## Abstract

We developed a method for measuring airway resistance (R_{aw}) in mice that does not require a measurement of airway flow. An analysis of R_{aw} induced by alveolar gas compression showed the following relationship for an animal breathing spontaneously in a closed box: R_{aw} = A_{bt}V_{b}/[V_{t} (V_{e} + 0.5V_{t})]. Here A_{bt} is the area under the box pressure-time curve during inspiration or expiration, V_{b} is box volume, V_{t} is tidal volume, and V_{e} is functional residual capacity (FRC). In anesthetized and conscious unrestrained mice, from experiments with both room temperature box air and body temperature humidified box air, the contributions of gas compression to the box pressure amplitude were 15 and 31% of those due to the temperature-humidity difference between box and alveolar gas. We corrected the measured A_{bt} and V_{t} for temperature-humidity and gas compression effects, respectively, using a sinusoidal analysis. In anesthetized mice, R_{aw} averaged 4.3 cmH_{2}O·ml^{−1}·s, fourfold greater than pulmonary resistance measured by conventional methods. In conscious mice with an assumed FRC equal to that measured in the anesthetized mice, the corrected R_{aw} at room temperature averaged 1.9 cmH_{2}O·ml^{−1}·s. In both conscious mice and anesthetized mice, exposure to aerosolized methacholine with room temperature box air significantly increased R_{aw} by around eightfold. Here we assumed that in the conscious mice both V_{t} and FRC remained constant. In both conscious and anesthetized mice, body temperature humidified box air reduced the methacholine-induced increase in R_{aw} observed at room temperature. The method using the increase in A_{bt} with bronchoconstriction provides a conservative estimate for the increase in R_{aw} in conscious mice.

- bronchoconstriction
- methacholine
- tidal volume
- gas conditioning
- conscious mice
- anesthetized mice

airway resistance (R_{aw}) in small animals is used to evaluate the efficacy of pharmacological agents in the treatment of allergic airway disease. In particular, the use of barometric plethysmography with conscious unrestrained mice in a closed box is particularly appealing because of its noninvasiveness and technical simplicity. Drorbaugh and Fenn (7) showed that tidal volume (V_{t}) is related to excursions in box pressure caused by changes in temperature and humidity of the airflow from the box to the alveolar space. This method for measuring V_{t} has been validated in studies of newborn infants (7), monkeys (12), and mice (22); however, its accuracy has been questioned (9). The measurement of airway responsiveness from the cyclic variations of box pressure (17, 19) has been criticized because of the compounding effects of alveolar gas compression and the effects of temperature and humidity (gas conditioning). Lundblad et al. (17) from experiments in anesthetized mice concluded that the contribution of gas compression to the calculated airway resistance was half that contributed by changes in temperature and humidity. Some investigators have estimated the effect of gas compression to be 15% of the box pressure excursions (19).

In early studies, Jaeger and Otis (13) measured alveolar pressure due to gas compression in humans rebreathing body temperature humidified air from a bag in a body plethysmograph, a method suggested by DuBois and coworkers (8). In Jaeger and Otis's study, airway resistance was calculated from the alveolar pressure and flow measured simultaneously with a pneumotachograph.

In the present study we proposed a method for measuring airway resistance in mice that eliminates the temperature-humidity effects from the changes in alveolar pressure but does not require the measurement of flow. First, we examined the issue of the contribution of the gas compression and temperature-humidity to airway resistance and V_{t}. We repeated the analysis of Lundblad et al. (17) using a somewhat different approach. Differential calculus was applied to the adiabatic gas law to relate the respiration-induced changes in pressure and volume of the alveolar gas to those of the box gas. We derived an equation for airway resistance due to alveolar gas compression in terms of the area under the box pressure-time curve, V_{t}, and functional residual capacity (FRC). We used sine waves to analyze the contributions of gas conditioning and gas compression to the box pressure excursions. In anesthetized and conscious mice, we measured the relation between these two contributions from studies with both room temperature and body temperature humidified box air. We used the analysis to estimate the increase in airway resistance in response to methacholine. Our results suggested that in both anesthetized and conscious mice, body temperature humidified box air reduced the methacholine-induced increase in airway resistance observed at room temperature.

## Glossary

- A
_{bt} - Area under the box pressure-time curve for inspiration or expiration, mean of the two areas (cmH
_{2}O·s) - alv
- Subscript, alveolar gas
- b
- Subscript, box gas
- c
- Subscript, corrected parameter
- °C
- Degree centigrade, unit of temperature.
- C
_{dyn} - Dynamic lung compliance (ml/cmH
_{2}O) - C
_{L} - Static lung compliance (ml/cmH
_{2}O) - d
- Prefix, differential of
- δ
- Prefix, amplitude of
- exp
- Subscript, peak expiratory
- f
- Respiratory frequency (cycles/s, Hz)
- F
- Fraction
- FRC
- Functional residual capacity (V
_{e}, ml) - g
- Subscript, gas compression effects
- γ
- Isentropic exponent of alveolar and box gas
- h
- Subscript, temperature-humidity effects
- insp
- Subscript, peak inspiratory
^{o}K- Degree Kelvin, unit of absolute temperature
- K
- A
_{bt}f ratio, intervention vs. control - K
_{o} - A
_{bt}f ratio, body temperature humidified box air vs. room temperature box air - K
_{1} - A
_{bt}f ratio, methacholine vs. control, both with room temperature box air - K
_{2} - A
_{bt}f ratio, methacholine vs. control, both with body temperature humidified box air - K
_{3} - A
_{bt}f ratio, body temperature box air with methacholine vs. room temperature control without methacholine - P
_{alv} - Absolute alveolar gas pressure (mmHg)
- P
_{ao} - Airway opening pressure relative to ambient (cmH
_{2}O) - P
_{b} - Absolute box gas pressure (mmHg)
- PEF
- Peak expiratory flow in enhanced pause method
- Penh
- Airway parameter defined by enhanced pause method, (PEF/PIF)(Te − Tr)/Tr
- P
_{es} - Intraesophageal pressure relative to ambient (cmH
_{2}O) - P
_{g} - Gas compression part of box pressure (cmH
_{2}O) - P
_{h} - Temperature-humidity part of box pressure (cmH
_{2}O) - PIF
- Peak inspiratory flow in enhanced pause method
- P
_{wa} - Water vapor pressure of alveolar gas (mmHg)
- P
_{wb} - Water vapor pressure of box gas (mmHg)
- Q
- Airway flow (ml/s)
- R
_{aw} - Airway resistance if viscous pressure loss were entirely laminar measured by gas compression part of box pressure (cmH
_{2}O·ml^{−1}·s) - R
_{L} - Pulmonary resistance measured by conventional methods (cmH
_{2}O·ml^{−1}·s) - R
_{l} - Laminar flow part of the total viscous pressure loss (cmH
_{2}O·ml^{−1}·s) - R
_{t} - Turbulent flow coefficient if viscous pressure loss were entirely turbulent (cmH
_{2}O·ml^{−2}·s^{2}) - R
_{T} - Turbulent flow part of the total viscous pressure loss (cmH
_{2}O·ml^{−2}·s^{2}) - T
_{alv} - Absolute alveolar gas temperature (°K)
- T
_{b} - Absolute box gas temperature (°K)
- Te
- Expiratory time in enhanced pause method
- Ti
- Inspiratory time in enhanced pause method
- Tr
- Relaxation time in enhanced pause method
- V
- Gas volume (ml)
- V
_{alv} - Alveolar gas volume (ml)
- V
_{b} - Box volume (ml)
- V
_{e} - End-expiratory lung volume (FRC, ml)
- V
_{i} - Inspired gas volume (ml)
- V
_{m} - Mean lung volume, V
_{e}+ 0.5V_{t}(ml) - V
_{t} - Tidal volume (ml)
- 1
- Subscript, control
- 2
- Subscript, intervention

## MATERIALS AND METHODS

In this study we used both conventional and barometric body plethysmography to study BALB/c mice (body wt 20–25 g, *n* = 77). In the following sections, the protocols for the experiments are first described and are followed by the details of the experimental procedures.

### Effect of Room Temperature and Body Temperature Humidified Box Air

We separated the effects of gas compression and temperature-humidity on the box pressure excursions in both anesthetized and conscious mice. Mice were anesthetized with pentobarbital sodium (70 mg/kg) administered intraperitoneally, and their tracheas were cannulated. With the anesthetized mouse placed inside a barometric (box) plethysmograph (Buxco Electronics, Troy, NY) filled with room temperature (21–22°C) unhumidified air, the box was exposed to saline aerosol for 30 s, the box was sealed, and the box pressure excursions were measured for 10–20 s. Then the seal of the barometric box was removed, and a bias flow of 1 l/min was drawn through the box. We measured the parameter Penh (see below) according to the method of Hamelmann et al. (11) after the mouse was exposed to saline aerosol for 30 s. Then, the anesthetized mouse was placed in a conventional body box (see below) with room temperature box air, and its lung resistance (R_{L}), dynamic lung compliance (C_{dyn}), V_{t}, and FRC were measured after saline aerosol exposure for 30 s. The entire experiment was repeated with the box air heated to body temperature (37–38°C) humidified (∼90% saturation) air. While the box was heated, it was partly opened to reduce the accumulation of CO_{2}. Box pressure excursions were measured for 10–20 s on sealing the box at 37–38°C. In a separate group of conscious mice, we measured box pressure-time excursions and Penh after saline aerosol exposure using room temperature and body temperature humidified box air in turn. V_{t} was measured from the box pressure excursions by the method of Drorbaugh and Fenn (7).

Aerosol was generated by placing a 10-ml saline or methacholine (25 mg/ml) solution in the cup of an ultrasonic nebulizer (DeVilbiss, Somerset, PA), and it was delivered via a connecting tube and a three-way connector to the inside of the barometric box or the inlet of the tracheal tube of the animal in the conventional box. The median size of the aerosol was ∼3 μm (range, 1–5 μm), according to the manufacturer's specification. The box air was heated to body temperature (this required 20–30 min) with an infrared lamp and humidified by placing wet tissue paper in the box.

### Effect of Methacholine in Anesthetized Mice at Room Temperature

In a group (*n* = 11) of anesthetized mice, we measured box pressure-time excursions with aerosolized saline exposure followed by methacholine aerosol exposure for 30 s. Then, with the anesthetized mouse in a conventional body box, R_{L}, C_{dyn}, V_{t}, and FRC were measured after saline aerosol exposure for 30 s and remeasured after methacholine aerosol exposure for 30 s.

### Effect of Methacholine and Body Temperature Humidified Box Air in Anesthetized and Conscious Mice

In a group (*n* = 11) of anesthetized mice, we studied the effects of temperature and humidity of the box air on the methacholine response. First, we used the sealed barometric box filled with room temperature air to measure the box pressure-time excursions for 10–20 s after the anesthetized mouse was exposed to saline aerosol for 30 s. The seal of the barometric box was removed, and a bias flow of 1 l/min was drawn through the box and the parameter Penh was measured. We then remeasured the box pressure-time excursions and Penh with the box filled with body temperature humidified air after methacholine aerosol exposure for 30 s. Subsequently, the anesthetized mouse was placed in the conventional body box with room temperature air, and its R_{L}, C_{dyn}, V_{t}, and FRC were measured after saline aerosol exposure for 30 s. These measurements were repeated with body temperature humidified box air after methacholine aerosol exposure for 30 s. In a group (*n* = 10) of conscious mice, we measured box pressure-time excursions and Penh with room temperature box air after saline aerosol exposure for 30 s, then repeated the measurements after methacholine aerosol exposure with body temperature humidified box air.

### Effect of Methacholine and Box Air Temperature in Conscious Mice

In a group (*n* = 12) of conscious mice, box pressure-time excursions and Penh were measured for 10–20 s after aerosol saline exposure for 30 s with room temperature box air and remeasured after aerosolized methacholine exposure for 30 s. The latter experiments were repeated in another group (*n* = 11) of mice with body temperature humidified box air.

### Determination of Area Under the Box Pressure-Time Curve Using the Barometric Plethysmograph

We used a commercially available barometric plethysmograph (Buxco Electronics). The plethysmograph consisted of two cylindrical chambers: the main or animal chamber (7.5 cm internal diameter and 5.5 cm height) and a reference chamber (7.5 cm internal diameter and 3.5 cm height). The reference chamber served to reduce perturbations in the room air caused, for example, by a person walking in the room. The plethysmographic (box) pressure relative to ambient was monitored with a differential pressure transducer. The transducer was calibrated by use of a water manometer made of a 50-cm-long glass tube inclined to a height of 1 cm that produced a change of 0.02 cmH_{2}O per centimeter horizontal distance. The pressure signal was amplified, digitized, and stored on a computer, and the pressure-time curve was plotted (BioSystem X, Buxco Electronics). The area under the box pressure-time curve (Fig. 1) was calculated over 10 cycles and averaged (Image-Pro Plus Version 3.0).

### Measurement of Penh by Barometric Plethysmography

The box pressure excursions caused by the animal breathing inside the box produced airflow through a port fitted with a wire screen in the wall of the box. The flow through the screen was monitored by measuring the pressure drop across the screen with a differential pressure transducer. The flow signal was amplified, digitized, and stored on a computer, and the desired respiratory parameters were calculated (BioSystem XA program, Buxco Electronics). We followed the method of Hamelmann et al. (11) and measured inspiratory time (Ti), expiratory time (Te), relaxation time (Tr), peak inspiratory flow (PIF), and peak expiratory flow (PEF). Tr was defined as the time at which the area under the pressure-time curve decayed to 36% of the total expiratory period. Penh, defined as Pause × PEF/PIF, where Pause = (Te − Tr)/Tr, was calculated.

### Measurement of R_{L}, C_{dyn}, V_{t}, and FRC Using Conventional Body Plethysmography in Anesthetized Mice

In anesthetized mice, we measured esophageal pressure (P_{es}), airway-opening pressure (P_{ao}), flow rate, FRC, and V_{t} as previously reported (15, 16). In brief, the anesthetized mouse was positioned in the plethysmograph with the tracheal cannula (1 cm length of an 18-gauge needle) connected to a port in the wall. Airway flow was monitored with a differential pressure transducer (Validyne DP45) as the pressure drop across three layers of wire screen (325-mesh) that covered a port in the wall. V_{t} was obtained by integration of flow. P_{ao} was measured with a pressure transducer (DTX/plus, Viggo-Spectramed). P_{es} was measured via a pressure transducer (DTX/plus) connected to a saline-filled PE-100 tube with its tip positioned in the lower third of the esophagus. Transpulmonary pressure was the difference between P_{ao} and P_{es}. During spontaneous breathing, R_{L} and C_{dyn} were determined by the method of Amdur and Mead (1). FRC was measured by neon dilution (15), and mean respiratory frequency (f) was measured from the box pressure-time excursions over several cycles.

To measure FRC by neon dilution (15), the lungs were inflated with air containing 0.5% neon via a syringe from FRC to ∼50% vital capacity. The total gas in the lungs, dead space of the instrument, and the syringe was mixed by repeatedly injecting and withdrawing gas 10–20 times. Then the neon concentration of the gas mixture was measured by gas chromatography. Total volume was calculated by a neon mass balance. FRC was total volume minus the instrumental dead space and syringe volume.

### Statistics

Data are reported as mean values ± SE. We used paired *t* and unmatched *t*-tests where appropriate to evaluate significant difference between two groups of data. We accept *P* < 0.05 to be significant.

## THEORY

### Airway Resistance Due to Alveolar Gas Compression

Consider the animal located in a sealed box of volume V_{b} and absolute gas pressure P_{b}. Lung (alveolar gas) volume is V_{alv} and alveolar pressure is P_{alv}. Initially, we neglect any effects caused by the differences in temperature and humidity between the box gas and alveolar gas. The contribution of temperature-humidity effects is evaluated separately. During inspiration, the contraction of the respiratory muscles causes an expansion (decompression) of the alveolar gas that obeys the adiabatic (isentropic) gas law: (1) Here P is the absolute gas pressure, V is the gas volume, and γ is the isentropic exponent. For an isothermal process, γ is equal to 1. From implicit differentiation of *Eq. 1*: (2) Applying *Eq. 2* to the alveolar gas volume, the change dV_{alv} due to gas decompression is given by: (3) Here P_{alv} is the absolute alveolar gas pressure. For an animal breathing spontaneously in a sealed box, dV_{alv} is equal to and opposite in sign to the change in box gas volume due to gas compression (−dV_{b}) given by an equation analogous to *Eq. 3*: (4) Substitution of dV_{b} = −dV_{alv} in *Eq. 4* and the use of *Eq. 3* result in the following equation: (5) Here we assume that P_{alv} equals P_{b} and that the same adiabatic process exists in both alveolar gas and box gas. The error produced by the latter assumption is evaluated in the discussion. The R_{aw} on inspiration is defined as: (6) Here (dP_{b} − dP_{alv}) is the difference between dP_{b}, the change in the airway opening pressure equal to the box pressure, and dP_{alv}. Q is the airway flow, equal to dV_{alv}/d*t*, the time (*t*) derivative of V_{alv}. Since dP_{alv} ≫ dP_{b} because V_{b} ≫ V_{alv} (in our experiments V_{b} = 220 ml, V_{alv} = ∼0.2 ml) from *Eqs. 3* and *4*, R_{aw} = −dP_{alv}/Q. dP_{alv}, the alveolar driving pressure for flow, is equal to the viscous pressure loss (R_{aw}Q). Substitution for dP_{alv} in *Eq. 5* results in the following equation: (7) In the barometric box method, dP_{b} is measured over the respiratory cycle. Integration of both sides of *Eq. 7* with respect to time results in the following equation: (8) Here, R_{aw} is assumed constant as if laminar flow conditions exist; the effect of turbulent flow conditions is treated in appendix a. The assumption of laminar flow provides only a lower limit for the viscous pressure loss estimated from A_{bt} that includes both laminar and turbulent flow contributions. V_{b} is treated as a constant because dV_{b} is negligibly small compared with dV_{alv}. The integral on the right-hand side of *Eq. 8* is equal to V_{alv}^{2}/2 and is evaluated on inspiration between the end-expiratory volume (V_{e}); that is, FRC, and the end-inspiratory volume equal to V_{e} + V_{t}. The integral on the left-hand side is evaluated during inspiration. R_{aw} is given as follows: (9) is the area (A_{bt}) under the dP_{b}−*t* curve for inspiration. A similar equation applies for expiration. We define the mean lung volume (V_{m}) during breathing as: (10) Substituting for V_{m} in *Eq. 9* results in the following equation: (11) Lundblad et al. (17) derived an approximation to *Eq. 11* with V_{m} replaced by V_{e}. The error inherent in this approximation is given in the discussion.

### Effects of Increased Temperature-Humidity and Gas Compression on Box Pressure Excursions

#### Sinusoidal analysis.

In the foregoing analysis, only gas compression effects were considered. However, an animal breathing room temperature box air produces a change in box pressure caused by the change in V_{alv} as the air on flowing into the airways becomes saturated with water vapor and heated to body temperature (7, 13, 14). We used sine waves to represent the effects of gas compression and temperature-humidity on box pressure. This was justified because a frequency analysis of box pressure-time curves showed that most of the energy (∼90%) occurred at the fundamental frequency (Fig. 1). This characteristic was verified in data from anesthetized mice exposed to aerosolized saline and methacholine by computing the absolute difference between the areas under sine waves with amplitudes equal to measured peak inspiratory and expiratory pressures and the measured areas. This difference as a fraction of the measured areas averaged 8.1 ± 2.2% (*n* = 10) for saline aerosol exposure and 9.3 ± 1.8% after exposure to aerosolized methacholine. Actual area differences rather than absolute differences averaged −1.1 ± 1.7% and 0.9 ± 11% for saline and methacholine exposure, respectively, and were not significantly different from zero, indicating no systematic variation from the sine wave.

The cyclic changes in box pressure (P_{b}) with an animal breathing room temperature air in a box are considered as the sum of the effects of gas compression and temperature-humidity. The change in box pressure due to gas compression of the alveolar gas (P_{g}) is in phase with the alveolar pressure (17) or flow and is represented as a sine wave with breathing frequency (f) and amplitude (δP_{g}) in Fig. 2. The change in box pressure contributed by the changes in temperature and humidity (P_{h}) is proportional to and in phase with inspired volume (Ref. 7, *Eq. A11* of appendix e) and is represented by a sine wave of amplitude (δP_{h}) that is 90° (π/2 radians) out of phase with the gas compression wave and displaced along the ordinate by δP_{h}. Then, P_{b} is given by: (12) *Eq. 12* can be rewritten as a single sine wave: (13a) (13b) Here tan Θ = δP_{h}/δP_{g}. This represents a phase difference that could be used to evaluate changes in V_{t} (see appendix b). In *Eq. 12* and Fig. 2, the baseline (zero) pressure is referenced to thebeginning of inspiration at P_{h} of 0 (FRC). In practice, the zero pressure is sometimes taken as the lowest point of the P_{b}-*t* curve. However, *Eqs. 12* and *13* indicate that because P_{b} is not in phase with the inspired volume (P_{h}), the minimum point of P_{b} cannot in general be used as the beginning of inspiration (Fig. 2). Nonetheless, shifts in the reference pressure are shown to have little effect on the measured area under the P_{b}-*t* curve, which is used in subsequent analysis to determine R_{aw} (see below and appendix c).

To simplify the analysis, we consider P_{b} − δP_{h} given by *Eq. 13* as the measured box pressure excursion with an amplitude δP_{b} = (δP_{g}^{2} + δP_{h}^{2})^{1/2}. The errors inherent in this approximation are discussed in a following section. The sum of the magnitudes of the areas under the inspiratory part and expiratory part of a sine wave of amplitude δP is 2δP/(πf). Accordingly, the areas under the gas compression, temperature-humidity, and box pressure curves are 2δP_{g}/(πf), 2δP_{h}/(πf), and 2δP_{b}/(πf), respectively. The area ratio of any two curves is equal to the ratio of their pressure amplitudes. Thus δP_{g}/δP_{b} is equal to the ratio of the area under the gas compression curve to the area under the measured box pressure curve, and from *Eq. 13b* is given by: (14) From measurements of δP_{b}, δP_{g} can be calculated if δP_{h}/δP_{g} is known. The parameter δP_{h}/δP_{g} was measured from an experiment with room temperature box air and body temperature humidified box air (see below).

#### Errors in V_{t}.

The amplitude δP_{b} has been used to calculate V_{t} based on the assumption that only δP_{h} contributes to δP_{b} (7, 12, 22). The equation relating changes in P_{b} to the inspired volume derived by Drorbaugh and Fenn (7) is given in appendix e (*Eq. A11*). The error in V_{t} with the assumption that only δP_{h} contributes to δP_{b} is evaluated as follows. δP_{h}/δP_{b} is related to δP_{g}/δP_{h} using *Eq. 13b*: (15) Thus if δP_{h}/δP_{g} is known, δP_{h} required to give the correct value for V_{t} in terms of the value calculated using δP_{b} can be determined.

#### Errors due to baseline shifts in P_{b}.

The use of a sine wave (*Eq. 13*) to represent box pressure rather than the actual box pressure (*Eq. 12*) produces a smaller area (sum of inspiratory and expiratory area magnitudes) under the box pressure curve and an underestimate of R_{aw}. An analysis showed that the error in using a sine wave to represent box pressure was <10% (see appendix c). Thus no correction for the area approximation or any adjustment of the box pressure baseline was deemed necessary. However, in practice, the error in area could be eliminated by shifting the pressure baseline to make peak inspiratory pressure (δP_{insp}) equal to peak expiratory pressure (δP_{exp}) before measuring the area.

### Application of Theory to Experiments

In the following sections, we applied the foregoing theory to experimental data to obtain corrected values for changes in airway resistance caused by an intervention such as a change in the box air conditions or exposure to methacholine aerosol. The equation used to evaluate experimental data is obtained using *Eq. 13b*: (16) Here subscript 1 represents the control condition and subscript 2 represents the intervention. With inspiratory area equal to expiratory area (A_{bt}), K is then the A_{bt}f ratio. In practice, we used the average of inspiratory and expiratory areas. Thus A_{bt} is equal to δP/(πf).

### Estimate of δP_{g}/δP_{h} for Room Temperature Box Air Conditions

In two groups of anesthetized and conscious unrestrained mice, we measured δP_{b} with room temperature box air (δP_{b1}) and with body temperature humidified box air (δP_{b2}). We assumed that with body temperature humidified box air δP_{h2} was zero and that δP_{g1}, the viscous pressure loss for flow, did not change. This behavior required that both the flow and R_{aw} did not change (*Eq. 6*). The error produced by this assumption is evaluated below (see discussion and appendix e). Thus, from *Eq. 16*, δP_{b2}/δP_{b1} equals the measured A_{bt}f ratio (K_{o}): (17) From *Eq. 17*, δP_{h1}/δP_{g1} for an animal breathing room temperature box air is: (18) The experiments (Table 1) produced average K_{o} values of 0.39 and 0.44 for anesthetized and conscious mice, respectively. However, the values for δP_{h1}/δP_{g1} computed using *Eq. 18* for each mouse and then averaging were 6.5 ± 2.9 (SE) and 3.1 ± 0.78 for the anesthetized and conscious mice, respectively. K_{o} calculated using these average values in *Eq. 18* was 0.15 and 0.31. From *Eq. 17*, because δP_{b2} = δP_{g1}, δP_{g1}/δP_{b1} = K_{o}. Thus the actual area under the gas compression curve averaged 15 and 31% of the area under the box pressure curve for the anesthetized and conscious mice, respectively. On this basis, A_{bt} due to gas compression effects at room temperature was overestimated by 6.5-fold and 3.1-fold for the anesthetized and conscious mice, respectively. δP_{h1}/δP_{g1} was calculated as if body temperature humidified box air (37–38°C) was actually at the same temperature and humidity as the alveolar gas (δP_{h2} = 0). The errors produced by a difference in temperature-humidity between the box air and alveolar gas are evaluated below (see discussion and appendix e).

Because the calculated V_{t} is proportional to δP_{b1}, for δP_{h1}/δP_{g1} of 6.5 and 3.1 in *Eq. 15*, the actual V_{t} based on the temperature-humidity curve (δP_{h1}) was 0.99 and 0.95 times those computed using the measured δP_{b1}. Thus V_{t} at room temperature was overestimated by 1 and 5% for the anesthetized and conscious mice, respectively. The corrected values of A_{bt} and V_{t} together with a measurement of FRC produce the correct value for R_{aw} in *Eq. 9*.

### Effect of Methacholine With Room Temperature Box Air

Let the ratio of A_{bt}f with aerosolized methacholine exposure (subscript 2) to that with a prior aerosolized saline exposure (subscript 1) be equal to K_{1} in *Eq. 16*. Then δP_{g2}/δP_{g1} due to gas compression effects is related to the V_{t} ratio, δP_{h2}/δP_{h1}, by the following equation: (19) K_{1} is the measured A_{bt}f ratio and δP_{h1}/δP_{g1} (6.5 and 3.1 for anesthetized and conscious mice, respectively) is known from the previous experiments (Table 1). δP_{g2}/δP_{g1} is a function of δP_{h2}/δP_{h1}, the actual box pressure amplitude ratio due to temperature-humidity effects. δP_{h2} cannot be determined by using room temperature and body temperature humidified box air as was done for δP_{h1} because airway resistance changed with the body temperature humidified box air conditions (see below). Thus we assumed that δP_{h2}/δP_{h1} was 1, that is, V_{t} was constant, a behavior that was measured in the anesthetized mice. Fig. 3 is a plot of δP_{g2}/δP_{g1} vs. δP_{b2}/δP_{b1}, the measured box pressure amplitude ratio or A_{bt}f ratio (K_{1}), for different δP_{h1}/δP_{g1} isopleths with δP_{h2}/δP_{h1} of 1. For the average A_{bt}f ratio (K_{1}) of 1.6 (Table 2) with δP_{h1}/δP_{g1} of 6.5 in the anesthetized mice and δP_{h2}/δP_{h1} of 1, δP_{g2}/δP_{g1} was 8.3 (Fig. 3). Alternatively, for each anesthetized mouse, a value of δP_{g2}/δP_{g1} was computed using the measured K_{1} and V_{t} (δP_{h2}/δP_{h1}) ratio in *Eq. 19*. The A_{bt} ratio was obtained by dividing the computed value of δP_{g2}/δP_{g1} by the measured f_{2}/f_{1} value and the R_{aw} ratio was calculated by using *Eq. 9* with the measured V_{t} and V_{e} values. R_{aw} ratio averaged 8.4 ± 1.4 for the anesthetized mice.

In the conscious mice (see Table 4), K_{1} averaged 2.2 and with δP_{h1}/δP_{g1} of 3.1 and δP_{h2}/δP_{h1} of 1, δP_{g2}/δP_{g1} was 6.5. Alternatively, for each conscious mouse, a value of δP_{g2}/δP_{g1} was computed using the measured K_{1} and δP_{h2}/δP_{h1} of 1 in *Eq. 19*. The A_{bt} ratio was obtained by dividing the computed value of δP_{g2}/δP_{g1} by the measured f_{2}/f_{1} value and was equal to the R_{aw} ratio with V_{t} and V_{e} assumed constant (*Eq. 9*). R_{aw} ratio averaged 7.5 ± 2.0 for the conscious mice.

*Eq. 19* also indicates that δP_{g2}/δP_{g1} is a relatively weak function of the V_{t} ratio (δP_{h2}/δP_{h1}) for K_{1} > 1.5. For V_{t} ratios between 0.5 and 1.5, the error in the prediction of δP_{g2}/δP_{g1} for K_{1} of 1.5 is ± 20% and diminishes for greater K_{1} values.

### Effect of Methacholine With Body Temperature Humidified Box Air

We consider the case of body temperature humidified box air conditions without and with methacholine in conscious mice (Table 4). δP_{h} does not contribute to the box pressure excursions that consist of only gas compression effects. The ratio of box pressure with methacholine (δP_{b2}) to that without methacholine (δP_{b1}) is given by *Eq. 16* with δP_{h1} = δP_{h2} = 0: (20) K_{2} is the measured A_{bt}f ratio. R_{aw} ratio was equal to the A_{bt} ratio (1.8 ± 0.74) with V_{t} and FRC assumed constant (*Eq. 9*), a behavior measured in the anesthetized mice. These A_{bt} ratios were not significantly different from 1 and were ∼25% of the corrected value (7.5) estimated at room temperature. These results suggest that body temperature humidified box air conditions reduced the methacholine-induced bronchoconstriction observed at room temperature. This behavior also indicates that body temperature humidified box air cannot be used to separate the contributions of temperature-humidity and gas compression to the box pressure under conditions of increased bronchoconstriction.

### Effect of Increased Temperature and Humidity With Methacholine

We consider the effect of methacholine under body temperature humidified box air conditions compared with room temperature box air conditions without methacholine (Table 3). To evaluate this case we start with *Eq. 16*, with the ratio of box pressure with methacholine (δP_{b2}) to box pressure without methacholine (δP_{b1}) given by: (21) K_{3} is the measured A_{bt}f ratio. Here δP_{h2} = 0 and δP_{g2} is the only contributor to δP_{b2}. With δP_{h1}/δP_{g1} equal to 6.5 and 3.1, the actual area ratio due to gas compression (δP_{g2}/δP_{g1}) is 6.6K_{3} and 3.3K_{3} for anesthetized and conscious mice, respectively (Table 3). The experiments showed average K_{3} values of 0.31 and 0.67 for anesthetized and conscious mice, respectively. Thus δP_{g2}/δP_{g1} was 2.0 and 2.2 for anesthetized and conscious mice, respectively. For the anesthetized mice, R_{aw} ratios based on A_{bt} ratio (δP_{g2}/δP_{g1} divided by f_{2}/f_{1}) with measured V_{t} and V_{e} values (*Eq. 9*) averaged 1.4 ± 0.45, much less than the value (8.4) estimated at room temperature. For the conscious mice, R_{aw} ratios based on (δP_{g2}/δP_{g1})/(f_{2}/f_{1}) and constant V_{t} and V_{e} averaged 1.4 ± 0.26. In the conscious mice, breathing frequency increased 30% and a simultaneous increase in V_{t} would produce a lower R_{aw} ratio. By contrast, if V_{t} were to decrease to maintain ventilation constant, a 30% increase in V_{t} from 0.2 to 0.26 ml with a constant V_{e} of 0.23 ml would produce an increase in R_{aw} of ∼40% (*Eq. 9*). Thus the corrected R_{aw} ratio would be 2.0, about 30% of the corrected value for R_{aw} ratio of 7.5 estimated with room temperature box air (Table 4). This suggests that in both anesthetized and conscious mice body temperature box air reduced the methacholine-induced increase in R_{aw} observed with room temperature box air.

## RESULTS AND ANALYSIS

The experiments were designed to test whether the airway resistance predicted from a theory based on alveolar gas compression would increase with airway exposure to the bronchoconstrictor methacholine in anesthetized and conscious mice. From theory (*Eq. 9*), R_{aw} is a function of the inspiratory or expiratory area (A_{bt}) under the pressure-time curve due to gas compression, end-expiratory lung volume (V_{e} or FRC), and V_{t}. In the analysis of experimental data, we averaged the absolute magnitudes of A_{bt} for inspiration and expiration to determine the mean airway resistance over the respiratory cycle. This procedure eliminated errors caused by baseline shifts in the box pressure.

First, we measured the relative contributions of gas compression and temperature-humidity effects to the box pressure excursion in both anesthetized and conscious mice. Table 1 summarizes the data. In the anesthetized mice, FRC, V_{t}, C_{dyn}, and R_{L} were measured by conventional methods. A_{bt} was measured from the box pressure excursions with both room temperature box air and body temperature humidified box air. An analysis (see theory section) showed that A_{bt} at room temperature due to gas compression was 15 and 31% of the values calculated by using the measured box pressure excursions for anesthetized and conscious mice, respectively. The actual V_{t} based on the temperature-humidity curve was 0.99 and 0.95 times that computed by using the measured box pressure excursions. R_{L} values averaged 0.80 and 0.84 cmH_{2}O·ml^{−1}·s for the anesthetized mice at room temperature and body temperature humidified box air, respectively, using conventional methods. Thus box air temperature per se had no effect on R_{L}. A similar behavior was observed for R_{aw} measured by using the present theory. However, the values of R_{aw} (4.2 ± 1.2 cmH_{2}O·ml^{−1}·s) was around fourfold greater than the values of R_{L}. The reasons for this discrepancy are speculative (see discussion).

The values of δP_{h1}/δP_{g1} allowed the correction for temperature-humidity effects in evaluating the increases in airway resistance due to gas compression effects with methacholine. Table 2 summarizes the results of the experiments using both aerosolized saline and methacholine exposure in anesthetized mice at room temperature. R_{aw} was calculated using the measured A_{bt} values corrected for temperature-humidity effects with the measured V_{t} and FRC values in *Eq. 9*. Box air volume V_{b} in *Eq. 9* was box volume (240 ml) minus the mouse volume (∼20 ml). The ratio of each parameter measured after methacholine exposure to that measured after saline exposure was tested against 1 to determine any significant increase with methacholine. The fractional change in any parameter is the difference between the ratio of the parameter values and 1. On the basis of this measure, methacholine significantly increased A_{bt} by 60% but had no significant effect on either FRC or V_{t}. An analysis (see theory section) showed that R_{aw} increased 8.4-fold with methacholine after correction for temperature-humidity effects. This increase in R_{aw} was about double that measured by R_{L} (R_{L} ratios averaged 3.5 ± 1.4, Table 2). By contrast, the Penh ratio increased by 50%.

Table 4 summarizes the effects of methacholine on A_{bt} and V_{t} measured in conscious mice at room temperature. We assumed that V_{t} and FRC remained constant and used the A_{bt} ratio to indicate a change in resistance with methacholine. The A_{bt} ratio (2.2) was significantly greater than 1. An analysis (see theory section) showed that the actual A_{bt} ratio due to gas compression effects was 7.5, equal to the R_{aw} ratio with the assumption of constant V_{t} and FRC. By contrast, with body temperature humidified box air the effect of methacholine produced A_{bt} ratios of 1.8 ± 0.74 that was not significantly different from 1 (Table 4). These A_{bt} ratios were equal to the R_{aw} ratios that represented only gas compression effects because temperature-humidity effects were absent. The insignificant R_{aw} ratios suggest that body temperature humidified box air reduced the methacholine-induced increase in airway resistance observed with room temperature unhumidified box air (R_{aw} ratio of 7.5). By contrast, methacholine significantly increased the Penh values with both room temperature unhumidified box air and body temperature humidified box air by 97 and 62%, respectively.

A result similar to that observed between room temperature and body temperature humidified box air was obtained when methacholine was added to the body temperature humidified box air (Table 3). The A_{bt} ratios averaged 0.38 and 0.72 for the anesthetized and conscious mice, respectively. An analysis (see theory section) showed that the actual A_{bt} ratio due to gas compression was 6.6 and 3.3 times the calculated A_{bt}f values, which produced R_{aw} ratios of 1.4 for both the anesthetized and conscious mice. In the anesthetized mice, the R_{aw} ratio (based on A_{bt} ratio) obtained by analyzing each animal then averaging were 1.4 ± 0.45 (Table 3), smaller than the values for R_{L} ratios of 2.5 ± 0.69. In the conscious mice R_{aw} ratio might be different than 1.4 with methacholine because breathing frequency significantly increased by 30%. Nevertheless, these methacholine-induced increases in R_{aw} for both the anesthetized and conscious mice were much less than the eightfold increase estimated at room temperature and indicated a reduction of a methacholine-induced bronchoconstriction by body temperature humidified box air.

In conscious mice at room temperature, R_{aw} estimated by using the corrected A_{bt} values, measured V_{t}, and assumed values for FRC in *Eq. 9* averaged 2.3, 1.5, and 2.0 cmH_{2}O·ml^{−1}·s in the three groups studied (Tables 1, 3, and 4). Pooled R_{aw} values averaged 1.9 ± 0.41 cmH_{2}O·ml^{−1}·s (*n* = 32). These values were about double the values for R_{L} (0.8–1.0 cmH_{2}O·ml^{−1}·s, Tables 1–3) for the anesthetized mice, and half the pooled values of 4.3 ± 0.77 cmH_{2}O·ml^{−1}·s (*n* = 31) for R_{aw} (4.2, 2.2 and 4.9 cmH_{2}O·ml^{−1}·s, Tables 1–3) based on gas compression effects in the same three groups of anesthetized animals. Thus, in the anesthetized mice, R_{aw} was two- to fourfold greater than R_{L} values. We speculate on the reasons for these differences in the discussion.

In anesthetized mice, methacholine had no significant effect on f at room temperature (Table 2). A similar behavior was observed in conscious mice at both room temperature and body temperature humidified box air conditions (Table 4). By contrast, in anesthetized mice, body temperature box air significantly increased f by 90% compared with room temperature conditions (Table 1). This effect was significantly reduced in the conscious mice (Table 1). The greater temperature-humidity induced increase in frequency in anesthetized than in conscious mice was also observed with the addition of methacholine to the body temperature humidified box air (Table 3). The smaller increase in frequency in conscious mice with body temperature humidified box air was associated with a reduced increase in R_{aw} estimated with methacholine exposure compared with room temperature box air. Thus anesthesia had the effect of increasing frequency under body temperature humidified box air conditions with both saline and methacholine aerosol exposure. These changes in f occurred in conjunction with an approximately twofold reduction in frequency and an ∼80% increase in V_{t} with anesthesia. Pooled values of f averaged 2.5 ± 0.31 Hz (*n* = 33) for anesthetized mice and 5.2 ± 0.24 Hz (*n* = 31) for conscious mice under control conditions breathing room air.

## DISCUSSION

The important results of this study are as follows. We presented a method based on an analysis of alveolar gas compression to determine changes in airway resistance in anesthetized and conscious unrestrained mice placed in a sealed box. We separated the contribution of gas compression from temperature-humidity in the box pressure excursions with an analysis using sinusoids. Experiments in anesthetized and conscious mice showed that gas compression effects were 15 and 31% of the pressure excursion values caused by changes in temperature and humidity, respectively. These data allowed the correction of A_{bt} to estimate R_{aw} at room temperature with and without methacholine exposure. In anesthetized mice at room temperature, methacholine increased R_{aw} by around eightfold, similar to the behavior measured in conscious mice with FRC and V_{t} assumed constant. In both conscious and anesthetized mice, body temperature humidified box air reduced the methacholine-induced increase in airway resistance observed at room temperature.

### Method

Our analysis of the data differed from that of other investigators in several aspects. First, our approach using differential calculus allowed the derivation of a more exact relationship for R_{aw} than the relationship derived by Lundblad et al. (17), who assumed that V_{t} was negligible compared with V_{e}. Measurements in the anesthetized mice showed V_{t} values that were about equal to V_{e} values (Table 1). Thus the neglect of V_{t} compared with V_{e} would overestimate R_{aw} by ∼50%. Second, the separation of the effect of gas compression from the effect of temperature-humidity was based on sinusoidal changes in box pressure. This was justified from a frequency analysis of the data and a comparison in area between the data and a sine wave description. Other investigators (17) have avoided this approach using sinusoids because they found that sine waves were not a good description of their data. The reason might be the low f (∼1 Hz) of the anesthetized mice studied (17). The present study showed frequencies in the range 2–3 Hz for anesthetized mice and 5–6 Hz for conscious mice (11, 23). Third, we used the method of Drorbaugh and Fenn (7) to measure V_{t} in the unrestrained conscious mice from the box pressure excursions at room temperature. No corrections were made for nasal temperature (9, 12), and these errors have been estimated to be less than 30%. An underestimate of V_{t} by 30% would produce an overestimate of R_{aw} by ∼30%.

### Limitations of the Method

Several characteristics of the box pressure excursions were at odds with the theory using sinusoids and thus proved unreliable for the estimation of R_{aw}. First, in theory the method based on A_{bt} can be used to measure airway resistance for both inspiration and expiration. However, small shifts in the baseline box pressure from the theoretical baseline value at FRC would produce unacceptable errors in the inspiratory and expiratory resistances. This was particularly true for animals breathing spontaneously in a sealed box because sealing the box produced changes in the baseline pressure established before sealing. Accordingly, we used the average area of the inspiratory and expiratory parts of the respiratory cycle to estimate R_{aw}. An analysis showed that the use of the average area with measured peak inspiratory-to-expiratory pressure ratio (δP_{insp}/δP_{exp}) of 0.5–2.5 reduced the errors due to shifts in the baseline pressure on A_{bt} to <10% (Fig. 6, appendix c).

Second, baseline shifts in box pressure produced values for δP_{insp}/δP_{exp} that were not a reliable measure of δP_{g}/δP_{h} (Fig. 5, appendix c). In addition, δP_{insp}/δP_{exp} did not decrease systematically with methacholine exposure, as would occur as gas compression effects increased with bronchoconstriction.

In anesthetized and conscious mice, the amplitudes of the box pressure-time curve due to gas compression were 15 and 31% of those due to temperature-humidity, respectively. The higher value was somewhat smaller than that measured for anesthetized mice in previous studies (17) whereas the lower value agreed with calculated estimates (19). Our estimates were based on the assumption that for control conditions (without methacholine) airway resistance did not change between the room temperature and body temperature humidified box air conditions. This behavior was supported by the measurements of R_{L} in anesthetized mice (Table 1).

In conscious mice, methacholine-induced changes in R_{aw} were based on a constant FRC and V_{t}. This was supported by two observations. First, in the anesthetized mice, neither FRC nor V_{t} measured by conventional methods changed significantly with methacholine at room temperature. This behavior was consistent with the constant f measured. Second, frequency remained constant in the conscious mice under most of the conditions imposed in the experiments. The only exception was the 30% increase in frequency observed with body temperature humidified box air with methacholine compared with room temperature box air without methacholine. Under these conditions, V_{t} or FRC might have decreased to maintain ventilation constant. The assumption of a constant V_{t} and FRC might be criticized because both parameters would change with increased bronchoconstriction (17). Indeed, increases in FRC and V_{t} have been reported in anesthetized mice (17) and rats (6) in response to an increase in airway resistance, and these observations have been used to question the use of A_{bt} as a satisfactory indicator of an increased airway resistance (17). However, our studies in the anesthetized mice showed no significant increase in either V_{t} or FRC with a methacholine-induced increase in R_{aw} that was similar to that estimated in the conscious mice with the assumptions of constant V_{t} and FRC.

### Comparison With Previous Results

The box pressure excursions used to estimate airway resistance was extremely small because of the large box volume (240 ml) relative to lung volume (0.2 ml), and the actual magnitude of airway resistance might be in error owing to inaccurate calibration. Such errors were avoided by the use of ratios to estimate changes rather than absolute values. Other errors due to three assumptions in the model that would tend to overestimate R_{aw} are discussed in the following two paragraphs. Nevertheless, on the basis of the corrected values of A_{bt} and V_{t} measured at room temperature in conscious mice (see theory and results and analysis sections) and an assumed value for FRC, R_{aw} based on three groups of mice (*n* = 30) averaged 1.9 cmH_{2}O·ml^{−1}·s. This value for conscious mice is comparable to the value for anesthetized mice of 1.7 cmH_{2}O·ml^{−1}·s measured by using airway occlusion (10) but is greater than the value of 0.5 cmH_{2}O·ml^{−1}·s by using the forced oscillation technique (17) and values of 0.8–1.1 cmH_{2}O·ml^{−1}·s (present study) measured by conventional methods. Our values of R_{aw} measured in the anesthetized mice (average value of 4.3 cmH_{2}O·ml^{−1}·s, Tables 1–3) were around fourfold larger than values of R_{L} measured by conventional methods (0.8–1.0 cmH_{2}O·ml^{−1}·s), and around twofold larger than R_{aw} estimated in conscious mice with an assumed FRC equal to that in the anesthetized mice. The reasons for these differences are speculative and discussed below.

First, the contribution of temperature-humidity relative to gas compression measured by δP_{h}/δP_{g} might be underestimated because of three assumptions made in the experiment with body temperature humidified box air. We assumed that box air temperature (37–38°C) was equal to actual body temperature that increased as the box air was heated. Additional experiments in nine conscious mice showed that heating and humidifying the box air from 21°C (unhumidified) to 37°C (humidified) increased body temperature measured by a rectal (lower colon) thermistor from 37 ± 0.17 to 40 ± 0.30°C. These errors would result in an overestimation of R_{aw} and underestimation of the R_{aw} ratio with methacholine exposure. An error (sensitivity) analysis (appendix e) showed that for a value of δP_{h}/δP_{g} of 6.5 estimated for the anesthetized mice, a 3°C difference between the box air and body temperature would produce a 20% overestimate of R_{aw} and a 30% underestimate for the R_{aw} ratio with methacholine. In the conscious mice, the errors in R_{aw} and R_{aw} ratio would be around fourfold smaller (∼5%). Another assumption was that water vapor saturation of the box air heated to 37–38°C was 100%. An error analysis (appendix e) showed that a 10% decrease in saturation would produce effects on R_{aw} and R_{aw} ratio similar in magnitude to those calculated from the 3°C increase in body temperature. Another assumption was that δP_{g} did not change with the body temperature humidified box air; that is, the viscous pressure loss due to flow remained constant. However, measurements in the anesthetized mice (Table 1) showed that flow amplitude (πV_{t}f) increased 30% with the increased temperature and thus δP_{g2} would increase with a constant R_{aw} (*Eq. 6*). An error analysis (appendix e) showed that δP_{g2}/δP_{g1} of 1.3 would overestimate R_{aw} by ∼20% and underestimate R_{aw} ratio by ∼35% for both the anesthetized and conscious mice. In summary, the errors involved with the foregoing three assumptions would tend to overestimate R_{aw} and underestimate R_{aw} ratio with methacholine.

Second, the expression for R_{aw} was developed on the assumption that the rate of gas compression was similar for both the alveolar gas and box gas and produced the same adiabatic process. However, the expression for R_{aw} (*Eq. 9*) would be multiplied by 1/γ for isothermal conditions in the alveolar gas and adiabatic conditions in the box gas; and for γ of 1.4 for adiabatic conditions, the calculated R_{aw} would be reduced by 31%. These changes would not affect the R_{aw} ratio with methacholine. These effects warrant further study.

Third, A_{bt} measured from the box pressure excursions might be in error because of differences between the inspiratory and expiratory excursions. A 30% difference has been reported in monkeys (12). These changes would result in a negligible change (<2%) in A_{bt} measured by averaging the inspiratory and expiratory areas (see Fig. 6) and in the R_{aw} calculated by using A_{bt}.

Fourth, the lower values of R_{L} might be caused by an underestimation of the changes in pleural pressure measured by the esophageal catheter with breathing. From the measured values of V_{t} (Tables 1, 3, and 4) in the conscious mice, the total change in lung static recoil with inspiration was ∼4 cmH_{2}O, based on the lung pressure-volume curve (16). Thus the amplitude of the change in lung static recoil (2 cmH_{2}O) is one-third the predicted value (6 cmH_{2}O) of the alveolar pressure amplitude (equal to R_{aw}δQ, where flow amplitude δQ = πfV_{t}) required to produce an R_{aw} of 1.9 cmH_{2}O·ml^{−1}·s in conscious mice (see appendix d). The alveolar pressure amplitude would be slightly smaller than the pleural pressure amplitude that includes the change in lung static recoil. Because the change in alveolar pressure bears the same relationship to the change in lung static recoil as the relationship between the gas compression and temperature-humidity curves (*Eq. 13*), the pleural pressure amplitude was (6^{2} + 2^{2})^{1/2} or 6.3 cmH_{2}O, only slightly greater than the alveolar pressure amplitude (6 cmH_{2}O). Thus changes in pleural pressure in mice reflected almost entirely the viscous pressure loss, that is, airway resistance. The relatively large value for the estimated total change in alveolar pressure (twice the alveolar pressure amplitude, 12 cmH_{2}O) fell within the range of the pleural pressure changes (∼7–20 cmH_{2}O) that have been reported with the use of an extraesophageal subpleural catheter in the anesthetized and conscious rat (21). Changes in pleural pressure are expected to be larger in mice than in rats on the basis of an allometric analysis (appendix d). The change in pleural pressure that reflects almost the total viscous pressure loss suggests that expiratory flow in mice is partly driven by forces of the respiratory muscles. Expiratory muscle activity was also suggested by an expiratory time of ∼0.1 s in conscious mice breathing at 5 Hz that was shorter than the passive time constants of 0.15–0.20 s measured in adult mice (25) and in newborn animals of all sizes (20), and much shorter than the time (twice the time constant) required to expire passively ∼90% of the V_{t}. These passive time constants (R_{aw}C_{L}) are consistent with a static lung compliance (C_{L}) of ∼0.05 ml/cmH_{2}O (16) and R_{aw} of ∼3 cmH_{2}O·ml^{−1}·s measured in the present study. The presence of expiratory muscle activity in mice needs more direct experimental support.

Fifth, R_{aw} was calculated from A_{bt} as if the flow were laminar, even though A_{bt} might contain contributions from both laminar and turbulent flows (*Eq. 6*). Thus R_{aw} might include a significant turbulent flow contribution that was not measured by the forced oscillation technique (17) that uses small-amplitude flow oscillations to produce laminar flow conditions. The assumption of turbulent flow with a sinusoidal V_{alv} produced a 27% greater viscous pressure loss or alveolar driving pressure for flow than the assumption of laminar flow (see appendix a). Thus the present method predicts a relatively narrow range for viscous pressure loss in airways whatever the fluid mechanical characteristics (laminar, transitional, and turbulent) of the flow.

Sixth, the flow resistance of the upper airway above the trachea might contribute to the greater R_{aw} measured. Although a contribution of upper airway resistance might be a potential problem in the conscious mice, it was eliminated in the anesthetized mice by a tracheostomy and thus cannot account for the greater values of R_{aw} measured. Although we did not observe any nasal secretions, increases in resistance due to upper airway secretions with body temperature humidified box air or methacholine exposure cannot be ruled out, and these effects warrant further study.

Seventh, we neglected the effects of tissue viscosity that would contribute to R_{L}, pulmonary resistance measured by conventional methods, but not to R_{aw}. Thus tissue resistance cannot account for the greater R_{aw} than R_{L} values measured in anesthetized mice, but might partly account for the higher R_{aw} ratios than R_{L} ratios measured with methacholine (8.4 vs. 3.5, Table 2).

Finally, R_{aw} in the conscious mice might be partly caused by acute changes in frequency and V_{t} associated with a reaction to the box environment. In the present experiments, there was no period of acclimatization for mice with experience in the box, but we allowed 15 min of acclimatization for mice with no experience in the box. The inability of the mice to adapt to the box environment might contribute to the R_{aw} measured in the conscious mice but cannot explain the larger R_{aw} measured in the anesthetized than conscious mice.

The use of the box pressure excursions to estimate V_{t} based on the assumption that only temperature-humidity effects contribute to the box pressure excursions has been validated in studies in mice (22), monkeys (12), and newborn infants (7). Our analysis showed that in both anesthetized and conscious mice V_{t} measured from box pressure excursions was correct within 5% error. However, the analysis also showed that the errors in V_{t} increase with bronchoconstriction as the effects of gas compression increase. On this basis, this method would not be reliable for measuring V_{t} with bronchoconstriction without suitable corrections for gas compression effects.

Our results were compared with those using the Penh method, an ad hoc approach based on a change in the breathing pattern (3, 5, 11). In some instances, the Penh method produced results qualitatively similar to those of the present method. In other instances, the Penh method indicated increases in airway resistance when changes in airway resistance were neither expected nor measured by the present method (see Tables 1 and 3). We agree with other investigators who have questioned the validity of the Penh method for evaluating airway resistance in mice (17–19, 23).

The eightfold increase in R_{aw} measured in mice with aerosolized methacholine in the present study was somewhat larger than the sixfold increase measured in rabbits during apnea (24) but much less than the 20-fold increase measured in rats with 2 cmH_{2}O airway pressure (2), with intravenous methacholine.

Our results suggest that in the anesthetized and conscious mice a methacholine-induced increase in airway resistance observed at room temperature was reduced by breathing body temperature humidified air. A similar behavior has been reported in humans with asthma (4). In the present study, the reduced response to methacholine might be secondary to an increase in body temperature that occurred on heating the box air. This response to breathing body temperature humidified air warrants further study.

### Concluding Remarks

The measurement of the A_{bt} ratio is proposed as a viable approach for studying the airway resistance response to bronchoconstrictor agents in conscious mice. The measured A_{bt} ratio at room temperature grossly underestimated the actual R_{aw} ratio due to gas compression but could serve as a conservative estimate for the increase in R_{aw}. The correction for temperature-humidity effects required an additional experiment with body temperature humidified box air. A deficiency of the present study was the absence of a measure of the changes in FRC and V_{t} in conscious mice. A change in breathing frequency was used as an indication of a change in V_{t} or FRC because conscious unrestrained animals would tend to maintain constant ventilation. The requirement of changes in FRC and V_{t} to evaluate changes in R_{aw} might be satisfied with length and surface area measurements of the lung from single projection roentgenograms taken at end expiration and end inspiration and referenced to the control V_{t} calculated using box pressure excursions (see appendix b).

## APPENDIX A

### Assumption of Turbulent Flow Conditions

In evaluating the airway resistance due to gas compression effects, R_{aw} was computed from A_{bt} as if the viscous pressure loss were due to laminar flow conditions alone (*Eqs. 6*–*9*). If only turbulent flow conditions were to exist, the alveolar driving pressure would be proportional to Q^{2} with a constant of proportionality R_{t}: (A1) *Eq. 8* becomes for turbulent flow conditions: (A2) The integral on the right-hand side of *Eq. A2* can be evaluated for a sine wave description for V_{alv}: (A3) Substitution for V_{alv} and dV_{alv}/d*t* = πV_{t} sin (2πf*t*) in *Eq. A2* and integration between the limits of *t* = 0 and *t* = 1/(2f) for inspiration result in an equation for turbulent flow conditions analogous to *Eq. 11*: (A4) R_{t} is equal to R_{aw} (*Eq. 11*) divided by the term πδQ/4, where δQ, the flow amplitude, is equal to πfV_{t}. The ratio of the viscous pressure loss for turbulent flow (R_{t}Q^{2}) to that for laminar flow (R_{aw}Q) is proportional to Q; that is, it changes cyclically like Q: (A5) For Q = δQ, the ratio of the maximum viscous pressure loss for turbulent flow to that for laminar flow is 4/π or 1.27. That is, the maximum viscous pressure loss equal to the alveolar pressure amplitude (δP_{alv}) is 27% greater with the assumption of turbulent flow conditions than with the assumption of laminar flow conditions. This relationship between turbulent and laminar flow-induced viscous pressure loss is independent of f, airway geometry, and the gas properties (viscosity and density). It defines the upper and lower limits for the maximum viscous pressure loss that is possible for a given A_{bt}. Although A_{bt} does not provide a unique value for viscous pressure loss, the narrow predictable range for the maximum viscous pressure loss provides a measure of airway resistance that could be of practical utility. These assumptions are implicit in the method of Amdur and Mead (1).

It is important to note that R_{aw} represents only a lower limit for the maximum viscous pressure loss during breathing and does not represent the separate contribution due to laminar flow. Thus if most of the viscous pressure loss were due to turbulent flow, R_{aw} computed as if the flow were laminar would be much greater than airway resistance measured by a method, such as the forced oscillation technique, that measures airway resistance under imposed laminar flow conditions.

The present method provides a conceptual framework for the Rohrer equation that equates the viscous pressure loss in pulmonary airways to the sum of separate contributions from laminar and turbulent flow. *Eqs. 4* and *A1* can be combined into the following (Rohrer) equation: (A6) Here R_{l} represents the laminar flow component that is equal to R_{aw} when R_{T} is zero, whereas R_{T} represents the turbulent flow component that is equal to R_{t} when R_{l} is zero. The solution for R_{l} and R_{T} analogous to *Eqs.* 11 and A4 is: (A7) This equation states that A_{bt} contains both laminar and turbulent flow contributions to the viscous pressure loss defined by *Eq. A6*. However, these two contributions cannot be separated without knowledge of the fluid mechanical characteristics of the flow that depend on airway geometry and gas properties.

## APPENDIX B

### Correction for Changes in V_{t} Ratios Based on Phase Differences Between V_{alv}-t and P_{b}-t Curves

Because the phase difference between P_{g} and P_{b} from *Eq. 13* is given by Θ = tan^{−1} (δP_{h}/δP_{g}) and because dV_{alv} is proportional to and in phase with P_{h} that is 90° out of phase with P_{g}, the phase difference (φ) between P_{b} and V_{alv} is: (A8) For the conscious mice under control conditions with δP_{h1}/δP_{g1} of 3.1, φ_{1} was 18°. In general, with bronchoconstriction due to methacholine: (A9) With the assumption that V_{t} was constant (δP_{h2}/δP_{h1} of 1), δP_{g2}/δP_{g1} was 6.5 and thus δP_{h2}/δP_{g2} was 0.48 and φ_{2} was 65°. In the event that V_{t} is not constant, the measurement of the phase differences between P_{b} and V_{alv} without and with methacholine provides through *Eqs. A8* and *A9* a value for (δP_{h2}/δP_{h1})(δP_{g1}/δP_{g2}). Thus a value for δP_{g2}/δP_{g1} can be calculated using the measured V_{t} ratio and compared with the value calculated from *Eq. 19*. The measurement of the FRC and V_{t} ratios from single projection roentgenograms [with the assumption that volume ∝ (length)^{3} and volume ∝ (area)^{3/2}] taken at different points of the respiratory cycle referenced to the measured P_{b}-*t* curve would provide a check on the validity of the analysis using sinusoids in addition to the correct R_{aw} ratio.

## APPENDIX C

### Errors Due to Baseline Shifts in P_{b}

We estimated the error in area (sum of inspiratory and expiratory area magnitudes) between a sine wave (*Eq. 13*) used to represent box pressure rather than the actual box pressure (*Eq. 12*) as follows. The areas under the actual box pressure (P_{b}-*t*) curve and its sine wave representation [(P_{b}−δP_{h})-*t*] are shown in Fig. 2. The (P_{b}−δP_{h})-*t* curve is obtained by an upward shift of the time axis by δP_{h} that is equivalent to a downward shift of the P_{b}-*t* curve by δP_{h}. The fractional difference between the two areas (filled areas in Fig. 2) is a function of δP_{g}/δP_{h}, is greatest for small δP_{g} relative to δP_{h}, and vanishes as δP_{g} becomes much greater than δP_{h} (Fig. 4).

The actual box pressure curve (*Eq. 12*) also produces a ratio of peak inspiratory pressure (δP_{insp}) to peak expiratory pressure (δP_{exp}) that is different from 1, the value for the ratio of the sine wave representation (*Eq. 13b*): (A10) δP_{insp}/δP_{exp} depends on δP_{g}/δP_{h} and decreases as δP_{g}/δP_{h} increases, tending toward 1 as δP_{g}/δP_{h} ≫ 1 (Fig. 5). The fractional area difference vs. δP_{insp}/δP_{exp} is shown in Fig. 6. In theory, the use of the measured values for δP_{g}/δP_{h} of 0.15–0.31 would predict errors in area of 34% and δP_{insp}/δP_{exp} values in the range >10. However, these values were never realized in practice because the baseline pressure at FRC is difficult to establish in practice and a shift in baseline always occurred on sealing the box with the animal breathing spontaneously (see discussion). Accordingly, measurements of δP_{insp}/δP_{exp} were unreliable as estimates of δP_{g}/δP_{h}. In the experiments, δP_{insp}/δP_{exp} was in the range 0.5–2.5 for which the error in using a sine wave to represent box pressure was <10%. This is illustrated in Fig. 6, a plot of error in area vs. δP_{insp}/δP_{exp}, which is obtained from the data in Figs. 4 and 5. Note that the error is undefined for δP_{insp}/δP_{exp} less than 1 because, in theory, δP_{insp}/δP_{exp} of 1 represents the lower limit when only gas compression contributes to the box pressure and a contribution of temperature and humidity can only increase δP_{insp}/δP_{exp}. In practice when δP_{insp}/δP_{exp} is less than 1 owing to shifts in the box pressure on sealing the box, the error from geometry is equal to that based on the inverse of δP_{insp}/δP_{exp}.

## APPENDIX D

### Estimate of Maximum Viscous Pressure Loss

In the conscious mice, δP_{alv} associated with an average R_{aw} of 1.9 cmH_{2}O·ml^{−1}·s and flow amplitude of ∼3 ml/s (πf V_{t} with V_{t} of ∼0.2 ml and f of ∼5 Hz) was 6 and 8 cmH_{2}O with the assumption of laminar and turbulent flow conditions, respectively. Higher values (10 and 13 cmH_{2}O) for δP_{alv} were estimated for the anesthetized mice with a higher R_{aw} (4.3 cmH_{2}O·ml^{−1}·s), greater V_{t} (0.30 ml), and reduced f (2.5 Hz). R_{t}, equal to 4R_{aw}/(πδQ), the coefficient for turbulent flow was ∼0.8 and ∼2.3 cmH_{2}O·ml^{−2}·s^{2}, for conscious and anesthetized mice, respectively.

In large animals such as humans, the viscous pressure loss during breathing is relatively small and most of the force generated by the respiratory muscles goes to expand the lung. However, in the smaller animals such as mice (20 g), with airway resistance that scales inversely with body mass (M), airway resistance is ∼3,500-fold greater than in humans (70 kg) and most of the force generated by the respiratory muscles is dissipated as a flow resistance. The relatively large airway resistance and changes in alveolar pressure estimated by the present technique in mice are in line with the following allometric prediction. Based on Poiseuille's law (R_{aw} ∝ *L*/*r*^{4}) with airway length *L* ∝ M^{−1/3} and airway radius *r* ∝ M^{−1/3}, R_{aw} ∝ M^{−1}. With R_{aw} of ∼1 cmH_{2}O·l^{−1}·s in humans, R_{aw} for mice is ∼3.5 cmH_{2}O·ml^{−1}·s, within the range measured in the present study (2–4 cmH_{2}O·ml^{−1}·s). The predicted R_{aw} is in line with the power regression analysis of typical values of 1 cmH_{2}O·l^{−1}·s for 70-kg humans, 0.02 cmH_{2}O·ml^{−1}·s for 3-kg rabbits (21), 0.5 cmH_{2}O·ml^{−1}·s for 0.3 kg rats (2) and 3.5 cmH_{2}O·ml^{−1}·s for 20-g mice: R_{aw} = 0.082M^{−1.03}, *R*^{2} = 0.99, *P* = 0.006. The alveolar pressure amplitude (δP_{alv} = R_{aw}δQ) with δQ ∝ V_{t}f, V_{t} ∝ M, and f ∝ M^{−1/4} results in δP_{alv} ∝ M^{−1/4}. Thus δP_{alv} is around eightfold greater in mice than in humans. With δP_{alv} of ∼1 cmH_{2}O for humans, δP_{alv} is ∼8 cmH_{2}O for mice, within the range of the values (6–10 cmH_{2}O) estimated in the present study.

## APPENDIX E

An error (sensitivity) analysis was used to determine the errors in the estimate of δP_{h1}/δP_{g1} and R_{aw} caused by the assumptions that were made in the experiment with body temperature humidified box air.

### Effect of Increased Body Temperature and Reduced Water Vapor Saturation

We used the equation of Drorbaugh and Fenn (7) to relate changes in the box pressure (dP_{b}) to the inspired volume (V_{i}): (A11) Here T and P are absolute temperature and pressure, respectively. P_{wb} and P_{wa} are water vapor pressure of box gas and alveolar gas, respectively. The term γP_{b}/V_{b} is the inverse of the box gas compliance (*Eq. 4*). First, we estimated the error in the box pressure excursion (dP_{b}) for an increase in body temperature of 3°C with the mouse breathing room air. We used typical values (19) for T_{b} of 294°K, T_{alv} of 310°K, P_{b} of 760 mmHg, P_{wa} of 47 mmHg, and P_{wb} of 10 mmHg in *Eq. A11*. An increase in T_{alv} of 3°C results in a 9% increase in dP_{b} for room air conditions. In addition, a change in P_{wb} equal to 10% of P_{wa} results in ∼10% change in dP_{b}.

To determine the error in the calculated δP_{h1}/δP_{g1}, we equate the expression for the measured K_{o} in terms of δP_{h1}/δP_{g1} (*Eq. 17*) with δP_{h2} of 0 to the exact expression (*Eq. 16*) with δP_{h2} = FδP_{hc}: (A12) Here δP_{hc}/δP_{g1} is the correct value for ratio of temperature-humidity to gas compression effects. F is the fraction of δP_{hc} attributed to failure of the box air to reach body temperature. Rearranging *Eq. A12* results in the following: (A13) *Eq. A13* is the fractional error in the calculated value of δP_{h1}/δP_{g1}. For the anesthetized mice with δP_{h1}/δP_{g1} of 6.5 and F of 0.1, the error is 34%, that is, δP_{h1}/δP_{g1} would be 8.7 instead of 6.5, K_{o} is reduced from 0.15 to 0.12, and R_{aw} is overestimated by ∼20%. After methacholine exposure with K_{1} of 1.6, δP_{g2}/δP_{g1} and R_{aw} ratio are both underestimated by ∼30%. These errors depend strongly on K_{o} and are reduced fourfold as K_{o} increases to 0.3 for the conscious mice. For the conscious mice with K_{o} of 0.3 and K_{1} of 2.2, R_{aw} is overestimated by 6% and R_{aw} ratio is underestimated by 5%. Thus the errors due to failure of the box air to reach actual body temperature would reduce the estimated R_{aw} in the anesthetized mice but would have little effect in the conscious mice. Similar errors are predicted for failure of the box air to reach 100% water vapor saturation. A 10% reduction from 100% saturation produces nearly the same errors as the 3°C change in body temperature.

### Effect of Increased Temperature on Viscous Pressure Loss Due to Flow

The error in δP_{h1}/δP_{g1} incurred by the assumption that δP_{g2} with body temperature humidified box air was equal to δP_{g1} at room temperature was evaluated as follows. To determine the error in the calculated δP_{h1}/δP_{g1}, we equate the expression for the measured K_{o} in terms of δP_{h1}/δP_{g1} (*Eq. 17*) with δP_{h2} of 0 and δP_{g2} equal to δP_{g1} to the exact expression (*Eq. 16*) with δP_{h2} of 0 and δP_{g2} as a variable. (A14) Here δP_{hc}/δP_{g1} is the correct value for the ratio of temperature-humidity to gas compression effects. Rearrangement of *Eq. A14* results in the expression for δP_{hc}/δP_{g1}: (A15) For the measured δP_{g2}/δP_{g1} of 1.3 (Table 1) and calculated δP_{h1}/δP_{g1} of 6.5 for the anesthetized mice, δP_{hc}/δP_{g1} is 8.5, R_{aw} is overestimated by ∼20%, and R_{aw} ratio with methacholine is underestimated by ∼30%. In the conscious mice, for the same δP_{g2}/δP_{g1} of 1.3 and calculated δP_{h1}/δP_{g1} of 3.1, δP_{hc}/δP_{g1} is 4.1 and R_{aw} is overestimated by ∼20% and R_{aw} ratio with methacholine is underestimated by ∼40%.

## GRANTS

This research was supported by Taiwan grants NHRI-EX91-9130NN and NSC89-2320-B002-136M41.

## Acknowledgments

The authors thank Yu-Cheng Lu for technical assistance.

## Footnotes

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- Copyright © 2005 the American Physiological Society