f
Estimating the parameters of aerobic function during exercise using an exponentially increasing work rate protocol Y. Fukuba I
K. Hara I
Y. Kimura I A. Takahashi I B. J. W h i p p 3
S.A. Ward 2
1Department of Exercise Science and Physiology, School of Health Sciences, I-liroshima Women's University, I-liroshima, Japan 2Centre for Exercise Science and Medicine, University of Glasgow, Glasgow, UK 3Department of Physiology, St George's Hospital Medical School, London, UK
new exercise protoco/ has been proposed, with respect to cardiopulmonary exercise testing, which starts at a low work rate (WR) and increases exponentially by a standard percentage of the previous work rate every minute: the test is termed STEEP (standardised exponential exercise protocol). The potential advantage of this protocol is that it can accommodate a wide range of subjects, since it allows a maximum to be attained with a relatively narrow variation of tolerance time, regardless of subjects exercise capacity. To date, only the VO2max has been compared with that from the current standard ramp protocol. The ramp, however, also allows other important parameters of aerobic function to be estimated: the anaerobic threshold (AT); the response time constant; and A(/O2/AWR. The aim of this study was, therefore, to clarify whether these aerobic parameters can be readily discerned from the responses to the STEEP protocol both from a theoretical and practical viewpoint. As a result of theoretical considerations, we demonstrated that the V02 time constant and A(/O2/AWR may not both be estimated uniquely. As a practical expedient, a procedure was proposed for estimating the parameter analogues. The preliminary results for six subjects between the STEEP and ramp protocols showed consistent positive correlation for VO2max(r = 0.997) and. A T - V O 2 (r = 0.980), whereas the correlation for the (/02 time constant and AVO2/AWR were not significant. Further study is needed to clarify the reason(s) for the discrepancies both from a theoretical and practical viewpoint.
Abstract--A
Keywords--Exercise testing, Exponential work rate forcing, (/02-kinetics, First-order dynamics Med. Biol. Eng. Comput., 2000, 38, 433-437
J
1 Introduction
THE RAMP incremental work rate protocol (WHIPPel at., 1981) has come into widespread use in human exercise physiology, as it allows the experimenter to determine three important parameters of aerobic function: oxygen uptake (f/O 2) at anaerobic threshold ( A T - f / 0 2 ) , time constant of VO 2 dynamics (Tc), and A ".VO2/Awork rate (AI/O2/AWR), in addition to maximum VO 2 ( VO2max or peak VO 2 ) to be determined simultaneously in a relatively short time. This test has been utilised to determine these parameters in a wide range of subjects, encompassing patients with limited exercise tolerance to elite athletes (TAVAZZI and DIPRAMPERO, 1986; WASSERMAN, 1984, WASSERMAN et al., 1994): the ramp slope being varied as needed to achieve the target test duration, i.e. 10-15 minutes. Correspondence should be addressed to Dr Y. Fukuba; emaih
[email protected] First received 14 March 2000 and in final form 15 May 2000 MBEC online number: 20003486 © IFMBE:2000
Medical & Biological Engineering & Computing 2000, Vol. 38
NORTHRIDGE et at.(1990), however, have proposed a novel exercise protocol termed STEEP (STandardized Exponential Exercise Protocol) designed to be suitable for a wide range of exercise capacity. The STEEP protocol starts at a low work rate and increases every minute by a fixed percentage of the previous work rate. This provides an exponentially increasing work rate profile. The proposed advantage of this protocol, is its suitability for a wide range of subjects, because, in the later stages of the test, the rate at which the power increment develops becomes progressively larger such that fitter subjects are brought rapidly to their limit of tolerance. This results in a relatively narrow range of test duration, regardless of the subject's fimctional capacity. However, it remains to be determined whether the STEEP protocol allows important parameters of aerobic function (i.e. A T - V 0 2 , A f / O 2 / A W R , and Tc) other than f/O2max to be validly determined. These additional parameters provide valuable information for the diagnosis of determinants of exercise intolerance (WASSERMAN et al., 1994). The aim of this study therefore was to establish an estimation technique both from theoretical and practical viewpoints for the aerobic parameters of 433
exercise using the dynamic gas-exchange responses to the STEEP protocol.
2 Theory and parameter estimation
The kinetics of 1/O2 during the transient phase of a constant moderate intensity work rate (WR) i.e. sub-AT, manifests a rapid increase lasting 15 to 25 seconds (phase 1 ) followed by a monoexponential increase (phase 2) to the new steady state level (phase 3) which is normally attained within 3 min in a healthy young subject (WHIPP, 1987). That is: Al/O2(t) = Al/O2ss*{1 - e x p ( - t / r c ) }
(1)
where Al/O2(t) is the increment of 1/O2 at any time t above the prior steady state level (lmin -1) of baseline WR (WRBL), AVO2ss is the steady state increment of 1/02, and Tc (min) is the time constant of the response. Even in patients with cardiovascular and/or pulmonary impairments, the V02 time course merely shows a slowed exponential increase without any essential kinetic difference to those in normal subjects (BAUERet al., 1999; NERY et al., 1982; SIETSEMAet al., 1994). That is, the f/O2 kinetics for moderate intensity exercise have usually been treated adequately as a 'first-order linear system' with a single time constant (Tc). Tc for 1/O2 has been recognised as an important dynamic parameter of the exercise test response, as it reflects the total adaptability, or integrative functioning, of the physiological systems linking cardio-respiratory and muscle metabolic functions (WHIPP, 1994).
From eqns 5 and 6, it can be seen that Al/O2(t) is a linear function which is shifted parallel to Al/O2ss(t) along the time axis. The time lag corresponds to Tc; therefore, Tc may be determined from the V 0 2 - t data set, conceptually at least up to the A T . A I / O 2 / A W R can be also determined as the slope in eqn 5, because A V O 2 / A W R is K. 2.2 STEEP protocol Since the 1/O2 response to the STEEP protocol (i.e. exponential work rate forcing) and its mathematical representation is the main focus of this study, we describe them in detail. First, considering an exponential change of work rate (WR(t)) as the input to the WR- V02 control system on the basis of the firstorder linear differential equation, i.e.:
where a is the rate constant of the exponential WR increment (a > 0). Solving eqn 7, WR(t) is derived as WR(t) = WRBL* exp(a*t)
AWR(t) = WRBL* e x p ( a * t ) - WRBL
As we assume the system of WR- 1/O z control to be linear and first-order as for the ramp (i.e. the eqn 3), the l/Oz response to the 5000
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720
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a
300
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(3) 4000 200 '7,
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A typical response curve during ramp exercise test is shown in Fig.la. The initial response is curvilinear during the transient phase after the onset of the ramp as the exponential term in eqn 4 does not approach zero until time t is greater than approximately 4*Tc. However, when time t is long compared to Tc (i.e. the exponential term is negligible), then A VO2 (t) in eqn 4 simplifies to
.>o
(5)
Here, Al/Ozss(t) is defined as the increment required for the instantaneous work rate corresponding to the steady state 1/O 2 in response to a step exercise:
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(4)
Al/Ozss(t) = K * A * t
3000
(2)
where K is the gain 1 W -1 of the control system. The 1/O2 response, considered as the output of the first-order system from the input ramp WR function, can be derived from the convolution of eqns 2 and 3, and expressed as
Al/Oz(t) = K * A * ( t - Tc)
300
v
where A is the rate constant (slope) of the continuous work rate increment (W min -1). Since the WR- 1/O2 control system in the exercise intensity domain below A T in humans can be treated as a linear first-order system, the time domain expression of the transfer function of the system (i.e. the impulse response) is given by
Al/O2(t) = K*A*[t - Tc*{1 - e x p ( - t / T c ) } ]
(9)
4000
As the detailed mathematical description of 1/O2 kinetics in response to a ramp-incremental exercise has been previously detailed (WHIPPet al., 1981, WHIPP, 1994), only the most salient arguments are presented here. During the ramp exercise, the A WR (W) increases at a constant rate continuously from WRBL at t = 0 (t; min), i.e.:
g(t) = ( K / T c ) * e x p ( - t / T c )
(8)
where WRBLis WR(t) at t = 0, i.e. the prior steady state baseline level. Accordingly, AWR(t) is given by
2.1 Ramp protocol
AWR(t) = A*t
(7)
d W R ( t ) / d t = a* WR(t)
(6)
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Fig. 1 Time-serial data of the (zO2 response for a single subject (subject TT) to the (a) ramp and (b). STEEP work rate forcing. - work rate forcing, o o V02 response data. Time '0' indicates the start of either the ramp or STEEP work rate protocol
Medical & Biological Engineering & Computing 2000, Vol. 38
exponential WR input (eqn 9) is (see details in the Appendix);
A~/O2(f)
3000
_
\(1 + r e * a ) /
\
- t + exp(a*t)) exp (-~c)
(10) A typical response curve is presented in Fig. lb. As for the ramp exercise, following an appropriate transient phase after the onset of the exercise, the term, e x p ( - t / T c ) approaches zero at t >>Tc. Accordingly, the 1/02 response can be shown to be Al/O2(t) = ( K * ~ B , L , ' ] * e x p ( a * t ) - - K*WRBL \ t t + 1c a U
,#
K*WRBL
'7.,=_ E 2000
AT-VO2= 1.
&l 0 0 .>
~ 1000
ill)
That is, the response of the AI/O 2 increase to the exponential work rate forcing would, after the transition phase, also be characterised as an exponential function. As a compatible equation to eqn 6 in ramp protocol, Al/O2ss(t) may be expressed as Al/O2ss(t) = K* WRBK* exp(a*t) -- K*WRBL
(12)
(13)
This means that the time lags are the same after the transient phase (t >>Tc) up to the A T , and determined by an unknown variable Tc and a given variable a. it would appear that Tc may be determined from eqn 13. However, because K may not be estimated in eqn 11 in the STEEP protocol (i.e. different from the ramp protocol; eqn 5), it is important to recognise that there is no way to determine simultaneously both parameters Tc and K uniquely.
Medical & Biological Engineering & Computing 2000, Vol. 38
I
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The conventional methods for the estimation of l/O2max and A T-I/Oz may be utilised during the STEEP protocol. The VOzmax (or peak [/O z) during a ramp protocol is usually determined as the highest attainable Oz uptake. This criterion is, naturally, also appropriate for the STEEP response. The Vslope method (BEAVER et al., 1986) and the additional criteria for estimating the A T - V O z (WHIPPetal., 1986) also seem as appropriate for the STEEP protocol as for the ramp protocol. In both cases, discarding the early transient phase and also the later hyperventilatory region, a plot of COz output (I/CO z) against VOz gives two effectively linear components, the intersection of which occurs at A T (Fig. 2). In addition to the V-slope method, the beginning of a systematic increase in pulmonary ventilation (I/E)~ VOz and end-tidal POz (PETO2! without end-tidal PCOz (PETCOz) decreasing as a function of VO z also provides a valid index of the A T in subjects with normal chemosensitivity and normal pulmonary mechanics (WHIPPet al., 1986). However, as the V-slope method relies on the accelerated I/CO z above the A T, it is expected to be valid even when I/E does not respond, as necessary for estimation using the ventilatory response variables. As presented in subsection 2.2, however, the remaining two important parameters, Tc and K may not be estimated uniquely in the STEEP protocol. Therefore, the following method is proposed. First, discarding the early transient region and also the supra-AT region, a plot of VO z against WR may give one linear regression (see Figs 3a and 3c}. The slope of this regression line provides an analogue of A V 0 2 / A WR equivalent to the ramp slope, that is K. i f K is actually characterized in eqn 11, then the only unknown parameter would be Tc. Therefore,
I
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3000
0
3 Practical estimation of the aerobic parameters from STEEP exercise protocol
I
1000
VO2,mlmin a
The time lag (A T) between both exponential curves of eqns 11 and 12 along the time axis is AT = [ln(1 + Tc*a)]/a
VO2max=2.43Imin-1
VO2max=2.48I min-1
, I 1000
I 2000 -1 ~/02, ml min b
I 3000
Fig. 2. A T - V O 2 estimation using the Gslope method for a single subject (subject TT) during (a) ramp and (b) STEEP protocols. The data for Kslope AT estimation discards the ealtv transient region and the later hyperventilatoly region (ooo). See text for discussion
Tc could be estimated from the plot of 1/02 against time utilising the same region of the exercise response, which may be appropriately regressed with respect to eqn 11 (i.e. if K is known) by non-linear least squares regression analysis, in this study, we used the Marquardt-Levenberg Algorithm* to estimate Tc (Figs 3b and 3d). We investigated the relationships with respect to the ventilatory and gas exchange responses to both the ramp (WRBL= 20W, S = 2 0 W m i n -1) and STEEP (WRBL=20W, a 0.2 min -1) protocols, performed to the limit of tolerance in six healthy young subjects (age: 18-25 years). The work rate generated by the cycle ergometer** was controlled second-bysecond using a dedicated PC. The ventilatory and pulmonary gas exchange variables were measured breath-by-breath during the test by means of an integrate.d computerised system'S. For the VO2max and A T - V 0 2 , the values in STEEP protocol were compatible with those obtained from the ramp protocol (Fig..2). That is, the correlation coefficients of VO2max and A T - V 0 2 for six subjects between the STEEP and ramp proto*SigmaPIot, Jandel Scientific, CA, USA *'232cXL, Combi, Japan -~Centaura-I, Chest, Japan
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control can be treated as a first-order system as is typically the case for the ramp protocol. Since we demonstrate that the T c and A V O z / A W R parameters of the VO2 dynamics could not be uniquely determined from a mathematical standpoint, we propose, as a practical expedient, a procedure for estimating the parameter analogues. However, further studies are needed to determine empirically the extent to which the parameters o f aerobic function can be estimated validly from the response to the STEEP protocol, at least to the extent that they are compatible with those from the ramp protocol. Acknowledgmen~This study was supported in part by Grant-inAid for Scientific Research from The Ministry of Education, Science, Sports and Culture in Japan (#12680048).
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Appendix
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The response o f a linear first-order system to an exponential input Jiunction In the present study, the exponential function as the work rate input for the STEEP protocol (i. e. x(t) = A WR and x 0 = WRBL) is expressed as x(t) = x0* exp(a*t) - x 0
Fig. 3 Examples fi'om two subjects (a attd b: sltbje([t FV and c attd d: subject TT) of tke straight line fit for the VO2 against work rate relationship (a attd (9 used to estimate A VO2/A WR attd tke exponential fimction (eqn 11)for A/zO2 versus time (b attd d) used to estimate ~VO 2 time constant (TO fi'om tke STEEP protocol. Tke data whick are discarded fi'om tke fit are shown as solid symbols (i.e. tke early transient phase attd tke later/tear3; supra-AT region). See text for tke details'. (a) A[zO2/AWR (ml mitt z W-Z)=10.02 0"=0.984): (b) To(s)=63.8 0"=0.991): (0 AVOg/AWR (ml mitt z V~Z)=11.57 0"=0.974): (d) To(s)=82.5 0"=0.974)
(A1)
As ordinary linear differential equations can be solved simply using the Laplace transform (L) in the complex domain, eqn A1 may be expressed in the complex domain using complex Laplace transform parameter s, X ( s ) = L[x(t)] = Xo/(S - a) - Xo/S
(A2)
Because the system transfer function o f WR - l/Ox control may be treated as a linear first-order system, its Laplace transform is; G(s) = L[g(t)] = K / ( 1 + Tc*s)
(A3)
The output function (Y(s)) in complex domain is then expressed as
cols were significantly positive (r = 0.997 and 0.980; p < 0.05) and the regression lines were very close to the identity line (l/O2max: STEEP = 1.08 - ramp-263, AT-V02: STEEP = 0.91 - ramp ÷ 144). The correlation coefficients of AI/O2/AWR and T c for both protocols, however, were not significant (p>0.05). For example, the data for estimating A V O 2 / A W R and T c in the STEEP protocol are shown for two representative subjects .(subjects TT and YN) in Fig. 3. In one subject (YN), the A V O 2 / A W R and Tc, estimated as described above, were 10.02 ml mill -1 W -1 and 64 s for the STEEP, and 10.96ml m i n - l w -1 and 60s for the ramp, respectively (Figs 3a and 3b). In this case, these two aerobic parameters were relatively consistent between the protocols. However, in the other subject (TT), the AI/O2/AWR and T c in STEEP protocol (11.57ml min - 1 W -1 and 83 s) were quite different from the values derived from the ramp protocol (9.50ml m i n - l w -1 and 41 s) (Figs 3c and 3d). The inconsistency o f A I / O 2 / A W R and T c between the STEEP and ramp protocols suggests that further investigation is needed, in subjects with a wider range o f functional capacities, to clarify the reason(s) for the discrepancies both from a theoretical and practical viewpoint.
4 Comments We describe in detail the analytical procedure for estimating the parameters o f aerobic function from pulmonary gas exchange data during an exercise test with special reference to the STEEP protocol. The derivation o f the VO2-kinetics in response to exponential work rate forcing was established using mathematical modelling assuming that the WR to 1/O 2
436
Y(s) = X(s)*G(s) = [(Xo/(S - a)) - Xo/S]*[K/(1 + f c'*s)] = {K*xo/(l+Tc*a)}*[{-1/(s+(1/Tc))}+{1/(s
+ [{l/s} - {1/(s + (1/Tc))}]
- a)}]
(A4)
Accordingly, the 1/O2 response in the time domain can be derived by means o f the inverse Laplace transform (L 1): y(t) = L l[g(s)] = K*xo*[Tc*a/(1 + Tc*a)]*[-exp(-t/Tc)] + K*xo*[1/(1 + Tc*a)]*[exp(a*t)] - K * x o (A5)
This allows eqn 10 to be obtained.
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SIETSEMA, K. E., BEN-DOM I., ZHANG, Y. Y., SULLIVAN, C. and WASSERMAN,K. (1994): 'Dynamics of oxygen uptake for submaximal exercise and recovery in patients with chronic heart failure', Chest, 105, pp. 693-700 TAVAZZI,L. and DIPRAMrERO, R E. (1986): 'The anaerobic threshold: physiological and clinical significance', A&: Cardiol., 35, pp. 1-155 WASSERMAN, K. (1984): 'Exercise testing in the dyspneic patient', Am. Re~: Respirat. Dis'., 129, pp. S1-S100 WASSERMAN, K., HANSEN, J. E., SUE, D. Y., WHIPR B. J. and CASABURI,R. (1994): 'Principles of exercise testing and interpretation', Lea and Febiger, Philadelphia
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WHIPR g. J. (1987): 'Dynamics of pulmonary gas exchange', Circulation, 76 (suppl VI), pp. 18-28 WHIRR B. J. (1994): 'The bioenergetic and gas exchange basis of exercise testing', Clin. Chest Med., 15, pp. 173-192 WHIRR B. J., DAVIS, J. A., TORRES, E and WASSERMAN,K. (1981): 'A test to determine parameters of aerobic function during exercise', J. Appl. Plo.'siol., 50, pp. 217-221 WHIRR B. J., WARD, S. A. and WASSERMAN, K. (1986): 'Respiratory markers of the anaerobic threshold', A&: Cardiol., 35, pp. 47-64
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