# Modeling Respiratory Mechanics

Studying the mechanics of the respiratory system means evaluating the parameters of a mathematical model that describes the position and motion of the respiratory system through a process called “inverse modeling” or “system identification” [4].

The models which are usually adopted are invariably rather simple, having few independent components and a small number of parameters. They do not consider detailed information of the structure of the airways, lung and chest wall, but only approximated versions, because such models have to be matched to experimental data, and usually only few variables can be measured.

Once the structure of an inverse model of the respiratory system has been defined, the so-called equations of motion are derived. These equations state how pressure is related to flow and volume within each component of the model and tell how the complete model will behave under every conceivable circumstance. The equations contain “variables”, i.e., measurable parameters, which usually vary with time and are typically pressures, flows, and volumes. Equations also contain parameters that characterize the “mechanical properties” of the overall system and of its different components, typically including resistance and compliance. The process of finding those parameter values that cause the model to behave like a particular real respiratory system is known as parameter estimation.

The total pressure applied to the respiratory system (PRS) is the pressure difference from airway opening (PAO) to body surface (PBS, usually equal to atmospheric pressure). In a ventilated patient, PRS is given by the sum of the pressure generated by the mechanical ventilator (PVENT) and the pressure developed by the respiratory muscles (Pmus) (Fig. 14.1):

PVENT, in turn, is the sum of the driving pressure necessary to move air into the thorax (AP) and the positive end-expiratory pressure (PEEP). PMUS is the pressure generated by the respiratory muscles, with the inspiratory muscles producing a decrease in intrathoracic pressure and expiratory muscles an increase.

Considering a mechanical model of the respiratory system, the pulmonary system (composed of the airways and the lung) and the chest wall have to be considered as structures arranged in parallel, while if an electrical (or hydraulic) model is considered, like the one shown in the figures of the present chapter, they are in series. This means that, at least as a first approximation, the pulmonary system and the chest wall share the same volume (V) and volume variations (dV/dt), and the total pressure applied to the respiratory system (PRS) is subdivided into the “transpulmonary” pressure (PTP, i.e., the pressure difference from airway opening to pleural space) and the “thoracic” pressure (PCW, i.e., the pressure difference from pleural space to body surface) (Fig. 14.2).

Fig. 14.1 Model of the entire respiratory system (see text for details). Pressures are indicated at specific points, circles represent pressure generators, rectangles represent passive components, and arrows indicate pressure differences. In this electric- or hydraulic-like representation, the total pressure acting on elements in series is the sum of the pressures acting on the single components, while the flow and volume changes are the same across the system. PBS body surface pressure, PAO airway opening pressure, PVent sum of the driving pressure necessary to move air into the thorax (Д? ), and the positive end-expiratory pressure (PEEP), PalV alveolar pressure, PPL pleural pressure PMus pressure generated by the respiratory muscles

Although the significance of Eq. 14.2 is very important, it is often disregarded and is barely considered in the clinical practice when managing ARDS patients. The relevant distending pressure for the lung is PTP [5], but also the chest wall, through the pressure in the pleural space (PPL), plays a significant role in determining lung expansion and stress as well [6, 7]. As shown in Eq. 14.2, if PPL is known, PTP and P CW can be determined. While measuring PAO is relatively simple (all that is required is a lateral pressure tap in a mouthpiece or a section of ventilator tubing, see below), however, measuring PPL in living patients is not. PPL has been estimated in the human physiology laboratory for more than 50 years from measurements of esophageal pressure (PES) [8, 9]. Most of the existing literature in which PES is used to estimate PPL, however, reports data obtained with the subject under analysis either in seating or standing position. When the analyzed subject is in supine or prone

Fig. 14.2 Model of the passive components of the entire respiratory system, divided into the pulmonary system (in turn, composed by airways and lung) and the chest wall. PPL pleural pressure, PALV alveolar pressure, PTP transpulmonary pressure, PCW thoracic pressure, RPULM pulmonary system resistance, RAW airway resistance, Rt, L. lung tissue resistance, Rt, CW chest wall tissue resistance, EPULM pulmonary system elastance, EL lung elastance, ECW chest wall elastance, CPULM pulmonary system compliance, CL lung compliance, and CCW chest wall compliance See legend of Fig. 14.1 for all the other symbols

position, artifacts caused by mediastinal weight and postural effects have to be considered.

Transpulmonary pressure is the sum of the pressure drop across the airways (PAW, i.e., the pressure difference from airway opening to alveolar space) and lung pressure (PL, i.e., the lung “transmural” pressure, pressure difference from alveolar space to pleural space):

Under dynamic conditions, namely, when airflow is present and dV/dt is different than zero, PAW is different than zero. Only under static conditions PAW = 0, and therefore PTP = PL.

PRS depends on the mechanical properties of the respiratory system, namely, resistance and elastance, while the “inertance,” i.e., the capability to store some energy in the kinetic, form is usually considered negligible, with the exception of the case of high-frequency forced ventilation, where it might play a significant role.

The elastic behavior of the respiratory system is everything that makes it spontaneously returning to an equilibrium position. It is described by elastance or its inverse, the compliance, and it depends mostly on the elastic properties of lung and chest wall tissues. Lung tissues consist mainly of elastin and collagen, structural proteins that have different responses to deformation. The former react to deformation as springs and, up to a certain degree of extension, show a linear behavior. The latter are sort of “safety cables,” which are entangled in a disordered way in the lung at rest. Collagen fibers do not play any role in the mechanical behavior of the lung at lung volume close to rest, while with increasing lung volume, more and more fibers reach their maximal length, determining a firm increase of lung elastance. The chest wall, namely, all the structures and tissues surrounding the lung and moving with it (rib cage and abdomen), also contributes to the elastance of the respiratory system [4]. The sum of all the elastic forces acting on the respiratory system is expressed by the elastic pressure.

Pel, which is proportional to volume V (deviation from the equilibrium position, i.e., deviation from equilibrium volume V0) by means of elastance ERS.

With resistance of the respiratory system (Rrs), all those phenomena that oppose to deformation of the respiratory system, thereby dissipating a given amount of energy, are considered. Dissipation means that the energy supplied to the system (work performed by the respiratory muscles) or stored as potential energy (e.g., during mechanical ventilation) is lost as heat. The main source of resistance corresponds to the flow of air through the airways. Due to the viscous effects, most energy provided to the system to ventilate is actually lost as heat in the air. Rrs, however, is due not only to the flow resistance of the airways but also contains a significant contribution from the lung (Rt, L) and chest wall (Rt, CW) tissues. When lung and chest wall tissues are stretched, in fact, their constituent fibers, cells, and fluids move against each other and this internal friction produces heat representing another form of dissipation [4].

The sum of all the resistive forces acting on the respiratory system is expressed by the resistive pressure PRES, which is proportional to volumetric flow rate dV/dt (also known as volume flow rate, rate of fluid flow, or volume velocity) by means of resistance Rrs.

In summary, the total pressure acting on the respiratory system, PRS, is given by the sum of the elastic pressure (PEL), resistive, (PRES) and inertial (PINeRT) components (Fig. 14.3). Being PINERT « 0, it follows that

Fig. 14.3 Model of the passive components of the entire respiratory system, divided into the elastic and resistive components. See text for details and legends of Figs. 14.1 and 14.2 for used symbols

Being the pulmonary system and the chest wall mechanically in parallel, total respiratory system elastance and resistance are equal to the sum of pulmonary and chest wall elastances and resistances:

Combining Eqs. 14.4 and 14.5, it follows that

Lung compliance (CL), defined as the change in lung volume per unit change in PL, is of particular importance in ARDS and is sometime referred as either “static” or “dynamic,” depending on the method of measurement adopted to obtain it (see below). Nevertheless, compliance represents a property that is only static, i.e., it expresses the distensibility of the lung. Lung compliance in ARDS is decreased due to diffused alveolar damage and loss of cellular integrity of the alveoli which are filled with proteinaceous edema fluid that also results in dilution and dysfunction of pulmonary surfactant, leading to alveolar collapse [10], and causes compliance to decrease in inverse proportion to the fraction of lung volume that has been lost by derecruitment, i.e., the fraction of the lung to become blocked from receiving ventilation.

Transpulmonary pressure is the relevant distending pressure for the lung [5]. This concept, however, is often disregarded, and the effect of the chest wall in determining lung expansion and stress is barely considered in the clinical practice [6, 7].

Nevertheless, in ARDS patients, it is common that not only the lung but also the chest wall compliance is abnormally low due to the presence of obesity, increased abdominal pressure, chest wall deformities, and resuscitation with large fluid volumes [11, 12, 13].

It must be noted again that only if the pressure in the pleural space (PPL) is estimated by measuring PES, CL and CCW can be determined separately. When PAO is measured, the parameters E and R pertain to the entire respiratory system, which includes both the lungs and the chest wall, and therefore it is impossible to identify the separate contribution of the lung and chest wall to the stiffer respiratory system.

Due to several mechanisms, the elastance and compliance of both the lung and the chest wall and, therefore, of the respiratory system are not the same at different lung volumes, in other words the elastic behavior is not linear. The presence and amount of surfactant molecules at the air-liquid interface cause surface tension to vary with lung volume in determining lung hysteresis (i.e., the pressure at a given lung volume is different during inflation compared to deflation) and significant nonlinear mechanical behavior.