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Computerized estimation of lactate threshold
COMPUTERS
AND
BIOMEDICAL
Computerized
RESEARCH
19, 481-486 (1986)
Estimation
of Lactate Threshold
MARTHA A. LUNDBERG,* RICHARD L. HUGHSON,? KENNETH H. WEISIGER,? RICHARD H. JONES,* AND GEORGE D. SwmsoNt *Department of Preventive Medicine and Biometrics, and TDepartment of Anesthesiology, School of Medicine, University of Colorado Health Sciences Center. Denver, Colorado 80262 Received April 9, 1986
Traditional approaches to estimating a lactate threshold during a progressive exercise test have utilized visual inspection of the data. We describe a computerized approach which utilizes a log-log transformation to yield two approximately linear segments. Linear regression lines are fit to these segments and the intersection of the two lines yields an estimate of the lactate threshold. An approximate 95% confidence interval is also generated. o 1986 Academic
Press. Inc.
Changes in blood lactate concentration during a progressive exercise test to exhaustion have suggested the possibility of a threshold-a point where blood lactate increases dramatically. Attempts to estimate a lactate threshold (LT) have utilized simple visual inspection of lactate data plotted against time or 02 consumption. This subjective approach has led to LT estimates that are dependent on the particular observer (6, 20) and has contributed in part to the controversy as to whether LT exists at all (3). Clearly a more objective approach is needed. Recently, Beaver et al. (2) have suggested that a log-log transformation on the data enhanced the appearance of LT. They described a method of estimating LT that utilized an observer to determine a division point such that the data to the left of this point belonged to a low slope linear segment and data to the right belonged to a high slope linear segment. Linear regression lines were determined for each of these two segments, with the intersection of the two lines being the estimate of the LT. Although this approach is less subjective than simple visual inspection of the raw data, LT estimate is directly influenced by the choice of the division point. The purpose of this paper is to describe a computerized approach for estimation of LT using the log-log transformation proposed by Beaver et al. (2). This computer method does not use an observer and is thus compktely objective. 481 00104809186 $3.00 Copyright 0 1986 by Academic Press, Inc. All rights of reproduction in any form reserved.
482
Furthermore, LT.
LUNDBERGETAL.
the method generates a confidence interval about the estimate of METHODS
The approach utilizes the log-log transformation suggested by Beaver et al. (2). We then utilize a computer algorithm (7) to fit two lines to the log-log data that join at a breakpoint. This breakpoint is varied within the range of the loglog data and LT estimate is determined as that breakpoint for which the overall residual sum of squares (RSS) is minimized. If x0 is the position of the unknown breakpoint, the equations of the two lines are : Y = PO + PlX
x 5 x0
Y
x
=
P2
+
Lll
P3x
>
x0
When x = x0 the lines must intersect. Thus, PO
+
PIXO
P2
=
PO
=
P2
+
/33x0,
El
-
P3xo.
[31
yielding The parameter
+
PIXO
p2 can now be eliminated
from [ 11 yielding.
Y = PO + PIX
Y
=
PO
+
r41 L ~>
PlXO
+
P3b
-
x
x0)
>
x0.
The method involves searching for the value of x0 that minimizes RSS. This is actually a four parameter regression involving three linear parameters, PO, /3t, and & and the nonlinear parameter x0. The breakpoint parameter x0 is varied over the range of the log-log data. For each value of x0 the linear parameters are determined by matrix linear regression methods (5). Thus, given a trial x0, from [4] the matrix equation is given by YI
1
Y2
lx:!
Y3
1x3
Yj Yj+l
Tz
0
Xl
0 0
. .
. .
1
Xj
1
X0
. . .
. . .
1 x0
7 PO
0 Xj+l
-
PI x0 P3
_
x, - no
where the jth data point is the last data point included ment. Equation [5] can be more compactly written as
in the lower line seg-
COMPUTERIZED
LACTATE
THRESHOLD
ESTIMATION
Yl = [Xl PI. The conditional
least-square
solution
of the parameter
p = (XTX)-‘XTY
483
[61 vector given x0 is [71
with RSS = YTY - /PXTY.
F31
Therefore, varying x0 over the range of the log-log data yields the value of x0 (XOmin) for which RSS is minimum (RSS,i”). This is LT estimate. The corresponding mean square error (MSE) is given by RSSnin MSE = n-4 where there are 12data points and four parameters. Since x0 is a nonlinear parameter, statistical tests on x0 are approximate rather than exact (4). An approximate confidence interval can be determined by determining the test statistic [91 where RSS is the residual sum of squares at a particular x0 and RSS,i” corresponds to the value of xomin. This test statistic is approximately distributed as an F statistic with one and n - 4 degrees of freedom (5). An approximate 95% confidence interval is then determined as that x0 lower than XOminand that x0 higher than xomin where the test statistic is equal to FlTn-4 at the 0.05 level. By plotting RSS as a function of x0, this 95% confidence interval can be graphically determined or alternatively determined via a computer algorithm. RESULTS
Figure 1 indicates the lactate versus 02 consumption data for subject number 1 from the Beaver et al. study (2). Note in Fig. la, there is a relatively flat response phase followed by a phase where the lactate concentration begins to rise. This characteristic is more evident in Fig. lb where the log lactate concentration versus log O2 consumption is plotted. Two approximately linear segments of data are now evident. The two segment linear regression model is shown superimposed on the data of Fig. lb. The intersection of the two lines for which the RSS is a minimum is the estimate of LT. Figure 2 indicates the plot of RSS increase as a function of x0. Note that the 95% confidence interval is determined when this increase reaches the appropriate F statistic. Table 1 summarizes the results for all 10 subjects from the Beaver et al. (2) study. Note that LT using our computerized technique yields results similar to the subjective approach of Beaver et al. (2) when an experienced observer is used. Furthermore, our computerized technique yields the approximate 95% confidence interval as indicated.
484
LUNDBERG
“02
ET AL.
Log
(L/mid
Vo,
(L/mid
FIG. I. Lactate versus O2 consumption data for subject number one of the Beaver et ul. study (2 ). (A) The linear plot. (B) The log-log plot.
DISCUSSION
At some point in a progressive exercise test, the ability to metabolize lactate lags behind the rate of lactate production and the blood lactate concentration begins to rise (3). This point is termed the lactate threshold (LT). Accompanying this change in blood lactate concentration is an equimolar change in hydrogen ion [H+] which is buffered primarily by bicarbonate. The resultant increase in CO* production from this buffering process is met by an increase in ventilation. A ventilation threshold (VT) has often been described at the point of increased CO2 production (9). LT and/or VT have been suggested to represent the upper limit for most normal daily activities in the typical adult (9). Selection of the thresholds for ventilation (VT) or blood lactate (LT) has been done primarily by subjective assessments of plotted data. These subjective estimates are dependent on the observer (6, 10) and tend to show considerable variation with repeated evaluation of a data set by the same observer (6). This uncertainty of the estimates has complicated attempts to characterize the asso-
“02
(Llmin)
FIG. 2. Plot of the change in residual sum of squares (RSS) as a function of the intercept of the line segments shown for the data of Fig. I. The approximate 95% confidence interval is indicated.
COMPUTERIZED
LACTATE
THRESHOLD
TABLE
ESTIMATION
485
1
COMPUTERIZED LACTATE THRESHOLD ESTIMATES WITH THEIR CORRESPONDING95% CONFIDENCE INTERVALS FOR THE 10 SUBJECTS OF THE BEAVER et al. STUDY (2) Subject No. 1 2 3 4 5 6 7 8 9 10 Average SD
LT-Objective (Urnin) 1.465 2.351 1.742 1.344 1.250 1.436 1.540 1.510 1.034 1.562 1.52 235
95% C.I. (L/min) 1.347 2.181 1.624 1.221 1.026 1.300 1.327 1.157 0.944 1.377
1.571 2.452 1.861 1.507 1.418 1.551 1.721 1.697 1.188 1.721
LT-Subjective (L/min) 1.54 2.28 1.72 1.39 1.29 I.35 1.49 1.64 1.09 1.58 1.54 +-32
Note. The subjective LT estimate using an experienced
ob-
server is also indicated. LT indicates lactate threshold. 95% C.I. indicates the approximate F test 95% confidence interval. LTObjective indicates the lactate threshold obtained by our computerized technique. LT-Subjective indicates the lactate threshold obtained by the technique of Beaver et a/. (2).
ciation between the LT and the VT (I). Furthermore, this uncertainty has contributed to the controversy as to whether LT exists at all (3). An objective computerized approach has previously been applied to the estimation of the VT (8). This method fits two or three linear segments to the ventilation versus oxygen consumption data. The VT was selected as the intersection of the first two linear segments. We have now developed a computerized approach for estimating LT. Our approach is completely computerized with the algorithm searching for the best intersection point. This is in contrast to the need for an observer in the method of Beaver et al. (2). The estimation of a confidence interval about LT is based on the “extra sum of squares” principle (5). When the estimated parameter is linear and the residuals are normally distributed, this approach yields an exact confidence interval (4). However, the breakpoint parameter is nonlinear and, thus, the confidence interval is approximate. This confidence interval appears to be conservative. Computer simulation studies indicate that the true confidence interval is bounded by the one we generate using the F test statistic. The incorporation of a confidence interval along with the estimate of LT during an exercise test is important because it yields a means of judging the
486
LUNDBERG
ET AL.
precision of LT. If the confidence interval is wide, it suggests that the corresponding estimate of LT is uncertain and thus repeated tests may yield a wide variation. Furthermore, conditioning programs that purport to increase LT can only be evaluated statistically if a means of encoding the uncertainty in objective estimates of LT is available. CONCLUSION
Our computerized approach to estimating LT removes the subjectivity of previous approaches. Furthermore, it yields an approximate 95% confidence interval so that the precision of the estimate can be judged and meaningful statistical comparisons can be made. ACKNOWLEDGMENTS The authors thank Brian J. Whipp for providing the lactate-O, consumption data. This study was partially funded by University of Colorado Office of Space Science and Technology through Health Sciences Center Space Medicine Committee.
REFERENCES 1.
2. 3. 4. 5. 6. 7. 8. 9.
ANDERSON, S. J., HUGHSON, R. L., SHERRILL, D. L., AND SWANSON, G. D. Determination of the “anaerobic threshold.” J. Appl. Physiol. 60, 2135-2136 (1986). BEAVER, W. L., WASSERMAN, K., AND WHIPP, B. J. Improved detection of lactate threshold during exercise using a log-log transformation. J. Appl. Physiol. 59, 1936-1940 (1985). BROOKS, G. A. Anaerobic threshold: Review of the concept and directions for future research. Med. Sci. Sports Exericse 17, 22-31 (1985). DONALDSON, J. R. “Computational Experience with Confidence Regions and Confidence Intervals for Nonlinear Least Squares.” M.S. thesis, University of Colorado, 1985. DRAPER, N. R., AND SMITH, H. “Applied Regression Analysis,” 2nd ed. Wiley. New York. 1981. GLADDEN, L. B., YATES, J. W., STREMEL, R. W., AND STAMFORD, B. A. Gas exchange and lactate anaerobic thresholds: Inter- and intravascular agreement. J. Appl. Physiol. S&20822089 (1985). JONES, R. H., AND MOLITORIS, B. A. A statistical method for determining the breakpoint of two lines. Anal. Biochem. 141,287-290 (1984). ORR, G. W., GREEN, H. J., HUGHSON, R. L., AND BENNETT, G. S. A computer linear regression model to determine ventilatory anaerobic threshold. J. Appl. Physiol. 52, 1349-1352 (1982). WASSERMAN, K. The anaerobic threshold measurement to evaluate exercise performance. Amer.
10.
Rev.
Respir.
Dis.
129, S35-S40
(1984).
YEH, M. D., GARDNER, R. M., ADAMS, T. D., YANOWITZ, F. G., AND GRAPO, R. 0. “Anaerobic threshold”: problems of determination and validation. J. Appl. Physiol. 55, 1178-1186 (1983).
AND
BIOMEDICAL
Computerized
RESEARCH
19, 481-486 (1986)
Estimation
of Lactate Threshold
MARTHA A. LUNDBERG,* RICHARD L. HUGHSON,? KENNETH H. WEISIGER,? RICHARD H. JONES,* AND GEORGE D. SwmsoNt *Department of Preventive Medicine and Biometrics, and TDepartment of Anesthesiology, School of Medicine, University of Colorado Health Sciences Center. Denver, Colorado 80262 Received April 9, 1986
Traditional approaches to estimating a lactate threshold during a progressive exercise test have utilized visual inspection of the data. We describe a computerized approach which utilizes a log-log transformation to yield two approximately linear segments. Linear regression lines are fit to these segments and the intersection of the two lines yields an estimate of the lactate threshold. An approximate 95% confidence interval is also generated. o 1986 Academic
Press. Inc.
Changes in blood lactate concentration during a progressive exercise test to exhaustion have suggested the possibility of a threshold-a point where blood lactate increases dramatically. Attempts to estimate a lactate threshold (LT) have utilized simple visual inspection of lactate data plotted against time or 02 consumption. This subjective approach has led to LT estimates that are dependent on the particular observer (6, 20) and has contributed in part to the controversy as to whether LT exists at all (3). Clearly a more objective approach is needed. Recently, Beaver et al. (2) have suggested that a log-log transformation on the data enhanced the appearance of LT. They described a method of estimating LT that utilized an observer to determine a division point such that the data to the left of this point belonged to a low slope linear segment and data to the right belonged to a high slope linear segment. Linear regression lines were determined for each of these two segments, with the intersection of the two lines being the estimate of the LT. Although this approach is less subjective than simple visual inspection of the raw data, LT estimate is directly influenced by the choice of the division point. The purpose of this paper is to describe a computerized approach for estimation of LT using the log-log transformation proposed by Beaver et al. (2). This computer method does not use an observer and is thus compktely objective. 481 00104809186 $3.00 Copyright 0 1986 by Academic Press, Inc. All rights of reproduction in any form reserved.
482
Furthermore, LT.
LUNDBERGETAL.
the method generates a confidence interval about the estimate of METHODS
The approach utilizes the log-log transformation suggested by Beaver et al. (2). We then utilize a computer algorithm (7) to fit two lines to the log-log data that join at a breakpoint. This breakpoint is varied within the range of the loglog data and LT estimate is determined as that breakpoint for which the overall residual sum of squares (RSS) is minimized. If x0 is the position of the unknown breakpoint, the equations of the two lines are : Y = PO + PlX
x 5 x0
Y
x
=
P2
+
Lll
P3x
>
x0
When x = x0 the lines must intersect. Thus, PO
+
PIXO
P2
=
PO
=
P2
+
/33x0,
El
-
P3xo.
[31
yielding The parameter
+
PIXO
p2 can now be eliminated
from [ 11 yielding.
Y = PO + PIX
Y
=
PO
+
r41 L ~>
PlXO
+
P3b
-
x
x0)
>
x0.
The method involves searching for the value of x0 that minimizes RSS. This is actually a four parameter regression involving three linear parameters, PO, /3t, and & and the nonlinear parameter x0. The breakpoint parameter x0 is varied over the range of the log-log data. For each value of x0 the linear parameters are determined by matrix linear regression methods (5). Thus, given a trial x0, from [4] the matrix equation is given by YI
1
Y2
lx:!
Y3
1x3
Yj Yj+l
Tz
0
Xl
0 0
. .
. .
1
Xj
1
X0
. . .
. . .
1 x0
7 PO
0 Xj+l
-
PI x0 P3
_
x, - no
where the jth data point is the last data point included ment. Equation [5] can be more compactly written as
in the lower line seg-
COMPUTERIZED
LACTATE
THRESHOLD
ESTIMATION
Yl = [Xl PI. The conditional
least-square
solution
of the parameter
p = (XTX)-‘XTY
483
[61 vector given x0 is [71
with RSS = YTY - /PXTY.
F31
Therefore, varying x0 over the range of the log-log data yields the value of x0 (XOmin) for which RSS is minimum (RSS,i”). This is LT estimate. The corresponding mean square error (MSE) is given by RSSnin MSE = n-4 where there are 12data points and four parameters. Since x0 is a nonlinear parameter, statistical tests on x0 are approximate rather than exact (4). An approximate confidence interval can be determined by determining the test statistic [91 where RSS is the residual sum of squares at a particular x0 and RSS,i” corresponds to the value of xomin. This test statistic is approximately distributed as an F statistic with one and n - 4 degrees of freedom (5). An approximate 95% confidence interval is then determined as that x0 lower than XOminand that x0 higher than xomin where the test statistic is equal to FlTn-4 at the 0.05 level. By plotting RSS as a function of x0, this 95% confidence interval can be graphically determined or alternatively determined via a computer algorithm. RESULTS
Figure 1 indicates the lactate versus 02 consumption data for subject number 1 from the Beaver et al. study (2). Note in Fig. la, there is a relatively flat response phase followed by a phase where the lactate concentration begins to rise. This characteristic is more evident in Fig. lb where the log lactate concentration versus log O2 consumption is plotted. Two approximately linear segments of data are now evident. The two segment linear regression model is shown superimposed on the data of Fig. lb. The intersection of the two lines for which the RSS is a minimum is the estimate of LT. Figure 2 indicates the plot of RSS increase as a function of x0. Note that the 95% confidence interval is determined when this increase reaches the appropriate F statistic. Table 1 summarizes the results for all 10 subjects from the Beaver et al. (2) study. Note that LT using our computerized technique yields results similar to the subjective approach of Beaver et al. (2) when an experienced observer is used. Furthermore, our computerized technique yields the approximate 95% confidence interval as indicated.
484
LUNDBERG
“02
ET AL.
Log
(L/mid
Vo,
(L/mid
FIG. I. Lactate versus O2 consumption data for subject number one of the Beaver et ul. study (2 ). (A) The linear plot. (B) The log-log plot.
DISCUSSION
At some point in a progressive exercise test, the ability to metabolize lactate lags behind the rate of lactate production and the blood lactate concentration begins to rise (3). This point is termed the lactate threshold (LT). Accompanying this change in blood lactate concentration is an equimolar change in hydrogen ion [H+] which is buffered primarily by bicarbonate. The resultant increase in CO* production from this buffering process is met by an increase in ventilation. A ventilation threshold (VT) has often been described at the point of increased CO2 production (9). LT and/or VT have been suggested to represent the upper limit for most normal daily activities in the typical adult (9). Selection of the thresholds for ventilation (VT) or blood lactate (LT) has been done primarily by subjective assessments of plotted data. These subjective estimates are dependent on the observer (6, 10) and tend to show considerable variation with repeated evaluation of a data set by the same observer (6). This uncertainty of the estimates has complicated attempts to characterize the asso-
“02
(Llmin)
FIG. 2. Plot of the change in residual sum of squares (RSS) as a function of the intercept of the line segments shown for the data of Fig. I. The approximate 95% confidence interval is indicated.
COMPUTERIZED
LACTATE
THRESHOLD
TABLE
ESTIMATION
485
1
COMPUTERIZED LACTATE THRESHOLD ESTIMATES WITH THEIR CORRESPONDING95% CONFIDENCE INTERVALS FOR THE 10 SUBJECTS OF THE BEAVER et al. STUDY (2) Subject No. 1 2 3 4 5 6 7 8 9 10 Average SD
LT-Objective (Urnin) 1.465 2.351 1.742 1.344 1.250 1.436 1.540 1.510 1.034 1.562 1.52 235
95% C.I. (L/min) 1.347 2.181 1.624 1.221 1.026 1.300 1.327 1.157 0.944 1.377
1.571 2.452 1.861 1.507 1.418 1.551 1.721 1.697 1.188 1.721
LT-Subjective (L/min) 1.54 2.28 1.72 1.39 1.29 I.35 1.49 1.64 1.09 1.58 1.54 +-32
Note. The subjective LT estimate using an experienced
ob-
server is also indicated. LT indicates lactate threshold. 95% C.I. indicates the approximate F test 95% confidence interval. LTObjective indicates the lactate threshold obtained by our computerized technique. LT-Subjective indicates the lactate threshold obtained by the technique of Beaver et a/. (2).
ciation between the LT and the VT (I). Furthermore, this uncertainty has contributed to the controversy as to whether LT exists at all (3). An objective computerized approach has previously been applied to the estimation of the VT (8). This method fits two or three linear segments to the ventilation versus oxygen consumption data. The VT was selected as the intersection of the first two linear segments. We have now developed a computerized approach for estimating LT. Our approach is completely computerized with the algorithm searching for the best intersection point. This is in contrast to the need for an observer in the method of Beaver et al. (2). The estimation of a confidence interval about LT is based on the “extra sum of squares” principle (5). When the estimated parameter is linear and the residuals are normally distributed, this approach yields an exact confidence interval (4). However, the breakpoint parameter is nonlinear and, thus, the confidence interval is approximate. This confidence interval appears to be conservative. Computer simulation studies indicate that the true confidence interval is bounded by the one we generate using the F test statistic. The incorporation of a confidence interval along with the estimate of LT during an exercise test is important because it yields a means of judging the
486
LUNDBERG
ET AL.
precision of LT. If the confidence interval is wide, it suggests that the corresponding estimate of LT is uncertain and thus repeated tests may yield a wide variation. Furthermore, conditioning programs that purport to increase LT can only be evaluated statistically if a means of encoding the uncertainty in objective estimates of LT is available. CONCLUSION
Our computerized approach to estimating LT removes the subjectivity of previous approaches. Furthermore, it yields an approximate 95% confidence interval so that the precision of the estimate can be judged and meaningful statistical comparisons can be made. ACKNOWLEDGMENTS The authors thank Brian J. Whipp for providing the lactate-O, consumption data. This study was partially funded by University of Colorado Office of Space Science and Technology through Health Sciences Center Space Medicine Committee.
REFERENCES 1.
2. 3. 4. 5. 6. 7. 8. 9.
ANDERSON, S. J., HUGHSON, R. L., SHERRILL, D. L., AND SWANSON, G. D. Determination of the “anaerobic threshold.” J. Appl. Physiol. 60, 2135-2136 (1986). BEAVER, W. L., WASSERMAN, K., AND WHIPP, B. J. Improved detection of lactate threshold during exercise using a log-log transformation. J. Appl. Physiol. 59, 1936-1940 (1985). BROOKS, G. A. Anaerobic threshold: Review of the concept and directions for future research. Med. Sci. Sports Exericse 17, 22-31 (1985). DONALDSON, J. R. “Computational Experience with Confidence Regions and Confidence Intervals for Nonlinear Least Squares.” M.S. thesis, University of Colorado, 1985. DRAPER, N. R., AND SMITH, H. “Applied Regression Analysis,” 2nd ed. Wiley. New York. 1981. GLADDEN, L. B., YATES, J. W., STREMEL, R. W., AND STAMFORD, B. A. Gas exchange and lactate anaerobic thresholds: Inter- and intravascular agreement. J. Appl. Physiol. S&20822089 (1985). JONES, R. H., AND MOLITORIS, B. A. A statistical method for determining the breakpoint of two lines. Anal. Biochem. 141,287-290 (1984). ORR, G. W., GREEN, H. J., HUGHSON, R. L., AND BENNETT, G. S. A computer linear regression model to determine ventilatory anaerobic threshold. J. Appl. Physiol. 52, 1349-1352 (1982). WASSERMAN, K. The anaerobic threshold measurement to evaluate exercise performance. Amer.
10.
Rev.
Respir.
Dis.
129, S35-S40
(1984).
YEH, M. D., GARDNER, R. M., ADAMS, T. D., YANOWITZ, F. G., AND GRAPO, R. 0. “Anaerobic threshold”: problems of determination and validation. J. Appl. Physiol. 55, 1178-1186 (1983).