Download A Propagation of Error Analysis of the Enzyme Activity Expression. A

CLIN.
CHEM.
22/9,
1438-1450
(1976)
A Propagation of Error Analysis of the Enzyme Activity
Expression. A Model for Determining the Total System
Random Error of a Kinetic Enzyme Analyzer
Thomas 0. Tiffany,1’2 Philip C. Thayer,2 Chris M. Coelho,2 and Gilbert B. Manning2’3
We present a total system error evaluation of random error,
based on a propagation of error analysis of the expression
for the calculation of enzyme activity. A simple expression
is derived that contains terms for photometric error, timing
uncertainty, temperature-control error, sample and reagent
volume errors, and pathlength error. This error expression
was developed in general to provide a simple means of
evaluating the magnitude of random error in an analytical
system and in particular to provide an error evaluation
protocol for the assessment of the error components in
a prototypeMiniatureCentrifugal Analyzer system. Individual system components of error are measured. These
measured error components are combined in the error
expression to predict performance. Enzyme activity
measurements are made to correlate with the projected
error data. In conclusion, it is demonstrated that this is one
method for permitting the clinical chemist and the instrument manufacturer to establish reasonable error limits.
Additional Keyphrases: centrifugal analyzer
source of
ment
#{149}
analytical
error
#{149}
enzyme
#{149}variation,
activity measure-
Recent attention has been focused on the error
components of spectrophotometric
analytical instrumentation and on the establishment of acceptable error
limits for such instrumentation
used in the clinical
laboratory (1, 2). These analytical instruments range
from manually operated photometric devices to fully
automated analyzers. The latter contain pipettors for
sample and reagent, timing sequencers, temperature
control modules, photometers, quantization modules,
and data processing modules, all acting in concert to
simply and rapidly produce results for the clinician.
‘Pathology
Associates,
Inc., Terminal
Box 2687, Spokane,
Wash.
99220. (To whom reprint requests
should be addressed.)
2 Micro Biochemical
Research
Corp., Spokane,
Wash. 99206.
Sacred
Received
‘
1438
Heart Medical Center,
May 3, 1976; accepted
CLINICAL CHEMISTRY,
Spokane,
Wash.
July 1, 1976.
Vol. 22, No. 9, 1976
99204.
The problem becomes how to establish and to define
the acceptable limits of accuracy (systematic error) and
precision (random error) for these more complex analytical systems. Photometric error and its relationship
to error in kinetic analyses has been considered by
several authors; Ingle (3) and Ingle and Crouch (4-6)
considered the precision of normal spectrophotometric
techniques and the signal-to-noise (S/N) ratio theory
of fixed time spectrophotometric
reaction rate measurements. Maclin (7) attempted a system analysis of
the GEMSAEC Centrifugal Analyzer as a kinetic enzyme
analyzer system. Maclin et al. (8) further defined the
relationships
between various instrumental
noise
sources and the performance of the instrumental system
which is designed to do enzyme kinetic assays. Pardue
et al. (9) developed an excellent comprehensive treatment of photometric error, which was to some extent a
compendium of the other efforts. All of these efforts
have produced in varying depths and usability an error
analysis of spectrophotometric
measurements related
to different forms of kinetic analyses. Ingle and Crouch
are rigorous and thorough in their treatment of photometric error. They begin from first principles to define
the various photometric error components, then they
methodically apply them to various types of photometric measurement techniques. They limit their discussion to photometric error. Macin extended the error
estimation to the total system concept to include errors
of pathlength, volume measurement, and temperature
fluctuation. He further developed the idea of an error
budget as it relates to system error. Pardue et al. effectively considered five categories of photometric error
and combined these derived error coefficients into
mathematical expressions that permit the calculation
of systematic and random error components. Furthermore, the random error analysis developed by this group
is useful because it allows the instrument user and
manufacturer to define the instrumental nature of the
photometric error. However, these authors did not
consider the other error terms associated with a more
complex automated kinetic photometric analyzer.
Therefore this paper sets out to develop a total system
error evaluation of random error based on a propagation
of error analysis of the expression for the calculation of
enzyme activity. The expression contains terms for
photometric error, timing uncertainty, temperaturecontrol error, sample- and reagent-volume errors, and
pathlength error. The error expression was developed
in general to provide a simple means to evaluate the
magnitude of random error in an analytical system and
in particular to provide an error evaluation protocol for
the assessment of the error components in a prototype
miniature centrifugal analyzer (MCA) system. The error
expression provides levels of sophistication for its use.
It will be demonstrated to be one means, by simple experimental technique, for the clinical laboratory practitioner to measure total random error and to determine
its magnitude in the various module components. It will
further permit the manufacturer to dissect by methods
similar to Pardue et al. the instrumental error components. Finally, such an approach is one method of permitting the clinical chemist and the instrument manufacturer to establish mutually acceptable error limits
that will provide a balance between good clinical enzymology and needed economy in instrumentation.
()3u
Su 2 =
(U/liter)
I
t.A
‘Vt’
(-)#{231}--)#{231}
I1\
e1(T]
A
(1)
where LA is the absorbance change during the measurement period, tmin is the elapsed time in minutes, V
is the total reaction volume, Vs is the sample volume,
B is the pathlength,
A isthe molar absorptivityof the
monitored reaction species and K isthe temperature
coefficientof the enzyme reaction.The exponential
term is added in this treatiseto include the obvious
temperature dependence of enzyme activity.The ad-
dition of this term in no way suggests that the use of
temperature
coefficients
to correct enzyme activity from
one temperature to another is sanctioned by this author.
For any given U where U = F (X, Y. . .), the basic
propagation of error theory expression for the uncertainty (variance) of U in terms of all the related error
source terms X, Y. . , is:
.
Sy2
(2)
+...
v4ance of enzyme activityas a function A,
V8, B, and T as follows:
S2=’_i_”
u
(MA)
S2
+
V,
Sr2
(3)
#{212}U2
1-”)
Stmin2
‘atmin,
+
(L)2sv2+
+
time,
(-)2sv2
()
dU2
SB2
+
aU2
(;)
Solving the partial derivatives of each variable, substituting the results in expression 3, and dividing both
sides by U2 results in expression 4:
S2
S
..A
Q .2
‘-‘tmin
2 +%/2
min
2
S2
The relative
variation
(CV)
standard
S
deviation
or coefficient
(4)
of
is defined as:
CVx=
A general approach to the development of a random
error instrumental evaluation protocol is to perform a
propagation of error analysis of the enzyme activity
expression as a reasonable model to develop the error
treatise (10).
The formula for the calculation of enzyme activity is
derived basically from Beer’s Law and is expressed
as:
=
(;)
3U2
Sx2 +
where S is the standard deviation.
Therefore the formula for the enzyme activity (U),
expression 1, can be operated on according to the simplified expression for the error (2) to determine the
Propagation of Error Analysis of
the Enzyme Activity Expression
Enzyme activity
2
(5)
Thus the variance or uncertainty of enzyme activity
as a function of the differenterrorcontributionsof the
analyticalsystem simply becomes:
CVenzymeactivity
=
[(CVA)2
+ (CVB)2
+ (CVtmin)2
+ (CVv5)2
+ (CVv)2
+ K2S2J1”2
(6)
The usefulness of expression 6 is in the simplicity with
which it can be used to predict the magnitude of the
percent relative standard deviation of enzyme activity
(CV) as a function of the magnitude of the uncertainty
in each of the individual error parameters. As a simple
example, if each term has an uncertainty or relative
standard deviation of (0.01) or 1% relative standard
deviation, the percent relative standard deviation of the
measured enzyme activitywillbe:
CV
=
100[6(.01)2]112
=
2.45%
(7)
Obviously, not allthe terms willcontribute to the
variance with the same magnitude and some of the
terms willbe much lessthan 1%. Therefore, it isthe
purpose of thisdiscussionto examine each of the error
terms of expression 6 as they relateto instrumental
system components, and to the MCA
prototype, to
make estimations of the expected magnitude of uncer-
tainty from each system component, to correlate these
errors with published data, and discuss the error parameters in terms of the actual enzyme analysis. It is
useful to relate each term of expression 6 with an analytical system component: C V
is related to the elecCLINICAL CHEMISTRY, Vol. 22, No. 9, 1976
1439
tro-optical noise of the transmission signal (T) from the
analytical module and the quantization error of the
analog-to-digital
converter (ADC) in the computer
module. CVv, CVv represent the uncertainty of volume measurement ofthe sample and reagentpipetting
system.C Vtmjn istheuncertaintyoftime when theactualabsorbancemeasurements are made by the computer or other timing device. For a computer, the uncertainty is of the order of 10 ms or less. The magnitude
of this error in an actual enzyme activity measurement
is of the order of 4 X 10-% for a 4-mm enzyme activity
assay. Therefore, this term can essentially be ignored
for computerized instrumentation
but not for manually
timed photometer
systems. CVB is the pathlength
variation.
This term is important
for multi-cuvette
centrifugalanalyzers.The K2S2 term is related to
temperature control. The significance of this term will
be explored in the next section.
To summarize this section, the expression for the
calculation of enzyme activity can serve as a good model
to examine the uncertainty in the determination
of
enzyme activity as this uncertainty is related to the
various analyticalcomponents of the instrumental
system. A simple and useable expression was obtamed:
CV(IU/L)
=
[(CV)
2
+ (CVv)
2
2
+ (CVv8)
+ K
2
+ (CVB) 2
2
P/2
(8)
which willbe usefulinthepredictionof the magnitude
of the system error (C17) from experimentallyqf
mathematicallyderivedindividualerrorcomponents.
nents and to be able to predict what their worst case
contribution will be to the overall noise or uncertainty
of the analytical system. This is important in the quality
assurance of the analytical system by the manufacturer.
Furthermore it is important to the establishment of
meaningful individualmodule performance characteristics
that arenecessarytothefinaldefinition
oftotal
analytical system performance by the various society
committees on instrumental standards.
The individual error components will be examined
in this section. A prototype Miniature Centrifugal Analyzer (11, 12) willbe used as a point of discussion.
Photometric
error (electrical and quantization),
total
volume error, sample volume error (the pipettor system), temperature
uncertainty
(temperature
control),
and pathlength
error components
will be considered.
There are two functions of this section. First, it serves
to establish individual error limits in a more rigorous
manner, thereby providing criteria by which to judge
the performance of analytical systems [see Pardue et al.
(9)]. Second, the individual error limits can be determined for each analytical system component and these
relative
standarddeviationscan then be insertedinto
the total error expression (expression 6) to demonstrate
the system performance for the analyticalrange of
conditionsproposed forthe instrument.Enzyme ac#{149}
tivity
assays can then be performed and the results
correlatedwith the predictederror.
Photometric
Error
Individual System Component Error Sources
The usualphotometricerrortreatment has been a
propagationoferroranalysisoftheBougert-Beer law.
The concentrationin terms of absorbance or trans-
The expression
for the total system error in the
measurement
of enzyme activity as derived in the pre-
mission error are evaluated, assuming
sorbance or transmission
error (13).
vious sectionsimply relates the total uncertainty of
enzyme activityto the relativestandard deviation
(coefficient
of variation)
of each error component.
Therefore,
one can determine
the magnitude
of the
relative standard deviation of each term (absorbance,
volume, temperature,
etc.) by experimental
measurement. The resulting measured uncertainty
of the mdividual analytical system components can then be used
to predictthe overalluncertainty(aswillbe shown in
thediscussion
section)
and toidentify
critical
individual
errorcomponents ofthe analytical
system.This isthe
levelatwhich the manufacturerneeds toassessthereliability
ofhisproduct (e.g., production quality control),
and itisthelevelatwhich theclinical
chemist can assess
the reliability
and continuedperformance ofthe analytical
system in his laboratory.
However,
the enzyme
values
for ab-
This type of
treatment of photometric error has limited use for
current instrumentation,
which is in general not quantization error limited.
However, the approach to the
evaluation of photometric
error as outlined by Pardue
et al. (9) is extremely useful and will serve as a base of
discussion. A direct result of their discussion was to
demonstrate
that the major error sources of a centrifugal analyzer, which has constant reference up date
(minimization
of source drift and dark current reading),
are:transmissionerrorcaused by electrical
noiseand
quantizationerror caused by the analog-to-digital
converter.
The standard deviation of absorbance (SA) with respecttotransmission(Tr) is:
0.434
STr
(1)
activity error expression derived in the previous section
Tr
did not give a further dissection of the individual error
components intotheirrespectiveinstrumentalor meHowever, this only relates the uncertainty of absorbance
chanicalerrorcomponents. For example, the uncer5A as a function of the transmission error STr. Nothing
taintyofabsorbancemeasurements (photometricerror) isindicatedabout the natureor magnitude ofthe error
can be further broken down into electrical noise,
components contained in the total transmission error
quantizationerror,source flickernoise,and stray
STr. Random error is expressed as:
light-dark current error. It is important to be able to
estimate the contribution of the actual error compoS = (5i2 + S22 + S32. . . SN2)’12
(2)
1440
CLINICAL CHEMISTRY,
Vol. 22, No. 9, 1976
where the total random error is the sum of the squares
of the individual error components (section 1). For this
discussion, two types of error (a) electrical noise and (b)
quantization error are of interest.
Electrical
I
I
0-
/
Noise
/
Variations in photon arrival rates at low intensities
(3) and shot noise-white noise (4) lead to transmittance
errorsthat are proportioned to the square root of
transmittance (9):
=
error from noise becomes,
o
/
z
4
SNA
=
-
/
/
/
/
/
/0;
(4)
/
I
0’
/‘\
4/
/1
Quantization
OS
/
,.,
zv
constant, SNT is the random noise or root mean square noise of the analog signal, and Tr is the transmission.
__________
S
&
I..
I,..
I.-
where B is a proportionality
M$SC (S4O
OITTID $40 TA
MS*C 1405)
/
4.
0.434 BSNT
Tr112
- ‘TIU04(TIC*l.
--
/
0’
and the absorbance
.-
x.-
‘s-
/
U.-
(3)
BTr”25N
‘
/
4-
0
Tr112SN
STr
/
/
I
I/
P0WTEOS
/
(io’
./
,,./
/
/
S
/
4/
Error
/
/
The quantization
error for a 12-bit ADC in terms of
absorbance has been shown (9) to be as follows:
A
/
#{149}
I
#{149}#{149},
z,
i
ABSORBANCE
‘5)
Fig. 1. Standard deviation of absorbance (SA) vs. absorbance
SQA
(7.05 X 10-s)
=
(5)
for quantization error and measured photometric error of the
MCA system
This was derived from the error expression for an N-bit
ADC as previously described by Pardue et al. (9).
Total Random Error of Absorbance
(MCA)
The total random error of absorbance SA for the
MCA photometric system can be related to the quantizationerrorand the electrical
noiseas follows:
SA
which becomes
=
0.434
=
[
[(SNA)2
BSNT
(T1/2)
+
(SQA)2]’12
+
(7.05 x
2
(6)
10_5)2]1/2
(7)
T,.
upon substitution
of expressions 4 and 5. This type of
two-term or multiple-term
photometric error expression
has been shown to be usefulforpredictionof photometric error over a wide range of operational absorbances, to be useful in showing what types of errors predominate for a given system at different
levels of absorbance, and to be useful for establishing
what can be
done to minimize instrumentally
the photometric
error
(9).
Measurement
of Photometric
Error
The test of expression 7 is how well it relates to the
actual generated photometric
error data and how useful
it is for the prediction
of what the major error components are.
A first approach in its use is to assume that the total
photometric
error contribution
is due to quantization
error. Therefore expression 7 reduces to,
=
3.05
X 105e23A
(8)
as indicated by Pardue et al. (9). Thus by simply inserting various absorbance values from 0 to 2.0 absorbance units into expression 8, the random error
caused by quantization
error can be obtained. This is
plotted in Figure 1 (long dashed lines).
The actual photometric
error can be obtained by the
measurement
of absorbance of solutions that are relevant to the clinical laboratory.
The absorbance of solutions of NADH and p-nitrophenol
(PNP) at 340 nm
and 405 nm, respectively,
can be made. Photometric
noise measurements
were made by pipetting solutions
of increasing concentrations
of NADH in Tris buffer
(0.1 mol/liter, pH 7.4) and of PNP in 0.6 mol/liter MAP
buffer into a calibrated
rotor. The solutions
can be
placed in order of increasing absorbance for convenience
and with the final solution concentrations
so chosen as
to provide several absorbances over the range of 0-2.0
absorbance units. The absorbance data are then acquired with a tabular absorbance program that gives the
mean absorbance measurements
and the standard absorbance for each cuvette. These data can then be
plotted as shown in Figure 1. The measured photometric error (SA) vs. absorbance as obtained from the
MBRC MCA system at 340 nm and 405 nm are listed
in Table 1. The data were fitted to an exponential curve,
which resulted in expressions; SA = 7.1 X 105e 1.64A and
= 7.7 X 10 5el69A
for the 340 nm and 405 nm data,
respectively.
Absorbance
data from the ENI SN1O9
analyzer obtained by Maclin (7) gave a similar expression of SA = 7.2 X 105e i.6OA The exponential
formu-
CLINICAL CHEMISTRY,
Vol. 22, No. 9, 1976
1441
Table 1. Measured Photometric Precision vs.
Absorbance of the MBRC MCA System at
340 nm and 405 nm
405 nm
340 nm
Absorbance
SA X iO
0.2273
0.5469
1.0711
1.0
0.3905
1.5
0.7361
2.8
5.3
1.0764
3.7
1.5824
10.8
1.4238
2.0616
18.2,
2.1421
A0=7.07X
b=1.65
r= 0.99
=
A0= 7,73x
b= 1.69
r= 0.98
10
7.07 x 10’
SA X iO
Absorbance
e1.65A
9=
7.73 x
1.8
7.3
34.7
1O
o
10e’
00
0*
0*
ci
0
0
14
Ia
IS
00
ABSORBANCE-
Fig. 2. Standard deviation of absorbance vs. absorbance as a
function of data averaging
lation of the error data as given above for the simple log
data plots of the data (Figure 1) provide two types of
information.
First, the photometric
error at any absorbance level can be readily obtained and, as will be
NA05-4
PWS
V-SiX
0
,.
0.59
ojar.i
tP.,
r-
shown later, can be used to calculate the photometric
contribution to the overall relative standard deviation
of enzyme activity. Second, as noted by Pardue et al. (9),
TUP.bI0
uM
‘A’#{176}’’
EuFF.
0.97
these data provide information
about the nature of the
photometric
error components. As an example, the data
plotted in Figure 1 reveal that the photometric
error is
more than quantization
error. The actual data, like the
data of Maclin as plotted by Pardue et al. (9), seem to
show less error than predicted by the quantization
error
expression at higher absorbances and more error than
predicted at lower absorbances. The reason for this is
the dual nature
(at least) of the error component
of the
photometric error, which is comprised of the transmission independent quantization error (Be23A) and
ABSORBANCE-340
FIg. 3. Standarddeviationof absorbancevs. absorbanceof clear
and turbid NADH solutions
the analog error, which in turn depends on the square
root of the transmission
signal (Be’ 15A). A log plot of
Figure 1 would reveal a slope of less than 1 for these
1442
CLINICAL CHEMISTRY,
Vol. 22, No. 9, 1976
A
223
data. If the above error was quantization error, only the
slope would be one, if the error were only transmission
noise (analog) the slope would be 0.5 (9).
Information generated in this manner is probably
more useful to the manufacturer than to the clinical
chemist in the laboratory. However, they do provide a
definition of the limits of error faced by the instrument
manufacturer in his effort to develop a reliable clinical
analyzer. It is easy to say that by digitally averaging the
data the error will be reduced by
As shown in
Figure 2, this improves the data, but digital averaging
has its limits. The effort by Pardue points out that one
can reduce quantization error by going to a higher resolution ADC or by imposing an auto-ranging device to
permit certain portions of the transmission signal to
operate over the full range of the ADC. As an example,
100-10% transmission could operate over 4096 bits, and
then the 10-0% could be ranged over the same 4096 bits.
Increased resolution
of the ADC will not be the total
answer if the analog signal is extremely noisy. Thus one
NM
(0)
o,I,,
.53
LiJ
L)
“
r-0999&
I20
5,c-O.0t45
aD
o
U)
so
05
<0
5
.030
0’S
I
I
I
.100
I’S
ISO
P-NITROPHENOLJ
13
.200
X tO MOL/LITER
z..
o
B
020
00) I’S
I iso
IS.’
Icc
A, - 3.15.6/.505.
A, -(0Z4XIO0
r - 0.9999
009
aD,s
Q50
.05
<C
590.0.
0
0.1
[NADHJ
0.t
0.3
04
03
00
0.1
X I0 MOL/LITER
Fig. 4. Optical linearity of the MCA system at 340 nm and 405
nm
may decrease the noise by grounding, by changing the
mode of detection
to photon counting, or by other
error-reducing
alternatives. A combination of the above
can reduce photometric
error by a factor of 10, as
pointed out by Pardue et al. (9). An additional error
component not mentioned by Macun or Pardue is the
increased error caused by turbidity of the sample. The
effect of serum turbidity on photometric error is shown
in Figure 3. Neglect of this component can negate the
effortsto reduce quantizationand analogsignalerror
components. The economic balance between further
reducing photometric
error and status quo can best be
defined by the current level of performance of the instruments, as indicated by the enzyme activity error
expression. The definition of acceptable level of performance of the instrument by the clinical chemist,
based on sample throughput and acceptable error of
activity measurement at critical enzyme levels, is a
necessary component to the establishment of good
clinical enzymology and economy in instrumentation.
Systematic
Photometric
Table 2. LinearityStray-LightData for NADH
Absorbance Measurements at 340 nm
NADH concn,
mmol/Iiter
Absorbance
0.064
0.2037
0.2097
0.2082
0.128
0.4 192
0.4183
0.4247
0.256
0.8336
0.8356
0.8332
0.384
1.2511
1.2533
1.2574
0.512
1.6502
1.6506
1.6444
0.640
2.008
2.015
2.020
coefficient
=
1. The expression for calculation
V’I,A
of volume:
1.lrrS
2. The error in volume caused by radius:
I dV\
(A)SVr*,,j)Sr
Svr
(I2lTf)Sr
=
(B)-T7#{176}’/irr2
2r
(C) SVr
(/2irr)
r
=
3. The error in volume caused
(A)Svj
Svi
(B)---=j
()Si11Tr2Sj
irr2
S
by Ieigth or linear motion:
=ri
4. The total random error in volume:
S,
=
/ Sp.\1’/
(S1f +SVr2)h/2 =
Error
Accurate preparation of NADH and PNP solutions
with absorbances over the range of 0-2.0 absorbance
units provides the tools for measuring photometric
error. Furthermore, the measured absorbance of these
solutions
can be used todeterminestraylight.
A plotof
the absorbancedata at405 nm and 340 nm is shown in
Figure4.The measured absorbancedata and amount
ofstraylightare listed in Table 2.The straylightcom-
Stray-light
Table 3. Random Error of Volume Caused by
Mechanical-Driven Syringes
0.16% (see Appendix
A)
ponent
at 340 nm and 405 nm was calculated
to be less
than 0.2%. The molar absorptivity of NADH at 340 nm
was calculated to be 6.24 X 10 liter mol’ cmt, which
is a 0.3% error. The method for calculation of stray light
for centrifugal
analyzers in general is shown in Ap-
pendix
A.
Sample-Reagent
Pipettor Error
The random error caused by the pipetting
and dispensing operation of a mechanically
driven piston or
syringe system can be determined
by a propogation
of
error analysis of the formula for volume, as outlined in
Table 3. The random error of volume displacement
(sample or reagent) can be considered as being com-
prised of an error term from radius variation-either
of the plunger
delimiter-and
tioning
or the barrel,
whichever
an uncertainty
of the plunger
or barrel
is the tolerance
in the repeatable posi(linear
motion),
which
can be called backlash.
The expression for the random error in volume (line
4, Table 3) can be used to defmethe limits of a pipetting
system for a given set of tolerances. A 1OO-0’1
syringe can
be chosen as an example. If the radius is 7.28 X 10-2 cm
and the variation in syringe radius SR = ±3.64 X 10
cm, or 0.5%, then the standard deviation in volume
(SVr) from radius error alone for pipetting a volume of
1 Ml is ±0.01 Ml, or a relative standard deviation of 1%
(line 2C, Table 3). If we state that the error in linear
positioning (backlash) of the syringe driving mechanism
can be no greater than S1 = ±2 X 10 inch, or ±5 X
10 cm, then the standard deviation of volume caused
by error in linear motion (S,1) for a volume of 1 Ml is
*8.2 X 10 Ml (line 3B, Table 3). The total error from
radiusand backlashuncertainty5,, is ±0.0131 Ml, or a
relative standard deviation of 1.3% at 1 Ml. These are
obviously very close tolerances for a pipettor system and
show the stringent requirement necessary to provide a
pipettor system capable of precisely dispensing the
CLINICAL CHEMISTRY,
Vol. 22, No. 9, 1976
1443
2
0
11J2
#{149}
XAMPLFI
*
00
0.5
-
5irF
uJ
0
.
d
111
(I
5100
>
A
Jo
RI5..UT
I-
LI)
VOLUME -
uJ
10
>
1
-.-.
--
B
0*
uJ
z
Q0
0--
Z
05
30
M7O59
VOLUME DELIVERED
-j.iL
TOTAL
LIJ
1._I .0
Fig. 6. Relative
standard deviation (in percent) vs. volume of
00
0*
lx
dispensed aliquots of water
00a
1/1
ID
#{176}
05
I
I
I
I
I
I
VOLUME -iL
Fig. 5. Relative standard deviation (in percent) and absorbance
of pipetted NADH solution vs. volume
Temperature
Uncertainty
The effect of temperature
on enzyme reactions has
been expressed in terms of the temperature
coefficient
micro-scale
volumesrequiredforcurrentkinetic
enzyme
analyzers. If one considers the error in pipetting 2, 5, or
100 Ml, the relative standard deviation drops to 0.65%,
0.26%, and 0.13%, respectively.
Evaluation
Several
of Pipettor Error
methods
can be used to evaluate
pipettor
performance, including measurement of absorbance of
pipetted optically absorbing solutions, the weighing of
dispensed volumes of de-ionized water, and measurement of the radioactivity
of dispensed radiolabeled
materials.
The first two methods have been used to measure
sample pipetting
precision and the precision of total
volume pipetting
(sample plus diluent),
respectively.
Concentrated
solutions of accurately prepared NADH
(Sigma Type lilA) in 0.1 mol/liter
Tris buffer, pH 7.4,
and concentrated
solutions of p -nitrophenol (PNP) in
0.6 mol.liter MAP buffer were pipetted into calibrated
rotors. The precision of pipetting several samples (n =
10) at various volumes in the range of 1 to 20 MI is shown
in Figure 5A. The pipettorprecisionfora pipetting
sample is better than 0.5% (relative standard deviation)
above 5 Ml. The relative standard deviation for pipetting
sample and diluent for volumes in the range of 50-75 Ml
is given in Figure 6. The overall error in pipetting
and
dispensing total volume is less than 0.5% in this volume
range.
Pipettor
Linearity
The linearity of pipetting and dispensing increasing
volumes of NADH is shown in Figure 5B. The regression
line of absorbance vs. volume is y = 0.04 + 1.02x, with
r = 1.00. The slope of the regression line for absorbance
vs. NADH concentration
yielded a molar absorptivity
of 6.28 liter mmol1 cm1 compared to a literature value
of 6.22, which is a 0.96% error. These data represent the
systematic error of the pipettor system.
1444
CLINICAL CHEMISTRY,
Vol. 22, No. 9, 1976
(Qio), the factor by which the velocity of the enzyme
reaction increases on increasing the reaction temperature by 10 #{176}C
(14). The temperature
coefficient
for
enzyme catalyzed reactions usually lies between 1 and
2, depending on the enzyme and on the degree of coupling used in the enzyme activity assay. The temperature coefficient decreases for coupled enzyme catalyzed
reactions.
The effect of temperature
on the reaction rate is
generally
related to the empirical
Arrhenius
equation:
K
=
(9)
e/RT
where K is the reaction rate constant, Ea is the energy
of activation
of the reaction, R is the universal
gas
constant, and T is the temperature
in K (degrees Kelvin). Enzyme activity
(V) can be expressed as follows:
V=K[E]
(10)
where K is the rate constant
of the enzyme reaction, [E]
is the enzyme concentration,
and V is the velocity of the
reaction or activity. The temperature-dependent
activity becomes:
(11)
Ve_E/RT[E1
The type of expression is limited by the thermal denaturation of the enzyme(s) in the activity assay. However,
over a limited temperature
range it is useful, for this
discussion, to obtain the temperature-related
activity
change and hence the uncertainty
caused by temperature fluctuation.
Henry
(15) has reported
temperature
in enzyme activity
experimentally
determined
correction factors for lactate dehydroge-
nase, alanine aminotransferase,
and aspartate aminotransferase (LDH, ALT, and AST). If one assumes an
activity of 100 U/liter for each of the three enzymes at
32 #{176}C,
then the activity of each enzyme can be calculated for any other temperature
from 25-40 #{176}C.
The
resulting velocity/temperature
data can be fitted to an
exponential
curve. The resulting expressions for LDH,
ALT, and AST were: v = be.0718T, r = 0.999; V =
beO649T, r = 0.999; and v = be0O629T, r = 0.999, respectively. The Qio temperature
coefficients
for these reactions, calculated
from the relationship
Qio = efiT,
were found to be 2.05, 1.91, and 1.88, respectively,
for
the three enzymes. The use of temperature
correction
factors for converting a measured activity at one temperature to a calculated activity at a second temperature
is not recommended
because of the obvious enzymedependent nature of the factor, the reaction dependence
of the factor, and the effect of temperature denaturation
of the enzyme. However, the above exercise is useful in
predicting
the experimental
magnitude of the activity
variation as temperature
fluctuates.
The propagation
of error analysis of an exponential
expression, v = AeBT, results in an expression for the
relative standard deviation of activity with respect to
the standard deviation of temperature:
(12)
where S,,/V is the relative standard deviation of activity
and B is the temperature
coefficient
of the respective
enzyme reaction. For LDH, ALT, and AST the error
expressions with respect to temperature
become:
SVLDH
=
(O.O7l8)ST
=
(O.O649)ST
=
(0.0629)ST
VLDH
5VALT
VALT
SV
VAST
If the temperature
variation
(Sr) for the reaction is
±0.1 #{176}C,
then the relative standard deviation
of the
three enzyme activity
and ±0.63%.
Experimental
assays will be ±0.72%, ±0.65%,
Measurement
to a computer.
13
w
lx
lx
u-i
ci.
w
I-
of Temperature
Error
This
permits
simulta-
neous within-cuvette
temperature and activity measurements to be made if so desired.
A plot of temperature vs. time for several rotor
transfers is shown in Figure 7. The time for temperature
equilibration
to be attained within the cuvette after
transfer of room temperature reagents (e.g., 21 #{176}C)
was
approximately 150 s, and it required approximately 90
s to come within 0.1 #{176}C
of equilibrium temperature for
these intial experiments. A rotor modification has re-
5
TIME
R0105’3g
4.
3
0
1
$
5
0 II
duced this equilibration
5
0.5144105’S
Fig. 7. Temperature vs. time at 30 #{176}C
for within-rotor
rotor-to-rotor thermistor-measured temperature variation
and
time. Enzyme assays such as
creatine kinase (CK), AST, and ALT require preincubation times of 150 s or longer before linearity of reaccaused by lag phase or endogenous substrate is sufficient to assure good assay conditions.
However, LDH-L and hydroxybutyrate dehydrogenase
do require absorbance measurements as early as 30 s
after initiation of reaction. This becomes the limiting
time for thermal equilibration
within a cuvette. The
temperature variation (ST) within a cuvette over a
10-mm reaction period after thermal equilibration was
±0.02 #{176}C.
The cuvette-to-cuvette
variation (ST) within
a rotor was ±0.07 #{176}C.
Data on equilibrium
temperature
variation
for10 runs duringfourdays provideda mean
cuvette temperature of 30.03 #{176}C
and a rotor-to-rotor
temperature variation of ±0.08 #{176}C.
In general, the initial reaction data would not be acquired before thermal equilibration of the cuvette retion velocity
action
There are two levels of temperature uncertainty to
be considered for a centrifugal analyzer, within-rotor
variation and rotor-to-rotor
temperature variation.
Temperature in the rotor can be measured either by the
use of a thermistor calibrated with an NBS traceable
thermometer or by the use of a thermal optical solution
(16). A thermistor
was chosen for this work. The
thermistor was placed in the cuvette during measurements and the analog signal from the thermistor circuit
was interfaced
0
contents.
Thus
the rotor
thermal
conductance
heatingconditionsare adjusted to
assure essential equilibration
of temperature of the
reaction mixture by the 30-s limit. However, it is of interest to examine the effect of temperature equilibration
on the variation of temperature (ST) during a reaction
under “worst-case”
conditions.
The curves in Figure 7
show that the temperature
is within 0.2 #{176}C
of equilibrium at 45s after transfer, within 0.1 #{176}C
at 90s, and at
±0.02 #{176}C
of equilibrium by 150-180s. If an initial data
point were obtained at 45 s and 11 more data points
acquired at 20-s intervals thereafter, the mean temperature variation of the reaction during this 3.75-mm
and characteristic
reaction
time would
be ±0.06
#{176}C,
an acceptable
tem-
peraturevariation.
Pathlength
Uncertainty
The final individual
error component that must be
discussed is pathlength
variation.
For multicuvette
photometers and systems with individual cuvettes such
as the Du Pont aca, the cuvette variation
is a random
error component
as wellas a systematicerrorcompo-
CLINICAL CHEMISTRY,
Vol. 22, No. 9, 1976
1445
nent. If the rotor is 1 cm in thickness, the rotor cannot
vary more than ±0.005 cm (±0.0002 inch) to maintain
a ±0.5% variation.
CVB==
CVT
The Projection of Enzyme Activity Random Error
An error expression was derived in the first section
of this paper that related the relative standard deviation
of enzyme activity to various random error parameters
associated with a spectrophotometric kinetic analyzer.
The individual error components were considered in the
next section, and these were discussed with reference
to a prototype centrifugal analyzer. This section will
utilize the individual error components to predict the
error in enzyme activity measurement by use of the
Miniature Centrifugal Analyzer (MCA) for a reaction
in which absorbance is decreasing (e.g., AST) and for
a reaction of increasing absorbance (e.g., LDH or
CK).
The expression developed in the first section is
CV(U/liter)
[(CV)2
+ (CVv1)2
Each of the individual error components was discussed,
and determined experimentally for the MCA system,
in the second section. The error of an enzyme activity
measurement in range of 50 U/liter, 150 U/liter and 500
U/liter can be considered for an enzyme reaction that
starts with a high absorbance (e.g., 1.2-1.6) and one that
starts low (e.g., 0.2-0.3). A sample volume of 20 Mlin 200
Ml total reaction volume can be assumed and the enzyme
reaction will be run for a total time of 4 mm. The temperature uncertainty within a rotor will be less than 0.1
#{176}C
(cf. second section) and the pathlength variation will
be 0.5%. The standard deviation and relative standard
deviation of photometric error and sample-reagent
pipetting can be obtained from Figures 1, 5, and 6, respectively. The activity error for LDH and AST at 50
U/liter can be calculated as follows.
(50 U/liter)
The uncertainty
in photometric
error: The relative
standard deviation of absorbance change is C V
=
SA/A,
where S
=
[(SAl)2 + (SA2)2]”2. For a
downward going reaction at 340 nm, with a starting
absorbance of 1.6 A, the uncertainty of absorbance is
±1.1 X 10 A. At 50 U/liter the absorbance A1 A2,
because the absorbance change over a 4-mm period is
0.0622 A. Therefore, the worst-case relative standard
deviation of absorbance (C V)
is:
CV
=
The remainder
lows:
SAV
=
1.1 X iOv’
0.0622
=
of the error components
0.0033
V5
CVVT
1448
=
CLINICAL CHEMISTRY,
T
=
0.0030
Vol. 22, No. 9, 1976
(0.0649)(.01)
=
0.0250
are as fol-
0.00649
=
[(0.250)2 + (.0033)2
+ (0.030)2 + (.005)2 + (.00649)211/2
CVactivity
=
(0.0266)(100)
=
2.7%
The relative standard deviation for an AST sample run
on the MCA system is 2.7%. The photometric error
contribution is 2.5% of the total error.
The above assumes the worst-case photometric
measurement, which is a data-averaged two-point kinetic analysis. During a routine activity assay, error will
improve because of the additional averaging provided
by the fitting routine. However, this improvement is of
diminishing returns and noise problems cannot be totally solved by statistical methods alone (9).
Error of LDH Activity
+ (CVv5)2
+ (CVB)2 + K2ST2I’12
Error of AST Activity
=
0.005
(50 U/liter)
The photometric error can be calculated in a similar
manner but the standard deviation of a reaction with
an initial absorbance of 0.25 A is about ±2 X 10 A or
about five times lower.
CVA
=
2
=
0.0045
The other error terms are the same as the previous example, except temperature, which is:
=
[(.0045)2 + (.0033)2
+ (.0030)2 + (.005)2 + (.00718)211/2
=
(0.0108)(100)
=
1.1%
The relative standard deviation decreases more than
twofold because of the fivefold reduction of photometric
error. In this example, all error components were contributing significantly to the total error. The greatest
contribution was due to temperature uncertainty. The
random error in activity for the two enzymes are calculated for different activities, temperature uncertainty,
and path length error to demonstrate the relative contributions of different errors on the overall uncertainty
of enzyme activity measurement (Table 4). The most
significant error contributions are photometric error for
AST at low activity and temperature when controlled
to ±0.2-0.3 #{176}C.
It should be noted from Table 4, for
example, that the establishment of stringent requirements which demand pipettors of pathlength tolerances
to be better than 0.5%, while the “state of art” temperature control for most instruments remains in the range
of 0.1-0.3 #{176}C
will result in an insignificant improvement
of system performance.
Experimental Measurement of
Enzyme Activity
Error
Verification of the above projected error estimates
of enzyme activity as measured by the MCA, was first
accomplished by use of premix experiments to eliminate
Table 4. Effect of Various Error Components on Projected Random Error of Enzyme Activity
Enzyme
assay
Activity,
U/liter
CV
CVv
%
%
CVVS*
%
CVB*
%
Calculated
% relative
standard
±#{176}C deviation a
S1
AST
LDH
AST
LDH
AST
LDH
50
50
150
150
500
500
2.5
0.5
0.8
0.2
0.3
0.1
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.5
0.5
0.5
0.5
0.5
0.5
±0.1
±0.1
±0.1
±0.1
±0.1
±0.1
2.7
1.1
1.1
1.0
0.9
1.0
LDH
LDH
50
50
0.2
0.2
0.3
0.3
0.3
0.3
0.5
0.5
±0.05
±0.1
0.9
1.6
AST
LDH
AST
AST
AST
500
50
500
500
50
0.3
0.2
0.3
0.3
2.5
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.5
0.5
0.5
0.5
±
0.05
±0.3
±0.2
±0.3
±0.3
0.8
2.3
1.5
2.1
3.2
LDH
50
0.2
0.3
0.3
1.0
±0.1
1.4
LDH
AST
AST
50
50
0.2
2.5
2.5
0.5
0.3
0.5
0.5
0.3
0.5
1.0
P.O
±0.1
±0.1
1.5
2.8
2.9
aCV(U,lite)
=
E(CVA)2
+
(CV Vt)2
+
(CV Vs)2
+
(CVB)2
+ K252J
Parameter varied
Photometric
(CVA)
Error
J
‘
I
Temperature
Photometric(Sr)
(CVA)and
Error
‘I
,
Pipettor (CVv1)
Pathlength
(CVB)
Pipettor (CVv3)
402
pipettor contributions. Second, the pipettor module was
included to obtain a measurement of the total system
random error.
The premix experiment was performed as follows.
Serum at AST and CK reagent ratios of 20 Ml to 200 Ml
of reagent was chosen and 400 Ml of control sera was
added to a 4 mol/liter concentration of enzyme reagent
and the two were gently mixed. The resulting reaction
mixture was rapidly pipetted into a rotor. The rotor was
then brought to a temperature of 30 #{176}C
and AST and
creatine kinase activity were determined. The important element of this experiment is that all reactions in
the rotor originated from a common pipetting and
mixing of sample and reagent. The reactions proceed
from the same initial reaction time and as long as substrate exhaustion is not allowed to occur, the precision
of the in-run electro-optical rotor system can be obtained. The data are shown in Table 5. The measured
CV (relative standard deviation) is listed prior to the
predicted CV. These data are in excellent agreement
with the previous projected random error data.
The data from the total system analysis of AST activity are given in Table 6. Again there is a close correlation between predicted and measured random error
Table 5. Enzyme Activity
Data (Premix Experiment)
sx
Sample
Reaction time
1. AST (SGOT),
patient serum
2. AST (SGOT),
turbid control
serum
3. Creatine kinase (CK)
(contrQl serum)
a Parameters used for predicting
i.A/time
285
(A)
Mean activity,
U/liter
0.075
105
(B) 210
(C) 450
240
5.4
(1 SD).
U/liter
CV, %
(n = 19)
0.4
7.2
Predicted
cv,
7.6
0.0268
54.6
1.6
2.9
3.4
0.0529
0.1125
0.2465
53.5
53.8
218
1.0
0.7
1.0
1.9
1.3
0.5
1.9
1.4
0.6
CV.
CVactjvjty
=
((CV)2
+ (CVB)2+
(0.0678)2
(A) Path length variation:
(measured variation)
AST,
CK3
(B) Std. deviation of absorbance:
(data obtained from photometric error data)
(C) Temperature
variation:
AST, = 0.4 X 103A
AST2=O.6X
103A
CK3=O.4X
103A
AST, 2 = 0.1 #{176}C
CK,
0.05#{176}C
+ ST2
I“
0.79%
0.49%
2 =
CLINICAL CHEMISTRY,
Vol. 22, No. 9, 1976
1447
Table 6. Enzyme Activity Data (Loaded Rotor)
Sample
Reaction time
AST
(control serum)
435s
Reaction
conditions,
a Note: Difference
.A/time
.1257
20-iI sample, 200-MI total; volume
between
premix experiment
CVenzyme
Parameters:
SD absorbance
% rel. std. dev.
SD of temp. =
% rel. std. dev.
% rel. std. dev.
=
52.4 U/liter
temperature,
activity
=
((CVA)2+
CV
= 19)
1.3%
0.5 cm; enzyme factor,
Predicted
cva
1.2%
3022.
(CVB)2
+
K2(ST)2]
0.79%
activity
CV enzyme activity = 1.1%
= ((CV)2
+ (CVB)2 + K#{176}(ST)2
+ (CV)2
+
(CV5)2]
“
0.6 X 103A
of pathlength = 0.79%
±0.1 #{176}C
of sample pipette = 0.33%
of total volume = 0.30%
=
of AST activity. Data on the difference between pipetted and premix experiments are also provided in Table
6, and show the insignificant contribution
of the pipettor error in comparison to photometric and temperature error.
Error
Estimation of systematic error has been included but
not discussed. The individual components of systematic
error (bias) are additive and can either be negative or
positive. As a final approximation,
if one sums the
random error components with the stray light, bandwidth, and pipettor linearity errors, then for AST at
borderline activities, the total error (system and random) is 5.0% while at high activities it is about 2.5%
when the limits are held to the tolerances of Table 4.
Summary and Conclusions
The major emphasis of this paper has been on the
derivation of a simple error expression that relates the
random error of enzyme activity as measured by a
photometric analytical system to the individual error
components of that system. The expression for the
calculation of enzyme activity has been the model, and
the approach used by Pardue et al. (9) for the evaluation
of random error components has been the guideline.
The thrust of the discussion has been to break the analytical system down into its component and subcomponent parts as they relate to random error. Measurement of the magnitude of these error components enables one to establish the relationship of total error to
these individual components. The relationship of total
error to the variation of tolerances of each of these
components under different conditions of performance
therefore permits the unimpassioned rational establishment of error limits and performance characteristics
for such analytical systems. The example used was the
projection of random error of enzyme activity for different levels of LDH and AST at different analytical
1448
(n
0.7 U/liter
30 #{176}C;
pathlength,
CV enzyme activity
Systematic
SD)
and loader rotor experiment
CVenzyme
Parameters: SD absorbance = 0.6 X 103A
% rel. std. dev. of pathlength
SD of temp.
±0.1 #{176}C
S(1
Mean activity
CLINICAL CHEMISTRY,
Vol. 22, No. 9, 1976
=
1.2%
systems tolerance limits. A prototype MCA system was
used as the point of discussion.
Two conclusions can be made from the discussions
in this paper. First, the error expression can be thought
of as a balance. On the left-hand side of the equation
(balance) is the relative standard deviation of enzyme
activity (random error) and on the right-hand side are
the various individual components that add up to the
overall error. The limit of acceptable random error
(CViiiter)
dictates what the limits of the individual
components must combine to be, to fall within acceptable limits of tolerance. The limit of acceptable total
random error must be established for the most meaningful dictation of what the limit of individual error
components must be. Maclin et al. (8) have introduced
the concept of error budget. This concept is reinforced
by the error expression developed herein. The society
committees must establish the acceptable error limits
for the relative standard deviation of enzyme activity.
Thus they establish the error budget. The manufacturer
then must meet that error budget by establishment of
proper error limits of the individual system components.
The user can easily verify whether this has been
achieved by simple measurement of relative standard
error of enzyme actmvity at various enzyme activities.
Second, temperature control will be a major error
source, followed by photometric error at low enzyme
activities of the aminotransferases or other reactions
monitored at higher absorbances. Long-term temperature control to better than ±0.1 #{176}C
for most analytical
systems is difficult to achieve and will more realistically
be ±0.2-0.3 #{176}C.
This means a limit of relative error for
enzyme activity measurements over a period of time of
0.7% to 2% without consideration of any other error
components. Therefore, as an example, a requirement
that the sample and reagentpipettorerrorlimitsfor
random error be reduced from 0.5% to 0.25% would have
an insignificant effect on the total random error (see
Table 4). The establishment of error limits without
careful consideration
error components
of the relationship
of individual
to the total random error can result
in significant escalation of system cost without noticeable improvement
in system performance.
The discussion of error as developed by Macun and
Pardue
and their colleagues
is useful for the proper es-
tablishment of limits of photometric instrument performance. It is hoped that the development of a simple
error expression based on the enzyme activity equation
will be useful for user and instrument maker alike in the
evaluation and establishment of performance limits.
counting
and direct current measurements
for quantitative
photometric
methods.
Anal. Chem. 44, 785 (1972).
6. Ingle,
molecular
Chem.
spectro-
J. D., Jr., and Crouch,
S. R., Evaluation
of quantitative
absorption
spectrophotometric
measurements.
Anal.
44, 1375 (1972).
7. Maclin, E., A systems
kinetic enzyme analyzer.
analysis
of GEMSAEC
precision
used as a
Clin. Chem. 17, 707 (1971).
8. Maclin, E., Rohlfing,
D., and Ansour, M., Relationships
between
variables
in instrument
performance
and results of kinetic enzyme
assays-a
system approach.
Clin. C/tern. 19,832 (1973).
9. Pardue, H. L., Hewitt, T. E., and Milano, M. J., Photometric
errors
in equilibrium
and kinetic analyses based on absorption
spectroecopy.
Clin. Chem. 20, 1028 (1974).
P. R., Data Reduction and Error Analysis for the
McGraw Hill, New York, N. Y., 1969, chap. 4.
10. Bevington,
Physical
Sciences,
References
11. Burtis, C. A., Mailen, J. C., Johnson,
W. F., et al., Development
of a Miniature
Fast Analyzer.
Clin. Chem. 18, 753 (1972).
1. Widdowson,
G. M., Performance
specifications
for instruments
used in enzyme activity measurements.
In Proceeding
of the Second
International
Symposium
on Clinical Enzymology.
N. W. Tietz and
A. Weinstock,
Eds., In press (AACC).
12. Burtis, C. A., Johnson,
W. F., Mailen, J. C., et al., Development
of an analytical
system based around a Miniature
Fast Analyzer. Clin.
C/tern. 19,895 (1973).
2. Sims, G. M., Alternative
viewpoints
for instruments
used in enzyme activity
for performance
measurements.
objectives
Ibid.
3. Ingle, J. D., Jr., Comparison
of the precision of normal and precision spectrophotometric
technique.
Anal. Chem. 45, 861 (1973).
4. Ingle, J. D., Jr., and Crouch, S. R., Signal-to
noise ratio theory of
fixed time spectrophotometric
reaction
rate measurements.
Anal.
Chem. 45, 333 (1973).
5. Ingle,
J. D., Jr., and Crouch
S. R., Critical
comparison
of photon
13. Ayres,
G. H., Evaluation
of accuracy
in photometric
analysis.
(1949).
14. Dixon, M., and Webb, E. C., Enzymes,
Academic Press, Inc., New
York, N. Y., chap. 4.
15. Henry, R. J., Clinical Chemistry: Principles and Techniques,
1st
ed., Hoeber, New York, N. V., 1964.
Anal. Chem. 21,657
16. Bowie, L,, Esters, F., Bolin, J., and Gochman,
N., Development
of an aqueous temperature-indicating
technique
and its application
to clinical laboratory
instrumentation.
Clin. Chem. 22,449 (1976).
total stray light intensity
=
=1#{176}
Appendix A. Computer Evaluation of Stray Light
true reference cuvette intensity
General
The
current
generation
of centrifugal
analyzers,
havingthe filter
between therotorand photomultiplier
tube, is inherently
insensitive
the rotor. When subsequent transmission
readings on
the same revolution
are reduced by this amount, the
result is to compensate completely for steady-state stray
light entering the system below the rotor. Stray light
entering the system above the rotor is, of course, additive with the source lamp and thus of no consequence.
The source of stray light which is not fully compensated
is that which enters the system from within the rotor
(e.g., transmitted via internal reflections within the
cover lens). The stray light represents identical additive
terms to both the “true” sample and reference intensities, i.e., for the reference cuvette,
=
It
+ Js
(1)
(2)
where the superscripts m, t, and s representmeasured,
true, and stray light respectively.
The stray light coefficient
is defined as:
(4)
Am=ac+log(1+s)_1og(1+s)
which the true absorbance has been set equal to cxc
where c is the concentration
of the absorbing material
and a is the product of the molar absorptivity and the
in
pathlength.
The stray light coefficient
may then be
determined
by comparing measured absorbances with
known absorbances of carefully prepared samples.
Determination
of a and S
An iterative method of nonlinear
regression is used
in determining
these quantities.
Briefly, the method
consists of estimating the quantities a and S, computing
the best estimate of correction terms, iXa and itS, and
finally repeating the process until a and S are determined within acceptable limits.
The correction terms are obtained from a multivariant, Taylor’s series expansion of A in which only terms
up to first order in LS and Za are retained. The expansion
and for a sample cuvette:
15m=15t+Is
The absorbance then takes the form:
to effects of stray light.
This situation arises from the use of self-compensating
dark-current readings taken on each revolution. The
dark-current
readings are taken on the darkened region
between two adjacent curvettes and thus are actually
a measure of the ambient stray light that exists below
Irm
Irt
takes the form:
(5)
AAo+Aaa+As.S,
where A0 is the function
A evaluated at initial values of
S and a, i.e.,
A0 = a0c + M ln(1 + S0)
-
M ln(1 + SoecIM).
CLINICAL CHEMISTRY,
Vol. 22, No. 9, 1976
(6)
1449
Act is
(Ai
having the form:
-
Aoi)Aai
z=1
Act
while A8 is
aS
Aa
c
=
1L1+SoecMJ1
1,
-
i=1
having the form:
A2
=
1
=
(A
(7)
M [(1 + S)(1 + SoeM)]
=
(8)
To obtain zS, za the least-squares difference between
the data, A4, and A1 is minimized.
The subscript i indicates that these values are to be associated with the
ith value of the uniformly distributed independent
variables C4. For n data points the least-squares difference is:
[ha]
iLSj
A8Ai
Agj
Since this is a two dimensional matrix, it can quite easily
be inverted to produce La and S. That is
a
-
=
(A
-
Aoj)AajAsj2
(A1
-
-
Aoj)AsjAsjAaj
D
(14)
-
(9)
j=1
The minimum
and
of this function occurs when both
and
--
t9&x
(10)
--
S
=
>(AI
-
Ao)A5A02
-
D
(A
-
are exactly equal to zero. That is, when
(A,
an d
(13)
-
A1)A81
0
Aoi)AaiAsiAai
(15)
(11)
where
= >Aaj2 new
a52estimates
(AsjAof a, ai)2
With theseD values,
and S are (16)
ob-
1=1
tained as follows:
(A4
-
A)A04
0
(12)
These lead to the following equations, expressed here
in matrix form:
1450
CLINICAL CHEMISTRY,
Vol. 22, No. 9, 1976
SR+1
=
aR
+ za,
(17)
=
SR + zS.
(18)
The process is then repeated.