On analytical methods and inferences for 2 x 2 contingency table data from medical studies.

Analysis of 2 x 2 contingency tables is not as trivial as it appears. The choice of the statistical test can affect the inferences resulting from data analysis, especially at small sample sizes. Canned statistical programs do not necessarily lead to an appropriate test. These points are demonstrated using examples from the literature.

Time domain analysis of oxygen uptake during pseudorandom binary sequence exercise tests.

Pseudorandom binary sequence (PRBS) exercise tests involve repeated switching between two work rates (WR) according to a computer-generated pattern. This paper presents an approach to analysis of O2 uptake (VO2) in the time domain. First, the autocorrelation function (ACF) of the input WR was recognized to be a triangular-shaped pulse that can be taken to be equivalent to a ramp increase followed by a ramp decrease in WR. Then the cross-correlation function of the input (WR) and the output (VO2) was treated as if it were the response to a triangular-shaped pulse. The cross-correlation function was analyzed by fitting a linear summation of the ramp form of a two-component exponential function to this triangular pulse. VO2 responses of eight subjects were obtained from two different PRBS tests, as well as step changes in WR. The first PRBS test consisted of 15 units, each 30 s in duration. Its ACF had a base width of 60 s. The ramp increase-ramp decrease model fit the data throughout the range of response. The second PRBS test had 63 units, each 5 s in duration; thus its ACF base width was 10 s. Again, the ramp model fit adequately. The data from the second PRBS test could be fit by the impulse form of the two-component exponential equation, although the fit in the first 30 s tended to be poorer. The time constants of VO2 dynamics estimated from step and PRBS tests were not significantly different. PRBS tests can be analyzed in the time domain, and the indicators of system dynamics reflect physiological properties similar to those investigated during step changes in WR.

Investigation of VO2 kinetics in humans with pseudorandom binary sequence work rate change.

The dynamic response of oxygen uptake (VO2) was investigated with two different cycle ergometer tests in which the work rate changed as a pseudorandom binary sequence (PRBS). One sequence had 15 units, each of 30-s duration for a total of 450 s (PRBS1). The second had 63 units, each of 5-s duration for a total of 315 s (PRBS2). The useful range of frequencies available for investigation of the dynamic characteristics of the VO2 response as described by their bandwidth were 0.002-0.013 Hz for PRBS1 and 0.003-0.089 Hz for PRBS2. Eight subjects each completed both PRBS tests. Data from four or five consecutive sequences were ensemble averaged to reduce the biological noise. A Fourier analysis was then conducted, with the range of frequencies investigated spanning those of the bandwidth for PRBS2. This was up to the 28th harmonic. For PRBS1, the VO2 response could be adequately reconstructed by including Fourier coefficients only up to the 5th harmonic. In contrast, for PRBS2, there was still a clear pattern in the residuals at the 5th harmonic. The data were not adequately reconstructed until higher-frequency components up to the 28th harmonic were included. Evidence for this came from analysis of the mean square error. The mean square error at the 28th harmonic was reduced to 83 +/- 8% of the mean square error at the 5th harmonic for PRBS1 and to 31 +/- 3% for PRBS2 (P less than 0.0001). These data obtained by Fourier analysis and reconstructed for comparison with the original VO2 response indicate the presence of a high-frequency component that was not apparent when a test with a smaller bandwidth was used as the work rate forcing.

Assembling control models from pulmonary gas exchange dynamics.

A model abstracts certain features of real systems, which are consistent with the purpose intended for the model. Therefore, models are classified with respect to intended purpose. A structural model predicts behavior based on a hypothetical physiological structure. An empirical model summarizes observed behavior. A functional model attempts to relate observed behavior to physiological structure. It differs from structural model since it includes only structural aspects that are essential for describing the system response. This model classification provides for the interaction among experimental data, underlying physiological hypotheses and experimental design. In this paper, we illustrate the utility of this modeling process for studying pulmonary gas exchange dynamic control processes during exercise. The modeling process is applied to the problem of estimating breath-by-breath gas exchange data, to the problem of selecting appropriate models for characterizing the response to dynamic work rate inputs, and finally to the problem of design of the dynamic aspects of the work rate input.

A probabilistic approach to the single-point, single-dose problem.

A general probabilistic approach is applied to the single-point, single-dose method for estimating individual infusion rates and serum drug concentrations. By using transformations of probability density functions, the effects of variations in the elimination rate constant upon pharmacokinetic variables may be studied and optimal sampling times may be chosen. Although this study treats the case of error-free sampling in a single-compartment model with a normal distribution of rate constants, the methods presented can be extended to more general situations.

Application of the general linear model for smoothing gas exchange data.

The precision of an interpretation of gas exchange records in progressive exercise is limited by the typical breath-to-breath variation in the data. Recently, two procedures have been proposed for minimizing the "noise" in the estimates of alveolar gas exchange time series data. One approach utilizes an estimate of pulmonary blood flow (Q) for smoothing purposes. The other approach utilizes an estimate of effective lung volume (V'L) for smoothing purposes. In this paper, we formulate the smoothing problem as a general linear model and demonstrate the concurrent estimates of both V'L and Q. Furthermore, we investigate the interaction between V'L and Q. Specifically, when a high value of lung volume is used (such as the subject's resting functional residual capacity) in the alveolar gas exchange algorithm, the estimate of Q is biased low and the result is a less effective smoothing of the data. In addition, we demonstrate how the Q estimate can be improved by utilizing more appropriate estimates of arterial carbon dioxide tension.

On the modeling and interpretation of oxygen uptake kinetics from ramp work rate tests.

Ramp work rate tests have been used to estimate aerobic parameters in exercise stress testing. Previous studies have suggested an assumption of a linear dynamic system for O2 uptake kinetics. The implication is that model parameters estimated from ramp tests should be similar to those estimated from other dynamic tests. In nine healthy subjects, we found that model parameters used to characterize O2 consumption ramp data were not consistent with those used to characterize step data, when the comparison was made on a subject-to-subject basis. Furthermore the ramp data model parameter values were highly dependent (P less than 0.0001) on the ramp slope. A linear dynamic system interpretation of the ramp data model does not appear to be appropriate, suggesting that caution is needed in the interpretation of ramp data aerobic parameters. The data may be better described by nonlinear or higher order function. Ramp exercise testing is not suitable for assessing dynamic control properties of the cardiorespiratory response to exercise.

On the estimation of pulmonary blood flow from CO2 production time series.

It may be possible to estimate a nominal pulmonary blood flow (Q) during an exercise stress test via the algorithm used to estimate breath-by-breath alveolar CO2 production. Recently it has been demonstrated that by relating breath-to-breath fluctuations in alveolar CO2 production to breath-to-breath fluctuations in end-tidal CO2, an optimizing parameter related to Q can be determined that can be used to process the CO2 production fluctuations and minimize their variation. However, the reported values of Q using this procedure appear to be biased low. Using a computer simulation of gas exchange, we demonstrate that the estimate of Q is biased low when the nominal lung volume used in the alveolar gas exchange algorithm is too large. Furthermore, alveolar CO2 transport is determined by an integral of alveolar CO2 over the breath time and, thus, is a path-dependent quantity. The use of end-tidal CO2 fluctuations to approximate fluctuations in this integral contributes to an error in the estimation of Q which yields estimates that are biased low. Alternatively, the use of mean alveolar CO2 fluctuations yield more appropriate Q estimates. These results suggest practical implications for estimating effective pulmonary blood flow during an exercise stress test by using breath-to-breath estimates of mean alveolar CO2.

Using smoothing splines to make inferences about the shape of gas-exchange curves.

Respiratory gas-exchange data from progressive exercise tests are typically interpreted by visual inspection. Attempts to objectify such interpretation have applied particular parametric models which limit the measures which can be studied and the inferences which can be made. We use a known spline-smoothing procedure which fits a continuous curve to such data, yielding confidence intervals for the curve and for its first and second derivatives. Rules can be made which use the derivatives to infer features of a curve's shape and to relate features from different curves in the same data set. In this way complex interpretations can be made objective, so that they may be adequately tested.

Kinetics of VO2 with impulse and step exercise in humans.

The constancy of the time course (i.e., dynamic linearity) of the O2 uptake (VO2) response to exercise was examined by testing the law of superposition on data from impulse and step work rate forcings. Two impulses (10 s at a 235-W increase above a 25-W base line, I-235; and 5 s at a 475-W increase above a 25-W base line, I-475), four steps (ST) (25-65 W, ST1; 65-105 W, ST2; 25-105 W, ST3; and 25-145 W, ST4), and the corresponding off-transient responses were performed six to eight times by each of five subjects. The integrated area (G) of the VO2 response for I-235 was similar to that of ST1 and ST2 (P greater than 0.05); the I-475 G was significantly greater (P less than 0.05). The time constant of VO2 during the step function on-transient response for the second exponential component was significantly faster for ST1 and significantly slower for I-235 and I-475 than for ST2, ST3, and ST4 (P less than 0.05). However, I-235 and I-475 time constants for VO2 were not different from the ST off-transient values. Attempts to superimpose the integral of the impulse on the ST data showed that the early rapid increase in VO2 in the ST was underpredicted by the impulse and that the impulse response lagged behind the ST at all points before steady state. It can be concluded that VO2 kinetics failed the test of superposition and are therefore described by a nonlinear dynamic system.

Estimate of mean tissue O2 consumption at onset of exercise in males.

A mathematical model has been developed that permitted the calculation of the flow-weighted mean tissue O2 consumption (VO2T) at the onset of a step increase in work rate. From breath-by-breath measurements of alveolar O2 consumption (VO2A) and cardiac output (Q) by impedance cardiography and assumptions about the site of depletion of O2 stores, the rate of change in O2 stores (VO2s) was determined. The sum of VO2A + VO2s = VO2T. Six very fit males performed six repetitions of each of two step increases in work rate. STlo was a transition from rest to 100-W cycling; SThi was a transition from 100- to 200-W cycling. For each work rate transition, the responses of VO2A and Q were averaged over the six repetitions of each subject and the model was solved to yield VO2T. The responses of VO2A, VO2T, and Q after the increase in work rate were fit with a monoexponential function. This function included a time constant and time delay, the sum of which gave the mean response time (MRT). In the STlo test, the MRT of VO2A (24.9 +/- 1.1 s, mean +/- SE) was longer than that of VO2T (15.3 +/- 1.3 s) and of Q (16.5 +/- 6.5 s) (P less than 0.05). The MRT of VO2T and Q did not differ significantly. Also for SThi, the MRT of VO2A (34.4 +/- 3.3 s) was significantly longer than that of VO2T (30.0 +/- 3.4 s) (P less than 0.05). The MRT of VO2T and Q (30.3 +/- 5.5 s) were not significantly different at this work rate either.(ABSTRACT TRUNCATED AT 250 WORDS)

Blood lactate concentration increases as a continuous function in progressive exercise.

The relationship between arterialized blood lactate concentration [( La-]) and O2 uptake (VO2) was examined during a total of 23 tests by eight subjects. Exercise was on a cycle ergometer with work rate incremented from loadless pedaling to exhaustion as a 50-W/min ramp function. Two different mathematical models were studied. One model employed a log-log transformation of [La-] and VO2 to yield [La-] threshold as proposed by Beaver et al. (J. Appl. Physiol. 59: 1936-1940, 1985). The other model was a continuous exponential plus constant of the form La- = a + b[exp(cVO2)]. In 21 of 23 data sets, the mean square error (MSE) of the continuous model was less than that of the log-log model (P less than 0.001). The MSE was on average 3.5 times greater in the log-log model than in the continuous model. The residuals were randomly distributed about the line of best fit for the continuous model. In contrast, the log-log model showed a nonrandom pattern indicating an inappropriate model. As an index of the position of the [La-]-VO2 continuous model, the VO2 at which the rate of increase of [La-] equaled the rate of increase of VO2 (d[La-]/dVO2 = 1) was determined. This VO2 was 2.241 +/- 0.081 l/min, which averaged 64.6% of maximal VO2. It is proposed that this lactate slope index could be used as a relative indicator of fitness instead of the previously applied threshold concept. The change in [La-] could be better described mathematically by a continuous model rather than the threshold model of Beaver et al.

Computerized estimation of lactate threshold.

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.

Importance of oscillations in alveolar gas concentrations in the analysis of rebreathing data.

Cyclic rebreathing of a soluble inert gas can be used to estimate lung tissue volume (Vt) and pulmonary blood flow (Qc). A recently proposed method for analyzing such cyclic data (Respir. Physiol. 48: 255-279, 1982) mathematically assumes that ventilation is a continuous process. However, neglecting the cyclic nature of ventilation may prevent the accurate estimation of Vt and Qc. We evaluated this possibility by simulating the uptake of soluble inert gases during rebreathing using a cyclic model of gas exchange. Under cyclic uptake conditions alveolar gases follow an oscillating time course, because gas concentrations tend to increase during inspiration and to decrease during expiration. We found that neglecting these alveolar gas oscillations leads to the underestimation of soluble gas uptake by blood, particularly during the early rebreathing breaths. When continuous ventilation is assumed Vt and Qc are overestimated unless rapid rebreathing rates, large tidal volumes, and gases of moderately low solubility are used. Under these conditions the amplitude of the cyclic oscillations is minimized, the alveolar time course more closely resembles that expected from continuous ventilation, and the resulting errors are minimized. Alternatively, when the effect of oscillating alveolar gas concentrations on mass transfer are considered, these estimation errors can be eliminated without restricting rebreathing rate or gas solubility. We conclude that failure to consider the effect of cyclic rebreathing on the time course of alveolar gas concentrations may result in significant errors when evaluating rebreathing data for Vt and Qc.

On-line computer estimation of carbon dioxide response curves.

Anesthesiologists are concerned with the effect of various anesthetics on a patient's central nervous ventilatory control. The most widely accepted method of determining the effect of a drug is to compare carbon dioxide response curves (delta VE/delta PETCO2, where VE = minute ventilation [in L/min] and PETCO2 = end-tidal carbon dioxide [in mm Hg]) measured before and after administration of the drug. Additional information concerning neuromechanical control can be obtained by also including a measure of the airway occlusion pressure (generally measured 100 ms after occlusion, i.e., P100). To facilitate these measurements we have developed a portable, computer-controlled data acquisition system. It includes an Apple II+ computer and measures VE, PETCO2, and P100. Each subject rebreathes exhaled carbon dioxide through a two-way breathing valve attached to a 9-liter reservoir, which is initially filled with 5% carbon dioxide and balance oxygen. Exhaled carbon dioxide concentrations are measured with an infrared medical gas analyzer on samples taken through a catheter connected at the mouthpiece. The exhaled flow is measured with a pneumotachograph in conjunction with a differential pressure transducer, and P100 is determined with a Validyne MP45 pressure transducer.