PII S0360-3016(98)00175-8

● Biology Contribution

MATHEMATICAL MODEL OF 5-[ 125I]IODO-2 *-DEOXYURIDINE TREATMENT:CONTINUOUS INFUSION REGIMENS FOR HEPATIC METASTASES

GEORGE SGOUROS, PH.D.,* JOSEPHA. O’DONOGHUE, PH.D.,* STEVEN M. LARSON, M.D.,†

HOMER MACAPINLAC, M.D.,† JUSTINE J. LARSON†

AND NANCY KEMENY, M.D.‡

*Department of Medical Physics;†Nuclear Medicine Service, Department of Radiology and‡Gastrointestinal and Solid Tumor Service,Department of Medicine, Memorial Sloan-Kettering Cancer Center, New York, NY

Purpose: Due to the cytotoxicity of DNA-bound iodine-125, 5-[125I]Iodo-2 *-deoxyuridine ([125I]IUdR), an analogof thymidine, has long been recognized as possessing therapeutic potential. In this work, the feasibility andpotential effectiveness of hepatic artery infusion of [125I]IUdR is examined.Methods: A mathematical model has been developed that simulates tumor growth and response to [125I]IUdRtreatment. The model is used to examine the efficacy and potential toxicity of prolonged infusion therapy.Treatment of kinetically homogeneous tumors with potential doubling times of either 4, 5, or 6 days is simulated.Assuming uniformly distributed activity, absorbed dose estimates to the red marrow, liver and whole-body arecalculated to assess the potential toxicity of treatment.Results: Nine to 10 logs of tumor-cell kill over a 7- to 20-day period are predicted by the various simulationsexamined. The most slowly proliferating tumor was also the most difficult to eradicate. During the infusion time,tumor-cell loss consisted of two components: A plateau phase, beginning at the start of infusion and ending oncethe infusion time exceeded the potential doubling time of the tumor; and a rapid cell-reduction phase that wasclose to log-linear. Beyond the plateau phase, treatment efficacy was highly sensitive to tumor activity concen-tration.Conclusions: Model predictions suggest that [125I]IUdR will be highly dependent upon the potential doublingtime of the tumor. Significant tumor cell kill will require infusion durations that exceed the longest potentialdoubling time in the tumor-cell population. © 1998 Elsevier Science Inc.

Iododeoxyuridine, 125Iodine, Hepatic artery infusion, Treatment planning, Modeling.

INTRODUCTION

Due to the cytotoxicity of DNA-bound iodine-125,5-[125I]iodo-29-deoxyuridine ([125I]IUdR), an analog of thy-midine, has long been recognized as possessing therapeuticpotential. The objective of this work was to determine thefeasibility and potential efficacy of [125I]IUdR, delivered byhepatic artery infusion, for the treatment of hepatic colorec-tal cancer metastases. Autopsy studies of patients with colo-rectal cancer have shown that, in approximately 40% ofcases, liver is the only site of metastasis (1). Blood supplyto hepatic metastases originates almost exclusively from thehepatic artery, whereas only 20% of the blood supply tonormal liver is derived from this source. Thus, hepaticartery infusion reduces the initial exposure of normal hepa-tocytes to therapeutic agents without compromising theexposure of target tumor cells. This approach makes itpossible to deliver a much greater concentration of drug totumor than would be possible using systemic administration

(2). In the 7 randomized trials reported to date, comparinghepatic artery with systemic infusion of chemotherapy, thehepatic artery treatment was considerably more effectivethan systemic treatment (3).

In a study of unlabeled IUdR, Spethet al. measured 1.7to 4.5% IUdR incorporation in hepatic metastases of colo-rectal cancer after intravenous (IV) infusion of a pharma-cologic dose of IUdR (1000 mg/m2/day by continuous in-fusion for 3 days). Hepatic artery infusion of the same doseyielded 3.8 to 10.5% thymidine replacement; thymidinereplacement in normal liver was,1% (9 of 10 patients) (4).The fraction of tumor cells labeled after the 3 day hepaticartery infusion ranged from 16 to 48% (median 32%); 2–9%(median 5%) of normal cells were labeled (4).

The cytotoxicity of 125iodine has been examined by anumber of investigators (5–10). Decay of this radionuclideproduces a highly localized ionizing cloud of Auger elec-trons whose biologic effect can exceed that of alpha parti-cles when the radionuclide is incorporated into nuclear

Reprint requests to: George Sgouros, Ph.D., Department ofMedical Physics, Memorial Sloan-Kettering Cancer Center, 1275York Avenue, New York, NY 10021.Acknowledgements—We thank Drs. John Humm, Farhad

Daghighian and C. Clifton Ling for many useful discussions.Supported, in part, by the Klearman Fund, Department of EnergyGrant DE-FG02-86ER60407 and NIH Grant R01 CA-62444.

Accepted for publication 23 April 1998.

Int. J. Radiation Oncology Biol. Phys., Vol. 41, No. 5, pp. 1177–1183, 1998Copyright © 1998 Elsevier Science Inc.Printed in the USA. All rights reserved

0360-3016/98 $19.001 .00

1177

DNA (10); if it is not incorporated into the DNA, thecytotoxicity of 125I is dramatically reduced and is similar tothat of X-rays (8). Because its utilization leads to DNAincorporation of125I, the thymidine analog, 5-[125I]iodo-29-deoxyuridine ([125I]IUdR), has long been recognized aspossessing therapeutic potential (6).

Mariani and colleagues (11) have identified the followingconditions for the effective use of [125I]IUdR in cancertreatment: 1. Targeting of tumor cells that exhibit rapidproliferation kinetics, and therefore, greater DNA incorpo-ration of [125I]IUdR relative to adjacent healthy tissue; 2.local-regional administration to circumvent the rapid sys-temic degradation of [125I]IUdR; and 3. prolonged exposureof the tumor cells to [125I]IUdR so that all tumor cellstraverse the DNA synthesis (S) phase of the cell cycle.Prolonged administration of [125I]IUdR has been recentlyexamined in an experimental animal model and in a spher-oid system (12, 13); studies performedin vitro have alsobeen used to derive strategies for clinical implementation of[125I]IUdR treatmentin vivo (14). In this work, a theoreticalanalysis of prolonged hepatic artery infusion of [125I]IUdRfor colorectal cancer metastatic to the liver is examined.

Using previously developed models of tumor-cell prolif-eration kinetics (15) and data from a prePhase 1 trial ofhepatic artery [125I]IUdR administration (16, 17), the po-tential efficacy of continuous infusion treatment is evalu-ated.

METHODS AND MATERIALS

Model of tumor growth and [125I]IUdR treatmentThe model used to describe tumor growth and [125I]IUdR

treatment is depicted in Fig. 1. The equations describing thismodel are provided in the Appendix. The model is derivedfrom a generalized model of tumor-cell proliferation andtreatment (15). Treatment with [125I]IUdR is represented asa loss of cells that is dependent on the rate of disintegrationper tumor cell and the probability of tumor-cell kill perdisintegration. The disintegration rate per tumor cell isdetermined instantaneously by the number of125I atomsincorporated in the DNA. This results from [125I]IUdRuptake from the local environment and will be time-depen-dent. In an individual tumor cell, part of the total is due toincorporation during the S-phase, if this has occurred, andpart due to radioactivity passed down the ancestral line fromits predecessors. For a time-invariant concentration of[125I]IUdR, incorporated activity will be the product of thelocal concentration and a time-dependent fractional tumoruptake. To simplify analysis, the assumption is made thatthe steady-state activity level in the whole body can be usedto scale the amount of [125I]IUdR incorporated into thetumor. This is reasonable because the steady-state levelreflects the rate of infusion of IUdR.

Whole-body radioactivity during infusionAssuming the whole-body clearance of125I is exponential

with a rate, M, and that the [125I]IUdR is administered with

an infusion rate,i, the radioactivity in the whole body,A, asa function of time is given by the differential equation:

dA

dt5 i 2 l z A, (1)

whose solution is:

A~t! 5i

lz ~1 2 e2lzt!. (2)

The maximum or steady-state activity in the patient is,therefore, given by:

A 5i

l. (3)

The infusion rate is selected so that the steady-state activitylevel meets red-marrow toxicity constraints.

Marrow and liver dosimetry[125I]IUdR in plasma is rapidly (half-life, 5 min) ca-

tabolized to [125I]iodouracil and further dehalogenated tofree [125I]iodide (11, 18, 19). The concentration of intact[125I]IUdR in plasma is, therefore, assumed negligible interms of its availability for incorporation in healthy tissue.Red marrow and liver exposure is, expected from125I that isnot DNA-incorporated. In addition to the very short-rangeAuger electrons,125I also emits electrons with energiesranging from 23 to 35 keV.125I also emits X-rays in theenergy range of 27—31 keV. These emissions are not

Fig. 1. Schematic representation of the mathematical model usedfor tumor growth and treatment response. The compartment on theleft contains cells that have not traversed a full cell cycle (pre-Sphase); the compartment on the right contains cells that havetraversed a full cell cycle (post-S phase). At the start of infusion,all cells are in the left compartment. Tumor cells in this compart-ment are subject to proliferation (a) and cell loss due to turnover(a z f(N)). Tumor cells are transferred from the left compartmentto the right by a linear process with a rate (k) that is inverselyproportional to the potential doubling time of the tumor cells. Therate is chosen so that, after a potential doubling time, all cellsinitially present in the left compartment would have been trans-ferred to the right compartment. Cells in the right compartment aresubject to proliferation and cell loss due to both turnover and[125I]IUdR therapy (R). The sum of the two compartment contentsyields the total number of tumor cells.

1178 I. J. Radiation Oncology● Biology ● Physics Volume 41, Number 5, 1998

usually considered in much detail because their cytotoxiceffect is negligible compared with the cytotoxicity of DNA-bound 125I. At high enough radioactivity concentrations,however, these emissions would also be cytotoxic. Ab-sorbed doses to the red marrow and to the liver werecalculated to assess the potential toxicity of treatment. Thecumulated activity (20), A˜ , in the whole body (WB) wasobtained by multiplying the steady-state activity level, i/l,by the infusion duration. A conventional S-factor—basedcalculation of absorbed dose (21) yields the following equa-tions for the absorbed dose, D, to liver (LI) and red marrow(RM) (20):

DLI 5 ~ fLI z SLI4LI 1 fRM z SLI4RM! z AWB 1 ARB z SLI4RB

(4)

DRM 5 ~ fLI z SRM4LI 1 fRM z SRM4RM! z AWB 1 ARB z SRM4RB

(5)

with,

ARB 5 AWB z ~1 2 fRM 2 fLI! (6)

SLI4RB 5 SLI4WB zmWB

mRB2 SSLI4RM z

mRM

mRB1 SLI4LI z

mLI

mRBD(7)

SRM4RB 5 SRM4WB zmWB

mRB2 SSRM4LI z

mLI

mRB1 SRM4RM z

mRM

mRBD(8)

mRB 5 mWB 2 mRM 2 mLI (9)

The parameters, fLI, fRM are equal to the fraction of whole-body cumulated activity that occurs in the liver and redmarrow, respectively. “S,” represents the S-factors for aparticular source-target organ combination, “m” is the mass,and “RB” denotes the “rest-of-body” (20). The latter termaccounts for radioactivity that is not specifically appor-tioned to an organ. S-factors and organ masses were ob-tained from Pamphlet 11, of the Medical Internal RadiationDose (MIRD) Committee (21).

Parameter valuesThe potential doubling time of colorectal cancer has been

observed to vary considerably, both among patients andwithin individual tumor biopsy samples (13, 22). Using thexenografted nude mouse model, Perezet al. obtainedTpot

values ranging from 1.3 to 10.5 days with a mean value of2.2 days (22). In a series of 30 patients with colon carci-noma, analysis of multiple tissue samples obtained fromsurgical specimens yielded median values ranging from 1.5to 12.35 days in individual patients (23). In this work,modeling analysis was performed for kinetically homoge-nous tumor cells with potential doubling times,Tpot, of 4, 5,or 6 days. The initial number of tumor cells,n(t 5 0), wasset to 109. A cell density, (rcell, of 108 cells/g is assumed,yielding an initial tumor burden of 10 grams. The tumor

activity concentration, C, is varied from 0.05% to 0.005% ofthe steady-state activity per gram (17, 24). A baseline valueof 0.009% is used in most simulations. The number ofdecays for 37% or e21 survival is approximately 110 for[125I]IUdR (5). The efficiency of cell kill per disintegration,E, is, therefore, set to 0.009. The limiting Gompertziantumor cell number,N`, for all simulations was set to 531010, corresponding to 500 grams. The transition from ex-ponential to Gompertz growth kinetics was set at 13 106

cells (N0).A whole-body clearance half-time, Tclr, of 13 h for125I is

used in these analyses (17). The infusion rate,i, chosen forall simulations is 197 MBq/h (5.33 mCi/h). This yields asteady-state activity level of 3.7 GBq (100 mCi).

The maximum concentration of125I in blood after hepaticartery infusion of [125I]IUdR in patients ranges from 2.8 to3.8%ID/L (16). Assuming a uniform distribution of theactivity within blood, this would also be the plasma con-centration. Approximately 40% of the red marrow is com-posed of water (25). Assuming this total fluid volume is inrapid equilibrium with125I in plasma, the concentration ofactivity in the red marrow would be 1.5%ID/L (26). Thisvalue was used to estimate the red marrow cumulated ac-tivity. The cumulated activity to liver was obtained byassuming that 30% of the whole-body cumulated activityremains in the liver. This is a conservative assumptionbecause the uptake of125I in normal hepatocytes has beenshown to be less than 10% of the injected dose after a 3-dayhepatic artery infusion of [125I]IUdR (4).

RESULTS

The reduction in tumor cell number for three differentkinetically homogeneous tumors is depicted in Fig. 2A, B.A plateau phase is observed that begins at the start ofinfusion and ends when the infusion time exceeds the po-tential doubling time of each population. This is followedby a rapid cell-reduction phase that is nearly log-linear. Theearly plateau phase represents the delay in achieving a lethaldisintegration rate per cell. This delay is associated with thetime required for all cells to traverse the S-phase of the cellcycle after the onset of [125I]IUdR infusion. At the start ofinfusion, cells that do not immediately enter the S phase willnot be affected by [125I]IUdR exposure. Correspondingly,the right panel (B) of Fig. 2, which shows the number ofcells in the tumor over the first 5 days of infusion, shows asmall increase in cell number at a very early time. After onecomplete cell cycle, all cells are labeled and the cell-kill ratebecomes largely independent of potential doubling time.The potential doubling time does, however, impact upon thetime at which the transition between the plateau phase andthe rapidly decreasing phase occurs.

Figure 3A, B depicts tumor cell kill for a population oftumor cells with a potential doubling time of 6 days. Curvesobtained using three different values for the fractional tumoruptake of [125I]IUdR are shown. Treatment efficacy ishighly dependent upon activity concentration within the

1179Modeling of [125]IUdR continuous infusion therapy● G.SGOUROSet al

tumor. At a high enough activity concentration (A5 3.7GBq, C 5 0.05% steady-state activity/g), 90% tumor con-trol of a 10-gram tumor is achieved with an infusion dura-tion that is only slightly greater than the maximum potentialdoubling time of the tumor. At the lowest activity concen-tration considered (A5 3.7 GBq, C5 0.005% steady-stateactivity /g), 90% tumor control requires 17 to 18 days ofcontinuous infusion. These simulations also predict a fun-damental limitation of continuous [125I]IUdR treatment.The infusion duration must exceed the potential doublingtime of the kinetically slowest tumor-cell population presentin the tumor. Tumors with very slowly proliferating sub-populations will not be treated successfully with a singlecontinuous infusion of [125I]IUdR.

With an infusion rate that yields a steady-state whole-body activity of 3.7 GBq, the absorbed dose rates to the redmarrow, liver, and total body were 0.097, 0.59, and 0.074Gy per day of infusion, respectively. The 17-day infusionthat was required to achieve 10 logs of cell kill when thetumor activity concentration was 0.005% ID/g resulted in ared-marrow absorbed dose of 1.65 Gy. This dose is belowthe 2-Gy level that is commonly associated with onset ofhematologic toxicity (26–28).

DISCUSSION

A model of tumor cell growth and response to continuousinfusion [125I]IUdR treatment has been developed. The

Fig. 2. The total number of tumor cells is plotted against time, in days. (A) Curves for homogeneous tumors withpotential doubling times of 4, 5, or 6 days are shown. A tumor uptake of 0.009% steady-state activity/g and a steady-stateactivity of 3.7 GBq are used in these simulations. The data are plotted on a log scale. (B) The data of panel (A), expandedto show the reduction in cell number over the first 5 days of infusion (linear scale).

Fig. 3. The total number of tumor cells is plotted against time, in days. (A) Curves for tumors with an uptake of 0.005,0.009, or 0.05 % steady-state activity/g are shown. A potential doubling time of 6 days and a steady-state activity of 3.7GBq are used in these simulations. The data are plotted on a log scale. (B) The data of panel (A), expanded to showthe reduction in cell number over the first 5 days of infusion (linear scale).

1180 I. J. Radiation Oncology● Biology ● Physics Volume 41, Number 5, 1998

model predicts a high (.90%) probability of tumor celleradication with 8 to 17-days of continuous [125I]IUdRinfusion. The results are highly dependent on the potentialdoubling time of the tumor. The infusion duration for asingle infusion treatment must exceed the potential doublingtime of the tumor. Assuming negligible incorporation of[125I]IUdR in the red marrow, tumors with potential dou-bling times of 6 days or less may be effectively treated withminimal red-marrow toxicity. Rapid and uniform exposureof tumor cells to infused [125I]IUdR is presumed. It isimportant to note that the results presented are for theidealized case of tumors with a kinetically homogenouspopulation of tumor cells. Three different potential doublingtimes are considered to examine the effect of potentialdoubling time on treatment; the values chosen represent anarrow range of the potential doubling times that may beobserved in tumorsin vivo. In a large population of tumorcells, some cells will have potential doubling times that aremuch longer than the 6 day value used in the analysis (13,22). As the model demonstrates, such tumor cells are likelyto escape sterilization.

As noted earlier, slowly proliferating cells were the mostdifficult to eradicate. These simulated results are consistentwith observations of diminished [131I]IUdR uptake inslowly vs. rapidly proliferating tumorsin vivo (29).

In developing the model, administered [125I]IUdR wasassumed uniformly distributed throughout all tumor cells.Depending upon the size and hepatic artery perfusion of themetastasis that is being targeted, a variability in tumor cellexposure to [125I]IUdR may be expected (24). Because theinfusion is prolonged over several days, the importance ofthis may not be as significant as in short-duration infusionprotocols. The model assumes that all tumor cells are cy-cling (growth fraction5 1). Recruitment of noncycling cellsinto a cycling phase is not incorporated into the model.

Absorbed dose estimates to liver and red marrow wereobtained using the MIRD S-factor formalism (20). Theseestimates provide the mean dose to the tissue volume andare based upon the assumption of a uniform distribution ofradioactivity in all source organs. As such, they should beregarded with caution and primarily for the purposes ofcomparing different treatment protocols. The biologicallyrelevant absorbed dose may differ significantly from thesevalues (30). If, for example, one only considers the absorbeddose to red-marrow cell nuclei and assumes that all of thedecays occur outside of the cell, then by excluding the dosecontribution from electrons with insufficient energy to reachthe nucleus, a 10-fold decrease in the biologically relevantred-marrow absorbed dose is obtained (16). Conversely, if asignificant fraction of plasma activity remains in the form of[125I]IUdR and is incorporated into the DNA of hematopo-etic cells, the biologically relevant dose will be muchgreater than that obtained using S-factors. It is also not yetclear that the red marrow will be the dose-limiting organ inhepatic artery infusion of [125I]IUdR. It is noteworthy, how-ever, that in a clinical trial of i.v.-administered125I-labeledA33 antibody, which is internalized, no hematological or

healthy organ toxicity was observed after administration ofactivities ranging from 7.4 to 29.6 GBq (200 to 800 mCi) (S.Welt, Personal communication, Memorial Sloan-KetteringCancer Center, New York, NY 1997).

Exposure to [125I]IUdR has been shown to increase thepotential doubling time of tumor cellsin vitro (8, 31). Thiseffect is not directly accounted for in this model and is alsoexpected to reduce treatment effectiveness unless the infu-sion duration is correspondingly increased. It is possible toincorporate such a phenomenon in the current model; dataare lacking, however, on the degree to which this occursinvivo.

The analysis provided in this work focuses on the use of125I-labeled IUdR.123Iodine has been suggested as a betteralternative due to its shorter half-life and the resultingpotential for a greater number of DNA-associated decayswithin a given time-interval (30, 32, 33). The rationale for123I is particularly strong when considering a single, short-duration infusion protocol. In such a case, after the DNA islabeled, the likelihood of delivering a lethal number ofdecays to a tumor cell before dilution due to cell division, isincreased as the decay rate increases. In the case of aprolonged infusion regimen in which tumor cells are pre-sumably exposed to an equilibrium level of activity, thisadvantage is lost.

The infusion rate chosen for the simulations was based onobtaining a maximum whole-body activity that did notexceed 3.7 GBq (100 mCi) of125I. This allowed for thepossibility of designing an outpatient treatment protocolbecause, given the low energy of the photons emitted by125I, the radiation exposure rate at 1 meter will be wellbelow the 5 mR/h limit.

These model results provide a theoretical guide to thedesign and evaluation of outpatient clinical trials usingintrahepatic infusion of [125I]IUdR. Treatment of liver me-tastases of colorectal cancer by hepatic artery infusion maymatch the conditions of the model. Such an implementationof [125I]IUdR therapy turns well-established, potential dif-ficulties associated with its use (i.e., rapid dehalogenation inplasma, short-range cytotoxic effect, dependence upon cell-replication kinetics) into potential advantages.

APPENDIX

During [125I]IUdR infusion, the rate of change of cellnumber is set to the sum of three terms. A positive termequal to the unconstrained growth rate of the tumor cellsand two loss terms due to [125I]IUdR treatment and cell-turnover. These terms are denoted by:a, R, andL, respec-tively.

dN

dt5 @a 2 ~R 1 L!# z N (A1)

Because one potential doubling time must elapse before allof the tumor cell are labeled with [125I]IUdR, a delay in the

1181Modeling of [125]IUdR continuous infusion therapy● G.SGOUROSet al

therapeutic effect would be predicted. To account for this, 2compartments are established, one containing cells thathave not yet traversed the S-phase (N2s) and one in whichthey have (N1s). Loss due to therapy is operative only onthe second compartment. Passage of tumor cells from thefirst compartment to the second is modeled as a linearprocess with a rate constant that is inversely proportional tothe potential doubling time,Tpot, of the tumor cells. This isrepresented as a linear loss in the first compartment and again in the second compartment, both of which are repre-sented by the parameter,k.

dN2S

dt5 @a 2 L# z N2S 2 k (A2)

dN1S

dt5 @a 2 ~L 1 R!# z N1S 1 k (A3)

k 5N0

Tpot(A4)

N2S~t 5 0! 5 N0 (A5)

N1S~t 5 0! 5 0 (A6)

During this period, the total number of cells,N, is N2s 1N1s. WhenN2s 5 0, the parameterk is set equal to zero andEq.A1 is used in the remainder of the simulation. Becausepassage of cells from one to the other compartment alsoimplies cell doubling, the expression used fork is a conser-vative approximation of a more complex process that de-pends upon the distribution of cells within the cell cycle, theduration of S phase relative toTpot, and the fraction of cellsthat are lost due to natural turnover (see Eq. A14).

The loss rate,R, due to therapy is modeled by firstderiving an expression for the disintegration rate,d, ofradiolabeled125I per cell. This, multiplied by the efficiencyof cell kill per disintegration,E, yields the loss rate.

R 5 E z d (A7)

The disintegration rate per cell is obtained by assuming thatthe first generation of cells that enter S-phase during IUdRexposure will take up some fraction,f, of the steady-stateactivity,A, that is achieved during a prolonged infusion. Thesecond generation of cells will have one half of this originalamount plus the additionalfA obtained during DNA synthe-sis. The activity in each generation of cells, therefore, willbe fA plus one half of the parent cell activity. This may berepresented by:

d 5 fA z S Oi50

g

~1⁄2! iD (A8)

where g is an index for elapsed time and represents thenumber of replications that have occurred during infusion.The maximum amount of activity that a particular cell willhave at the end of an infinite number of replications will be

2 fA. Given its long, 60-d half-life, isotopic decay of125Iinherited from previous cell generations is considered neg-ligible for the Tpot values considered in the simulations.

Expression A8 is equivalent to:

d 5 2 fa z ~1 2 ~1⁄2!~t/Tpot!!. (A9)

The parameterg is replaced byt/Tpot, where Tpot is thepotential doubling time of the tumor cells. Equation A9may, in turn, be expressed as:

d 5 2 fA z ~1 2 e2at!. (A10)

As in Eq. A1, the parametera is the tumor cell growth rateand is equal to ln(2)/Tpot. The fraction,f, of steady stateactivity that may be accumulated by a tumor cell after onesynthesis (S) phase is equal to the activity concentration perunit administered activity per unit mass in the tumor,C,divided by the density of cells in the tumor:

f 5C

rcell(A11)

Given an infusion rate,i, and a whole-body clearance rate,l, the steady-state activity,A, is (see Methods):

A 5i

l(A12)

Using Eqs. A10 through A12, the loss rate due to therapy(Eqn. A7) is:

R 5 E z 2C

rcell

i

lz ~1 2 e2at! (A13)

The loss rate due to cell turnover,L, may be expressed interms of the exponential growth rate,a, and the cell lossfactor,f. For Gompertzian growth, the cell loss factor maybe related to the total number of tumor cells at a particulartime (N), the initial total number of cells (N0), and the totalnumber of cells when the tumor has reached its limitingmaximum volume (N`):

L 5 a z w~N! (A14)

w~N! 5 S lnS N

N0DY lnSN`

N0DD (A15)

The derivation of this relationship is presented elsewhere(15). Growth from a single cell to a macroscopic tumor maybe modeled by settingN0 equal to the number of cells whenthe tumor stops exhibiting exponential growth. Correspond-ingly, whenN , N0, f(N) is set to 0:

L 5 H 0a z w~N!

N , N0

N $ N0(A16)

1182 I. J. Radiation Oncology● Biology ● Physics Volume 41, Number 5, 1998

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1183Modeling of [125]IUdR continuous infusion therapy● G.SGOUROSet al