(CANCER RESEARCH 49. 3823-3828. July 15, 1989]

Kinetic Analysis of Cell Killing Effect Induced by Cytosine Arabinoside andCisplatin in Relation to Cell Cycle Phase Specificity in Human Colon Cancerand Chinese Hamster Cells'

Shogo Ozawa,2 Yuichi Sugiyama, Junko Mitsuhashi, and Makoto Inaba'

Cancer Chemotherapy Center, Japanese Foundation for Cancer Research, Kami-lkehukuro, Toshima-ku, Tokyo 170 [S. O., J. M., M. l.J; and Faculty of PharmaceuticalScience, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113 [Y. S.J

ABSTRACT

We presented a cell kill pharmacodynamic model for cell cycle phase-specific antitumor agents as well as that for cell cycle phase-nonspecificagents. The former was based upon assumptions that a cell populationcould be divided into a drug-sensitive population having sensitive (orspecific) cell cycle phase(s) and a drug-resistant population in resistantcell cycle phases. This model revealed that cell killing action of antitumoragents with sufficient cell cycle phase specificity was time dependent.This time dependence was actually observed with cytosine arabinosideby colony-forming inhibition assays using Chinese hamster V79 cells.

On the other hand, the present analysis indicated that cell killingeffects of drugs lacking cell cycle phase specificity are dependent on Cx T (concentration-time product). We analyzed cell kill kinetics ofcisplatin and showed C x T dependence of cell killing action of the drugusing human colon cancer WiDr cells. These results indicate that cellkilling action of antitumor agents can be kinetically analyzed in terms ofcell cycle phase specificity.

INTRODUCTION

On the basis of the voluminous data from in vitro colonyforming assays for the determination of cell killing activity,Shimoyama (1) classified cell cycle phase-nonspecific agents (2)as "concentration-dependent" drugs, and cell cycle phase-specific agents (2) as "time-dependent" ones. In our previous report

(3), we attempted a kinetic analysis of cell killing effects of cellcycle phase-nonspecific agents and established mathematicalequations that describe the relationships of cell survival to drugconcentration and exposure time. For cell cycle phase-nonspecific agents, it is assumed that a cell population is homogeneousand cells in any cell cycle phase are equally sensitive to thedrug. These equations incorporate drug degradation, since anumber of drugs undergo rapid decomposition in the culturemedium. These relationships were experimentally validated byshowing that the cell kill kinetics of MMC,4 ACNU, and HN2

obey the equations. In agreement with the results of Shimoyamaand coworkers, we concluded that the cell killing action of cellcycle phase-nonspecific agents depends on the C x r value orAUC (3). This indicates that cell killing action of cell cyclephase-nonspecific agents is dependent equally on concentrationand time for demonstration of a certain cell killing effect.

On the other hand, for cell cycle phase-specific agents suchas antimetabolites and Vinca alkaloids, several investigators

Received 12/15/88; revised 4/10/89; accepted 4/19/89.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 inaccordance with 18 U.S.C. Section 1734 solely to indicate this fact.

' This study was supported in part by Grants-in-Aid for Cancer Research from

the Ministry of Education, Science and Culture, and for New Drug DevelopmentResearch from the Ministry of Health and Welfare. Japan.

2Present address: Department of Pharmacology. School of Medicine. KeioUniversity, Shinano-machi. Shinjuku-ku. Tokyo 160. Japan.

1To whom requests for reprints should be addressed.4The abbreviations used are: MMC. mitomycin C; IC^o. drug concentration

for 90% cell kill; ACNU, l-(4-amino-2-methylpyrimidine-5-vl)-methyl-3-(2-chlo-roethyl)-3-nitrosourea hydrochloride: HN2. nitrogen mustard: C x T, concentration-time product; AUC. area under concentration-time curve; DDP. cisplatin:BCNU, l,3-bis(2-chloroethyl)-l-nitrosourea; ara-C. cytosine arabinoside.

presented the following characteristics of cell killing kinetics:(a) cytostatic action, (b) concentration-independent action, and(c) time-dependent action (1,4). Jusko presented a pharmacodynamic model for cell cycle-specific chemotherapeutic agents(5). The investigator assumed that the growing cell populationwas susceptible to cell cycle-specific agents whereas the restingone was not.

In the present report, for the purpose of proposing a cell killkinetic model for cell cycle phase-specific agents, it is assumedthat there exist both drug-sensitive and less sensitive cell populations. Especially S-phase-specific agents such as ara-C areexamined in this report, and in the case of S-phase-specificagents, cells in S-phase are thought to be the most sensitive(i.e., sensitive population). Making such an assumption, wehave designed a phramacodynamic cell kill kinetic model forcell cycle phase-specific agents. According to this model, thesurviving fraction after drug exposure can be mathematicallyrelated to concentration of the drug, exposure time, a physiological cell degradation rate constant to represent loss of cellviability in the absence of drug, cell cycle phase traverse rateconstants for both the sensitive and resistant cell cycle phases,irreversible cell death rate constants for sensitive and resistantcell cycle phases, and initial number of cells in sensitive andinsensitive cell cycle phases. We have simulated the relationshipbetween drug concentration and exposure time for 90% cellkilling effect with a fixed cell death rate constant for sensitivephase(s) and a variable cell death rate constant for resistantphases. We have also quantified the cell killing effect of ara-Cusing Chinese hamster V79 cells in order to determine whethertime-dependent cell killing action of the drug is produced.

DDP is a member of new class of antitumor agents and is ofgreat clinical use. High response rates to DDP have beenachieved in a variety of cancers (6). However, analysis of cellkilling action of DDP has not yet been presented on a kineticbasis. In this report, considering differences between cell cyclephase-nonspecific and -specific agents, we have attempted tomake a more kinetically based analysis of the cell killing effectof DDP, and to determine which type of cell killing action ofDDP shows [i.e., that of cell cycle phase-nonspecific agents likeACNU, HNi, and MMC (3) or of cell cycle phase-specificones]. We studied the cytotoxic effects of DDP on a humancolon cancer, WiDr cell line, and compared the relations ofIGm and exposure time observed to the curve anticipated fromthe kinetic model of the cell killing effect of cell cycle phase-nonspecific and -specific agents.

MATERIALS AND METHODS

Chemicals. ara-C and DDP were kindly provided by Nippon Shiny-aku Co., Ltd., Kyoto, and Bristol-Myers Research Institute, Tokyo,Japan, respectively. All other chemicals were of analytical grade.

Cell Culture. The human colon cancer cell line WiDr was providedby Dai-nippon Seiyaku Co., Ltd., Osaka. Japan, from the AmericanType Culture Collection. WiDr cells were maintained in minimum

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KINETIC ANALYSIS OF ara-C AND DDP ACTIONS

essential medium supplemented with 10% fetal bovine serum. Chinesehamster V79 cells were grown in RPMI 1640 medium supplementedwith 10% fetal bovine serum. Both cell lines were cultured in thepresence of 100 ^g/ml kanamycin at 37°Cin a humidified atmosphere

of 5% CO2 and 95% air.Estimation of Cell Killing Effect of Drugs. After cell number was

determined by a model ZBI Coulter counter, WiDr and V79 cells wereseeded at a cell density of 100-12,800 cells into 60-mm dishes containing 3.0 ml of culture medium. Dishes receiving 100 or 200 cells wereused as controls or for relatively low drug concentrations, and dishescontaining more cells were used for higher drug concentrations. Thirty»ilof drug solution was added to each dish on the day after seeding,and the cultures were then incubated for various periods. Because ofdifficulty in preparing DDP solution at 600 ^g/ml or higher in physiological saline, DDP was dissolved in the culture medium and addedto the culture at the desired concentrations. All assays were done intriplicate or quadruplicate. At the end of the drug exposure, the platewas washed twice with 3 ml of Hanks' balanced salt solution, and theHanks' solution was replaced with 3 ml of the culture medium and

further incubated. On the fifth day (V79 cells) or 10th day (WiDr cells)after seeding, the dishes were washed once with phosphate-bufferedsaline, fixed with 10% formalin, and stained with 0.05% crystal violet.Colonies were counted by a CA-7 colony analyzer (Oriental InstrumentsLtd., Tokyo, Japan). The plating efficiency of the drug-treated disheswas normalized to that of the untreated controls, which was about 50%or higher for WiDr cells and more than 80% for V79 cells.

The IG»values of the drugs for each exposure time (the concentration that reduced the surviving fraction to 10% that of the control) weredetermined from the dose-response curves.

Degradation of Drugs during Incubation in the Culture Medium. Todetermine the decomposition of drugs in the culture medium duringthe incubation period, the concentration of DDP was measured bybioassay using V79 cells. For the decomposition of DDP, drug wasadded to the culture medium in the presence of WiDr cells (1.3 x 10'

cells/ml) and incubated. The initial concentration of DDP was 10 Mg/ml. After incubation for the desired time, a few milliliters of the mediumwas removed and stored at -80°C until bioassay was performed. For

the bioassay, dose-response curves of DDP for V79 cells were madeafter a 96-h exposure, to be used as titration curves.

An appropriate volume of each stored sample was added to the V79culture (total 3.0 ml) so that the surviving fraction of the cells wouldbe in the range of the titration curve. The residual drug concentrationof the sample was estimated as the average of three or more determinations, and the degradation kinetics of DDP were derived.

RESULTS

Model Analysis

According to a basic pharmacodynamic model of chemother-apeutic effects by Jusko (5, 7), we tried to describe a cell killingkinetic model of antitumor agents making a comparison between cell cycle phase-specific and -nonspecific agents.

Cell Cycle Phase-Nonspecific Agents. As shown in Fig. 1, weassumed that the cell population is kinetically homogeneousfor this class of drugs and that cells and the drug react in abimolecular reaction, where C/ = density of treated cells; t =time of drug exposure; Cm= drug concentration in the medium;C, = intracellular concentration of drug; k = drug-inducedirreversible cell death rate constant; AC.,= cell proliferation rateconstant; kr = rate constant of physiological degradation ofcells; and k^t = first-order drug degradation constant.

From the above assumptions, the rate of change in cell densitywas expressed by differential equation and the relationshipbetween ICW (C„,.TO)and exposure time (f) was obtained aspreviously reported (3):

Culturemedium

Ad,

2.3kK

(A)

Intracellularfluid

O;Deathof cells

Fig. 1. Schemes of cell killing effect by cell cycle phase-nonspecific agents. Itis assumed that intra- and extracellular drug rapidly reach equilibrium and thatintracellular drug and cell react in a bimolecular reaction.

10=

10'

10

10'

ICf 10"' 1 10 \02

Exposure time (t)

IO3

Fig. 2. Log-log relationships between Cm.m(i.e., 1C«,)and exposure time (t)simulated from a kinetic model for cell killing effect of antitumor agents withvarious drug-degradation constants (Adc,).kK is assumed to be about 5.

where K is an equilibrium constant between intra- and extracellular drug concentrations. Cm.w- t curves on a log scale forvarious degradation constants, A:dt.8values, can be simulated asasymptotic as shown in Fig. 2, where kK is set to be about 5.Considering Equation A, C x T for 90% cell kill is expressedas follows:

CxT =f< , 2.3= —¿�(constant) (B)

KA

If the drug is stable under the culture condition (kdcltthen Equation A becomes:

x t = 2.3kK

0),

(C)

This means that cell killing action of cell cycle phase-nonspecific agents is C x 7-dependent, and log Cm.90-logt curve islinear with a slope of —¿�1.

Cell Cycle Phase-Specific Agents. For the analysis of cellkilling action of cell cycle phase-specific agents, it is assumedthat the cell population is divided into sensitive and less sensitive (resistant) ones (Fig. 3), where Cs = density of cells sensitiveto drug; CK = density of cells resistant to drug; k, = drug-induced irreversible cell death rate constant for drug-sensitivecells; k2 = drug-induced irreversible cell death rate constant fordrug-resistant cells; kXK,kKS= cell cycle traverse rate constants;and kr = physiological degradation rate constants of the cells.

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KINETIC ANALYSIS OF ara-C AND DDP ACTIONS

DeathofcellsFig. 3. Scheme of cell killing effect by cell cycle phase-specific agents. It is

assumed that cell populations are divided into drug-sensitive and -resistant onesand that drug acts on both cell populations. As for S-phase-specific agents, Csrepresents cell density in S phase (i.e., drug-sensitive phase).

In this case a drug can act on both drug-sensitive and-resistant phase cells and A:,is larger than k2 because of differential sensitivity to drugs. Cells in S phase are assumed to bethe most sensitive to S-phase-specific agents. Since cell numbermust be doubled when cells traverse from resistant phases(including G2, M, and d) into S phase, the rate of increase incell number is expressed as 2 kKSCK.As a bimolecular reactionbetween drug and cell is assumed, the rate of change in celldensity of sensitive and resistant phase can be written as follows:

^ = -(ksR + k,)Cs -k,-K-Cm-Cs + 2kxs-C„dt

, —¿�ksK ' Cs ~ (k/iat

kr)-CK -

(D)

(E)

Mathematical procedures for derivation of Equations F fromD and E are shown in the "Appendix" for simplicity.

CV= Ae-" + Be «A'e-'" + B'e~"' (F)

where A ' and B' are A and B in which C,,, is taken as zero. A,B, A', B', a, and ßinclude kSK, kKX,kr, C„„k¡,and k2. kXR,kKX,

kr are fixed constants for a certain cell line and they are relatedas Equation S. We have simulated the relationship between SFand exposure time t with various Cm (assuming a K value of1.0), k, and A2.As already mentioned, CS/CR = 1/3 and Cx andC°Kare fixed to be 100 and 300, respectively. kSK, kKX,kr, are

intrinsic constants of tumor cells which relate to cell doublingtime. From Equation T, if kXKis set as 0.5, kKXbecomes 0.095.k, is assumed to be 0.001. From these mathematical applications, we obtained the relationship between Cm and t for 90%cell kill (SF= 0.1) as shown in Fig. 4 with a fixed k, of 1.0 andvarious k2 values. As shown in this figure, in the case of k, =1.0 and k2 = 1.0, the log C,,, —¿�log t curve proved to be linearwith a slope of —¿�1 which is typical for cell cycle phase-nonspecific agents (Fig. 2). On the other hand, when k, and k2 are setto be 1.0 and 0.02 or less, respectively, very high concentrationis necessary to kill 90% cell at the relatively short exposuretime and the slope of the curve becomes steeper than —¿�1.

Indeed, if a drug does not act on cells in resistant phases at all(k2 = 0), the slope around time of 100 is almost infinite. Thisimplies that the cell killing action of cell cycle phase-specificagents is not dependent on C x T value, but rather dependenton exposure time. Therefore, our model analysis for cell cyclephase-specific antitumor agents reproduces the time-dependentcell killing action of this class of agents. When we set variousk2 values with one fixed k¡,the smaller k2 becomes, the greateris the difference in drug sensitivity between sensitive and resist-

100

Exposure time (t)Fig. 4. Log-log relationship between C„M(i.e., IO»)and exposure time (t)

simulated from a kinetic model for cell killing effect of cell cycle phase-specificantitumor agents with A of 1.0 and various A values. K '.„,values obtained from

the dose-response curves anticipated according to Equation F (see text) wereplotted against exposure time on a log scale. kst, A«s,k„K, C¡and O were set to be 0.5, 0.0953. 0.001. 1. 100, and 300. respectively. Numbersbeside lines, each line's A3value.

ant phases. This implies that the difference between A:,and k2expresses degree of cell cycle phase specificity of the drug. Onthe contrary, when k2 was fixed to be 1.0, time dependence wasobserved when A:,was set to be as large as 10,000 (data notshown). These results suggest that the time dependence of cellkilling action is closely related to the degree of phase specificityof the drug in question.

Cell Kill Kinetics of ara-C against V79 Cells. Dose-responsecurves for ara-C against V79 cells for various exposure timeswere made, and ICW values were assessed from the curves. 1C«,values were plotted against exposure time on a log scale in Fig.5. Similar to the results of our model analysis for cell cyclephase-specific antitumor agents shown in Fig. 4, ara-C showedtime-dependent cell killing action, as has been reported by anumber of investigators (1,4). The steepness of the slope of theconcentration-time curve for short exposure time of ara-Ccorresponds to that of the curve in Fig. 4 where k: of 0.001 isgiven. Therefore, ara-C shows typical time-dependent cell killing action, indicating that very high concentration (greater than100 i/g/ml) is necessary for cytotoxicity at the exposure time of10 h or shorter. The curve in Fig. 5 also looks asymptotic. Thismay reflect degradation of ara-C after relatively long exposure.Strictly considering, drug degradation also should be includedin the equation for cell cycle phase-specific agents. However,drug degradation is not of much significance because steepnessof the slope for relatively short exposure (i.e., loss in activity ofdrug is not so serious) is important for distinction betweencycle phase being specific or nonspecific. Thus, although thereare difficulties in giving a precise experimental support for themodel analysis, the time dependence of cell killing action ofara-C can be explained by differential sensitivity within a cellcycle (i.e., cell cycle phase specificity).

Degradation Kinetics of DDP in the Culture Medium. TheDDP concentration in the culture medium in the presence ofWiDr cells after incubation at 37°Cfor various periods was

measured by bioassay. The kinetics of decay of DDP are shown3825

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100

10

1.0

0.1

o.oi

KINETIC ANALYSIS OF ara-C AND DDP ACTIONS

100

1 10 100

Exposuretime (h)

Fig. 5. Log-log relationship between ICWand exposure time for ara-C in V79cells. 1C«,values obtained from the dose-response curves for ara-C using V79cells were plotted against exposure time on a log scale. Each point is the mean ofthree experiments with standard deviations (bars).

§

20 40Incubation time

60

(h)

80

Fig. 6. Degradation of cisplatin in the culture medium. DDP was added at aconcentration of 10 pg/ml to the culture of WiDr cells and incubated at 37'C. Its

concentrations were determined by bioassay. Each point is the mean of threedeterminations with standard deviations (bars).

in Fig. 6. The rate constant of degradation was estimated to be(3.10 ±0.14) x IQ-- h. The half-life of DDP under our culture

conditions was 22.4 h.Cell Kill Kinetics of DDP against WiDr Cells. Dose-response

curves for DDP against WiDr cells for various exposure timeswere constructed, and IC9ovalues were assessed from the curves.Both the experimental data thus obtained and an asymptoticcurve simulated from Equation A with a AdeRvalue of 3.10 x10~2/h were plotted on a log-log graph of IC90 and exposure

time (Fig. 7). The two curves showed good coincidence, indicating that when decomposition of DDP is taken into consideration, the relationship between exposure time and IC90of thedrug is described by Equation A. Therefore, the cell killingaction of DDP is dependent on the C x rvalue or AUC.

i.o 10Exposure time (h)

100

Fig. 7. Log-log relationship between IC«and exposure time to DDP in WiDrcells. 1C™values obtained from the dose-response curves for DDP using WiDrcells were plotted against exposure time on a log scale. The curve was drawn bya simulation according to Equation A, using an A,,,,.value of 3.10 x 10 'h.

Fitting was done by nonlinear regression analysis (8), and the best fit to theexperimental data was obtained at a kK value of 1.15 x 10"' ml/^g/h. Each point

is the mean value of three experiments with standard deviations (bars).

DISCUSSION

In order to quantitatively analyze cell killing action of variouskinds of antitumor agents, it is necessary to establish as simplebut rational a cell kill kinetic model as possible for cell cyclephase-specific agents as well as for cell cycle phase-nonspecificagents. In this report, we tried to establish a cell kill kineticmodel for cell cycle phase-specific agents and showed time-dependence (not C x T dependence) of cell killing action whichis the major character of cell killing effect induced by this classof agents. Furthermore, this time dependence was actuallyobserved for ara-C using our colony-formation assay. We alsoinvestigated on DDP to determine whether DDP-induced cellkilling action is cell cycle phase-specific or not. DDP platínalesDNA, and the major adduct is a DNA cross-link (9). ( ¡rallareported formation of blocking lesions at identical DNA sequences by DDP and ACNU, a cell cycle phase-nonspecificagent (10). These observations led to the hypothesis that thecell killing action of DDP is of the C x 7-dependent type. Asa result of colony assay, cell killing action of DDP was foundto obey Equation A and show C x T dependence characteristicof cell cycle phase-nonspecific agents.

Several results concerning the cell killing action of DNA-interacting agents have been reported so far. However, thereseems to be some discrepancies among them. We found thatACNU, HN2, MMC, and DDP showed cell killing actioncharacteristic of cell cycle phase-nonspecific agents. Skipperalso classified cyclophosphamide, BCNU, daunomycin, andactinomycin D as cell cycle phase-nonspecific agents (2). Incontrast, Kim reported that tumor cells in S phase were themost sensitive to daunomycin and cells in G, phase were themost resistant according to their age-response studies (11).Similar to daunomycin, Adriamycin was reported to be S phase-specific (12). As for MMC, Djordjevic showed its G, specificity(13). These results indicate that DNA-interacting agents possess a certain degree of cell cycle phase specificity.

In this regard, the simulation demonstrated in Fig. 4 arequite suggestive, where log-log relationships between Cm.90andexposure time were depicted corresponding to various k,/k2ratios between 1 and infinity. The drugs with the ratio of 1 aresubstantially cell cycle phase-nonspecific ones and exhibit typical linear relationship with the slope of —¿�1. In this study, the

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KINETIC ANALYSIS OF ara-C AND DDP ACTIONS

curve produced by DDP (Fig. 7) corresponds to this pattern.The greater k,/k2 ratios are, the steeper becomes the slope.

However, when the ratio is 5 or 10, the slope was not sodifferent from that of the ratio of 1, indicating that the drugshaving such relatively low phase specificity express almost C x^-dependent cell killing effect. This suggests that the above-mentioned phase specificity observed with DNA-interactingagents seems to be not so great as to express time dependence.

Furthermore, when the ratio of k,/k2 is 100 or more, theslope of curve becomes remarkably steep. The curve producedby ara-C seems to correspond to this pattern, indicating significant time dependence of its cell killing action. It seems truethat time dependent cell killing action needs significantlygreater phase specificity. Together with experimental datashown in Fig. 5 and 7, the simulation in Fig. 4 indicates thattypical cell cycle phase-specific and -nonspecific agents areeasily distinguished by the slope of log Cm.mand log t curve.

Drug decomposition should be considered especially withagents of the C x 7-dependent type. With regard to degradationof DDP in aqueous media, several investigators reported thatin blood or plasma, the half-life of ultrafiltrable DDP was 1.5-2.5 h (14-16). On the contrary, the drug was quite stable for afew days at 25 or 37°Cin 0.9% sodium chloride solution (16,

17). We determined the degradation of the drug in culturemedium containing fetal calf serum as low as 10% and foundthe half-life to be 22.4 h. Ali-Osman et al. reported that thehalf-life of DDP in minimum essential medium supplementedwith 20% fetal calf serum was 15.1 h (18). Judging from theseresults, the half-life of DDP determined by us may be fairlyreasonable.

The model analysis for cell cycle phase-nonspecific agentsrevealed that if IC9o-exposure time relationship of a drug wasexpressed by Equation A, the extent of cell killing of the drugwas determined by C x T. We found that for DDP, the experimentally obtained relationship between IC9I, and exposureduration was well matched by Equation A. Therefore, DDPshowed C x 7-dependent cell killing action. This conclusion isvery significant for human tumor clonogenic assays and forprediction of the antitumor effectiveness of administration ofDDP. In human tumor clonogenic assays, the results of ourstudy can be used to select a reasonable concentration andduration of drug exposure, based on the enormous data ofclinical AUC values of DDP. With regard to obtaining a morephysiological testing or patient tumor cell sensitivity to anti-cancer drugs in the human tumor clonogenic assay, Ali-Osmanet al. reported application of in vivo and in vitro pharmacoki-

netics for physiologically relevant drug exposure in these assays(18). Such attempts are very important for establishment ofrational human tumor clonogenic assay protocols. Furthermore, the effectiveness of these drugs can be predicted bycomparing the C x. T value for a definite cell kill of a certaintype of tumor with the C x T achieved in the extracellular spaceof the tumor or in the plasma. The clinical efficacy of DDPtreatment may be enhanced through efforts to achieve higherAUC in the region of tumors, since AUC is shown to be thedetermining factor of the cell killing action of these drugs. Nowour efforts are directed toward establishment of a reliableparameter for the antitumor effectiveness of cell cycle phase-specific agents based on our cell kill kinetic analysis.

APPENDIX

In order to solve these simultaneous differential equations Laplacetransform is applied. Laplace transform of Cv and CR are Cv and C«,

respectively. For simplification, alterations of parameters were madeas follows and Equations D and E were solved:

Cr-K2) + 2C'x-k,,

(i + K,)(s + K2) -

Cx-kXK + C°R(s-i

(s + K,)(s + K2) - :

(G)

(H)

where C.v and C«are the initial density of cells in sensitive and resistantphases, respectively, and

K2 = k„,+ k,+ k2-K-C„,

Hence the inverse Laplace transform of Equations G and H gives:

= A2-e~' B2e

vhere

A, =

B, =•- ß)

+ C«(K,- a)

B2 =

(Q), - 2kXK-k„

(D

(J)

(K)

(L)

(M)

(N)

(O)

(P)

(R)

where a and ßare solutions of a quadratic equation, x~ - (Ki + K2)x +K,-K2-2kXR-kKX = 0.

The ratio of Cx to Ca must not be changed in the control (C„,= 0).Assuming that for a certain cell line, the period of S phase is one-fourthof the total cell cycle time (CS/CK is 1/3) in the control,

3 xdCx

dtdC«

dtand —¿�'- 7 kKX+ k,

4 k*«+ 3 *,

should be satisfied. From Equation S,

knx —¿�„¿�.

(S)

(T)

is derived. Number of cells (CT; CT = Cv + GO treated with a drug at aconcentration of Cmis expressed as

CT = Ae~"' + Be-

where A = A, + A2 and B = B,number of cells (C'r) is written as

C'T = A'e-' + B'e-

(U)

+ B2. In the control (C„= 0), the

(V)

Therefore the surviving fraction (SF) can be expressed as Equation F.

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KINETIC ANALYSIS OF ara-C AND DDP ACTIONS

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1989;49:3823-3828. Cancer Res   Shogo Ozawa, Yuichi Sugiyama, Junko Mitsuhashi, et al.   Specificity in Human Colon Cancer and Chinese Hamster CellsArabinoside and Cisplatin in Relation to Cell Cycle Phase Kinetic Analysis of Cell Killing Effect Induced by Cytosine

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