Sensory Irritation Response in Rats II: Recovery and Dose-Dependence

Abstract

Inhaled irritants can cause respiratory depression by simulating trigeminal nerves in the nasal cavity. This decrease in inhalation rate results in a decrease in the rate of the irritant gases flowing to the stimulated nerves, creating a complex feedback response. Previously, a model was created to describe how the presence of formaldehyde affects respiration in the rat. This ordinary differential equation model incorporated a model of the physiology of the upper respiratory tract of the rat and a model of the neurological control of the respiration rate due to signaling from the stimulated nerves in the nasal cavity. However, an optimal fit to data was not fully established. In the current study, the fit of the previously established model is reevaluated while incorporating the recovery of the ventilation rate after the end of exposure. Additionally, the dose-dependence of the adaptation time allowed by the previous model is more fully quantified, and the updated model predicts formaldehyde data well. Not only are the results of the previous study improved, the model is also shown to predict ventilation decrease in response to other irritants, specifically acrolein, ammonia, and sulfur dioxide. The model is expected to translate to predictions of other irritants with minor parameter changes.

Appendices

Appendix A: The Sensory Irritant Model Symbols

Compartments:

NV:

The nasal vestibule compartment

DR:

The respiratory compartment of the dorsal flow

DO1:

The first olfactory compartment of the dorsal flow

DO2:

The second olfactory compartment of the dorsal flow

VR1:

The first respiratory compartment of the ventral flow

VR2:

The second respiratory compartment of the ventral flow

NP:

The nasopharynx compartment

LRC:

The lower respiratory compartment

States:

C _i :

The concentration of formaldehyde in the air flow through compartment i (μmol/cm³)

Q :

The ventilation rate, i.e., air flow rate through the respiratory tract (cm³/s)

Parameters:

S _i :

The surface area of compartment i (cm²)

V _i :

The volume of the air in compartment i (cm³)

$K_{\mathrm{gc},i}^{j}$ :

Mass transfer coefficient for compartment i for chemical j (cm/s)

C _inh :

The concentration of formaldehyde being inhaled (μmol/cm³)

r(C _i ):

Function representing the response due to chemical concentration that depends on occupancy chemical receptors on the nerve receptor surface (dimensionless)

K _D :

The receptor-chemical dissociation constant (μmol/cm³)

r ₀ :

The threshold parameter for r(C _i ) (dimensionless)

τ _r :

An adaptation time constant for the firing rate n (s)

k ₀ :

The intrinsic sensitivity of the receptor to a chemical (dimensionless)

w _i :

A weighting factor for each compartment that will depend on nerve receptors (dimensionless)

δ :

Small positive constant in order to achieve a nontrivial solution to the differential equation system (dimensionless)

n :

Total firing rate relative to the maximum (dimensionless)

Δn _i :

Firing rate addition from receptors in the mucosa of compartment i (dimensionless)

Q _steady :

Sigmoidal function to represent the nerve change in response to formaldehyde as a function of n (cm³/h)

τ _Q :

Time constant characterizing the time it takes for the nervous response to take effect. The value of τ _Q is denoted $\tau_{Q_{1}}$ during the initial decrease in ventilation and is denoted $\tau_{Q_{2}}$ during the recovery to steady state (s)

Q _min,Q _max :

The maximum and minimum values of ventilation (cm³/s)

p :

Parameter controlling the steepness of the sigmoid Q _steady (dimensionless)

α :

Positive constant in Q _steady dependent on data (dimensionless)

τ ₀ :

Positive constant in the function for τ _r (s/(μmol/cm³))

c ₁ :

Positive constant in the function for τ _r (s)

MW _j :

Molecular weight of chemical j

Appendix B: The Sensory Irritant Model Equations

Equations (8)–(27) were used in the original model in Yokley et al. (2008). Equations (26) and (27) were used in a slightly modified way in the current study as was discussed in Sect. 3.2. Equation (28) was added in the current investigation.

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

(26)

(27)

(28)

Appendix C: Parameter Values

The fixed parameters used in the model are contained in Tables 3 and 4. The choice of model parameters is more fully described in Yokley et al. (2008).

Table 3 Values of compartmental surface areas (S _i ), compartmental volumes (V _i ), and receptor dissociation constants used in the sensory irritant model. All areas and volumes are from Frederick et al. (1998) except for the surface area and volume for the lower respiratory compartment, which were based on Yeh et al. (1979). All dissociation constants were calculated using values from Frederick et al. (1998) converted as described in Yokley et al. (2008) and shown in Eq. (17) using the values from the bottom row of the table

Table 4 Values of fixed parameters used in the sensory response portion of the sensory irritant model. Note that a different Q _max value was used for NH₃ and SO₂ predictions as was discussed in Sect. 5.2