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 Divisionw Division
&&
Institute of Nuclear Physics,Institute of Nuclear Physics,
Polish Academy of Sciences, Krakow,Polish Academy of Sciences, Krakow,
POLANDPOLAND

Modern conformal radiotherapy usesModern 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

uniformuniform distributiondistribution of surviving cellsof surviving cells 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 12
1E-3
0.01
0.1
1
Data: Tillyet al, 1999
* Stenerlöwet al, 1995
Model Parameters:
m= 2.14
E0
= 2.13*10
4
erg/cm
3
0
= 5.15*10
-7
cm
2
 = 1100
Katz Model
Co 60 &N-ions, V79 cells
Survival
Dose (Gy)
Co 60
N76.6 eV/nm
N121 eV/nm*
N159 eV/nm
Range & Dose (LET) RRange & Dose (LET) RadioadioBBiologicaliological EEffectivenessffectiveness
OOxygenxygen EEnhancementnhancement RRatioatio

A complicated dependence of cell survival, RBE andA 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: FurusawaData: 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 6
1E-3
0,01
0,1
1
survivingfraction
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 κ
all data points can be
represented using
values best fitted to
all data points:
m = 2.35
D0 = 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 6
1E-3
0,01
0,1
1
0 1 2 3 4 5 6
1E-3
0,01
0,1
1
0 1 2 3 4 5 6
1E-3
0,01
0,1
1
0 1 2 3 4 5 6
1E-3
0,01
0,1
1
0 1 2 3 4 5 6
1E-3
0,01
0,1
1
0 1 2 3 4 5 6
1E-3
0,01
0,1
1 B
survivingfraction
Dose[Gy]
38keV/m
55keV/m
84keV/m
91keV/m
94keV/m
98keV/m
D
survivingfraction
Dose[Gy]
30keV/m
44keV/m
58keV/m
77keV/m
105keV/m
127keV/m
156keV/m
184keV/m
C
survivingfraction
Dose[Gy]
45keV/m
59keV/m
77keV/m
105keV/m
132keV/m
158keV/m
177keV/m
E
survivingfraction
Dose[Gy]
55keV/m
59keV/m
69keV/m
113keV/m
145keV/m
173keV/m
214keV/m
F
survivingfraction
Dose[Gy]
200keV/m
260keV/m
300keV/m
350keV/m
400keV/m
A
survivingfraction
Dose[Gy]
13keV/m
19keV/m
38keV/m
54keV/m
64keV/m
73keV/m
76keV/m
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 cellThis complicated dependence of cell survival,survival, RBE andRBE and
OER on LETOER on LET can be modelledcan be modelled for ion beamsfor ion beams
Data: TsuruokaData: 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. LET
Carbon-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
3
1000
100
10
0,01
0,1
1
survivingfraction
LET keV/m
D
ose
[G
y]
0
2
4
6
8
1E-3
0,01
0,1
1
1
2
31000
100
10
0,01
0,1
1
survivingfraction
LET keV/m
D
ose[G
y]

This complicated dependence of cell survivalThis complicated dependence of cell survival,, RBERBE andand
OER on LETOER on LET can be modelledcan be modelled for ion beamsfor ion beams
Data: TsuruokaData: 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
RBE
Data: Tsuruoka et al. (2005)
C 290 MeV/u
C 135 MeV/u
Ne 230 MeV/u
Ne 400MeV/u
Si 490 MeV/u
Fe 500 MeV/u
m = 2.35
D0 = 1.10 Gy
0 = 14.2 um2
k = 1230

In photon beam radiotherapy, uniformIn 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, RBELET, RBE
and OERand OER varvaryy widely along the depth ofwidely along the depth of
the beam and dependthe beam and depend onon ::
 the physical characteristics of the ion beamthe physical characteristics of the ion beam,,
 the radiobiologicalthe radiobiological characteristicscharacteristics of tumoof tumouurr
and healthy tissue cell linesand healthy tissue cell lines..
NO!
uniformuniform distributiondistribution of surviving cellsof surviving cells 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 theIn the ““αα--ββ -- aproachaproach””,, i.e.i.e.
SS = 1= 1-- expexp –– ((αα DD ++ ββ DD 22 )),,
wherewhere SS == N/NN/N00 is the number ofis the number of
cells surviving of a population ofcells surviving of a population of
NN00 cells exposed to a dosecells exposed to a dose DD ofof
radiationradiation, we, we evaluate the RBE ofevaluate the RBE of
thesethese ““highhigh--LETLET”” modalities andmodalities and
calculate acalculate a distribution ofdistribution of
““biologically equivalent dosebiologically equivalent dose””::
D = DD = Dbiolbiol == RBERBE** DDphysphys..
“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 ACCOUNT
FOR VARIATION OF RBE
WITH S AND ION LET !

uniformuniform distributiondistribution of surviving cellsof surviving cells in the tumourin the tumour
regionregion……
But NOT for ion beams!But NOT for ion beams!
0 2 4 6 8 10 12 14 16 18 20
1E-4
1E-3
0.01
0.1
1
60
Co
1
H
4
He
11
B
12
C
14
N
20
NeV79 cells
Survival
Dose (Gy)
0.0 0.2 0.4 0.6 0.8 1.0
0.01
0.1
1
10
100
V79 cells
1
H
4
He
11
B
12
C
14
N
20
Ne
RBE
Survival
At the same dose of different
ions, survival will differ …
(this is what RBE is all about)
…but RBE will also depend on
the level of survival, S !

uniformuniform distributiondistribution of surviving cellsof surviving cells in the tumourin the tumour
regionregion……
But NOT for ion beams!But NOT for ion beams!
1 10 100 1000 10000
0
1
2
3
4
5
6
7
8
9
Data: Tilly et al, 1999
* Stenerlöw et al, 1995
Model Parameters:
m = 2.14
E0
= 2.13*10
4
erg/cm
3
0
= 5.15*10
-7
cm
2
 = 1100
Katz Model
RBE0.1
, V79 cells
Track Segment
RBE0.1
LET (MeV/cm)
H
He
RBE for He*
B
N
RBE for N
Ar
Fe
1 10 100 1000
0
1
2
3
4
5
6
7
8
9
10
Model Parameters:
m = 2.5
E0
= 2.23 Gy
0
= 5.7 *10
3
nm
2
 = 876
Data: Furusawa et al, 2000
V79 cells, RBE0.1
RBE0.1
LET [keV/m]
3
He
12
C
20
Ne
……and RBE depends on LET, of course…….

Our proposal:Our proposal:
We propose that in order to transfer theWe propose that in order to transfer the
experience of conventional radiotherapy to ionexperience of conventional radiotherapy to ion
beam radiotherapy, a direct comparison bebeam radiotherapy, a direct comparison be
made, in clinically relevant conditions, betweenmade, in clinically relevant conditions, between
the survival of cells in the tumour volume afterthe survival of cells in the tumour volume after
their irradiation bytheir irradiation by ““conventionalconventional”” photon orphoton or
electron beams, and after their irradiation by ionelectron beams, and after their irradiation by ion
beams.beams.
WWe further propose to base our comparisons one further propose to base our comparisons on
data fromdata from in vitroin vitro cell cultures. We wish tocell cultures. We wish to
investigate more closely theinvestigate more closely the particle fluenceparticle fluence
rather thanrather than particle doseparticle dose approachapproach to ionto ion
radiotherapy, to circumvent the doseradiotherapy, to circumvent the dose--relatedrelated
concept of RBE inherent in theconcept of RBE inherent in the ““αα--ββ –– formulaformula””

Some relevant questionsSome relevant questions::
In conventional radiotherapyIn conventional radiotherapy
(60 Gy in 30 fractions of 2 Gy each):(60 Gy in 30 fractions of 2 Gy each):
What fraction of cells surviveWhat fraction of cells survive 2 Gy2 Gy ??
aboutabout ½½
What fraction of cells surviveWhat fraction of cells survive 60 Gy60 Gy ??
aboutabout ((½½ ))3030 ~~ 1010 --1010
There are someThere are some 1010 1010
cells incells in 11 cmcm33
of tumour volumeof tumour volume
We assume that similar (We assume that similar (~~1010 --1010
)) survival is also requiredsurvival is also required
for ion radiotherapy beamsfor ion radiotherapy beams

We apply tWe apply the cellular track structure theoryhe cellular track structure theory
((Katz and coKatz and co--workersworkers, 1967, 1967……..)..)..
ThisThis fourfour--parameter analytical modelparameter analytical model hashas
been extremely successful in quantitativelybeen extremely successful in quantitatively
describing and predicting RBE for cellulardescribing and predicting RBE for cellular
survivalsurvival 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--6060
gamma rays.gamma rays.
THE CELLULAR TRACK STRUCTURETHE CELLULAR TRACK STRUCTURE
MODEL CALCULATIONMODEL CALCULATION

Cell Parameters:Cell Parameters: mm ,, EE00 ,, 00 ,, 
Ion Parameters:Ion Parameters: chargecharge zz ,, fluencefluence FF ,, speed (speed ()),,
tracktrack--segmentsegment LETLET ((z,z,))
MODEL FORMULATIONMODEL 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 beam
of 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 Parameters
Radiat. 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 12
1E-3
0.01
0.1
1
Data: Tilly et al, 1999
* Stenerlöw et al, 1995
Model Parameters:
m = 2.14
E0
= 2.13*10
4
erg/cm
3
0
= 5.15*10
-7
cm
2
 = 1100
Katz Model
Co 60 & N-ions, V79 cells
Survival
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
V79V79 (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 performedThe calculation is performed forfor waterwater
by following the variation of energyby following the variation of energy
of aof ann ion of chargeion of charge ZZ and initial energyand initial energy EEinin
(or speed,(or speed, inin ), as it slows down (CSDA),), as it slows down (CSDA),
in consecutivein consecutive track segments of lengthtrack 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 RBEss areare thus calculatedthus calculated,,
vs.vs. range of ionrange of ion (cm).(cm).

TheThe dosedose (in water) of a beam of ions is(in water) of a beam of ions is
calculated as thecalculated as the product of the ion fluenceproduct of the ion fluence
F (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),, thethe survivingsurviving
fractionfraction of cells is calculatedof cells is calculated in consecutivein consecutive
track segments from Katztrack segments from Katz’’s cellular tracks 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 followingIn the following figuresfigures areare shownshown::
-- surviving fractionsurviving fraction, S,, S, of V79of V79 & AA& AA cellscells
vs.vs. ddepth in waterepth in water, for different, for different ions,ions,
-- RBERBEss vs. depthvs. depth, where, where RBERBEss , the, the RBERBE
at the level of survival at a given depth,at the level of survival at a given depth, SSii ,,
is calculated as the ratio of theis calculated as the ratio of the CoCo--6060 dosedose
required to obtainrequired to obtain SSii and theand the ““ion doseion dose””,,
DDii = F= F  LET(LET(ii)) at theat the ii--th track segmentth track segment
at that depthat that depth,,
-- LETLET vs. depthvs. depth..

DEPTH DISTRIBUTIONS OFDEPTH DISTRIBUTIONS OF
LET, SURVIVAL AND RBELET, SURVIVAL AND RBESS
0 5 10 15 20 25 26
0,01
0,1
1
10
100
1000
Dose
0.25 Gy
1 Gy
V79 cells
12
C
Survival
RBES
LET
Survival,RBES
,LET(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-4
0,01
0,1
1
10
1 Gy
12
C
Survival
RBES
LET
Survival,RBES
Residual range (cm)
Cells
V79
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-3
1E-4
1E-3
0,01
0,1
1
Dose Cell Lines
V79 AA
0.25 Gy
0.5 Gy
1 Gy
Survival12
C
Survival
Residual range (cm )
The Fluence
Problem
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 cm2
AA: 0 = 3.3*10-7 cm2

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-4
1
2
3
4
5
Bragg Peak
12
C
RBES
Residual range (cm)
Dose Cell Lines
V79 AA
0.25 Gy
0.5 Gy
1 Gy
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 0,0
1
2
3
4
V79 cells
20
Ne
14
N
12
C
11
B
7
Li
4
He
1
H
1 GyRBES
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 Irradiation12 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.
0,00,51,01,52,02,53,0
0,0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1,0
0.2 Gy
12
C + 1.8 Gy 
0.6 Gy
12
C
1.8 Gy 
0.2 Gy
12
C
Survival
Residual range (cm)

CONCLUSIONSCONCLUSIONS
 Cell survival is the common denominatorCell survival is the common denominator
betweenbetween ““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 photonlevels of survival encountered in photon
radiotherapy,radiotherapy, the clinical experiencethe clinical experience
gained fromgained from ““cconventionalonventional”” radiotherapyradiotherapy
can be transferred to ion beamcan be transferred to ion beam
radiotherapy.radiotherapy.

 The presentedThe presented fluencefluence tracktrack--segmentsegment
approach enables survivalapproach enables survival--depthdepth
dependences to be calculated directlydependences to be calculated directly
for different ion speciesfor different ion species, obviating the use, obviating the use
of doseof dose--related concepts, such asrelated concepts, such as RBERBE
oror ““biologicalbiologically equivalently equivalent dosedose””..

 The plotted values ofThe plotted values of RBERBEss represent therepresent the
value of RBE at the actual level of survival atvalue 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 speciesthe light ion species (H(H -- Ne)Ne), for, for
cellular parameters representing V79cellular parameters representing V79 and AAand AA
cells,cells, ourour values ofvalues of RBERBEss appearappear to rangeto range
around 2around 2--3.3. Note that aNote that a ““clinical RBEclinical RBE”” ofof
about 3about 3 is usedis used forfor 1212 C radiotherapyC radiotherapy
( Chiba, Japan).( Chiba, Japan).

 The presented oneThe presented one--dimensional track segmentdimensional track segment
fluencefluence approachapproach could becould be representative of therepresentative of the
variable energy treatment technique.variable energy treatment technique. AnAn
example ofexample of „„ion boostion boost”” (mixed X+ion(mixed X+ion
radiotherapy) has been shown. Work is inradiotherapy) has been shown. Work is in
progress on including range straggling and theprogress 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 radiotherapy
the physical specification of the irradiationthe 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 areCellular track structure calculations are
readily available for mixed fields (ionreadily available for mixed fields (ion--ionion
and ionand ion--photon combinations) and arephoton 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 interstitialFrom the perspective of interstitial
brachytherapybrachytherapy, is achieving uniform, is achieving uniform
isoiso--survival over the target volumesurvival over the target volume
a necessary requirement for ion beama necessary requirement for ion beam
radiotherapy?radiotherapy?

Special thanks to:Special thanks to:
•• Irena GudowskaIrena Gudowska –– Associate Professor,Associate Professor,
Department of Medical Physics, Karolinska Institutet andDepartment of Medical Physics, Karolinska Institutet and
Stockholm University, Stockholm, SwedenStockholm University, Stockholm, Sweden
•• Malin HollmarkMalin 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 KorcylMarta 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 FORMULATIONMODEL 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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