## Abstract

The aim of the present study is to determine the distribution of residual circumferential strains along the duodenum in anesthetized guinea pigs. A silicone elastomer was allowed to harden in the duodenal lumen under a pressure of 0.7 kPa. The duodenum was excised with the cast and photographed. The zero-stress state was obtained by cutting rings of duodenum radially. The geometric configuration at the zero-stress state is of fundamental importance, because it is the basic state with respect to which the physical stresses and strains are defined. A basic piece of information is the way the tangent vector rotates from one end of the circumference to the other. In the duodenum at zero-stress state, the total rotation of the tangent from one tip to the other is −500 to −850°, with the lowest absolute value in the proximal duodenum. In other words, the duodenum usually turns itself inside out on changing from a loaded state to the zero-stress state. The serosal circumference, the duodenal wall thickness, and the ratio of wall thickness to mucosal circumference decreased in the distal direction. In the pressurized state, the serosal Cauchy strain was tensile and increased in the distal direction; the mucosal Cauchy strain was compressive in the proximal half of the duodenum and tensile in the distal half. The large circumferential residual strains must be taken into account in a study of physiological problems in which the stresses and strains are important, e.g., the bolus transport function.

- biomechanics
- mucosal compression
- residual strain
- small intestine
- tangent rotation angle

the function of the duodenum is mechanical to a large degree. Contents received from the stomach are propelled further down the intestine and mixed with secreted fluids to digest and absorb the food constituents. Data in the literature pertaining to the mechanical aspects of duodenal function are concerned with the contraction patterns (4, 16, 22), the length-tension relationship in circular and longitudinal tissue strips (32), fluid mechanics (29), and the compliance and the stress-strain relationship of the walls (26). However, the constitutive equation, which is the mathematical description of the relationship between physical stresses and strains in three dimensions, is basically unknown. The three-dimensional distribution of stress and strain in the duodenum and the effect of stress and strain on the biology of the mucosa, submucosa, and longitudinal and circumferential muscles have not been studied in detail. It is the objective of this study to make a beginning. For this purpose, the overall scheme must be briefly described. Because the active contraction of the muscles is the most important feature of the peristaltic movement of the duodenum, the method of handling the muscles must be described first. We shall use the “three-element model” of Nobel Laureate A. V. Hill from 1939. The model considers the tissue to be composed of a “contractile element” connected with a “series elastic element” to describe the active contraction of the muscle and a “parallel element” to describe the connective tissue. This model has been applied extensively to the mechanics of the heart, lung, blood vessels, and kidney. No competing model had better success. With regard to the duodenum, we consider the mucosa, the submucosa, and the quiescent muscles as parallel elements, whereas the contractile and series elastic elements belong to the active muscles. Continuum mechanics will link all the elements together. This study deals with the parallel elements exclusively.

Strain describes the mechanical deformation of a material. In biomechanics, strains express dimensionless fractional changes in dimensions and are related to stresses. In a distensible biologic tube like the duodenum, the principal strain of interest is the circumferential strain. To compute the strains under physiological or pathophysiological conditions, one must know the configuration at which the stress is zero. In recent years, it has become evident that in blood vessels, airways, and the heart wall the zero-stress state is very different from the no-load state, in which all the external loads are removed. A simple way to approach the zero-stress state is to cut the organ under study transversely into a series of short ring-shaped segments and then to cut each ring once radially (5, 6, 27), causing it to open into a sector (Fig.1
*B*). The closed ring with zero transmural pressure is the no-load state (Fig.1
*A*), but it may have locked-in stresses called residual stresses in this state. These residual stresses can be characterized by an opening angle (Fig.1
*B*). The strain differences between the zero-stress state and the no-load state are called the residual strains. Current literature on the zero-stress state contains data on the opening angles of systemic and pulmonary arteries of normal, hypertensive, and diabetic rats (8, 9, 12, 17), normal systemic arteries of pigs and rabbits (12, 28), systemic veins of rats (30), the left ventricle of rats (21), and the trachea of pigs and dogs (13). This listing of references for these organs is not comprehensive, but we are not aware of any such study aimed specifically at the gastrointestinal (GI) tract. We found that the duodenum has a greater opening angle than any other organs known today (see below).

We measured the circumferential strains with respect to the zero-stress state in the guinea pig duodenum and correlated these measurements with morphometric data. New experimental data obtained here allow the computation of the circumferential strain directly, taking the residual strains into account without invoking the unknown constitutive equation. In blood vessels the zero-stress state can be obtained by one radial cut and can be characterized by an opening angle. In the duodenum we found that the same method did not work. Cutting of the duodenum required elaborate preparation, and the measurement of the opening angle, which in the duodenum is much larger than 360°, was difficult. Consequently, a tangent rotation angle is introduced, instead of the traditional opening angle, as a measure of the residual strain. The mucosal layer is shown to be under compression in the circumferential direction in the resting state and at low pressures. A large variation along the length of the duodenum is demonstrated for most of the biomechanical and morphometric parameters.

The effect of the zero-stress state on physiology may not be obvious until the full Hill’s model is analyzed. It is clear, however, that it is a fundamental feature of the parallel element. The function of the contractile element depends on the parallel element. For example, the length of the muscle cell depends on the strain of the parallel element; hence, whether the muscle can achieve the optimum length of the length-tension relationship for its contraction depends on the zero-stress state of the parallel element in principle.

## MATERIALS AND METHODS

Seven guinea pigs (*Cavia porcellus*, 800 g) of both sexes were used in this study. The experiments were carried out in accordance with the guidelines of the American Physiological Society and had been approved by the Animal Subject Committee of the University of California, San Diego. The animals were anesthetized with ketamine (25 mg/kg im) and xylazine (0.25 mg/kg im). Atropine, a muscarinic receptor blocker, was administered intramuscularly before surgery. After the attainment of surgical anesthesia, a short midline abdominal incision was performed, and a small incision was made in the proximal part of the jejunum. A soft polyvinylchloride (PVC) tube (4.8 mm OD, 3.2 mm ID) was inserted in the proximal direction and fixed by ligation without damaging the adjacent vessels. Care was taken not to ligate the duodenal blood vessels. A small incision was then made in the stomach, and a soft PVC catheter (2.4 mm OD, 1.3 mm ID) was inserted and gently guided through the pylorus into the duodenal bulb. A small incision was made in the mesenterium, and the pylorus was ligated without affecting the adjacent blood vessels. At this point the gallbladder was punctured to empty its contents, since pilot studies showed that bile would otherwise flow from the bile tract into the duodenum and interfere with the hardening of the silicone elastomer. The stomach tube was connected to a fluid container, and the duodenal lumen was perfused under low pressure (inlet pressure of ∼0.5 kPa) with calcium-free Krebs solution containing 6% dextran and 2 mM ethylene glycol-bis(β-aminoethyl ether)-*N*,*N*,*N*′,*N*′-tetraacetic acid (EGTA) to clear the lumen of its contents and to relax the smooth muscles. No further contractile activity was observed in the duodenum in situ and ex situ in the organ bath (see below). After ∼10 min the Krebs solution was replaced with 30 ml of a catalyzed silicone elastomer solution (Microfil CP-101, Flow Tek, Boulder, CO) at an inlet pressure of 0.7 kPa. The use of silicone elastomer makes it possible to reconstruct organ geometry (15). After perfusion for 5 min the outlet from the proximal jejunum was clamped. The silicone elastomer was catalyzed with 5% tin octate and 25% ethyl silicate (15) to harden in ∼20 min. After hardening of the cast, the whole duodenum, including its mesenterium, the cast, and the most adjacent part of the catheters, was quickly dissected free and immersed in an organ bath containing the Krebs solution with dextran and EGTA aerated with 95% O_{2}-5% CO_{2} at pH 7.4 and room temperature. Within a short time, the duodenal surface was cleaned and dissected free of the mesenterium, and the free ends of the tubing were cut. Figure 2 shows the isolated duodenum with the cast inside. At this time, the duodenum with the cast inside was cut into approximately seven smaller samples. These cuts were made transversely through the structure in specific locations, so that the individual samples were straight. This facilitated photographing the samples from two directions corresponding to the major and minor diameters of the duodenum with the cast (loaded-state serosal diameters), the cast itself (loaded-state mucosal diameters), and the duodenum without the cast inside (no-load state). The duodenum in its no-load state and the cast were approximately equal in length, showing that longitudinal strain in the loaded state was negligible. Unfortunately, the no-load state of the samples could not be evaluated further, since the sections tended to twist and did not conform to any simple geometric shape. At this stage, two to four short rings from each specimen were cut radially to obtain the zero-stress state. The axial locations of these rings were noted. The width (1–2 mm) was chosen on the basis of pilot experiments (shorter rings tend to curl, and wider rings heel in the longitudinal direction to impede movement in the circumferential direction).

#### Characterization of the zero-stress state.

The traditional method for characterizing the zero-stress state is based on measurement of the opening angle (Fig. 1*B* in Ref. 5) denoted by α and defined as the angle subtended by two radii drawn from the midpoint of the inner wall to the inner tips of two ends of the specimen in the zero-stress state. When this method is applied to the duodenal specimens, one has to follow the tips as they open. When a section becomes completely inside out, the opening angle is 360°. The duodenum often has α > 360° at zero-stress state. In fact, most rings turned the mucosa to the outside even before the first radial cut, indicating large compressive forces in the mucosa in the no-load state. This can be prevented by taking longer segments. For short rings, however, it was not possible to obtain valid data on the no-load state, and the strain data in this study are therefore limited to the zero-stress and the loaded state. When the opening angle is 360°, the cut ends come in contact again after an inside-out deformation. For α > 360°, it is simpler to make two to three cuts in each circumference to establish the zero-stress state. In this case, it is easier to use the tangent rotation angle, denoted by φ in Fig.1
*B*, to describe the zero-stress state. To define φ, one fixes a point on the outer rim of the cross section, such as *point 0* in Fig.1
*A*, and measures the length of the outer rim *s* from the*origin 0* and uses*s* as a curvilinear coordinate along the rim. Define a unit tangent vector**T** on the outer rim. As*s* increases, the tangent rotates. For a complete circle, the tangent rotates 360° when*s* returns to the origin. When a section opens, the total angle of rotation from one tip to the other is <360°. The angle of rotation can be measured in segments. If the specimen is cut radially in three places to produce three nonoverlapping pieces, as shown in Fig.1
*C*, and the tangents of the segments rotate by angles φ_{a}, φ_{b}, and φ_{c}, then the total angle of rotation of the whole specimen is φ = φ_{a} + φ_{b} + φ_{c}. This resultant value of φ is independent of where the cuts are made in the original ring and does not require the original or final shapes to be circular (seeappendix in Ref. 31). The tangent rotation angle and the mucosal and serosal edge lengths in the zero-stress state were measured from photographs of the cut-open specimens. The photographs were taken when steady state was reached 30 min after the radial cuts to allow the viscoelastic phase to subside.

The geometry of the duodenum at the zero-stress state can also be characterized by the angle between the last tangent of the tip and the first tangent of the tip, ψ (Fig.1
*B*). ψ may be called the angle between the tip tangents. Clearly, φ = 2π − ψ. We use all three measures, α, φ, and ψ, to characterize the zero-stress state. Figure 3 depicts the relationship between the zero-stress state and measurements of α, φ, and ψ.

#### Data analysis.

The length of the duodenum, as well as the length of the cranial, descending, transverse, and ascending sections of the duodenum according to the classification by Cooper and Schiller (3), varied considerably among animals. This necessitated normalization of data within appropriate subdivisions of the duodenum. In this study we found that it was fairly easy to identify the proximal and distal ends of the duodenum as well as two intermediate locations corresponding to major duodenal bends (Fig. 4). Therefore, the duodenum was subdivided into proximal, middle, and distal segments in this study (Fig. 4). The lengths of these segments are given in Table1. Because of the large variation in length among the three segments and among animals, the data were normalized in terms of a local dimensionless curvilinear coordinate, with the orad and aborad ends of each segment assigned 0% and 100%, respectively. Consequently, the 100% location in the proximal segment is identical to the 0% location in the middle segment, and so on. Because it was only possible to obtain rings at exactly the 0% and 100% locations, the intermediate data points were allocated to 25% intervals. This interval was selected because larger intervals give a poorer resolution and smaller intervals would contain too few data in a given segment.

The morphometric data were measured from the photographs of the segments in the zero-stress and no-load states. The negatives were illuminated and displayed on a separate monitor using frame-grabbing software. The measurements were done using Optimas software. The resolution was 0.17° and 0.01 mm for angle and length measurements, respectively. The loaded-state data were measured on the photographs of the duodenum with the cast (data on the outer wall of the serosa) and of the cast itself (data on the inner wall of the mucosa). Measurements were done at locations corresponding precisely to the locations where the rings were cut out. The cross sections were not exactly round. The photographs allowed measurements of the major (*D*
_{ma}) and minor (*D*
_{mi}) diameters of the cross section of the loaded state. If one could assume the cross section to be elliptical, then it is possible to compute the circumferences of the mucosa and serosa,*C*
_{l mucosa}and*C*
_{l serosa}, respectively. The exact formula is
Equation 1where*E*(*k*) is the complete elliptical integral of the second kind and*k* is a parameter defined by
Equation 2If the ratio*D*
_{mi}/*D*
_{ma}is very close to 1, then *Eq. 1
* may be approximated by
Equation 3For example, if*D*
_{ma}/*D*
_{mi}= 1.15, then the*C*
_{l} calculated by*Eq. 3
* is 0.12% of that given by*Eq. 1
*. The use of *Eq.3
* involves two assumptions:*1*) that the cross section is elliptical and *2*) that*D*
_{mi}/*D*
_{ma}is quite close to 1. To test these assumptions, we sliced the casts. From photographs of these slices, we measured*D*
_{ma},*D*
_{mi}, and*C*
_{l}. Figure5 shows the circumference of the cast slices measured directly from photographs and the circumference computed from measurements of the major and minor radii from the same photographs using *Eq. 3
*. The regression line of Fig. 5 can be expressed in the form
Equation 4where*A* and*B* are empirical constants, 1.17 and 0.297 mm, respectively. The correlation coefficient was 0.98. We may regard *Eq. 4
* as an improvement of*Eqs. 1
* and *
3
* for a duodenum with a cross section that is not exactly elliptical and*D*
_{ma}/*D*
_{mi}not equal to 1. The serosal circumferences also needed correction and were calculated on the basis of *Eq. 4
*and the wall thickness. First, the mean wall thickness (*h*) in the loaded state was calculated as
Equation 5where OD and ID denote the outer (serosal) and inner (mucosal) diameters, respectively. The serosal circumference based on the correction factor was computed as
Equation 6To assess the physiological significance of the tangent rotation angle and the strains, the circumferences at zero-stress and homeostatic conditions must be measured on the same duodenal specimens to obtain the desired accuracy. The tangent rotation angle (or the opening angle when applicable) is a measure of the residual strain (7, 8). The strain at the loaded state is called homeostatic strain. For a continuum subjected to finite deformation, strains can be defined in several different ways in relation to the deformation gradient (7). The Cauchy strain ε, relative to the zero-stress state, is
Equation 7where subscripts l and z refer to the loaded and zero-stress states, respectively. The Cauchy strain is especially useful in linearized theory of elasticity, which is valid when ε is infinitesimal. For finite deformation, the strain defined by Green is more conveniently related to stress (7). Green’s strain in the circumferential direction of the duodenum is
Equation 8Thus, from the circumferential lengths at the zero-stress state and the loaded state, we can obtain the circumferential strain at the mucosal and serosal surfaces in the sense of Cauchy or in the sense of Green.

The data are representative of a normal distribution, and accordingly the results are expressed as means ± SE. Student’s*t*-test and analysis of variance were used to detect differences along the length of the duodenum and between groups of data. Spearman’s correlation test was used to investigate association between variables. The results were regarded as significant if *P* < 0.05.

## RESULTS

Figure 6 depicts a cross-sectional view of a specimen from the most proximal part of the duodenum cut into two pieces to obtain the zero-stress state. On reduction of the no-load state to the zero-stress state by cutting the ring, the two pieces expanded themselves into a configuration with a tangent rotation angle of about −640°. The pieces literally turned inside out. The outside of each specimen represents the mucosa and the inside the serosa.

The ratio*D*
_{ma}/*D*
_{mi}of the duodenum in the loaded state was ∼1–1.15 without significant axial variations. The distribution of the mucosal and serosal circumferences in the zero-stress state and the loaded state and the wall thickness as a function of the axial location in the proximal, middle, and distal segments of the duodenum are shown in Fig.7. The mean and standard error of the variables are functions of the normalized distance within each segment. A point labeled 26–50% means that this point is located at 26–50% of the length of that particular segment of the duodenum from the proximal end.

Because the duodenal specimens turned inside out, the mucosal circumferences were larger than the serosal circumferences in the zero-stress state. The serosal circumference was largest in the proximal duodenum and decreased in the distal direction (*F* = 4.1,*P* < 0.001), whereas the mucosal circumference showed only borderline significance in the axial direction (*F* = 1.6; 0.10 >*P* > 0.05). In the loaded state the serosal circumference was ∼20–25 mm in the duodenum with little axial variation (*F* = 1.6,*P* > 0.10). The mucosal circumference was ∼12–18 mm in the loaded state with no axial variation (*F* = 1.2,*P* > 0.2). However, examination of juxtaposed locations revealed some significant differences; e.g., the data point at the duodenal bend between the middle and the distal segments of the duodenum was significantly higher than the data point immediately distal to it (*P* < 0.05). The wall thickness in the proximal duodenum was approximately twice that in more distal parts (*F* = 3.8,*P* < 0.001).

The distribution of the tangent rotation angle in the zero-stress state, the mucosal and serosal strains, and the wall thickness-to-mucosal circumference ratio as functions of the axial location in the proximal, middle, and distal segments of the duodenum are shown in Fig. 8. The absolute value of the tangent rotation angle was lower in the proximal duodenum than in more distal parts (*F* = 3.8,*P* < 0.001). The lowest absolute value of 537 ± 28° was found in the first 25% of the proximal segment. The highest absolute values of ∼850° were found close to the major bends and most distal in the duodenum. In the loaded state the serosal Cauchy strain was tensile with values of ∼1 in the proximal part and a significant increase toward the distal direction (*F* = 3.0,*P* < 0.005). The mucosal Cauchy strain was compressive in the proximal half of the duodenum, with values of approximately −0.1 to −0.4, whereas it was between 0.05 and 0.4 in the distal half. Figure 8 depicts both Cauchy and Green strains. Statistically significant axial variation was found (*F* = 2.6,*P* < 0.01). The wall thickness-to-mucosal circumference ratio also varied in the axial direction (*F* = 4.0,*P* < 0.001), with the highest values in the proximal segment of the duodenum.

The tangent rotation angle did not correlate directly with the Cauchy strains at 7 cmH_{2}O or directly with any other morphometric parameters. These variables are tied together by mechanics, which can be validated after the stress-strain relationship is determined. The mucosal and serosal strains were negatively related to the wall thickness (*r* = −0.63,*P* < 0.01 and*r* = −0.31,*P* < 0.01, respectively). Because the wall thickness-to-mucosal circumference ratio is a determinant of the wall stress, the correlation between this ratio and mucosal strain was studied, and a significant correlation was found (*r* = −0.77,*P* < 0.01).

## DISCUSSION

The major function of the duodenum is to transport gastric contents further down the intestine and to mix the contents with secreted fluids to break down and absorb the food constituents. The evaluation of duodenal mechanical function traditionally has focused on the peristaltic contraction patterns and the bolus transport that results from the sequence of duodenal muscle contraction and relaxation and gastric emptying (4, 18, 22). It is well known that a number of factors such as the volume and viscosity of the bolus and the pressures generated by the peristaltic contractions determine the transport function of the duodenum. Classic biomechanical theory on the transport of fluid boli in distensible tubes indicates the importance of considering the geometry and elasticity of the organ under study. Furthermore, the wall of the GI tract is stretched passively in the vicinity of a bolus (25), indicating that the luminal dimension, the function of the contractile element, and the elastic properties of the duodenum in the low stress range determine the resistance to flow. To compute the stress and strain in tissue and muscle under physiological or pathophysiological conditions, one must know the configuration when the stress is zero. It is well recognized that in the blood vessels, the heart, and the airways the zero-stress state is different from the no-load state, where all the external loads are removed but the organ is intact (8, 9, 12, 13, 17, 21, 28, 30). To the best of our knowledge, previous studies of duodenal distensibility have not considered the zero-stress state. This study was confined to the analysis of strain, since stress analysis requires the stress-strain relationship (constitutive equation), which has not been determined for the duodenum. The strain analysis was based on experimental measurements of the duodenal geometry in the zero-stress and loaded states and was simplified by assuming elliptical geometry with an empirical correction. The whole length of the duodenum is not necessarily in the same horizontal plane, and therefore the applied hydrostatic pressure will vary slightly along the length. The difference in hydrostatic pressure will be at most 0.5–1 cmH_{2}O, i.e., <15% of the applied pressure. Furthermore, because the surrounding tissues have an approximate density of that of the silicone elastomer, the transmural pressure likely did not vary significantly along the duodenal length.

A detailed biomechanical analysis of GI physiology requires a complete data base on GI geometry. The present study constitutes the first attempt to reconstruct the geometry of the entire duodenum at zero-stress level and a specified pressurized condition. Some basic morphometric data are presented. Previous studies have mainly been descriptive. Gabella (10) described characteristics of the whole guinea pig GI tract. This provided important data on differences in histomorphology along the small intestine, but unfortunately only one data point was from the duodenum. In this study we obtained a good resolution for the entire duodenum and avoided the tissue shrinkage from the use of fixative by making optical measurements with in vivo casting. Gabella (10) found a decrease in wall thickness from the duodenum to the terminal ileum. Our results on the decrease in duodenal wall thickness toward the distal direction are thus in line with those of Gabella (10).

A principal result of the present study is the clear demonstration of the presence of large residual strains in the duodenum and a gradient of this parameter along the length of the duodenum. The large residual strain, due to the difference between the “no-load” state and the “zero-stress” state, is an inherent property of the duodenum. It can be influenced by the duodenal smooth muscles. In the present determination, we have made every possible effort to abolish muscle contractile activity by using a calcium-free medium containing EGTA to chelate intracellular calcium stores under otherwise physiological experimental conditions. The absence of the smooth muscle activity was consistent with the observation that whereas duodenal muscle contractions are mainly phasic, we did not see any contractile activity during the studies. The residual strain in the mucosa and submucosa was found to be compressive in the circumferential direction. The mucosa, of course, is not a smooth solid layer, since the villi are physically connected only via the base to the submucosal layer. The villi are, to a small degree, free to move away from each other, allowing fluids to move between the villi: more movement if the circumferential strain is tensile, less if the strain is compressive. Mucosal cell edema was avoided by using a Krebs solution containing high-molecular-weight dextran. We have observed that the duodenum springs open when cut longitudinally in vivo; thus it is very unlikely that our observations were caused by fluid movements. To further elucidate whether volume was conserved, we analyzed a few of the best no-load-state photographs and compared the whole wall cross section from the no-load state with that from the zero-stress state. This analysis revealed no difference in cross section between the no-load and the zero-stress state, indicating that volume was indeed conserved in these experiments. Thus we believe that the mucosa in the no-load state was under compression.

Because a large residual strain was observed in the duodenum, we may speculate about the function of this prestress. Prestressing the duodenum may be nature’s way of efficiently resisting luminal pressures in a manner similar to the prestressing of gun barrels and other mechanical devices. For arteries, it has been demonstrated that residual stress reduces the stress concentration at the inner wall of the artery at normal blood pressure (2). The present data also demonstrate that the prestressing effect causes compressive mucosal strains in the proximal duodenum at a physiological pressure of 0.7 kPa and in the mucosa of the whole duodenum at resting conditions. Thus the compressed mucosa may be better protected against injury from the flow of luminal contents than the uncompressed mucosa would otherwise be. These protection mechanisms could be important when unphysiologically high pressures are reached, e.g., in mechanical obstruction. The residual strain affects the whole stress and strain distribution, since large errors in strain calculation will result from ignoring the residual strain. Because peristaltic bolus transport bulges out the intestinal wall in the vicinity of the bolus (1, 25), the residual strain would likely influence the resistance to bolus flow under normal conditions.

It is worthwhile to notice that the duodenum differs from the blood vessels, trachea, bronchi, and esophagus in the sense that the residual strain is much larger and the wall is thinner. This study opens many new interesting questions, some of them hypothetical because of the lack of precise data. Because residual strains exist, the duodenal mucosa will inevitably be under compressive stresses and strain in the no-load condition, i.e., where the transmural pressure is zero. In phases I and II of the interdigestive state, the small intestine is at the no-load condition most of the time, since phasic contractions are only infrequently present (16). We may ask therefore if there could be functions of mucosal compression other than those mentioned above. It is well known that the mucosa in the small intestine is one of the tissues with the fastest turnover rate (14). The fast growth on the mucosal surface could easily cause mucosal compression and by itself explain the large residual strains found in this study. We know from studies on the intestine and other organs that mechanical stresses are factors regulating gene expression and growth (9; unpublished observations). A well-known example is the cardiac hypertrophy caused by hypertension. In bone, Wolff’s law indicates that stress can induce bone hypertrophy. In other words, the magnitude of compression may affect the rate of growth and vice versa. Absorption of luminal contents may theoretically also be affected by the compression. It is well known that a gradient in the height of the mucosal villi exists along the small intestine, with the highest villi found in the proximal duodenum (10), and that small intestinal absorption depends on the luminal pressure (19). Thus there may exist a correlation between the residual strain gradient and the gradient in the height of the villi. These interesting aspects of small intestinal function certainly need attention in future studies. Other important questions pertain to the function of the mechanoreceptors (nerve endings of the sensory afferent nerves) in the intestinal wall. Previous studies have shown that the receptors are located in the submucosa and in the muscle layers and that the receptors can be classified on the basis of their thresholds to pressure (24). From this study, we learn that a gradient in strain exists from the mucosa to the serosa of the wall under physiological conditions; hence, the mechanoreceptors from the different layers are exposed to different strains. This suggests that the receptors from different locations in the wall may have different zero settings and respond differently to the same stimuli because of the variation in the magnitudes of tensile stresses and strain during distension and in the magnitudes of compressive stresses and strain during muscle contraction. A difference in the responses of mucosal and muscle receptors to stimuli has been observed (11). Furthermore, the present results emphasize the importance of studying the receptor kinematics by using experimental models on the intact organ close to the in vivo conditions. Yuan et al. (33) studied the effect of mucosal compression on the peristaltic reflex. Compression was found to stimulate the reflex, but unfortunately the experiments were performed on excised flat sheets of tissue.

The most important future work is to complete Hill’s three-element model so that we can understand the interaction between the parallel element and the contractile element. Further work is also needed to determine how the residual strains affect responses to physical and chemical stimulations in pathophysiological states and to determine whether species variations exist. Because the duodenum is essentially a muscular tube, alteration of the activity of the smooth muscle cells by agents acting on the muscles directly or on their innervation will change the zero-stress state of this organ. Hence, the effects of these agents on the duodenum and other parts of the small intestine need to be investigated to accurately model the normal bolus transport and pathological situations of blockage or disease.

#### Mathematical discussion.

The opening angle is defined as the angle between two radius vectors connecting the midpoint of the inner wall to the inner tips of the ends. Designated α, its measurement focuses attention only on these two vectors and disregards the shape of the curve. It gives no information on the curvature of the wall.

The tangent vector **T** is a unit vector field associated with any space curve. If*s* is a curvilinear coordinate of points on the curve, with significance of length measured along the curve from an origin on the curve, then the angle between**T**(*s*) and**T**(*o*) is the angle of the tangent between *s*and *o*. The derivative of rotation d**T**(*s*)/d*s*is the curvature of the curve, which is a quantity of major importance in the mechanics of the curve representing a shell.

The angle between the tangents at the tips is the angle between**T**(*c*) and **T**(*o*), where *c* is the circumference. It is the integral of the curvature from *s* =*o* to*s* =*c*. Writing φ for the angle of rotation of the tangent from *s* =*o* to*s* = *c*and ψ for the angle between the tip tangents, we have
or
The change in α, φ, and ψ with the opening of a cut specimen is shown in Fig. 3.

#### Concluding remarks.

Measurements of the tangent rotation angles in the guinea pig duodenum at the zero-stress state show substantial residual strains and stresses in the duodenum, an organ that experiences low luminal pressures in resting conditions. Several radial cuts are needed to allow measurement of the residual strain in a segment. The tangent rotation angle is very large and varies along the length of the duodenum. The residual strain affects the strain distribution in the duodenum under resting conditions and influences the stress-strain distribution in the duodenum during bolus transport. It could potentially affect mucosal growth and absorption. Data on the zero-stress state in the duodenum have physiological and clinical relevance, since the biomechanical and morphological properties affect flow, and they change in various diseases of the GI tract, e.g., in mechanical obstruction of the small intestine, in diabetes, and after partial resection of the small intestine (20, 23).

The work to be done next is quite clear: the constitutive equations, i.e., the three-dimensional stress-strain relationships of the duodenal tissues, must be determined. With the constitutive equations, the distribution of physical stresses and strains in the duodenum in vivo and the function of bolus transport and fluid movement in the intestine can be analyzed by methods of continuum mechanics. The analytic results will relate the stress and function of the duodenum with the duodenal structure represented by the geometric parameters, such as the circumferential length, wall thickness, and tangent rotation angle, presented here. Actually, the job of determining the constitutive equations is not as daunting as it seems, because the mathematical form of most soft tissues is known (6, 7), including the vascular smooth muscles of the arterioles (6a). If the hypothesis that the constitutive equations of the duodenal tissues are similar to those of other soft tissues can be confirmed, then it remains only to identify the mathematical constants of the duodenal tissues. Instruments are available, and this task is being pursued in our laboratory.

## Acknowledgments

H. Gregersen was supported by the Danish Medical Research Association, the Karen Elise Jensens Foundation, and the University of Aarhus (Aarhus, Denmark).

## Footnotes

Address for reprint requests: H. Gregersen, Institute of Experimental Clinical Research, Skejby University Hospital, Brendstrupgaardsvej, DK-8200 Aarhus N, Denmark.

↵† Deceased 17 August 1997.

H. Gregersen was on leave from the University of Aarhus.

- Copyright © 1997 the American Physiological Society