Radiation induced lung damage - Appendices

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Radiation induced lung damage

Seppenwoolde, Y.

Publication date

2002

Link to publication

Citation for published version (APA):

Seppenwoolde, Y. (2002). Radiation induced lung damage.

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Appendixx I

Thee General Parallel Model

Thee organ is divided into voxels (n), each voxel has a relative contribution (c„) to the functioningg of the whole organ with

11 N XTT ^ J n

Forr lung d can be derived from SPECT lung perfusion imaging. For healthy lung tissue c^ doess not depend on the location in the lung. Furthermore there is a dose distribution with a dosee Dn in each voxel. The biological effect of this dose distribution on the functionality of the organn can be either relative functional changes with a continuous nature or be incidences of seriouss complications. Examples of functional changes are relative changes in pulmonary functionn tests. The incidence of a grade 3 radiation pneumonitis is an example of a serious complication. .

Inn the general parallel model the total effect or the response is found by adding up ail local effectss (W(Dn). This sum is defined as the Overall Response Parameter (ORP, Boersma 1905)

ORP^lJw(Da) )

Nextt to this ORP we define the function weighed corresponding OfRP

W(D)) is the dose-weighting function that depends on the local dose and can be every positive relation.. The function-weighting factor is c„. The function-weighting factor depends only on the locationn and has to be normalized on the total. For W(D) we can define a more specific shape likee a sigmoid dose effect relation E(D). In that case E(O) varies between 0 and 1 and a biologicall interpretation of the model is possible: c,, is the relative amount of functional sub unitss (FSUs) in each voxel and E(D) reflects the dose-effect relation for FSU-kill. The OfRP is thenn equal to the relative reduction of the total number of FSUs, (refer to Appendix III). Instead off ORPs in the Volume space' we can define an overall parameter in the 'dose space' by using thee inverse function W~1. this is possible as long as W{0) is not a step function. Using this inversee function we can define the Effective Uniform Dose (EUD)

AppendixAppendix I

andd the Effective function-weighted Uniform Dose (EfUD)

EfUDD = W"

1

(OfRP) = W ^ j l j c . -W(D

0

)J

(1-5) ) Usingg these EUDs and a sigmoid relation between the complication probability and a uniform dosee to the whole organ, the complication probability of each EUD can be calculated. When W(Dn)) is equal to a power-law function, we get the function-weighted Lyman-Kutcher-Burman

(1KB)(1KB) model (Lyman 1965, Kutcher 1989)

EfUDD = | l | j c

B

. ( D J

1

4 -EfUD,„

Iff Cn is homogeneous (or omitted) we get the LKB model and the associated EUD

(1-6) )

E U D =

{ ^ £ (

D

« )

I / B

}} =

E U D

LKB B (1-7) )

Thiss EUD is equal to the Den defined by Mohan et al. (1992), D ^ according to Damen et ai (1994)) and the EUD defined by Niemierko in 1999. If W(Dn) is a linear function (with the power-laww parameter equal to 1), the Mean function-weighted Lung Dose (MfLD) appears

EUDD =

fe£

D

*}

=MLD

EfUD = W

C l |

. ( D

n

) } =MfLD

(1-8,9) ) tff W(Dn) is a sigmoid function E(Dn) we get the 'classical' parallel model or the critical volume model,, according to Niemierko and Gotein (1993), Jackson et al. (1993) and Yorke et al. (1993)) with the corresponding EUD and EfUD

EfUDD = E~

I

jlf>,

B

)J = EfUD

|

(1-10) )

(1-11) )

Iff the sigmoid relation E(D) is defined by a logistic function, the EUD and EfUD become

V

11

1 *

E U D ^ ^ D ^ ^

11 v-» 1

- 1 1

(1-12) )

Appendix! Appendix!

EWD^-D,, ,

\f\fXX N C f l

(NSI+CDJ/DJ J

_{J J}

- 1 1

(1-13) )

Whenn E(D) is a step function, calculation of E*1 is not possible and we cannot calculate an EUDD and EfUD. But we can still calculate ORPs. In case

fOforD,, < D

E(D

n

)) = ©(D - D

M

) =

nJnJ ^v « so; |

l f o r D

^

D 50 0 50 0 (1-14) ) wee get

ORPP = l | ; e ( D

n

- D

5 0

) = ^

D>D50 0 (1-15) ) tot t 11 N fV (1-16) )

Usingg these ORPs and a sigmoid relation between the complication probability and the volume irradiatedd above a threshold dose, the complication probability of a certain ORP can be calculated. .

Appendixx II

Normalizationn of lung perfusion measured with SPECT

Wee assume that SPECT lung perfusion scans provide 3D local information about the distributionn of Ch that we interpret as the distribution of parallel FSUs (refer to Appendix III). Wee have a SPECT lung perfusion image with voxels (n,N) and a number of counts (Ctsn) in voxell n. Because the exact injected activity and thus the number of counts is not known, one cann normalize the SPECT scan on the average number of counts per voxel

n O T m ll =

^ Z

C t s

n C

B

E a- T^Li

Ca

=

l

<

I M

*

2

'

3

)

NN ~ nonnl N ~

Thiss Cn has the same properties of the earlier defined cn. The capital is to indicate that we use perfusion.. When the patient has a homogeneous perfusion, in each voxel the number of countss will be equal to Cts and

11

N

Cts

nannll =—YCts = Cts C = = 1 fli-4.5)

NN „^ norml

AA second normalization is made on the well-perfused regions because these regions are consideredd to be healthy and react normal to applied dose. In these regions the number of FSUss wiH be on average equal to F0. Well-perfused voxels are those for which the number of countss is larger mat 60% of the maximum number of counts in a voxel in the lungs

Cts1*' ' >> 0.6 (11-6)

MAXCCtsf) )

11 5

" W P P

S

C t s

. .

(11-7,8) )

norm22 = — 2 < ^ = ^ r ^ Perf

B

=

norm2 2

Thiss normalization can also be obtained by directly calculating norm2' from the original SPECT perfusionn image

nann2'== "fcts

NN **

11 *

(l"-9) ) 'WPP n=l

AppendixAppendix II

Becausee norm1*norm2=norm2':

Cts. .

Perf.. = ^

-norm2'' norm 1 - norm2 norm 1

(11-10) )

Normalizationn of follow-up SPECT lung perfusion scans

Too be able to compare pre- and post-RT images, normalization is required because the blood floww pattern through the lungs is altered due to treatment and because the breathing level of thee patient can be changed over time. Also the exact amount of injected radioactivity that reachedd the lung capillaries is not known a priori. We assume that in the lung regions receiving aa low dose (LD: less than 8 Gy) no changes in blood flow resistance would occur over time. Thee definition of the well-perfused region is made in the first scan and is kept constant for follow-upp scans.

Becausee the post-RT scans have to be normalized on the well-perfused low dose regions (WPLD),, we will also normalize the pre-RT scans on the same region. The difference between normalizingg on WP alone or on WPLD is minimal (Figure 1).

100 0

200 40 60 80 PHpreWPLD(%)(=1/rtorm2preWPLD) )

100 0

FigureFigure 1. The PHpre, normalized on perfused regions, plotted as a function of the PHpre, normalized on well-perfusedperfused low dose regions.

Becausee of this small difference and because normalization on WPLD enables us to compare pre-- and post-RT scans, we used the WPLD normalization throughout this thesis

norm2'= =

1 1

N N

•wn.rj j

ICts, ,

1 1

WPLDD n=l

N N

" W P L D D

£ P e r f

n

= l l

(11-11,12) ) WPLDD n=l 150 0

AppendixAppendix II

Thee homogeneity of perfusion throughout the tungs can be expressed in a parameter catted perfusionn homogeneity: the average perfusion throughout the lungs, expressed as a percentagee of the average perfusion in the well-perfused (low dose) lung regions.

11

N

PHH = j - i W =

_{™WfU> >}

(=—

l

—) (IM3)

1-PHH is called the Perfusion deficiency (PD).

inn chapter 3 the PD is defined a tfttie differently because lung perfusion is not totalfy homogeneous,, even for patients with healthy lungs, because the perfusion is less at the lung edges.. Therefore the average lung perfusion of a reference patient group is calculated and averagedd over all patiënte to provide a reference value. The perfusion of an individual patient ii is compared to the homogeneity of the reference group P (81 malignant lymphoma and breastt cancer patients), according to

(Perfusionn deficiency). = 1 - - — z — — — ...

A

..

Forr patients with a homogeneous perfusion:

norm22 = — Y C = 1 Perf = = 1 PH = 1 (11-15.16) N ^^ nonn2

Thee perfusion deficiency can be calculated from pre-RT scans, post-RT scans or the post-RT perfusionn homogeneity can be predicted from the pre-treatment perfusion scan, using a dose-effectt relationship.

Measuredd follow-up perfusion scan

Pre-treatmentt quantities will get the label 'pre* if they can be confused with post treatment parameters.. The WP voxels are determined from the pre-radiotherapy perfusion scan. The normalizationn factor for the post-RT scan is obtained by calculating the average perfusion in thee voxels that were well-perfused in the pre-RT scan and got a low dose

nornüT"*nornüT"*

00

** = — — ^ C t s ^ *

0 -

Perf

n mMBpo

'

t

=

Cts

mc

"

spo

* *

WPLDD a=l

'' Ï S S T C'-

17

.

18

)

Thenn of course

AppendixAppendix tl

^ -

N

£ ° p

e

r f , r * *

0 8 ,

= ll (M-19)

W L DD B"! and d (II-20) ) PHmeaspost=-- TPaf,°c"w°" = " " 1 = !

Ntff " 1 V Q - , . I nam*""-"'

Wee define Measured Perfusion Loss (MPL) and Measured Perfusion Reduction (MPR).

11 N

MPLL = — 2 (P e r fT - PeriT"*'*) = PHpre- PHmeaspost (11-21)

INN n_t

and d

__ MPL PHmeaspost

M P RR = — = 1 — - = (11-22)

PHprcc PHprc

Predictionss follow-up perfusion scan

Thee PHpost can also be predicted based on the pre-treatment perfusion scan, the local dose distributionn Dn and a local dose-effect relationship E(Dn).

Wee define Perfusion Homogeneity "post RT as predicted" as

P H p r e d p o s t - i ^ P e r f r ^^ - i r X P ^ f r - i l - E C a ) } (||-23) ii N a =i

Withh a homogeneous perfusion (PH and Perf are equal to 1) the overall response parameters (Appendixx I) become

11 N I N

OpRPP = ORP = - 2 , E ( a ) and OpRP = ORP = - £ E ( D » , ) (11-24,25)

Iff E(D) is linear and MLD is the mean lung dose:

PHpredpostt = 1 - ORP (H.26)

AppendixAppendix II

Withh MpLO the mean perfusion-weighted lung dose, the predicted perfusion loss wit) be

cc

N

PPLL = ^ X

P e r f

, r -Dk ^ c - M p L D

Y

(II.271)

andd the predicted perfusion reduction

N N

_{MpLDr r}

PPR=^£cr.D

a

-c.MpLDD = c ^ ^ (,.-28)

NJJJ PHpre

PPRR =

P H p r e

"

P H p r e d p 0 S t

(II-29)

PHpre e

PPLL = PHpre- PPR (H-30)

PHpred|)ostt = PHpre(l-PPR) = P H p r e - P P L

( t

j_

3 1 )

Reperfusion n

Thee reperfusion (REP) follows from the fact that PHmeaspost is not equal to PHpredposL If wee introduce a local reperfusion factor (RPn)

11

N

PHmeaspostt = — y P e r f

ap r c

. { l - E ( D

I l

) + R P

a

} (H-32)

N

n=1 1

11

N

REPP s — Y Perf;T RP

n

= PHmeaspost - PHpredpost = PPL - MPL fli-33)

andd the relative reperfusion (RREP)

RFP P

RREPP s = PPR - MPR (11-34)

PHpre e

Appendixx III

Functionall Sub Units

Iff the number of functioning FSUs in voxel n is given by:

F

n

s F

0

- P e r f

DD

(HM)

then,, as we expected, in the "well perfused low dose" voxels the average number of FSUs is equall to F0.:

—L__ £

Fa=

_*<L_ J p n ^

=Fo

J_£F«=^£Perf

a sFo

(in-2,3)

Nw>u>>

n

^ N

WFLD B

«j N w

o J

N

w

„,!

Thee total number of FSU's in the lung is:

Thee average number of FSU's per voxel in the lung is equal to:

F F

—— = F

0

- P H (IM-5)

NN °

Thee average number of FSU's per voxel relative to the ideal number of FSUs per voxel (F0) is:

F F

N - F0 0

== PH (HI-6) Afterr irradiation the number of FSUs decreases according to a dose-effect relation E(Dn):

Fr*""" =Frd-E(DJ) (w-7)

Forr the voxels in the WPLD regions, the average number of FSUs stays equal to F0. Thee predicted number of FSUs post-RT in the lung will be:

AppendixAppendix UI

11 N

withh FBsF0-Pterfft and OpRPy =— Y P ^ f f5 -B(Dn) (Appendix I)2:

Fpr«ip«tt = N.F o. { p H p r e - O p R Py} (IH-9) andd the predicted change in the total number of FSU's in the whole lung:

AFpwdportt = N - F0 {OpRPY } (111-10)

Thee predicted average number of FSU's per voxel relative to the 'ideal' number of FSU's per voxell (F0) is:

-—— - P H p r e - O p R Py = PHpredpost (111-11)

Thiss means that (as expected) PHpredpost (see Eq. 14) can be interpreted as the predicted averagee number of FSU's per voxel relative to the 'ideal' number of FSU's per voxel (F0) Thee predicted change in the average number of FSU's per voxel relative to the 'ideal' number off FSU's per voxel:

Appredpaet t

== OpRPY (111-12)

N - F0 0

ORPYY is interpreted as the Predicted Perfusion Loss (PFL). Withh a linear dose-effect relationship this becomes:

^ppredpwt t

== c M p L Dy (tll-13)

NF0 0

Thee predicted change in average number of FSU's per voxel relative to the pre-RT number of FSU'ss per voxel (=F0*PHpre)

1 ^ ^^ = OpRP (111-14)

OpRPP is the Predicted Perfused FSU Reduction or the Predicted Perfusion Reduction (PFR). Withh a linear dose-effect relation:

A Fp P C d p 0 S t t

——— = cMpLD (IIM5) 156 6

AppendixAppendix til

Thee follow-up FSU parameters are analogous to the pre-RT FSU parameters, only with measpostt instead of pre:

Thee number of functioning FSU's in voxel n is given by:

F r "

P 0 M

= F

0

- P e r f r

w o

"" (IIM6)

Thiss implies that in the "well perfused low dose" voxels the average number of FSU's is equal too F0, as we expected:

,, ^ _ y,**-"

B

- F

f l

(HI-17)

NWPLDD mwi a N^JJ, ^

Thee total number of FSU's in the whole lung becomes:

P-ncspos..

=

£jr«-*«*

= F o

,

N

. PHmeaspost

<IIM8)

D=I I

Thee average number of FSU's per voxel relative to the "ideal" number of FSU's per voxel is:

PHmeaspostt (in-19)

N-F

0 0

Thiss means that PHmeaspost is interpreted as the measured average perfused FSU's per voxell post-RT relative to the "ideat" number of FSU's per voxel.

Furthermoree (PHpre-Phmeaspost) can be interpreted as Measured Perfused FSU Loss and thuss as Measured Perfusion Loss (MPL)

Inn mis case the MPL has to correlate best with the absolute reduction in PFT.

Thee predicted average number of FSU's per voxel relative to the average number of FSUs per voxell preRT (=F0*PHpre):

F

mc

"

p0Btt =

PHmeaspost

N - F

1

"" ~ PHpre

andd as last the measures change in the average number of FSU's per voxel relative to the averagee number of FSU's per voxel preRT (=F0*PHpre) are given by:

A F ^ ^^ PHmeaspost

N - F ^^ PHpre

(,

""

21)

Thiss formula means that the (1-PHmeaspost/PHpre) can be interpreted as Measured Perfused FSUU Reduction and thus as Measured Perfusion Reduction (MPR).

22