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Airway resistance due to alveolar gas compression measured by barometric plethysmography in mice

¹Center for Biomedical Engineering, University of Kentucky, Lexington, Kentucky and ²Department of Physiology, National Taiwan University College of Medicine, Taipei, Taiwan

Submitted 10 August 2004 ; accepted in final form 20 January 2005

	ABSTRACT

TOP ABSTRACT Glossary MATERIALS AND METHODS THEORY RESULTS AND ANALYSIS DISCUSSION APPENDIX A APPENDIX B APPENDIX C APPENDIX D APPENDIX E GRANTS ACKNOWLEDGMENTS REFERENCES

 
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_btV_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 cmH₂O·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 cmH₂O·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

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

TOP ABSTRACT Glossary MATERIALS AND METHODS THEORY RESULTS AND ANALYSIS DISCUSSION APPENDIX A APPENDIX B APPENDIX C APPENDIX D APPENDIX E GRANTS ACKNOWLEDGMENTS REFERENCES

 

A_bt

Area under the box pressure-time curve for inspiration or expiration, mean of the two areas (cmH₂O·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/cmH₂O)

C_L

Static lung compliance (ml/cmH₂O)

d

Prefix, differential of

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

Isentropic exponent of alveolar and box gas

h

Subscript, temperature-humidity effects

insp

Subscript, peak inspiratory

^oK

Degree Kelvin, unit of absolute temperature

K

A_btf ratio, intervention vs. control

K_o

A_btf ratio, body temperature humidified box air vs. room temperature box air

K₁

A_btf ratio, methacholine vs. control, both with room temperature box air

K₂

A_btf ratio, methacholine vs. control, both with body temperature humidified box air

K₃

A_btf 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 (cmH₂O)

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 (cmH₂O)

P_g

Gas compression part of box pressure (cmH₂O)

P_h

Temperature-humidity part of box pressure (cmH₂O)

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 (cmH₂O·ml^–1·s)

R_L

Pulmonary resistance measured by conventional methods (cmH₂O·ml^–1·s)

R_l

Laminar flow part of the total viscous pressure loss (cmH₂O·ml^–1·s)

R_t

Turbulent flow coefficient if viscous pressure loss were entirely turbulent (cmH₂O·ml^–2·s²)

R_T

Turbulent flow part of the total viscous pressure loss (cmH₂O·ml^–2·s²)

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

TOP ABSTRACT Glossary MATERIALS AND METHODS THEORY RESULTS AND ANALYSIS DISCUSSION APPENDIX A APPENDIX B APPENDIX C APPENDIX D APPENDIX E GRANTS ACKNOWLEDGMENTS REFERENCES

 
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 CO₂. 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 cmH₂O 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).

View larger version (34K): [in this window] [in a new window]   Fig. 1. Examples of box pressure excursions in conscious mice measured after saline and methacholine aerosol exposure. Positive pressure represents expiration. Negative pressure represents inspiration. Area ratio was 1.3. Filled areas represent expiratory part and inspiratory part of the curves.

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 x 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

TOP ABSTRACT Glossary MATERIALS AND METHODS THEORY RESULTS AND ANALYSIS DISCUSSION APPENDIX A APPENDIX B APPENDIX C APPENDIX D APPENDIX E GRANTS ACKNOWLEDGMENTS REFERENCES

 
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 is the isentropic exponent. For an isothermal process, 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/dt, 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_awQ). 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_alv²/2 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 (P_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 (P_h) that is 90° (/2 radians) out of phase with the gas compression wave and displaced along the ordinate by P_h. Then, P_b is given by:

(12)

Eq. 12 can be rewritten as a single sine wave:

(13a)

(13b)

Here tan = P_h/P_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).

View larger version (54K): [in this window] [in a new window]   Fig. 2. Box pressure (Pb, thick line) variation during a respiratory cycle is represented as the sum of 2 sine waves. One (Pg, dashed line) has an amplitude (Pg) of 0.4 units owing to gas compression. The other (Ph, thin line) has an amplitude (Ph) of 0.5 units owing to temperature-humidity effects that is 90° out of phase with Pg and displaced by Ph above the baseline pressure-time axis. Baseline (zero) pressure is at the start of inspiration (Ph = 0). The period is 0.2 s and frequency is 5 Hz. Thick dashed horizontal line is a shift in the time axis to a pressure equal to the Ph from which Pb becomes a sine wave. Pinsp and Pexp are peak inspiratory and peak expiratory pressures of the Pb-t curve. A: filled areas under the inspiratory part and expiratory part of the Pb-t curve. B: filled area of the (Pb–Ph)-t curve, the sine wave representation of the Pb-t curve.

(14)

From measurements of P_b, P_g can be calculated if P_h/P_g is known. The parameter P_h/P_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 P_b has been used to calculate V_t based on the assumption that only P_h contributes to P_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 P_h contributes to P_b is evaluated as follows. P_h/P_b is related to P_g/P_h using Eq. 13b:

(15)

Thus if P_h/P_g is known, P_h required to give the correct value for V_t in terms of the value calculated using P_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 (P_insp) equal to peak expiratory pressure (P_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_btf ratio. In practice, we used the average of inspiratory and expiratory areas. Thus A_bt is equal to P/(f).

Estimate of P_g/P_h for Room Temperature Box Air Conditions

In two groups of anesthetized and conscious unrestrained mice, we measured P_b with room temperature box air (P_b1) and with body temperature humidified box air (P_b2). We assumed that with body temperature humidified box air P_h2 was zero and that P_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, P_b2/P_b1 equals the measured A_btf ratio (K_o):

(17)

From Eq. 17, P_h1/P_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 P_h1/P_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 P_b2 = P_g1, P_g1/P_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. P_h1/P_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 (P_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).

Effect of Methacholine With Room Temperature Box Air

Let the ratio of A_btf with aerosolized methacholine exposure (subscript 2) to that with a prior aerosolized saline exposure (subscript 1) be equal to K₁ in Eq. 16. Then P_g2/P_g1 due to gas compression effects is related to the V_t ratio, P_h2/P_h1, by the following equation:

(19)

K₁ is the measured A_btf ratio and P_h1/P_g1 (6.5 and 3.1 for anesthetized and conscious mice, respectively) is known from the previous experiments (Table 1). P_g2/P_g1 is a function of P_h2/P_h1, the actual box pressure amplitude ratio due to temperature-humidity effects. P_h2 cannot be determined by using room temperature and body temperature humidified box air as was done for P_h1 because airway resistance changed with the body temperature humidified box air conditions (see below). Thus we assumed that P_h2/P_h1 was 1, that is, V_t was constant, a behavior that was measured in the anesthetized mice. Fig. 3 is a plot of P_g2/P_g1 vs. P_b2/P_b1, the measured box pressure amplitude ratio or A_btf ratio (K₁), for different P_h1/P_g1 isopleths with P_h2/P_h1 of 1. For the average A_btf ratio (K₁) of 1.6 (Table 2) with P_h1/P_g1 of 6.5 in the anesthetized mice and P_h2/P_h1 of 1, P_g2/P_g1 was 8.3 (Fig. 3). Alternatively, for each anesthetized mouse, a value of P_g2/P_g1 was computed using the measured K₁ and V_t (P_h2/P_h1) ratio in Eq. 19. The A_bt ratio was obtained by dividing the computed value of P_g2/P_g1 by the measured f₂/f₁ 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.

View larger version (16K): [in this window] [in a new window]   Fig. 3. Ratio Pg2/Pg1 (Abtf ratio due to gas compression effects) plotted vs. Pb2/Pb1 (measured box pressure Abtf ratio) at different isopleths of Ph1/Pg1. Ph2/Ph1 (Vt ratio) was 1.

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). P_h does not contribute to the box pressure excursions that consist of only gas compression effects. The ratio of box pressure with methacholine (P_b2) to that without methacholine (P_b1) is given by Eq. 16 with P_h1 = P_h2 = 0:

(20)

K₂ is the measured A_btf 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 (P_b2) to box pressure without methacholine (P_b1) given by:

(21)

K₃ is the measured A_btf ratio. Here P_h2 = 0 and P_g2 is the only contributor to P_b2. With P_h1/P_g1 equal to 6.5 and 3.1, the actual area ratio due to gas compression (P_g2/P_g1) is 6.6K₃ and 3.3K₃ for anesthetized and conscious mice, respectively (Table 3). The experiments showed average K₃ values of 0.31 and 0.67 for anesthetized and conscious mice, respectively. Thus P_g2/P_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 (P_g2/P_g1 divided by f₂/f₁) 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 (P_g2/P_g1)/(f₂/f₁) 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

TOP ABSTRACT Glossary MATERIALS AND METHODS THEORY RESULTS AND ANALYSIS DISCUSSION APPENDIX A APPENDIX B APPENDIX C APPENDIX D APPENDIX E GRANTS ACKNOWLEDGMENTS REFERENCES

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 cmH₂O·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 cmH₂O·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 P_h1/P_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_btf 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 cmH₂O·ml^–1·s in the three groups studied (Tables 1, 3, and 4). Pooled R_aw values averaged 1.9 ± 0.41 cmH₂O·ml^–1·s (n = 32). These values were about double the values for R_L (0.8–1.0 cmH₂O·ml^–1·s, Tables 1–3) for the anesthetized mice, and half the pooled values of 4.3 ± 0.77 cmH₂O·ml^–1·s (n = 31) for R_aw (4.2, 2.2 and 4.9 cmH₂O·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

TOP ABSTRACT Glossary MATERIALS AND METHODS THEORY RESULTS AND ANALYSIS DISCUSSION APPENDIX A APPENDIX B APPENDIX C APPENDIX D APPENDIX E GRANTS ACKNOWLEDGMENTS REFERENCES

 
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 (P_insp/P_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).

View larger version (10K): [in this window] [in a new window]   Fig. 6. Fractional difference (error) in sum of inspiratory and expiratory area magnitudes between the actual box pressure curve and representation using a sine wave vs. Pinsp/Pexp (peak inspiratory-to-peak expiratory pressure ratio). Note that the error was <10% for values of 0.5–2.5 for Pinsp/Pexp measured in the experiments.

View larger version (10K): [in this window] [in a new window]   Fig. 5. Ratio of peak inspiratory box pressure (Pinsp) to peak expiratory box pressure (Pexp) vs. Pg/Ph.

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 cmH₂O·ml^–1·s. This value for conscious mice is comparable to the value for anesthetized mice of 1.7 cmH₂O·ml^–1·s measured by using airway occlusion (10) but is greater than the value of 0.5 cmH₂O·ml^–1·s by using the forced oscillation technique (17) and values of 0.8–1.1 cmH₂O·ml^–1·s (present study) measured by conventional methods. Our values of R_aw measured in the anesthetized mice (average value of 4.3 cmH₂O·ml^–1·s, Tables 1–3) were around fourfold larger than values of R_L measured by conventional methods (0.8–1.0 cmH₂O·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 P_h/P_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 P_h/P_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 P_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 (V_tf) increased 30% with the increased temperature and thus P_g2 would increase with a constant R_aw (Eq. 6). An error analysis (APPENDIX E) showed that P_g2/P_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/ for isothermal conditions in the alveolar gas and adiabatic conditions in the box gas; and for 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 cmH₂O, based on the lung pressure-volume curve (16). Thus the amplitude of the change in lung static recoil (2 cmH₂O) is one-third the predicted value (6 cmH₂O) of the alveolar pressure amplitude (equal to R_awQ, where flow amplitude Q = fV_t) required to produce an R_aw of 1.9 cmH₂O·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 (6² + 2²)^1/2 or 6.3 cmH₂O, only slightly greater than the alveolar pressure amplitude (6 cmH₂O). 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 cmH₂O) fell within the range of the pleural pressure changes (7–20 cmH₂O) 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_awC_L) are consistent with a static lung compliance (C_L) of 0.05 ml/cmH₂O (16) and R_aw of 3 cmH₂O·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 cmH₂O 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

TOP ABSTRACT Glossary MATERIALS AND METHODS THEORY RESULTS AND ANALYSIS DISCUSSION APPENDIX A APPENDIX B APPENDIX C APPENDIX D APPENDIX E GRANTS ACKNOWLEDGMENTS REFERENCES

 
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 Q² 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/dt = V_t sin (2ft) 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 Q/4, where Q, the flow amplitude, is equal to fV_t. The ratio of the viscous pressure loss for turbulent flow (R_tQ²) to that for laminar flow (R_awQ) is proportional to Q; that is, it changes cyclically like Q:

(A5)

For Q = Q, the ratio of the maximum viscous pressure loss for turbulent flow to that for laminar flow is 4/ or 1.27. That is, the maximum viscous pressure loss equal to the alveolar pressure amplitude (P_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

TOP ABSTRACT Glossary MATERIALS AND METHODS THEORY RESULTS AND ANALYSIS DISCUSSION APPENDIX A APPENDIX B APPENDIX C APPENDIX D APPENDIX E GRANTS ACKNOWLEDGMENTS REFERENCES

 
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 = tan^–1 (P_h/P_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 () between P_b and V_alv is:

(A8)

For the conscious mice under control conditions with P_h1/P_g1 of 3.1, ₁ was 18°. In general, with bronchoconstriction due to methacholine:

(A9)

With the assumption that V_t was constant (P_h2/P_h1 of 1), P_g2/P_g1 was 6.5 and thus P_h2/P_g2 was 0.48 and 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 (P_h2/P_h1)(P_g1/P_g2). Thus a value for P_g2/P_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

TOP ABSTRACT Glossary MATERIALS AND METHODS THEORY RESULTS AND ANALYSIS DISCUSSION APPENDIX A APPENDIX B APPENDIX C APPENDIX D APPENDIX E GRANTS ACKNOWLEDGMENTS REFERENCES

 
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–P_h)-t] are shown in Fig. 2. The (P_b–P_h)-t curve is obtained by an upward shift of the time axis by P_h that is equivalent to a downward shift of the P_b-t curve by P_h. The fractional difference between the two areas (filled areas in Fig. 2) is a function of P_g/P_h, is greatest for small P_g relative to P_h, and vanishes as P_g becomes much greater than P_h (Fig. 4).

View larger version (13K): [in this window] [in a new window]   Fig. 4. Fractional difference (error) in summed area between the actual box pressure curve and representation using a sine wave vs. Pg/Ph. Note that the error was 34% for Pg/Ph of 0.15–0.3 estimated in the experiments at room temperature.

(A10)

P_insp/P_exp depends on P_g/P_h and decreases as P_g/P_h increases, tending toward 1 as P_g/P_h >> 1 (Fig. 5). The fractional area difference vs. P_insp/P_exp is shown in Fig. 6. In theory, the use of the measured values for P_g/P_h of 0.15–0.31 would predict errors in area of 34% and P_insp/P_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 P_insp/P_exp were unreliable as estimates of P_g/P_h. In the experiments, P_insp/P_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. P_insp/P_exp, which is obtained from the data in Figs. 4 and 5. Note that the error is undefined for P_insp/P_exp less than 1 because, in theory, P_insp/P_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 P_insp/P_exp. In practice when P_insp/P_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 P_insp/P_exp.

	APPENDIX D

TOP ABSTRACT Glossary MATERIALS AND METHODS THEORY RESULTS AND ANALYSIS DISCUSSION APPENDIX A APPENDIX B APPENDIX C APPENDIX D APPENDIX E GRANTS ACKNOWLEDGMENTS REFERENCES

 
Estimate of Maximum Viscous Pressure Loss

In the conscious mice, P_alv associated with an average R_aw of 1.9 cmH₂O·ml^–1·s and flow amplitude of 3 ml/s (f V_t with V_t of 0.2 ml and f of 5 Hz) was 6 and 8 cmH₂O with the assumption of laminar and turbulent flow conditions, respectively. Higher values (10 and 13 cmH₂O) for P_alv were estimated for the anesthetized mice with a higher R_aw (4.3 cmH₂O·ml^–1·s), greater V_t (0.30 ml), and reduced f (2.5 Hz). R_t, equal to 4R_aw/(Q), the coefficient for turbulent flow was 0.8 and 2.3 cmH₂O·ml^–2·s², 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/r⁴) with airway length L M^–1/3 and airway radius r M^–1/3, R_aw M^–1. With R_aw of 1 cmH₂O·l^–1·s in humans, R_aw for mice is 3.5 cmH₂O·ml^–1·s, within the range measured in the present study (2–4 cmH₂O·ml^–1·s). The predicted R_aw is in line with the power regression analysis of typical values of 1 cmH₂O·l^–1·s for 70-kg humans, 0.02 cmH₂O·ml^–1·s for 3-kg rabbits (21), 0.5 cmH₂O·ml^–1·s for 0.3 kg rats (2) and 3.5 cmH₂O·ml^–1·s for 20-g mice: R_aw = 0.082M^–1.03, R² = 0.99, P = 0.006. The alveolar pressure amplitude (P_alv = R_awQ) with Q V_tf, V_t M, and f M^–1/4 results in P_alv M^–1/4. Thus P_alv is around eightfold greater in mice than in humans. With P_alv of 1 cmH₂O for humans, P_alv is 8 cmH₂O for mice, within the range of the values (6–10 cmH₂O) estimated in the present study.

	APPENDIX E

TOP ABSTRACT Glossary MATERIALS AND METHODS THEORY RESULTS AND ANALYSIS DISCUSSION APPENDIX A APPENDIX B APPENDIX C APPENDIX D APPENDIX E GRANTS ACKNOWLEDGMENTS REFERENCES

 
An error (sensitivity) analysis was used to determine the errors in the estimate of P_h1/P_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 P_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 P_h1/P_g1, we equate the expression for the measured K_o in terms of P_h1/P_g1 (Eq. 17) with P_h2 of 0 to the exact expression (Eq. 16) with P_h2 = FP_hc:

(A12)

Here P_hc/P_g1 is the correct value for ratio of temperature-humidity to gas compression effects. F is the fraction of P_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 P_h1/P_g1. For the anesthetized mice with P_h1/P_g1 of 6.5 and F of 0.1, the error is 34%, that is, P_h1/P_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 K₁ of 1.6, P_g2/P_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 K₁ 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 P_h1/P_g1 incurred by the assumption that P_g2 with body temperature humidified box air was equal to P_g1 at room temperature was evaluated as follows. To determine the error in the calculated P_h1/P_g1, we equate the expression for the measured K_o in terms of P_h1/P_g1 (Eq. 17) with P_h2 of 0 and P_g2 equal to P_g1 to the exact expression (Eq. 16) with P_h2 of 0 and P_g2 as a variable.

(A14)

Here P_hc/P_g1 is the correct value for the ratio of temperature-humidity to gas compression effects. Rearrangement of Eq. A14 results in the expression for P_hc/P_g1:

(A15)

For the measured P_g2/P_g1 of 1.3 (Table 1) and calculated P_h1/P_g1 of 6.5 for the anesthetized mice, P_hc/P_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 P_g2/P_g1 of 1.3 and calculated P_h1/P_g1 of 3.1, P_hc/P_g1 is 4.1 and R_aw is overestimated by 20% and R_aw ratio with methacholine is underestimated by 40%.

	GRANTS

TOP ABSTRACT Glossary MATERIALS AND METHODS THEORY RESULTS AND ANALYSIS DISCUSSION APPENDIX A APPENDIX B APPENDIX C APPENDIX D APPENDIX E GRANTS ACKNOWLEDGMENTS REFERENCES

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

	ACKNOWLEDGMENTS

TOP ABSTRACT Glossary MATERIALS AND METHODS THEORY RESULTS AND ANALYSIS DISCUSSION APPENDIX A APPENDIX B APPENDIX C APPENDIX D APPENDIX E GRANTS ACKNOWLEDGMENTS REFERENCES

 
The authors thank Yu-Cheng Lu for technical assistance.

	FOOTNOTES

 

Address for reprint requests and other correspondence: S. J. Lai-Fook, Center for Biomedical Engineering, Wenner-Gren Research Laboratory, Univ. of Kentucky, Lexington, KY 40506-0070 (E-mail: laifook{at}uky.edu)

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

	REFERENCES

TOP ABSTRACT Glossary MATERIALS AND METHODS THEORY RESULTS AND ANALYSIS DISCUSSION APPENDIX A APPENDIX B APPENDIX C APPENDIX D APPENDIX E GRANTS ACKNOWLEDGMENTS REFERENCES

 

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