Slope-only glomerular filtration rate and single-sample glomerular filtration rate as measurements of the ratio of glomerular filtration rate to extracellular fluid volume


Peters, A.Michael.; Glass, D.M.; Bird, N.J.

Nephrology 15(3): 281-287

2010


The Jacobsson single-sample equation for measuring glomerular filtration rate (GFR) after bolus injection is based on two factors of questionable theoretical validity for correcting the single-compartment assumption. The aims were to redevelop a more transparent equation, show its fundamental similarity with 'slope-only' GFR and compare it with the original equation and with slope-only GFR. The modified Jacobsson equation is k = (1/t).ln[V(t)/V(0)], where k is the rate constant of the terminal exponential and V(0) and V(t) are distribution volumes at times 0 and t. V(0) exceeds extracellular fluid volume (ECV): that is k' = (1/t).ln[V(t)/ECV], where k' > k. Moreover, [GFR/ECV] >k (= k + [15.4.k(2)]). The ratio k/k' was determined in 476 patients to calculate single-sample k (3 or 4 h post-injection). Slope-only and single-sample GFR/ECV were measured using Cr-51-EDTA in 105 further studies, multiplied by ECV (estimated from weight), scaled to 1.73 m(2) and compared with GFR/1.73 m(2) from the original Jacobsson equation against reference multi-sample GFR/1.73 m(2) simultaneously and independently measured with iohexol. The relation between k and k' was linear. k/k' was 0.827 at 3 h and 0.864 at 4 h. There was no difference in bias or precision between the original Jacobsson and modified equations. In both, precision was better than slope-only GFR/BSA. When GFR remained scaled to ECV, slope-only GFR showed marginally better precision against reference GFR/ECV. Single-sample and slope-only techniques give GFR as k. Although the theory of the modified Jacobsson equation is more transparent than the original equation, it gives the same result. It is, however, easier to use.

KEY
WORDS:
Cr-51-EDTA,
extracellular
fluid
volume,
glomerular
filtration
rate,
iohexol,
single-sample.
Correspondence:
Professor
A
Michael
Peters,
Audrey
Emerton
Building,
Royal
Sussex
County
Hospital,
Eastern
Road,
Brighton
BN2
5BE,
UK.Email:
Accepted
for
publication
15
October
2009
Accepted
manuscript
online
18
November
2009.
doi:10.1111/j.1440-1797.2009.01252.x
SUMMARY
AT
A
GLANCE
This
paper
shows
that
the
single-sample
method
of
Jacobsson
for
measuring
glomerular
filtration
rate
and
the
rate
constant
of
the
terminal
exponential
in
multi-sample
GFR
both
give
GFR
per
unit
extracellular
fluid
volume
(that
is
mUmin/L).
A
modified
Jacobsson
equation
is
presented
that
is
theoretically
more
transparent
than
the
original
equation.
Both
equations,
however,
give
almost
identical
values
of
GFR.
NEPHROLOGY
APSN
Nephrology
15
(2010)
281-287
Original
Article
Slope-only
glomerular
filtration
rate
and
single-sample
glomerular
filtration
rate
as
measurements
of
the
ratio
of
glomerular
filtration
rate
to
extracellular
fluid
volume
A
MICHAEL
PETERS,'
DAPHNE
M
GLASS
2
and
NICHOLAS
J
BIRD'
Departments
of
Nuclear
Medicine,
'Royal
Sussex
County
Hospital,
Brighton,
2
Harley
St
Clinic,
London
and
3
Addenbrooke's
Hospital,
Cambridge,
UK
ABSTRACT:
Aims:
The
Jacobsson
single-sample
equation
for
measuring
glomerular
fil-
tration
rate
(GFR)
after
bolus
injection
is
based
on
two
factors
of
question-
able
theoretical
validity
for
correcting
the
single-compartment
assumption.
The
aims
were
to
redevelop
a
more
transparent
equation,
show
its
funda-
mental
similarity
with
'slope-only'
GFR
and
compare
it
with
the
original
equation
and
with
slope-only
GFR.
Methodology:
The
modified
Jacobsson
equation
is
k
=
(1/0.1n[V(t)/V(0)],
where
k
is
the
rate
constant
of
the
terminal
exponential
and
V(0)
and
V(t)
are
distribution
volumes
at
times
0
and
t.
V(0)
exceeds
extracellular
fluid
volume
(ECV):
that
is
k'
=
(1/t).1n[V(t)/ECV],
where
k'
>
k.
Moreover,
[GFR/
ECV]
>k
(=
k+
[15.4.k
2
]).
The
ratio
k/k'
was
determined
in
476
patients
to
calculate
single-sample
k
(3
or
4
h
post-injection).
Slope-only
and
single-
sample
GFR/ECV
were
measured
using
Cr-51-EDTA
in
105
further
studies,
multiplied
by
ECV
(estimated
from
weight),
scaled
to
1.73
m
2
and
compared
with
GFR/1.73
m
2
from
the
original Jacobsson
equation
against
reference
multi-sample
GFR/1.73
m
2
simultaneously
and
independently
measured
with
iohexol.
Results:
The
relation
between
k
and
k'
was
linear.
k/k'
was
0.827
at
3
h
and
0.864
at
4
h.
There
was
no
difference
in
bias
or
precision
between
the
original
Jacobsson
and
modified
equations.
In
both,
precision
was
better
than
slope-
only
GFR/BSA.
When
GFR
remained
scaled
to
ECV,
slope-only
GFR
showed
marginally
better
precision
against
reference
GFR/ECV.
Conclusions:
Single-sample
and
slope-only
techniques
give
GFR
as
k.
Although
the
theory
of
the
modified
Jacobsson
equation
is
more
transpar-
ent
than
the
original
equation,
it
gives
the
same
result.
It
is,
however,
easier
to
use.
INTRODUCTION
Glomerular
filtration
rate
(GFR)
measured
from
a
single
blood
sample
following
bolus
intravenous
injection
of
a
filtration
marker
(indicator),
such
as
Cr-
51-
ethylenediaminetetraacetic
acid
(EDTA)
or
iohexol,L
2
and
slope-only
GFR,
which
is
based
exclusively
on
the
terminal
exponential
of
the
plasma
clearance
curve,
3-5
are
both
intended
to
simplify
the
measurement
of
GFR.
The
first
aim
of
this
study
is
to show
that
the
single-sample
technique
primarily
generates
the
rate
constant,
k,
of
the
terminal
exponential
(from
2
h
post-injection)
and
therefore,
like
the
slope-only
technique,
gives
GFR
that
is
scaled
to
extracellular
fluid
volume
(ECV).
The
second
aim
is
to
determine
which
of
these
two
methods
of
obtaining
k
is
the
more
accurate
for
the
determination
of
GFR
in
comparison
with
conventional
multi-sample
GFR.
THEORY
Three
simplified
techniques
were
studied,
all
of
which
gen-
erate
k,
as
shown
in
the
following
theory.
©
2010
The
Authors
Journal
compilation
©
2010
Asian
Pacific
Society
of
Nephrology
281
Am
Peters
et
al.
Slope-only
GFR
Slope-only
GFR
is
the
rate
constant,
k
of
the
terminal
expo-
nential
of
the
plasma
clearance
curve
corrected
for
the
one-
compartment
assumption.
3-5
It
is
a
measure
of
GFR
that
is
scaled
to
ECV
(GFR/ECV);
That
is,
GFR/ECV
=
k
x
f
l
(1)
where
f
1
is
the
correction
factor
(>1)
for
the
one-
compartment
assumption
(see
below).
Jacobsson
single-sample
equation
Two
theoretical
approaches
to
the
single-sample
technique
have
been
described.
One,
described
by
Christensen
and
Groth,
is
empirically
based
on
a
comparison
with
multi-
sample
GFR
1
while
the
other,
developed
by
Jacobsson,
uses
a
theoretical
approach
based
on
k.
2
It
is
the
second
of
these
that
is
considered
in
the
current
study.
In
the
Jacobsson
equation,
k
is
given
as
equal
to
GFR/ECV
(although,
as
already
shown
in
Eqn
1,
they
are
not
identical).
The
equation
is
re-arranged
for
GFR
and
multiplied
by
an
estimate
of
ECV.
A
correction
(the
term
0.0016)
for
non-
immediate
mixing
of
indicator
is
introduced
to
give
GFR
1
x
ln
[V
(t)/eECV]
(t/eECV)
+
0.0016
where
eECV
is
an
estimate
of
ECV
made
from
body
weight
and
V(t)
is
the
apparent
volume
of
distribution
of
the
filtra-
tion
marker
(equal
to
the
amount
of
administered
marker
divided
by
the
plasma
concentration
at
time
t).
The
eECV
is
corrected
for
non-uniform
distribution
(ern-
correction'
2
)
using
a
two-stage
process.
First,
GFR
is
calcu-
lated
from
Equation
2
(as
GFR
1
)
and
scaled
to
a
body
surface
area
(BSA)
of
1.73
m
2
.
Second,
the
correction
factor
m
is
calculated
from
the
equation
originally
described
by
Brochner-Mortensen
6
to
correct
'slope-intercept'
GFR
for
the
assumption
of
a
single
compartment:
m
=
0.991—
(GFR
i
x
0.00122)
(
3
)
and
GFR
recalculated
from
Equation
1
replacing
eECV
with
eECV/m.
Modified
Jacobsson
equation
The
theory
described
by
Jacobsson
in
the
development
of
Equation
2
can
be
made
more
transparent,
as
follows.
Assuming
a
single
compartment
C(t)=C(0)•
e
k
t
(4)
Where
C
is
plasma
marker
concentration.
Re-arranging
x
In
[C
(0)/C
(01
=
k
(
5
)
and
converting
concentrations
to
distribution
volumes
x
In
[V
(t
)/V
(0)]
=
k
(6)
With
respect
to
Equation
6,
V(0)
is
higher
than
ECV
because
it
is
based
on
the
assumption
of
a
single
compartment.
Equa-
tion
6
can
therefore
be
re-written
as
x
ln
[V
(t)/ECV]=
k'
(
7
)
where
k'
>
k
As
there
is
no
a
priori
knowledge
of
GFR,
an
aim
of
the
current
study
was
to
derive
retrospectively
an
equation
describing
the
relation
between
k'
and
k
from
which
k
could
be
prospectively
determined
from
k'
using
Equation
7
(see
below).
METHODS
Patients
Two
groups
of
patients
were
studied;
the
first
to
determine
the
ratio
k/k'
with
which
to
derive
prospectively
k
from
k'
and
the
second
to
test
the
modified
Jacobsson
technique
in
comparison
with
the
origi-
nal
Jacobsson
equation
and
slope-only
GFR
against
a
reference
multi-sample
technique.
Group
1
The
ratio
k/k'
was
derived
from
476
patients
(age
>13;
240
were
male)
undergoing
routine
three-sample,
slope-intercept
measure-
ment
of
GFR
using
Cr-51-EDTA.
Of
these
patients,
320
(142
were
male)
had
cancer
and
were
referred
prior
to
or
following
chemo-
therapy,
and
156
(98
were
male)
were
referred
for
a
variety
of
indications
unrelated
to
cancer.
Group
2
The
second
group,
in
which
the
simplified
methods
were
tested,
comprised
60
patients
(40
men)
routinely
referred
for
measurement
of
GFR
and
20
healthy
subjects
(seven
men)
with
no
history
of
allergy
to
iodine-containing
contrast
agents
who
were
all
studied
twice
or
three
times
on
separate
days.
Of
the
60
patients,
36
were
diabetics,
10
had
cancer,
13
had
skin
disease
(and
were
receiving
or
being
considered
for
cyclosporine
treatment)
and
one
was
referred
from
the
nephrology
service.
In
each
participant,
multi-sample,
two-
compartment
GFR
was
simultaneously
and
independently
measured
with
both
Cr-51-EDTA
and
iohexol.
With
regard
to
the
Cr-51-EDTA
data,
only
the
single-sample
values
at
3
h
and
4
h
were
used
in
the
current
study.
All
group
2
participants
gave
written
informed
consent.
The
study
was
approved
by
the
local
research
ethics
com-
mittee
and
by
the
Administration
of
Radioactive
Substances
Advi-
sory
Committee
of
the
UK.
Procedure
Group
1
Accurately
timed
cubital
venous
blood
samples
were
obtained
at
approximately
120,
180
and
240
min
after
injection
of
2
mL
Cr-51-
EDTA
(2
MBq;
GE
healthcare,
Bucks,
UK)
and
counted
in
a
well-
counter
with
background
correction.
(2)
©
2010
The
Authors
282
Journal
compilation
©
2010
Asian
Pacific
Society
of
Nephrology
Simplified
GFR
Group
2
Two
millilitres
of
Cr-51-EDTA
(2
MBq)
and
19
mL
iohexol
(contain-
ing
5.7
g
iodine;
Omnipaque
300;
GE
healthcare)
were
separately
injected
into
each
ante-cubital
fossa.
Ten
millilitre
samples
were
drawn
bilaterally
20,
40,
60,
120,
180
and
240
min
after
injection.
Each
marker
was
assayed
in
samples
drawn
contralateral
to
the
side
injected,
iohexol
by
X-ray
fluorescence
(XRF;
Oxford
Instruments,
Oxford,
UK)
and
Cr-51
by
well-counting
with
background
correc-
tion.
The
accuracy
of
XRF
depends
on
plasma
iohexol
concentration
and
therefore
on
GFR.
In
our
hands,
it
is
<5%
in
the
normal
GFR
range
and
reported
in
detail
elsewhere.'
Data
analysis
Group
1
Slope-only
GFR.
GFR/ECV
was
calculated
from
k,
the
rate
constant
of
the
exponential
fitted
to
the
120-240
min
samples.
The
equation
described
by
Bird
et
al.'
was
used
to
correct
for
the
one-compartment
assumption:
GFR/ECV
=
k
+
(15.4.
k
2
)
(8)
Measurement
of
'true'
ECV.
Three-sample
GFR
was
measured
using
the
standard
slope-intercept
equation
for
'one-
compartment'
plasma
dearance,
scaled
to
a
BSA
(estimated
from
height
and
weight
using
the
equation
of
Haycock
et
al.
8
)
of
1.73
m
2
and
corrected
for
the
one-compartment
assump-
tion
using
the
equation
of
Brochner-Mortensen
6
to
give
GFR/BSA.
GFR/BSA
was
divided
by
GFR/ECV
(from
Eqn
8),
from
which
GFR
cancels
out,
to
give
'true'
ECV
(scaled
to
1.73
m
2
).
5
Using
BSA,
it
was
converted
back
to
unscaled
ECV.
Data
supporting
the
validity
of
this
method
of
calculating
ECV
have
recently
been
published.
9
Modified
Jacobsson
equation:
derivation
of
factor
for
prospective
conversion
of
k'
to
k.
In
group
1
patients,
k
was
calculated
from
V(t)
and
V(0)
using
Equation
6
and
k'
was
calculated
from
V(t)
and
ECV
or
eECV
using
Equation
7.
As
in
the
study
of
Gaspari
et
al.,'
eECV
(mL)
was
estimated
from
body
weight
(W;
kg),
separately
for
men
and
women,
as
follows.
eECV
(men)
=
(166
x
W)+
2490;
eECV
(women)
=
(95
x
W)
+
6170
(9)
Mean
values
of
k/k'
were
then
calculated
for
3
h
and
4
h
samples.
EDTA
samples
at
3
h
and
4
h
and
multiplied
by
the
ratio
k/k'
derived
from
group
1
patients
to
give
k,
which
was
then
corrected
for
the
one-compartment
assumption
using
Equa-
tion
8
to
give
GFR/ECV.
Slope-only
GFR.
As
in
group
1
patients,
GFR/ECV
was
cal-
culated
from
k
and
corrected
for
the
one-compartment
assumption
using
Equation
8.
Measurement
of
multi-sample
(reference)
GFR
and
GFR/
ECV.
Plasma
iohexol
dearance
curves
were
resolved
into
two
exponentials
with
intercepts
A
and
B
and
corresponding
rate
constants
a,
and
a
2
using
a
two-stage
curve-stripping
procedure,
as
previously
described,"
from
which
multi-
sample (reference)
GFR
was
calculated
and
scaled
to
a
BSA
of
1.73
m
2
:
GFR
injected
activity
or
dose
(A/a
1
)+(B/a
2
)
(10)
Multi-sample
(reference)
GFR/ECV
was
calculated
from
the
equation
described
by
Nosslin
that
derives
mean
marker
transit
time
through
its
distribution
volume
12
:
GFR
(A/c(
i
)
+
(B/a2
)
ECV
(A/[aif)+(B/[azj
2
)
In
five
participant
studies
(involving
one
normal
subject
and
four
patients),
the
iohexol
dearance
curve
could
not
be
resolved
into
two
exponentials
so
these
studies,
along
with
the
corresponding
Cr-51-EDTA
studies,
were
exduded,
leaving
105
group
2
studies
with
which
to
test
the
simplified
methods.
Comparison
of
single-sample
and
slope-only
techniques
with
refer-
ence
technique.
GFR/ECV
values
calculated
for
each
simpli-
fied
technique
from
the
105
Cr-51-EDTA
dearances
recorded
in
group
2
were
compared
with
the
corresponding
values
of
multi-sample
GFR/ECV
measured
with
iohexol.
For
each
simplified
technique,
GFR/ECV
was
multiplied
with
eECV
to
give
GFR,
which
was
then
scaled
to
a
BSA
of
1.73
m
2
and
compared
with
multi-sample
GFR/BSA
measured
with
iohexol.
Two
GFR
values
were
assessed
for
each
single-
sample
method:
one
from
the
3
h
sample
and
one
from
the
4
h
sample.
Group
2
Single-sample
GFR
from
the
Jacobsson
equation.
The
Statistics
m-corrected
GFR
value
(Eqn
3)
was
calculated
(Eqn 2)
using
eECV
calculated
from
Equation
9
and
scaled
to
a
BSA
of
1.73
m
2
.
The
unscaled,
m-corrected
GFR
value
was
divided
by
eECV
to
give
a
'Jacobsson'
measurement
of
GFR/ECV.
Single-sample
GFR
from
the
modified
Jacobsson
equation.
Using
Equation
7
and
eECV,
k'
was
calculated
from
single
Cr-51-
Correlation
coefficients
(r)
were
quantified
using
Pearson's
regres-
sion
analysis.
Significance
of
differences
in
ECV
between
patients
with
and
without
cancer
was
quantified
using
unpaired
Student's
t-testing.
Differences
between
simplified
GFR
and
reference
GFR
were
expressed
as
the
mean
(bias)
and
SD
of
mean
(precision).
Differences
of
bias
from
zero
were
tested
using
Student's
paired
t-test.
Precisions
were
compared
using
the
F-test.
©
2010
The
Authors
Journal
compilation
©
2010
Asian
Pacific
Society
of
Nephrology
283
AM
Peters
et
al.
RESULTS
Group
1
The
ratio
of
eECV
to
ECV
was
higher
in
patients
without
cancer
compared
with
those
with
cancer
(Table
1)
in
both
men
and
women.
In
line
with
this,
ECV
per
unit
weight
in
cancer
patients
(0.196
[0.040]
L/kg)
was
higher
than
in
patients
without
cancer
(0.181
[0.030]
L/kg;
P
<
0.001).
Relations
between
k'
calculated
from
eECV
and
k
were
consistently
linear
(Fig.
1).
The
mean
ratio
of
k/
k'
based
on
Table
1
Ratio
of
eECV
to
ECV
(SD)
in
patients
from
group
1
with
or
without
cancer
Patients
Men
Women
All
Without
cancer
1.10
(0.16)
1.10
(0.15)
1.10
(0.16)
With
cancer
1.045
(0.17)
1.02
(0.18)
1.03
(0.17)
*P<
0.02
0.001
0.001
*Cancer
versus
no
cancer.
ECV,
extracellular
fluid
volume;
eECV,
estimate
of
ECV.
ECV
was
almost
identical
between
cancer
and
non-cancer
patients
for
both
3
h
and
4
h
samples
but
when
based
on
eECV,
it
was
higher
in
the
non-cancer
versus
the
cancer
patients
(Table
2),
in
line
with
the
higher
ECV/weight
in
cancer
patients.
On
the
other
hand,
the
relations
between
k
and
k'
were
almost
identical
between
men
and
women,
both
in
cancer
and
non-cancer
patients
(Fig.
1).
Group
2
The
three
simplified
techniques
were
applied
to
Cr-51-EDTA
data
in
group
2
patients
and
compared
with
multi-sample
iohexol
clearance
as
the
reference
technique.
In
particular,
correction
factors
of
0.827
and
0.864
(the
averages
of
the
values
in
cancer
and
non-cancer
patients,
respectively,
derived
in
group
1
using
ECV
rather
than
eECV;
Table
2;
see
Discussion)
were
used
to
convert
k'
to
k
at
3
h
and
4
h,
respectively,
that
is
k
(3
h)
=
0.827
x
(1/t)
x
ln[V(t)/eECV]
and
k
(4
h)
=
0.864
x
(1/t)
x
ln[V(t)/eECV].
The
respective
agreements
with
six-sample
iohexol
clearance
given
by
Jacobsson
and
modified
Jacobsson
k
/min
0.000
0.000
••
0
o
o
4
9
o
°
0•
0.005
0.010
0.000
lA
0
.0
0
o
°
0.005
0.010
0.010
0.008
0.006
0.004
0.002
2
,
t
O
S
0.012
O
0
0
c'o
00•-•
.0
•0
ci
)
0.008
o
°
0
k
0
8
/min
0
0
O
O
0.000
0.000
0.007 0.014
0.000
0.007
0.014
k'
(3
h)/min
k'
(4
h)/min
Fig.
1
Relations
between
k
(calculated
using
Eqn
6)
and
k'
(calculated
using
Eqn
7
and
eECV)
for
indicator
concentrations
at
3
h
(left
panels)
and
4
h
(right
panels)
following
injection
of
Cr-51-EDTA
in
group
1
patients
without
(upper
panels;
n
=
156)
or
with
(lower
panels;
n
=
320)
cancer
(men
closed
circles;
women
open
circles).
Regression
lines
for
men
and
women
are
essentially
superimposed
in
all
four
panels.
Regression
equations
for
all
subjects
as
follows:
upper
left:
k
=
+
0.00086/min;
r
=
0.907;
lower
left:
k=
+
0.00094/min;
r
=
0.869;
upper
right:
k=
+
0.00054/min;
r
=
0.942;
lower
right:
k=
+
0.00052/min;
r=
0.918.
0,
men;
0,
women.
o
B
o
0.004
O O
O
0
0
o
°
O
©
2010
The
Authors
284
Journal
compilation
©
2010
Asian
Pacific
Society
of
Nephrology
150
12
00
0
10
o
8
8
0
o
4,83
80,
00
8
-
g
60
00
00
-
0
0
30
2
0
0
0
2
4
6
8
10
12
12
150
0
30
60
90
120 150
0
0
0 0
10
8
0
120
0
90
8,)
8
C.
60
4
30
0
0
0
30
60
90
120 150
GFR/BSA
(mUmin
per
1.73m
2
)
Fig.
2
Relations
between
GFR
based
on
six-sample
iohexol
clearance
and
simpli-
fied
techniques
for
measuring
GFR.
Lines
are
identity.
Left
panels:
reference
tech-
nique
based
on
Equation
10
and
scaled
to
body
surface
area
of
1.73
m
2
.
Right
panels:
reference
technique
based
on
Equation
11
and
therefore
scaled
to
extracellular
fluid
volume
(mUmin
per
litre).
Correlation
coef-
ficients
are
given
in
Table
3.
GFR,
glomeru-
lar
filtration
rate.
0
2
4
6
8
10
12
GFR/ECV
(mUmin
per
litre)
Simplified
GFR
Table
2
Mean
ratios
of
k/k'
(SD)
where
k'
is
based
on
true
ECV
or
eECV
at
3
h
and
4
h
in
patients
with
or
without
cancer
Cancer
Non-cancer
3h
4h
3h
4h
Based
on
ECVf
0.825
0.862
0.829
0.866
Based
on
eECV
0.856
(0.167)
0.882
(0.131)
0.950
(0.292)
0.949
(0.180)
fSD
not
applicable
because
ECV
and
V(0)
are
not
independent
(see
Discussion).
ECV,
extracellular
fluid
volume;
eECV,
estimate
of
ECV.
150
0
co
ci)
120
0
C
D
90
-o
r)
60
8
9
E
30
00
0
0
12
10
8
613
0
08
0
0
0
30
60
0
90
120 150
0
2
4
6
8
10
12
methods
for
measuring
GFR/BSA
were
almost
identical
at
both
3
h
and
4
h
(Fig.
2;
Table
3a).
Agreements
were
mar-
ginally
better
at
4
h.
Although
its
bias
was
almost
zero,
GFR/BSA
based
on
the
slope-only
method
gave
a
lower
precision
and
poorer
correlation
with
the
reference
tech-
nique
(Fig.
2).
When
GFR
values
from
simplified
methods
were
left
scaled
to
ECV
instead
of
BSA,
agreement
with
the
reference
©
2010
The
Authors
Journal
compilation
©
2010
Asian
Pacific
Society
of
Nephrology
285
AM
Peters
et
a
method
(six-sample
GFR/ECV
measured
with
iohexol)
was
generally
lower
(Fig.
2).
In
contrast
to
GFR/BSA,
it
was
best
with
the
slope-only
method
(Table
3b),
although
not
significantly.
DISCUSSION
Although
not
immediately
obvious
from
Equation
2,
but
quite
clear
from
Equation
6
(the
modified
Jacobsson
equa-
tion),
the
original
Jacobsson
equation
primarily
generates
GFR/ECV.
Indeed,
the
derivation
of
the
modified
equation
demonstrates
the
fundamental
similarity
of
single-sample
GFR
measurement
and
the
slope-only
technique,
which
also
Table
3a
Differences
(SD)
between
simplified
GFR/BSA
and
reference
GFR/
BSA
(Eqn
10)
in
patients
of
group
2;
correlation
coefficients
(r)
between
refer-
ence
and
simplified
methods
are
also
shown
Technique
Difference
(mL/min
per
1.73
m
2
)
J3/BSA
3.37
(7.69)
-
t
0.964
J4/BSA
2.01
(6.94)1
-
0.969
GFR/BSA3
3.84
(7.80)1
-
0.963
GFR/BSA4
1.84
(6.91)1
-
0.968
GFR/BSA(S0)
-0.32
(9.664)
0.935
To
calculate
GFR/BSA
based
on
the
modified
Jacobsson
equation,
k'
was
con-
verted
to
k
using
the
appropriate
conversion
factor,
corrected
for
the
one-
compartment
assumption
using
Equation
8,
multiplied
by
eECV
(Eqn
9)
and
scaled
to
BSA
of
1.73
m
2
.
For
GFR/BSA
based
on
slope
only,
k
was
corrected
using
Equation
8,
multiplied
by
eECV
and
scaled
to
1.73
m
2
.
13/BSA,
14/BSA:
original
Jacobsson
equation
at
3
h
and
4
h
scaled
to
BSA;
GFR/BSA3,
GFR/BSA4:
modified
Jacobsson
equation
at
3
h
and
4
h
scaled
to
BSA;
GFR/ECV(S0):
slope-only
GFR
multiplied
by
eECV
and
scaled
to
BSA.
1
-
Significantly
different
from
0.
*significantly
different
from
SD
of
other
simplified
measures
of
GFR/BSA.
BSA,
body
surface
area;
ECV,
extracellular
fluid
volume;
eECV,
estimate
of
ECV;
GFR,
glomerular
filtration
rate.
Table
3b
Differences
(SD)
between
simplified
GFR/ECV
and
reference
GFR/
ECV
(Eqn
11)
in
patients
of
group
2;
correlation
coefficients
(r)
between
refer-
ence
and
simplified
methods
are
also
shown
Technique
Difference
(mUmin
per
litre)
J3/eECV
0.357
(1.14)1
-
0.863
J4/eECV
0.249
(1.06)-t
0.880
GFR/ECV3
0.381
(
1
.
11
)
-
t
0.868
GFR/ECV4
0.229
(1.03)-t
0.883
GFR/ECV(S0)
0.060
(0.95)
0.901
Simplified
GFR/ECV
is
k-corrected
for
the
one-compartment
assumption
(Eqn
8).
13/ECV,14/ECV:
GFR
from
Jacobsson
equation
at
3
h
and
4
h
divided
by
eECV;
GFR/ECV3,
GFR/ECV4:
k
from
modified
Jacobsson
equation
at
3
h
and
4
h
corrected
for
the
one-compartment
assumption
(Eqn
8);
GFR/ECV(S0):
kfrom
slope-only
corrected
for
the
one-compartment
assumption.
1
-
Significantly
different
from
0.
ECV,
extracellular
fluid
volume;
eECV,
estimate
of
ECV;
GFR,
glomerular
filtra-
tion
rate.
generates
GFR/ECV.
In
the
original
Jacobsson
method,
the
ratio
GFR/ECV
is
essentially
converted
to
GFR
by
multipli-
cation
with
the
same
m-corrected
estimate
of
eECV
that
substitutes
for
V(0).
The
current
study
confirms
the
robustness
of
single-
sample
GFR
measurement
using
the
Jacobsson
method.
The
modified
version
of
the
Jacobsson
equation
functions
equally
to
the
original
equation.
However,
it
uses
corrections
that
are
more
rational
and
transparent
than
the
original
equation.
Two
corrections
are
involved:
one
for
the
inequality
of
V(0)
and
eECV
and
the
other
for
the
assumption
of
one
compart-
ment.
In
the
original
Jacobsson
equation,
the
term
0.0016
appears
to
correct
for
one,
while
the
Brochner-Mortensen
equation
appears
to
correct
for
the
other.
However,
the
Brochner-Mortensen
equation
is
inappropriate
as
it
is
was
developed
to
correct
slope-intercept
GFR
and
is
not
appli-
cable
to
the
corrections
either
of
ECV
to
V(0)
or
of
k
to
GFR/ECV.
Nevertheless,
the
original
equation
ends
up
gen-
erating
a
value
for
single-sample
GFR
almost
identical
to
that
from
the
modified
equation.
The
relation
between
k
and
k'
might
be
expected
to
be
dependent
on
GFR
and
therefore
non-linear.
Instead,
it
was
linear,
leading
to
the
constant
conversion
factors
reported
in
Table
2.
This
linearity
can
be
explained
by
the
logarithmic
transformations
in
Equations
6
and
7.
Of
the
conversion
factors
in
Table
2,
we
applied
those
based
on
ECV
rather
than
eECV
because
ECV/weight
was
significantly
different
between
patients
with
and
without
cancer.
This
difference
is
presumably
the
result
of
generally
less
body
fat
in
patients
with
cancer
and
explains
the
difference
between
cancer
and
non-cancer
patients
in
eECV/ECV.
The
correlation
between
k
and
k'
based
on
ECV
is
expected
to
be
very
close
because
ECV
and
V(0)
are
not
independent
and
the
relation
between
k
and
k'
reflects
the
magnitudes
of
the
one-compartment
corrections
of
GFR
and
GFR/ECV.
The
mean
ratio,
klk'
,
nev-
ertheless
is
valid
as
a
conversion
factor
as
it
informs
the
relation
between
V(0)
and
true
ECV,
uninfluenced
by
defi-
ciencies
in
the
equation
used
for
estimating
eECV.
Single-sample
methods
rely
on
adequate
estimation
of
eECV,
usually
based
on
body
weight.
In
the
current
study,
the
mean
ratio
k/k'
when
based
on
ECV
was
very
similar
between
patients
with
and
without
cancer.
The
ratio
based
on
eECV
was
also
similar
to
that
based
on
ECV
in
patients
with
cancer
but
it
was
appreciably
higher
in
patients
without
cancer,
presumably
because
ECV/weight
was
lower
in
such
patients.
Both
k/k'
ratios,
however,
were
almost
identical
between
men
and
women
(Fig.
1).
Single-sample
GFR
might
have
performed
better
were
it
not
for
one
relative
weakness
of
the
reference
method,
namely
the
use
of
XRF
to
assay
iohexol,
rather
than
high
performance
liquid
chromatography
(HPLC),
which
is
prob-
ably
superior!'
HPLC
was,
however,
not
available
to
us
at
the
start
of
the
study.
This
study
shows
that
GFR/ECV
derived
from
the
slope-
only
method
and
then
converted
to
GFR
by
multiplication
©
2010
The
Authors
286
Journal
compilation
©
2010
Asian
Pacific
Society
of
Nephrology
Simplified
GFR
with
eECV
was
outperformed
by
the
single-sample
methods,
but
when
left
as
GFR/ECV
performed
better
than
the
single-
sample
methods.
This
might
be
explained
by
variations
in
ECV
that
would
have
equal
impact
on
k
measured
from
slope-only
GFR
and
reference
multi-sample
GFR/ECV,
but
a
quantitatively
different
impact
on
k
measured
from
a
single
sample
and
one
that
would
depend
on
sample
timing
and
the
prevailing
filtration
function.'
In
conclusion,
a
simple
equation
based
on
the
Jacobsson
equation
and
developed
from
first
principles
is
described
for
measurement
of
single-sample
GFR.
Although
it
has
not
improved
the
accuracy
of
the
original
equation,
it
incorpo-
rates
two
simple
and
easy-to-understand
correction
factors.
It
also
illustrates
the
fundamental
similarity
of
the
single-
sample
technique
with
slope-only
GFR,
which
it
outper-
formed
when
scaled
to
BSA
but
not
when
scaled
to
ECV.
REFERENCES
1.
Christensen
AB,
Groth
S.
Determination
of
'Tc-DTPA
dearance
by
a
single
plasma
sample
method.
Clin.
Physiol.
1986;
6:
579-88.
2.
Jacobsson
L.
A
method
for
the
calculation
of
renal
dearance
based
on
a
single
plasma
sample.
Clin.
Physiol.
1983;
3:
297-305.
3.
Peters
AM.
Expressing
glomerular
filtration
rate
in
terms
of
extracellular
fluid
volume.
Nephrol.
Dial.
Transplant.
1992;
7:
205-10.
4.
Peters
AM.
The
kinetic
basis
of
glomerular
filtration
rate
measurement
and
new
concepts
of
indexation
to
body
size.
Eur.
J.
Nucl.
Med.
Mol.
Imaging
2004;
31:
137-49.
5.
Bird
NJ,
Henderson
BL,
Lui
D,
Ballinger
JR,
Peters
AM.
Indexing
glomerular
filtration
rate
to
suit
children.
J.
Nucl.
Med.
2003;
44:
1037-43.
6.
Brochner-Mortensen
J.
A
simple
method
for
the
determination
of
glomerular
filtration
rate.
Scand.
J.
Clin.
Lab.
Invest.
1972;
30:
271-4.
7.
Bird
NJ,
Peters
C,
Michell
AR,
Peters
AM.
Suitability
of
a
simplified
technique
based
on
iohexol
for
de-centralised
measurement
of
glomerular
filtrstion
rate.
Scand.
J.
Urol.
Nephrol.
2008;
42:
472-80.
8.
Haycock
GB,
Schwarz
GJ,
Wisotsky
DH.
Geometric
method
for
measuring
body
surface
area:
A
height-weight
formula
validated
in
infants,
children
and
adults.
J.
Pediatr.
1978;
93:
62-6.
9.
Bird
NJ,
Michell
AR,
Peters
AM.
Accurate
measurement
of
extracellular
fluid
volume
from
the
slope/intercept
technique
after
bolus
injection
of
a
filtration
marker.
Physiol.
Meas.
2009;
30:
1371-9.
10.
Gaspari
F,
Guerini
E,
Perico
N,
Mosconi
L,
Ruggenenti
P,
Remuzzi
G.
Glomerular
filtration
rate
determined
from
a
single
plasma
sample
after
intravenous
iohexol
injection:
Is
it
reliable?
J.
Am.
Soc.
Nephrol.
1996;
7:
2689-93.
11.
Bird
NJ,
Peters
C,
Michell
AR,
Peters
AM.
Reproducibilities
and
responses
to
food
intake
of
GFR
measured
with
chromium-51-
EDTA
and
iohexol
simultaneously
and
independently
in
normal
subjects.
Nephrol.
Dial.
Transplant.
2008;
23:
1902-9.
12.
Nosslin
B.
Determination
of
dearance
and
distribution
volume
with
the
single
injection
technique.
Acta
Med.
Scand.
1965;
442:
97-101.
13.
James
TJ,
Lewis
AV,
Tan
GD,
Altmann
P,
Taylor
RP,
Levy
JC.
Validity
of
simplified
protocols
to
estimate
glomerular
filtration
rate
usingiohexol
dearance.
Ann.
Clin.
Biochem.
2007;
44:
369-76.
14.
Bird
NJ,
Peters
C,
Michell
AR,
Peters
AM.
Effect
of
extracellular
fluid
volume
on
single
sample
measurement
of
glomerular
filtration
rate.
Nephrol.
Dial.
Transplant.
2009;
24:
104-8.
©
2010
The
Authors
Journal
compilation
©
2010
Asian
Pacific
Society
of
Nephrology
287