Appendix: Equations and parameters
Model equations
The following equations describe the transport of labeled (indexed with *) and
unlabeled peptide to the receptor sites, its binding, internalization, degradation,
excretion and radioactive decay. The peptide was injected as a bolus for the
pretherapeutic measurements and as a 30 min infusion for therapy. The variables are
defined in Table A. Index “i” refers to the corresponding organ “i”.
Liver, spleen, tumor, kidney and rest
Constraint for total sst2 receptors R0,i
R0, i  Ri  RPi  RPi*
(1)
Internalized peptide
d
Pintern i  int  RPi  deg, i  Pintern i   phy  P*intern i
dt
d *
P intern i  int  RP* i  deg, i  RP*intern i   phy  P*intern i
dt
Bound peptide on cell surface
R
d
RPi  kon  Pi  i  (koff  int )  RPi   phy  RPi*
dt
Vi
R
d
RPi*  kon  Pi*  i  (koff  int )  RPi*   phy  RPi*
dt
Vi
Liver, spleen, tumor and rest
Free peptide
R
F
F
d
Pi   k on  Pi  i  k off  RPi  i  PC  i  Pi   phy  Pi*
dt
Vi
VC
Vi
R
F
F
d *
Pi   k on  Pi *  i  k off  RPi *  i  PC *  i  Pi *   phy  Pi*
dt
Vi
VC
Vi
1
Kidneys
Trapped peptide in kidney cells
Model A
P
P
d
*
Pintra,K  int,K  ( F fil  Fex )  intra,K  ( F fil  Fex )   phy  Pintra,
K
dt
Vint,K
Vintra,K
*
Pint,
P*
d *
K
*
Pintra,K 
 ( F fil  Fex )  intra,K  ( F fil  Fex )   phy  Pintra,
K
dt
Vint,K
Vintra,K
Model B
P
d
*
Pintra,K  int,K  ( F fil  Fex )  Pintra,K  deg,REST   phy  Pintra,
K
dt
Vint,K
*
Pint,
d *
K
*
*
Pintra,K 
 ( F fil  Fex )  Pintra,
K  deg,REST   phy  Pintra,K
dt
Vint,K
Free peptide vascular spaces
Model A
P
P
F
d
PK  K  ( kon  RK  F fil  FK )  koff  RP K  K  PC  intra,K  ( F fil  Fex )   phy  PK*
dt
VK
VC
Vintra,K
F
d * PK*
P *intra,K
PK 
 ( kon  RK  F fil  FK )  koff  RP K  K  PC* 
 ( F fil  Fex )   phy  PK*
dt
VK
VC
Vintra,K
Model B
P
F
d
PK  K  ( k on  RK  F fil  FK )  k off  RPK  K  PC   phy  PK*
dt
VK
VC
F
d * PK*
PK 
 ( k on  RK  F fil  FK )  k off  RPK  K  PC*   phy  PK*
dt
VK
VC
Model C and D (for Model D Fex = Ffil)
P
P
F
d
PK  K  ( k on  RK  F fil  FK )  k off  RP K  K  PC  int, K  ( F fil  Fex )   phy  PK*
dt
VK
VC
Vint , K
P*
F
d * PK*
PK 
 ( k on  RK  F fil  FK )  k off  RP K  K  PC*  int , K  ( F fil  Fex )   phy  PK*
dt
VK
VC
Vint , K
2
Free peptide interstitial spaces
Model A, B, C and D (for Model D Fex = Ffil)
F fil
P
P
d
P K ,int   K ,int  Fex  K ,int  ( F fil  Fex ) 
 P K   phy  P*K ,int
dt
VK ,int
VK ,int
VK
F fil *
d *
P*K ,int
P* K ,int
P K ,int  
 Fex 
 ( F fil  Fex ) 
 P K   phy  P*K ,int
dt
VK ,int
VK ,int
VK
Main vascular compartment
F
F
d
PC   i  PC   i  Pi   phy  Pi*
dt
VC
Vi
F
F
d
PC *   i  PC *   i  Pi*   phy  Pi*
dt
VC
Vi
3
TABLE A Parameter definition
Variable
Value
Unit
l·nmol-1·min-1
Source
kon
association rate
kon = koff / KD
koff
dissociation rate
0.013
min-1
KD
dissociation constant
5.57
nmol·l-1
(Edwards et al., 1994)
F
flow total plasma
VP ·1.23a
l·min-1
(Leggett and Williams, 1995)
FL
flow liver total
0.25·F
l·min-1
(Leggett and Williams, 1995)
FS
flow spleen
0.19·F
l·min-1
(Leggett and Williams, 1995)
FK
flow kidneys
0.03·F
l·min-1
(Leggett and Williams, 1995)
FINT
flow to interstitial spaces of rest body
3.6 ·10-5 BW · Pep/ Pep
l·min-1
(Thomas et al., 1989)
FTU
flow tumor
estimatedb
l·min-1
Pep/ Pep
ratio permeability peptides/antibodies
Ffil
filtration
Fex
excretion
fex
excretion/filtration
θOctreoscan /
θCr-51-EDTA
ratio of sieving coefficients
VL
volume of distribution liver
VS
volume of distribution spleen
VK
volume of vascular spaces kidney
Vint,K
volume of interstitial spaces kidney
Vintra,K
volume intracellular kidney
VINT
volume of distribution interstitial rest
VTu
volume of distribution tumor
VP
volume of total body serum
male 2.8·(1-hemato)·BSA
female 2.4·(1-hemato)·BSA
l
VC
volume of readily distribution
VP+ Vtotal,L · fint,L+Vtotal,S
·fint,S+ Vtotal,Tu ·fint,Tu+300mle
l
Vtotal,L
volume total liver
0.722 · BSA -1.176
l
(Johnson et al., 2005)
Vtotal,S
volume total spleen
BSA ·(278·age-0.36)
l
(Harris et al., 2010)
Vtotal,K
volume total kidney
l
(Snyder et al., 1975)
Vtotal,TU
volume total tumor
l
(Velikyan et al., 2010)
fvas,L
fraction vascular liver
fint,L
fraction interstitial liver
fvas,S
fraction vascular spleen
50
GFRmeasured · θOctreoscan /
θCr-51-EDTA c
= Ffil · fex
estimated
0.7
Vtotal,L · (fvas,L+ fint,L)
Vtotal,S · (fvas,S+ fint,S)
Vtotal,K · fvas,K
Vtotal,K · fint,K
unity
l·min-1
unity
unity
l
l
l
3·VP - Vtotal,L · fint,L- Vtotal,S·
fint,S- Vtotal,Tu · fint,Tu -300mle
l
Estimated, assumingf
R0,tu = 34 nmol/l
0.13
0.13
0.1
(Schmidt and Wittrup, 2009)
l
l
0.310
(Schmidt and Wittrup, 2009)
l·min-1
(Vtotal,K - Vint,K - VK) ·2/3d
Vtotal,Tu · (fvas,Tu+ fint,Tu)
(Ferl et al., 2009)
l
unity
(Covell et al., 1986)
unity
(Covell et al., 1986)
unity
(Covell et al., 1986)
4
fint,S
fraction interstitial spleen
fvas,K
fraction vascular kidney
fint,K
fraction interstitial kidney
fvas,Tu
fraction vascular tumor
fint,Tu
fraction interstitial tumor
0.09
0.2
0.19
0.1
unity
(Covell et al., 1986)
unity
(Covell et al., 1986)
unity
(Covell et al., 1986)
unity
unity
(Baxter et al., 1995)
(Schmidt and Wittrup, 2009)
0.4
R
receptors free
nmol
R0
receptors total number
RPi
peptide bound
nmol
Pintern
peptide internalized
nmol
Pi
peptide free
nmol
Pv,K
peptide vascular kidney
nmol
Pint,K
peptide interstitial kidney
nmol
Pintra,K
peptide interacellular kidney
nmol
λdeg
degradation and release from sst2 cells
min-1
λint
internalization rate sst2
estimated
nmol
(Hofland and Lamberts,
0.0037
min-1
2003)
λphy
physical decay 111In
aFor
1.72 · 10-4
min-1
the average normal adult (blood) F = 6500 ml/min and V = 5300 ml. Therefore, a factor of
1.23 was assigned to account for the changes in total serum flow due to volume changes.
bA
Bayesian term (0.2±0.1· VTU) was used to determine the blood flow to the tumor
cFor
patient 2 this value was estimated as the time interval between GFR measurement and
dosimetry was too large.
dIt
is assumed that 2/3 of the total intracellular volume of the kidney is represented by the
proximal tubular cells
eThe
interstitial spaces of the red marrow (approximately 300ml (Baxter et al., 1995)) are added
to the readily accessible volume
fIn
(Velikyan et al., 2010) is was found that the SUV of tumor and spleen were about the same.
Thus we set the R0,TU to the value of R0,S derived from the first 5 patients (34 nmol)
5
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