Page 1

Radioimmunotherapy with radioactive nanoparticles: Biological doses

and treatment efficiency for vascularized tumors with or without

a central hypoxic area

V. Bouchata?and V. E. Nuttens

Research Center in Physics of Matter and Radiation (PMR), Laboratoire d’Analyses par Réactions

Nucléaires (LARN), University of Namur (FUNDP), Rue de Bruxelles 61, B-5000 Namur, Belgium

C. Michiels

Unité de Recherche en Biologie Cellulaire (URBC), University of Namur (FUNDP),

Rue de Bruxelles 61, B-5000 Namur, Belgium

B. Masereel

Department of Pharmacy (DP), University of Namur (FUNDP), Rue de Bruxelles 61,

B-5000 Namur, Belgium

O. Feron

Unité de Pharmacothérapie (FATH), Université Catholique de Louvain (UCL),

Avenue Mounier 53, B-1200 Brussels, Belgium

B. Gallez

Laboratoire de Resonance Magnétique Biomédicale (CMFA), Université Catholique de Louvain (UCL),

Avenue Mounier 73, B-1200 Brussels, Belgium

T. Vander Borght

Center for Molecular Imaging and Experimental Radiotherapy (IRME), Université Catholique de Louvain

(UCL), Dr. G. Therasse 1, B-5530 Yvoir, Belgium

S. Lucas

Research Center in Physics of Matter and Radiation (PMR), Laboratoire d’Analyses par Réactions

Nucléaires (LARN), University of Namur (FUNDP), Rue de Bruxelles 61, B-5000 Namur, Belgium

?Received 7 September 2009; revised 23 February 2010; accepted for publication 1 March 2010;

published 29 March 2010?

Purpose: Radioactive atoms attached to monoclonal antibodies are used in radioimmunotherapy to

treat cancer while limiting radiation to healthy tissues. One limitation of this method is that only

one radioactive atom is linked to each antibody and the deposited dose is often insufficient to

eradicate solid and radioresistant tumors. In a previous study, simulations with the Monte Carlo

N-Particle eXtended code showed that physical doses up to 50 Gy can be delivered inside tumors

by replacing the single radionuclide by a radioactive nanoparticle of 5 nm diameter containing

hundreds of radioactive atoms. However, tumoral and normal tissues are not equally sensitive to

radiation, and previous works did not take account the biological effects such as cellular repair

processes or the presence of less radiosensitive cells such as hypoxic cells.

Methods: The idea is to adapt the linear-quadratic expression to the tumor model and to determine

biological effective doses ?BEDs? delivered through and around a tumor. This BED is then incor-

porated into a Poisson formula to determine the shell control probability ?SCP? which predicts the

cell cluster-killing efficiency at different distances “r” from the center of the tumor. BED and SCP

models are used to analyze the advantages of injecting radioactive nanoparticles instead of a single

radionuclide per vector in radioimmunotherapy.

Results: Calculations of BED and SCP for different distances r from the center of a solid tumor,

using the non-small-cell lung cancer as an example, were investigated for90Y2O3nanoparticles.

With a total activity of about 3.5 and 20 MBq for tumor radii of 0.5 and 1.0 cm, respectively, results

show that a very high BED is deposited in the well oxygenated part of the spherical carcinoma.

Conclusions: For either small or large solid tumors, BED and SCP calculations highlight the

important benefit in replacing the single ?-emitter90Y attached to each antibody by a90Y2O3

nanoparticle. © 2010 American Association of Physicists in Medicine. ?DOI: 10.1118/1.3368599?

Key words: shell control probability ?SCP?, biological effective dose ?BED?, radioimmunotherapy,

dosimetry, Monte Carlo, tumor model, nanomedicine

18261826Med. Phys. 37 „4…, April 20100094-2405/2010/37„4…/1826/14/$30.00© 2010 Am. Assoc. Phys. Med.

Page 2

I. INTRODUCTION

In radioimmunotherapy ?RIT?, cancer cells are killed thanks

to potential induction of immune response and to ionizing

radiation delivered by single radionuclides coupled to anti-

bodies. The radiation efficacy is mainly influenced by the

choice of the radioactive atom linked to each antibody as

well as the biokinetics and the biodistribution of radiolabeled

antibodies used to target a human tumor antigen. Clinical

studies have, however, shown that doses higher than 60–70

Gy inside tumors are required for treating solid and poorly

vascularized cancers.1Such doses are not easily obtained

with such an approach, in spite of numerous efforts made to

increase antibody accumulation and penetration inside the

tumor.2–5

In a previous paper, we have proposed improving the tu-

mor dose deposition, and therefore, the treatment response

by replacing the single radioactive atom bound to each

monoclonal antibody ?mAb? by a 5 nm diameter inorganic

nanoparticle composed of numerous radioactive and nonra-

dioactive atoms.6The aim of such a treatment is to deliver a

much higher dose to the tumor. Moreover, inorganic nano-

particles can contain different types of radionuclides ??, ?,

?, or x-ray emitters? suited to both diagnostic and therapeutic

applications.7The possible mix between different radiations

will thus represent a practical tool for theragnostics.8This

kind of radiolabeled antibody do not exist yet, but the devel-

opment of both organic and inorganic nanoparticles that spe-

cifically target tumor cells or cancer vasculature has received

considerable interest these past few years. Progress has been

made to improve the stability of these nanoparticles within a

biological microenvironment.7For example, by creating

spherical radioactive nanoparticles whose diameter does not

exceed 5 nm, the radiolabeled antibodies are small enough to

reduce the opsonization process by the reticuloendothelial

system ?RES? responsible for their rapid clearance from

blood circulation.9To avoid that any radioactive atom come

off of the nanoparticle, it is possible to cover the nanopar-

ticles with a thin layer of inert and biocompatible matter,

such as Au or C. These biocompatible surfaces could also be

coated by polymeric macromolecules, such as polyethylene

glycol. This type of macromolecule enhances the chemical

stability of nanoparticles in an aqueous environment and can

act as a protective layer against the RES.7,9–13Finally, nano-

particles of 5 nm diameter possess a large surface capable of

accommodating large number of functional groups so that

more than one antibody can be conjugated per particle. The

more antibodies per nanoparticles there are, the higher will

be the biological half-life and uptake. Indeed, in vivo imag-

ing on animals and humans has shown that targeted nanopar-

ticles can be preferentially distributed in tumor mass after

injection and relatively low accumulation of these nanopar-

ticles is observed in other organs such as the spleen or liver,

which means that antibodies can well be used to target the

nanoparticles to specific anatomical sites.7,14–16

Beta-emitting radioactive nanoparticles have already been

created and investigated with promising results. For ex-

ample, researchers at the University of Missouri-Columbia

have developed an efficient methodology to synthesize radio-

active gold nanoparticles ?12–18 nm? containing

?-emitters ??max=0.96 MeV; half-life of 2.7 days?. Coated

with gum arabic ?polysaccharide glycoprotein?, they did not

observe any aggregation or decomposition of these

nanoparticles in saline solution. Moreover, glycoprotein mol-

ecules have receptors in the liver and biodistribution studies

performed on mice clearly showed a significant localization

of198Au nanoparticles in liver.17,18Finally, therapeutic effi-

cacy was tested on mice bearing a model of human prostate

cancer. An uptake of NPs in prostate cancer cells and a de-

crease in the tumor volume are clearly observed after a direct

injection into the solid tumor.19More recently, Wu and

co-workers20proposed synthesizing radioactive nanopar-

ticles of90Y by using a protein cage of apoferritin. First tests

were performed by diffusing89Y and phosphate ions into the

cavity of apoferritin. The diameter of these phosphate/

apoferritin nanoparticles was around 8 nm after functional-

ization.

Despite these promising results to produce radioactive

nanoparticles for radioimmunotherapy, a question remains:

What would be the benefit of such configuration involving

antibodies labeled with several

nanoparticles in term of dose and tumor control. This paper

attempts to answer to that question with the help of Monte

Carlo simulations and a simple90Y configuration.

Monte Carlo N-Particle eXtended ?MCNPX? simulations

have been used to evaluate the physical dose around and

throughout a spherical solid tumor. Dosimetry calculations

were performed for the beta-emitting radionuclide90Y2O3

and preliminary results showed that viable tumor cells re-

ceive physical doses of up to 50 Gy everywhere inside the

tumor. This observation is still valid even for large nonuni-

form distribution of the total activity inside the tumor. More-

over, dose deposited around the tumor remains sufficiently

weak to avoid affecting the surrounding healthy tissues.6

However, the efficacy of a treatment against cancer does not

depend only on the physical absorbed doses but also on other

radiobiological parameters such as the radiosensitivity of the

targeted tissues, the doubling time of the cancer cells, the

repair process of sublethal damage taking place between two

irradiations, and the effect of hypoxia for which the radiosen-

sitivity changes according to whether cells are anoxic or well

oxygenated.21–23All these biological effects are taken into

account in the biological effective dose ?BED? and tumor

control probability ?TCP? models. Consequently, they offer

interesting tools to evaluate, at least theoretically, the effi-

cacy of a specific treatment. Moreover, these mathematical

models are often used to compare different radiotherapy

techniques and to predict the most appropriate treatment for

individual patients.24–28

In this work, BED and control probability for successive

concentric spherical shells inside and around the tumor ?shell

control probability ?SCP?? have been calculated for the non-

small-cell lung cancer ?NSCLC?. This type of tumor is a

good example of carcinomas for which patients have poor

survival.29,30Indeed, this cancer is relatively radioresistant

and displays important hypoxic areas. Doses larger than

198Au

198Au

90Y atoms configured as

1827Bouchat et al.: BED and SCP for radioactive nanoparticles1827

Medical Physics, Vol. 37, No. 4, April 2010

Page 3

60–70 Gy are thus required to cure NSCLC.1,30,31In radio-

immunotherapy, doses up to 60 Gy can be delivered within

the tumor after several injections of radiolabeled antibodies

over several days or weeks. Unfortunately, NSCLC is also

well-known to be fast growing, with a cell doubling time

ranging from 2.5 to 3.3 days.1So it is better to deliver a high

total dose during a single injection rather than in small frac-

tions over a longer time. In external beam therapy, such

doses can be deposited at the tumor in one session but a great

part of the surrounding healthy tissues will be also irradiated.

In these conditions, RIT using radioactive nanoparticles

coupled to an antibody seems to be a good method for treat-

ing solid carcinomas such as the NSCLC, since a single in-

jection will be able to deliver doses larger than 60–70 Gy to

the targeted tumor. It is, however, important to verify

whether doses to healthy lung tissue remain lower than 30

Gy to avoid late complications like pneumonitis or

fibrosis.1,22,29In the present paper, the physical doses D?r?

previously simulated with MCNPX 5.0 are converted into

BED and SCP distributions. The main objective is, then, to

analyze the cell cluster-killing efficiency resulting from the

use of antibodies coupled to radioactive nanoparticles and

compare these results to BED and SCP calculations when a

single ?-emitter90Y is coupled to each antibody.

II. BED AND SCP CALCULATIONS

II.A. Tumor model

Absorbed doses as a function of distance from the center

of the tumor were simulated according to the MCNPX 2.5.0

code. This Monte Carlo software requires accurate informa-

tion on the tumor structure and the microscopic distribution

of radioactivity delivered by the radiolabeled antibodies. In

order to accurately provide such information, we developed a

new model for a spherical vascularized tumor in which the

antibody distributions inside the tumor are uniform or het-

erogeneous. This tumor model is described in detail in a

previous publication and consists of a set of spherical cell

clusters of 250 ?m radius arranged in a simple cubic lattice

structure as illustrated in Fig. 1.6The spherical tumor of 0.5

cm radius can contain a maximum number of 3591 cell clus-

ters. When the spherical tumor has a radius of 1.0 cm, the

number of cell clusters increases to a value of 31 071 ?Table

I?. The matter around each cell cluster represents the vascu-

larized stroma containing a pre-existing blood network or

new vessels created during angiogenesis. The radiolabeled

antibodies can penetrate inside the tumor through this vascu-

lature to surround the various cell clusters. The distribution

of radiolabeled antibodies inside the tumor is uniform if all

cell clusters in the cubic lattice have the same probability to

be reached by an antibody. However, it is well-known that

during angiogenesis, the outer region of the tumor is better

vascularized than the center.32–34The density of blood ves-

sels decreases toward the center and the heterogeneity in

blood flow thus generates a nonuniform distribution of radio-

activity in tumors.35In order to take this decrease in vascu-

lature into account, linear and exponential distributions of

radiolabeled antibodies inside the tumor have been model-

ized by subdividing the tumoral sphere into concentric shells

of 0.05 cm thickness, with differing probabilities to be

reached by an antibody. In our model, the probability that an

antibody reaches a cell cluster located near the tumor radius

hits a maximum value of 1.0. Inversely, the probability de-

creases to 0.1 for antibodies which reach the central cell

cluster of the tumor. Between those values, the probability

may decrease linearly ?Eq. ?1?? or exponentially ?Eq. ?2??.

LP?m? = 0.1+?

1.0− 0.1

NbShell− 1?? ?m − 1?,

?1?

(b)(a)

FIG. 1. Spherical tumor of 0.5 cm radius subdivided into 3591 cell clusters.

?a? The first picture is a three-dimensional arrangement of these cell clusters

when 25% of the cell clusters are poorly oxygenated at the center of the

tumor. Picture ?b? corresponds to a cross-section of the center of this mod-

elized tumor with cell clusters inside a cubic lattice of 500 ?m width.

TABLE I. Parameters used for calculating the total activity inside tumors of different radii ?0.5 and 1.0 cm? when

antibodies are linked with radioactive nanoparticles or with single90Y atoms.

ParametersUnits

¯

cm

¯

h

h

mAb/cm2

¯

MBq

RT=0.5 cm

RT=1.0 cm

NP of90YSingle90YNP of90YSingle90Y

Total # of cell clusters

Hypoxic radius

# of nonhypoxic cell clusters

Physical half-life

Biological half-life

Covering fraction

# of90Y per MAb

Calculated total activity

3591

0.32

2702

64.1

72

3.8?107

1000

3.5

3591

0.32

2702

64.1

72

1010

1

0.9

31 071

0.64

23 342

64.1

72

2.5?107

1000

20

31 071

0.64

23 342

64.1

72

1010

1

7.5

1828Bouchat et al.: BED and SCP for radioactive nanoparticles1828

Medical Physics, Vol. 37, No. 4, April 2010

Page 4

EP?m? = exp?ln?0.1? +?ln?1.0? − ln?0.1?

NbShell− 1?? ?m − 1??,

?2?

where NbShell represents the total number of shells obtained

by subdividing the tumor radius by the thickness of concen-

tric shells. “m” may vary from 1 for the cell cluster located at

the center of the tumor to Nbshell for the cell clusters located

in the vicinity of the tumor surface.

Furthermore, it has to be noted that the center of large

tumors often displays poorly oxygenated or hypoxic cells

caused by the lack of vasculature. This situation can lead to

the formation of a necrosed core. Central hypoxic areas are

introduced in our tumor model by simply considering that

the probability that an antibody reaches this region is null.

The radius of the hypoxic core can vary widely with time

and tumor type. In this work, we have chosen radii of 0.32

and 0.64 cm for both tumor radii of 0.5 and 1.0 cm, respec-

tively. In this case, numbers of hypoxic cell clusters are 889

for the first tumor and 7729 for the second, meaning that

about 25% of the total number of cell clusters are not oxy-

genated ?dark gray cell clusters in Fig. 1?. Radii of hypoxic

cores and numbers of normally oxygenated cell clusters are

given in Table I for tumors of 0.5 and 1.0 cm radii.

II.B. Absorbed dose calculations

This tumor model was used to calculate the deposited

dose inside and around cancer cells as a function of the dis-

tance from the center for two tumor radii ?0.5 and 1.0 cm?. In

these simulations, tumors were irradiated by90Y2O3nano-

particles of 5 nm diameter. A 5 nm diameter nanoparticle of

90Y2O3can contain a maximum of 1.73?103atoms of

yttrium-90, but we assume that the delay time between the

antibodies’ injection and their binding with tumoral antigens

is 2 days, which means that only 60% of radioactive atoms

inside each nanocluster are still radioactive when the nano-

clusters reach the cancer cells.

Energy deposition for different distances “r” from the tu-

mor center was determined by using SMESH tally proposed

in the MCNPX code. The latter is capable of studying the

electron transport through matter by taking into account the

loss of energy, multiple scattering angles and “bremsstrah-

lung.” All these physical processes are considered by using

the photon-electron mode and the default PHYS cards for

electron and photons with a cutoff energy at 0.005 MeV for

both particles. The number of histories ?NPs? was set to

50?106particles and ? spectral data for yttrium-90 was

taken from tables on the RADAR site ?www.doseinfo-

radar.cp/RADARDecay.html?. The SMESH tally builds vir-

tual three-dimensional spherical grids superimposed on the

geometry of our tumor model and gives the energy deposi-

tion, in MeV/g per emitted particle, into each concentric

spherical shell independently of the composition or density

of each material used to modelize the tumor and its sur-

rounding tissues. This energy deposition is then converted

into deposited doses D?r?, in Gy, by the formula ?3? pro-

posed by Nuttens36

D?r? = 21.34? E?r? ? p ? A˜/?eff,

?3?

where E?r? is the energy deposition at a distance r from the

center of the tumor, p is the average number of electrons

emitted per disintegration, ?eff?=ln?2?/T1/2

decay constant, assuming a monoexponential decay.37?effis

given by the sum of the physical decay and the biological

clearance rate constants ??eff=?phy+?biol?.6,36The physical

half-life of yttrium-90 is well-known and corresponds to 64.1

h.38–40Based on pharmacokinetic studies on90Y, Wiseman

and co-workers41estimated that the biological blood half-life

of the antibody varies between 22 and 140 h. For our simu-

lation, we supposed a biological half-life for healthy tissues

of about 72 h.42,43Clearance of the antibodies inside the

tumor must be slower than in healthy tissues. But due to the

short physical half-life of yttrium-90, the biological half-life

in the tumor is mainly determined by the physical half-life of

yttrium-90. So, for our simulations, we also supposed a bio-

logical half-life of 72 h for tumor tissues. In these conditions,

the effective decay constant ??eff? was evaluated at

2.04?10−2h−1both for healthy tissues and tumors. The to-

tal activity A˜was calculated according to the expression ?4?

given in a previous publication6

eff? is the effective

A˜= ?phys? nu? na? nmAb,

?4?

where nuand narepresent, respectively, the number of non-

hypoxic tumor cell clusters and the number of radioactive

atoms per nanoparticle ?Table I?. nmAbis the quantity of

monoclonal antibodies which surrounds each cell cluster. Its

value was calculated by multiplying the surface of the cell

cluster by the covering fraction defined as the number of

bound mAbs per unit of tumor surface. The covering fraction

was adjusted to ensure a maximal dose of 30 Gy around the

tumor surface. In our model, this value is reached if covering

fractions are 3.8?107and 2.5?107mAb/cm2for solid tu-

mors of 0.5 and 1.0 cm radii, respectively. These values are

clearly lower than typical values for covering fractions given

for monoclonal antibodies coupled to a single radionuclide,

which range from 108to 1010mAb/cm2.39With parameters

listed in Table I, we calculated the total activity according to

Eq. ?4? above. The latter represents the activity that would be

injected directly into tumors. When the radioactive nanoclus-

ters are used, the total activities are about 3.5 MBq for tumor

of 0.5 cm radius and 20 MBq for tumor of 1.0 cm radius.

These values are very large if we compare them to cumulated

activities obtained when a single90Y atom is linked to each

antibody. Indeed, despite of a maximal covering fraction of

1010mAb/cm2, total activities for single90Y per mAb de-

crease to 0.9 and to 7.5 MBq for tumor radii of 0.5 and 1.0

cm, respectively. These total activities ?Table I? will be used

to compute the TCP and SCP.

The thickness between each spherical mesh was reduced

to 0.05 cm to increase the number of shells inside and around

the tumor. This value is less than 1 mm, which is the minimal

thickness normally imposed by the software

2.5.0.44,45In order to test the influence of shell thicknesses on

deposited energies simulated by

MCNPX

MCNPX, a comparison

1829Bouchat et al.: BED and SCP for radioactive nanoparticles1829

Medical Physics, Vol. 37, No. 4, April 2010

Page 5

among three different segmentations of concentric spherical

shells ?0.5, 1.0, and 2.0 mm? was performed. As shown in

Fig. 2 for tumors of 0.5 cm radius, the three curves repre-

senting the energy deposition for the three different thick-

nesses display great disparities. When the activity is uni-

formly dispersed throughout the tumor, deposited energies

simulated with MCNPX tend to decrease in tumoral and nor-

mal tissues with the number of concentric spherical shells.

For linear distribution of antibodies, predicted deposited en-

ergy inside the tumor is diminished when the number of

segmentations decreases, whereas it increases in healthy tis-

sues. Finally, curves plotted for the exponential distribution

of antibodies show an increase and a shift in the peak toward

the right when the thickness between meshes decreases. It is

worth noting that these differences in simulated energy depo-

sition according to the choice of the segmentation disappear

for a larger tumor radius. Indeed, as illustrated in Fig. 2 for a

tumor with a radius of 1.0 cm, curves of deposited energies

are similar whatever the thickness of concentric spherical

shells, especially for uniform and linear antibody distribu-

tion. When the antibodies are distributed exponentially, an

increase in the maximal dose deposited near the tumor sur-

face is observed when the thickness between concentric

spherical shells decreases. Consequently, the nature of the

mesh chosen to simulate energy deposition may have an in-

cidence on the shape of the predicted absorbed dose distri-

bution curve for small tumor radii. The segmentation of 0.05

cm was chosen to provide more points on the energy depo-

sition curve and because linear or exponential probability

decreases for each cell cluster to be reached by antibodies

have also been done by steps of 0.05 cm. The latter corre-

sponds to the diameter chosen for cell cluster constituting the

tumor.6

II.C. BED„r…, error, and SCP„r… calculations

BED distribution and TCP are adequate methods to ana-

lyze the radiobiological effects of treatment resulting from

radioactive nanoparticles. Both BED and TCP distributions

are mathematical models often used to predict the response

to irradiation of normal and tumor tissues, which is very

useful for evaluating the best treatment for each individual

patient.25,42,46Absorbed doses D?r? were thus converted into

BED according to Eq. ?5? given by

BED?r? =D?r?

q

??eff+ ??

+

?eff

D?r?2

q2?/?.

?5?

In our model, a spherical tumor is subdivided in cell clusters

and BED?r? is defined as the dose required for killing a cell

unit located at a distance r from the center of the tumor. D?r?

is the absorbed dose computed by our MCNPX simulations.

?eff, ?, ?/?, and q are biological parameters ?see hereunder?.

This equation is valid for both tumors and healthy tissues,

but differences exist between radiobiological factors for tu-

moral and normal tissues. The values of these parameters for

the three antibody distributions are presented in Table II. The

choice of these values is explained in the following two para-

graphs.

All terms in Eq. ?5? have a biological significance. Ioniz-

ing radiation can affect living tissues on a cellular level by

FIG. 2. Effect of the thickness of concentric spherical shells on energy depo-

sition ?per gram and per emitted particle? for tumors with radii of 0.5 and

1.0 cm: 0.5 ?solid lines?, 1.0 ?dotted lines?, and 2.0 mm ?dashed lines?.

Uniform, linear, and exponential distributions of antibodies are taken into

account.

TABLE II. Biological parameters and their references used for BED and SCP calculations for tumor and healthy

tissues.

Biological factors

TumorHealthy tissues

Range values Tumoral tissuesErrorRange valuesHealthy tissuesError

Tbio

?/? ?Gy?

? ?h−1?

? ?#/cm3?

? ?Gy−1?

1/2?h?

22–140a

?5b–d

0.3–2.5b,i

107–108

0.1–1.0l

72

?24

?5

?0.5

¯

¯

22–140a

?5b–d

0.3–2.5b,i

107–108

¯

72

?24

?2

?0.5

…

…

10e–g,b,h

1.39b,h

5?107 k

0.35e

3e–g,b,h

0.46g,b,j

5?107 k

0.031m

aReference 41.

bReference 49.

cReference 58.

dReference 59.

eReference 1.

fReference 22.

gReference 42.

hReference 51.

iReference 53.

jReference 54.

kReference 60.

lReference 37.

mReference 61.

1830Bouchat et al.: BED and SCP for radioactive nanoparticles 1830

Medical Physics, Vol. 37, No. 4, April 2010

Page 6

breaking chemical bonds within DNA molecules. The linear

term in expression ?5? refers to single ionizing events, which

directly provoke double-strand breaks in DNA. This kind of

damage,alsocalledtype A damage

co-workers,47,48is necessarily lethal because it is not repair-

able. Inversely, the quadratic component describes the cellu-

lar death as a consequence of two separated sublethal dam-

ages ?type B damage? and these damages can be repaired

when the lapsed time between the two hits is long

enough.26,37,47,49–52The ratio ?eff/??eff+?? was incorporated

in the expression to take into account the reduction in cell

destruction due to repair of sublethal damage during continu-

ous irradiation.26,47,51,52This term is deduced from the dose

protraction factor for continuous irradiation ?T=?? and var-

ies between 0 and 1. ??=ln?2?/T1/2

constant that quantifies the rate of sublethal damage repair.

The half-time of DNA repair ?T1/2

tissues may range from a few minutes to several hours.49,53

In our simulation, calculations have been performed with a

T1/2

time most widely adopted by authors.42,49,54The correspond-

ing repair constant ? is then 0.46 h−1. For a solid tumor, a

lower repair half-time of 0.5 h is generally proposed for

simulations, giving a repair constant ? of 1.4 h−1.49,51This

higher value of ? limits the chance of producing lethal dam-

age by interaction with a second hit, thus reducing the over-

all treatment efficacy. Also called cellular radiosensitivity, ?

and ? are tissue specific parameters expressed in Gy−1and

Gy−2, respectively. Both parameters are determined by a fit

of the cell survival curves.27,49,55–57The ?/? ratio gives an

indication of the relative importance of the linear and qua-

dratic terms and determines the shape of the cell survival

curve. It is well recognized that ?/? ratios less or equal to 5

Gy are generally observed for late-responding tissue. In-

versely, values higher than 5 Gy are observed for a majority

of tumors.49,58,59For our simulations, values of 3 and 10 Gy

were chosen for normal and tumoral tissues, respectively,

because they are the most often used values.1,22,42,49,51

In our tumor model, we have introduced the possibility of

having a hypoxic core. In these conditions, all cell clusters at

the center of the tumor are less radiosensitive than those

which are well oxygenated.21–23,55So, to differentiate ? and

? values for poorly and normally oxygenated cell clusters,

the “q” parameter was introduced. This parameter, also

called hypoxia reduction factor, was first proposed by Jones

and co-workers62to calculate BED for hypoxic cells. They

supposed that the radiosensitivity coefficient ? and ? for a

specific tissue are reduced, respectively, by q and q2when

cells are hypoxic. Indeed, experimental studies have shown

that poorly vascularized areas are up to three times more

resistant to ionizing radiation than proliferating cells.30,62So,

the simplest model is to assume that q is equal to 1 when

cells are well oxygenated and to 3 when cells are hypoxic.

The biological parameters are strongly dependent on the

structure, the shape, and the cellular composition of a tumor.

Experimental evaluations of these radiobiological parameters

often lead to variable values. For our simulations, we simply

byDale and

Rep? is the exponential rate

Rep? for normal and tumor

Repof 1.5 h for healthy tissues, which is the repair half-

took the most recurrent values proposed in the literature. In

order to evaluate the impact of variation in these biological

parameters on BED distributions, errors in BED were calcu-

lated according to Eq. ?6?

?BED=?BED

???/?????/?? +?BED

??

?? +?BED

?eff

??eff

+?BED

?D

?D.

?6?

The last term ??BED/?D??D is negligible because D is com-

puted to the point that ?D is minimal. Since ??eff=??bio

+??physand ??physis negligible compared to ??bio, then

??eff= −ln 2

Tbio

2?Tbio.

?7?

Values for ???/??, ??, and ?Tbioused to calculate ?BED

are presented in Table II ?column 4 for tumor and column 7

for healthy tissues?. Partial derivatives are determined by

?BED

???/??=

?

? + ?.D˙0

2

?2.

− 1

??/??2,

?8?

?BED

??

=

− ?

?? + ??2.

1

??/??.D˙0

2

?2,

?9?

?BED

??eff

BED?r? distributions calculated according to Eq. ?5? will

be used to determine SCP?r?, which is defined as the prob-

ability that no cell cluster inside each spherical mesh located

at a distance r from the center of the tumor survives irradia-

tion. SCP?r? is based on the TCP-NTCP concept. Indeed,

SCP?r? is capable of predicting the survival rate of cell clus-

ters after irradiation inside the tumor but, also, in the sur-

rounding healthy tissues. The control probability for a single

cell cluster inside a spherical shell located at a distance r

from the center of the tumor defined by the SMESH tally

was first calculated. Cell clusters are assumed to be dead

when all clonogenic cells inside the aggregate are killed by

single or double hit events. As TCP, this probability can be

calculated using the Poissonian form expressed by28,37,49

= −D˙0

2

?2+

− 1

??2+ ???2.

1

??/??. D˙0

2. ?2? + ??.

?10?

CCP?r ? ?r? ? exp?− N · exp?− ? · BED?r???,

?11?

where CCP?r??r? is the cluster control probability of a cell

cluster located at a distance r??r from the center of the

tumor, ?r representing the small variations between the dis-

tance separating the cluster from the tumor center and the

real distance r of the spherical shell defined by the SMESH

tally. N is the number of cells before irradiation inside each

cube defined by the cubic lattice of 500 ?m length sides

used to subdivide the tumor sphere. The cell density is as-

sumed to be uniform throughout the tumor and healthy tis-

sues. So, with a value of 5?107cell/cm3, the number N of

cells in each cube of the cubic lattice corresponds to 6250.60

1831 Bouchat et al.: BED and SCP for radioactive nanoparticles 1831

Medical Physics, Vol. 37, No. 4, April 2010

Page 7

Equation ?11? requires a defined value of ? and not only

the ?/? ratio. This is probably the most difficult radiobio-

logical factor to determine because it is clearly dependent on

the patient and the type of tumor. Atthey and co-workers37

proposed that plausible values for ? may range from 0.1 to

1.0 Gy−1. For non-small-cell lung cancer, ? parameter may

vary between 0.3 and 0.4 Gy−1.1So, we have chosen a mean

value of 0.35 Gy−1for calculating BED and CCP. A lower

value of 0.031 Gy−1was used for the ? parameter of healthy

lung tissues according to Dubray and co-workers.61Finally,

shell control probability for the three different radiolabeled

antibody distributions ?uniform, linear, and exponential? is

defined by the following product:

SCP?r? =?

i

K

CCPi?r ? ?r?,

?12?

where K represents the number of cell clusters in each

spherical shell defined by the SMESH tally. The TCP can be

then calculated by the product of all SCP?r? values, with r

varying from 0 ?center of the tumor? to rT?surface of the

tumor?.

TCP=?

r=0

r=rT

SCP?r?.

?13?

III. RESULTS AND DISCUSSION

Results presented in this work highlight the advantages of

replacing a single radionuclide per antibody with inorganic

nanoparticles containing a high number of radioactive atoms.

BED?r? and SCP?r? were determined according to Eqs. ?5?

and ?12?, which take into account the effects of inhomoge-

neous dose distributions, cellular repair effects between two

single-hit events, and the influence of a radioresistant hy-

poxic core within the tumor. We investigated whether the

nature of the spatial distribution of antibodies and the pres-

ence of a hypoxic center affect the level of damage caused to

cancer cells and to the cells in the surrounding tissue. Effects

of repopulation in normal and tumor tissues will be ignored

because they are less important than damage repair effects

during continuous irradiation.63BED was calculated for two

spherical tumors of 0.5 and 1.0 cm radii irradiated by 5 nm

diameter nanoparticles of90Y2O3. Results are summarized

in Table III for tumor radius of 0.5 cm and in Table IV for

tumor radius of 1.0 cm. They present values for physical

doses D?r? computed with MCNPX, biological effective doses

calculated with Eq. ?5?, and errors on BED?r? due to varia-

tions in biological parameters, obtained from Eq. ?6? and

expressed in %.

Tables III and IV resume the most important values for

the physical and biological doses inside and outside the tu-

mor. The first are the D and the BED values at the center of

the tumor, written down “Cent” in the second column of

tables. When this center is well irradiated with doses of up to

60 Gy, the complete destruction of the tumor would be pos-

sible. In the second column, we can also find “Max” which

gives the maximum values, which can be deposited within

the tumors. To analyze the impact of the radiolabeled nano-

particles on healthy tissues, the value of D and BED at the

surface of the tumor ?Surf? and 1 mm beyond the tumor

surface ?Out? are also given. The best situation is to obtain a

high physical or biological dose at the tumor surface where

the proliferation is probably the most important and a mini-

mum value for D or BED in the surrounding healthy tissues.

Finally, values of D and BED at the center, at the surface and

1 mm beyond the tumor surface, as well as the maximum

physical and biological doses deposited inside the tumor are

TABLE III. Calculated values of D ?in Gy?, BED ?in Gy?, and ?BED ?in %? for a tumor of 0.5 cm radius with or without hypoxia according to data in Tables

I and II. For each antibody distribution ?uniform, linear, and exponential?, the table gives the maximum doses deposited inside the tumor ?Max?, doses at the

center ?Cent?, doses at the surface ?Surf?, and 1 mm beyond the surface ?Out? of the tumor. Data are calculated for a single90Y atom labeled antibody and for

a radioactive nanoparticle labeled antibody.

Tumor of 0.5 cm radius

Single radionuclide total activity: 0.9 MBqRadioactive nanoparticle total activity: 3.5 MBq

No hypoxia

BED

?Gy?

25% hypoxic

BED

?Gy?

No hypoxia

BED

?Gy?

25% hypoxic

BED

?Gy?

D

?Gy?

?BED

?%?

D

?Gy?

?BED

?%?

D

?Gy?

?BED

?%?

D

?Gy?

?BED

?%?

Uniform Cent

Max

Surf

Out

Cent

Max

Surf

Out

Cent

Max

Surf

Out

44

45

11

5

28

34

12

5

28

34

12

5

47

47

11

5

29

35

12

5

29

35

12

5

6

6

2

6

4

5

2

7

4

5

2

7

16

35

12

5

13

33

13

6

12

32

14

6

51

5

2

7

1

5

2

8

1

4

2

8

173

173

42

18

110

132

45

19

67

122

53

22

217

217

45

22

128

157

48

24

73

144

57

28

18

18

6

20

12

14

6

21

8

14

7

23

63

135

48

20

51

127

53

22

48

126

55

23

21

162

51

26

17

150

57

29

16

149

59

30

3

15

6

21

2

14

7

23

2

14

7

23

36

12

6

4

34

14

6

12

34

14

6

Linear

Exponential

1832Bouchat et al.: BED and SCP for radioactive nanoparticles1832

Medical Physics, Vol. 37, No. 4, April 2010

Page 8

calculated for a tumor with or without a hypoxic core and for

the three types of antibody distributions: Uniform, linear, and

exponential.

III.A. BED distributions for tumors without hypoxia

III.A.1. Nanoparticle labeled antibody

Figures 3 and 4 show a comparison between absorbed

?solid gray lines? and biological ?solid black lines? dose dis-

tributions for nonhypoxic tumors with a radius of 0.5 or 1.0

cm after injection of radioactive nanoparticles of 5 nm diam-

eter containing90Y2O3distributed uniformly, linearly, or ex-

ponentially. A comparison between the three graphs of Figs.

3 and 4 shows that types of antibody distribution ?uniform,

linear, or exponential? in tumors affect noticeably the shape

of absorbed and biological dose distributions, generating

lower or higher doses at different distances from tumor cen-

ter. However, the overall shape of the physically absorbed

dose curve is similar to that of biological dose and this is true

for each antibody distribution. The conversion of physical

doses into BED shows higher values for doses inside the

tumor, especially at the center when antibodies are dispersed

uniformly. At this position, a difference of about 45 and 30

TABLE IV. Calculated values of D ?in Gy?, BED ?in Gy?, and ?BED ?in %? for a tumor of 1.0 cm radius with or without hypoxia according to data in Tables

I and II. For each antibody distribution ?uniform, linear, and exponential?, the table gives the maximum doses deposited inside the tumor ?Max?, doses at the

center ?Cent?, doses at the surface ?Surf?, and 1 mm beyond the surface ?Out? of the tumor. Data are calculated for a single90Y atom labeled antibody and for

a radioactive nanoparticle labeled antibody.

Tumor of 1.0 cm radius

Single radionuclide total activity: 7.5 MBq Radioactive nanoparticle total activity: 20 MBq

No hypoxia

BED

?Gy?

25% hypoxic

BED

?Gy?

No hypoxia

BED

?Gy?

25% hypoxic

BED

?Gy?

D

?Gy?

?BED

?%?

D

?Gy?

?BED

?%?

D

?Gy?

?BED

?%?

D

?Gy?

?BED

?%?

UniformCent

Max

Surf

Out

Cent

Max

Surf

Out

Cent

Max

Surf

Out

51

51

13

6

25

45

15

7

17

50

22

11

54

56

13

7

26

48

15

12

17

54

23

12

7

7

2

8

4

6

2

9

3

6

3

13

31

55

17

9

0

54

20

11

0

59

25

14

1

7

3

132

135

33

16

49

117

38

19

44

131

57

28

157

162

34

20

52

137

40

24

46

156

62

39

14

15

5

18

6

13

5

20

6

14

7

27

721

14

6

22

?1

14

6

25

?1

15

8

29

51

16

8

1

50

19

10

1

54

24

12

132

42

21

3

131

50

25

2

141

62

30

158

45

27

1

156

54

34

1

170

67

43

10

1

6

3

12

1

7

4

14

Linear

Exponential

FIG. 3. Comparison of D?r? and BED?r? profiles for90Y2O3nanoparticle

labeled antibodies and for single90Y labeled antibodies, in Gray unit, as a

function of the distance r from center for tumor with a radius of 0.5 cm and

for three different antibodies distributions: Uniform, linear, and exponential.

The tumors do not have hypoxic cores and dotted vertical lines represent the

tumor radius. Solid black curves for all spectra are BED?r? and solid gray

curves are D?r? doses. Values for BED?r? were determined according to

parameters given in Table II and Eq. ?5? with q=1 for both tumor and

healthy tissues.

FIG. 4. Comparison of D?r? and BED?r? profiles for90Y2O3nanoparticle

labeled antibodies and for single90Y labeled antibodies, in Gray unit, as a

function of the distance r from center for tumor radius with a radius of 1.0

cm and for three different antibodies distributions: Uniform, linear, and

exponential. The tumors do not have hypoxic cores and dotted vertical lines

represent the tumor radius. Solid black curves for all spectra are BED?r? and

solid gray curves are D?r? doses. Values for BED?r? were determined ac-

cording to parameters given in Table II and Eq. ?1? with q=1 for both tumor

and healthy tissues.

1833Bouchat et al.: BED and SCP for radioactive nanoparticles1833

Medical Physics, Vol. 37, No. 4, April 2010

Page 9

Gy is observed between the absorbed dose and the biological

dose for tumors of 0.5 and 1.0 cm radii, respectively. This

increase is clearly due to the presence of the quadratic com-

ponent of the calculated BED. For the linear or exponential

distributions of antibodies, maximum peaks are observed at

0.25 or 0.40 cm for a tumor radius of 0.5 cm and at 0.70 or

0.90 cm when the tumor radius reaches 1.0 cm. These maxi-

mum doses are clearly located near the tumoral surface

where the cell proliferation is the most important. For both

tumor radii, BED?r? at 1 mm beyond the tumoral surface

?i.e., in healthy tissue? remains inferior to 30 Gy, except for

the exponential distribution of antibodies for the 0.1 cm ra-

dius tumor where the biological dose reaches 39 Gy. This

last result highlights how significantly radiolabeled nanopar-

ticles penetrate deeply inside the tumor rather than staying in

the vicinity of the surface. Indeed, the more the distribution

is exponential, the more that dose deposition inside healthy

tissues is important. It is also possible to use radionuclides

with shorter emission ranges to increase the sharpness of the

falloff near the surface of the tumor on the absorbed dose

and the biological dose curves.

As there exist uncertainties in the values for different bio-

logical parameters proposed in literature, error bars on bio-

logical dose curves have been added to see how the variation

in ? and ?/? can influence initial BED values ?Figs. 3 and

4?. The difference between biological and absorbed doses is

strongly dependent on the choice of radiobiological factors.

The decrease or increase in BED values is mainly due to

variations in the repair half-time ?T?? parameter. For a tumor

of 0.5 cm radius, an increase in T?values from 0.5 to 1.5 h

generates an increase in BED maximal values from 217 to

301 Gy, from 157 to 205 Gy, or from 144 to 185 Gy for

uniform, linear, or exponential antibody distributions, respec-

tively. Such an observation makes sense because cells with

higher repair half-time for sublethal damage have a higher

probability of undergoing cell death after an interaction with

a second radiation. BED values are also weakly reduced

when ?/? increases. If a ?/? ratio of 15 Gy is used in the

model instead of 10 Gy, the maximal BED value for a tumor

of 0.5 cm radius changes from 217 to 203 Gy, from 157 to

148 Gy, and from 144 to 136 Gy for uniform, linear, or

exponential antibody distributions, respectively. The fact that

BED values are reduced is logical because a diminution in

the ?/? ratio induces a decrease in the quadratic term of Eq.

?5?. However, BED values in healthy tissues are lower than

BED inside the tumor, although values of ? and ?/? are

about three times less than in normal tissues.

All these results demonstrate how important it is to know,

with accuracy, the biodistribution of antibodies and the bio-

logical factors when BED is calculated inside the tumor.

However, with values up to 60 Gy everywhere inside the

tumor, biological doses required to treat a cancer remain suf-

ficient even if ?/? or ? varies within a plausible range of

values. Moreover, in most cases, BED values lower than 30

Gy are observed in surrounding healthy tissues indicating

that they will be spared. However, all these values are prob-

ably underestimated because the presence of antibodies in

healthy tissues in this model is assumed to be null. Further-

more, error bars for BED in healthy tissues are small, which

means that large variations in biological parameters induce

small differences of BED. Such findings are important to

ensure the patient safety.

III.A.2. Single atom labeled antibody

Previous results have to be compared to dosimetry calcu-

lations when only a single radionuclide

each antibody. To determine absorbed doses, the maximal

covering fraction of 1010mAbs/cm2has been considered.39

With such a value, total activities correspond to about 0.9

and 7.5 MBq for tumors of 0.5 and 1.0 cm radii, respectively.

As illustrated in Fig. 3 for a tumor of 0.5 cm radius, differ-

ences between D and BED curves are smaller than those

observed for radioactive nanoparticles. Moreover, absorbed

and biological doses are inferior to 60 Gy, which means that

the activity deposited by radiolabeled antibodies is insuffi-

cient to correctly treat solid tumors such as non-small-cell

lung carcinomas. This result remains true independently of

the type of antibody distribution. Similar results are obtained

for a tumor with a radius of 1.0 cm ?Fig. 4? despite the larger

total activity. Without hypoxia, values of absorbed and bio-

logical doses vary between 20 and 60 Gy for the three dif-

ferent antibody distributions, which is again insufficient to

obtain good treatment outcome.

90Y is coupled to

III.B. BED distributions for tumors with hypoxia

The presence of a large central hypoxia in the tumor can

greatly influence clinical outcome in targeted radiotherapy.

As explained earlier, our tumor model is capable of taking

into account two types of cells with different radiosensitivity:

Normally oxygenated ?q=1? and hypoxic cells ?q=3?. BEDs

were modelized for tumors of 0.5 and 1.0 cm radii contain-

ing a hypoxic core ?Figs. 5 and 6?. The percentage of hy-

FIG. 5. Comparison of D?r? and BED?r? profiles for90Y2O3nanoparticle

labeled antibodies and for single90Y labeled antibodies, in Gray unit, as a

function of the distance r from center for a tumor radius with a radius of 0.5

cm, for three different antibodies distribution: Uniform, linear, and exponen-

tial. The gray parts symbolize the hypoxic core of 0.32 cm radius and dotted

vertical lines represent the tumor radius. Values for BED?r? were determined

according to parameters given in Table II and Eq. ?5? with q=1, with the

exception that the value of q changes from 1 to 3 for the hypoxic core.

1834Bouchat et al.: BED and SCP for radioactive nanoparticles1834

Medical Physics, Vol. 37, No. 4, April 2010

Page 10

poxic cell clusters compared to the total number of cell ag-

gregates was chosen to be 25%, corresponding to a spherical

hypoxic core of 0.32 radius for the smaller carcinoma and of

0.64 cm radius for the largest ?cf. Table I?. The absorbed and

biological dose profiles are calculated with the same total

activity as the tumor model without central hypoxia, namely,

3.5 and 20 MBq for 0.5 and 1.0 tumor radii, respectively.

When a single radioactive atom of

antibody, the total activities decrease to values of 0.9 and 7.5

MBq for 0.5 and 1.0 cm tumor radii, respectively. Distribu-

tions of radiolabeled antibodies in the normally oxygenated

part of the spherical solid tumor can be uniform, linear, or

exponential.

90Y is linked to each

III.B.1. Nanoparticle labeled antibody

Figures 5 and 6 show the impact of poorly oxygenated

cells on BED calculation. For the three distributions of ra-

diolabeled antibodies, the shapes between physical dose and

BED curves are different. Indeed, biological doses are higher

than absorbed doses in the normally oxygenated region of

the tumor but lower in the hypoxic core. Maximum values

for BED?r? in a tumor of 0.5 cm radius are 162 Gy for

uniform distribution and 150 Gy for linear or exponential

distribution of radiolabeled antibodies. The aforementioned

biological doses are located at a distance varying between

0.30 and 0.40 cm from the tumor center where the prolifera-

tion of cells is still important. Higher values for maximal

BED are observed when the tumor reaches 1.0 cm radius

with 158, 156, and 170 Gy for uniform, linear, and exponen-

tial distributions, respectively. These values are localized at

0.70 cm from the tumor center for uniform or linear distri-

bution and at 0.90 for the exponential distribution of activity.

As for Figs. 4 and 5, error bars have been added on BED

curves and wide variations in biological doses are observed

only in the living part of the tumor. Changes in BED are

essentially due to fluctuations in the repair half-time since an

increase of 40–55 Gy is observed when T?is 1.5 h rather

than 0.5 h for both tumor radii. Once again, important varia-

tions for ?/? and ? biological parameters do not affect BED

very much: BED?r? remains high inside the nonhypoxic area

of the tumor with doses up to 50 Gy, which are values high

enough to ensure good treatment outcomes. Inversely, BED

values in the hypoxic core are always inferior to 50 Gy with

a minimal value of about 20 Gy at the center of the core for

the tumor of 0.5 cm radius. The latter is still reduced to

values close to zero for larger spherical tumors. BED?r? val-

ues in this region are significantly smaller than physical

doses when we take into account that hypoxic cells are more

radioresistant than oxic cells. Finally, BED values in healthy

tissue are about 30 Gy at a distance of 1 mm from the tumor,

except for the exponential distribution. In this last case, BED

value is higher than 40 Gy, which is too high if we wish to

avoid important damage to the lung tissues ?or surrounding

tissues? in close proximity.

III.B.2. Single atom labeled antibody

Once again, previous results have been compared to D

and BED calculations when only a single atom is linked to

each antibody. For the tumor of 0.5 cm radius in which 25%

of the cell clusters are poorly oxygenated at the center ?Fig.

5?, maximal doses range from 30 to 35 Gy at 0.35 cm from

the tumor center for uniform distribution and at 0.40 cm

from the tumor center for linear or exponential antibody dis-

tribution. Biological doses lower than 10 Gy are also calcu-

lated in the hypoxic core. All these BED values are much too

low to eradicate completely the tumor. For a larger tumor of

1.0 cm radius ?Fig. 6?, maximal doses reached also remain

inferior to 60 Gy whatever the antibody distribution and the

biological doses are close to zero at the tumor center. It has

to be noted that values for absorbed and biological doses

FIG. 7. SCP profiles, in %, as a function of distance r from center of a tumor

of 0.5 cm radius ??a?–?c?? and 1.0 cm radius ??d?–?f?? when 5 nm diameter

nanoparticles containing90Y2O3molecules ?black curves? or when single

90Y ?gray curves? are linked to antibodies distributed uniformly, linearly and

exponentially. Solid and dashed curves correspond, respectively, to BED

calculations applied for tumors without hypoxia ?solid curve? and when 25%

of total cell clusters are hypoxic at the tumor center ?dashed curves?.

FIG. 6. Comparison of D?r? and BED?r? profiles for90Y2O3nanoparticle

labeled antibodies and for single90Y labeled antibodies, in Gray unit, as a

function of the distance r from center for a tumor with a radius of 1.0 cm,

for three different antibodies distribution: Uniform, linear, and exponential.

The gray parts symbolize the hypoxic core of 0.64 cm radius and dotted

vertical lines represent the tumor radius. Values for BED?r? were determined

according to parameters given in Table II and Eq. ?5? with q=1, with the

exception that the value of q changes from 1 to 3 for the hypoxic core.

1835Bouchat et al.: BED and SCP for radioactive nanoparticles 1835

Medical Physics, Vol. 37, No. 4, April 2010

Page 11

remain inferior to 15 Gy in healthy tissues for both tumor

radii and for the three activity distributions, which means

that the treatment remains harmless for healthy tissues.

III.C. SCP distributions

Up to now, it is not clear which antibody distribution

would be the most efficient to treat spherical solid cancers,

such as NSCLC. The only important difference between the

three distributions we envisaged is the localization of the

maximum of the biologically effective dose inside the tumor

that could influence treatment outcome. For uniform distri-

bution, this maximum is located at the tumor center while

this maximum shifts near the tumor surface when the distri-

bution becomes exponential. Figure 7 shows shell control

probabilities in relation to the distance r for two spherical

tumors with radii of 0.5 cm ?cf. Figs. 7?a?–7?c?? or 1.0 cm

?cf. Figs. 7?d?–7?f??. SCP?r? has been plotted for non-small-

cell lung cancer ??=0.350 Gy−1? surrounded by lung tissues

??=0.031 Gy−1?. Uniform ?Figs. 7?a? and 7?d??, linear ?Figs.

7?b? and 7?e??, or exponential ?Fig. 7?c? and 7?f?? distribu-

tions caused by uniform or nonuniform antibody uptake are

considered.

III.C.1. Nanoparticle labeled antibody

When a NSCLC tumor of 0.5 cm radius does not present

a hypoxic core, and when it is treated with 5 nm diameter

nanoparticles of90Y2O3, the SCP curve displays a plateau

from the center of the tumor toward its surface and this,

independently of the antibody distribution. These results

mean that the probability to observe a cell cluster surviving

irradiation is very weak throughout the whole tumor. The

only difference between the three distributions is the value of

SCP at the tumor surface. Indeed, SCP is maximal for expo-

nential antibody distributions and is a little bit lower when

antibodies are distributed linearly ?SCP of 94%? or uniformly

?SCP of 83%?. Finally, SCP rapidly decreases to zero in lung

healthy tissues, for the three distributions. Tumor control

probabilities for uniform, linear, and exponential distribu-

tions are, respectively, 83%, 94%, and 100%.

Similar results are obtained when the spherical tumor

reaches a radius of 1.0 cm ?Figs. 7?d?–7?f??. SCP values of

100% are calculated inside the tumor, except for the expo-

nential antibody distributions for which a small decrease in

SCP from 100% to 97% is observed halfway between the

center and the surface of the tumor. Inversely, SCP values of

0% are observed in healthy tissues. At the tumor surface,

SCP varies from 100% for the exponential activity to 0%

when the radionuclides contained in the nanoparticles are

distributed uniformly through the tumor. In general, simu-

lated curves plotted on Fig. 7 indicate that a good treatment

outcome could be obtained with the use of radioactive nano-

particles coupled to each antibody, independently of the ac-

tivity distribution ?uniform, linear, or exponential?. With

87%, the best TCP is obtained for exponential distribution of

antibodies.

Figure 7 also shows values of SCP plotted against r when

spherical carcinomas present a hypoxic core, i.e., 25% of the

total number of cell clusters is poorly oxygenated at the cen-

ter of the tumor. In this case, SCP decreases from 100% to

0% in the region of the hypoxic core. This result indicates

that the use of 5 nm diameter nanoparticles containing90Y is

not efficient enough to irradiate three times more radioresis-

tant cell clusters when the antibodies cannot penetrate inside

this part of the tumor.

We have also investigated the variation in SCP when val-

ues of the different biological parameters ??, ?/?, N, and ??

vary. Study was made of a plausible range of values, similar

to those proposed by Atthey and co-workers.37For small

tumors of 0.5 cm radius, variation in ?/? and ? does not

modify SCP values, which means that treatment outcome

does not change even with a reasonable fluctuation of BED.

For a larger tumor of 1.0 cm radius, similar variations in ?/?

and ? provoke only very small differences of SCP at the

tumor surface. When the cell density varies from 107to 108,

N changes, but SCP values remains identical for both tumors.

The fact that SCP values remain unchanged when ?/?, ?,

and N vary is the consequence of the high activity released

by the nanoparticles. However SCP at the tumor surface in-

creases when ? varies from 0.20 to 0.5 Gy−1. For ? larger

than 0.5 Gy−1, SCP reaches a maximum of 100% at the tu-

mor surface, where cellular proliferation is very important.

On the other hand, when ? values are lower than 0.20 Gy−1,

SCP values at the tumor surface remain zero.

III.C.2. Single atom labeled antibody

As presented by the SCP curves in Fig. 7, when a single

90Y is linked to each antibody, the sizes of tumors, antibody

distributions, and the presence of hypoxic cells greatly influ-

ence SCP distribution. For example, no cell clusters will be

killed when the tumor has a radius of 0.5 cm and presents a

central hypoxia of 0.32 cm radius ?Figs. 7?a?–7?c??. Indeed,

SCP values of 0% are calculated for all measures of r. Simi-

lar results are observed when the tumor has no hypoxic core

and an exponential antibody distribution. Consequently,

when tumor has a radius of 0.5 cm and no central hypoxia,

better therapeutic effects are obtained when antibodies are

distributed uniformly or linearly. In these last cases, a SCP of

100% is reached at the tumor center and decreases progres-

sively through the tumor to a value of 0% at 2 mm before the

surface. These results lead to the conclusion that the most

highly proliferating cells are not sufficiently irradiated to

have a chance of a cancer cure.

Figure 7 shows that SCP results for larger tumors of 1.0

cm radius are totally different. Without hypoxia, the shape of

SCP curves varies according to the types of antibody distri-

bution. When antibodies are distributed uniformly ?Fig.

7?d??, SCP displays a plateau in the center of the tumor and

then decreases near the tumor surface which is the region

where the number of proliferating cells is the most impor-

tant. For linear and exponential distributions ?Figs. 7?e? and

7?f??, the maximal values of SCP are located, respectively, at

a distance of 0.70 and 0.90 cm from the center of the tumor,

with a narrower and higher peak for the exponential distri-

bution. Inversely, the shape of SCP curves for the three dif-

1836Bouchat et al.: BED and SCP for radioactive nanoparticles1836

Medical Physics, Vol. 37, No. 4, April 2010

Page 12

ferent antibody distributions is similar when the tumor has a

hypoxic core. A maximal SCP of 100% is reached in the

normally oxygenated part of the tumor, with a deeper radial

position within the tumor when the antibodies are uniformly

distributed through the tumor. Moreover, the more the distri-

bution is exponential, the more the peak is narrow. A better

shell control probability seems to be obtained when the

quantity of antibodies decrease linearly through the nonhy-

poxic part of the 1.0 cm radius tumor.

Variations in the maximal value for SCP were evaluated

for a wide range of biological parameters. Figure 8 shows

that maximal SCP values obtained for a tumor of 1.0 cm

radius without hypoxia decrease when the ratio ?/?, the cel-

lular repair constant and the cell density ? increase. Results

also indicate that shell control probability decreases rapidly

when the intrinsic radiosensitivity ? is lower than 0.35 Gy.

Consequently, due to the lower doses deposited inside the

tumor when single radionuclides coupled to each antibody is

used rather than radioactive nanoparticles, values of SCP de-

pend more on the choice of biological parameters and the

antibody distributions.

IV. CONCLUSIONS

In RIT, dose deposition inside the tumor is performed by

the use of monoclonal antibodies labeled with only one ra-

dioactive atom. In a previous work, the advantage of using

radioactive nanoparticles containing hundreds of radioactive

atoms rather than single radionuclides was modelized.6Do-

simetry calculations were performed with the MCNPX Monte

Carlo code by introducing a new model of tumor in which

radiolabeled antibodies can be uniformly, linearly, or expo-

nentially distributed. However, these deposited doses did not

take into account the radiosensitivity and cellular repair ef-

fects of the targeted tumor and surrounding tissue. Therefore,

BED and SCP formulas were applied in this work to estimate

more properly the tumor response and treatment outcome.

BED and SCP distributions were calculated for advanced

non-small-cell lung cancer for which no efficacious therapy

exists. When only one ?-emitter90Y is coupled to each an-

tibody, the calculated BED is lower than 60 Gy in the overall

tumor. SCP values show clearly that these biological doses

are insufficient to correctly treat NSCLC. When the single

?-emitter is replaced by a 5 nm diameter nanoparticle con-

taining approximately 100090Y atoms, sufficiently high bio-

logical doses can be obtained to completely kill the nonhy-

poxic part of a NSCLC, while limiting radiation in healthy

lung tissues. SCP calculations confirm these results by reach-

ing a maximal value of 100% inside the normally oxygen-

ated part of the tumor and a minimal value of 0% in the

surrounding healthy tissues. However, BED and SCP values

are influenced by the choice of the geometrical factors used

to describe the tumor ?i.e., morphology of the tumor and cell

cluster dimensions, radioactivity distribution throughout the

tumor, and volume of hypoxic core? by the model used to

calculate BED?r? ?i.e., LQ model, cellular repair, and hy-

poxic and repopulation effects? and by biological parameter

values selected for NSCLC and lung tissues. Variability of

both SCP and BED distributions have been analyzed for a

wide range of biological parameter values and the results

confirm their important impact on BED distributions inside

the tumor. Inversely, it is interesting to note that despite large

differences in BED curves, shell control probability remains

relatively unaffected when radioactive nanoparticles are

used. This last result is probably due to the high activity

deposited inside the tumor by the numerous radioactive at-

oms contained in the nanoparticles. To conclude this paper,

for either small or large solid tumors, BED and SCP calcu-

lations clearly confirm the efficacy of radioimmunotherapy

when using radioactive nanoparticles rather than a single ra-

dionuclide coupled to each antibody.

(a)

(b)(c)(d)

FIG. 8. Variation in maximal SCP values, in %, with the different radiobiological factors: ?a? The ratio ?/?, ?b? the intrinsic radiosensitivity ?, ?c? the

exponential cellular repair constant ?, and ?d? the cell density ?. SCP calculations were performed for a tumor of 1.0 cm without hypoxia and for single90Y

labeled antibodies. Activity inside the tumor may be distributed uniformly ?dotted curves?, linearly ?dashed curves?, or exponentially ?solid curves?.

1837Bouchat et al.: BED and SCP for radioactive nanoparticles1837

Medical Physics, Vol. 37, No. 4, April 2010

Page 13

ACKNOWLEDGMENTS

This research ?Targan Project–Convention 0516071? was

supported by the Walloon Region ?Belgium?. Olivier Feron is

senior research associate of FNRS ?Fonds National de la Re-

cherche Scientifique, Belgium?.

a?Author to whom correspondence should be addressed. Electronic mail:

virginie.bouchat@fundp.ac.be; Telephone: 0032-81-725479; Fax: 0032-

81-725474.

1M. Mehta, R. Scrimger, R. Mackie, B. Paliwal, R. Chappell, and J.

Fowler, “A new approach to dose escalation in non-small-cell lung can-

cer,” Int. J. Radiat. Oncol., Biol., Phys. 49, 23–33 ?2001?.

2W. A. Bethge and B. M. Sandmaier, “Targeted cancer therapy using ra-

diolabeled monoclonal antibodies,” Technol. Cancer Res. Treat. 4, 393–

405 ?2005?.

3S. V. Govindan, G. L. Griffiths, H. J. Hansen, I. D. Horak, and D. M.

Goldenberg,“Cancertherapy with

conjugated antibodies,” Technol. Cancer Res. Treat. 4, 375–391 ?2005?.

4R. M. Sharkey and D. M. Goldenberg, “Perspectives on cancer therapy

with radiolabeled monoclonal antibodies,” J. Nucl. Med. 46, 115s–127s

?2005?.

5D. M. Goldenberg, “Advancing role of radiolabeled antibodies in the

therapy of cancer,” Cancer Immunol. Immunother 52, 281–296 ?2003?.

6V. Bouchat, V. E. Nuttens, S. Lucas, C. Michiels, B. Masereel, O. Feron,

B. Gallez, and T. V. Borght, “Radioimmunotherapy with radioactive

nanoparticles: First results of dosimetry for vascularized and necrosed

solid tumors,” Med. Phys. 34, 4504–4513 ?2007?.

7M. J. Welch, C. J. Hawker, and K. L. Wooley, “The advantages of nano-

particles for PET,” J. Nucl. Med. 50, 1743–1746 ?2009?.

8F. Pene, E. Courtine, A. Cariou, and J. P. Mira, “Toward theragnostics,”

Crit. Care Med. 37, S50–S58 ?2009?.

9S. M. Moghimi, A. C. Hunter, and J. C. Murray, “Nanomedicine: Current

status and future prospects,” FASEB J. 19, 311–330 ?2005?.

10S. M. Moghimi and J. Szebeni, “Stealth liposomes and long circulating

nanoparticles: Critical issues in pharmacokinetics, opsonization and

protein-binding properties,” Prog. Lipid Res. 42, 463–478 ?2003?.

11D. E. Owens III and N. A. Peppas, “Opsonization, biodistribution, and

pharmacokinetics of polymeric nanoparticles,” Int. J. Pharm. 307, 93–102

?2006?.

12R. Gref, M. Luck, P. Quellec, M. Marchand, E. Dellacherie, S. Harnisch,

T. Blunk, and R. H. Muller, “‘Stealth’ corona-core nanoparticles surface

modified by polyethylene glycol ?PEG?: Influences of the corona ?PEG

chain length and surface density? and of the core composition on phago-

cytic uptake and plasma protein adsorption,” Colloids Surf., B 18, 301–

313 ?2000?.

13Y. Yi, J. H. Kim, H. W. Kang, H. S. Oh, S. W. Kim, and M. H. Seo, “A

polymeric nanoparticle consisting of mPEG-PLA-Toco and PLMA-

COONa as a drug carrier: Improvements in cellular uptake and biodistri-

bution,” Pharm. Res. 22, 200–208 ?2005?.

14J. Chen, H. Wu, D. Han, and C. Xie, “Using anti-VEGF McAb and

magnetic nanoparticles as double-targeting vector for the radioimmuno-

therapy of liver cancer,” Cancer Lett. 231, 169–175 ?2006?.

15G. Kaul and M. Amiji, “Biodistribution and targeting potential of poly-

?ethylene glycol?-modified gelatin nanoparticles in subcutaneous murine

tumor model,” J. Drug Target. 12, 585–591 ?2004?.

16J. D. Woodward, S. J. Kennel, S. Mirzadeh, S. Dai, J. S. Wall, T. Richey,

J. Avenell, and A. J. Rondinone, “In vivo SPECT/CT imaging and bio-

distribution using radioactive ?CdTe?-Te-125m/ZnS nanoparticles,” Nano-

technology 18, 175103 ?2007?.

17R. Kannan, V. Rahing, C. Cutler, R. Pandrapragada, K. K. Katti, V. Kat-

tumuri, J. D. Robertson, S. J. Casteel, S. Jurisson, C. Smith, E. Boote, and

K. V. Katti, “Nanocompatible chemistry toward fabrication of target-

specific gold nanoparticles,” J. Am. Chem. Soc. 128, 11342–11343

?2006?.

18K. V. Katti, R. Kannan, K. Katti, V. Kattumori, R. Pandrapraganda, V.

Rahing, C. Cutler, E. J. Boote, S. W. Casteel, C. J. Smith, J. D. Robertson,

and S. S. Jurrison, “Hybrid gold nanoparticles in molecular imaging and

radiotherapy,” Czech. J. Phys. 56, D23–D34 ?2006?.

19N. Chanda et al., “Radioactive gold nanoparticles in cancer therapy:

Therapeutic efficacy studies of ?198?AuNP-GA nanoconstruct in prostate

tumor-bearing mice,” Nanomedicine ?in press?.

radiolabeled and drug/toxin-

20H. Wu, J. Wang, Z. Wang, D. R. Fisher, and Y. Lin, “Apoferritin-

templated yttrium phosphate nanoparticle conjugates for radioimmuno-

therapy of cancers,” J. Nanosci. Nanotechnol. 8, 2316–2322 ?2008?.

21D. D. Dionysiou and G. S. Stamatakos, “Applying a 4D multiscale in

vivo tumor growth model to the exploration of radiotherapy scheduling:

The effects of weekend treatment gaps and p53 gene status on the re-

sponse of fast growing solid tumors,” Cancer Inform. 2, 113–121 ?2006?.

22R. Ruggieri, “Hypofractionation in non-small cell lung cancer ?NSCLC?:

Suggestions from modeling both acute and chronic hypoxia,” Phys. Med.

Biol. 49, 4811–4823 ?2004?.

23D. Levin-Plotnik and R. J. Hamilton, “Optimization of tumour control

probability for heterogeneous tumours in fractionated radiotherapy treat-

ment protocols,” Phys. Med. Biol. 49, 407–424 ?2004?.

24M. G. Stabin and G. D. Flux, “Internal dosimetry as a tool for radiation

protection of the patient in nuclear medicine,” Biomed. Imaging Interv. J.

3, e28 ?2007?.

25M. G. Stabin, M. Tagesson, S. R. Thomas, M. Ljungberg, and S. E.

Strand, “Radiation dosimetry in nuclear medicine,” Appl. Radiat. Isot. 50,

73–87 ?1999?.

26R. Barone, F. O. Borson-Chazot, R. Valkerna, S. Walrand, F. Chauvin, L.

Gogou, L. K. Kvols, E. P. Krenning, F. Jamar, and S. Pauwels, “Patient-

specific dosimetry in predicting renal toxicity with Y-90-DOTATOC: Rel-

evance of kidney volume and dose rate in finding a dose-effect relation-

ship,” J. Nucl. Med. 46, 99s–106s ?2005?.

27M. Astrahan, “Some implications of linear-quadratic-linear radiation

dose-response with regard to hypofractionation,” Med. Phys. 35, 4161–

4172 ?2008?.

28B. Warkentin, P. Stavrev, N. Stavreva, C. Field, and B. Fallone, “A TCP-

NTCP estimation module using DVHs and known radiobiological models

and parameter sets,” J. Appl. Clin. Med. Phys. 5, 50–63 ?2004?.

29P. Thirion, O. Holmberg, C. D. Collins, C. O’Shea, M. Moriarty, M.

Pomeroy, C. O’Sullivan, S. Buckney, and J. Armstrong, “Escalated dose

for non-small-cell lung cancer with accelerated hypofractionated three-

dimensional conformal radiation therapy,” Radiother. Oncol. 71, 163–166

?2004?.

30E. Kodym, R. Kodym, A. E. Reis, A. A. Habib, M. D. Story, and D. Saha,

“The small-molecule CDK inhibitor, SNS-032, enhances cellular radi-

osensitivity in quiescent and hypoxic non-small cell lung cancer cells,”

Lung Cancer 66, 37–47 ?2009?.

31M. Engelsman, E. M. Damen, K. De Jaeger, K. M. van Ingen, and B. J.

Mijnheer, “The effect of breathing and set-up errors on the cumulative

dose to a lung tumor,” Radiother. Oncol. 60, 95–105 ?2001?.

32B. Döme, S. Paku, B. Somlai, and J. Timar, “Vascularization of cutaneous

melanoma involves vessel co-option and has clinical significance,” J.

Pathol. 197, 355–362 ?2002?.

33J. A. M. Beliën, P. J. van Diest, and J. P. A. Baak, “Relationships between

vascularization and proliferation in invasive breast cancer,” J. Pathol.

189, 309–318 ?1999?.

34C. Schlueter, H. Weber, B. Meyer, P. Rogalla, K. Roser, S. Hauke, and J.

Bullerdiek, “Angiogenetic signaling through hypoxia-HMGB1: An angio-

genetic switch molecule,” Am. J. Pathol. 166, 1259–1263 ?2005?.

35J. L. Humm and L. M. Cobb, “Nonuniformity of tumor dose in radioim-

munotherapy,” J. Nucl. Med. 31, 75–83 ?1990?.

36V. E. Nuttens, A. C. Wera, V. Bouchat, and S. Lucas, “Determination of

biological vector characteristics and nanoparticle dimensions for radioim-

munotherapy with radioactive nanoparticles,” Appl. Radiat. Isot. 66, 168–

172 ?2008?.

37M. Atthey, A. E. Nahum, M. A. Flower, and V. R. McCready, “Effects of

cellular repair and proliferation on targeted radionuclide therapy: A mod-

eling study,” Phys. Med. Biol. 45, N15–N20 ?2000?.

38M. Cremonesi, M. Ferrari, L. Bodei, G. Tosi, and G. Paganelli, “Systemic

and locoregional dosimetry in receptor radionuclide therapy with pep-

tides,” Q. J. Nucl. Med. Mol. Imaging 50, 288–295 ?2006?.

39R. W. Howell, D. V. Rao, and K. S. R. Sastry, “Macroscopic dosimetry

for radioimmunotherapy-nonuniform activity distributions in solid tu-

mors,” Med. Phys. 16, 66–74 ?1989?.

40R. M. Sharkey and D. M. Goldenberg, “Use of antibodies and immuno-

conjugates for the therapy of more accessible cancers,” Adv. Drug Deliv-

ery Rev. 60, 1407–1420 ?2008?.

41G. A. Wiseman, C. A. White, R. B. Sparks, W. D. Erwin, D. A. Podoloff,

D. Lamonica, N. L. Bartlett, J. A. Parker, W. L. Dunn, S. M. Spies, R.

Belanger, T. E. Witzig, and B. R. Leigh, “Biodistribution and dosimetry

results from a phase III prospectively randomized controlled trial of Zeva-

1838Bouchat et al.: BED and SCP for radioactive nanoparticles1838

Medical Physics, Vol. 37, No. 4, April 2010

Page 14

lin ?TM? radioimmunotherapy for low-grade, follicular, or transformed

B-cell non-Hodgkin’s lymphoma,” Crit. Rev. Oncol. Hematol. 39, 181–

194 ?2001?.

42R. W. Howell, S. M. Goddu, and D. V. Rao, “Application of the linear-

quadratic model to radioimmunotherapy-further support for the advantage

of longer-lived radionuclides,” J. Nucl. Med. 35, 1861–1869 ?1994?.

43K. Tobinai, Y. Kobayashi, M. Narabayashi, M. Ogura, Y. Kagami, Y.

Morishima, T. Ohtsu, T. Igarashi, Y. Sasaki, T. Kinoshita, and T. Murate,

“‘Feasibility and pharmacokinetic study of a chimeric anti-CD20 mono-

clonal antibody ?IDEC-C2B8, rituximab? in relapsed B-cell lymphoma,’

The IDEC-. C2B8 Study Group,” Ann. Oncol. 9, 527–534 ?1998?.

44N. Reynaert, H. Palmans, H. Thierens, and R. Jeraj, “Parameter depen-

dence of the MCNP electron transport in determining dose distributions,”

Med. Phys. 29, 2446–2454 ?2002?.

45D. R. Schaart, J. T. Jansen, J. Zoetelief, and P. F. de Leege, “A compari-

son of MCNP4C electron transport with ITS 3.0 and experiment at inci-

dent energies between 100 keV and 20 MeV: Influence of voxel size,

substeps and energy indexing algorithm,” Phys. Med. Biol. 47, 1459–

1484 ?2002?.

46B. Brans, L. Bodei, F. Giammarile, O. Linden, M. Luster, W. J. G. Oyen,

and J. Tennvall, “Clinical radionuclide therapy dosimetry: The quest for

the ‘Holy Gray’,” Eur. J. Nucl. Med. Mol. Imaging 34, 772–786 ?2007?.

47R. G. Dale, “The application of the linear-quadratic dose-effect equation

to fractionated and protracted radiotherapy,” Br. J. Radiol. 58, 515–528

?1985?.

48R. Dale and A. Carabe-Fernandez, “The radiobiology of conventional

radiotherapy and its application to radionuclide therapy,” Cancer Biother.

Radiopharm. 20, 47–51 ?2005?.

49R. K. Bodey, P. M. Evans, and G. D. Flux, “Application of the linear-

quadratic model to combined modality radiotherapy,” Int. J. Radiat. On-

col., Biol., Phys. 59, 228–241 ?2004?.

50C. Chiesa, F. Botta, E. Di Betta, A. Coliva, M. Maccauro, G. Aliberti, S.

Bavusi, L. Devizzi, A. Guidetti, E. Seregni, A. M. Gianni, and E. Bom-

bardieri, “Dosimetry in myeloablative Y-90-labeled ibritumomab tiuxetan

therapy: Possibility of increasing administered activity on the base of

biological effective dose evaluation. Preliminary results,” Cancer Biother.

Radiopharm. 22, 113–120 ?2007?.

51S. Baechler, R. F. Hobbs, A. R. Prideaux, R. L. Wahl, and G. Sgouros,

“Extension of the biological effective dose to the MIRD schema and

possible implications in radionuclide therapy dosimetry,” Med. Phys. 35,

1123–1134 ?2008?.

52J. Z. Wang, X. A. Li, W. D. D’Souza, and R. D. Stewart, “Impact of

prolonged fraction delivery times on tumor control: A note of caution for

intensity-modulated radiation therapy ?IMRT?,” Int. J. Radiat. Oncol.,

Biol., Phys. 57, 543–552 ?2003?.

53G. G. Steel, J. M. Deacon, G. M. Duchesne, A. Horwich, L. R. Kelland,

and J. H. Peacock, “The dose-rate effect in human-tumor cells,” Radio-

ther. Oncol. 9, 299–310 ?1987?.

54M. Guerrero and X. A. Li, “Halftime for repair of sublethal damage in

normal bladder and rectum: An analysis of clinical data from cervix

brachytherapy,” Phys. Med. Biol. 51, 4063–4071 ?2006?.

55A. Dawson and T. Hillen, “Derivation of the tumour control probability

?TCP? from a cell cycle model,” Comput. and Math. Meth. in Medicine 7,

121–141 ?2006?.

56N. Matsufuji, T. Kanai, N. Kanematsu, T. Miyamoto, M. Baba, T. Ka-

mada, H. Kato, S. Yamada, J. E. Mizoe, and H. Tsujii, “Specification of

carbon ion dose at the National Institute of Radiological Sciences

?NIRS?,” J. Radiat. Res. ?Tokyo? 48, A81–A86 ?2007?.

57R. G. Dale, “Dose-rate effects in targeted radiotherapy,” Phys. Med. Biol.

41, 1871–1884 ?1996?.

58J. F. Fowler, “Development of radiobiology for oncology-a personal

view,” Phys. Med. Biol. 51, R263–R286 ?2006?.

59R. Dale and C. Deehan, “Brachytherapy,” in Radiobiological Modeling in

Radiation Oncology, edited by R. G. D. B. Jones ?The British Institute of

Radiology, London, 2007?, pp. 113–137.

60S. Webb and A. E. Nahum, “A model for calculating tumour control

probability in radiotherapy including the effects of inhomogeneous distri-

butions of dose and clonogenic cell density,” Phys. Med. Biol. 38, 653–

666 ?1993?.

61B. Dubray, M. Henryamar, J. H. Meerwaldt, E. M. Noordijk, D. O.

Dixon, J. M. Cosset, and H. D. Thames, “Radiation-induced lung damage

after thoracic irradiation for Hodgkin’s disease—The role of fraction-

ation,” Radiother. Oncol. 36, 211–217 ?1995?.

62B. Jones, A. Carabe-Fernandez, and R. Dale, “The oxygen effect,” in

Radiobiological Modeling in Radiation Oncology, edited by R. G. D. B.

Jones ?The British Institute of Radiology, London, 2007?, Vol. 1, pp.

138–157.

63R. D. Stewart and R. J. Traub, “Radiobiological modeling in voxel con-

structs,” presented at the Proceedings of the Monte Carlo 2000 Meeting,

Lisbon, Portugal, 2000, edited by F. B. A. Kling, M. Nakagawa, L.

Tavora, and P. Vaz ?Springer-Verlag, Berlin, 2001?, pp. 285–290.

1839 Bouchat et al.: BED and SCP for radioactive nanoparticles1839

Medical Physics, Vol. 37, No. 4, April 2010