HOME HELP FEEDBACK SUBSCRIPTIONS ARCHIVE SEARCH TABLE OF CONTENTS		QUICK SEARCH:   [advanced] Author: Keyword(s): Year:  Vol:  Page:

This Article Abstract Full Text (PDF) Free All Versions of this Article: 99/5/1885    most recent 00450.2005v1 Submit a response Alert me when this article is cited Alert me when eLetters are posted Alert me if a correction is posted Citation Map Services Email this article to a friend Similar articles in this journal Similar articles in ISI Web of Science Similar articles in PubMed Alert me to new issues of the journal Download to citation manager Citing Articles Citing Articles via HighWire Citing Articles via ISI Web of Science (1) Citing Articles via Google Scholar Google Scholar Articles by Lambert, R. K. Articles by Wilson, T. A. Search for Related Content PubMed PubMed Citation Articles by Lambert, R. K. Articles by Wilson, T. A.

Smooth muscle dynamics and maximal expiratory flow in asthma

¹Institute of Fundamental Sciences, Massey University, Palmerston North, New Zealand; and ²Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, Minnesota

Submitted 21 April 2005 ; accepted in final form 17 June 2005

	ABSTRACT

TOP ABSTRACT MODEL RESULTS DISCUSSION REFERENCES

 
A computational model for maximal expiratory flow in constricted lungs is presented. The model was constructed by combining a previous computational model for maximal expiratory flow in normal lungs and a previous mathematical model for smooth muscle dynamics. Maximal expiratory flow-volume curves were computed for different levels of smooth muscle activation. The computed maximal expiratory flow-volume curves agree with data in the literature on flow in constricted nonasthmatic subjects. In the model, muscle force during expiration depends on the balance between the decrease in force that accompanies muscle shortening and the recovery of force that occurs during the time course of expiration, and the computed increase in residual volume (RV) depends on the magnitude of force recovery. The model was also used to calculate RV for a vital capacity maneuver with a slow rate of expiration, and RV was found to be further increased for this maneuver. We propose that the measurement of RV for a vital capacity maneuver with a slow rate of expiration would provide a more sensitive test of smooth muscle activation than the measurement of maximal expiratory flow.

mathematical model; mechanics; constricted lungs; flow-volume curve

More recently, Anafi and Wilson (1) have presented a mathematical model for the dynamic length-tension behavior of airway smooth muscle. This model consists of a set of differential equations relating muscle force, length, and the time derivatives of force and length. Solutions to these equations for a given oscillatory length history match the observed force-length behavior of smooth muscle for different amplitudes of length excursions and a wide range of oscillatory frequencies (3, 20). Anafi and Wilson also incorporated this model into the model of Gunst et al. (5) for the mechanics of an intact airway containing smooth muscle and subjected to a transmural pressure that includes the forces exerted by the attachments to the surrounding parenchyma, the interdependence force. They calculated the time course of muscle shortening and airway narrowing that would occur after a deep breath, and this agreed well with the observed time course of the increase in airway resistance after a deep breath (18).

Here, we describe a computational model for maximal flow in constricted lungs. The model was generated by introducing the Anafi model for constricted airways into the Lambert model for expiratory flow. Flow-volume curves were calculated for different levels of muscle activation. The computed flow-volume curve for maximal muscle activation agrees well with data for flow in maximally constricted normal subjects, and the model provides insight into the mechanics of flow in constricted lungs.

	MODEL

TOP ABSTRACT MODEL RESULTS DISCUSSION REFERENCES

 
All of the elements of the model have been described in detail in earlier papers, and these descriptions will be reviewed briefly here.

Airway geometry and mechanics.   As in the original Lambert model (11), the conducting airways are represented by the trachea (z = 0) plus 16 generations of a symmetric dichotomously branching network (z = 1–16) extending peripherally from the trachea. Thus each generation consists of 2^z identical airways in parallel.

The model for the mechanics of the airways is a combination of the models proposed by Lambert et al. (11) and Anafi and Wilson (1). A schematic diagram of an airway cross section is shown in Fig. 1. The radius and area of the lumen of the airway are denoted r and A, respectively. The airway wall consists of a layer of incompressible tissue surrounded by a thin band of smooth muscle with radius r_m. The airway is embedded in the surrounding parenchyma.

View larger version (21K): [in this window] [in a new window]   Fig. 1. Schematic diagram of the cross section of an airway embedded in the parenchyma. The radius of the ring of muscle at the outer edge of the airway wall is denoted rm, and the radius and area of the lumen are denoted r and A, respectively.

(1)

The terms on the right side of Eq. 1 are functions of A, and these functions are described in the following paragraphs.

The relation between Pel and A is the same as that used by Lambert et al. (11).

(2)

The parameters in Eq. 2, maximal area (A_max), normalized area when Pel = 0 (₀), P₁, P₂, n₁, and n₂, have different values for each generation in the bronchial tree, and the values that are used here are the same as those listed in Ref. 8.

For the intraparenchymal airways (z = 3–16), the relation between Pint and A is the one proposed by Lai-Fook (7) and subsequently used in models of airway mechanics (1, 2, 5).

(3)

In this equation, PL denotes lung recoil pressure, and x describes the mismatch between the outer radius of the airway and the radius of the hole in the undeformed parenchyma. It is assumed that, for the conditions, A = A_max, r_m equal to its corresponding maximal value, denoted r₀, and lung volume equal to total lung capacity (TLC), airway radius matches the radius of the hole in the undeformed parenchyma. At other lung volumes, the radius of the hole in the undeformed parenchyma is assumed to equal r₀v^1/3, where v denotes the ratio of lung volume to volume at TLC. The variable x in Eq. 3 is defined as the difference between the radius of the airway and the radius of the undeformed hole, nondimensionalized by the radius of the undeformed hole, and x is therefore given by the following equation.

(4)

To complete this description of the relation between Pint and A, the relation between r_m and A is required. The area of the tissue inside the muscle is assumed to be 16% of the area inside the muscle at r_m = r₀. This value for the fractional area occupied by tissue is consistent with the data of James et al. (6). The tissue is assumed to be incompressible, and, therefore, A and r_m are related by the following equation.

(5)

Finally, the relation between Pm and A is the same as that proposed by Gunst et al. (5). The active force in the muscle layer per unit distance in the axial direction along the airway (transverse to the plane of Fig. 1) is denoted F. That is, F equals the active stress in the muscle times the thickness of the layer. The compressive pressure exerted by the muscle layer is given by the ratio of F to r_m.

(6)

The value of F depends on the level of muscle activation, muscle length, and length history. The muscle layer is assumed to have adapted to generate maximal force (F₀) when r_m = r₀. Data for the relation between isometric force (Fiso) and muscle length (4) are approximated by the following equation.

(7)

During dynamic muscle shortening, F is smaller than Fiso by the factor f. That is,

(8)

Combining Eqs. 6–8 yields the following expression for Pm.

(9)

The factor F₀/r₀ in this equation describes the amount of smooth muscle in the airway and the degree of activation of the muscle. Based on the data of Gunst and Stropp (4), for maximal muscle activation, this factor is assigned the value of 40 cmH₂O. Finally, the factor f in Eq. 9 depends on muscle history. For muscle shortening, the factor f is determined as the solution to the following empirical differential equation, where k and a are constants and the superscripted dot denotes the time derivative.

(10)

The values of k and a that were obtained by fitting solutions of this equation to the data of Shen et al. (20) are k = 28 and a = 0.12 s^–1. Thus the first term on the right side of Eq. 10 describes a very sharp decrease in f with decreasing r_m, and the second term describes a relaxation of f toward the value 1 with a time constant of 8 s.

The relation between A and P for airways with active smooth muscle is, therefore, given by the solution to Eq. 1 for different values of P, using Eqs. 2, 3, and 9, together with the auxiliary Eqs. 4, 5, and 10 to evaluate the terms on the right side of Eq. 1. The relation between A and P depends on the history of the maneuver.

An example of a particular A-P relation is shown in Fig. 2. This example shows the airway-area curve for the airways in generation 6 at a value of PL = 20 cmH₂O. In Fig. 2, the value of A as a function of Pel is shown by the dashed line. This is the airway-area curve that was used in the original computational model (8). The curve shown by the dotted dashed line is the curve A vs. Pel + Pint. This is the airway-area curve with interdependence forces included. Interdependence has little effect near A/A_max = 1, but it has a significant effect for A/A_max small. The magnitude of Pint depends strongly on PL and hence on lung volume; at lower lung volumes, Pint is considerably smaller. Finally, the curve for P = Pel + Pint + Pm for F₀/r₀ = 40 cmH₂O is shown by the solid line. Because muscle force depends on the history of the maneuver, the history for this example must be specified. The example shown depicts the A-P curve at the moment that maximal flow begins, and the previous history is the inhalation that precedes forced expiration. During that inhalation, the lung and airways expand, and the smooth muscle lengthens. Data show that, during either the lengthening phase of cyclic length changes (20) or lengthening from an initial isometric state (3), force rises slightly above the isometric value. We, therefore, assume that, at the end of inspiration, f = 1, and airway area is given by the solution to Eq. 1 with f = 1. For generation 6 with P = 20 cmH₂O and f = 1, A/A_max = 0.32 and r_m/r₀ = 0.66. Thus, for maximal smooth muscle force, airway area is significantly reduced. As flow begins, P drops below 20 cmH₂O, the airways narrow, and muscle shortens rapidly. For rapid muscle shortening, the first term on the right side of Eq. 10 dominates, and the solution to Eq. 10 is f = exp[k(r_m/r₀ – 0.66)]. The curve shown in Fig. 2 is the solution to Eq. 1 for this relation between f and r_m/r₀. Thus this curve describes the locus of possible values of A and P for the initial flow. Because the force drops rapidly with decreasing muscle length, the curve for Pm 0 merges with the curve for Pm = 0 as airway area decreases.

View larger version (12K): [in this window] [in a new window]   Fig. 2. Ratio of lumen area (A) to maximal area (Amax) vs. lumen pressure (P) for the airways in generation 6 at the onset of expiratory flow (solid line). The contributions of the different components of transmural pressure in Eq. 1 are described by the curves that include only the elastic component (Pel; dashed line) and only the elastic plus interdependence components (Pel + Pint; dotted dashed line).

The pressure gradient in the airways is described by the same equation used in the modeling of maximal flow in human lungs (8).

(11)

In this equation, s denotes distance along the airway in the flow direction, Re denotes the Reynolds number of the flow, µ and denote gas viscosity and density, respectively, and denotes volume flow rate in an airway. The first two terms on the right side of Eq. 11 describe dissipative pressure losses in the airways, and the numerical coefficients in these terms were obtained from Reynolds's data on pressure losses in flow through a cast of a human bronchial tree (19). The first of these is proportional to flow and gas viscosity, and this term is like the term that describes the pressure loss in laminar flow in a cylindrical tube. The second is like the term that describes the turbulent pressure loss in a smooth cylindrical tube. The third term on the right side of Eq. 11 describes the convective pressure decrease in flow through a tube with decreasing cross-sectional area.

Computational method.

The computation of maximal expiratory flow was begun by calculating flow for PL = 20 cmH₂O. The A-P curves for this initial flow were calculated as described above. A small value of was chosen, and the pressure distribution in the bronchial tree was computed by using the same method that was used before (11). First, the Bernoulli drop from the stationary alveolar gas to the flowing gas in the entrance to generation 16 was calculated. Then Eq. 11 was integrated along the airway. Then the Bernoulli drop from the end of generation 16 to the entrance to generation 15 and the pressure drop through generation 15 were calculated, and so on through the bronchial tree to the exit of the trachea. In all steps, the dependence of airway area on pressure was taken into account.

The value of was successively increased, and the calculation was repeated until a value of was reached for which flow speed equaled wave speed at some point in the bronchial tree, or P at the end of the trachea reached –100 cmH₂O. The flow for which one of these two conditions was met was taken as maximal flow at the given lung volume. Flow increments were reduced as maximal flow was approached.

For a small time step, the changes in lung volume and PL were calculated from the value of maximal flow at PL = 20 cmH₂O, and the change in f was computed from the second term on the right side of Eq. 10. New A-P curves were calculated, and a value of maximal flow for this volume was obtained. These calculations were repeated for successive time steps until a volume was reached for which maximal flow was essentially zero. In this process, the time step was adjusted to be appropriate to the rate of muscle shortening. The initial time step was 5 ms, followed by 20 ms, and 100 ms late in expiration.

	RESULTS

TOP ABSTRACT MODEL RESULTS DISCUSSION REFERENCES

 
The MEFV curves calculated for different values of F₀/r₀ are shown in Fig. 3. For values of F₀/r₀ < 20 cmH₂O, smooth muscle activation has little effect on flow. For F₀/r₀ > 20 cmH₂O, maximal flow decreases, and residual volume (RV) increases with increasing F₀/r₀. For maximal smooth muscle activation, maximal flow is less than one-half that in the control state at all lung volumes, and RV is increased by 16% of VC.

View larger version (14K): [in this window] [in a new window]   Fig. 3. Maximal expiratory flow (), in liters per second, vs. expired lung volume [maximal expiratory flow-volume (MEFV curves)], as a percentage of vital capacity (VC), computed from the model for different values of smooth muscle activation (F0/r0, where F0 is maximal force and r0 is maximal value of rm), from no activation (control) up to maximal activation (F0/r0 = 40 cmH2O). Lines are shown for F0/r0 = 0 cmH2O (solid line), 20 cmH2O (dashed line), 30 cmH2O (dotted line), and 40 cmH2O (dotted dashed line). The points at 50 and 30% VC are reduced from the control flows by the same percentage as the average maximum reduction observed in normal subjects with constricted lungs (15).

View larger version (13K): [in this window] [in a new window]   Fig. 4. vs. pressure at the opening of the trachea (Pao) relative to pleural pressure (Ppl) at lung volumes of 75, 50, and 25% VC.

	DISCUSSION

TOP ABSTRACT MODEL RESULTS DISCUSSION REFERENCES

The major results of the modeling are the computed MEFV curves shown in Fig. 3. The relationship between the shapes of the curves for the constricted and control states shown in this figure is like those in examples reported in the literature (e.g., Refs. 10, 16). The most extensive study of maximal expiratory flow in constricted normal lungs is that of Moore et al. (15). These authors measured the dose-response relationship of MEFV curves to inhaled methacholine in 73 normal subjects of both genders and a wide range of ages. Although not all of the subjects reached a plateau in response to increasing doses of methacholine, many did, and one might expect that muscle activation was near maximal at the maximal response. Moore et al. reported the fractional decrease in flow at 30 and 50% VC. The average maximal response of the subjects was a 53% reduction in flow at 50% VC and a 66% reduction at 30% VC. Points with these percentage decreases in flow are shown in Fig. 3. These points lie close to the model prediction for maximal muscle activation. Also, Moore et al. report that the average maximal decrease in forced expiratory volume in 1 s (FEV₁) in their subjects was 24%, and, for maximal muscle activation in the model, the decrease in FEV₁ is essentially the same, 26%. Thus the quantitative agreement between the predicted flows and the data is good. We would like to emphasize the fact that the values of the parameters in the current model were all set by previous matching to independent anatomic and physiological data, and no parameter values were adjusted to match the data for flow in constricted lungs.

A computational model constitutes a quantitative expression of our understanding, and that is its primary use. To the extent that predictions of a model can be compared with data, it provides a quantitative test of our understanding. Also, with a model, quantities that are inaccessible to measurement can be evaluated, and the results of impossible experiments can be calculated, and these exercises can extend our understanding. In the following paragraphs, the information that the model provides about the mechanisms of flow limitation in constricted lungs is discussed.

One notable feature of the curves shown in Fig. 3 is the insensitivity of the MEFV curve to lower levels of smooth muscle activation. For F₀/r₀ = 20 cmH₂O, the MEFV curve and FEV₁ are nearly the same as those for the control state. For this value of F₀/r₀, A/A_max for the peripheral airways is 0.7. As a result, the pressure losses in the periphery, which are proportional to A^–2, are twice those in the control state. However, the peripheral losses in the control state are small, and, with an increase by a factor of 2, they remain small. More importantly, at lower levels of activation, the A-P curve for the constricted airway merges with the A-P curve for the unconstricted airway at positive pressures, rather than at negative pressures, as shown in Fig. 2. As a result, for negative values of P, the areas of the constricted airways are nearly the same as the areas of unconstricted airways. Thus the A-P curve in the region where flow limitation occurs is nearly the same as in the control case. Because FEV₁ is insensitive to lower levels of muscle activation, one might expect the dose-response curve of FEV₁ to methacholine (15) to be narrower than the dose-response curve of smooth muscle force in vitro (13), and this is the case.

At higher values of F₀/r₀, this drop in muscle force with decreasing muscle length also diminishes the effect of muscle activation on the MEFV curve. To illustrate this point, an MEFV curve for F₀/r₀ = 40 cmH₂O, and no effect of length on force (k = 0 in Eq. 11) was calculated, and this curve is shown in Fig. 5. In this case, muscle force remains equal to the isometric force for maximal muscle activation as the muscle shortens. Flow is drastically reduced at higher lung volumes and drops sharply with decreasing volume. The airways close at a lung volume for which PL = 10.4 cmH₂O. This is consistent with the observation that, under static conditions, maximally activated smooth muscle closes the airways for values of PL is less than 10 cmH₂O (5).

View larger version (11K): [in this window] [in a new window]   Fig. 5. Effects of muscle dynamics on flow. The MEFV curves for the control (dotted line) and maximal smooth muscle activation states (solid line) are compared with those calculated for no dynamic effect of shortening (constant k = 0 in Eq. 11, dotted dashed line) and no force recovery (a = 0 in Eq. 11, dashed line). Also shown is the flow-volume curve for expiration limited to 200 ml/s (solid line at low flow). The value of residual volume (RV) for this slow expiration is considerably higher than that for the forced VC maneuver.

The second term on the right side of Eq. 10, the equation that governs smooth muscle dynamics, also plays a significant role in determining the MEFV curves shown in Fig. 3. This term describes the recovery of force over time that occurs when muscle force is depressed below the isometric force. To illustrate the effect of this term, an MEFV curve for F₀/r₀ = 40 cmH₂O and no force recovery (a = 0 in Eq. 10) was calculated, and this curve is shown in Fig. 5. It can be seen from Fig. 5 that this term affects the MEFV curve at later times in the expiration, and it is the term that determines the magnitude of the increase in RV in the constricted lung.

It can be seen from Fig. 3 that, at F₀/r₀ = 40 cmH₂O, flow is decreasing rapidly with increasing F₀/r₀. Therefore, muscle hypertrophy would be expected to have a big effect on flow. To model muscle hypertrophy, the MEFV curve for a 25% increase in F₀/r₀ from 40 to 50 cmH₂O was calculated, and the result is shown in Fig. 6. The increase in F₀/r₀ caused a nearly parallel shift of the MEFV curve, and FEV₁ decreased to 46% of the control value. The effect of inflammation was modeled by increasing the amount of tissue in the wall. Maximal flow was calculated for F₀/r₀ = 40 cmH₂O and a 25% increase in the tissue volume, from 16% of the area inside the muscle ring to 20%. The result is shown in Fig. 6. The change in wall area also caused a nearly parallel shift in the MEFV curve, but the magnitude of the shift was less than one-half that for the model of muscle hypertrophy. In asthma, muscle hypertrophy and inflammation would be expected to be nonuniform, whereas, in the modeling, the changes in airway properties were applied uniformly. Therefore, these results only indicate the potential magnitude of the effects of muscle hypertrophy and inflammation.

View larger version (12K): [in this window] [in a new window]   Fig. 6. Effects of inflammation and muscle hypertrophy on flow. MEFV curves are shown for the following cases: the base model with maximal muscle activation (solid line); inflammation, modeled by increasing the volume of wall tissue by 25% (dotted line); muscle hypertrophy, modeled by increasing muscle force by 25% (dashed line); and combined inflammation and muscle hypertrophy (dotted dashed line).

View larger version (13K): [in this window] [in a new window]   Fig. 7. Percent change from control in expired volume in 1 s (FEV1) for the forced VC maneuver vs. level of activation (F0/r0), and change in RV from control as a fraction of VC for a VC maneuver with slow expiration.

	FOOTNOTES

 

Address for reprint requests and other correspondence: T. A. Wilson, 107 Akerman Hall, 110 Union St. SE, Minneapolis, MN 55455 (e-mail: wilson{at}aem.umn.edu)

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

	REFERENCES

TOP ABSTRACT MODEL RESULTS DISCUSSION REFERENCES

 

1. Anafi RC and Wilson TA. Empirical model for dynamic force-length behavior of airway smooth muscle. J Appl Physiol 92: 455–460, 2002.
2. Anafi RC, Beck KC, and Wilson TA. Impedance, gas mixing, and bimodal ventilation in constricted lungs. J Appl Physiol 94: 1003–1011, 2003.
3. Gunst SJ. Contractile force of canine airway smooth muscle during cyclical length changes. J Appl Physiol 55: 759–769, 1983.
4. Gunst SJ and Stropp JQ. Pressure-volume and length-stress relationships in canine bronchi in vitro. J Appl Physiol 64: 2522–2531, 1988.
5. Gunst SJ, Warner DO, Wilson TA, and Hyatt RE. Parenchymal interdependence and airway response to methacholine in excised dog lobes. J Appl Physiol 65: 2490–2497, 1988.
6. James AL, Hogg JC, Dunn LA, and Pare PD. The use of the internal perimeter to compare airway size and to calculate smooth muscle shortening. Am Rev Respir Dis 138: 136–139, 1988.
7. Lai-Fook SJ. A continuum mechanics analysis of pulmonary vascular interdependence in isolated dog lobes. J Appl Physiol 46: 419–429, 1979.
8. Lambert RK. Sensitivity and specificity of the computational model for maximal expiratory flow. J Appl Physiol 57: 958–970, 1984.
9. Lambert RK. Bronchial mechanical properties and maximal expiratory flows. J Appl Physiol 62: 2426–2435, 1897.
10. Lambert RK and Beck KC. Airway area distribution from the forced expiration maneuver. J Appl Physiol 97: 570–578, 2004.
11. Lambert RK, Wilson TA, Hyatt RE, and Rodarte JR. A computational model for expiratory flow. J Appl Physiol 52: 44–56, 1982.
12. McNamara JJ, Castile RG, Ludwig MS, Glass GM, Ingram RH Jr, and Fredberg JJ. Heterogeneous regional behavior during forced expiration before and after histamine inhalation in dogs. J Appl Physiol 76: 356–369, 1994.
13. Mehta D, Wu MF, and Gunst SJ. Role of contractile protein activation in the length-dependent modulation of tracheal smooth muscle force. Am J Physiol Cell Physiol 270: C243–C252, 1996.
14. Mele MFV, Harris RS, Layfield JDH, and Venegas JG. Topographic basis of bimodal ventilation-perfusion distributions during bronchoconstriction in sheep. Am J Respir Crit Care Med 171: 714–721, 2005.
15. Moore BJ, Hilliam CC, Verburgt LM, Wiggs BR, Vedal S, and Pare PD. Shape and position of the complete dose-response curve for inhaled methacholine in normal subjects. Am J Respir Crit Care Med 154: 642–648, 1996.
16. Pellegrino R, Violante B, and Brusasco V. Maximal bronchoconstriction in humans. Relationship to deep inhalation and airway sensitivity. Am J Respir Crit Care Med 153: 115–121, 1996.
17. Pellegrino R, Violante B, Selleri R, and Brusasco V. Changes in residual volume during induced bronchoconstriction in healthy and asthmatic subjects. Am J Respir Crit Care Med 150: 363–368, 1994.
18. Pellegrino R, Wilson O, Jenouri G, and Rodarte JR. Lung mechanics during induced bronchoconstriction. J Appl Physiol 81: 964–975, 1996.
19. Reynolds DB. Steady expiratory flow-pressure relationship in a model of the human bronchial tree. J Biomech Eng 104: 153–158, 1982.
20. Shen X, Wu MF, Tepper RS, and Gunst SJ. Mechanisms for the mechanical response of airway smooth muscle to length oscillations. J Appl Physiol 83: 731–738, 1997.

This article has been cited by other articles:

				R. Pellegrino, V. Brusasco, R. O. Crapo, and G. Viegi From the authors Eur. Respir. J., May 1, 2006; 27(5): 1070 - 1070. [Full Text] [PDF]

This Article Abstract Full Text (PDF) Free All Versions of this Article: 99/5/1885    most recent 00450.2005v1 Submit a response Alert me when this article is cited Alert me when eLetters are posted Alert me if a correction is posted Citation Map Services Email this article to a friend Similar articles in this journal Similar articles in ISI Web of Science Similar articles in PubMed Alert me to new issues of the journal Download to citation manager Citing Articles Citing Articles via HighWire Citing Articles via ISI Web of Science (1) Citing Articles via Google Scholar Google Scholar Articles by Lambert, R. K. Articles by Wilson, T. A. Search for Related Content PubMed PubMed Citation Articles by Lambert, R. K. Articles by Wilson, T. A.