## Lung Ventilation Modeling for Lung Disease Diagnosis

CONTENTS 5.1 Scope ........................................................................................................... 146 5.2 Ventilatory Function Represented by a First-Order Differential

Equation Model......................................................................................... 146 5.2.1 Simulating Clinical Data.............................................................. 147 5.2.2 Derivation of the Governing Differential Equation

for Lung Volume (V) Response to Driving Pressure.............. 148 5.3 Lung Ventilation Performance Using the Linear First-Order

Differential Equation Model ................................................................... 149 5.4 Ventilatory Index ...................................................................................... 153

5.4.1 Noninvasively Determinable Ventilatory Index...................... 154 5.5 Work of Breathing .................................................................................... 155 5.6 Second-Order Model for Single-Compartment Lung Model ............ 157 5.7 Two-Compartmental First-Order Ventilatory Model ......................... 159

5.7.1 Two-Compartmental Lung Ventilatory Model Analysis ....... 162 5.7.2 Simulation of a Stiff Right Lung

(with Compliance Problems) ...................................................... 162 5.7.3 Simulation of a Right Lung with Flow-Rate

Resistance Problems ..................................................................... 164 5.7.4 Determining Left and Right Lung Parameters

and Indices without Requiring Intubation of the Patient ...... 165 Appendix A: Solution of Equation 5.4 ........................................................... 165 Appendix B: Solution of Equation 5.28.......................................................... 167 Reference ............................................................................................................. 171

This chapter dealswithmodeling of lungventilation, first, in the formof a firstorder differential equation of the lung volume (V) response to lung driving pressure (PN), in terms of the lung compliance (C) and resistance to airflow rate (R). The solution of this equation is derived in terms of compliance (C) and the resistance to flow rate (R), which can then be employed to diagnose lung disease states. The parameters R and C are also combined to formulate a nondimensional index, whose ranges of valueswould differ with lung disease states. Thus, this index makes it more convenient to diagnose lung diseases. The determination of the model parameters R and C from lung volume

and driving pressure requires intubation of the patient. However, the solution of the governing equation also contains terms involving a combination of pressure and compliance as well as of t (¼RC). Hence, when this solution is made to match the monitored lung volume response, we can evaluate these terms without requiring to know the driving pressure independently; this avoids intubation of the patient. We then formulate another corresponding nondimensional index involving these terms, and demonstrate that this index in fact involves the pressure terms, as well as R and C independently. This provides validation of this index, based on the lung volume response to driving pressure in terms of R and C. We next formulate a second-order differential equation for lung volume

response to lung driving pressure. We demonstrate how the new parameters of this governing equation can be determined. These parameters also involve R and C, and hence can also be employed to diagnose lung disease states. Now, it is possible that one lobe of the lung be normal and the other

diseased. For this purpose, we develop a two-lobe lung model in terms of the response of their volumes to lung driving pressure. This two-lobe model is formatted using the first-order differential equation model. The model involves the compliance and flow resistances of the two lobes. We then demonstrate how this two-lobe model can be employed to separately evaluate the parameters of the two lobes, and hence assure the normality or diseased states of the two lobes separately. This chapter is developed along similar lines to our previous Chapter 4 in

Ref. [1], and the figures employed are adopted from this chapter in the afore-mentioned book.*

Lung mechanics involves inhalation and exhalation pressure and volume changes. Three pressures are involved in the ventilatory function, namely

atmospheric pressure or pressure at the mouth (Pm), alveolar pressure (Pa), and pleural pressure (Pp). The pressure gradient between the atmospheric and alveolar pressures causes respiration to occur. During inspiration, Pa<Pm, and air enters the lungs. During expiration, Pa>Pm, and air is expelled out of the lungs passively. This pressure differential between Pm and Pp provides the driving pressures (PL) for gas flow, in terms of the elastic recoil pressure of the lumped alveolar chamber and the pressure differential between Pm and Pa (expressed as R _V). Thus, the assessment of respiratory mechanics involves the measurements of flows, volumes, pressure-gradients, and their dynamic interrelationships. The lung ventilatory model (LVM) then enables computation of lung compliance (C) and airway resistance-to-airflow (R), which are the parameters of the governing equation. Lung ventilatory dysfunction due to various diseases is characterized by the altered values of R and C, or in terms of an index involving a combination of R and C. Hence, the LVM can be employed to detect and characterize lung disease states. The lung ventilation model is based on the equilibrium differential equa-

tion, expressing lung volume response to driving pressure across the lung. This dynamic relationship includes lung compliance (C) and the resistanceto-flow (R) offered by the airways during inspiration and expiration. In this model, the pressure generated by the respiratory muscles on the chest wall, namely the mouth pressure minus the pleural pressure, represents the driving force for the operation of the respiratory pump (for lung filling and expiration), as depicted in Figure 5.1.