fluence vs. dose approach waligorski

Fluence vs. Dose ApproachFluence vs. Dose Approach in Radiobiological Modellingin Radiobiological Modelling of Ion Beam Radiotherapyof Ion Beam Radiotherapy

Michael P.R. WaligMichael P.R. Waligóórskirski

National Atomic Energy Agency, WarsawNational Atomic Energy Agency, Warsaw& &

The Maria SkThe Maria Skłłodowskaodowska--Curie Centre of Oncology, Curie Centre of Oncology, KrakKrakóów Division w Division

&&Institute of Nuclear Physics, Institute of Nuclear Physics,

Polish Academy of Sciences, Krakow,Polish Academy of Sciences, Krakow,

POLANDPOLAND

Modern conformal radiotherapy uses Modern conformal radiotherapy uses MegaMega--Volt photon beamsVolt photon beams

Uniform dose distribution over the target volume impliesUniform dose distribution over the target volume implies uniformuniform

distributiondistribution

of surviving cellsof surviving cells

in the tumour regionin the tumour region




in the tumour regionin the tumour region……

Is this also true for ion beam radiotherapy?Is this also true for ion beam radiotherapy?

0 2 4 6 8 10 121E-3

0.01

0.1

1

Data: Tilly et al, 1999 * Stenerlöw et al, 1995

Model Parameters:m = 2.14E0 = 2.13*104 erg/cm3

0 = 5.15*10-7 cm2

= 1100

Katz ModelCo 60 & N-ions, V79 cells

Surv

ival

Dose (Gy)

Co 60 N 76.6 eV/nm N 121 eV/nm* N 159 eV/nm

Range & Dose (LET) RRange & Dose (LET) RadioadioBBiologicaliological

EEffectivenessffectivenessOOxygenxygen

EEnhancementnhancement

RRatioatio

A complicated dependence of cell survival, RBE and A complicated dependence of cell survival, RBE and OER on LET is observed for ion radiotherapy beamsOER on LET is observed for ion radiotherapy beams

Data: Furusawa Data: Furusawa et al. Radiat. Reset al. Radiat. Res. . 154154, 485, 485--496 (2000)496 (2000)

Survival of V79 cells in vitro vs. LET of a Carbon-12 beam:Aerated cells Anoxic cells

The Cell Survival Curve(cell cultures in vitro)

Note that „high-LET”

(i.e. densely ionising radiation, such as neutrons or heavy ions) are more effective cell killers per dose –

as given by Relative Biological Effectiveness (RBE)

Survival curve formulae: αD + βD2

m -

target

Does in matter how we fit the survival curve?Does in matter how we fit the survival curve?S = exp –

(αD+βD2) or S = 1 –

(1-exp(-

D/D0

)m

?

At high doses: poor fit (beta term dominates )

good fit (linear = exponential)

At low doses : linear

= exponential (alpha term) zero initial slope

a = 0.478 /Gy

b = 0.028 /Gy2

m = 2.35

D0

= 1.10 Gy

0 1 2 3 4 5 61E-3

0,01

0,1

1

surv

ivin

g f

ract

ion

Dose [Gy]

Data: Tsuruoka et al. 200 keV X-ray TST model: D0 = 1.10 Gy, m = 2.35 = 0.478 = 0.048

It does matter:

Alpha and beta terms are fitted individually to each curve, while with two additional parameters:σ0

and κall data points can be represented using values best fitted to all data points:

m = 2.35D0

= 1.10 Gy

= 14.2 um2

= 1230

Survival of Normal Human Skin Fibroblasts after irradiation by ions, Tsuruoka et. al., J.Rad.Res. (2005),163, 494-500.

0 1 2 3 4 5 61E-3

0,01

0,1

1

0 1 2 3 4 5 61E-3

0,01

0,1

1

0 1 2 3 4 5 61E-3

0,01

0,1

1

0 1 2 3 4 5 61E-3

0,01

0,1

1

0 1 2 3 4 5 61E-3

0,01

0,1

1

0 1 2 3 4 5 61E-3

0,01

0,1

1 B

surv

ivin

g fra

ctio

n

Dose [Gy]

38 keV/m 55 keV/m 84 keV/m 91 keV/m 94 keV/m 98 keV/m

D

surv

ivin

g fra

ctio

n

Dose [Gy]

30 keV/m 44 keV/m 58 keV/m 77 keV/m 105 keV/m 127 keV/m 156 keV/m 184 keV/m

C

surv

ivin

g fra

ctio

n

Dose [Gy]

45 keV/m 59 keV/m 77 keV/m 105 keV/m 132 keV/m 158 keV/m 177 keV/m

E

surv

ivin

g fra

ctio

n

Dose [Gy]


F

surv

ivin

g fra

ctio

n

Dose [Gy]

200 keV/m 260 keV/m 300 keV/m 350 keV/m 400 keV/m

A

su

rviv

ing

fract

ion

Dose [Gy]


C-290 MeV/u C-135 MeV/u

Ne-230 MeV/u C-290 MeV/u

Si-490 MeV/u

Fe-500 MeV/u

This complicated dependence of cell This complicated dependence of cell survival,survival,

RBE and RBE and OER on LET OER on LET can be modelled can be modelled for ion beamsfor ion beams

Data: Tsuruoka Data: Tsuruoka et al. J. Radiat. Reset al. J. Radiat. Res. . 163163, 494, 494--500 (2005)500 (2005)

Survival of normal human skin fibroblast cells in vitro vs. LETCarbon-12 ions Iron-56 ions Korcyl & Waligorski, Int. J. Radiat. Biol. 85, 1101-1113 (2009)

0

2

4

6

8

1E-3

0,01

0,1

1

1

2

31000

100

10

0,01

0,1

1

surv

ivin

g fra

ctio

n

LET keV/m

Dose [Gy]

0

2

4

6

8

1E-3

0,01

0,1

1

1

2

31000

100

10

0,01

0,1

1

surv

ivin

g fra

ctio

n

LET keV/m

Dose[Gy]

This complicated dependence of cell survivalThis complicated dependence of cell survival,,

RBERBE

and and OER on LET OER on LET can be modelled can be modelled for ion beamsfor ion beams

Data: Tsuruoka Data: Tsuruoka et al. J. Radiat. Reset al. J. Radiat. Res. . 163163, 494, 494--500 (2005)500 (2005)

RBE at 10% survival vs. LET for normal human skin fibroblast cells in vitro, for C-12, Ne-20, Si-28 and Fe-56 ions

Korcyl & Waligorski, Int. J. Radiat. Biol. 85, 1101-1113 (2009)

10 100 1000

LET [keV/m]

1

2

3

4

5

RB

E

Data: Tsuruoka et al. (2005)C 290 MeV/uC 135 MeV/uNe 230 MeV/uNe 400MeV/uSi 490 MeV/uFe 500 MeV/u

m = 2.35D0

= 1.10 Gy0

= 14.2 um2

k = 1230

In photon beam radiotherapy, uniform In photon beam radiotherapy, uniform dosedose distribution over the target volume is recommendeddistribution over the target volume is recommended……

(ICRU(ICRU--50) 50)

Is this also true for ion radiotherapy?Is this also true for ion radiotherapy?

Spreading out the Bragg peak by: varying the absorber depth magnetic beam scanning

In ion radiotherapy beamIn ion radiotherapy beams,s,

LET, RBE LET, RBE and OERand OER

varvaryy

widely along the depth of widely along the depth of

the beam and dependthe beam and depend

on on ::

the physical characteristics of the ion beamthe physical characteristics of the ion beam,,

the radiobiological the radiobiological characteristicscharacteristics

of tumoof tumouurrand healthy tissue cell linesand healthy tissue cell lines..

NO!




in the tumour regionin the tumour region……

Is this also true for ion beam radiotherapy?Is this also true for ion beam radiotherapy?

So, what do we do about it?So, what do we do about it?In the In the ““αα--ββ

--

aproachaproach””,,

i.e. i.e.

SS = 1= 1-- exp exp –– ((αα DD + + ββ DD 22 )), ,

wherewhere

SS = = N/NN/N00 is the number of is the number of cells surviving of a population of cells surviving of a population of NN00 cells exposed to a dose cells exposed to a dose D D of of radiationradiation, we , we evaluate the RBE of evaluate the RBE of these these ““highhigh--LETLET””

modalities and modalities and

calculate a calculate a distribution of distribution of ““biologically equivalent dosebiologically equivalent dose””::

D = DD = D

biolbiol

= = RBERBE* * DD

physphys

..

“clinical RBE”-

is usually the number

by which the “physical

dose”

(DDphysphys , absorbed dose in tissue, in Gy) applied to the target region should be divided

in

order to correctly treat a given type of tumour.

The Clinical Solution:

BUT WE HAVE TO ACCOUNTFOR VARIATION OF RBEWITH S AND ION LET !




in the tumour in the tumour

regionregion……

But NOT for ion beams!But NOT for ion beams!

0 2 4 6 8 10 12 14 16 18 201E-4

1E-3

0.01

0.1

1

60Co 1H 4He 11B 12C 14N 20NeV79 cellsS

urvi

val

Dose (Gy)0.0 0.2 0.4 0.6 0.8 1.0

0.01

0.1

1

10

100

V79 cells

1H 4He 11B 12C 14N 20NeR

BE

Survival

At the same dose of differentions, survival will differ …(this is what RBE is all about)

…but RBE

will also depend onthe level of survival, S !




in the tumour in the tumour

regionregion…… But NOT for ion beams!But NOT for ion beams!

1 10 100 1000 100000

1

2

3

4

5

6

7

8

9


Model Parameters:m = 2.14E

0 = 2.13*104 erg/cm3

0 = 5.15*10-7 cm2

= 1100

Katz ModelRBE0.1, V79 cells

Track Segment

RB

E 0.1

LET (MeV/cm)

H He RBE for He* B N RBE for N Ar Fe

1 10 100 10000

1

2

3

4

5

6

7

8

9

10

Model Parameters:m = 2.5E0 = 2.23 Gy

0 = 5.7 *103 nm2

= 876

Data: Furusawa et al, 2000V79 cells, RBE0.1

RB

E 0.1

LET [keV/m]

3He

12C

20Ne

……and RBE depends on LET, of course…….

Our proposal:Our proposal:We propose that in order to transfer the We propose that in order to transfer the experience of conventional radiotherapy to ion experience of conventional radiotherapy to ion beam radiotherapy, a direct comparison be beam radiotherapy, a direct comparison be made, in clinically relevant conditions, between made, in clinically relevant conditions, between the survival of cells in the tumour volume after the survival of cells in the tumour volume after their irradiation by their irradiation by ““conventionalconventional””

photon or photon or

electron beams, and after their irradiation by ion electron beams, and after their irradiation by ion beams.beams.WWe further propose to base our comparisons on e further propose to base our comparisons on data from data from in vitroin vitro cell cultures. We wish to cell cultures. We wish to investigate more closely the investigate more closely the particle fluenceparticle fluence

rather than rather than particle doseparticle dose approachapproach

to ion to ion radiotherapy, to circumvent the doseradiotherapy, to circumvent the dose--related related concept of RBE inherent in the concept of RBE inherent in the ““αα--ββ

––

formulaformula””

Some relevant questionsSome relevant questions::In conventional radiotherapy In conventional radiotherapy

(60 Gy in 30 fractions of 2 Gy each):(60 Gy in 30 fractions of 2 Gy each):What fraction of cells survive What fraction of cells survive 2 Gy 2 Gy ??

about about ½½What fraction of cells survive What fraction of cells survive 60 Gy 60 Gy ??

aboutabout ((½½

))30 30 ~~

1010 --1010

There are someThere are some 1010 1010 cells incells in 11 cmcm3 3 of tumour volumeof tumour volume

We assume that similar (We assume that similar (~~1010 --1010 ) ) survival is also required survival is also required for ion radiotherapy beamsfor ion radiotherapy beams

We apply tWe apply the cellular track structure theory he cellular track structure theory ((Katz and coKatz and co--workersworkers, 1967, 1967……..)..)..

ThisThis

fourfour--parameter analytical model parameter analytical model has has been extremely successful in quantitatively been extremely successful in quantitatively describing and predicting RBE for cellular describing and predicting RBE for cellular survival survival in vitroin vitro after heavy ion bombardmentafter heavy ion bombardment, , whereby RBE is referred to a beam of Cowhereby RBE is referred to a beam of Co--60 60 gamma rays. gamma rays.

THE CELLULAR TRACK STRUCTURE THE CELLULAR TRACK STRUCTURE MODEL CALCULATION MODEL CALCULATION

Cell Parameters:Cell Parameters: mm ,, EE00 , ,

00 , , Ion Parameters:Ion Parameters: charge charge z z ,,

fluence fluence FF , , speed (speed ()),,

tracktrack--segmentsegment

LETLET

((z,z,))

MODEL FORMULATION MODEL FORMULATION --

TRACK SEGMENTTRACK SEGMENT(Katz et al. 1994 (Katz et al. 1994 Radiat. Res. Radiat. Res. 140, 356140, 356--365)365)

Survival curves after a dose from a beamof heavy ions (specified by the charge, energy and fluence of these ions) can be calculated, once the four parameters have been simultaneously fitted to a set of experimentally measured cellular survival curves.

Model parameters are fitted from experimental dataModel parameters are fitted from experimental data

R.A. Roth, S.C. Sharma and R. Katz, Systematic evaluation of cellular radiosensitivity parameters,Phys. Med. Biol. 21, 491-503 (1976)

R. Katz, R. Zachariah, F.A. Cucinotta and C. Zhang, Survey of Cellular Radiosensitivity ParametersRadiat. Res. 140, 356-365 (1994).

For a given cell line, cell survival depends on ion dose (fluence),ion charge,and ion energy.

0 2 4 6 8 10 121E-3

0.01

0.1

1


Model Parameters:m = 2.14E0 = 2.13*104 erg/cm3

0 = 5.15*10-7 cm2

= 1100

Katz ModelCo 60 & N-ions, V79 cells

Sur

viva

l

Dose (Gy)

Co 60 N 76.6 eV/nm N 121 eV/nm* N 159 eV/nm

The cellular parameters of the modelThe cellular parameters of the model representrepresentinging

V79 V79 (Chinese Hamster) (Chinese Hamster) cellscells..AA (human melanoma) AA (human melanoma) celcell parametersl parameters

were fittedwere fitted

from experimental datafrom experimental data..

CELL PARAMETERSCELL PARAMETERS

The calculation is performed The calculation is performed forfor

waterwaterby following the variation of energyby following the variation of energyof aof ann

ion of charge ion of charge ZZ and initial energy and initial energy EEinin

(or speed, (or speed,

inin ), as it slows down (CSDA),), as it slows down (CSDA),in consecutive in consecutive track segments of length track segments of length xxii ((

ii ), over which LET(), over which LET(

ii ) is constant) is constant..

For each ion species, For each ion species, tracktrack--segment LETsegment LET, , survival, and RBEsurvival, and RBEs s areare

thus calculatedthus calculated, ,

vs. vs. range of ionrange of ion (cm).(cm).

THE CELLULAR TRACK STRUCTURE THE CELLULAR TRACK STRUCTURE MODEL CALCULATIONMODEL CALCULATION


The The dosedose

(in water) of a beam of ions is (in water) of a beam of ions is calculated as the calculated as the product of the ion fluenceproduct of the ion fluenceF (no. of particles/cmF (no. of particles/cm22) and LET) and LETinin

= LET(= LET(

inin

), ), represented as the entrance (represented as the entrance (””skinskin””) values) values..

As the beam particles slow downAs the beam particles slow down

(no range (no range straggling nor fluence loss)straggling nor fluence loss),,

the the surviving surviving

fractionfraction

of cells is calculatedof cells is calculated

in consecutive in consecutive track segments from Katztrack segments from Katz’’s cellular track s cellular track structure modelstructure model..

ION PARAMETERS (BEAM DATA)ION PARAMETERS (BEAM DATA)

The CSDA range of all ion beams is R = 26.0 cm, in water

In the following In the following figuresfigures are are shownshown::

--

surviving fractionsurviving fraction, S,, S, of V79 of V79 & AA & AA cellscellsvs. vs. ddepth in waterepth in water, for different, for different

ions,ions,

--

RBERBEss vs. depthvs. depth, where, where

RBERBEss , the , the RBERBEat the level of survival at a given depth, at the level of survival at a given depth, SSii ,,is calculated as the ratio of the is calculated as the ratio of the CoCo--6060 dosedoserequired to obtain required to obtain SSii and the and the ““ion doseion dose””,,DDii = F= F

LET(LET(

ii )) at the at the ii--th track segmentth track segment

at that depthat that depth,,

--

LETLET vs. depthvs. depth..


DEPTH DISTRIBUTIONS OF DEPTH DISTRIBUTIONS OF LET, SURVIVAL AND RBELET, SURVIVAL AND RBESS

0 5 10 15 20 25 260,01

0,1

1

10

100

1000

Dose 0.25 Gy 1 Gy

V79 cells12C

Survival

RBES

LET

Sur

viva

l, R

BES, L

ET

(keV

/m

)

Depth in tissue (cm)

DEPTH DISTRIBUTIONS OF LET, DEPTH DISTRIBUTIONS OF LET, SURVIVAL AND RBESURVIVAL AND RBESS

26 10 1 0,1 0,01 1E-3 1E-40,01

0,1

1

10

1 Gy

12C

Survival

RBES

LET

Sur

viva

l, R

BE S

Residual range (cm)

CellsV79 AA

10

100

1000

LET

(keV

/m

)

Depth distributions of ion LET, cell survival and RBES for V79 and AA cells in a beam of 12C ions of initial energy 385.2 MeV/amu, delivering an entrance dose of 1.0

Gy.

DEPTH DISTRIBUTIONS OF SURVIVALDEPTH DISTRIBUTIONS OF SURVIVAL FOR V79 & AA CELLSFOR V79 & AA CELLS

26 10 1 0,1 0,01 1E-31E-4

1E-3

0,01

0,1

1

Dose Cell Lines V79 AA0.25 Gy 0.5 Gy 1 Gy

Survival12C

Sur

viva

l

Residual range (cm )

The FluenceProblem

V79 and AA cell survival-depth dependences in a beam of 12C ions of initial energy 385.2 MeV/amu, delivering an entrance dose of 0.25, 0.5 or 1

Gy

ION PARAMETERS (BEAM DATA)ION PARAMETERS (BEAM DATA)

The CSDA range of all ion beams is R = 26.0 cm, in water

V79: 0 = 5.7*10-7 cm2AA: 0 = 3.3*10-7 cm2

DEPTH DISTRIBUTIONS OF SURVIVALDEPTH DISTRIBUTIONS OF SURVIVAL FOR V79 & AA CELLSFOR V79 & AA CELLS

26 10 1 0,1 0,01 1E-31E-4

1E-3

0,01

0,1

1


Survival12C

Sur

viva

l

Residual range (cm )

The FluenceProblem

V79 and AA cell survival-depth dependences in a beam of 12C ions of initial energy 385.2 MeV/amu, delivering an entrance dose of 0.25, 0.5 or 1

Gy

DEPTH DISTRIBUTIONS OF RBEDEPTH DISTRIBUTIONS OF RBESS FOR V79 AND AA CELLSFOR V79 AND AA CELLS

26 10 1 0,1 0,01 1E-3 1E-41

2

3

4

5

Bragg Peak

12C

R

BE

S

Residual range (cm)


RBES

-depth dependences for

V79 and AA cells in a beam of 12C ions of initial energy 385.2 MeV/amu, delivering an entrance dose of 0.25, 0.5 or 1 Gy. The residual range of the Bragg peak maximum is also shown.

DEPTH DISTRIBUTIONS OF RBEDEPTH DISTRIBUTIONS OF RBESS (V79 CELLS) FOR LIGHT ION BEAMS(V79 CELLS) FOR LIGHT ION BEAMS

1,0 0,8 0,6 0,4 0,2 0,01

2

3

4V79 cells

20Ne

14N12C

11B

7Li4He

1H

1 Gy

RBE

S

Residual range (cm)

RBES -depth dependences of V79 cells over the last 1 cm of residual ion ranges, for light ion beams of range 26 cm, delivering an entrance dose of 1 Gy. Aerobic V79 cells are represented by parameters fitted to the data of Furusawa

et al. (2000)

Mixed CoMixed Co--60 and C60 and C--12 Irradiation 12 Irradiation Survival vs. Depth (V79 Cells)Survival vs. Depth (V79 Cells)

Calculated V79 (data of Furusawa

et al.) cell survival-residual range dependences, following mixed-field irradiation. A modelled 3 cm-thick “target volume”

of cells at the distal end of the beam was “uniformly irradiated”

by a 1.8 Gy

dose of Co-60 γ-rays, resulting in 77% survival (dotted line). Full line: mixed-field irradiation by 1.8 Gy

of Co-60 γ-rays and 0.2 Gy

of carbon ions representing a “high-LET boost”

after “conventional”

radiotherapy. Dashed-dotted line –

0.2 Gy, carbon beam only, dashed line –

0.6 Gy, carbon beam only.

0,00,51,01,52,02,53,00,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

0.2 Gy 12C + 1.8 Gy

0.6 Gy 12C

1.8 Gy

0.2 Gy 12C

Sur

viva

l

Residual range (cm)

CONCLUSIONSCONCLUSIONS

Cell survival is the common denominator Cell survival is the common denominator between between ““cconventionalonventional””

(photon) and ion (photon) and ion

beam radiotherapy.beam radiotherapy.By estimating, within the presented model, By estimating, within the presented model, levels of survival encountered in photon levels of survival encountered in photon radiotherapy,radiotherapy,

the clinical experience the clinical experience

gained from gained from ““cconventionalonventional””

radiotherapy radiotherapy can be transferred to ion beam can be transferred to ion beam radiotherapy.radiotherapy.


The presented The presented fluencefluence

tracktrack--segment segment approach enables survivalapproach enables survival--depth depth dependences to be calculated directlydependences to be calculated directlyfor different ion speciesfor different ion species, obviating the use , obviating the use of doseof dose--related concepts, such as related concepts, such as RBE RBE or or ““biologicalbiologically equivalently equivalent

dosedose””..


The plotted values of The plotted values of RBERBEss

represent the represent the value of RBE at the actual level of survival at value of RBE at the actual level of survival at a given deptha given depth, for a 2 Gy fraction, for a 2 Gy fraction..OvOverer

the light ion species the light ion species (H (H --

Ne)Ne), for , for

cellular parameters representing V79 cellular parameters representing V79 and AA and AA cells, cells, our our values of values of RBERBEss

appear appear to range to range around 2around 2--3.3.

Note that aNote that a

““clinical RBEclinical RBE””

of of

about 3 about 3 is used is used for for 12 12 C radiotherapyC radiotherapy( Chiba, Japan).( Chiba, Japan).


The presented oneThe presented one--dimensional track segment dimensional track segment fluencefluence

approach approach could becould be

representative of the representative of the

variable energy treatment technique.variable energy treatment technique.

An An example of example of „„ion boostion boost””

(mixed X+ion (mixed X+ion

radiotherapy) has been shown. Work is in radiotherapy) has been shown. Work is in progress on including range straggling and the progress on including range straggling and the SpreadSpread--out Bragg Peak (SOBP) out Bragg Peak (SOBP) techniqutechnique, e, following earlier work by Katz & Sharma (1974). following earlier work by Katz & Sharma (1974).

(Katz and Sharma 1974, (Katz and Sharma 1974, Phys. Med. Biol.Phys. Med. Biol. 19, 41319, 413--435)435)..


In reporting ion beam radiotherapyIn reporting ion beam radiotherapythe physical specification of the irradiation the physical specification of the irradiation field, in terms of initial energyfield, in terms of initial energy--fluencefluence

spectra, should be considered.spectra, should be considered.


Cellular track structure calculations are Cellular track structure calculations are readily available for mixed fields (ionreadily available for mixed fields (ion--ion ion and ionand ion--photon combinations) and are photon combinations) and are extremely fastextremely fast, so could be included in ion , so could be included in ion transport codestransport codes..


From the perspective of interstitial From the perspective of interstitial brachytherapybrachytherapy, is achieving uniform, is achieving uniformisoiso--survival over the target volumesurvival over the target volumea necessary requirement for ion beam a necessary requirement for ion beam radiotherapy?radiotherapy?

Special thanks to: Special thanks to:

•• Irena Gudowska Irena Gudowska ––

Associate Professor,Associate Professor,Department of Medical Physics, Karolinska Institutet and Department of Medical Physics, Karolinska Institutet and Stockholm University, Stockholm, SwedenStockholm University, Stockholm, Sweden

••

Malin Hollmark Malin Hollmark ––

Ph.D. Department of Medical Physics, Ph.D. Department of Medical Physics, Karolinska Institutet and Stockholm University, Stockholm, Karolinska Institutet and Stockholm University, Stockholm, SwedenSweden

•• Marta Korcyl Marta Korcyl ––

Ph.D. Student, Jagiellonian University , KrakowPh.D. Student, Jagiellonian University , Krakow

•• Urszula SrokaUrszula Sroka

––

student AGH, Krakstudent AGH, Krakóóww

•• Leszek MalinowskiLeszek Malinowski

––

student AGH, Krakstudent AGH, Krakóóww

MODEL FORMULATION MODEL FORMULATION -- TRACK SEGMENTTRACK SEGMENT(Katz et al. 1994 (Katz et al. 1994 RadiatRadiat. Res. . Res. 140, 356140, 356--365)365)

fluence vs. dose approach waligorski

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