mixture data uing the NEK ell (enlarged) Table Tet reult for teting the hypothei of additivity, a well a the hypothee that the -fator interation do not exit, =,,4 ypothei F p-value Overall tet for additivity : = = 4 = 799 (, 6) < Individual tet A : = 95 (, 65) < B : = 654 (, 64) < C : 4 = 49 (, 6) Uing ohberg orretion () for multiple omparion, thee tet are aoiated with a ignifiant -fator interation uing an overall 5% tet Signifiant with an overall ignifiane level of % 98 VOLUME SUPPLEMENT 6 DECEMBER Environmental ealth Perpetive

5 Chemial Mixture Mixture of hemial along fixed ratio ray Table 4 Ratio (%) of ompound to be ued for the four additional experiment, whih are baed on the LD 5 mixing ratio (77 µm, 49 µm, 6 µm, and µm) A Cr Cd Pb Total doe Four-fator ombination Original experiment µm Three-fator ombination Experiment # µm Experiment # µm Experiment # µm Experiment # µm, indiate the metal i not ued in the mixture experiment Table alo preent the ingle-parameter tet aoiated with eah of the -fator interation, =, 4 Uing ohberg () orretion for multiple omparion, all three parameter are ignifiantly different from zero when the overall ignifiane level i et at 5% owever, if we onider the ae an overall ignifiane level of % i ued, only the two mallet p value are ignifiant uing ohberg orretion () Therefore, we onlude that a three-fator interation exit, implying that the two-fator interation are not ontant Now it i of interet to determine whih three metal are interating with one another Thi an be aomplihed by performing 4 = 4 additional experiment at the ame ratio of metal that were ued in the original ray Table 4 give the ratio of ompound along with the orreponding total doe to be ued for the four additional experiment Thi approah limit the inferene of the original experiment, a well a the four additional experiment, to be made about the partiular mixing ratio ued in the experiment Conluion It wa hown that the lai methodology ued in evaluating an interation require ingledrug/hemial data In New Methodology it wa hown that the evaluation of interation ould be aomplihed with a ray deign that did not generate ingle-drug and inglehemial data When a ray deign i ued, departure from additivity i aoiated with higher-order polynomial term in a linear model Additivity, or abene of interation, i deribed by a imple linear model in term of total doe A a reult, we have hown that we an obtain information about departure from additivity from data olleted along a fixed ratio ray Thi reult i important in that it permit a redution in the total experimental effort for tudying a ombination when ompared with that aoiated with a traditional fatorial deign Additionally, by inorporating the approah taken by Jone and Mithell (), we have preented methodology for determining optimal level along the fixed ratio ray (total doe) to be onidered in the experiment for deteting model inadequay REFERENCES AND NOTES Genning C, Carter W Jr, Campain J, Bae D-S, Yang RS Statitial analyi of interative ytotoxiity in human epidermal keratinoyte following expoure to a mixture of four metal J Agri Biol Environ Stat 7():58 7 () ATSDR 997 CERCLA Priority Lit of azardou Subtane That Will Be the Subet of Toxiologial Profile & Support Doument Atlanta, GA:Ageny for Toxi Subtane and Dieae Regitry, 997 Gener PK, Cabana BE A tudy of the interation of the hypnoti effet and of the toxi effet of hloral hydrate and ethanol J Pharmaol Exp Ther 74:47 59 (97) 4 Carter W Jr, Genning C, Staniwali J, Campbell E, White K A tatitial approah to the ontrution and analyi of iobologram J Am Coll Toxiol 7:96 97 (988) 5 Kelly C, Rie J Monotone moothing with appliation to doe-repone urve and the aement of ynergim Biometri 46:7 85 (99) 6 Genning C, Carter W Jr Utilizing onentration-repone data from individual omponent to detet tatitially ignifiant departure from additivity in hemial mixture Biometri 5:64 77 (995) 7 Dawon KS, Carter W Jr, Genning C A tatitial tet for deteting and haraterizing departure from additivity in drug/hemial ombination J Agri Biol Environ Stat 5:4 59 () 8 Fraer TR An experimental reearh on the antagonim between the ation of phyotigma and atropia Pro R So Edinb 7:56 5 (87 87) 9 Fraer TR The antagonim between the ation of ative ubtane BMJ ii: (87) Loewe S, Muihnek Kombination-Wirkungen : Mitteilung: hilfmittle der fragetel-lung Arh Exp Pathol Pharmaol 4: 6 (96) Loewe S The problem of ynergim and antagonim of ombined drug Arzneim Forh :85 9 (95) Berenbaum MC Criteria for analyzing interation between biologially ative agent J Am Stat Ao 78:9 98 (98) Gener PK The iobolographi method applied to drug interation In: Drug Interation (Morelli PL, Garattini S, Cohen S, ed) New York:Raven Pre, 974; Weinger WD Approahe to the tudy of drug interation in behavioral pharmaology Neuroi Behav Rev : (976) 5 Berenbaum MC What i ynergy? Pharmaol Rev 4:9 4 (989) 6 Genning C, Carter W Jr, Campbell, Staniwali JG, Martin TJ, Martin BR, White KL Jr Iobolographi haraterization of drug interation inorporating biologial variability J Pharmaol Exp Ther 5:8 7 (99) 7 Martin JT The problem of the evaluation of rotenoneontaining plant VI: The toxiity of -elliptone and of poion applied ointly, with further obervation on the retonone equivalent method of aeing the toxiity of derri root Ann Appl Biol 9:69 8 (94) 8 Mantel N An experimental deign in ombination hemotherapy Ann NY Aad Si 76:99 (958) 9 Finney DJ Probit Analyi London:Cambridge Univerity Pre, 964,97 Bruden M, Vidmar T Optimal fatorial and -ray deign for drug interation experiment with dihotomou repone Commun Stat 8(8): (989) Meadow SL Deign Optimality and Sample Size Planning for Mixture Studie [PhD Thei] Rihmond, VA:Virginia Commonwealth Univerity, Jone ER, Mithell TJ Deign riteria for deteting model inadequay Biometrika 65:54 55 (978) ohberg Y A harper Bonferroni proedure for multiple tet of ignifiane Biometrika 75:8 8 (988) 4 Yang RS, ed Toxiology of Chemial Mixture: Cae Studie, Mehanim, and Novel Approahe San Diego, CA:Aademi Pre, SAS SAS/STAT (GLM proedure), Verion 8 Cary, NC:SAS Intitute, In Environmental ealth Perpetive VOLUME SUPPLEMENT 6 DECEMBER 98