# Evidence of extranuclear cell sensitivity to alpha-particle radiation using a microdosimetric model. I. Presentation and validation of a microdosimetric model.

**ABSTRACT** A microdosimetric model that makes it possible to consider the numerous biological and physical parameters of cellular alpha-particle irradiation by radiolabeled mAbs was developed. It allows for the calculation of single-hit and multi-hit distributions of specific energy within a cell nucleus or a whole cell in any irradiation configuration. Cells are considered either to be isolated or to be packed in a monolayer or a spheroid. The method of calculating energy deposits is analytical and is based on the continuous-slowing-down approximation. A model of cell survival, calculated from the microdosimetric spectra and the microdosimetric radiosensitivity, z(0), was also developed. The algorithm of calculations was validated by comparison with two general Monte Carlo codes: MCNPX and Geant4. Microdosimetric spectra determined by these three codes showed good agreement for numerous geometrical configurations. The analytical method was far more efficient in terms of calculation time: A gain of more than 1000 was observed when using our model compared with Monte Carlo calculations. Good agreements were also observed with previously published results.

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- Seminars in nuclear medicine 05/2014; 44(3):172-178. · 3.96 Impact Factor
- S Gouard, A Pallardy, J Gaschet, A Faivre-Chauvet, F Bruchertseifer, A Morgenstern, C Maurel, E Matous, F Kraeber-Bodéré, F Davodeau, M ChérelNuclear Medicine and Biology 05/2014; 41S:e30-e35. · 2.52 Impact Factor
- SourceAvailable from: Nicolas ChouinVincent Boudousq, Laure Bobyk, Muriel Busson, Véronique Garambois, Marta Jarlier, Paraskevi Charalambatou, André Pèlegrin, Salomé Paillas, Nicolas Chouin, François Quenet, Patrick Maquaire, Julien Torgue, Isabelle Navarro-Teulon, Jean-Pierre PougetPLoS ONE 01/2013; 8(7):e69613. · 3.53 Impact Factor

Page 1

657

RADIATION RESEARCH 171, 657–663 (2009)

0033-7587/09 $15.00

? 2009 by Radiation Research Society.

All rights of reproduction in any form reserved.

Evidence of Extranuclear Cell Sensitivity to Alpha-Particle Radiation

Using a Microdosimetric Model. I. Presentation and Validation of a

Microdosimetric Model

N. Chouin,aK. Bernardeau,aF. Davodeau,aM. Che ´rel,a,bA. Faivre-Chauvet,aM. Bourgeois,aC. Apostolidis,c

A. Morgenstern,cA. Lisbonaa,dand M. Bardie `sa,b,1

aInserm, U892, Nantes, F-44093 France;

France;

bUniversite ´ de Nantes, Faculte ´ de Me ´decine (UFR Me ´decine et Techniques Me ´dicales), Nantes, F-44093

cInstitute for Transuranium Elements, Karlsruhe, Germany; and

dCLCC Nantes-Atlantique ‘‘Rene ´ Gauducheau’’, St. Herblain, France

Chouin, N., Bernardeau, K., Davodeau, F., Che ´rel, M., Fai-

vre-Chauvet, A., Bourgeois, M., Apostolidis, C., Morgenstern,

A., Lisbona, A. and Bardie `s, M. Evidence of Extranuclear Cell

Sensitivity to Alpha-Particle Radiation Using a Microdosi-

metric Model. I. Presentation and Validation of a Microdos-

imetric Model. Radiat. Res. 171, 657–663 (2009).

A microdosimetric model that makes it possible to consider

the numerous biological and physical parameters of cellular

?-particle irradiation by radiolabeled mAbs was developed.

It allows for the calculation of single-hit and multi-hit distri-

butions of specific energy within a cell nucleus or a whole cell

in any irradiation configuration. Cells are considered either

to be isolated or to be packed in a monolayer or a spheroid.

The method of calculating energy deposits is analytical and is

based on the continuous-slowing-down approximation. A

model of cell survival, calculated from the microdosimetric

spectra and the microdosimetric radiosensitivity, z0, was also

developed. The algorithm of calculations was validated by

comparison with two general Monte Carlo codes: MCNPX

and Geant4. Microdosimetric spectra determined by these

three codes showed good agreement for numerous geometrical

configurations. The analytical method was far more efficient

in terms of calculation time: A gain of more than 1000 was

observed when using our model compared with Monte Carlo

calculations. Good agreements were also observed with pre-

viously published results.

? 2009 by Radiation Research Society

INTRODUCTION

Alpha-particle radioimmunotherapy is a promising treat-

ment method that has been actively investigated for more

than two decades (1). The first clinical trials were carried

out for the treatment of acute myeloid leukemia using an

anti-CD33 antibody radiolabeled with213Bi (2). Since then,

different clinical trials using211At (3),213Bi (4, 5) and223Ra

(6) have shown promising results. Alpha particles possess

the following characteristics that make them of interest for

1Address for correspondence: Inserm U892, 44093 Nantes Cedex,

France; e-mail: manu@nantes.inserm.fr.

the treatment of small targets such as micrometastases (7):

(1) a short range in water [84 ?m for the 8.376 MeV ?

particle of213Po (8)]; which limits the nonspecific irradia-

tion of tissues close to the tumor, (2) a high LET, which

renders them extremely cytotoxic; and (3) a cytotoxicity

that does not depend on the absorbed dose rate or oxygen-

ation state of the tumor cells (9).

To better understand the radiobiological effects of ? par-

ticles on cells, studies are needed to correlate the physics

of ?-particle irradiation (mean absorbed dose, mean num-

ber of hits received by cellular targets, percentage of hit

cells, etc.) and the visible damage (cell survival, apoptosis

rate, micronucleus formation, etc.). The absorbed dose (ex-

pressed in Gy) is a relevant parameter for expressing the

level of damage caused in a biological structure by ionizing

radiation (10). Dosimetry should thus provide an index to

compare in vitro studies carried out in different conditions.

The calculation of energy deposits within cells is a complex

task and requires the development of dedicated dosimetric

models. Energy deposits must be quantified at the cellular

or subcellular level. The cell nucleus is traditionally con-

sidered as the critical target (11, 12). Due to the high cy-

totoxicity of ? particles, low activities are used, and as a

result, a small number of energy deposition events is ex-

pected in each cell. The stochastic nature of energy deposits

must be considered (13). In its 1983 report (14), the ICRU

recommended that microdosimetry should be used if ‘‘the

standard relative deviation of the frequency mean specific

energy was more than 20%’’. This criterion, which is often

verified for in vitro studies, marks out the zone within

which the statistical fluctuations of energy deposits are sig-

nificant and should be taken into account. Many microdos-

imetric models have been developed since Roesch’s initial

work (15). They can be divided into two groups: those

based on analytical calculations (16) and those using Monte

Carlo-based calculations (17–19). Different refinements

have recently been added to the original models, including

the diffusion of daughter radionuclides in a decay chain

(20) and the consideration of distributions of cellular radii

Page 2

658

CHOUIN ET AL.

or distributions of activity bound to the cells among a cell

population (21).

We developed a microdosimetric model based on the

method published by Stinchcomb (16) that makes it pos-

sible to consider the numerous biological and physical pa-

rameters of cellular ?-particle irradiation by radiolabeled

mAbs. Different comparative studies with two Monte Carlo

codes (MCNPX and Geant4) were carried out to verify the

algorithm of calculations.

MATERIAL AND METHODS

Microdosimetric Model

We developed a microdosimetric approach based on analytical mod-

eling. This takes into account the wide range of source–target distribu-

tions encountered in in vitro irradiation experiments. Cells and nuclei are

modeled as two concentric homogeneous spheres of density 1 g/cm3.

Energy deposits are calculated according to the continuous-slowing-down

approximation (CSDA). Tables of the stopping power of ? particles in

water published by the ICRU (8) were used. Stinchcomb (16) showed

that, if every emitter emits only one particle, the multi-hit distribution of

specific energy (MHDSE), f(z) within the target considered could be de-

termined with the fundamental equation

[

where n ¯ is the mean number of hits to the target, f1(z) is the single-hit

distribution of specific energy (SHDSE), and F{} is the Fourier transform

of the expression between the brackets. The Fourier transform and the

inverse Fourier transform are calculated with a Fast Fourier Transform

algorithm. The microdosimetric spectrum, f(z), is then derived from the

mean number of hits to the target and the f1(z) spectrum. These two

spectra, f1(z) and f(z), can be determined for different targets. The follow-

ing will consider two targets: the whole cell or the cell nucleus. As de-

scribed by Stinchcomb (16), space can be divided into subspaces in which

the distribution of activity can be considered as homogeneous. Five dif-

ferent subspaces are thus defined: (1) the surface of the target cell, (2)

the cytoplasm of the target cell, (3) the nucleus of the target cell, (4) the

medium surrounding the cells, and (5) the surface of the cells close to

the target cell.

The f1(z) distribution can be expressed as

?n ¯[1 ? F{f (z)}] ,

F{f (z)} ? exp (1)

1

]

1

i

1

f (z) ?

1

n ¯ · f (z) ,

i

(2)

?

i

[]

n ¯ ?

i

i

where n ¯iis the mean number of particles emitted in the subspace i that

hit the target and is the SHDSE deposited by particles coming from

1

the subspace i. The mean number of hits to the target is

?

if (z)

n ¯ ?

n ¯ . (3)

i

i

The model can be used in different cell configurations: single cell, cell

monolayer, and cell spheroids.

n ¯iandfrom each subspace are needed to calculate the MHDSE,

1

f(z). The following paragraphs provide details on the way they are cal-

culated in each subspace.

if (z)

Contribution of Radiolabeled mAbs Bound to the Surface of the Target

Cell

The nucleus of the cell or the whole cell can be considered as the

target. In the first case, the probability that a particle emitted from the

cell surface hits the nucleus is equal to the solid angle from which the

nucleus is seen from a point on the surface divided by 4? steradians. If

the whole cell is the target, the probability is 0.5. Multiplying this prob-

ability by the number of particles emitted from the surface gives the mean

number of hits to the target. The calculation of the SHDSE is imple-

mented as follows: For a particle emitted according to an angle ? with

respect to the line linking the cell center to the emission point, one should

calculate the ?-particle track length within the target. Alpha particles are

assumed to travel in straight lines. The specific energy deposited by an

? particle within the target is calculated as follows (we assumed unit

density here):

?

m

t1

where m is the target mass, dE/dx is the energy deposited by the ? particle

by unit of length according to ICRU tables (8), and t1and t2are the

coordinates of the entrance and exit points of the particle track within

the target. Calculations are carried out for each angle ?icomprised be-

tween ? and ?max(the maximum angle at which the particle track inter-

sects the target). The interval [?; ?max] was divided into 400 bins. This

choice of 400 bins is a compromise between accuracy and calculation

time. For each angular bin, the emission probability of the particle was

calculated.was calculated from the 400 couples (specific energy

f

(z)

1

deposited; probability).

t2

1

z ?

(dE/dx) dx, (4)

surf

Contribution of Radiolabeled mAbs Internalized in the Nucleus of the

Target Cell

The mean number of hits to the target is equal to the number of ?

particles emitted from the nucleus. The

?

V

nucl

f

(z) ?

1

?

distribution is given by

nucl

1

f

(z)

d(s)· f (z, s) dV

1

,(5)

d(s) dV

V

where d(s) is the activity density at a radial distance s from the center of

the cell and V the volume of interest (VOI), which corresponds to the

volume containing all the particles that can hit the target. In this case, it

is the cell nucleus, modeled as a sphere. For each distance and for each

angle ?icomprised between 0 and 2?, calculations are conducted per-

taining to the chord length of the particle track inside the target and the

probability that a particle is emitted according to this angle. f1(z, s), the

SHDSE deposited by a particle emitted at a distance s from the cell center,

can subsequently be derived.

Contribution of Radiolabeled mAbs Free in the Medium or Internalized

within the Cytoplasm

Thedistribution can be calculated using the following equation:

?

V

vol

f

(z) ?

1

d(s) dV

?

V

vol

1

f

(z)

d(s)· f (z, s) dV

1

. (6)

The VOI is a sphere of radius R0? Rn(R0is the CSDA range of the ?

particle in water and Rnis the radius of the cell nucleus) or R0? Rc(Rc

is the radius of the cell) if the target is the whole cell. f1(z, s) is calculated

as described below. The mean number of hits is

?

V

n ¯

?

vol

?

V

d(s)?(s) dV

, (7)

d(s) dV

where ?(s) is the probability that a particle emitted at a distance s from

the cell center hits the target. Calculations for sources within the cyto-

Page 3

659

MICRODOSIMETRIC ANALYSIS OF IN VITRO ?-PARTICLE IRRADIATION. I

plasm are carried out in the same way. As stated previously, only the

VOI is different. For a cell monolayer, the VOI is a half-sphere of radius

R0? Rn(or R0? Rc). For this configuration and that of a cluster, the

VOI is systematically corrected for the volume occupied by the cells.

Contribution of Radiolabeled mAbs Bound to the Surface of Neighbor

Cells

The distribution and the value of n ¯othare determined as described

previously. In the present study, the VOI is made up only of cells that

are inside the sphere of radius R0? Rn(or R0? Rc). For each infinitesimal

element of the integral (7), the intersection in the space of a segment of

a sphere of thickness ds with the surface of cells surrounding the target

cell is calculated. The density of activity at a distance s from the center

of the target cell is the ratio of the intercepted cell surface to the total

surface of a cell multiplied by the number of decays on a cell surface.

oth

1

f

(z)

Intercellular Variations of Cell Radii and Activity Uptake

Among a cell population, cells have different radii. In addition, the

uptake of activity is variable among cells (22). The distribution of cell

radii and the uptake of activity for a representative sample of cells there-

fore must be measured. Using these parameters, precise microdosimetric

spectra can be calculated (21).

1

given by

1 ??

00

distributions of each subspace i are

if (z)

??

ii

1

f (z) ?

p(a)p(R , R ) f (z, R , R ) da dR dR ,

cn

(8)

cncn

where p(Rc, Rn) is the joint probability density function of Rcand Rn, p(a)

is the activity distribution, and

1

of radii Rcand Rnand from particles emitted in the subspace i. The mean

number of hits to the target is

i ? ?

00

is the SHDSE within a cell

if (z, R , R )

cn

??

n ¯ ?

p(a)p(R , R )n ¯ (R , R ) da dR dR ,

cnic

(9)

ncn

where n ¯i(Rc, Rn) is the mean number of hits for a cell of radii Rcand Rn.

The MHDSE within the target can still be evaluated according to Eq.

(1).

Model of Cell Survival after an ?-Particle Irradiation

Survival curves represented as a function of mean absorbed dose after

an ?-particle irradiation are single exponential (23). Cell survival can be

expressed as

?(D/D )

0

S ? e

,(10)

where D0represents the mean absorbed that leads to 37% cell survival.

It is a macrodosimetric description of the dose/survival relationship. It

does not take into account the large variability of doses delivered among

a cell population. It is straightforward to consider microdosimetric spec-

tra, which represent the distribution of doses delivered to cells within a

population, to model survival. This approach was investigated by nu-

merous authors (24–27). They expressed cell survival as a function of

the integral of the distribution of specific energy weighted by an expo-

nential survival function, exp(?z/z0):

?

Survival ?

exp(z/z )· f (z) dz ? exp{?n ¯[1 ? T (z )]},

0

(11)

10

where T1(z0) is the Laplace transform of the SHDSE and n ¯ is the average

number of hits to the target. The microdosimetric radiosensitivity, z0,

corresponds to the specific energy delivered to a cellular target that leads

to a survival of 37%. Whereas the D0parameter depends on the config-

uration of the irradiation, z0depends only on the intrinsic radiosensitivity

of the cell and on the average LET of the radiation. A sensitivity factor

that takes into account the impact of LET on cell survival can be added

to Eq. (11). No bystander effect is considered here.

Comparisons with Monte Carlo Codes

Our algorithm was compared with two different Monte Carlo codes,

MCNPX (28) and Geant4 (29).

MCNPX represents a major extension of MCNP (30) that makes it

possible to track all types of particles, in particular ? particles. This code

enables a relatively easy specification of complex geometries and sources.

Atomic electron interactions will cause an ? particle to release its energy

along the track length (ionization). Energy loss straggling is implemented

using the Vavilov model. Multiple scattering of charged particles is also

considered. However, there is no ‘‘?-ray’’ production of knock-on elec-

trons for ? particles. The default cut-off energies can be set to 1 keV for

? particles. The so-called ptrac output file of MCNPX gathers all the data

concerning the transport of each simulated particle, in particular the en-

ergy and the direction of particles at each interface of the geometry. If a

sufficient number of particles have been simulated, the SHDSE within a

cell of interest can be calculated.

Geant4 is a Monte Carlo code developed by the CERN laboratory.

Like MCNPX, it allows for transporting ? particles. This code offers the

possibility to use different physical models to simulate the transport of

particles. The continuous energy losses along the track of the particle

range are modeled by the Bethe-Bloch equation, which is modified by

taking into account various corrections (31). Straggling is approximated

by a Gaussian distribution with Bohr’s variance (32). Multiple scattering

of charged particles is also considered as well as knock-on electrons

productions. The default cut-off energies for ? particles were set as 1

keV.

During the simulation process, the energy delivered by each ? particle

and secondary electrons was recorded if energy depositions occurred in

the target considered. A histogram representing the SHDSE was produced

in an output file.

For both codes, the MHDSE is derived from Eq. (1). The mean number

of hits to the target is calculated as the ratio of the number of particles

that hit the target to the total number of simulated particles. Different

comparisons were carried out by varying the parameters of the simulation

(geometry of irradiation, radionuclide). The SHDSE and MHDSE plus

the mean absorbed dose and the mean number of hits to the nucleus were

calculated both with our model and with MCNPX and Geant4 for the

following cases:

1. Single cell: five decays of211At in the nucleus, in the cytoplasm, on

the cell surface, and in the sphere of interest surrounding the target

cell (Radius ? R0,max? Rn). R0,maxis the CSDA range of the 7.450

MeV ? particle of211Po.

2. Cell monolayer: five decays of213Bi on the surface of each cell.

3. Closely packed cluster of 13 cells: five decays of211At on the surface

of each cell surrounding the target cell.

Cells and their nuclei are spheres of radii 10 ?m and 5 ?m. Another

calculation was carried out for a single cell labeled on its surface with

five atoms of213Bi. Instead of an average radius, a Gaussian distribution

of cell radii (mean cell radius 10 ?m; standard deviation 1 ?m) was

considered. The nucleocytoplasmic ratio was constant and was equal to

0.5. The Gaussian distribution was sampled (nine samples), and MCNPX

(or Geant4) calculations were repeated for these nine values. For211At,

the two main ?-particle energies were considered, i.e., a 5.867 MeV ?

particle (42% probability per211At decay) released instantaneously upon

the decay of211At, and a 7.450 MeV (58%) ? particle from the decay of

the daughter211Po. Ten million particles were simulated with the MCNPX

(or Geant4) code to obtain spectra with satisfactory statistics in each

energy bin (bin size: 1 mGy).

Comparisons with Methods Published Previously

Another comparison was carried out with results published by Roeske

(33). This method allows for Monte Carlo simulation of ?-particle trans-

port using a standard spreadsheet. To determine the accuracy of this ap-

proach, Roeske provided values of ?z1? and ? ? (the average and average

2z1

Page 4

660

CHOUIN ET AL.

FIG. 1. Single-hit (panel A) and multi-hit (panel B) distributions of

specific energy within the nucleus of a cell at the center of a closely

packed cluster of 13 cells, calculated by our model, by MCNPX and by

Geant4. The 12 surrounding cells are labeled by five atoms of211At. A

total of 107particles were simulated for MCNPX and Geant4 calculations.

Panel C: Single-hit distribution of specific energy within the nucleus of

a single cell. Five atoms of213Bi decayed on its surface. A distribution

of cell radii was taken into account in the calculations carried out by our

model, by MCNPX and by Geant4. A total of 9 ? 107particles were

simulated for MCNPX and Geant4 calculations.

square, in Gy and Gy2, respectively, of the single-hit specific energy) for

four source/target geometries for a broad range of ?-particle emitters.

Thus, with our method, we calculated ?z1? and ? ? within the cell nucleus

for sources located in the cell nucleus, in the cytoplasm, on the cell

surface, or in the volume surrounding the cell for three ?-particle energies

(3.97 MeV, 5.867 MeV and 8.37 MeV).

2z1

RESULTS

Comparisons with Monte Carlo Codes

Comparisons between the analytical model and the Mon-

te Carlo codes were carried out on more than ten different

source-target configurations. Only a few examples are re-

ported here. Figure 1 shows the f1(z) and f(z) distributions

obtained by both methods of calculations in the case of one

target cell at the center of a small cluster of 13 cells labeled

on their surface by five atoms of211At. A good agreement

was observed between the three codes (panels A and B).

A peak can be seen for low specific energies for the Geant4

results that corresponds to energy depositions by secondary

electrons produced outside of the target.

A good agreement was also seen for macrodosimetric

parameters. The mean number of hits calculated with

MCNPX was equal to that calculated with the model (1.06)

and very similar to that calculated with Geant4 (1.07). Fig-

ure 1C shows the f1(z) distribution for a single cell irradi-

ated by ? particles coming from213Bi atoms bound to its

surface. This calculation was carried out for a Gaussian

distribution of cell radii (mean radius ? 10 ?m, SD ? 1

?m). The nucleocytoplasmic ratio was kept constant and

was equal to 0.5. The distribution was sampled, and nine

samples were used for the calculation. A sound agreement

was again observed, except for the peak at low specific

energies present only for the Geant4 results. The mean

numbers of hits to the nucleus calculated by the three codes

were also similar: 0.34 for MCNPX and our code, 0.35 for

Geant4. Table 1 provides all the macrodosimetric results

obtained for the different source–target configurations. Both

methods of calculation provided similar results. The imple-

mentation of our algorithm for calculation of energy de-

posits was then considered as validated. It is useful to com-

pare the time of calculation for both methods. Ten million

particles were simulated to obtain f1(z) and f(z) with a bin

size of 1 mGy with MCNPX and Geant4. For the cluster

of 13 cells, 3 h of calculations was necessary with MCNPX.

Similar calculation times were observed with Geant4. The

same calculation was made in less than a minute with the

model on the same computer (Power Mac G5, 2.2 GHz).

Calculation time increased when a distribution of cell radii

was considered. For each sample of the Gaussian distri-

bution corresponding to a given couple (Rn; Rc), 10 million

particles have to be simulated for each radius bin. Thirty

hours was necessary with MCNPX or Geant4 to obtain the

distributions in Fig. 1C. On the other hand, it took only a

few minutes to run the analytical calculations.

Page 5

661

MICRODOSIMETRIC ANALYSIS OF IN VITRO ?-PARTICLE IRRADIATION. I

TABLE 1

Absorbed Dose and Mean Number of Hits Received by the Nucleus of a Target Cell in

Five Different Configurations

MCNPX

Absorbed

dose (Gy)

Mean number

of hits

Geant4

Absorbed

dose (Gy)

Mean number

of hits

Analytical model

Absorbed

dose (Gy)

Mean number

of hits

Single cell

Activity in the nucleus

Activity in the cytoplasm

Activity on the cell surface

Activity in the medium

Cell monolayer

Activity on the cell surface

0.4321

0.0984

0.0554

0.0030

5.005

0.640

0.341

0.012

0.4333

0.1002

0.0561

0.0031

5.012

0.651

0.350

0.013

0.4332

0.0984

0.0555

0.0031

5.009

0.644

0.343

0.013

0.26111.531 0.26111.532 0.25941.514

Notes. Calculations were carried out with MCNPX, Geant4 and the developed model. For single cells, five decays

of211At were considered. For the cell monolayer, five decays of213Bi on the surface of each cell were considered.

Comparisons with Methods Published Previously

Good agreement was found between our approach and

the results presented by Roeske (33). Table 2 presents the

values of ?z1? and ? ? calculated by the two methods for

three energies and for different source locations. Differenc-

es are less than 3% for ?z1? values between the two calcu-

lation methods. For ? ?, the largest difference is 5%. These

results confirmed the relative accuracy of our calculation

algorithm. As already mentioned by Roeske (33), these dif-

ferences are attributable to methods of interpolation or nu-

merical integration. Roeske used an interpolating polyno-

mial fit of the energy-range relationship, whereas we used

a numerical integration of stopping power data.

2z1

2z1

Evaluation of the Uncertainty in the MHDSE Associated

with the Analytical Approach

The uncertainty in the MHSDE associated with the an-

alytical approach was evaluated according to two methods

as described by Stinchcomb and Roeske (16).

In this paper, it was stated that the area under the

MHSDE plus the area of the ? function (

unity. In all simulated cases, we found that this sum was

between 0.9999 and 1.0002. It appears that there are no

appreciable artifacts caused by the Fourier transforms.

The second internal check uses Kellerer’s theorem (34),

which establishes that the mean specific energy equals the

mean number of events times the mean single-event spe-

cific energy, ?z? ? n ¯?z1?. We checked that this also agreed

(to within 0.01%) in all cases simulated for this work.

) should be

?n ¯

e

DISCUSSION

We present here a newly implemented microdosimetric

model for ?-particle irradiation. This model incorporates

different methods and formalisms published previously.

The algorithm of microdosimetric calculations was com-

pared with two Monte Carlo codes (MCNPX and Geant4).

These tools are based on different calculation methods and

present small differences in the physical treatment of ?-

particle transport. Energy straggling and multiple scattering

algorithms are used in MCNPX and Geant4, but they are

not incorporated into the analytical model. The creation of

knock-on electrons is considered only in Geant4. Despite

these differences, microdosimetric distributions obtained

with MCNPX, Geant4 and our model showed a sound

agreement for every cellular configuration. The only sig-

nificant divergence between the distributions is the peak at

low specific energies when calculated with Geant4. It cor-

responds to the contribution of low-energy secondary elec-

trons produced outside the target, and it is not believed to

play an important role in the toxicity of an ?-particle ir-

radiation. Kellerer (35) had already shown that the effects

of straggling and ? rays were insignificant compared with

the Poisson distribution of the number of hits and the dis-

tribution of path lengths in the target for ? particles of less

than 10 MeV and targets larger than 0.1 ?m. The different

parts of our numerical calculations were then validated. In

addition, this comparison highlighted the differences in cal-

culation times between the two methods. A factor of more

than 1000 was found in favor of our model for the same

achievement in the precision of the calculations. In many

studies, calculation time with a Monte Carlo code appar-

ently proves to be prohibitive.

CONCLUSION

A microdosimetric model based on analytical calcula-

tions was developed, and its algorithm was validated with

a comparative study with all-purpose Monte Carlo codes

and with results published previously. This model enables

the multi-hit distribution of specific energy to be calculated

within a nucleus or a cell deposited by ? particles in nu-

merous source–target configurations encountered during in

vitro experiments. The major outcome of this study is the

description of a validated analytical model that can describe

any in vitro irradiation configurations using ?-particle-emit-

ting radionuclides to derive precise dosimetry calculations

and requires much less computation power than the Monte

Carlo approach.

Page 6

662

CHOUIN ET AL.

TABLE 2

Values of ?z1? and ?z ? (in Gy and Gy2) for Sources Located in the Cell Nucleus, Cytoplasm, Cell Surface or

1

Surrounding Medium

2

3.97 MeV

?z1??z ?

1

2

5.867 MeV

?z1??z ?

1

2

8.37 MeV

?z1??z ?

1

2

Nucleus

RN2 ?m 0.760

(2.4)

0.126

(2.3)

0.034

(0)

0.830

(4.7)

0.023

(4.2)

1.73 ? 10?3

(0)

0.573

(?2.0)

0.093

(?1.1)

0.024

(0)

0.466

(?3.1)

0.012

(0)

8.56 ? 10?4

(?2.5)

0.440

(2.4)

0.072

(1.4)

0.018

(0)

0.275

(4.8)

0.007

(0)

4.82 ? 10?4

(0)

RN5 ?m

RN10 ?m

Cytoplasm

RC5 ?m

RN4 ?m

RC10 ?m

RN6 ?m

RC15 ?m

RN8 ?m

Cell surface

RC5 ?m

RN4 ?m

RC10 ?m

RN5 ?m

Volume

RC10 ?m

RN5 ?m

RC15 ?m

RN8 ?m

0.323

(2.1)

0.162

(0.6)

0.102

(?2.0)

0.123

(4.6)

0.032

(3.0)

0.012

(0)

0.237

(?1.7)

0.113

(0)

0.067

(2.9)

0.066

(?1.5)

0.015

(0)

0.005

(0)

0.180

(1.6)

0.085

(1.2)

0.049

(0)

0.038

(5.0)

0.009

(0)

0.003

(0)

0.342

(2.3)

0.261

(?0.8)

0.136

(3.5)

0.078

(?2.6)

0.249

(?0.8)

0.176

(0.6)

0.072

(?2.9)

0.035

(0)

0.188

(2.1)

0.129

(0.8)

0.041

(2.4)

0.019

(0)

0.282

(0.7)

0.099

(2.9)

0.104

(?2.0)

0.014

(0)

0.247

(?1.2)

0.095

(?1.1)

0.078

(?4.0)

0.011

(0)

0.203

(?1.0)

0.079

(?2.6)

0.054

(?1.9)

0.007

(0)

Notes. Each column represents a given ?-particle energy. Rows correspond to the cell nucleus radius RN(for sources located in the nucleus) or the

couple cell radius RC– nucleus radius RN(for sources located in the cytoplasm, on the cell surface, in the volume surrounding the cell). The percentage

differences between these values and those from Roeske (33) are given in parentheses.

ACKNOWLEDGMENTS

This work was supported by the Comite ´ de Loire-Atlantique de la

Ligue Nationale de Lutte Contre le Cancer and European FP7-HEATH-

2007-A project TARCC.

Received: February 22, 2008; accepted: November 18, 2008

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