CO MMU NICAT1ONS
Instrument Response Time in an Analytical System €or Continuous Glucose Measurement C. Robert Baillodl and William C. Boyle Department of Civil Engineering, University of Wisconsin, Madison, Wis. 53706
To interpret a continuous record of glucose concentration plotted by the Technicon AutoAnalyzer, it is necessary to assess the effect of the instrumental lag on the recorded data. The effect of this lag is demonstrated by comparing the output record produced in response to a known input function with the input function itself. The known input function consisted of a 0 to 10 mg. per liter step function followed by a n approximate zero order decrease in glucose concentration. Under these conditions, the concentration by which the output record lagged the input record approached an asymptotic value, which may be estimated according to the principle of superposition. H
C
ontinuous monitoring of glucose concentrations in microbiological culture medium by means of the Technicon AutoAnalyzer has been employed by Jeris and Cardenas (1966) and by Baillod (1968) to measure glucose uptake rates by biological systems. In the interpretation of these data, some consideration must be given to the response time limitations of the instrument. The effect of instrument lag time on the recorded glucose concentration profile is illustrated for the case where the uptake rate is approximately zero order.
H H
(e) = 0 , for e < o (e) = 1, for e = 0
H
(e) =
1 - k , for B
>
(no glucose present) (glucose added a t time zero, C,) 0 (zero order glucose disappearance)
(3)
where H (0)
= CjC,, dimensionless input concentration glucose concentration in the sample stream, mg. per liter Ca = glucose concentration in the same stream at time zero, mg. per liter 0 = t / t o ,dimensionless time (used to refer to input functions) t = time, minutes t o = the time constant of the instrument, minutes K = at ,/Co, a dimensionless rate constant u = volumetric rate of glucose disappearance in the sample stream, mg. per liter minute
C
=
To simulate this signal, a baseline sample was drawn from a stirred beaker containing phosphate buffer (1 .O gram per liter of K2HP04,0.5 gram per liter of K H j P 0 4 , pH = 7.1). At time zero, a quantity of glucose was added to the beaker in order to give a concentration, Co.Also, at time zero, addition of phosphate buffer to, and withdrawal of, solution from the stirred beaker was begun (Figure 2). A second AutoAnaljzer
Experimentul
The method of glucose analysis employed in this study was specific for P-D-glucose, and was based on the oxidation of glucose to gluconic acid by glucose oxidase, and the subsequent reduction of H20, by a peroxidase, according to the following simplified reaction scheme. glucose
H201
+ 0, $- H2O-$'';i;Z+
H?Oz
+ reduced chromogen
+ gluconic acid (1)
f
oxidized chromogen (color)
+ H20
(2)
The AutoAnalyzer flow diagram of Jerk and Cardenas (1966) was extensively modified in order t o measure glucose concentrations of from 0 to 10 mg. per liter. The modified flow diagram is shown in Figure 1. The enzyme reagent was prepared by adding two vials of chromogen (Worthington Biochemicals, Freehold, N.J., cat. no. CHGX4) and two vials of Glucostat (Worthington, cat. no. X4G) to 400 ml. of distilled water. The continuous-flow filter and dialyzer were necessary to eliminate bacterial cells from the sample stream during the actual glucose uptake experiments (Baillod, 1968). To test the response speed of the instrument, the response to a known input signal was observed. The input signal supplied t o the system can be stated as Present address, Department of Civil Engineering, Michigan Technological University, Houghton, Mich. 4993 1
I G i r a r g e expander
Figure 1. AutoAnalyzer flow diagram for continuous glucose analysis DMC MC ps
= = =
double mixing coil mixing coil pulse suppressor
F.
c=o
Figure 2. Continuous stirred reactor simulation Volume 3, Number 11, November 1969
1205
tion is known. This mathematical analysis, based on the principle of superposition, has been used by Mueller et al. (1967) to assess the effect of instrument lag o n oxygen uptake rates measured by a galvanic cell oxygen probe. Applied to a linear instrumental system, this analysis predicts the instrument output record produced in response to a known input function as
Asymlotic lag = t,a : 0086 mg/l
2 ’.
where
G(T)= dimensionless output concentration plotted by the 0
I
5
I
C
I
I5
I
20 25 Time, mindtes
30
35
40
Figure 3. AutoAnalyzer response to linear input signal
pump was employed for this. For the arrangement depicted in Figure 2, an unsteady state material balance (Baillod, 1968) gives the concentration in the vessel as a function of time as
recorder, C/Co B(r) = dimensionless output concentration produced in response to a step input function H(0) = general dimensionless input concentration 8 = dimensionless time-integrating variable, used for input functions T = dimensionless time, t/to,used for output functions H(0) = H(0) evaluated at 0 = 0. When the instrument was subjected to a step input function of 0 to 10 mg. per liter, it was observed that the instrument output function could be described by B(T) = 1
where
V O = volume of fluid in the vessel at time zero, ml. Fo = rate of inflow, ml. per minute Fl = rate of outflow, ml. per minute. Equation 4 indicates that, if FI = 2F0, then C/Cowill be a linear function of time. Slight variations between different pieces of pump tubing, however, resulted in values of Fl slightly less than 2Fo, so that the rate of change of concentration in the test vessel decreased slightly with time. The actual input function therefore, calculated with Fo = 2.45 ml. per minute, Fl = 4.73 ml. per minute, Co = 10 mg. per liter, and Vo = 100 ml., was C/Co = (1
- 0.0228t)’~oi5
Results
The solid lines of Figure 3 show the input function given by Equations 3 and 5, along with the output record plotted by the recorder. T o compare the input and output functions directly with the same time scale, the transport lag through the AutoAnalyzer was subtracted from the output time scale. The data indicate that the measured output function consistently lags the input function. However, the concentration by which the output lags the input appears to approach an asymptotic value of approximately 0.1 mg. per liter. In experiments with a pure culture of Zoogloea ramigera (Baillod, 1968), the glucose uptake rate usually decreased as the glucose concentration fell below 2 to 5 mg. per liter. For these cases, the concentration profile plotted by the recorder will tend to converge with the actual input concentration profile, and the concentration by which the output function lags the input function will decrease even further. Therefore, under appropriate conditions, the effect of instrument lag may be neglected when interpreting the final portion of the glucose concentration profile plotted by the AutoAnalyzer recorder. Discussion
It is possible to describe mathematically the instrument output function produced in response to a known continuous input function if the instrument response to a step input func1206 Environmental Science & Technology
-
e-‘
(7)
The instrument time constant, t o , was determined by fitting Equation 7 to the recorder output produced in response to a step input function. The value of t o was approximately 0.35 minute. Combining Equations 3, 6, and 7 predicts the transient output function as G(r) = (1 which, in the case of
7
+ K ) (1 - e-’)
- Kr
(8)
>> 0, reduces to
G(r) = 1
+ K - KT
(9)
The concentration by which the output lags the input is obtained by subtracting the input signal from the output signal at T >> 0. G(T) - H(T)
=
C,,,/Co
=
K
(10)
Thus, the asymptotic lag concentration is given by Clag=
at0
(1 1)
The dashed line in Figure 3 shows the output function predicted by the principle of superposition, as given by Equation 8. By comparing the predicted output function with the measured output function, it can be seen that the recorder “overshoot” is somewhat greater than that predicted by the principle of superposition. However, the asymptotic concentration of 0.086 mg. per liter predicted by the principle of superposition appears to agree quite well with that given by the recorder output. Therefore, it is reasonable to conclude that, under these conditions, Equation 11 gives a valid estimate of the concentration by which the AutoAnalyzer output function lags the input function. Literature Cited Baillod, C. R., Ph.D. thesis, University of Wisconsin, Madison, Wis., 1968. Jeris, J. S., Cardenas R. R., Jr., Appl. Microbiol. 14, 857-64 (1 966). Mueller, J. A., Boyle, W. C.. Lightfoot, E. N., Microbiol. 15,674-6 (1967). Receiced for reciew April I , 1969. Accepted July 22, 1969.
