Introduc)on to PET Pharmacokine)cs: Irreversible Models

Transcription

1 Inroduc)on o PET Pharmacokine)cs: Irreversible Models

2 TWO TISSUE IRREVERSIBLE COMPARTMENT MODEL K 1 RE k 3 C M dc PRE d dc M d = K 1 ( )RE = k 3 RE

3 dre d dc MET d = K 1 ( )RE = k 3 RE RE >

4 dc MET /d > dre d dc MET d = K 1 ( )RE = k 3 RE RE > For Irreversible Case: No Thermal Equilibrium

5 Seady Sae: RE consan = K 1 ( )RE RE = K 1 dc M d = k 3 RE = K 1 k 3 Seady Sae Upake Rae K in

6 Solving he sysem Impulse Response Func)on h = K 1 ( k + e (+k 3 ) ) 3 C T = RE + C M = K 1 k 3 = K in (τ)dτ + K k 1 2 C + k P e +k 3 3 (τ)dτ + K k 1 2 C + k P e +k 3 3 ( ) ( )

7 Solving he sysem C T = RE + C M = K 1 k 3 (τ)dτ + K k 1 2 C + k P e +k 3 3 ( ) Nonlinear Leas Squares Approaches: Es)mae K 1, and k 3 Es)mae K in, K 1 /( +k 3 ) and +k 3 Basis func)on wih, { exp(- +k 3 ) j )} j

8 Irreversible 2TC Arerial Plasma K 1 RE k 3 C M ime (min) ime (min) ime (min) ime (min)

9 Irreversible 2TC Arerial Plasma K 1 RE k 3 C M ime (min) ime (min) ime (min) ime (min)

10 Irreversible 2TC Arerial Plasma K 1 RE k 3 C M ime (min) ime (min) ime (min) ime (min)

11 Irreversible 2TC Arerial Plasma K 1 RE k 3 C M ime (min) ime (min) ime (min) ime (min)

12 Irreversible 2TC Arerial Plasma K 1 RE k 3 C M ime (min) ime (min) ime (min) ime (min)

13 Palak Plo C T = K in (τ)dτ + K k 1 2 C + k P e +k 3 3 ( ) Palak Showed: for large ( > *) and large class of C T = K in (τ)dτ + K k C 1 2 P e ( +k 3 ) Regress C T / agains CP( )d / for > * ; Slope = K in

14 Palak Plo Regress Here

15 Palak Plo FDG in human DLPFC 2TC Nonlinear Leas Squares Palak Plo K in =.32 ml cm -3 min -1 K in =.3 ml cm -3 min -1

16 Furher Simplifica)ons C T = K in (τ)dτ + K k 1 2 C + k P e +k 3 3 ( ) Following Bolus Injec)on, exp(-( +k 3 )) goes o for large RE Pure Convolu)on C M Convolu)on+ Inegral ime (min) ime (min) ime (min)

17 Furher Simplifica)ons (Bolus Inpu) C T = K in (τ)dτ + K k 1 2 C + k P e +k 3 3 C T ( > *) (τ)dτ C T = K in = K in (τ)dτ For > * ( ) Rhodes e al, Ann. Neurol, 1983 Palak e al, JCBFM, 1985

18 Furher Simplifica)ons C T = K in (τ)dτ + K k 1 2 C + k P e +k 3 3 For > * ( ) K in = C ( > *) T (τ)dτ = Toal Trapped Toal Delivered Rhodes e al, Ann. Neurol, 1983 Palak e al, JCBFM, 1985

19 Furher Simplifica)ons C T ( > *) (τ)dτ = K in Rhodes e al, Ann. Neurol, 1983 Palak e al, JCBFM, 1985 Replace d wih Injeced Dose per uni body mass: C T ( > *) injeced aciviy body weigh DUR (differen)al upake ra)o) Kuboa e al JNM 1985 Adler e al, JNM 1991 OR SUV (sandard upake value) BUT: ID/weigh is no a perfec predicor of Cpd, especially in popula)ons wih compromised clearance mechanisms (liver, kidneys)

20 Furher Simplifica)ons Reference )ssue version: C T C REF = K in V T (C REF ) C REF (τ)dτ C REF + erms ending o consan

21 Example: FDOPA (Subsrae for AADC) K in aken as indicaor for capaciiy for DA synhesis

22 In Plasma Crosses BBB Can be Inhibied FDOPA yes * OMFD yes yes F-18 DA no yes* In Brain Crosses BBB Can be meabolized FDOPA yes yes OMFD yes * F-18 DA no yes Holden e al, JNM 1997

23 Fae of FDOPA in plasma BBB (COMT) FDOPA OMFD FDOPA OMFD (AADC) FDA

24 Fae of FDOPA in brain BBB FDOPA (AADC) FDA (COMT) OMFD (MAO) (COMT) (MAO) DOPAC HVA

25 Comprehensive Model (overparameerized) Cumming and Gjedde, 1998

26 Simplified Model (parameers can be esimaed) From Sossi e al, 21 OMFD assumed o be pre-correced BBB peneraing brain meabolies folded ino single-rae process (k loss )

27 CONFOUND I: OMFD In PLASMA AND BRAIN

28 COMT is acive in periphery and brain OMFD crosses bbb by same roue as FDOPA OMFD in brain from periphery >> OMFD formed in brain OMFD kineics assumed o be uniform in brain Tradiional Sraegy: Subrac C REF from C T C M C T - C REF ime (min)

29 Correcion for OMFD in brain Holden e al JCBFM 1997

30 PATLAK PLOT WITH approximae C M FROM C T -C REF : 2 {C T -C REF }/C p C T / (C T -C REF )/ C p (')d'/c p

31 Approach of Kumakura e al: Measure plasma concenraion of FDOPA AND OMFD Assume 1TCM for OMFD wih same V e as FDOPA and fixed rae consan raio: V e = K 1 / = K 1M /M (V e is like V ND ) and K 1M = 1.5K 1 *** Compue brain OMFD as plasma OMFD K 1M exp(-m ) Subrac esimaed OMFD curve from ROI daa *** limied daa available on how accurae or variable across he populaion his assumpion is.

32 Approach of Kumakura e al: Cerebellum Es)maed OMFD Kumakura e al, 25

33 Confound II: Meabolism of [ 18 F] FDA In Brain

34 Fae of FDOPA in brain BBB FDOPA (AADC) FDA (COMT) OMFD (COMT) (MAO) (MAO) 18 F-DOPAC 18 F-HVA

35 LEAKY Irreversible Comparmen Model Arerial Plasma K 1 RE k 3 C M k loss dre d dc M d = K 1 ( )RE = k 3 RE k loss C M

36 LEAKY Irreversible Comparmen Model Arerial Plasma K 1 RE k 3 C M k loss dc M = k d 3 RE k loss C M C M = k 3 RE exp( k loss ) Could do 4 parameer NLLS fi o 2TC model, or.

37 LEAKY Irreversible Comparmen Model Arerial Plasma K 1 RE k 3 C M k loss Sraegy: Replace regressor / in Palak plo (τ)dτ e k loss = (τ)e k loss( τ ) dτ

38 PATLAK PLOT WITH K loss hr daa wih k loss, wihou k loss C T /C p 1 5 C T /C p C p (')d'/c p C p (')d'/c p C T /C p 15 1 C T /C p C p (')d'/c p C p (')exp(-k loss (-')d'/c p 2 hr daa w/ k loss and righ srech ime

39 PATLAK PLOT WITH K loss Form convoluion curves wih a range of possible k loss values Fi he daa using he resuling srech ime from each Selec he k loss ha provides he sraighes curve (Acquired 4 hr daa o esimae k loss!) Holden e al, JNM 1997

40 EDV: Composie parameer for Overall FDOPA reenion K in describes seady sae rae of FDOPA meabolism relaive o C p ; mos is convered o FDA k loss describes rae of loss of FDA from brain due o meaolism in brain Define EDV = K in /k loss ; 1/EDV approximaes he effecive urnover rae for DA (Sossi e al, 22) Kumakura e al re-labeled Sossi's parameer as EDV 1 ; called heir esimaor (using heir OMFD mehod) EDV 2

41 Mulilinear fiing procedure, Kumakura 26 (Similar in flavor o MA1: Inegrae he diff equaions) dc M d = k 3 RE k loss C M = K 1 k 3 - k loss C M (afer equilibraion) C M = K 1 k 3 (')d' k loss C M (')d' C M = C T RE = C T Subsiue for C M : C T K 1 = K 1 k 3 rearrange : K 1 K C T (')d' = EDV 2 1 (')d' k loss C T (') (')d' 1 k loss C T + K 1 C + k P (') d' 3 K 1 ( ) k loss

42 Mulilinear fiing procedure, Kumakura 26 C T (')d' = EDV 2 K 1 (')d' 1 Can measure each of hese; STRATEGY: Muliple regression k loss C T + K 1 ( ) k loss C T agains C T, and ge 3 coefficiens : EDV 2 K 1 ( ), 1 k loss and EDV 2 k loss, and in erms of hese K 1 ( ) k loss ; solve for

43 FDG Imaging Glucose ranspored across BBB AND cell membranes by GLUT1 Hexokinase Glucose Glucose-6-P Glucose-6-Phosphae Isomerase Glucose-6-P Frucose-6-P 2-deoxy-2-[18F]fluoro-D-glucose (FDG) is an analog of glucose FDG ranspored across BBB AND cell membranes by GLUT1 Hexokinase FDG FDG-6-P FDG-6-P NOT a subsrae for Glucose-6-Phosphae Isomerase FDG-6-P is a rapped sae (irreversible) FDG ranspor and phosphoryla)on occurs in parallel wih glucose

44 FDG Imaging FDG K in give some indica)on of he local rae of Glu meabolism BUT Rae consans for glucose and FDG are no he same IF plasma glucose is in seady sae hen Rae of glucose upake and phosphorylaion = K in (glucose) (glucose) K in * = upake rae for FDG K in = upake rae for glucose LC= K * in K in CmRGlu = 1 LC K * C in P (glucose) (.6 o.8 in lieraure)

45 Appendix: Deriva)on of he 2TC Irreversible Model and Palak Plo dre d dc M d = K 1 ( )RE = k 3 RE Equa)on for RE is in sandard form, wih solu)on: RE = K 1 e ( +k 3 ) And herefore C M equals ( )τ C M = K 1 k 3 e +k 3 (τ)dτ

46 Appendix: Deriva)on of he 2TC Irreversible Model and Palak Plo The equa)on for C M can be inegraed by pars d (τ)e ( +k 3) ( τ) dτ = d e ( +k 3) ( ) = = 1 e ( +k 3 ) ( (τ)e +k 3) τ 1 e +k 3 ( ) + ( ) dτ = 1 C + k P ( τ)dτ 3 + d + ( (τ)e +k 3) ( τ) dτ 1 e ( +k 3 ) ( C + k P ( )e +k 3) 3 C T = RE + C M = K 1 k 3 = K 1 k 3 Adding his o RE gives he full expression for C T : k (τ)dτ + K k 2 + k 3 C P e +k 3 (τ)dτ + K k 1 2 C + k P e +k 3 3 ( ) ( ) = K in (τ)dτ + K k 1 2 C + k P e +k 3 3 ( )

47 Appendix: Deriva)on of he 2TC Irreversible Model and Palak Plo Noe ha for a bolus inpu, he convolu)on erm evenually decays o. Palak addressed a slighly more general case in which he inpu could be represened as a sum of decaying exponen)als wih he smalles rae consan less han he eigenvalue in he convolu)on erm (e-value = -( +k 3 ) in he 2TC case): C p = m B j e β j, j=1 < β m < β m 1! < β 1 β m < In par)cular, for large, say > * for some *, C p B m e β m, > * e ( +k 3) = Inser)on ino he convolu)on inegral yeilds: m ( )( τ) = B j e β j τ+ +k 3 dτ = B j e +k 3 j=1 m B j e ( +k 3) e β j ( +k 3) j=1 + k 3 ( ) β j ( ) 1 = m m j=1 B j ( ) ( )τ e β j ( +k 3) e ( ) β j e +k 3 j=1 β j dτ ( ( ) )

48 Appendix: Deriva)on of he 2TC Irreversible Model and Palak Plo For > *, all erms will decay away excep he one involving he smalles eigenvalue: e ( +k 3) = m B j ( ( ) ) e ( ) β j e +k 3 j=1 β j B m e ( ) βm, > * β m Therefore, dividing hrough by makes his erm end o a consan e ( +k 3) B m e ( ) βm β m B m e β m 1 ( ) β m and he complee equa)on becomes C T = K in (τ)dτ + cons., for > *

49 Appendix: Deriva)on of he 2TC Irreversible Model and Palak Plo Palak plo wih k loss erm dre = K d 1 ( )RE dc M = k d 3 RE k loss C M RE = K 1 e ( +k 3) C M = k 3 RE e k loss As before, inegrae C M by pars; le +k 3 = λ and k loss = k for noa)onal simpliciy

50 1 K 1 k 3 C M = e k( τ) τ ( )e λ(τ ) d dτ = e k ( e k λ τ )τ C P ( )e λ d dτ 1 = e k k λ e ( k λ τ )τ ( )e λ d τ= = e k = k λ e k λ ( )e λ d 1 k λ { ( ) C } P (τ)ekτ dτ { } 1 k λ e λ e k ( )τ (τ)e λτ dτ e k λ This is one erm convolved wih e -kloss and one erm convolved wih e -(k2+k3) C T = K 1 k 3 C k P e k loss + cons. e ( +k 3 ) loss As before, he second convolu)on evenually becomes propor)onal o, and herefore C T = K 1 k 3 e kloss k loss + consan, > * Noe he slope is greaer han K in, bu close if >> k loss