Abstract
A system is said to be "instantaneous" when for a given constant input an equilibrium output is obtained after a while. In the meantime, the output is changing from its initial value towards the equilibrium one. This is the transient period of the system and transients are important features of openrespirometry systems. During transients, one cannot compute the input amplitude directly from the output. The existing models (e.g., first or second order dynamics) cannot account for many of the features observed in real openrespirometry systems, such as time lag. Also, these models do not explain what should be expected when a system is speeded up or slowed down. The purpose of the present study was to develop a mechanistic approach to the dynamics of openrespirometry systems, employing basic thermodynamic concepts. It is demonstrated that all the main relevant features of the output dynamics are due to and can be adequately explained by a distribution of apparent velocities within the set of molecules travelling along the system. The importance of the rate at which the molecules leave the sensor is explored for the first time. The study approaches the difference in calibrating a system with a continuous input and with a "unit impulse": the former truly reveals the dynamics of the system while the latter represents the first derivative (in time) of the former and, thus, cannot adequately be employed in the apparent timeconstant determination. Also, we demonstrate why the apparent order of the output changes with volume or flow.
openrespirometry systems; thermodynamic model
Braz J Med Biol Res, August 2000, Volume 33(8) 967982
Further analysis of openrespirometry systems: an acompartmental mechanistic approach
J.G. ChauiBerlinck and J.E.P.W. Bicudo
Departamento de Fisiologia, Instituto de Biociências, Universidade de São Paulo, São Paulo, SP, Brasil
References
Correspondence
Abstract
A system is said to be "instantaneous" when for a given constant input an equilibrium output is obtained after a while. In the meantime, the output is changing from its initial value towards the equilibrium one. This is the transient period of the system and transients are important features of openrespirometry systems. During transients, one cannot compute the input amplitude directly from the output. The existing models (e.g., first or second order dynamics) cannot account for many of the features observed in real openrespirometry systems, such as time lag. Also, these models do not explain what should be expected when a system is speeded up or slowed down. The purpose of the present study was to develop a mechanistic approach to the dynamics of openrespirometry systems, employing basic thermodynamic concepts. It is demonstrated that all the main relevant features of the output dynamics are due to and can be adequately explained by a distribution of apparent velocities within the set of molecules travelling along the system. The importance of the rate at which the molecules leave the sensor is explored for the first time. The study approaches the difference in calibrating a system with a continuous input and with a "unit impulse": the former truly reveals the dynamics of the system while the latter represents the first derivative (in time) of the former and, thus, cannot adequately be employed in the apparent timeconstant determination. Also, we demonstrate why the apparent order of the output changes with volume or flow.
Key words: openrespirometry systems, thermodynamic model
Introduction
The dynamics of a respirometry system is an essential feature of such a system and investigators must be fully familiarized with it before experiments begin. This is due to the fact that respirometry systems have transients, i.e., periods of time during which the output amplitude (the signal) of the system is below the amplitude of the input. Therefore, during transients, one cannot compute the amplitude of the input (e.g., oxygen consumption, carbon dioxide production, etc.) directly from the output. Two general solutions are possible. The first is to work with very fast systems which behave as zeroorder systems (in which transients are not detected). However, in most cases this is not usually a feasible task for the wellknown limits of openrespirometry systems (i.e., flow, volume, the biological system under investigation, etc.). The second solution is to transform the output into instantaneous readings by some mathematical procedure (1), i.e., to obtain output amplitudes linearly related to the input amplitudes. Three branches of such solutions are found in the literature: a) singlechamber firstorder models (210), b) twochamber secondorder models (11), and c) unknown number of chambers freeoforder models (12,13). The firstorder models analyze the output as coming from a single volume of dilution, and the Ztransformation of the output is the mathematical procedure used to obtain instantaneous readings (4) even though adding the output to its first derivative in relation to time is another possibility (79). Freeoforder models employ deconvolution of the output by a transfer function obtained during calibration procedures. The putative number of volumes diluting the input is irrelevant to this transformation, and the procedure is simply instrumental in this respect.
Frappell et al. (11) proposed a twochamber model that resulted in a secondorder solution. They stated the superiority of their model in relation to firstorder ones based on the fact that the secondorder function profile is much more similar to empirical output profiles than are firstorder function profiles. Several flaws can be detected in their model, and we list them in Appendix A. One of these flaws (perhaps the most relevant one) is the statement that the profile of the output will approach a firstorder dynamics if the first chamber is large in relation to the second one (or viceversa). Empirically, one can demonstrate that the larger the volume of a chamber the less the output is close to a firstorder model (see Figure 5A). This phenomenon is not related to an a priori predetermined number of chambers counted from an anthropocentric point of view.
Let us state the problem in another way. Consider a given system (Footnote 1) Syst_{1} with an output that apparently obeys a given order (e.g., N order). Then, increasing the volume of such a system or decreasing the convective flow through it (i.e., slowing down the system) would result in a) a system (Syst_{2}) that is of the same N order as Syst_{1} but with different values for the time constants that describe it, or b) a change in the order of Syst_{2} in relation to the apparent N order describing Syst_{1}. Mutatis mutandis, the question holds true for the opposite, i.e., when a system is speeded up. Empirically, we already know that a change in the order of the apparent function describing the output will occur. However, none of the present models explains why and how this happens to be so. These models also cannot account for the socalled "time lag" (i.e., a finite amount of time between the input taking place and some alteration occurring in the sensor cell) (Footnote 2). All these problems are present in such models because the latter are not truly describing the mechanics of the systems. Instead, they are fitting a preset of differential equations to the observed output.
In the present study we developed an approach to openrespirometry system dynamics employing basic concepts of thermodynamics and statistical mechanics. This approach allows an unequivocal understanding of the principles underlying the relationship between volume and flow in openrespirometry systems, thereby clearly explaining why and how there is a change in the apparent order of the output when a system is speeded up or slowed down. We also explore the role of the sensor cell itself in the system and the difference between calibrating an openrespirometry system (i.e., how to obtain its dynamics) by a continuous input and by a unit impulse.
Methods and Results
The basic experimental setup
Appendix B contains a list of symbols and definitions. We describe here the main features of openrespirometry systems. These systems consist of tubes connected to each other in series. The system is said to initiate at the L_{0} point, i.e., the point where the system starts physically (before L_{0} is the outside environment). The series of connected tubes reaches the entrance of the sensor cell, S, at L_{S}. The sensor cell has a given volume and ends at a point called L_{S+Ds}. Finally, the system ends at a point called L_{f}: beyond L_{f} there is, again, the outside environment. There is a convective flow, , of ambient gas through the system from L_{0} to L_{f}. determines a sense of convective flow within the system, defined as the x+ (x positive) sense. The input into the system occurs at a point called L_{0IN}. For practical reasons we will simply state that L_{0IN}>L_{0}, i.e., the input occurs beyond the entrance to the system, in the x+ sense. The connected tubes consist of a volume which, from the "output point of view", extends from L_{0IN} to L_{S+Ds} (not just to L_{S}) as we will demonstrate below. The gas molecules of the input flow through the system from L_{0IN}. Figure 1 illustrates the main features of openrespirometry systems. For the sake of simplicity, but without loss of generality, the following constraints are part of this approach: a) temperature is constant; b) the crosssection of the connected tubes is constant; c) the sensor cell has the same crosssection as the other tubes; d) the external environment acts as an infinite pool and sink; e) all gas molecules have the same mass and diameter, and collisions are elastic; f) the system is in a steadystate of pressure, i.e., the transients of the convective flow have finished, and g) there is no input loss (this is why L_{0IN}>L_{0}) and turbulence will not be accounted for in the approach.
Main symbols and definitions
If all molecules had the same velocity in the x+ sense of the system the output of any given input would look like a square pulse. However, in real systems, this is not the case, because the input molecules do not have all the same velocity, as will be demonstrated below.
Real velocities of the particles  total and the X component
At any given temperature (greater than 0 K) the distribution of velocities in a population of gas molecules follows a Maxwell distribution (14):
where M is the molar mass of the molecules, R is the gas constant, T is temperature and v is the specified velocity of a population f of molecules (Figure 2A). At any Lj point along the system (L_{0}<Lj<L_{f}) the profile of velocities described by equation 1 is expected to be found. Consider now a step change in the input that occurs at t_{0} (t_{0} is the exact moment when a transition in the amplitude of a former constant input occurs). This new input is composed of a set of molecules whose velocity profile is described by equation 1 as well, and this set of molecules shows this profile at any observation time (given that we are considering a conservative system, item (a) above). However, this does not mean that each molecule in the set is locked in a fixed velocity. On the contrary, each time a molecule collides its velocity changes. Thus, each molecule tends to reach a mean velocity due to collisions. Over time, the set of these molecules can be described as having a mean velocity (the mean velocity of the Maxwell distribution: although the profile of the set (given by equation 1) is the same at any particular time. This mean velocity has a standard deviation, , where sd_{mxw }is the standard deviation of the original (Maxwell) distribution divided by the square root of the number of collisions of the particle from t_{0} to t (the result for is simply the Law of Large Numbers or central limit theorem (15)). Due to the huge number of collisions a particle suffers within short periods of time (see below) we will simply use and instead of and, respectively. Therefore, a gaussian curve adequately describes the velocity profile of the particles in a given initial set:
We insist in the difference between equations 1 and 2: the profile of velocities is the one described by equation 1 at any given moment of observation of the set, and the gaussian curve of equation 2 is the net result of the changes in velocity of the particles along time. Given that each molecule in a gas phase under ordinary conditions collides about 10^{9} times per second, the mean velocity of each particle tends to be determined in the very first second after the input takes place. Figure 2 illustrates these points.
Thus, it was shown that the total velocity, v, tends to reach a mean value for each particle as time goes by. This total velocity is the vector composed by the three orthogonal components of velocity:
These orthogonal components are independent of each other, i.e., changes in one of them do not imply changes in the other ones (as long as the temperature is constant and, therefore, the total kinetic energy of the initial set is constant (14)). We are concerned with the travelling rate in the x direction, i.e., with the speed of the particles going from L_{0IN} to L_{S+Ds}. The distribution of velocities on a given axis (e.g., x) is:
With time, each molecule in the initial set attains a mean velocity on such an axis, and, as seen before, the set can be described as having a mean velocity on the axis () and a standard deviation of this mean ():
It should be noted that equation 5 describes the mean velocity in the x direction (or any other orthogonal axis), but is blind to the sense of the motion (i.e., it represents both the positive and the negative components in the given direction). We will now consider what happens when convective flow is present.
Apparent velocities of the particles
Gas particles are in a constant rocking motion, coming and going all the time. They collide with each other and, at such collisions, they can change the sense of their motion (e.g., from a positive x sense to a negative one). When convective flow is imposed on the system, there is nothing that really pushes or pulls each particle. Instead, there is a higher probability that a particle once moving in the positive sense of the flow (as defined above) will continue in such a sense longer than when it is moving in the opposite one. Given that changing in the sense of the motion only occurs when particles having opposite senses collide, we can write the difference in probability that a particle would change sense at a given point Lj of the system as:
where z stands for the collision frequency that a particle experiences at Lj when travelling in the positive (z_{+}) and in the negative (z_{}) sense of the x axis. Thus, when collision frequency is the same in both senses (no convective flow is present) L = 0, i.e., only the probability of diffusion affects the movement of particles from one point to another. A good approximation for L (as we justify in Appendix C) is:
The proportion of time that a particle spends travelling in the positive sense of the flow
The frequency of collisions (z) of a particle is (14):
where s is the collision crosssection of the molecules, N is the density of particles in a given volume and
where = 2^{.}
as stated above, a particle will change its sense of motion when it collides with a particle coming from the opposite sense. Thus, let us define a function L that measures (or estimates) the proportion of displacement that a particle makes in a specified sense in relation to the opposite displacement. Therefore, L ends up estimating the amount of time a particle spends travelling in a given sense in relation to the amount of time travelling in the opposite sense.
Convective flow is generated either by compression or by suction. When compression is the causal factor there is an increase in the collision frequency in the negative sense of the flow (i.e., upstream). On the other hand, when suction is the causal factor, there is a decrease in the collision frequency in the positive sense of the flow (i.e., downstream). Both the increase and decrease just cited are related to the velocity of the moving wall of a piston, Dw. Consider, for example, a particle moving towards the wall of the piston (in a suction setup) with velocity v_{x} and at a distance d from the wall. If the wall is not moving, after a time of d/v_{x} the particle is expected to be found colliding with the wall. If the wall is moving with a speed of Dw (Dw<<v_{x}), after a time of d/v_{x} the particle would not have reached the wall, which is Dw^{.}d/v_{x} units of length ahead. Therefore, a second particle expected to collide with the first one (when the latter would be returning after an elastic shock with the still wall) experiences a decrease in the collision frequency, proportional to the difference v_{x}  Dw when the wall is moving. Therefore, we will write the L function for suction and compression as, respectively:
Notice that L is independent of the Lj position in the system where it is being evaluated. Also, notice that Dw<<Dw/
gives a good estimate of the proportion of time that particles spend travelling in the positive sense of convective flow.
where Dw is the absolute value of the velocity of the moving piston causing convective flow and is the mean velocity of the particles in the x direction. Notice that L is independent of the Lj point of the system. The apparent velocity (in the x direction) of a particle will be the product of its mean velocity and L. Therefore, the whole set has a mean apparent velocity indicated by the following equation:
A note on variance: many factors were not taken into account in this study. Some of them are turbulence, inhomogeneities of the gas medium, changes in the geometry of the tubing, and the variance in L. Analysis of such factors is beyond the scope of the present study. However, it is very important to note that these added variances make a considerable contribution to the final spreading of velocities around the mean value.
We will not propagate error and the standard deviation of the above apparent mean velocity will be simply described as
^{ .}L, where the superscript + indicates that added components of variance should be included. Normality is preserved (Footnote 3) and the set of particles can be adequately described by a gaussian function as:
Equation 9 is the foundation of this study. It tells us that any given initial set of particles will attain a mean apparent velocity on the axis of the convective flow imposed on the system, and that particles of such a set will have apparent velocities normally distributed around the mean velocity of the set. This obviously explains, unequivocally, the existence of the time lag between an input and the beginning of the corresponding output: it is the length of time that the faster particles of the input take to travel from L_{0IN} to L_{S} (even though a gaussian function ranges from ¥ to +¥, for practical purposes the entire population can be considered to lie between 3sd and +3sd around the mean, so there will exist a minimum time required for the faster particles to travel). In the next section we will explore the output dynamics that results from equation 9.
Output dynamics of openrespirometry systems
The input. Most of the arguments of the functions will be omitted for the sake of simplicity (e.g., G instead of ). Let us consider a set A of gas molecules as an unitary amplitude input into the system. Therefore, such a set has particles which have velocities distributed according to function G (equation 9). Obviously, set A has a total number of molecules corresponding to the integral H of G (because the velocities in this case are equal to or greater than zero, the integral will be evaluated only in the positive range):
The total amount of molecules in a continuous input into the system at time t is the integral of A in relation to time, from t_{0} = 0 to t:
The output. Let us define a function E_{(t)} representing the state of the sensor at time t. The state of a sensor is a function (linear, in general) of the occupancy level of the sensor by the subject of measurement of that sensor. The entry rate into the sensor is the amount of molecules (subject of measurement) that cross the entrance to the sensor cell at L_{S}. The exit rate is the amount of molecules leaving the sensor cell at L_{S+Ds}:
Therefore, the output dynamics of an openrespirometry system, equivalent to the state E_{(t)} of the sensor, is only a function of the velocities of the particles of the input travelling a linear distance to reach the entrance and then the exit of the sensor. The apparent mixing or dilutional features of these systems are just the result from the difference in the time of arrival, at the sensor, of the particles of a given input set (see below). This was never taken into account before. By examining equation 12 we can see that when
i.e., roughly speaking, when the exit rate equals the entry rate, and so, E_{(t) }= constant. At that time, the output stabilizes at a level, and such a level is the same as the amount of molecules coming from a corresponding time interval at the input (see equation 10). In other words, the output becomes linearly related to the input.
Another point that was also never appreciated before is that the leaving rate of the gas molecules from the sensor is crucial to the output dynamics. To stress such a point, consider a very long sensor (in the x direction), in a way that
at a time t when
i.e., the entry rate attains a stable value at a time when the exit rate still equals zero. Roughly speaking again, the slowest subset of molecules enters the sensor while the fastest subset has not reached the exit of the measuring device. Under these conditions, the output becomes a linear function of time (from equation 11, E_{(t)} = t.H) (Footnote 4)! This would be impossible to understand using other current models interpreting the output of openrespirometry systems. Even when the sensor is not that long some linear component can be present, arising from the slower subset(s) of molecules.
Particular solution for a small measuring device. Let us consider a very short sensor (in relation to the convective flow employed), where L_{S}@ L_{S+Ds}, and let us state without proof that this happens to be the most usual finding in openrespirometry systems. Thereby, the integral of equation 12 becomes (see Appendix D):
Justification of the use of equation 13 to describe the output of a "very short sensor"
The integration of equation 12 over time has no analytic solution. However, the problem of the length of the sensor can be demonstrated in another way. Let us define a as a subset of A composed of a class of molecules that has the same apparent velocity in the x direction, say v_{a}. Thus, in A, a contains G_{(v}_{a}_{)} particles. The constant input constituted by a begins at t_{0}. Thus, a time lag t_{m1}_{a} until the initial a molecules reach the entrance to the sensor will exist:
The same molecules will reach the outlet boundary of the sensor at a time t_{m2}_{a}:
Therefore, for the homogeneous input a:
where en_{(t)} and ex_{(t)} stand for the entry and exit rates, respectively. By integrating both rates of equation D3 in relation to time we obtain the state of the sensor:
Notice that for t_{m1}_{a}£t£t_{m2}_{a} a linear increase with time occurs in the occupancy level of the sensor by molecules of the specified class. For t>t_{m2}_{a} the occupancy level of the sensor by the a class attains a plateau. Such a plateau value is of the general form T_{i}^{.}G_{(vi) }where T_{i} = t_{m2i } t_{m1i} . As can be seen, each class of molecules will have a different T due to the different time each class takes to cross the sensor cell. The important point to be noted is that linearity with time should be expected in the output while t<t_{m2i}.
Let us define a very short sensor as one for which T_{slowest}®0, i.e., the time the slowest subset of A takes to cross the sensor can be neglected on practical grounds. Note that this is not a limit, because L_{S+}_{D}_{s } L_{S} is measurable. Obviously, T_{i} for all the other classes can be neglected as well. Therefore, we may consider that each class of velocity of A is represented at the sensor proportionally to the other classes, and such a proportionality is given only by each G_{(vi)} (because the different T_{i}s are overlooked). Thus, under the assumption of a very short sensor, to obtain E_{(t)} integrating equation 12 requires integrating G_{(vi)}, and this is H, as found in equation 13.
Equations 12 and 13 are useful to approximate the real output. Thus, the derivative of the output of the system is the distribution function of the apparent velocities of the particles (equation 9), and the output itself is the integral of that gaussian function. Figure 3 illustrates numerical solutions of equation 12 and equation 13 for constant input in systems with different volumes (notice that volume is the linear distance from L_{0IN} to L_{S}) under the same convective flow (apparent velocity of the particles). Notice how the profiles of the output suggest the existence of dilutional spaces (volumes) dictating the dynamics of the systems. Increasing the volume of the system (i.e., increasing the linear distance from L_{0IN} to L_{S}) appears to increase the number of "volumes" diluting the output, and this will be discussed below. However, before approaching this subject we will examine another common calibration procedure, the "unit impulse". In this technique, a set of particles A makes up a unique input which occurs within an interval of "zero" units of time. Therefore, as in equation 10,
Thus, the output, when integrated over time, results in H, and as a consequence (Footnote 5):
where the superscript ui stands for "unit impulse". Notice that the output of a unit impulse (equation 14) corresponds to the first derivative in relation to time of a continuous input (equation 12). When the time constants of a given system are determined using an exponential decay model (as the current models) there would be little distortion in such constants, because the exponents in those decays are the same in both the integral and its derivative. However, authors overlook the ascending portion of (Footnote 6) (see Figure 3A and next section) and look only at the descending ("exponential decay") part of the curve. Time constants are then adjusted to this descending portion of the curve. This can lead to serious distortions in the reconstituted experimental inputs.
Change in output "order" with the slowing down or the speeding up of openrespirometry systems
This section is devoted to the problem stated in the Introduction of this study: increasing the volume of a Syst_{1} system or decreasing the convective flow through it (i.e., slowing down the system) would result in a Syst_{2} system that is of the same order as Syst_{1} but with different values of the time constants describing it, or would there be a change in the order of Syst_{2} in relation to the apparent order describing Syst_{1}? We will demonstrate that the latter alternative is the case. Firstly, we will approach the causes leading to a change in the apparent order of the output dynamics occurring in a system that is being described by a single time constant, i.e., a putative firstorder output. Then, we will extend the conclusion to any given apparent order of openrespirometry systems. Throughout the analysis it is necessary to bear in mind that we are considering functions which tend to an asymptotic value as time goes by. Thus, any pair of vectors representing such functions will tend to have norm®0 as t®¥ simply because they both tend to stabilize at a certain value. Therefore, in the following reasoning, we are concerned only with the period of time within which transients are present. In a general form: 0 = t_{0}£t£T, where t_{0} = 0 is the time when the output begins to be detected and T is a later time when the total amplitude of the output is attained, on practical grounds. Note that t_{0} in this section has a different meaning from that used in the preceding ones, because it is not the initial time at which the input takes place.
Consider a Syst_{1} system where Y1_{(t)}^{.} H1_{(t)} = e_{1} (where w_{1}.w_{2} is the norm between two vectors w_{1 }andw_{2}), and one tends to accept function Y as a good approximation to H within an e_{1} error. The Y function has the general form shown below (equation 15).
Proposition: given a system with output H1_{(t)} (H_{(t)} is found in equation 13) for which an approximation Y1_{(t) }@ H1_{(t)} (within an error e_{1}) is considered valid, the progressive increase of L_{S} (Footnote 7) or decrease of will always result in Y2_{(t)}^{.}H2_{(t)}>e_{1} with H2_{(t)}<<Y2_{(t)} for all t<t_{cross} (to be defined below). That is, at the beginning of the output, the approximating (new) function Y_{(t)} has values progressively greater than those of the real H_{(t)} function as a given system is slowed down.
Demonstration. Let functions f_{1} and f_{2} be:
where l is the inverse of a time constant. One intends to minimize (let this function be f_{3}) by varying l. The traditional method to do so is to find the value of l for which Resolving the square and taking the first derivative of f_{3} in relation to l, one obtains the following result:
Therefore, because there are no squares in either term of equation 16, it becomes implicit that part of the values of Y_{(t)} must be higher than the corresponding H_{(t)} and another part must be lower for the sum to result in zero (unless Y_{(t)}º H_{(t)}, see below). If Y_{(t)}º H_{(t),} it follows that , but:
invalidating the putative Y_{(t)}º H_{(t)}.
The crossing time of G_{(t)} and l^{.}e^{l.t} , t_{cross¶}, is:
The initial conditions are:
So:
Let us define t_{cross} (t_{cross}>0) as the time when Y_{(t)} and H_{(t)} cross each other after their initial crossing at t_{0} = 0 (equation 19). It follows that Y_{(t)}>H_{(t)} for all t<t_{cross}, that is, the solution of the minimization (equation 16) implies that the initial part of the approximating function Y_{(t)} has higher values than the real function H_{(t)}. Because t_{cross¶} is directly proportional to L_{S} (equation 18) and equation 16 is a minimization, then Y2_{(t)} (for all t<t_{cross}) is progressively higher than H2_{(t)} as L_{S} is progressively increased (and the same is valid for decreasing ), irrespective of wether the computed value l_{2} (Footnote 8) satisfies equation 16 or any other metric minimization procedure. In other words, as an initial system 1 in which it is accepted that Y1_{(t) }@ H1_{(t)} is slowed down and becomes system 2, the approximating function Y2_{(t)} has greater values than the real function H2_{(t)} from t = t_{0} until their crossing at a later time (t_{cross}). The vectors representing the functions fall progressively apart. At the beginning of the output, the error e_{2} is greater than the formerly accepted e_{1} for the same period. This phenomenon, under compartmental analysis, emerges as the necessity of adding more "dilutional volumes" to a sum of exponentials (i.e., more time constants in order to "slow down" the initial part of the approximate function).
Consider any output H_{(t)} being described (by approximation) by a function Y_{(t)} in the general form Any l_{i} represents a relationship between a putative "dilutional volume" and flow. Thus, the above reasoning extends to any number of time constants employed to describe the output of a given system. This reasoning demonstrates the obligatory change in the apparent order of the system output, and, therefore, the impossibility of a given order be the same as openrespirometry systems are speeded up or slowed down. Note that the definition of volume (L_{S}) is not anthropocentric, i.e., we are not "counting chambers". Instead, L_{S} is a continuum measure, blind to "chamber" definitions.
Some experiments were done in order to illustrate the points demonstrated above: a) the change in the apparent order of output as volume and flow are changed, and b) that this is not linked to the number of "chambers" from an anthropocentric viewpoint. Figure 4 shows the system employed. Note that there is a single chamber whose volume can be increased or decreased on the parallel axis to the convective flow, which in turn can be increased or decreased as well. Figure 5 presents the empirical results obtained by changing volume or flow in the system described above. Also, numerical evaluations of integrals of the general form of equation 13 are shown. Notice the fine adjustment between empirical data and the predicted functions (see legend).
Transients are important features of openrespirometry systems. Whether they are detected or not in real data acquisition procedures is a question that has to be known beforehand by the researcher. In cases where transients are detected, and the input is to be evaluated during such transients, the dynamics of the output is a major problem. In this study we employed basic concepts of thermodynamics and statistical mechanics to develop an approach to the problem of the output dynamics of openrespirometry systems.
The main result of this mechanistic approach is that molecules of the input travel a linear distance from the place where such an input takes place to the sensor, with an apparent velocity normally distributed around a mean value (function G in equation 9). All the important features of the output dynamics are determined by this travelling rate of the molecules which end up reaching and leaving the sensor cell at different times. This is the observed dynamics. Once this is recognized and quantified, time lag (the time between the beginning of input and the time where something begins to be detected at the sensor) becomes easily and unequivocally explained. Other current models cannot account for such an explanation. This study also revealed that the rate at which the molecules leave the sensor cell is an important part of the observed dynamics, a fact never considered before. This leads to a particular solution to the output for "very short sensors" (equation 13). Unless such a particular solution can be applied to a real experimental setup, the presence of linear components in the output should be expected, and the current exponential approaches would be inherently inappropriate.
By applying the concepts developed, it was demonstrated that exponential decays are only functions approximating the real output one. The difference between calibrating the system with a continuous input and a "unit impulse" is stressed when one recognizes that the latter is the first derivative of the former. The resulting output is, therefore, the velocity distribution function G instead of its integral H. The distribution function G is composed of an ascending portion that cannot be simply ignored when computing the time constants of the approximating function to the output.
Finally, why and how the putative order of a system will change as the system is slowed down or speeded up is also presented. The main point is that at the beginning of the detected signal the approximate exponential function tends to have values progressively greater than the real output function as a system is slowed down. Therefore, more time constants need to be included in the approximate function for an adequate description of the output. This last result implies that neither a firstorder model nor a secondorder one can be taken for granted based on a system formerly evaluated or on the number of chambers counted from an anthropocentric point of view.
A very important point to be noted is that the present study was not intended to develop a new kind of signal reconstitution procedure for transients in openrespirometry systems. Therefore, under clinical conditions or in any other situations, this model should not be directly applied in order to obtain the input. However, this model opens the way to new approaches to the recovery of the input based on this mechanistic view of the process.
Discussion
Acknowledgments
We would like to thank Dr. R. Ranveaud, Dr. M.V. Baldo and two anonymous reviewers for their criticisms, which improved the quantification analysis presented here.
Footnotes
Footnote 1) This means all the chambers and subchambers, tubing, sensor cells, the convective flow, the absorbents, etc.
Footnote 2) Authors simply take for granted the existence of the time lag, without any further consideration.
Footnote 3) And this is true because these errors are normally distributed as well.
Footnote 4) Until molecules reach the outlet boundary and thus begin to leave the sensor. Note that this linearity is completely different from a linear relationship between the input and the output, as discussed above. This is a linear increase of the output in relation to time.
Footnote 5) We are still working with the constraint of a very short sensor.
Footnote 6) Do they consider that the peak of the curve corresponds to the total amplitude of the unit impulse input itself?
Footnote 7) The linear distance from the input to the entrance of a very short sensor (see above) directly related to the volume of the system.
Footnote 8) The time constant of Syst_{2}, a slowed down system in relation to Syst_{1}.
Address for correspondence: J.G. ChauiBerlinck, Departamento de Fisiologia, Instituto de Biociências, USP, 05508900 São Paulo, SP, Brasil. Fax: +55118187422. Email: jgcb@usp.br
Research supported by a FAPESP grant to J.E.P.W. Bicudo. Part of a PhD thesis presented by J.G. ChauiBerlinck to the Departamento de Fisiologia, Instituto de Biociências, Universidade de São Paulo, and supported by a CAPES fellowship. Received June 23, 1999. Accepted March 10, 2000.
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Appendix B
: mean velocity of the Maxwell distribution : mean velocity of the particles in the x direction: mean relative velocity (total): mean relative velocity of the particles on the x axis: mean apparent velocity of a set of molecules in the: mean apparent velocity of a particle on the positive: standard deviation of the mean velocity: standard deviation of the mean velocity in the x direction: standard deviation of the mean apparent velocity in the : mean velocity of the Maxwell distribution
 : mean velocity of the particles in the x direction
 : mean relative velocity (total)
 : mean relative velocity of the particles on the x axis

: mean apparent velocity of a set of molecules in thepositive sense of the x direction

: mean apparent velocity of a particle on the positivesense of the x direction
 : standard deviation of the mean velocity
 : standard deviation of the mean velocity in the x direction

: standard deviation of the mean apparent velocity in thepositive sense of the x direction (with added components ofvariance also taken into account)
Dw: the velocity of a piston wall causing convective flow
E_{(t)}: the state of a sensor at time t
G: the gaussian function of velocity distribution
H: the integral of G
l: a time constant
L: difference in probability that a particle would change
sense in the sense of the convective flow
L_{0}: the entry of an openrespirometry system
L_{0IN}: the place where input takes place in an
openrespirometry system
L_{f}: the exit of an openrespirometry system
Lj: any point between L_{0IN} and L_{S}
L_{S}: the entry of the sensor
L_{S+}_{D}_{s}: the exit of the sensor
t_{0}: time when a new input begins (or the beginning of
the output corrected for the time lag, as in section
"Change in output "order" with the slowing down or the
speeding up of openrespirometry systems"
Y_{(t)}: an exponential function approximating the output of
an openrespirometry system
z: collision frequency
Appendix C
Appendix D
Publication Dates

Publication in this collection
31 July 2000 
Date of issue
Aug 2000
History

Received
23 June 1999 
Accepted
10 Mar 2000