Abstract
We have developed a mathematical model for a tracheal ring that consists of a “horseshoe” of cartilage with its tips joined by a membrane. The ring is subjected to a uniform transmural pressure (Ptm) difference. The model was used to calculate the crosssectional area (A) of the trachea. Whereas the mechanics of the deformation of the cartilage were analyzed using elastica theory, the posterior membrane was treated as a simple membrane that is inextensible under changes in Ptm. The membrane can be specified to be of any length less than baseline and thus can represent a posterior membrane under tension. The cartilage can have specifiable nonuniform unstressed curvature as well as nonuniform bending stiffness. We have investigated the effect on the tracheal APtm curve of posterior membrane length and tensile force in the membrane, cartilage shape and elasticity, and localized weakening of the cartilage. The model predictions are in good agreement with magnetic resonance imaging data from rabbit tracheas and show that the shape of the horseshoe as well as the posterior membrane force are important determinants of tracheal compliance.
 tracheomalacia
 sabersheath trachea
 chronic obstructive pulmonary disease
 airway resistance
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 crosssectional area of the airway. An analysis of the interaction requires a knowledge of the areatransmural pressure (APtm) relationship of the airway and how this changes with diseaseinduced 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 welldefined 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 APtm 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 trachealAPtm 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
Our conceptual model of the trachea consists of a horseshoeshaped 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 cartilagemembrane 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.
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, 1012, 16). A brief derivation of a form of the equations more general than published to date is given in the
for completeness. This analysis yieldsEq. 1
, which is the normalized differential equation describing the deformation of the cartilage; θ 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 (λ).
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 constantκ
_{0}). To accommodate this observation, we allowed κ
_{0} to vary around the arc, as in a previous study (12)
There is no known equation for θ as a function of λ that satisfiesEq. 1 . Therefore, numerical methods were used to obtain solutions (see ).
RESULTS
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(λ) = 1, Eq.2 ]. Figure 2 A 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 2 B shows the effect of an external pressure field alone on tracheal collapse forL = L _{I} (the posterior membrane is inextensible) at four values of q.
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 isD/wR
^{3} where D = E_{C}
I
_{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}
=(π/4)ab
^{3}, where a andb 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
To investigate the effect of shortening of the posterior membrane on tracheal mechanics, we computed the APtm curves for the initially semicircular cartilage for four lengths ≤ L _{I}. These are shown in Fig.3. A _{0} is the area at Ptm = 0. In Fig. 3 A, 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. 3 B, the curves are normalized on their own value of A _{0}, which enables comparison of the specific compliances. The APtm curves have been computed from Ptm = 20 cmH_{2}O 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.
Because the lengthtension 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/R
^{2}. With the values from Begis et al. (3) given above, the dimensioned membrane tensile force (F) is given by
The importance of the APtm 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 APtm curves given in Fig. 3.
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 forf(λ) given below (Eqs. 5
,
6
, and
7
). In these, the magnitude of the stiffness is controlled by choosing a value for the parameter α between 0 and 1; α = 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), λ takes values between 0 (at the line of symmetry) and π/2 at the cartilage tip
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.7 A). Deformations of the profiles shown in Fig. 7 A were computed with constantD (α = 0 in Eqs. 57 ). A comparison of the APtm curves for L = L _{I}is presented in Fig. 7 B. The curves are selfnormalized; that is, A _{0} is the zeropressure area for each case.
APtm 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 cmH_{2}O (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.
Finally, we used the model to study the mechanics of two airway pathologies: sabersheath trachea (6) and lunate trachea (18). The results for these are given in Fig.9 in which Fig. 9 A 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.
DISCUSSION
We have developed an analysis of tracheal mechanics that yields predictions of the APtm characteristics of the tracheal cartilageposterior membrane system. We analyzed the trachea rather than other cartilaginous airways because its welldefined 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 theAPtm curve of the trachea and those of the membranous bronchioles for which models are starting to be formulated (13,24).
We used a wellestablished 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 thicknesstoradius 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 normalizedAPtm 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 tensionfree 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 25kg 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. 4 A 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. 4 B). 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 modelAPtm 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 cmH_{2}O. 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: sabersheath trachea and lunate trachea (Fig. 9). The APtm curves in Fig. 9 Billustrate the complex interaction between profile and F. The unstressed lunate trachea is lifethreateningly 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 sabersheath trachea is almost entirely restricted to positive values of Ptm. This shape is very stiff. The clinical implications are clear. Treatment of sabersheath 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 lifethreateningly compliant.
Acknowledgments
This study was supported in part by the Palmerston North Medical Research Foundation.
Footnotes

Address for reprint requests and other correspondence: R. K. Lambert, Institute of Fundamental SciencesPhysics, Massey Univ., Palmerston North, New Zealand 5331 (Email:R.Lambert{at}massey.ac.nz).

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 Copyright © 2001 the American Physiological Society
Appendix
The analysis of tracheal cartilage deformation is based on elastica theory as was the case in a previous study (12). The main features of the analysis are as follows.
The freebody diagram for a small section of a ring of arc length ds and constant width w is shown in Fig.10. S is the shear force, M is the bending moment, and θ is the angle between the tangent to wall and the xaxis, Ptm is the applied pressure difference (internal pressure minus external pressure), and s is the arc length. The requirements of static equilibrium lead to Eqs.EA1
–
EA3
Four boundary conditions are required to solve Eq. EA7
. Symmetry provides one of these; namely, S at the symmetry axis must be zero. The other conditions can be elucidated by applying the conditions of static equilibrium to the deformed cartilage as a whole. Deformation is caused either by the trachealis muscle shortening and generating a force, T_{M}, or an external, uniform, pressure field, Ptm, is applied.
Equation EA7
had to be solved numerically. This was done as follows. At λ = 0, θ and θ" are zero. Because the other two derivatives of θ are unknown, initial guesses were made as to their values and Eq. EA7
integrated numerically to the tip of the halfring (λ = λ_{f}) using a BulirschStoer algorithm (21). At this point, the boundary conditions given in Eqs. EA9
and
EA10
apply. However, these conditions could not be used directly because they contain the unknown force and bending moment applied by the attached membrane. Therefore, the forces in the ring were evaluated and equated to the force applied to the tip by the membrane. M_{x} and M_{y} were calculated from this. Because the membrane force must be tangential to the membrane at the point of attachment and the membrane has the same axis of symmetry as the cartilage, the curvature and thus theL of the membrane could be calculated. In all of the results presented here, the membrane was held at some specified constant length. The difference between the computed and required lengths (L − L
_{comp}) was calculated. From these calculations, a cost function (CF) was evaluated