BGRT: Relative Biologically Effectiveness (RBE) and Oxygen ... · RBE f D DD n n γ γγ γγ αβ αβ αβ ≡= −++ +⎨ ⎜⎟⎜⎟⎬ ⎪⎩⎭ ⎝⎠⎪ physical parameters:

Biologically Guided Radiation Therapy (BGRT) Biologically Guided Radiation Therapy (BGRT)

Relative Biologically Effectiveness Relative Biologically Effectiveness g yg y(RBE) and Oxygen Effects in Particle (RBE) and Oxygen Effects in Particle

TherapyTherapy

Rob Stewart, Ph.D.Rob Stewart, Ph.D.Rob Stewart, Ph.D.Rob Stewart, Ph.D.Associate Professor and Assistant Head, School of Health SciencesDirector, Radiological Health Science ProgramSchool of Health SciencesPurdue UniversityPurdue [email protected]://rh.healthsciences.purdue.edu/faculty/rds.html

University of WashingtonUniversity of WashingtonDepartment of Radiation OncologyDepartment of Radiation Oncology

School of Health SciencesSchool of Health Scienceshttp://healthsciences.purdue.edu

p gyp gyFriday April 23, 2010Friday April 23, 2010

Purdue University School of Health SciencesPurdue University School of Health SciencesSlide Slide 22/71/71

Physics → Chemistry → Biology → ClinicPhysics → Chemistry → Biology → ClinicChemicalChemical

RepairRepair1010--33 ss CorrectCorrectRepairRepair

1 1 GyGy ~ 1 in 10~ 1 in 1066

RadiationRadiation

Ionization Ionization ExcitationExcitation

Enzymatic RepairEnzymatic Repair((BER, NER, NHEJ, …))

pp

101022 ss

1010--66 ss

101044 ss

RadiationRadiation

DNA damageDNA damage1010--1818 to 10to 10--1010 ss

Incorrect orIncorrect orAngiogenesis andE l Eff tE l Eff t 66 Incorrect or Incorrect or Incomplete RepairIncomplete Repair

Cell DeathCell Death

101033 ss 101055 ssLoss of Function Loss of Function and Remodelingand Remodeling

Angiogenesis and Vasculogenesis

S lf l d

Inflamatory Responses

Early EffectsEarly Effects((erythema, …))

Late EffectsLate Effects

101066 ss

101044 ss 101055 ss

77CancerCancer

HeritableHeritableEffectsEffects Germline

101055 ss

Self renewal and Differentiation101088 ss((fibrosis, …))

Neoplastic Neoplastic TransformationTransformation

Somaticcells

SmallSmall-- and largeand large--scale mutationsscale mutations((point mutations and chromosomal aberrations))

ClonalClonalExpansionExpansion

101077 ss

101088 ss


RBE Effects in the SOBPRBE Effects in the SOBP

Delivered dose in the spreadDelivered dose in the spread--out Bragg peak (SOBP) due to a out Bragg peak (SOBP) due to a mixture of low, intermediate and high energy protonsmixture of low, intermediate and high energy protons

Source: ICRU 78 (2007)


Effect of Radiation Quality (Effect of Radiation Quality (1st level))Effect of Radiation Quality (Effect of Radiation Quality (1st level))

500500 VV l tl t

““Simple DSBSimple DSB” Opposed strand ” Opposed strand breaks within breaks within about 10 about 10 bpbp

500 500 eVeV electronelectron

““Complex DSBComplex DSB” composed 2 strand ” composed 2 strand breaks with collateral base damage in breaks with collateral base damage in

same or opposed strandssame or opposed strands

Complex SSBComplex SSB –– one or more strand one or more strand

Segment of a 4 Segment of a 4 MeVMeV αα particle (particle (44HeHe2+2+))

ppbreaks (same side) with collateral breaks (same side) with collateral

abasicabasic sites and base damage in the sites and base damage in the opposed strandopposed strand


Relative SSB and DSB inductionRelative SSB and DSB induction

1.2 3.5

Yiel

d 1.0

Yiel

d

3.0electron alpha

lativ

e S

SB

Y

0.8

ativ

e D

SB

Y

2.0

2.5

l h

electron

Re

0.6 Rel

a

1.5

alpha

Increasing LET Increasing LET

Kinetic Energy (MeV)10-4 10-3 10-2 10-1 100 101 102 103

0.4

Kinetic Energy (MeV)10-4 10-3 10-2 10-1 100 101 102 103

1.0

Ratio of SSB per DSB decreases with increasing LETRatio of SSB per DSB decreases with increasing LET

V.A. Semenenko and R.D. Stewart. Fast Monte Carlo simulation of DNA damage formed by electrons and light ions. Phys. Med. Biol. 5151(7), 1693-1706 (2006)


DSB complexityDSB complexityDSB complexityDSB complexity

D bl d b k9

Double-strand breaks

r Clu

ster

6

7

8

alphaalpha

Lesi

ons

per

4

5

6protonproton

umbe

r of L

2

3electronelectron

10-4 10-3 10-2 10-1 100 101 102 103

N

0

1 Increasing LET

Particle Energy (MeV)



SSB and DSB yieldSSB and DSB yieldSSB and DSB yieldSSB and DSB yield) 200

250

totaltotal totaltotalprotonprotonelectronelectron

Yiel

d (G

y-1 G

bp-1

100

150

200 totaltotal totaltotal

>> 2 lesions2 lesions >> 2 lesions2 lesions

pp

SSB

Y

0

50

30

>> 4 lesions4 lesions>> 4 lesions4 lesions

>> 6 lesions6 lesions

Gy-1

Gbp

-1)

15

20

25

30

Total (2 or more)Total (2 or more) Total (2 or more)Total (2 or more)

DSB

Yie

ld (G

5

10

15

>> 6 6 lesionslesions

>> 4 lesions4 lesions>> 4 lesions4 lesions

Electron Energy (MeV)

10-4 10-3 10-2 10-1 100 101 102 1030

Proton Energy (MeV)

10-1 100 101 102 103



DNA and ChromatinDNA and Chromatin

Induction of individual and clustered DNA lesions by ionizing Induction of individual and clustered DNA lesions by ionizing di ti l t d t h ti t ddi ti l t d t h ti t dSpatial scale < 10 nm (Spatial scale < 10 nm (100 to 1000 nucleotides))Time scales < 10Time scales < 10--33 ss

radiation related stochastic events and processes onradiation related stochastic events and processes on

6 6 -- 6.5 nm6.5 nm

11 n11 n0

nm0

nm nmnm3030

NucleosomeNucleosome(146 (146 ++ 1 bp)1 bp)


Radiation Quality and Fragment Size Radiation Quality and Fragment Size Distributions (Distributions (2nd level))Distributions (Distributions (2nd level))

~ 85 bp (nucleosome)~ 85 bp (nucleosome)~ 1000 bp (fiber)

××××××

××××××

~ 1000 bp (fiber)~ 1000 bp (fiber)××××

~ 85 bp (nucleosome)

Holley WR, Chatterjee A. Clusters of DNA induced by ionizing radiation: formation of short DNA fragments. I. Theoretical modeling. Radiat Res. 145(2):188-99 (1996). Rydberg B. Clusters of DNA damage induced by ionizing radiation: formation of short DNA fragments. II. Experimental detection. Radiat Res. 145(2):200-9 (1996).


RMDS and DSB Proximity RMDS and DSB Proximity Significance?Significance?RMDS and DSB Proximity RMDS and DSB Proximity –– Significance?Significance?

DSB in close spatial proximity are DSB in close spatial proximity are more likely to interact (more likely to interact (less likely to y (y ( ybe correctly repaired) than DSB ) than DSB separated by large distances separated by large distances

HighHigh--LET radiation is more effective at producingLET radiation is more effective at producing spatially and spatially and temporally temporally correlated DSB than low LET radiationcorrelated DSB than low LET radiation

On average DSB formed by highOn average DSB formed by high LET radiation are more likely toLET radiation are more likely toOn average, DSB formed by highOn average, DSB formed by high--LET radiation are more likely to LET radiation are more likely to cause harm than DSB from lowcause harm than DSB from low--LET radiationLET radiation


Damage Distribution Among Cells (3rd level)Damage Distribution Among Cells (3rd level)

0.08

SB-1

)

0.06

0.07

1 Gy absorbed dose

ency

(DS

0.04

0.05

1 Gy absorbed doseLow LET (1 keV/μm)40 DSB cell-1

Freq

u

0.02

0.03 490.2 tracks490.2 tracks0.0816 DSB track-1

0 20 40 60 80 1000.00

0.01 2.04 mGyFz =

Number of DSB0 20 40 60 80 100


DSB distribution (1 Gy 2 MeV α particle)

0.025

DSB distribution (1 Gy, 2 MeV α particle)B-1

) 0.020

1 Gy absorbed dose

Not hitNot hit

ency

(DS

0.015

1 Gy absorbed doseHigh LET (162.5 keV/μm)139.64 DSB cell-11 hit1 hit

3 hit3 hit2 hit2 hit

Freq

ue

0 005

0.010 3.02 tracks3.02 tracks46.3 DSB track-14 hit4 hit

0 50 100 150 200 250 3000.000

0.005

331.49 mGyFz =

Nucleus 5 μm in diameter

Number of DSB0 50 100 150 200 250 300

μ


Effects of Radiation Quality on Effects of Radiation Quality on αα and and αα//ββEffects of Radiation Quality on Effects of Radiation Quality on αα and and αα//ββ

1/Fz

α θθ κ

Σ⎛ ⎞= Σ +⎜ ⎟⎝ ⎠

ΣΣ = number of DSB Gy= number of DSB Gy--11 cellcell--11 ((estimated usingMonte Carlo simulations))

// 2 Fz θ κα β ⎛ ⎞= +⎜ ⎟Σ⎝ ⎠

θθ and and κκ are dimensionless, cellare dimensionless, cell--specific biological specific biological constants that are independent or a weak constants that are independent or a weak function of LET up to about 80function of LET up to about 80--100 keV/100 keV/μμmm

20.204FLETzd

≅ frequencyfrequency--mean specific energy (Gy)mean specific energy (Gy)


Proton RBE (experimental)Proton RBE (experimental)

2.25

1.75

2.00

In vitroIn vivo

In vitro: 1.2 In vitro: 1.2 ++ 0.2 (0.2 (0.8, 1.5))

roto

n R

BE

1.25

1.50 In vivo: 1.1 In vivo: 1.1 ++ 0.1 (0.1 (0.8, 1.3))

P

0.75

1.00 ICRU 78 (2007) ICRU 78 (2007) recommends an RBE = 1.1recommends an RBE = 1.1

Fraction Size (Gy)10-1 100 101 102

0.50

P tti H Ni i k A A ki i M G k LE G it i MPaganetti H, Niemierko A, Ancukiewicz M, Gerweck LE, Goitein M, Loeffler JS, Suit HD. Relative biological effectiveness (RBE) values for proton beam therapy. Int J Radiat Oncol Biol Phys. 53(2), 407-421 (2002)


RBE for RBE for ClonogenicClonogenic SurvivalSurvivalRBE for RBE for ClonogenicClonogenic SurvivalSurvival

100

Low LET (e-, γ-rays, …)

7.6 Gy 1.74.4 Gyp

DRBE

Dγ≡ ≅ =

ng F

ract

ion

10-2

10-1( , γ y , )

Higher LET protons

yp

Sur

vivi

10-3

0 1 2 3 4 5 6 7 8 9 1010-4

Absorbed Dose (Gy)


Equivalent Dose of the Reference RadiationEquivalent Dose of the Reference RadiationEquivalent Dose of the Reference RadiationEquivalent Dose of the Reference Radiation

Outcome reference radiation = Outcome mixture of protonsOutcome reference radiation = Outcome mixture of protonsn

∏1 2 31

( ) ( ) ( ) ( ) ( ) ( )n ii

S D S D S D S D S D S Dγ=

= =∏2 2exp exp i i i i iD G D D G Dγ γ γ γ γα β α β

⎡ ⎤⎡ ⎤− − = − −⎢ ⎥⎣ ⎦

⎣ ⎦∑

Take logarithm, apply quadratic formula Take logarithm, apply quadratic formula and rearrange termsand rearrange terms

p p i i i i ii

γ γ γ γ γβ β⎢ ⎥⎣ ⎦⎣ ⎦∑

( / ) 41 1 1

2 ( / ) ( / )i i

i i

G G DD DG

γ γγ

α βα

α α β α β

⎧ ⎫⎛ ⎞⎪ ⎪= − + + +⎨ ⎬⎜ ⎟⎝ ⎠⎪ ⎪⎩ ⎭

∑

Formula has Formula has explicitexplicit corrections for total dose, fraction size, and dose rate effects corrections for total dose, fraction size, and dose rate effects Effect of radiation qualityEffect of radiation quality implicitimplicit in the biological parametersin the biological parameters

2 ( / ) ( / )iiGγ γ γα α β α β⎝ ⎠⎪ ⎪⎩ ⎭

ααii //ααγγ, , ((αα//β)β)ii , , ((αα//ββ))γγ

Effect of radiation quality Effect of radiation quality implicitimplicit in the biological parameters.in the biological parameters.


RBE for a Mixture of ProtonsRBE for a Mixture of ProtonsRBE for a Mixture of ProtonsRBE for a Mixture of Protons

,n

p iD D≡∑/ ,i if γα α≡DefineDefine andand1 ,Gγ ≅

1iG ≅

( / ) 4D n Dα β ⎧ ⎫⎛ ⎞⎪ ⎪∑

1

,p ii=∑,i if γ ,

nγγ

ipn

( / ) 41 1 12 ( / ) ( / )

ii i

p p p ii

D n DRBE f DD D n n

γ γ γ

γ γ

α βα β α β

⎛ ⎞⎪ ⎪≡ = − + + +⎜ ⎟⎨ ⎬⎜ ⎟⎪ ⎪⎝ ⎠⎩ ⎭∑

physical parameters: physical parameters: nnγγ, , nnpp, , DDpp, , DDii

biological parameters: biological parameters: ffii, (, (αα//ββ))γγ, (, (αα//ββ))ii

Similar RBE formula derived by Wilkens and Oelfke (2004) and references therein.

Wilkens JJ, Oelfke U, A phenomenological model for the relative biological effectiveness in therapeutic proton beams. Phys Med Biol. 49(13), 2811-2825 (2004).


Possible proton RBE rangePossible proton RBE rangePossible proton RBE rangePossible proton RBE range3.0

(α/β) = 1.5 Gy2.25

In vitro

2.0

2.5

(α/β)γ 1.5 Gy

1.75

2.00

In vitroIn vivo

RB

E

1.5

Prot

on R

BE

1.25

1.50

0.5

1.0

0.75

1.00

Fraction Size (Gy)0 1 2 3 4 5 6 7 8 9 10

0.0Fraction Size (Gy)

10-1 100 101 1020.50

P tti H Ni i k A A ki i M G k LE G it i

αγ = 0.1 Gy-10.35 Gy-1 > αp > αγ

100 Gy > (α/β)p > (α/β)γSampled using MC methods

Paganetti H, Niemierko A, Ancukiewicz M, Gerweck LE, Goitein M, Loeffler JS, Suit HD. Relative biological effectiveness (RBE) values for proton beam therapy. Int J Radiat Oncol Biol Phys.53(2), 407-421 (2002)


Potential explanation for RBE < 1Potential explanation for RBE < 1Potential explanation for RBE < 1Potential explanation for RBE < 12.5

2.0

tion

100

proton

RB

E

1.5

urvi

ving

Fra

ct

10-1

proton

1.0

Su

photon

Fraction Size (Gy)0 1 2 3 4 5 6 7 8

0.5

Fraction Size (Gy)0 1 2 3 4 5 6 7 8

10-2

αγ = 0.1 Gy-1

(α/β)γ = 1.5 Gyαp = 0.25 Gy-1

(α/β)p = 100 Gy


Is proton RBE constant?Is proton RBE constant?

Any RBE in range from 0.8 to 1.5 consistent y gwith available experimental dataRBE = 1.1 unlikely to be useful (accurate) fory ( )• Comparing the efficacy of photon vs proton treatments• Guiding the selection of appropriate prescription dose and

tolerance doses for normal tissuestolerance doses for normal tissues

Constant RBE inconsistent with widely accepted biophysical mechanismsaccepted biophysical mechanisms underpinning clonogenic survival, tumor control, …control, …


lnS vs Aberrations ( processed DSB)-lnS vs Aberrations (= processed DSB)

One to one correlation between One to one correlation between lethal exchanges (lethal exchanges (dicentrics and

centric rings) implies this form of ) implies this form of damage is a major reason cells are damage is a major reason cells are

killed by radiationkilled by radiation

AC 1522 normal humanAC 1522 normal human

Source: Cornforth and Bedford, Rad. Res., 111, p 385-405 (1987).

AC 1522 normal human AC 1522 normal human fibroblasts irradiated by xfibroblasts irradiated by x--raysrays


Breakage and Reunion TheoryBreakage and Reunion TheoryBreakage and Reunion TheoryBreakage and Reunion Theory

DSBDSB

Chromosome Chromosome

I tI t

Chromosome (Chromosome (= DNA molecule))

Incorrect Incorrect Rejoining Rejoining

R.K. Sachs and D.J. Brenner, Chromosome Aberrations Produced by Ionizing Radiation: Quantitative R.K. Sachs and D.J. Brenner, Chromosome Aberrations Produced by Ionizing Radiation: Quantitative Studies http://web.ncbi.nlm.nih.gov/books/bv.fcgi?call=bv.View..Studies http://web.ncbi.nlm.nih.gov/books/bv.fcgi?call=bv.View..ShowTOC&ridShowTOC&rid=mono_002=mono_002


RBE for DSB inductionRBE for DSB inductionRBE for DSB inductionRBE for DSB induction

DSB induction is closely linked to (DSB induction is closely linked to (but not identical to) a ) a

RBE for DSB induction: dose of a reference radiation

major cell killing mechanism for ionizing radiationmajor cell killing mechanism for ionizing radiation

RBE for DSB induction: dose of a reference radiation needed to produce the same # of DSB as another radiation (“isoeffect calculation”)

{dose of photons}{d f }

pDRBE

Dγ Σ

≡ = =Σ{dose of proton} pD γΣ

ΣΣ = number of DSB Gy= number of DSB Gy--11 cellcell--11ΣΣ number of DSB Gy number of DSB Gy cellcell


Monte Carlo Damage Simulation (MCDS)Monte Carlo Damage Simulation (MCDS)Monte Carlo Damage Simulation (MCDS)Monte Carlo Damage Simulation (MCDS)

MCDS successfully reproduces induction of DNA damage predicted by track structure simulations• Electrons (> 80 eV), protons (> 105 keV), α (> 2 MeV) up to about 1

GeV (Semenenko and Stewart 2006)( )• Photons from about 80 eV up to 1 GeV (Hsaio and Stewart 2008)

Reasonable agreement with measured data• Number of SSB and DSB per Gy per cell• Number of SSB and DSB per Gy per cell

Computationally efficient and freely available• Damage configurations for 100,000 cells exposed to 1 Gy can be

simulated on 2.8 Ghz Pentium in about 1.5 minutes•• http://rh.healthsciences.purdue.edu/mcds/http://rh.healthsciences.purdue.edu/mcds/

Y Hsiao, R.D. Stewart, Monte Carlo Simulation of DNA Damage Induction by X-rays and Selected Radioisotopes. Phys. Med. Biol. 53, 233-244 (2008) . V.A. Semenenko and R.D. Stewart. Fast Monte Carlo simulation of DNA damage formed by electrons and light ions. Phys. Med. Biol. 51(7), 1693-1706 (2006)


DSB yields DSB yields Track Structure and MCDSTrack Structure and MCDS200

DSB yields DSB yields –– Track Structure and MCDSTrack Structure and MCDS

Open symbols NikjooOpen symbols Nikjoo et alet al (1994 1997 1999 2001 2002)(1994 1997 1999 2001 2002)

cell-1

)

150

Open symbols Nikjoo Open symbols Nikjoo et al.et al. (1994, 1997, 1999, 2001, 2002)(1994, 1997, 1999, 2001, 2002)Filled symbols Friedland Filled symbols Friedland et al.et al. (2003, 2005)(2003, 2005)Solid Line Semenenko and Stewart (2004, 2006)Solid Line Semenenko and Stewart (2004, 2006)

ld (G

y-1

100

Diamonds Diamonds –– electronselectronsSquares Squares –– protonsprotonsTriangles Triangles –– αα particlesparticles

DS

B Y

iel

50L LET (L LET (hi h KEhi h KE))

100 101 102 103 104

D

0

High LET (High LET (low KElow KE))Low LET (Low LET (high KEhigh KE))

Zeff2/β2

10 10 10 10 10

Figure adapted from V.A. Semenenko and R.D. Stewart. Fast Monte Carlo simulation of DNA damage formed by electrons and light ions. Phys. Med. Biol. 5151(7), 1693-1706 (2006)


Fragmentation Analysis vs Theory (Fragmentation Analysis vs Theory ( t k t k))Fragmentation Analysis vs Theory (Fragmentation Analysis vs Theory (per trackper track))

102

Tsao et al (2007)

-1

101

Tsao et al. (2007)Lobrich et al. (1996)Newman et al. (1997, 2000)Hoglund et al. (2000)Belli et al. (2002)R db t l (2002)

ck-1

Gbp

-100

Rydberg et al. (2002)Radulescu et al. (2004)Meijer et al. (2005)Esposito et al. (2005)Dini et al. (2005)

DSB

trac

10-1

protonsprotons

10-2Lines denote damage yields for Lines denote damage yields for l t (l t (bl kbl k) t () t (blbl ))

protonsprotons

LET (keV/μm)10-1 100 101 102 103

electrons (electrons (blackblack), protons (), protons (blueblue), ), and and αα particles (particles (redred) predicted ) predicted by MCDSby MCDS


Proton RBE for DSB induction

102 3.5

Proton RBE for DSB induction

M t C l i l ti (S k d S 2006)

m) 101

uctio

n

2.5

3.08.9 μm (0.5 MeV)

75 μm (2 MeV)2 41

Monte Carlo simulation (Semenenko and Stewart 2006)

LET

(keV

/μm

100 E fo

r DS

B in

du

1 5

2.0

1.2 mm

2.41

1.5510

RBE

1.0

1.5(10 MeV) 1.13

ICRU 78 (2007)

0.5 MeV electronelectron

Kinetic Energy (MeV)10-1 100 101 102 103

10-1

Kinetic Energy (MeV)10-1 100 101 102 103

0.5

f S i iRBE for DSB induction computed using the MCDS


Model for Model for αα and and αα//ββ ((conceptual basis))Model for Model for αα and and αα//ββ ((conceptual basis))

DSBDSB

Chromosome Chromosome

I tI t

1/Fz

α θθ κ

Σ⎛ ⎞= Σ +⎜ ⎟⎝ ⎠

Incorrect Incorrect Rejoining Rejoining

// 2 Fz θ κα β ⎛ ⎞= +⎜ ⎟Σ⎝ ⎠θθ andand κκ are dimensionless cellare dimensionless cell--specific biologicalspecific biological

ΣΣ = number of DSB Gy= number of DSB Gy--11 cellcell--11 ((estimated usingMonte Carlo simulations))

20.204FLETzd

≅

θθ and and κκ are dimensionless, cellare dimensionless, cell--specific biological specific biological constants that are independent or a weak constants that are independent or a weak function of LET up to about 80function of LET up to about 80--100 keV/100 keV/μμmm


Radiosensitivity Parameters for ProtonsRadiosensitivity Parameters for Protons

Other Radiation(same θ and θ/κ)

Analyze clinical or laboratory Analyze clinical or laboratory data for a reference radiationdata for a reference radiation(60Co γ-rays, 6 MV x-rays, …)

1/Fz

α θθ κ

Σ⎛ ⎞= Σ +⎜ ⎟⎝ ⎠

/θ κ⎛ ⎞

(same θ and θ/κ)

20.204FLETzd

≅ Σ αR, (α/β)R

// 2 Fz θ κα β ⎛ ⎞= +⎜ ⎟Σ⎝ ⎠

d R, ( β)R

Σ20.204F

LETzd

≅

21( )

R F

R

zαθα β

⎛ ⎞= −⎜ ⎟Σ ⎝ ⎠

[ ]( ) 2 2R Fzθ κ α β= Σ − Other RadiationOther Radiation( )( )Rα β⎝ ⎠ (e-, p, α, γ)

Reference Radiation


Human Kidney T1 cells (in vitro)Human Kidney T1 cells (in vitro)

1.8

2.0 0.30

1 Fzα θ

Σ⎛ ⎞= Σ +⎜ ⎟2κβ Σ

1 ) 1.2

1.4

1.6

2 )

0.20

0.251

/α θ

θ κ= Σ +⎜ ⎟

⎝ ⎠ 2β = Σ

α (G

y-1

0.6

0.8

1.0

β (G

y-2

0.10

0.15

10-1 100 101 1020.0

0.2

0.4

10-1 100 101 1020.00

0.05

LET (keV/μm) LET (keV/μm)

Radiosensitivity parameters for human kidney cells irradiated in vitro by x-rays, 2H1+ and 4He2+ ions. Filled symbols: estimates of α and β reported by Carlson et al. (2008) for x-rays (filled green triangles), 2H+ (filled red circles) and 4He2+ (filled blue squares). Error bars denote the 95% CI. Solid lines: predicted LQ parameter for

3protons with kinetic energies between 0.1 MeV to 1 GeV (θ = 5.27×10-3 and θ/κ = 249.5). Estimate of θ and θ/κcomputed from LQ parameters for x-rays with Σ = 50 DSB Gy-1 cell-1, α = 0.265 Gy-1, α/β = 10 Gy.

Carlson DJ, Stewart RD, Semenenko VA, Sandison GA, Combined use of Monte Carlo DNA damage simulations and deterministic repair models to examine putative mechanisms of cell killing. Rad. Res. 169, 447-459 (2008)


Maximum (Maximum (low dose) RBE) RBEMaximum (Maximum (low dose) RBE) RBE

When When DD << << αα//ββ, , SS((DD) ) ≅≅ exp(exp(--ααDD) and) and

( / ) 41 1 12 ( / ) ( / )

ii i

p p ii

n DRBE f DD n n

γ γ

γ γ

α βα β α β

⎧ ⎫⎛ ⎞⎪ ⎪= − + + +⎜ ⎟⎨ ⎬⎜ ⎟⎪ ⎪⎝ ⎠⎩ ⎭∑

reduces to 1i i

P i

RBE f DD

= ∑

⎛ ⎞

where

( )1( )/ 1

/1

F i ii

i i F i i ii

z LETz LETf

zγ γγ γ γγ

α θ κα θ κ

Σ⎛ ⎞Σ +⎜ ⎟ Σ Σ Σ⎛ ⎞⎝ ⎠≡ = ≅ + ≥⎜ ⎟Σ Σ Σ⎛ ⎞ ⎝ ⎠Σ +⎜ ⎟/γ θ κ⎜ ⎟⎝ ⎠

Σi (protons) and Σγ (reference radiation) determined using the MCDS (S k d St t 2006 H i d St t 2008)(Semenenko and Stewart 2006, Hsaio and Stewart 2008)

Y Hsiao and R.D. Stewart, Monte Carlo Simulation of DNA Damage Induction by X-rays and Selected Radioisotopes. Phys. Med. Biol. 53, 233-244 (2008). V.A. Semenenko and R.D. Stewart. Fast Monte Carlo simulation of DNA damage formed by electrons and light ions. Phys. Med. Biol. 51(7), 1693-1706 (2006)


Maximum Proton RBEMaximum Proton RBE

12

13

Maximum RBE increases as

9

10

11

12

α/β =1.5 Gyα/β =3.1 Gyα/β =5 Gyα/β =10 GyRBE for DSB induction

Maximum RBE increases as α/β decreases

RBE for cell survival is always

xim

um R

BE

6

7

8

RBE for DSB induction RBE for cell survival is alwaysgreater than or equal to RBE for DSB induction

Max

3

4

5 1.93 to 3.00 (1 MeV)1.13 to 1.18 (10 MeV)1.05 to 1.10 (25 MeV)

10-1 100 101 102 1030

1

21.05 to 1.10 (25 MeV)

< 1.01 (> 100 MeV)ICRU 78 (2007)

Kinetic Energy (MeV) For comparison, ICRU 78 For comparison, ICRU 78 recommends an RBE = 1.1recommends an RBE = 1.1


Effective RBE in the SOBP

EUDγ

Effective RBE in the SOBP

Proton Dose Proton Dose Equivalent Equivalent Photon DosePhoton Dose EUDEUD

p

URBE

EUDγ≡DistributionDistribution Photon Dose Photon Dose

DistributionDistributionEUDEUDγγ

EUDEUDpp Niemierko, Med. Phys.24(1) 103 110 (1997)24(1), 103-110 (1997)

ProximalProximal3.0

3.5

ProximalProximal

125 MeV + 0.3 MeV protons incident on water phantom

(1 cm)

DistalDistal(1 )

Absorbed D

1 5

2.0

2.5(1 cm)

DistalDistal(1 )Pencil beam CSDA algorithm

(NIST stopping power)EntranceEntrance

(5 mm “skin”)

(1 cm)ose (Gy)

0.5

1.0

1.5

EntranceEntrance(5 mm “skin”)

(1 cm)

Depth into Water (cm)0 1 2 3 4 5 6 7 8 9 10 11 12

0.0


RBE in the SOBP (4 cm target 2 Gy fraction)RBE in the SOBP (4 cm target, 2 Gy fraction)

3.0

3.5

ProximalProximal3.0

3.5

ProximalProximal α = 0.039 Gy-1, α/β = 10 Gy

rbed

Dos

e (G

y)

1.5

2.0

2.5(1 cm)

DistalDistal(1 cm)rb

ed D

ose

(Gy)

1.5

2.0

2.5(1 cm)

DistalDistal(1 cm)

Region EUDp EUDγ RBESkin 0.96 0.97 1.00

Proximal 1.95 2.03 1.04

0 1 2 3 4 5 6 7 8 9 10 11 12

Abso

r

0.0

0.5

1.0


0 1 2 3 4 5 6 7 8 9 10 11 12

Abso

r

0.0

0.5

1.0


Middle 2.00 2.24 1.12 Distal 1.81 2.81 1.57

Target Avg. 2.00 2.24 1.12 Depth into Water (cm)Depth into Water (cm)

Region EUD EUD RBEα = 0.039 Gy-1, α/β = 1.49 Gy

Region EUDp EUDγ RBESkin 0.96 0.97 1.01

Proximal 1.95 2.06 1.06 Middle 2 00 2 34 1 17

RBE increases as RBE increases as photonphoton αα//ββ decreasesdecreasesMiddle 2.00 2.34 1.17

Distal 1.81 3.27 1.83 Target Avg. 2.00 2.34 1.17

photon photon αα//ββ decreasesdecreases


Effect of Target and Fraction SizeEffect of Target and Fraction Size

1.5

1.4

Photon: α = 0.039 Gy-1, α/β = 1.49 Gy

RBE increases as target

ctiv

e R

BE

1.3

0.01 Gy

RBE increases as target size decreases

RBE d f ti

Effe

1.1

1.2

ICRU 78 (2007)

0.01 Gy

25 Gy2 Gy

RBE decreases as fraction size increases

Width of SOBP (cm)0 1 2 3 4 5 6 7 8 9 10

1.0

ICRU 78 (2007)

Width of SOBP (cm)


Summary and ConclusionsSummary and ConclusionsSummary and ConclusionsSummary and Conclusions

Entrance RBE (e.g., skin) close to 1.0Entrance RBE (e.g., skin) close to 1.0RBE in distal edge of SOBP higher than proximal edge (1 05 → 1 8)proximal edge (1.05 → 1.8)• Potential for biological hot and cold spots

Average RBE in target (1 1 to 1 5)Average RBE in target (1.1 to 1.5)• Increases as α/β decreases (tissue and tumor specific)• Increases as target size decreasesg• Increases as fraction size decreases• For large fractions, RBE < 1

R bl t i t l b ti (0 8 t 1 5)• Range comparable to experimental observations (0.8 to 1.5)

Monte Carlo Simulation of the Effects of Oxygen on Clustered DNAyg

Lesions Formed by Ionizing Radiation

Rob Stewart, Ph.D.Rob Stewart, Ph.D.Associate Professor and Assistant Head of Health SciencesDirector Radiological Health Science ProgramDirector, Radiological Health Science ProgramSchool of Health SciencesPurdue [email protected]://rh.healthsciences.purdue.edu/faculty/rds.html

Presented at

2008 RRS Annual Meeting (2008 RRS Annual Meeting (Sept 21-25, 2008))Boston, MABoston, MABoston, MABoston, MA

Stochastic Track ModelsStochastic Track ModelsMonday September 22, 2008Monday September 22, 2008

School of Health SciencesSchool of Health Scienceshttp://healthsciences.purdue.edu

2:30 to 4:30 pm2:30 to 4:30 pm


Effects of oxygen on DNA damage inductionEffects of oxygen on DNA damage inductionEffects of oxygen on DNA damage inductionEffects of oxygen on DNA damage inductionLong known that oxygen is a powerful modulator of DNA Long known that oxygen is a powerful modulator of DNA damage induction and closely related endpoints such asdamage induction and closely related endpoints such as

Cells irradiated under reduced oxygen sustain less damage and are much

damage induction and closely related endpoints, such as damage induction and closely related endpoints, such as clonogenicclonogenic death and mutationdeath and mutation

more likely to survive (factor ~ 2-4)

Models to predict the effects of oxygen on cluster induction Models to predict the effects of oxygen on cluster induction

Reduced oxygen levels at the time of irradiation decreases cluster complexity (number

p ygp ygcould be used to help test hypotheses such ascould be used to help test hypotheses such as

of DNA lesions per cluster) and reduces the overall cluster yield per cell and per unit dose.

The rate and fidelity (efficiency) of cluster repair increases as cluster complexity y ( ff y) p p ydecreases. Enhanced repair of clusters increases cell survival.


Chemical Basis of the Oxygen EffectChemical Basis of the Oxygen EffectChemical Basis of the Oxygen EffectChemical Basis of the Oxygen Effect

Competition between oxygen fixation and chemical repair is Competition between oxygen fixation and chemical repair is the prevailing hypothesis (the prevailing hypothesis (von Sonntag 2006))the prevailing hypothesis (the prevailing hypothesis (von Sonntag 2006))

((1) DNA + ionizing radiation ) DNA + ionizing radiation →→ DNA lesion (DNA lesion (biochemical repair required))

((2) DNA + ionizing radiation ) DNA + ionizing radiation →→ DNADNA⋅⋅ ((various))

((3) ) DNADNA⋅⋅ + O+ O22 →→ DNADNA--OO22⋅⋅ ((“oxygen fixation” – biochemical repair required))

((5) ) DNADNA⋅⋅ →→ DNA lesion (DNA lesion (biochemical repair required))((4) ) DNADNA⋅⋅ + RSH + RSH →→ DNA (DNA (“chemical repair” – restoration of the DNA*))

** Von Sonntag notes that donation of a proton to a DNA radical may or may Von Sonntag notes that donation of a proton to a DNA radical may or may not restore the original chemical structure of the DNA. But, the chemical not restore the original chemical structure of the DNA. But, the chemical repair process evidently converts the DNA radical (repair process evidently converts the DNA radical (or cluster of radicals?) into a ) into a

Clemens von Sonntag, Free-Radical-Induced DNA Damage and its Repair – A chemical perspective. Springer-Verlag, New York, NY (2006)

form that is more amenable to biochemical repair…form that is more amenable to biochemical repair…


MCDS Modification MCDS Modification Step 1Step 1MCDS Modification MCDS Modification –– Step 1Step 1

Use the original MCDS to simulate the location of DNA Use the original MCDS to simulate the location of DNA di ldi lradicalsradicals

At this stage of the simulation…At this stage of the simulation…At this stage of the simulation…At this stage of the simulation…All lesions formed in the original MCDS treated as a radical that may undergo fixation or chemical repair

No distinction made between radicals formed through direct and indirect mechanisms

Preserves the ability of the MCDS to simulate lesion (radical) clustering y ( ) geffects without introducing any additional parameters into the modeling process.


MCDS Modification MCDS Modification Step 2 (Step 2 (conceptual basis))MCDS Modification MCDS Modification –– Step 2 (Step 2 (conceptual basis))Oxygen is uniformly distributed near the DNA and able to interact with all Oxygen is uniformly distributed near the DNA and able to interact with all DNA radicals with equal probabilityDNA radicals with equal probabilityDNA radicals with equal probabilityDNA radicals with equal probability

OO22

OO22OO22

OO22

OO22OO22OO22 OO22 OO22

If If oxygen fixationoxygen fixation occurs (DNAoccurs (DNA⋅⋅ + O+ O22 →→ DNADNA--OO22⋅⋅), radical converted to a strand ), radical converted to a strand break or damaged base break or damaged base as in the original MCDSas in the original MCDS –– cluster yields for normoxic cluster yields for normoxic conditions same as original MCDS.conditions same as original MCDS.

DNADNA⋅⋅ →→ DNA lesion (DNA lesion (biochemical repair))

“Fixation” implicitly includes other processes that convert DNA radicals to a form “Fixation” implicitly includes other processes that convert DNA radicals to a form requiring biochemical repair.requiring biochemical repair.

If If chemical repairchemical repair occurs (DNAoccurs (DNA⋅⋅ + RSH + RSH →→ DNA), DNA is restored to its DNA), DNA is restored to its original (original (undamaged) state ) state –– lesion not createdlesion not created


Chemical repair or fixation?Chemical repair or fixation?Chemical repair or fixation?Chemical repair or fixation?Define Define ff as the fraction of the DNA radicals that undergo chemical repairas the fraction of the DNA radicals that undergo chemical repairF ti (1F ti (1 ff ) f th DNA di l “fi d” b) f th DNA di l “fi d” bFraction (1Fraction (1--f f ) of the DNA radicals are “fixed” by oxygen) of the DNA radicals are “fixed” by oxygen

OO22

OO22

OO22OO22

OO22OO22OO22 OO22 OO22

[ ]( ) [ ][ ]

22 1

O Kf O

O M K+

= −+ ⋅

[O[O22] = oxygen concentration] = oxygen concentrationat time of irradiationat time of irradiation[ ]2O M K+

Formula is derived from (Formula is derived from (related torelated to) oxygen) oxygen--effect formula of Alper and effect formula of Alper and H dH d Fl d (1956)Fl d (1956)

Introduces two adjustable parameters (Introduces two adjustable parameters (MM and and KK) into the modeling process.) into the modeling process.

HowardHoward--Flanders (1956)Flanders (1956)

Alper T, Howard-Flanders P. Role of oxygen in modifying the radiosensitivity of E. coli B, Nature (London) 178, 978–979 (1956).


Eff t f OEff t f O fi ti d h i l i fi ti d h i l i

1.0

Effect of OEffect of O22 on fixation and chemical repairon fixation and chemical repair

0.8 [ ]( ) [ ][ ]

22 1

O Kf O

O M K+

= −+ ⋅

11--f f ((fixation)) normoxicnormoxic

obab

ility 0.6

[ ]2O M K+ ⋅

MM = 1.6574= 1.6574KK = 0.5209= 0.5209

ffmaxmax = 1 = 1 –– 1/M1/MMM determines maximum fraction of determines maximum fraction of radicals fixed at 0% Oradicals fixed at 0% O22

Pro

0.2

0.4

chemical chemical repairrepair

K K = oxygen concentration at which= oxygen concentration at whichff equals 1/equals 1/MMff at 1/at 1/MM

%% 22

Oxygen Concentration (%)10-3 10-2 10-1 100 101 102

0.0

KK

Oxygen Concentration (%)


OER for Damage Induction (OER for Damage Induction (low LET))4.0

V79 cellsCHO ll

4.0Georgakalis (2008 - unpublished)Frankenberg-Schwager et al. (1991)

OER for Damage Induction (OER for Damage Induction (low LET))

3.0

3.5CHO cellsHuman kidney T-1 cells

3.0

3.5

Frankenberg Schwager et al. (1991)Nygren and Ahnstrom (1996)Hirayama et al. (2005)Whitaker McMillan (1992)Prise et al. (1992)Botchway et al. (1997)

OE

R

2.0

2.5

OE

R

2.0

2.5

MCDSMCDS

1.0

1.5

1.0

1.5MCDSMCDS

(1 MeV (1 MeV ee--))

DSBDSB C ll S i lC ll S i l


0.5


0.5

DSBDSB Cell SurvivalCell Survival

Trends and estimates of OER for DSB induction and cell survival Trends and estimates of OER for DSB induction and cell survival comparable to values predicted by MCDS (comparable to values predicted by MCDS (KK = 0.5209, = 0.5209, MM = 1.6574= 1.6574).).

Carlson DJ, Stewart RD, Semenenko VA, Effects of oxygen on intrinsic radiation sensitivity: A test of the relationship between aerobic and hypoxic linear-quadratic (LQ) model parameters. Med Phys. 33(9), 3105-3115 (2006).


Interplay between oxygen and LETInterplay between oxygen and LETInterplay between oxygen and LETInterplay between oxygen and LET

((1) DNA + ionizing radiation ) DNA + ionizing radiation →→ DNA lesion (DNA lesion (biochemical repair required))((2) DNA + ionizing radiation ) DNA + ionizing radiation →→ DNADNA⋅⋅ ((various))

((4)) DNADNA⋅⋅ + RSH+ RSH →→ DNA (DNA (“chemical repair” restoration of the DNA*))((3) ) DNADNA⋅⋅ + O+ O22 →→ DNADNA--OO22⋅⋅ ((“oxygen fixation” – biochemical repair required))

((5) ) DNADNA⋅⋅ →→ DNA lesion (DNA lesion (biochemical repair required))((4) ) DNADNA⋅⋅ + RSH + RSH →→ DNA (DNA ( chemical repair – restoration of the DNA ))

If ionization does not substantially alter the local cellular If ionization does not substantially alter the local cellular environment (e.g., “environment (e.g., “oxygen-in-track hypothesis”), might expect ”), might expect reactions (reactions (3))--((5) to be same for low and high LET radiation.) to be same for low and high LET radiation.

[ ]( ) [ ][ ]

22 1

O Kf O

+= −

(( )) (( ) g) g

Assume Assume MM and and KK are independent of are independent of di ti litdi ti lit[ ]( ) [ ]2

2

1f OO M K+ ⋅ radiation quality.radiation quality.


OER with OER with ff independent of LETindependent of LET

DSB Yield (%)Number of

OER with OER with ff independent of LETindependent of LET4.0

1 - - 2 16.033 5.148 3 21 070 9 163

NormoxicNumber of Lesions Anoxic

3.0

3.5

αα3 21.070 9.163 4 18.900 10.914 5 14.523 11.180 6 10.345 10.490 7 6 929 9 408

OE

R

2.0

2.5

2 MeV 2 MeV ααee--

7 6.929 9.408 8 4.558 8.143 9 2.937 6.921

10 1.850 5.716 1.0

1.5

Cell survival (Carlson et al. 2006)Frankenberg-Schwager et al. (1991)Prise et al. (1989)Prise et al. (1990)Hirayama et al (2005)

((162.5 keV/μm))

ff = 39.6%= 39.6%

11 1.142 4.711 12 0.676 3.823 13 0.418 3.059 14 0.254 2.467

LET (keV/μm)10-1 100 101 102

0.5Hirayama et al. (2005)

Model cannot easily explain theModel cannot easily explain the 15 0.150 1.939 > 16 0.214 18.207

Avg per DSB 4.711 7.859

Model cannot easily explain the Model cannot easily explain the decrease in the OER as LET decrease in the OER as LET increasesincreases


Possible explanation for LET effect?Possible explanation for LET effect?Possible explanation for LET effect?Possible explanation for LET effect?

((1) DNA + ionizing radiation ) DNA + ionizing radiation →→ DNA lesion (DNA lesion (biochemical repair required))((2) DNA + ionizing radiation ) DNA + ionizing radiation →→ DNADNA⋅⋅ ((various))

((4)) DNADNA⋅⋅ + RSH+ RSH →→ DNA (DNA (“chemical repair” restoration of the DNA*))((3) ) DNADNA⋅⋅ + O+ O22 →→ DNADNA--OO22⋅⋅ ((“oxygen fixation” – biochemical repair required))

((5) ) DNADNA⋅⋅ →→ DNA lesion (DNA lesion (biochemical repair required))((4) ) DNADNA⋅⋅ + RSH + RSH →→ DNA (DNA ( chemical repair – restoration of the DNA ))

Ionization does not substantially alter the local cellular Ionization does not substantially alter the local cellular environment (environment (reactions 3-5 same for low and high LET radiation), but reaction ), but reaction 1 is enhanced relative to reaction is enhanced relative to reaction 2),),

[ ]( ) [ ][ ]

22 1

O Kf O

+= −

KK isindependentisindependent of radiation quality and of radiation quality and MMbecomes function of radiation qualitybecomes function of radiation quality[ ]( ) [ ]2

2

1( )

f OO M l K+ ⋅

becomes function of radiation quality becomes function of radiation quality ((convenient way to implement effect into model))


OER with OER with ff a function of radiation qualitya function of radiation qualityOER with OER with ff a function of radiation qualitya function of radiation quality

( ) [ ]2O K+4.0

[ ]( ) [ ][ ]

22

2

1( )

O Kf O

O M l K+

= −+ ⋅

( )1M − 3.0

3.5

( )00

2

1( )

1 /

h eff

MM l M

q l

Zl

= −+

⎛ ⎞⎜ ⎟

OE

R

2 0

2.5 αα

where efflβ

≡ ⎜ ⎟⎝ ⎠

( )2/31 exp 125effZ Z Zβ −⎡ ⎤= − −⎣ ⎦

1.5

2.0

Cell survival (Carlson et al. 2006)Frankenberg-Schwager et al. (1991)

ee--

( )

( )220

p

111 /

eff

T m c

β

β

⎣ ⎦

= −+

LET (k V/ )10-1 100 101 102

0.5

1.0g g ( )

Prise et al. (1989)Prise et al. (1990)Hirayama et al. (2005)

( )0/ m c

MM00 = 1.658326, = 1.658326, qq = 817.2638, = 817.2638, KK = 0.5209= 0.5209

LET (keV/μm)


Effect of Oxygen on Cluster ComplexityEffect of Oxygen on Cluster ComplexityEffect of Oxygen on Cluster ComplexityEffect of Oxygen on Cluster Complexity

DSB Yield (%)

N iNumber of A i

DSB Yield (%)Number of

1 - - 2 6.969 4.987 3 11.678 8.856


1 - - 2 64.844 49.407 3 25.854 30.495


4 13.178 10.730 5 12.721 10.969 6 11.266 10.388 7 9.600 9.347 8 7 799 8 125

4 7.117 12.936 5 1.700 4.786 6 0.385 1.603 7 0.081 0.532 8 0 015 0 1668 7.799 8.125

9 6.264 6.924 10 4.928 5.809 11 3.819 4.796 12 2.945 3.881

8 0.015 0.166 9 0.004 0.054

10 - 0.016 11 - 0.004 12 - 0.0019 5 3 88

13 2.214 3.168 14 1.694 2.555 15 1.279 2.020

> 16 3.645 7.446

12 0.001 13 - 0.001 14 - 0.000 15 - -

> 16 - -

2 MeV 2 MeV αα ((162.5 keV/μm))

Avg per DSB 6.856 7.994 OER 1.217

Avg per DSB 2.473 2.813 OER 2.585

1 MeV 1 MeV ee-- ((0.186 keV/μm))


OER for SSB and DSB inductionOER for SSB and DSB induction3.0

OER for SSB and DSB inductionOER for SSB and DSB induction3.0

1 MeV electron1 MeV electron

2.5

DSBDSB2.5 DSBDSB

1 MeV electron1 MeV electron

OE

R

1.5

2.0

SSBSSB

OE

R

1.5

2.0

SSBSSB

1.0

SSBSSB

1.0

SSBSSB

M d OERM d OER 22 4 (4 (l LET))


0.5

LET (keV/μm)10-1 100 101 102

0.5

Measured OERMeasured OERssbssb ~ 2~ 2--4 (4 (low LET))

OEROERdsbdsb ≅≅ ((OEROERssbssb))2 2 ≅≅ MM((l))22


SummarySummarySummarySummary

Predicted trends in the OER for DSB induction consistent with estimates derived from measured data for DSB induction andcell survival (OER ~ 2 to 3)

• Three new parameters introduced into the modeling process to account for oxygen concentration (M and K) and to correct for the effects of radiation quality (q)

• As LET increases, the probability chemical repair occurs per initial DNA radical may decrease (oxygen fixation increases and/or lesions are directly created)may decrease (oxygen fixation increases and/or lesions are directly created)

Predicted OER for SSB induction approximately equal to the square root of the OER for DSB induction (OER ~ 1.4 to 1.7)• Most of the measured data suggest a higher OER for SSB induction• Most of the measured data suggest a higher OER for SSB induction

Average cluster complexity increases as the oxygen concentration increases•• 11 MeVMeV ee-- (2 5 lesions/DSB 0% O and 2 8 lesions/DSB 21% O )•• 1 1 MeVMeV ee-- (2.5 lesions/DSB 0% O2 and 2.8 lesions/DSB 21% O2)•• 2 2 MeVMeV αα (6.8 lesions/DSB 0% O2 and 8.0 lesions/DSB 21% O2)

Documents

BGRT: Relative Biologically Effectiveness (RBE) and Oxygen ... · RBE f D DD n n γ γγ γγ αβ αβ αβ ≡= −++ +⎨ ⎜⎟⎜⎟⎬ ⎪⎩⎭ ⎝⎠⎪ physical parameters: