Estimation of Instantaneous Gas Exchange in Flow-Through Respirometry Systems: A Modern Revision of Bartholomew's Z-Transform Method

Abstract

Flow-through respirometry systems provide accurate measurement of gas exchange over long periods of time. However, these systems have limitations in tracking rapid changes. When an animal infuses a metabolic gas into the respirometry chamber in a short burst, diffusion and airflow in the chamber gradually alter the original signal before it arrives at the gas analyzer. For single or multiple bursts, the recorded signal is smeared or mixed, which may result in dramatically altered recordings compared to the emitted signal. Recovering the original metabolic signal is a difficult task because of the inherent ill conditioning problem. Here, we present two new methods to recover the fast dynamics of metabolic patterns from recorded data. We first re-derive the equations of the well-known Z-transform method (ZT method) to show the source of imprecision in this method. Then, we develop a new model of analysis for respirometry systems based on the experimentally determined impulse response, which is the response of the system to a very short unit input. As a result, we present a major modification of the ZT method (dubbed the ‘EZT method’) by using a new model for the impulse response, enhancing its precision to recover the true metabolic signals. The second method, the generalized Z-transform (GZT) method, was then developed by generalizing the EZT method; it can be applied to any flow-through respirometry system with any arbitrary impulse response. Experiments verified that the accuracy of recovering the true metabolic signals is significantly improved by the new methods. These new methods can be used more broadly for input estimation in variety of physiological systems.

Recovering signals in physiological systems with large datasets

“…The methods are reasonably straightforward and have only one parameter to be tuned. This feature makes them less complicated to use than the generalized Z-transform (GZT) method, recently introduced by our group ( Pendar and Socha, 2015 ), which requires the use of multiple parameters that must be determined experimentally. Therefore, tuning of the Tikhonov method is simpler.…”

“…The Z-transform method, also known as the Bartholomew method, has perhaps been most widely used to correct gas exchange signals, but the temporal resolution of this method is not always sufficient to study fast changes in respiratory signals ( Pendar and Socha, 2015 ). We compared the results of the grasshopper experiment using our new methods with those using the Z-transform method ( Fig.…”

“…To calibrate the methods, it is necessary to find the impulse response of the system; this has been explained in detail previously in Pendar and Socha (2015). Briefly, to find the impulse response in a flow-through respirometry system, a short pulse of CO 2 should be injected into the respirometry chamber, and this injection should be recorded concurrently with the data recorded by the gas analyzer.…”

“… is the impulse response of the system, which can be found experimentally ( Fig. 1 ) ( Pendar and Socha, 2015 ). u is the true gas exchange signal from the animal, which we aim to recover.…”

Improving temporal accuracy of human metabolic chambers for dynamic metabolic studies

“…5 mins) is often desired. The third school modeled the instantaneous MR signal as a linear combination of recorded gas concentration data and its derivative [ 11 ] using the Z-Transform method, hence transforming the inverse problem to a direct problem [ 12 ] [ 13 ] [ 14 ] [ 15 ]. The mathematical simplicity of this method is helpful for system diagnosis, and it also enables real-time computation of MR [ 13 ].…”