[CANCER RESEARCH 58, 5749-5761. December 15. 1998]

Modeling of the Time-Dependency of in Vitro Drug Cytotoxicity and Resistance1

Laurence M. Levasseur, Harry K. Slocum, Youcef M. Rustum, and William R. Greco2

Departments of Biomathemalics [L M. L. W. R. G.¡and Expérimental Therapeutics [L M. L, H. K. S., Y. M. R.. W. R. G.¡,Raswell Park Cancer Institute. Buffala. New York.1426}

ABSTRACT

For potential clinical extrapolation of in vitrofindings, it is of interest torelate the measured effect of an anticancer agent to concentration andexposure time. The Hill model (A. V. Hill, J. Physiol., 40: iv-vii, 1910) iscommonly used to describe pharmacodynamic (PD) effects, includingdrug-induced growth inhibition of cancer cells in vitro.The lt\" x T = k

relationship, in which I( \ is the concentration of agent required to reducecell growth by .V'<. / is the exposure time, and n and k are estimable

parameters, was first applied to bacterial disinfectant action and then wassuccessfully used to model anticancer drug potency as a function ofexposure time (D. J. Adams, Cancer Res., 49: 6615-6620,1989). Our goalwas to create a new global PD modeling paradigm to facilitate the quantitative assessment of the growth-inhibitory effect of anticancer agents asa function of concentration and exposure time. Wild-type human ovarianA2780 and ileocecal HCT-8 carcinoma cells and sublines that were resistant to cisplatin (A2780/CP3), doxorubicin (A2780/DX5B), and raltitrexed(RTX) (HCT-8/DW2) were exposed to various anticancer agents, cisplatin,doxorubicin, paclitaxel, trimetrexate, RTX, methotrexate, and AG2034,for periods ranging from 1 to 96 h. Cell growth inhibition was measuredwith the sulforhodamine B protein dye assay. Patterns of time-dependencyof drug potency, slope of the concentration-effect curves, and relativedegree of resistance were characterized. Empirical mathematical expressions were built into a global concentration-time-effect model. The globalPD model was then fit to the concentration-time-effect data with itera-tively reweighted nonlinear regression. Under specific treatment conditions, the examination of the slope and the shape of the concentration-effect curves revealed a large heterogeneity in drug response, e.g., shallowconcentration-effect curve or double or triple Hill "roller coaster" concentration-effect curve. These patterns, which were observed at intermediate exposure times in parental and resistant cells for paclitaxel andtrimetrexate or only in resistant HCT-8/DW2 cells for RTX, methotrexate,and AG2034, revealed mechanistic insights for the former cases butpossible methodological artifacts for the latter cases. The comprehensivePD modeling of the cytotoxic effect of anticancer agents showed that it waspossible to modulate drug effect, response heterogeneity, and drug resistance by altering the time of exposure to the agents. This approach will beuseful for: (a) describing complex concentration-time-effect surfaces; (b)refining biological interpretations of data; (c) providing insights on mechanisms of drug action and resistance; and (d) generating leads for clinicaluse of anticancer drugs.

INTRODUCTION

The AUC3 (1) has been successfully used as a predictor of clinical

anticancer drug efficacy (2) and drug toxicity (3-7). The effects of

both drug concentration and exposure time should be well characterized in vitro to guide the further development of a drug towardoptimal therapeutic effect. There have been many in vitro studiespublished for which concentration-effect curves are simply displayed

Received 3/20/98; accepted 10/15/98.The costs of publication of this article were defrayed in part by the payment of page

charges. This article must therefore be hereby marked advertisement in accordance with18 U.S.C. Section 1734 solely to indicate this fact.

1This work was supported by NIH Grants RR10742. CA65761. and CAI6056.2 To whom requests for reprints should be addressed. Phone: (716)845-8641; Fax:

(716)845-8467: E-mail: rosgreco@acsu.buffalo.edu.3 The abbreviations used are: AUC. area under the concentration-time curve; PD,

pharmacodynamic; DDP. cisplatin; PTX. paclitaxel; MTX, methotrexate; DOX, doxorubicin; TMQ. trimetrexate; RTX. raltitrexed: MDR. multidrug resistance; Pgp. P-glyco-protein; FPGS. folylpolyglutamate synthetase; SRB. sulforhodamine B: PK. pharmaco-kinetic.

for different exposure times without any mathematical modeling (8,9), and there have been fewer in vitro studies that model the effect ofexposure time on the concentrations of drug required for specificeffect levels (10, 11). In vitro studies that simultaneously model bothdrug concentration and exposure time are rare (12). In this study, weintroduce a comprehensive modeling paradigm for quantitatively assessing the contributions to m vitro growth inhibition of drug concentration and exposure time. We provide representative examples ofconcentration-time-effect modeling for seven anticancer compoundsagainst both parental and drug-resistant cell lines. The new paradigm

offers the opportunity for gaining quantitative insights into the mechanisms of drug action, drug resistance, and drug response heterogeneity that are hidden from the current approaches. In addition, possible methodological artifacts can be revealed that will refine thebiological interpretations of the data.

A common choice for a concentration-effect model is the Hillequation (13), also called the sigmoid-£max model (14) or logistic

model (15). In Fig. IA, the thin black curve illustrates the Hill model.The curve was simulated from Eq. 2a in Table 1 with the set ofparameters listed in the legend of Fig. 1. The Eccmparameter is thecontrol effect at zero drug concentration, and B is the backgroundeffect at infinite drug concentration. The inflection point on the curveis the IC50, a measure of the drug potency, which is the drug concentration (or dose) inducing a 50% decrease in the maximal effect(£con- B). The slope parameter of the curve is y. Note that when y

has a negative sign, the curve falls with rising concentrations, indicating an inhibitory effect of the drug. The larger the absolute valueof y is, the steeper the curve is, indicating a highly homogeneous drugresponse. The gray curve represents the inverted tolerance distributionof the Hill model. It was simulated from the first-order derivative of

the Hill function versus the natural logarithm of concentration (Eq. 3ain Table 1). Note that, to enhance the visibility in Fig. 1, this simulateddistribution function has been raised up to the background level. Themaximum of the function is at the IC50, and its width is directlyrelated to the value of y. As drug response heterogeneity increases, thegray bell-shaped curve becomes wider, and the black concentration-

effect curve becomes shallower.The black "roller coaster" curve in Fig. IB was constructed by

simulating Eq. 2b in Table 1, which is a combination of two Hillmodels, which we will refer to as the "double Hill model." Again, the

thick gray curve is the tolerance distribution function (Eq. 3b in Table1), which reveals "two populations of cells" with different sensitivities

(IC50a and IC50h) and heterogeneities of drug response (ya and yh).Because the parameter Econb from Eq. 2b in Table 1 stands for theplateau level observed in the middle of the concentration-effect profile(Fig. Iß),(£con—£conb)represents the size of the first population ofcells that can be affected by the drug, and (£c()nh —B) represents the

size of the second population that can be affected.Anticancer drug potency has been related to exposure time with the

PD model ICX" X T = k (10), in which ICX is the concentration of

agent required to reduce cell growth by X%, T is the exposure time,and n and k are estimable parameters. This principle was first used todescribe bacterial disinfectant action (16). It was observed that ICXwas linearly related to T on log-log coordinates for a given effect

level. The coefficient n, the reciprocal of the slope of the log(ICx)versus log(7") plot, indicates the relative importance of drug concen-

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TIME COURSE OF DRUG CYTOTOXICITY

B

Fig. 1. Schematic representation of a single Hillconcentration-effect curve (A} and a double Hillconcentration-effect curve (ß)simulated from Eqs.

2a and 2h in Table I. Thick gray line, the associatedtolerance distribution function simulated from Eqs.3a and 3b in Table I, subtracted from the background parameter. H. The parameters used for simulating the concentration-effect curve in A were:£con= 100. IC«,= 0.5 /IM, f = -2. and B = 10;the parameters used in ßwere: £con = 100,1C».. = 0.01 MM. -y, = -1.5. £c,m.b= 40.le».,, = 3 UM. yb = -3, and B = 10.

0 0.001 0.01

Drug concentration, f

10 0 0.001 0.01 0.1 1 10 100

Drug concentration, uM

tration and exposure time for the specific effect level. When n > 1,concentration is of greater importance; when n < 1, time is moreimportant: and finally, when n = 1, concentration and time are ofequal importance.

Although differences in the shapes of concentration-effect curvesfor different cell lines, different agents, and different conditions havebeen anecdotal ly noted by many researchers, there have been only afew published studies in which the shapes of individual concentration-effect curves were examined in a systematic manner, with attempts tomake mechanistic inferences based on shapes (17-19). These studiesall focused on log(cell survival) versus linear concentration plots, withunderlying exponential concentration-effect models from the field ofradiation oncology. In contrast, we have accented linear effect versuslog(concentration) plots, with underlying Hill concentration-effectmodels. To our knowledge, our study is the first report on theexposure time-dependency of the slope parameter y.

We report the results of a new modeling approach and investigatethe influence of exposure time on parameters, such as drug potency,response heterogeneity, and drug resistance. The study addressed thefollowing set of questions: (a) is the Hill model a universal model fordescribing drug effect in a population growth inhibition assay, forpairs of parental and resistant cells exposed to various anticanceragents for various exposure durations?; (b) if the simple Hill model isnot appropriate, what is the significance of a more complex concentration-effect model in terms of cellular response or drug action?; (c)what is the pattern of change of drug potency, the slope parameter ofHill concentration-effect curves and degree of resistance with exposure time?; (d) is it possible to modulate drug effect, cellular heterogeneity and drug resistance by altering the duration of exposure to thedrug?; (e) under what conditions can methodological artifacts berevealed, such as the effects of long-term incubation with residualdrug remaining in a culture vessel after a short-term drug incubationand subsequent washing steps, and how can the biological interpretations of the data then be refined?

The modeling approach presented here enables a complete description of the three-dimensional concentration-time-effect surface. Thisnew paradigm can provide insights into mechanisms of drug actionand resistance, suggestions for looking at drug action frugally in moreintimate, sophisticated, and expensive assays, and leads for the optimal clinical use of anticancer agents.

MATERIALS AND METHODS

Chemicals. RPMI 1640 cell culture medium, dialyzed FBS. saline. PBS.versene, and trypsin were obtained from Life Technologies. Inc. (Grand Island.

NY); FBS was from Atlanta Biologicals (Norcross. GA); methanol was fromFisher Scientific (Fair Lawn, NJ); and DDP. PTX (Taxol), and MTX werefrom Sigma Chemical Co. (St. Louis, MO). DOX was provided by AdriaLaboratories (Columbus, OH); TMQ was provided by Parke Davis Pharmaceutical (Ann Arbor, MI); and RTX (Tomudex) was provided by ZenecaPharmaceuticals (Wilmington, DE) and AG2034 by Agouron Pharmaceuticals(La Jolla, CA). DDP was dissolved in saline at 7.5 HIM(20). Stock solution ofPTX at 4 mM was made in methanol, and stock solution of DOX was made at5 mg/ml in PBS. MTX. TMQ, RTX, and AG2034 were dissolved in water at1. I. 5, and 10 mM, respectively. Stock of drugs were kept frozen at -80°C.

Cell Lines. Parental A2780 (21) and resistant A2780/CP3 and A2780/DX5B human ovarian carcinoma cells were maintained in RPMI 1640 cellculture medium supplemented with 10% FBS and 1% glutamine at 200 mM.The A2780/CP3 cells are resistant to DDP due to an elevated cellular glutha-

thione content (22). The A2780/DX5B line was obtained by cloning theDOX-resistant subline A2780/DX5 (23), and it expresses the MDR phenotypedue to a high level of expression of the Pgp P-170. The parental humanileocecal carcinoma cell line HCT-8 was obtained from the American TypeCulture Collection (Rockville. MD). The FPGS-deficient subline HCT-8/DW2

was made resistant to RTX by sequential exposure to escalating doses (24).HCT-8 and HCT-8/DW2 were cultured in RPMI 1640 supplemented with 10%

dialy/.ed FBS and 10 mM sodium pyruvate. All cells were maintained asmonolayer cultures in 25-cnr flasks and incubated at 37°Cin a humidified

atmosphere containing 5% CO2. Cells were passed twice weekly by trypsiniza-

tion. Cultures were tested bimonthly for Mycoplasmu. The cell cycle timeswere found to be 18, 20. and 29 h for the human ovarian carcinoma cellsA2780, A2780/DX5B, and A2780/CP3. respectively, and 16 and 18 h for thehuman ileocecal carcinoma cells HCT-8 and HCT-8/DW2. respectively.

Cell Growth Inhibition Assay. Exponentially growing cells were rinsedwith PBS (A2780 and derivatives) or versene (HCT-8 and derivative),

trypsinized, counted, suspended in complete medium, and seeded in 88 wellsof 96-well microtiter plates at a cellular density of 100 cells/155 /il. Theremaining eight wells were filled with cell-free medium. Because a shorter24-h attachment time was found to result in unacceptable cell losses during the

washing steps, cells were allowed to attach for 48 h. Each experiment included8-14 plates. Drug dilutions were freshly made in RPMI 1640 and seriallydiluted over a 105-107-fold range of concentration. Drug concentrations were

randomly assigned among wells (25). Cells were drug-treated (45 /¿I/wellofdrug-containing medium was added, resulting in a total incubation volume of

200 /il/well) for time periods ranging between 1 and 96 h. Medium wasremoved by turning the plate onto a sterile gauze with gentle tapping. For thestandard assay, plates were washed twice with RPMI 1640 (200 /xl/well, 37°C)

and incubated in complete medium (200 ¿tl/well)up to the end of the experiment. For a small number of experiments, four washes were used, as indicatedwhen appropriate in the text. For each wash step, medium was gently pipettedalong the sides of the wells in each column with an eight-channel multipipette

and then the plate was turned onto a sterile gauze and gently tapped.In experiments performed to characterize the assay (with A2780/DX5B),

it was found that the standard two-wash procedure resulted in an 11% loss

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TIME COURSE OF DRUG CYTOTOX1CITY

Table 1 Equations of the pharmacotlynamic models"

A. Variance model

(1) 4 =

B. Structural model

Single Hill model

<2a) E = ß+ (Eco„-

Double Hill model

(2h) E = B + (£„„-

Triple Hill model

(2c) £= fi + (£„„-

C"+ (£c„».t-

C*

C" C*

C. Tolerance distribution function

Single Hill model

dE(3a) - - = (£„,„din C

Double Hill model

dE

- B)(C- + KV)2

j i rd In C

D. Pattern of change of the parameters of the structural model rersH.vtime

1C50parameter

(4fl) Constant 1CW

(4fc) /C/-Tmodel

/C50 = constant

With dilution artifact

(4c) Sigmoidmodel /CM>= IC,,,.tak + (IC50.mlx- IC5(1.b,ck)

1C»= Ofi

7s

(—)""

\96 - T/

7*-

Slope parameter y(5a) Constant y

(5b) Linear modelWith dilution artifact

(5c) Polynomial modelWith dilution artifact

Plateau parameters Econ.hand Ec,(6a) Constant EC™, £„,„»= constant(6b) Exponential model £„„.>= Eme~K'T

Degree of resistance R(7a) Constant K R = R„

(7b) Linear relation R = fi„+ R,T(7c) Exponential relation R = Ra + R,e":T

y = y«y = yo+ y iTy = y»+ y,(96 - T)y= ya+ yJ + yJ2y = y„+ y,(96 - T) + y2(96 - T)2

" J~E,variance for a set of replicate measured effects: XE. mean response for a set of replicate measured effects; $-, and «¿3.variance parameters:E. measured effect (absorbance

read at wavelength 570 nm); C. drug concentration; T.exposure duration to a drug; dE/dlnC. tolerance distribution function; first derivative of £versus InC;£L.on.control effect observedat zero drug concentration; ß.background effect observed at infinite drug concentration for an inhibitory drug; £conh, plateau level in a double Hill model; upper plateau level in atriple Hill model: £,.<„,c. lower plateau level in a triple Hill model: IC50.concentration of drug resulting in 50*^ inhibition of the maximal effect [£con~ ß]:IC^-.concentration ofdrug resulting in X% inhibition of the maximal effect [Econ - B]: IC50a, concentration of drug resulting in 50% inhibition of the effect [£,.„„- £con.b]for a double or a triple Hillmodel: IC50b. concentration of drug resulting in 50% inhibition of the effect [£conb - fi] for a double Hill model or [£conb - £con.Jfor a triple Hill model; IC5(U..concentrationof drug resulting in 50% inhibition of the effect [£conc.~ B] for a triple Hill model: y, slope parameter of the concentration-effect curve; ya. slope parameter of the upper part of theconcentration-effect curve: yb, slope parameter of the lower pan of a double Hill concentration-effect curve, slope parameter of the median part of a triple Hill concentration-effectcurve; yc, slope parameter of the terminal part of a triple Hill concentration-effect curve; k, exposure constant; n. concentration coefficient; DF. drug dilution factor during artifactuallong-term exposure: IC^ohack-background ICS{)observed at infinite exposure time to the drug; IC50max,extrapolated 1C50at exposure time zero to the drug; 7"50.exposure duration

resulting in 50% decrease of the IC50;j, slope parameter of the time-IC50 curve; yn. y¡.and y2. coefficients of the polynomial relation for the slope of concentration-effect curves;Kp. rate constant of decrease of £t.,,n,b:R. relative degree of resistance: R0. /?,, and R2. coefficients of the linear or exponential relation for the degree of resistance.

of cells, whereas the four-wash procedure was estimated to result in a 29%

cell loss. The volume of the original incubation medium remaining after thetwo standard washing steps was found to be 0.0754 /xl/well. These dilutionfactors of 2650 for two washes and ~7 X 10°(i.e., 26502) for four washes

influence the interpretation of some of our results. Our standard two-wash

procedure was a compromise among several factors: minimization of cellloss, maximization of the dilution of the original incubation medium,convenience, and expense.

Cell growth inhibition was assessed 144 h after cell seeding with the SRBprotein dye assay (26). Briefly, medium was removed from the wells: plateswere fixed with 10% trichloroacetic acid for 1 h, washed with water in a platewasher, and stained with SRB for 15 min. Unbound dye was removed bywashing the plates with 1% acetic acid, and protein-bound dye was extracted

with IO mM Tris base. The absorbance was read at 570 nm in a plate reader.Data were written to a floppy disk for importation into Excel (Version 5.0:

Microsoft) and derandomization.

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TIME COURSE OF DRUG CYTOTOXICITY

Data Analysis. A list of all of the equations used in the elaboration of thefinal global PD models is provided in Table I. along with definitions of thevariables. The variance structure ot the data were assessed by fitting Eq. 1 fromTable I (25) to sets of mean-variance data from replicate measurements toestimate the variance parameter </>,.Concentration-effect models (Eqs. 2a and

2b in Table I ) were fit to data for each individual exposure time, with nonlinearregression, ¡teratively reweighted by the reciprocal of the predicted effectraised to the power <t>t.The pattern of change with time of parameters of Eqs.2a (IC,0 and y) and 2b (IC50a, IC50h, ya, yh, and £con,b)was studied andmodeled with Eqs. 3-6 in Table 1. Suggested empirical mathematical terms forIC50s, ys, and £conh were then substituted, as described in the "Results"

section, in the final structural models Eqs. 2a-2c. A resultant custom global PDmodel was then fit (o all of the concentration-time-effect data at once for eachindividual experiment (704-1232 data points per experiment). The pattern of

change with lime of the relative degree of resistance, defined as the ratio of theIC50 of the resistant cell line divided by the IC5(>of the parental cell line, wasmodeled with Eqs. 7a-7c. All models were fit to data with iteratively re-

weighted nonlinear regression, performed with Proc NLIN under ihe Windowsversion of SAS (Version 6.12: Ref. 27). The models were selected from Table1 on the basis of a graphic examination of concentration-effect profiles,

patterns of change of parameters, goodness of fit, and the precision of theestimâtes of the model parameters. Statistical significance of the parameterestimates was assessed by forming a 95% confidence interval around theestimate and determining if the interval encompassed zero. All graphics wereprepared with Sigmaplot (Version 4.0: SPSS Inc.. Chicago. IL). Software wasrun on Pentium-based microcomputers.

RESULTS

Results presented in this section are from one representative experiment for each of 14 drug/cell line combinations. However, eachexperiment was performed in duplicate, triplicate, or quadruplicatewith comparable results. The Hill model (Eq. 2a in Table 1) was usedto describe the concentration-effect relationship of DDP, DOX, andTMQ for all of the exposure periods investigated, for both wild-type

and resistant cells. For PTX. a double Hill model (Eq. 2b in Table 1)was used for wild-type and resistant cells exposed to the drug for

T s 10 h; for longer times, the data followed the single Hill model.For MTX and AG2034, a double Hill model was used to fit dataobtained from treating HCT-8/DW2 cells for intermediate times (10h £7"£24 h and 6 h s 7£ 10 h, respectively); for shorter and longer

7 and for wild-type cells, the single Hill model was used. Finally, forRTX-induced cytotoxicity, data followed the single Hill model forHCT-8 cells. An unusually high background level (40% of £c()n)wasobserved with HCT-8/DW2 cells for all of the exposure times inves

tigated, which in subsequent repeated experiments (data not shown),when a wider range of concentrations (up to 1 HIM) was studied,appeared to be the plateau of an apparent double Hill pattern.

IC50 and slope (y) of the concentration-effect curves as a functionof time are displayed in Fig. 2. left and middle, for wild-type and

resistant cell lines, respectively. The associated relative degree ofresistance, R, is shown in Fig. 2, right. The curves are the fits ofappropriate models from the set of Eqs. 4-7 in Table 1, to the

parameters IC5(), y. and R. The patterns of change of the parameterscan be identified with the letters "w" and "r" for wild-type and

resistant cells, respectively, and a subscript, when necessary, to indicate a particular component of a complex concentration-effect curve.

Log-log scales were used for ICs,,s to linearize the ICV" X T = k

(Eq. 4b in Table 1) relationship in Fig. 2. Eq. 4b was appropriate todescribe the pattern of change with time of: (a) ICM,for DDP (A2780and A2780/CP3), DOX (A2780 and A2780/DX5B), RTX. MTX, andAG2034 (HCT-8) and (b) IC5(l.afor PTX (A2780 and A2780/DX5B),RTX, MTX, and AG2034 (HCT-8/DW2). In contrast, there was nosignificant time-dependence of ICM,h: a constant model (Eq. 4a in

Table 1) was fit to the parameter estimates. The pattern of change of

IC50 for TMQ exhibited a sigmoid-type decrease with time; drug

potency was level for the first 10 h of exposure, then sharply decreased, and finally plateaued. A sigmoid relation (Eq. 4c) was usedin which .vwas the slope parameter of the time-IC,,, curve and T50was

the exposure time necessary to achieve a 50% drop in IC5().Slope parameters -ysof the concentration-effect curves mostly fol

lowed quadratic polynomial patterns of change (Eq. 5c in Table 1).However, on occasion, the second-order polynomial coefficient was

not significant, and a simple linear relation (Eq. 5b in Table 1) wasused. For drugs inducing a double Hill concentration-effect curve, the

pattern of change of the ratio of the plateau level versus £,,,„,(data notshown) was constant (RTX in HCT-8/DW2), or exponentially de

creased with time (PTX in A2780 and A2780/DX5B, MTX, andAG2034 in HCT-8/DW2). Eqs. 6a and 6b, respectively, in Table 1

were used.To help decide on the best structural form of the concentration-

time-effect model and to determine whether the time-dependent man

ifestations of response heterogeneity (shallow slope and double Hillpattern) were originated by methodological artifacts related to thegrowth inhibition assay, such as an insufficient washing of the drugfrom the wells that would result in a long-term exposure to residualdrug concentration (dilution factor from the two-wash procedure:

2650), drug treatments were explored in a limited number of experiments with our usual 96-well plate assay, but for which the washing

procedure was altered to ensure a thorough drug dilution through fourwashes (dilution factor: 7 X 10''). Information provided by these

experiments, which included PTX against A2780 and TMQ, RTX,MTX, and AG2034 against HCT-8/DW2, were used in the creation of

the final global PD model as presented in detail below.For each individual experiment, terms characterizing the evolution

of IC50s, ys, and £con.hswith T were substituted into Eqs. 2a-2c as

specifically described below, and the resultant global PD modelE = f(C;7) describing the cell growth inhibitory effect as a function

of drug concentration and exposure duration was fit to all of the dataat once. Parameter estimates of the global model are presented inTable 2 with their associated SEs. In Figs. 3 and 4 are displayedfamilies of concentration-effect curves, which show the global modeling of the growth inhibition of the human wild-type and resistant

ovarian carcinoma A2780 cells (Fig. 3) and ileocecal carcinomaHCT-8 cells (Fig. 4) induced by DDP, DOX, and PTX and by TMQ,

RTX. MTX. and AG2034. respectively. Curves, which are slicesthrough the three-dimensional concentration-time-effect surface, weresimulated from the global model with the best-fit parameters from

Table 2. The superimposed symbols are the averages of four replicatemeasurements. Note that the drug concentrations are displayed on alogarithmic scale. Concentration units are shown in the lower rightcorner of each panel. To facilitate comparisons between cell lines anddrugs, a standard 7-log length scale was used for drug concentration,

and effect was expressed as percentage of the control growth. Goodness of fit, as judged by the relative precision of the parameterestimates in Table 2 and by the visual concordance of the fitted globalmodels with the data points in Figs. 3 and 4, was excellent.

Fig. 5 shows the simulated pattern of change of the literal IC50 andIC9„with exposure time, as well as the corresponding degrees ofresistance. fl5(, and /?90, predicted from the global PD model. Note thatthe literal ICSI>and 1C,,,, are the drug concentrations inducing a 50%or 90% decrease in the maximal effect (£c<m- B), regardless of theconcentration-effect model. For the Hill model, the literal IC5(,and the

IC5I, parameter are identical; however, for the double or triple Hillmodel, the literal IC5(>(or 1C,,,,) is a function of all of the modelparameters (£con,£con.h,£c,,n.c-B, IC50.,, ya, DF, IC5(U., and yj andis calculated with the aid of a one-dimensional root-finding numericalprocedure (28). For drugs exhibiting an apparent single Hill concen-

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TIMI-I COL-RSI-; 01 DRUG CYTOTOXICITY

IC50,uM1000

100ADDP M!

1

0.110

B 'DOX °'1

0.01

0.0011000

100101

0.10.01

0.0010.0001

1

PTX

DTMQ

ERTX

FMTX

0.1

0.01

0.001

0.00011000

100101

0.10.01

0.00110

10.1

0.01

0.001100

w

AG203 4

10

1

0.1

0.01

-1

-2

-30

-l

-2

-3O

-l

-2

-3

-4O

-l

-2

-30

-1

-2

-3

-4-1

-2

-3

-40

-1

-2

-3

Slope y

f

Degree of resistance100

10

1100

10

l . i \ 1 li. 1.1. i. i.i.i

10000

1000

1001

0.1

10000010000

1000

100

101

100

10

1

0.11000

100

10

1

®

© ®

1 10 100 0 10 20 30 40 50 60Exposure time, h Exposure time, h

0 10 20 30 40 50 60Exposure time, h

Fig. 2. Pattern of change of ICM1(left), slope y (middle), and relative degree of resistance (right) with incubation time for A2780 and A2780/CP3 cells exposed to DDP (A). A2780and A2780/DX5B cells exposed to DOX IB) or PTX (O. and HCT-8 and HCT-8/DW2 cells exposed to TMQ (/)). RTX (E). MTX (F). or AG2034 (C). Data points (left and middle).actual parameter estimates obtained from the individual fitting of Eq. 2a or 2b (Table 1) to data. Data points (right), ratios of the 1CS()estimates for the resistant cell lines over theIC,,, estimates for the parental lines. Curves represent the fits of appropriate equations among Eqs. 4-7 (Table 11to the points. •.IC50. IC5I,... y. or ya for wild-type cells (w or wa):O. 1C.;,). IC50a, y, or -y.,for resistant cells (r or r(;); 0, R or Ktl: A. IC5l)h. IC50i:, yh, or yt. for wild-type cells (vv/,.or vv(.);A. IC50h. IC50c. yh, or yt. for resistant cells (rh or r(.); A.

Rh or A,..

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TIMK COUKSI: 01 DRUG CYTOTOXICITY

Table 2 Parameters of the final global pharmacodynamic models"

DrugDDPDOXPTXTMQRTXMTXAG2034CellsA2780A2780/CP3A2780A2780/DX5BA2780A2780/DX5BHCT-8HCT-8/DW2HCT-8HCT-8/DW2HCT-8HCT-8/DW2HCT-8HCT-8/DW2ModelEq.2a2a2a2a2b2b2b2b2a2c2i2b2a2bEq.4b4b4b4b4b4b4b4b4b4I/14ac4b4b4b4bIC50ParameterestimatesitnknknknknDFknDFknDFknDFknknDFIC,0

cit'nknDFknknDF10.6

±0.200.883±0.01086.7

±4.70.874±0.0160.697

±0.0310.798±0.0173.70

±0.121.07±0.028(8.60

±535) X10~"2.35

±1.1266±5713.2±4.52.34±0.93551±1530.511

±0.270.699±0.081338

±530.614±0.160.548

±0.036477±830.560±0.0430.755±0.0183.28±0.740.695

±0.112930±1713677

[fixed]1.85±0.0400.687±0.00790.165

±0.141.19±0.16717±832.96

±0.120.598±0.0157.61

±1.70.813±0.1511

16 ±327Eq.5c

%Ti725c

7o7i725c

7o7i725c

7o7iy5c

TO7i735c

%7i725c

Tb7i725c

-y07i725a

7o5a*

7o5bc7o7i5c

7o7i72Sc

To7i725c

7o7i725c

7o7i727Parameter

estimates-1.56±0.018-0.0549±0.00450.00232±0.00018-1.34±

0.036-0.0189±0.00700.00

124±0.00023-3.10±0.180.213±0.026-0.00727±0.00090-1.60±0.074-0.0339±0.0130.00250±0.00045-0.339

±0.18-0.0845±0.0270.000676±0.00030-0.00601±0.12-0.0843±0.0200.000521

0.00045-0.8930.24-0.00702

0.013-0.00003810.00013-1.650.300.0346

0.013-0.0004010.000124-1.760.092-1.20

±0.15-2.12(fixed]-0.0324

[fixed]-2.62

±0.110.0449±0.010-0.000447

±0.000170.0879±0.16-0.0923

±0.0220.000607±0.00025-1.11

±0.11-0.0384±0.0130.000739

±0.00025-1.70±0.660.0146

±0.031-0.000143±0.00027con'

' c-M,. ™«£ton.Eq.

Parameterestimates£conߣconߣconß¿conߣconß6b

Ke£COI1ß6b

K,¿conB6b

K,.£conß6b

KeEc„„B£conß6bc

Kr6a£con.c£conߣeonß6b

Kr£conߣconß6b

Ke2.74

±0.0210.00408±0.000571.92

±0.00110.0461±0.00241.03

±0.0160.0167 ±0.00110.768

±0.00740.0222±0.00121.88

±0.0330.00589±0.0380.313

±0.0441.14±0.0100.00589±0.0360.132

±0.0171.60±0.0140.0774

±0.0350.0808±0.0140.996

±0.00820.0612±0.0170.0856±0.0132.27±0.0160.467±0.0140.917±0.00800.1

30[fixed]0.304±0.0630.343±0.0272.15

±0.0180.0616±0.00261.70

±0.0170.0948±0.0360.0598±0.00342.12

±0.0180.365±0.00970.844

±0.0110.104±0.0340.0784±0.012

" Units: k is expressed in |XM"•unitless. A. ". and c, upper, lower,

h; ICj0iC.IC50back.and IC50maxare expressed in /IM;and terminal components, respectively, of a double or

7"50is expressed in h; Ke is expressed in h ; n, DF, s, y0, y}, •

triple Hill profile..c<ancl" are

tration-effect model for all exposure conditions (e.g., DDP, DOX, andTMQ in both wild-type and resistant cell lines and RTX, MTX, and

AG2034 in sensitive cell lines), the predicted IC5I) pattern in Fig. 5compares favorably with the observed IC5(, pattern in Fig. 2. R5tlplotsalso showed similar trends. However, for drugs displaying an apparentdouble Hill model for some/all exposure durations, the trend in thepredicted literal IC50 and R5(tcannot be directly compared to panels inFig. 2. because the literal IC5() is a complex function of many parameters. The thin gray dotted lines show the simulated patterns of changeof the true biological IC50, IC90, R5U,and Rw that would have beenobserved in absence of a drug dilution artifact.

The pattern of change of IC50 followed the ICX" X T model for

DDP, with an almost parallel decline of IC5()with T for wild-type and

resistant cells (Fig. 2) and, therefore, similar estimates of n (0.883 forA2780 and 0.874 for A2780/CP3: Table 2). Because n was fairly closeto 1 in both cell lines, drug concentration and exposure time are ofequal importance for the cytotoxic effect. A2780/CP3 cells exhibitedshallower concentration-effect curves (Figs. 2 and 3), but the slope y

became steeper as T increased and reached the steepness level ofA2780 for T = 24 h (Fig. 2), suggesting a more homogeneous cellular

response to DDP. There was a slight linear increase of the degree ofresistance with T (R rose from 11.1 to 14.9 between 1 and 24 h ofexposure; Fig. 2).

The IC/ X T model was also used for DOX (Fig. 2). The rate ofdecrease of IC50 with T was faster in A2780 (n = 0.798) than in

A2780/DX5B (n = 1.07), resulting in an exponential increase of R

(Eq. 7c in Table 1). The MDR cell line A2780/DX5B reached a49.3-fold degree of resistance at 7" = 24 h. The large increase in

resistance with T predicted by the model (/?90; Fig. 5) was originallydescribed for the noncloned DOX-resistant cell line A2780/DX5 (23).For A2780, the concentration-effect curves became shallower as Tincreased. Similar estimates of y were obtained in wild-type and

resistant lines for T a 10 h. Our observations suggest that clinicaladministration of DOX leading to the exposure to small concentrations over long time periods (long-term infusion) may, in fact, result

in the expression of higher tumor resistance.For PTX, a double Hill pattern was observed for T ^ 10 h in

resistant and parental cell lines (Fig. 3), suggesting a non-Pgp-medi-

ated phenomenon for the roller coaster pattern. The lower componentof the double Hill curves exhibited a constant 1C» whereas itsproportion, defined as the ratio £L.onb/£con,decreased with exposuretime and became negligible when exposure to PTX went beyond 10 h.Although the four IC50 curves for PTX in Fig. 2 appear to be flat, thebest fitted model for the pattern of change of each IC50., over time wasthe ICy X T relation. The slope of this relation was nearly identicalfor the two cell lines (n = 2.35 for A2780: 2.34 for A2780/DX5B),

and its extreme shallowness indicated a minor change in IC50 a as Tbecame longer. Similar slope parameters of the concentration-effect

curves were observed in the two cell lines, with an increased steepnessof 7Uwith exposure time. The degree of resistance of each component

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Wild-type cells Resistant cells

ADDP

o -~ 0.001 0.01 0.1 1 10 100 1000 jiM 0.001 0.01 0.1 1 10 100 1000 u\l

«S 0.00001 0.001 0.01 0.1 l 10 0.0010.01 0.1 l 10

0.0010.01 0.1 l 10 100 1000 n\l 0.0010.01 0.1 l 10 100

Drug concentration

Exposuretime, hOo123A

A4

56AA7.5 89a

a10 1213T

T

V14

1517T18•

19• 21.5o

24«48

Fig. 3. Families of concentration-effect cunes for a l-48-h exposure of the human ovarian cells A2780 and A2780/CP3 to DDP (A} or A2780 and A2780/DX5B to DOX (B) orPTX (O. The sets of curves in each panel are the best fits of specific global models (o all of the data in each panel, as explained in detail in the text, nata ¡tttinis,averages of fourreplicates. Note that the effect was transformed and expressed as percentage of the control growth. Drug concentrations (fiM or nvi) are plotted on a logarithmic scale.

was scattered between 488- and 1893-fold, but there was no significant time-dependency. /?a and Rh were not significantly different. PTX

was also studied against A2780 cells for 2, 4, 8, 16, 24. and 96 h withthe four-wash procedure. The results were very similar to these foundwith the two-wash procedure (Fig. 3). It is, therefore, unlikely that

there is a drug dilution artifact from insufficient well washing, butrather the second component of growth inhibition for intermediate

exposure time is due to residual intracellular PTX bound to cellularstructures, such as microtubules. The final global PD model was basedupon a double Hill concentration-effect (Eq. 2b in Table 1) in whichan "apparent dilution factor," DF, was introduced in the expression of

IC5ah (Eq. 4b in Table 1) and yh (Eq. 4c in Table 1). Estimates of DFwere 266 and 551 for A2780 and A2780/DX5B cells, respectively.Our model predicts a large benefit in terms of homogeneity of re-

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ATMQ

BRTX

CMTX

0.001 0.01 0.1 1 10 fi\l 0.00010.001 0.01 0.1 1 10

DAG 2034

O -

0.00001

HCT-8/DW2

0.001 0.01 0.1 l 10 n\l 0.001 0.01 0.1

Drug concentrationl 10 100 |iM

Exposure *

time, h ^1 23A A4

56AA7.5 810a

14a15

si 17TTV18

1920v

24• 48« 60o

96

Fig. 4. Families of concentration-effect curves for a I-96-h exposure of the human ¡leocecalcells HCT-8 and HCT-8/DW2 to TMQ (A}. RTX (B). MTX (O, or AG2034 (D). Thesels of curves in each panel are Ihe best fits of specific global models to all of the data in each panel, as explained in detail in the text. Data points, averages of four replicates. Notethat the effect was transformed and expressed as percentage of the control growth. Drug concentrations (/¿Mor RM)are plotted on a logarithmic scale.

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ADDP

BDOX

CPTX

DTMQ

ERTX

FMTX

GAG2034

10000

1000

100

10

1

0.1100

10

10.1

0.01

0.0011000

100101

0.10.01

0.0010.0001

10

10.1

0.01

0.001

0.0001100001000

100101

0.10.01

0.001100

10

10.1

0.01

0.0011000

100

10

10.1

0.01

IC50, uM IC90, uM100

10

1100

10

1100000

10000

1000

1001

0.1100000

10000

1000

100

101

100

10

1

0.11000

100

10

110 1001 10

Exposure time, b100

R« R90

20 40 60 0 20 40 60Exposure time, h

Fig. 5. Simulated pattern of change of the literal 1C,,, and IC„,(left) and associated relative degree of resistance (right} with incubation time for A2780 and A2780/CP3 cells exposedto DDP (A), A2780 and A2780/DX5B cells exposed to DOX (fi) or PTX (O. and HCT-8 and HCT-8/DW2 cells exposed to TMQ (D). RTX (E), MTX I/O. or AG2034 (G). Predictedcurves were simulated from the best fit of specific global models to data. Thick gray lines, wild-type cells; Mack lines, resistant cells; thin gray dotted lines (RTX, MTX, and AG2034)actual literal "biological" 1C,,, and 1C«,and corresponding R50 and fig,, that would be observed in absence of a drug dilution artifact.

spense and pattern of resistance (Fig. 5) when cells are treated withPTX for at least one cell cycle period.

TMQ induced a unique sigmoid-type decrease of IC50 with T (Eq.

4c in Table 1); the major change in IC50 was observed

10 h s T < 48 h. The concentration-effect curves became extremely

shallow during this time frame (Figs. 2 and 4), suggesting a highlyheterogeneous cell response to the drug, yet there were no obvious

for double Hill patterns. Similar patterns of change of 1C,,, and y were

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seen for HCT-8 and HCT-8/DW2. However, the HCT-8/DW2 cell

line, which has a lower pool of endogenous methylenetetrahydrofolateand tetrahydrofolate (24), was more sensitive to TMQ (R = 0.282) as

a result of the decreased competition between the lipophilic TMQ,which is not a substrate of FPGS, and endogenous folates at the levelof the target enzyme dihydrofolate reducíase(29). As was found forPTX, TMQ studied against HCT-8/DW2 cells for 2, 8, and 96 h withthe four-wash procedure induced cytotoxicity profiles that were verysimilar to those found with the two-wash procedure. Therefore, thelikelihood of a well-washing artifact for TMQ is small. The apparent

drug dilution factors introduced in the global PD model were 338 and477 for HCT-8 and HCT-8/DW2 cells, respectively. It is clear fromthe simulations (Fig. 5) that a long-term exposure to TMQ is advan

tageous in terms of drug potency.For RTX, the concentration-effect curves exhibited a single Hill

model in the parental HCT-8 cells, whereas apparent double Hillcurves were observed at all exposure times for the FPGS-deficient

subline with a uniform plateau at 37.4% of £con(Fig. 4). For theglobal modeling of the effect of RTX against HCT-8/DW2 cells, the

parameters IC5()C. yc, and B defining the terminal portion of theconcentration-effect curves were fixed (Table 2) from a subsequent

repeated experiment (data not shown) focusing on the high concentration region. IC50 and IC5()a for HCT-8 and HCT-8/DW2, respectively, followed the ICX" X T model (Fig. 2). The slope of the relation

was slightly steeper for the resistant line, leading to an exponentialdecrease of /?a from 48.0 to 10.4 for T ranging from 2 to 48 h.Therefore, for RTX-sensitive cells, there may be an advantage in

terms of cytotoxicity (n < 1) and resistance profiles to administeringless drug, but for a longer duration. A gain in drug retention inside thecells through the formation of cellular RTX polyglutamates (30)would also be observed with such a protocol.

An experiment with 2-h RTX treatment against HCT-8/DW2 cells

and four washes showed virtually no growth inhibition at concentrations that showed strong inhibition with the two-wash procedure.

When treatment was extended, double Hill curves appeared that weresomewhat different from the profiles observed with the two-washprocedure. Indeed, the apparent plateau in the four-wash experiment

was not constant for T 2 16 h but rather decreased from 88 to 36% of£c()nbetween 2 and 16 h. The apparent background followed a parallelprofile with a decrease from 66 to 7%, so that the size of the terminalcomponent (£conc—B) remained fairly constant; this observation

correlates with the constant plateau £c,,n_cobserved with the two-wash

procedure. Because response heterogeneity was suggested by theextreme shallowness of ya for short-term exposure (Fig. 2) andbecause the concentration-effect curves were greatly altered by the

washing procedure, it is likely that two Hill profiles constitute theapparent upper component observed in the two-wash procedure (Fig.

4). Thus, the final global PD model (Table 2) was based upon a tripleHill model (Eq. 2c in Table 1), in which components A and C form theactual cytotoxicity profile for the studied exposure time and component B reflects the effect of long-term exposure to residual drug

concentration. The estimated DF was 2930. Interestingly, IC50l. wasconstant over the range of exposure time investigated and between thewashing procedures, resulting in a large increase of the associateddegree of resistance with T (Fig. 5).

The remaining cells from a 96-h continuous exposure of HCT-8/

DW2 cells to RTX were examined closely with a light microscope.There was no evidence of cell division after drug treatment wasinitiated. Only relatively large cells [on average, 9.8-fold larger in area

than cells grown under control conditions (no drug)] with largeamounts of protein (on average. 12.6-fold more than in control cells)were observed in wells for the region of the concentration-effect curvewith the thick background response level. Evidently, the HCT-8/DW2

subline is extremely resistant to RTX at short exposure times (Fig. 5,thin gray dotted line), but the lack of time-dependency of the plateau

level £concsuggests that the terminal component of the triple Hillpattern is not associated with the FPGS-deficiency of HCT-8/DW2

but was rather due to a special response of this cell line to RTX. Wehave found this pattern of growth inhibition of HCT-8/DW2 with

other thymidylate synthase inhibitors, both polyglutamylatable andnonpolyglutamylatable (data not shown).

MTX induced double Hill concentration-effect curves in HCT-8/DW2 cells for intermediate T (10 h < T < 24 h: Fig. 4). IC50 andIC50a for wild-type and resistant cells, respectively, were describedwith the ICX" X T relationship and were of similar magnitude for the

two lines (Fig. 2). There was a slight linear increase in /?a from 0.436to 1.01 when 7"ranged from 10 to 48 h. IC50h was time-independent

over the range of exposure time investigated, resulting in a largeincrease of /?h with T. Slopes of the concentration-effect curves were

steeper at shorter T (Fig. 2). The double Hill pattern reported forHCT-8/DW2 cells exposed to MTX with the standard two-washprocedure disappeared with the four-wash procedure (2, 4, 8, 16, 24,and 96 h). MTX was totally ineffective for short-term exposure(T < 4 h). When the treatment was extended, the cytotoxicity curves

followed a regular single Hill pattern but showed an exponentialdecay of the background level similar to the decline of the plateaulevel £,.„„,,.Thus, the profiles for the four-wash procedure were

almost identical to the top component of the double Hill curves for thetwo-wash procedure (Fig. 4).

Fig. 5 (thin gray dotted lines) shows the dramatic gain in MTXcytotoxicity (IC50 and IC9())with increasing exposure time, as well asthe extreme resistance (RSHand /?9())of the FPGS-deficient cell line

for short exposures. The parameter DF was 716. It is clear from themodel prediction that a long-term exposure to MTX enhances drug

potency and decreases the magnitude of resistance. Indeed, the impactof the FPGS-deficiency is diminished as exposure is extended because

the formation of MTX polyglutamates is no longer the limiting factorfor drug accumulation (31 ), and polyglutamylation of MTX has littleeffect on its dihydrofolate reductase-inhibitory potential (32).

AG2034 also yielded double Hill response curves at intermediate Tin HCT-8/DW2 cells. IC50 a followed the 1Cx" X T principle, and the

line of the relationship was parallel to that obtained for the parentalHCT-8 cells, but up-shifted (Fig. 2). In contrast with MTX, the

polyglutamates of AG2034 are more potent than the monoglutamateparent compound, so that HCT-8/DW2 cells, which undoubtedly formless and shorter polyglutamate species than the wild-type cells, were8-fold resistant for long-term exposure (33). IC50-hwas constant with

T, leading to a large increase of Rb, together with an exponentialdecrease of the ratio £cllll̂f^,,,. Slopes of the concentration-effectcurves were steeper in HCT-8/DW2 than in HCT-8 cells for short

exposure times but became shallower as T increased and finallyreached the steepness level of the parental cell line at T = 48 h.Similarly to MTX, the double Hill patterns observed with the two-wash procedure disappeared with the four-wash procedure, andAG2034 was also found totally ineffective for short-term exposure(T ^ 4 h). For longer treatment periods, single Hill concentration-

effect curves were observed, as well as an exponential decline of thebackground level that coincided with the top component of the doubleHill curves for the two-wash procedure (Fig. 4). AG2034 dramatically

gained in terms of cytotoxicity from the increase in exposure timeagainst HCT-8/DW2 cells, as evidenced by Fig. 5 (thin gray dotted

lines). The estimated drug dilution factor was 1116.In summary, the most appropriate structural PD models describing

the concentration-time-effect relationships observed in the 14 drug/

cell conditions were: (a) the single Hill model, 7 cases; (b) the doubleHill model, 6 cases; and (c) the triple Hill model, 1 case. In the latter

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7 cases, the model was corrected for an apparent or real drug dilutionartifact. Patterns of change of IC50 were described with the ICX" X T

relation in 14 cases, whereas a constant term was more suitable in 1case (RTX against HCT-8/DW2; IC50 c). Patterns of change of the

slope parameter y were characterized with: (a) a polynomial, 12 cases;(b) a linear relation, 1 case; and (c) a constant term, 2 cases.

DISCUSSION

The PD modeling paradigm presented in this report offers manyunique features: (a) the investigation of a wide time frame and a largenumber of drug concentrations enables a complete mapping of thethree-dimensional concentration-time-effect surface, uncovering subtle but possibly important patterns: (b) the fitted three-dimensional

surface facilitates good predictions of effects at concentrations andtimes not studied, via interpolation and cautious extrapolation; (c) theinformation obtained through the detailed examination of the shapesof the concentration-effect curves (e.g., slope parameter y, double Hillpatterns) and the changes in the concentration-effect parameters with

exposure time, can be used to formulate hypotheses regarding drugaction, resistance, collateral sensitivity, and methodological artifacts;and (d) interpolated parameters (e.g., IC90 and /?9())may be used, inprinciple, for making careful inferences from in vitro data to the invivo level, or even to the clinical level. However, we cannot predictpatterns of selective toxicity, because our lack of data regardingnormal cell toxicity makes impossible the derivation of models describing drug effect on tumor versus normal cells.

The apparent cell growth inhibition measured in this PD assay is thecombination of various contributors, including cell killing (e.g., necrosis and apoptosis), immediate or delayed growth cessation, growthslow-down, and cell volume expansion. Total growth assays, such as

the SRB assay, provide an average measurement of the drug effectagainst the cell population. However, a total growth assay with anappropriate experimental design and modeling approach, as describedin this report, may provide clues regarding heterogeneity in response,drug action, and resistance. Questions raised through our detailed PDmodeling exercise have initiated subsequent experiments to isolate thebiological from the artifactual contributors to in vitro growth inhibition. It is clear that our hypotheses could be tested through morespecific methodologies, including biochemical, biomolecular, or image-based assays [e.g., individual colony forming assay (34) andtime-lapse video (35)].

Shallow slopes of the concentration-effect curves (small y in absolute value) and complex concentration-effect curves (double/triple

Hill phenomenon) were used in this study as markers of large response heterogeneity. On the basis of the assumption of a highlyhomogeneous cellular population, we may presume that responseheterogeneity implies the existence of two or more cellular targets forthe drug; each target has its own mean sensitivity to the drug.However, because the cell population is more likely heterogeneous,we propose that patterns in the concentration-effect curves relate to acertain tolerance distribution of the above-mentioned contributors to

in vitro cell growth inhibition. Therefore, a single Hill curve refers toa single cellular population, but each cell behaves slightly differentlyfrom the next one, and cellular heterogeneity within this single population becomes more pronounced as the slope of the concentration-

effect curve becomes shallower. Interestingly, when the concentration-effect curves for a drug against each of two populations are both

quite shallow and the respective IC50 a and 1C,,, b are not too far apart,then a single Hill curve may hide two populations (e.g., TMQ againstHCT-8 and HCT-8/DW2).

The experimental and modeling results from our study showed thatextreme manifestations of response heterogeneity, such as a double or

triple Hill curve, can be caused by many factors: (a) the inhibition ofcell division with the expansion of cell volume and an increase inprotein content (e.g., RTX at high concentrations against HCT-8/

DW2); (b) inadequate removal and washing of residual drug from theextracellular medium and the subsequent exposure of the cells to lowlevels of drug for long exposure times (e.g., RTX, MTX, and AG2034against HCT-8/DW2); and (c) rapid uptake and strong binding of

lipophilic drugs to cellular structures, the retention of intracellulardrug after adequate removal and washing procedures, and the subsequent action of intracellular drug for long exposure times (e.g., PTXagainst A2780 and A2780/DX5B). Note that the second explanation isa methodological artifact that can be overcome by changing thewashing procedure, whereas the third explanation involves a truebiological phenomenon that is critical for drug action. Both explanations include one population that is sensitive to relatively high concentrations of an agent during a short exposure and another populationthat is sensitive to relatively low concentrations of the agent for longexposures. It is likely that the distribution of cells in the cell cycle andthe different sensitivities of cells in the different cell cycle phasescomprise the root cause of our observed extreme response heterogeneity. As the initial drug exposure time increases, a higher proportionof the total cell population enters the sensitive phases of the cell cycleand is exposed to high drug concentrations. Interestingly, both thesecond and third explanations can be quantified in a similar fashion,with the use of a dilution factor (DF) parameter, but that the inferencefrom the estimate of the parameter is very different.

On the basis of the knowledge that the top component of the doubleHill concentration-effect curves represents the actual biological effectfrom short-term drug treatment, whereas the bottom component rep

resents the artifactual effect of residual drug concentrations, it ispossible to use the PD model to isolate the true biological effect (Fig.5, thin gray dotted lines). In opposition to our initial observations andto previously reported data (29), the simulations show that RTX,MTX, and AG2034 given for short periods are totally ineffective ininducing growth inhibition of the FPGS-deficient line, resulting, inturn, in an infinite degree of resistance of the HCT-8/DW2 cells. We

may speculate from our observations that polyglutamylatable inhibitors of folate metabolism will induce time-dependent double Hillconcentration-effect curves when potency is examined against poly-glutamylatable-deficient cells in 96-well plate total growth assays

with inadequate washing of residual drug.It was found, with the same basic assay used in this paper, that

adding a 4-h drug-free medium incubation step, followed by two extrawashes, after a 4-h TMQ incubation increased the IC50 —10-fold,

implying that significant TMQ had been released from binding duringthe extra 4-h drug-free incubation step.4 In addition, a recently published study on the kinetics of intracellular [3H]PTX retention (36)

supports our explanation of the behavior of PTX. It was shown that (a)<4% of the intracellular PTX was released during washing; (b) 11.2%of the PTX applied dose remained in the cells after washing: and (c)three washings, each separated by a 3-h equilibration period, were

necessary to reduce the intracellular PTX concentration to <0.5% of

the applied dose.Although the double Hill phenomenon has not been described in

detail in the literature, it is possible to observe concentration-effect

plots with a suggestion of a plateau (9, 17). Such a pattern is moreeasily recognized when the concentration scale (X axis) is logarithmicand the effect scale ( Yaxis) is linear, than for the reverse convention.Recently, the shape of concentration-effect curves was extensively

discussed as a predictor of resistance mechanism (19). These authors

4 H. Faessel. persona] communication.

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speculated that a change in the slope of the concentration-effect curveplotted on log-linear coordinates suggests the appearance of resist

ance, the saturation of a resistance mechanism, or a combination ofdifferent mechanisms of resistance. As discussed above, our interpretation of the double Hill and the shallow slope phenomena is differentand implies either the existence of a true heterogeneous cell population regarding drug action or the existence of an artifactual long-term

exposure of cells to residual drug concentration.Fig. 5 attests to the need of clear definitions of the parameters

"relative degree of resistance" and the underlying IC50. Drug

potency or IC5() may be defined as the drug concentration responsible of a 50% reduction of either £CI)I1(estimated at E = 50%) or£c,,n- B (f-K- estimated at E = 607, if B = 20%). It is clear that

if the background response level is high, the two methods ofcalculation will yield discordant IC50 estimates. Thus, the background level, being an important marker of drug potency, is arelevant parameter to report when referring to IC5()s. We havedemonstrated through this study that IC50 is not the sole contributor to drug resistance and that K is also defined by the slope y ofthe concentration-effect curve and the background level B. Finally,

the pattern of change of the degree of resistance with time isclosely related to the parameters ;;. k, y,,, y¡,y-,, Kc, DF, and B.

Investigators of clinical PK/PD studies are interested in relating PKvariables, e.g., peak plasma levels and systemic exposure, to PDparameters of efficacy and toxicity. For instance, the AUC (C X 7")

has been successfully used to predict the WBC or platelet count aftertreatment with irinotecan (3, 4), anthracyclines (5, 6), and etoposide(7) and to predict the neurotoxicity occurring after therapy withvincristine (7). AUC was also used as a predictor of antitumor activityin a study in which mice bearing human xenografts were exposed toDDP (11). The relationship between systemic exposure and therapeutic outcome has not been so clearly established for drugs such asirinotecan (3) and carboplatin (2). Overall, the lack of such relationships may result from several factors, including the measurement ofthe inappropriate drug moiety (active metabolites), measurement ofthe drug in the inappropriate biological fluid (biophase), inisevalua-

tion of the systemic exposure (limit of quantification of the analyticalsystem, sampling time), difficulty in objectively measuring tumorresponse, the selection of an inappropriate population of patients(resistant tumor), molecular tumor variables (oncogenes), and so on(7, 37). In addition, perhaps AUC is an inappropriate dose metameterfor some drugs. We propose using C X Tl/n, instead of C X T, as a

predictor of efficacy or toxicity in future PK/PD studies. Interestingly,as mentioned above, our study showed that, for drugs in which pastclinical studies have shown PK/PD relationships using AUC, ourestimate of the parameter n was close to 1, e.g., for DDP and DOX.Perhaps, in PK/PD studies in which AUC failed to predict toxicityand/or efficacy, C X T"" would have performed better.

In conclusion, we demonstrated that the common Hill equation isnot a universal PD candidate for modeling the growth inhibitory effectof anticancer drugs. Under particular conditions of exposure, weobserved large response heterogeneity, eventually revealing a plateauin the concentration-effect curves, which was successfully described

with a double or a triple Hill model. We speculated that this phenomenon was initiated by heterogeneity at the cellular level or by methodological artifacts. The extensive investigation of the interrelationships among time, concentration and effect showed that it waspossible to modulate drug effect, cellular heterogeneity, and drugresistance by altering the conditions of exposure to the drug. Thismathematical approach provided a complete empirical characterization of drug resistance at various effect levels, revealed heterogeneityin terms of cell response, suggested artitactual mechanisms of action,and provided insights for examining drug action in more fundamental

assays. Quantitatively characterizing possible artifacts: (a) enables thesubtraction of the artifacts from the total effect to reveal the truebiological effect; (b) suggests new experiments to verify the existenceof the artifacts; (c) suggests new approaches to eliminate the artifacts;and (d) provides insights for refining the biological interpretation ofexperimental results. Finally, bearing in mind the limitations of invitro/in vn-0/clinical extrapolations, the model can be used to supply

leads for the clinical use of drugs. We believe that the modelingparadigm presented here has the potential to generate useful information for the development of anticancer agents over and above thelimited information commonly derived from growth inhibition assays.We offer new ideas for the analysis of data from clinical PK/PDstudies of anticancer agents.

ACKNOWLEDGMENTS

We thank Elva Winslow and Carol Wr/.osek for technical assistance andDrs. Amarnath Sharma. Kun Lu. HélèneFaessel. Roy Kisliuk. John J.McGuire. Ralph J. Bernucki, and David J. Adams for their helpful commentsduring the preparation of this manuscript.

REFERENCES

1. Gibaldi. M.. and Perrier. D. Pharmacokinelics. Ed. 2. Vol. 15. pp. 445-449. NewYork: Marcel Dekker. 1982.

2. Belani. C. P., Kim, K.. Bonomi, P., and Johnson. D. Retrospective estimation ofcarboplatin exposure by Culvert's and Chatelut's formulae and correlation with

pharmacodynamic effects in metastalic non-small cell lung cancer (NSCLC). Proc.Am. Soc. Clin. Oncol. Annu. Meet.. 15: Al 119, 1996.

3. Canal, P., Gay, C., Dezeuze. A.. Douillard. J. Y.. Bugal, R.. Brunei. R.. Adenis. A..Herait. P.. Lokiec. F.. and Mathieu-Boue. A. Pharmacokinetics and pharmacodynam-ics of irinotecan during a Phase II clinical tria! in coloréela!cancer. Pharmacology andMolecular Mechanisms Group of the European Organization for Research and Treatment of Cancer. J. Nail Cancer Inst. (Betnesda), 14: 2688-2695, 1996.

4. Sasaki, Y.. Hakusui, H.. Mizuno, S.. Morila, M., Miya, T., Eguchi. K.. Shinkai, T.,Tamura. T.. Ohe. Y., and Saijo. N. A pharmacokinetic and pharmacodynamic analysisof CPT-11 and its active metabolite SN-38. Jpn. J. Cancer Res.. 86: 101-110, 1995.

5. Rushing, D. A., Raber, S. R., Rodvold. K. A.. Pisciteli!. S. C.. Plank. G. S., andTewksbury. D. A. The effects of cyclosporinc on the pharmacokinetics of doxorubicinin patients with small cell lung cancer. Cancer (Phila.l, 74: 834-841. 1994.

6. Robert. J.. Monnier. A.. Poulignat. N.. and Herait. P. A pharmacokinelic and phar-macodynamic study of the new anlhracycline piraruhicin in breast cancer patienls.Cancer Chemother. Pharmacol.. 2<i: 75-79. 1991.

7. Evans, W. E. Clinical pharmacodynamics of anlicancer drugs: a basis for extendingthe concept of dose-intensity. Blut. 56: 241-248, 1988.

8. Greco. W. R.. and Hakala. M. T. Biochemical pharmacology of lipophilic diamin-opyrimidine antifolates in mouse and human cells HI vitro. Mol. Pharmacol., 18:521-528. 1980.

9. Rupniak. H. T.. Whelan. R. D., and Hill. B. T. Concentration and time-dependentinter-relationships for amitumour drug cytotoxicities against tumour cells in vitro. Int.J. Cancer. J2: 7-12, 1983.

10. Adams. D. J. In vitro pharmacodynamic assay for cancer drug development: application to crisnatol, a new DNA intercalato)-. Cancer Res.. 49: 6615-6620, 1989.

11. Kurihara. N.. Kubota, T., Hoshiya. Y.. Otani. Y.. Watanabe. M.. Kumai. K., andKitajima. M. Antitumour activity of i-ij-diamminedichloroplatinum 111)against hu

man tumour xenografts depends on its area under the curve in nude mice. J. Surg.Oncol.. 61: 138-142, 1996.

12. Kalns. J. E.. Millenbaugh. N. J.. Wientjes. M. G.. and Au, J. L. Design and analysisof in vitro antitumor pharmacodynamic studies. Cancer Res., 55: 5315-5322. 1995.

13. Hill, A. V. The possible effects of the aggregation of the molecules of haemoglobinon its dissociation curves. J. Physiol., 40: iv-vii, 1910.

14. Holford. N. H.. and Sheiner. L. B. Understanding the dose-effect relationship: clinicalapplication of pharmacokinelic-pharmacodynamic models. Clin. Pharmacokinet.. 6:429-453. 1981.

15. Waud, D. R., and Parker. R. B. Pharmacological estimation of drug-receptor dissociation constants. Statistical evaluation. II. Competitive antagonists. J. Pharmacol.Exp. Ther.. 177: 13-24, 1971.

16. Chick, H. An investigation of the laws of disinfeclion. J. Hyg.. 8: 92-158. 1908.17. Hill, B. T.. and Whelan, R. D. Assessments of the sensitivities of cultured human

neuroblastoma cells lo anti-tumour drugs. Pediatr. Res., 15: 1117-1122, 1981.18. Bruce. W. R.. Meeker. B. E., and Valeriote. F. A. Comparison of the sensitivity of

normal hcmatopoictic and transplanted lymphoma colony-forming cells to chemo-therapeutic agents adminislered IH vivo. Ì.Nail Cancer Insl. (Belhesda). 37: 233-245.1966.

19. Stewart. D. J.. Raaphorst. G. P., Yau. J.. and Beauhien, A. R. Active vs. passiveresistance, dose-response relationships, high dose chemotherapy, and resistance modulation. A hypothesis. Invest. New Drugs. 14: 115-130. 1996.

5760

Research. on January 19, 2021. © 1998 American Association for Cancercancerres.aacrjournals.org Downloaded from

TIME COURSE OF DRUG CYTOTOXICITY

20. Greene, R. F.. Chatlerji. D. C., Hiranaka, P. K.. and Gallelli. J. F. Stability of cisplalinin aqueous solution. Am. J. Hosp. Pharm.. 36: 38-43, 1979.

21. Rogan, A. M.. Hamilton, T. C.. Young, R. C.. Klecker. R. W.. Jr.. and Ozols, R. F.Reversal of Adriamycin resistance by verapamil in human ovarian cancer. Science(Washington DC), 224: 994-996. 1984.

22. Masuda. H.. Ozols. R. F., Lai. G. M., Fojo. A.. Rothenberg. M.. and Hamilton, T. C.Increased DNA repair as a mechanism of acquired resistance to c/.v-diamminedichlo-roplatinum (II) in human ovarian cancer cell lines. Cancer Res., 48: 5713-5716. 1988.

23. Alaoui Jamah, M. A., Yin, M. B., Mazzoni, A., Bankusli, I., and Rustum. Y. M.Relationship between cytotoxicity. drug accumulation, DNA damage and repair ofhuman ovarian cancer cells treated with doxorubicin: modulation by the tiapamilanalog ROI 1-2933. Cancer Chemother. Pharmacol., 25.- 77-83. 1989.

24. Lu. K.. Yin. M. B., McGuire. J. J.. Bonmassar. E.. and Rustum. Y. M. Mechanismsof resistance to /V-|5-|/V-(3,4-dihydro-2-methyl-4-oxoquinazolin-6-ylmethyl)-/V-methylamino|-2-thenoyl]-L-g!utamic acid (ZD1694), a folate-based thymidylate syn-thase inhibitor, in the HCT-8 human ileocecal adenocarcinoma cell line. Biochem.Pharmacol., 50: 391-398, 1995.

25. Levasseur, L.. Faessel, H., Slocum. H. K.. and Greco. W. R. Precision and pattern in96-well plate cell growth experiments. Proc. Biopharm. Sect. Am. Stat. Assoc.,227-232, 1995.

26. Skehan, P.. Storeng, R.. Scudiere, D., Monks, A., McMahon. J.. Vistica, D., Warren,J. T.. Bokesch. H., Kenney. S., and Boyd. M. R. New colorimetrie cytotoxicity assayfor anticancer-drug screening. J. Nail Cancer Insl. (Bethesda), 82: 1107-1112. 1990.

27. SAS Institute Inc. SAS/STAT User's Guide. Version 6. Ed. 4, Vol. 2, pp. 1135-1193.

Gary, NC: SAS Institute Inc., 1989.28. Thisted, R. A. Algorithm. Elements of Statistical Computing, p. 170. New York:

Chapman and Hall. 1988.29. Lu, K., Faessel, H.. Slocum, H. K., Greco, W. R., and Rustum. Y. M. Role of

intracellular reduced folate pools on cellular sensitivity to trimelrexate (TMQ) andmethotrexate (MTX) in a folylpolyglutamate synthetase (FPGS)-deficient HCT-8

subline. Proc. Am. Assoc. Cancer Res.. 38: A1087, 1997.

30. Jackman. A. L.. Farrugia. D. C.. Gibson, W.. Kimbell. R.. Harrap. K. R..Stephens. T. C., Azab. M., and Boyle. F. T. ZDI694 (Tomudex): a new thymidylate synthase inhibitor with activity in colorectal cancer. Eur. J. Cancer., 3ÃŒA:1277-1282, 1995.

31. Takemura. Y., Kobayashi. H., Gibson, W.. Kimbell. R.. Miyachi. H.. and Jackman.A. L. The influence of drug-exposure conditions on the development of resistance lomethotrexate or ZD1694 in cultured human leukaemia cells. Int. J. Cancer.. 66:29-36. 1996.

32. Drake, J. C., Allegra. C. J.. Baram. J.. Kaufman. B. T.. and Chabner. B. A. Effects ondihydrofolate reducíaseof methotrexate metabolites and ¡ntracellular iblatcs formedfollowing methotrexate exposure of human breast cancer cells. Biochem. Pharmacol..36: 2416-2418. 1987.

33. Faessel. H.. Lu. K., Slocum, H. K., Rustum. Y. M.. and Greco. W. R. Comparisonsof the synergistic growth inhibition by trimetrexatc (TMQ) + AG2034 andTMQ + Tomudex (ZDI694) of human ileocecal HCT-8 cells and DW2. a subline

deficient in folylpolyglutamale synlhela.se (FPGS). Proc. Am. Assoc. Cancer Res.. 38:A656. 1997.

34. Slocum. H. K.. Malmberg, M., Greco, W. R., Parsons, J. C.. and Rusium. Y. M. Thedetermination of growth rales of individual colonies in agarose using high-resolutionautomated image analysis. Cytometry. //: 793-804. 1990.

35. Grunwald, J. Time-lapse video microscopic analysis of cell proliferation, motility and

morphology: applications for cytopathology and pharmacology. Biotechniques. 5:680-687. 1987.

36. Au. J. L., Li. D., Can, Y.. Gao. X.. Johnson, A. L., Johnston, J., Millcnhaugh. N. J..Jang. S. H.. Kuh. H. J.. Chen. C. T., and Wientjes. M. G. Pharmacodynamics ofimmediate and delayed effects of paclitaxel: role of slow apoptosis and intracellulardrug retention. Cancer Res., 58: 2141-2148, 1998.

37. Perkins, A. S.. and Stern, D. F. Molecular biology of cancer: oncogenes. In: T. T. J.DeVita, S. Hellman, and S. A. Rosenberg (eds.l. Cancer. Principles and Practices ofOncology. Ed. 5. pp. 79-102. Philadelphia: Lippincott-Raven. 1997.

5761

Research. on January 19, 2021. © 1998 American Association for Cancercancerres.aacrjournals.org Downloaded from

1998;58:5749-5761. Cancer Res   Laurence M. Levasseur, Harry K. Slocum, Youcef M. Rustum, et al.   and Resistance

Drug Cytotoxicityin VitroModeling of the Time-Dependency of

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