INTRODUCTION

TOP ABSTRACT INTRODUCTION MODEL RESULTS DISCUSSION APPENDIX REFERENCES

THE MAJOR CONTRIBUTION TO expiratory resistance during quiet breathing comes from the large central airways. Despite being the stiffest airways of the bronchial tree, they nonetheless undergo significant deformation at physiological pressures, and thus a coupling exists between flow in the airway and the cross-sectional area of the airway. An analysis of the interaction requires a knowledge of the area-transmural pressure (A-Ptm) relationship of the airway and how this changes with disease-induced alterations in tissue properties, particularly the stiffness of the cartilage and, in the trachea, the elasticity and muscular tone of the posterior membrane. These considerations have relevance to chronic obstructive pulmonary disease in which weakening of the wall structures of large airways has been suggested (1). In addition, cartilage geometry and elasticity appear to play roles in tracheomalacia (18, 19).

The trachea is an attractive airway for mechanical analysis because of its reasonably regular and well-defined structure. However, only one attempt at analyzing the mechanics of this airway has been published (3). This study was limited by the available computer power. There have been many roentgenographic, computerized tomography, and fiberoptic studies of tracheal behavior (e.g., Refs. 1, 5, 6, 18, 19), but few have produced A-Ptm data (e.g., Refs. 1, 19).

We have developed a model of tracheal mechanics using elastica theory. This is the same theory that was used in analyzing the mechanics both of a Starling resistor (4, 10, 16) and of folding of the airway epithelial basement membrane (11) as well as in an experimental study of tracheal cartilage elasticity (12). Using this model, we have investigated the effect on the tracheal A-Ptm curve of posterior membrane tension and length, cartilage shape and elasticity, and localized weakening of the cartilage. The model results show that unstressed cartilage geometry as well as membrane tension are important determinants of tracheal compliance.

	MODEL

TOP ABSTRACT INTRODUCTION MODEL RESULTS DISCUSSION APPENDIX REFERENCES

Our conceptual model of the trachea consists of a horseshoe-shaped cartilage "ring," a membrane containing smooth muscle that joins the tips of the horseshoe [separation = membrane length (L) at Ptm = 0], and intercartilage connective tissue as shown in Fig. 1. We chose a simple base case in which the undeformed cartilage is a semicircle. The unstressed posterior membrane is connected to the ends of this semicircle. The cartilage-membrane system can be deformed in one of two ways. Either the muscle shortens (L < L_I, where L_I is the maximal, fully relaxed length of the posterior membrane) and generates tension that forces the cartilage tips together and thus deforms the cartilage, or Ptm causes the posterior membrane to invaginate (Ptm < 0) into the lumen of the cartilage or to expand out of the lumen (Ptm > 0). The invagination or expansion of the posterior membrane can happen at any state of muscle shortening (L  L_I). We will ignore the contribution of the intercartilage membrane in this analysis.

View larger version (17K): [in this window] [in a new window]   Fig. 1.   Model tracheal cross sections under 4 conditions. The thick curves indicate cartilage, and the thin lines indicate the posterior membrane. L, length of the posterior membrane; LI, maximal length; Ptm, transmural pressure difference.

The governing equations describing the deformation of the cartilage are based on the physics of a shell and have already been presented several times elsewhere (4, 10-12, 16). A brief derivation of a form of the equations more general than published to date is given in the APPENDIX for completeness. This analysis yields Eq. 1, which is the normalized differential equation describing the deformation of the cartilage;  is the angle the cartilage makes with a reference direction, and subscript 0 indicates the zero stress state. A prime indicates differentiation with respect to normalized arc length ().

(1)

In developing the analysis, three issues had to be taken into account. First, for maximal generality of the results, it is convenient to work with normalized quantities. Thus, in Eq. 1, arc length is normalized on the radius (R) of the initially semicircular cartilage and q is the nondimensional pressure. A second issue is the likely lack of uniformity in the mechanical and geometrical properties of real cartilage. This is handled by the use of a function f(), which allows the introduction of inhomogeneities through their effect on the flexural rigidity (also known as bending stiffness) of the cartilage (D). D is a measure of the stiffness of the cartilage and depends on the cartilage Young modulus (E) and the dimensions of the cartilage cross section. We assume that D can vary along the length of the arc of the cartilage and express D as the product of a constant stiffness (D₀) and f(), which is a nondimensional function of arc length (Eq. 2)

(2)

f() is chosen such that it can take values only between zero and one. Thus D₀ is the maximal value of the flexural rigidity. The pressure scale for q in Eq. 1 is then D₀/wR³ (w is the width of the cartilage horseshoe).

A third issue is the effect of the unstressed cartilage shape on tracheal compliance. It is uncommon for an unstressed cartilage horseshoe to be exactly semicircular (that is, to have constant ₀). To accommodate this observation, we allowed ₀ to vary around the arc, as in a previous study (12)

(3)

where C is the curvature at the midpoint of the horseshoe, and B is related to how rapidly ₀ changes with . The posterior membrane is modeled as a membrane (as opposed to a shell) penetrating into the lumen (q < 0) or expanding out of the lumen (q > 0) (Fig. 1). Because it is a membrane subjected to a uniform Ptm, its cross-section perpendicular to the tracheal axis must be part of a circle. The radius and the subtended angle of the circular segment bounded by the membrane can then be calculated from geometry. L was assumed to be constant regardless of the value of q. The entire A of the lumen is the area enclosed by the cartilage (A_C) plus (q > 0) or minus (q < 0) the area (A_M) of the circular segment bounded by the membrane and the straight line joining the cartilage tips.

(4)

x and y are the nondimensional Cartesian coordinates of a point on the cartilage horseshoe. The origin of this coordinate system is the intersection of the cartilage and its axis of symmetry ( = 0, Fig. 10). The subscript f denotes the final coordinate; that is, the cartilage tip.

There is no known equation for  as a function of  that satisfies Eq. 1. Therefore, numerical methods were used to obtain solutions (see APPENDIX).

	RESULTS

TOP ABSTRACT INTRODUCTION MODEL RESULTS DISCUSSION APPENDIX REFERENCES

Figure 2 shows calculated profiles of the initially semicircular cartilage at several stages of deformation, assuming a constant stiffness along the circumference of the cartilage [f() = 1, Eq. 2]. Figure 2A shows the cartilage when the posterior membrane is shortened to 50% of its maximal length (L_I) and q = 0, that is without any pressure difference across the walls. Figure 2B shows the effect of an external pressure field alone on tracheal collapse for L = L_I (the posterior membrane is inextensible) at four values of q.

View larger version (18K): [in this window] [in a new window]   Fig. 2.   Calculated tracheal cross sections. q, Nondimensional Ptm. A: q is zero. Thin lines, zero tensile force in posterior membrane; thick lines, posterior membrane shortened to 50% of length at LI. B: L remains at LI; 4 values of q (1, 0, 1, 2.5) are shown.

Whereas the most general presentation of results is achieved using q, the presentation is more intelligible if dimensioned pressure is used. Therefore, we chose to convert q to Ptm by calculating a representative conversion factor. The normalization factor for the pressure difference across the cartilage horseshoe is D/wR³ where D = E_CI_A and E_C is the Young modulus for the cartilage and I_A is the second moment of area of the cross section. We approximated the cross section as an ellipse. Thus I_A = (/4)ab³, where a and b are the lengths of the semimajor and semiminor axes of the ellipse, respectively. We used the following data for human tracheal cartilage taken from Begis et al. (3): a = 1.8 mm, b = 0.8 mm, and R = 10 mm, where R is the average radius of the trachea. We chose 10 MPa as a representative value for E (22). The conversion is then

Thus, for simplicity, we will use Ptm = 20q cmH₂O to represent the case of human tracheal cartilage.

To investigate the effect of shortening of the posterior membrane on tracheal mechanics, we computed the A-Ptm curves for the initially semicircular cartilage for four lengths  L_I. These are shown in Fig. 3. A₀ is the area at Ptm = 0. In Fig. 3A, the curves are normalized on the area of the base case and thus give the relative sizes of the airway lumens and compliances. In Fig. 3B, the curves are normalized on their own value of A₀, which enables comparison of the specific compliances. The A-Ptm curves have been computed from Ptm = 20 cmH₂O to the value of Ptm (< 0) at which the posterior membrane touched the inner wall of the cartilage. At this point, the boundary conditions on the problem change, and it is not clear to us how to model the changed conditions.

View larger version (17K): [in this window] [in a new window]   Fig. 3.   Simulated human lumen area (A)-Ptm curves at 4 values of L. A: area normalized on area of base case (L = LI, Ptm = 0). B: area of each curve normalized on area at Ptm = 0 for that curve's value of L.

Because the length-tension relationship of the trachealis is unknown under conditions of changing Ptm, we investigated the interrelationship between membrane length, membrane tensile force, and pressure for the deformation of an initially semicircular cartilage. The nondimensional tensile force (T) in the posterior membrane was dimensioned using the force conversion factor D/R². With the values from Begis et al. (3) given above, the dimensioned membrane tensile force (F) is given by
0.072 N is the weight of 7.3 g. F was calculated as a function of Ptm at four stages of shortening of the posterior membrane (Fig. 4A). These data have been added to and replotted in Fig. 4B to show how F depends on membrane length at constant Ptm for values of L/L_I  0.25. This value was chosen as a lower limit because the muscle is unlikely to shorten further in vivo (9).

View larger version (18K): [in this window] [in a new window]   Fig. 4.   Relationship between Ptm and L and tensile force (simulated human). A: membrane force as a function of Ptm at 4 values of (constant) membrane length. B: membrane force as a function of length at constant pressure difference. Ptm measured in cmH2O.

The importance of the A-Ptm curves lies in the fact that they determine the resistance of the trachea as a function of Ptm. We evaluated the expiratory flow resistance using the empirically derived formula (23) used in other models of expiratory flow (15, 17) and normalized the results on the resistance of our base case at Ptm = 0. The results are shown in Fig. 5 for the A-Ptm curves given in Fig. 3.

View larger version (22K): [in this window] [in a new window]   Fig. 5.   Tracheal resistance normalized on the resistance of the base case (unstressed posterior membrane at 0 Ptm) at 4 values of membrane length. Ptm scale is from human data. Raw, airway resistance.

Cartilage is unlikely to have uniform stiffness around its circumference. To investigate this, we studied three cases of variable stiffness by choosing the periodic functions for f() given below (Eqs. 5, 6, and 7). In these, the magnitude of the stiffness is controlled by choosing a value for the parameter  between 0 and 1;  = 0 corresponds to constant stiffness. Because the radius of the undeformed semicircular cartilage is used as the length scale of the problem and the problem is solved over half the cartilage length (the cartilage is assumed to be symmetrical),  takes values between 0 (at the line of symmetry) and /2 at the cartilage tip

(5)

(6)

(7)

Each function puts the point of maximum weakness in a different location: Eq. 5 at the line of symmetry ("center"), Eq. 6 halfway between the line of symmetry and the tip ("mid-arc"), and Eq. 7 at the tip. Results from these investigations are presented in Fig. 6 in which  was kept constant at a value of 0.5 and the unstressed cartilage was semicircular.

View larger version (18K): [in this window] [in a new window]   Fig. 6.   A-Ptm curves showing the effect of variable bending stiffness (D) along the cartilage length (simulated human). See text for details.

We used Eq. 3 to study other profiles: curvature that is constant but greater than that for a semicircle (B = 0, C = 1.5) and the cases of semicircular curvature at the line of symmetry but the curvature increasing (B = 0.3, C = 1.0) or decreasing (B = 0.3, C = 1.0) toward the cartilage tips (Fig. 7A). Deformations of the profiles shown in Fig. 7A were computed with constant D ( = 0 in Eqs. 5-7). A comparison of the A-Ptm curves for L = L_I is presented in Fig. 7B. The curves are self-normalized; that is, A₀ is the zero-pressure area for each case.

View larger version (20K): [in this window] [in a new window]   Fig. 7.   Effect of initial cartilage profiles on A-Ptm curves (simulated human). A: cartilage profiles for 4 sets of values of B and C. B: A-Ptm curves corresponding to profiles in A. Area of each curve normalized on own area at Ptm = 0.

A-Ptm curves for rabbit tracheal cartilage rings obtained using magnetic resonance imaging (MRI) were published several years ago (14). These data are compared with our predictions in Fig. 8 in which the following values for rabbit tracheal cartilage were used: E_C = 10 MPa, b = 0.25 mm, and R = 4 mm, which yield Ptm = 960q Pa or 10q cmH₂O (approximately); b is the length of the semiminor axis of the elliptical cross section. The value of E_C is the same as that used in our calculations for human data for want of rabbit data, whereas the values of b and R were estimated from the MRI images.

View larger version (25K): [in this window] [in a new window]   Fig. 8.   Comparison of rabbit model results with magnetic resonance imaging (MRI) data from rabbit tracheas.

Finally, we used the model to study the mechanics of two airway pathologies: saber-sheath trachea (6) and lunate trachea (18). The results for these are given in Fig. 9 in which Fig. 9A shows the profiles with both a relaxed posterior membrane and a membrane shortened to 75% of its unstressed length. The relaxed semicircular case is included for reference.

View larger version (20K): [in this window] [in a new window]   Fig. 9.   Simulation of 2 tracheal disease conditions: lunate trachea and saber sheath trachea (simulated human). A: model cartilage profiles. Thick curve, base case (refer to Fig. 2); dotted lines, lunate trachea; thin lines, saber sheath trachea. In all cases, Ptm is zero, outer profile is with zero tension in posterior membrane, and inner profile has membrane shortened to 75% of its unstressed length. B: lumen A-Ptm curves. Lumen area is normalized on that of the base case. Human pressure scale is used.

	DISCUSSION

TOP ABSTRACT INTRODUCTION MODEL RESULTS DISCUSSION APPENDIX REFERENCES

We have developed an analysis of tracheal mechanics that yields predictions of the A-Ptm characteristics of the tracheal cartilage-posterior membrane system. We analyzed the trachea rather than other cartilaginous airways because its well-defined geometry permits the formulation of a relatively straightforward model. We expect other cartilaginous airways to be more compliant and the compliance to increase as the quantity of cartilage in the wall decreases. Thus it should be possible to interpolate between the A-Ptm curve of the trachea and those of the membranous bronchioles for which models are starting to be formulated (13, 24).

We used a well-established analytical technique to develop the model. The technique is valid for this situation of large displacement but small strain and has been used successfully in the past to study the mechanics of deformable tubes (4, 10, 11, 16) and cartilage rings (12). The small strain requirement can be gauged by the thickness-to-radius ratio (t/R) of the deformed structure. This is of the order of 0.1 for our model cartilage ring. Whereas this does not strictly satisfy the required smallness criterion (t/R 1), it leads to the relatively small error of 3% in the predicted displacement of the cartilage tips (12). We believe that this is sufficiently small for the analysis to have reasonable quantitative validity. This is borne out by the comparison with the MRI data (Fig. 8). Further validation would require us to use our rabbit model to predict results under other conditions, for instance, after the trachea had been challenged with a muscle agonist. We do not have access to appropriate experimental results.

Our treatment of the posterior membrane as being inextensible is a simplification. The membrane probably does change length with changes in the pressure field, especially for Ptm < 0 (20), but the length changes will be dependent on the tone in the smooth muscle, its previous stress history, and the time course of the pressure change, since the tissue is viscoelastic. We have assumed that the membrane inserts into the cartilage tips. This is a reasonable assumption for humans but is not valid for all species. In particular, rabbit posterior membrane inserts on the outside of the cartilage horseshoe away from the tips. However, the results in Fig. 8 suggest that this is not significant except, perhaps, at very negative values of Ptm.

It is likely that the cartilage is inhomogeneous in both its elastic and geometric properties. Whereas there is some experimental evidence for this, the experiments did not quantify the inhomogeneity (12). Our attempt to simulate inhomogeneity (Fig. 6) by making the stiffness of the cartilage change by 50% along the circumference produced only a very modest change to the normalized A-Ptm curves. A 50% decrease in stiffness could be generated in many ways. Two examples are a 50% decrease in the Young modulus of the cartilage or a 14% decrease in thickness. It appears that the force in the posterior membrane (Fig. 3) and the shape of the horseshoe (Fig. 7) are more important determinants of tracheal stiffness.

It is apparent that the MRI data match the theoretically predicted compliance curve for a trachea in which the posterior membrane is shortened to 50% of its tension-free length at zero Ptm (Fig. 8). That the agreement is with a case that has a prestressed posterior membrane is in accord with our (unpublished) observation that the mucosal membrane in rabbits is in tension. It is also in accord with observations of porcine and canine tracheas (8). The predictions of the magnitude of the posterior membrane tensile force (Fig. 4) are in accord with tracheal data from 20- to 25-kg mongrel dogs (7) and with bronchial data from adult humans (9). The graphs show that, for positive Ptm, F increases with Ptm, as one would expect; the trachea is being blown up like a balloon. For negative Ptm, it is not quite as obvious what will happen. If the pressure difference were applied to the cartilage alone, the membrane would become slack. However, the membrane must also support the pressure difference and meet the boundary conditions even as its radius of curvature changes. Thus there must be tensile force in the membrane, but, as Fig. 4A shows, this force depends not only on the pressure but also on the initial length of the membrane. For the case with the shortest membrane (L = 25% L_I), F steadily reduces as the pressure difference is increased, whereas the longer membranes all require greater F (Fig. 4B). The case of constant F appears to be between 25% and 50% L_I.

Under normal conditions of quiet breathing, tracheal Ptm is positive and the overall resistance of the total human airway system is low, with the intrathoracic trachea accounting for ~20% of that resistance. However, when the flow is elevated, as it is during exertion and cough, tracheal Ptm can become negative with possibly large increases in tracheal resistance. We used our model A-Ptm curves to estimate the changes in tracheal resistance under changing Ptm (Fig. 5). It can be seen that the resistance of the highly compliant base case increases very rapidly with increasingly negative values of Ptm and exceeds the resistance of the less compliant cases for Ptm less than 4 cmH₂O. Resistance at negative Ptm is reduced by a shortening of the posterior membrane but at the cost of increased resistance when Ptm > 0.

Our ability to alter the cartilage profile (Fig. 7) showed that unstressed cartilage shape is an important determinant of compliance. In particular, it is apparent that greater separation of the cartilage tips results in greater airway compliance. This led us to investigate two documented pathological profiles: saber-sheath trachea and lunate trachea (Fig. 9). The A-Ptm curves in Fig. 9B illustrate the complex interaction between profile and F. The unstressed lunate trachea is life-threateningly compliant near Ptm = 0. However, some tension in the posterior membrane (L < L_I) greatly stiffens the airway at the cost of decreased area for Ptm > 0. The effect of F on the saber-sheath trachea is almost entirely restricted to positive values of Ptm. This shape is very stiff. The clinical implications are clear. Treatment of saber-sheath trachea with bronchodilators will achieve very little because the shape is so intrinsically stiff. On the other hand, bronchodilator treatment of a lunate trachea in which there is some trachealis tension and a weakened mucosal membrane would make the trachea life-threateningly compliant.