During development, the heart undergoes dramatic changes in form. To help visualize these changes, geometric models can be constructed, for example, from serial sections of fixed tissue or sections generated by confocal microscopy. These methods have serious limitations, however, as fixed sections cannot be used to study morphogenesis at multiple time points in a single embryo, while confocal microscopes cannot penetrate the entire thickness of the embryonic heart. Another useful technique is micro-computed tomography (Butcher et al., 2007), which provides resolutions of approximately 10 μm, but the heart must be perfused with casting polymers. Optical coherence tomography (OCT) is a relatively new imaging technique that overcomes these limitations (Huang et al., 1991; Boppart et al., 1997; Yelbuz et al., 2002; Jenkins et al., 2006). By measuring backscattered light as a function of depth, OCT provides subsurface imaging (up to 2 mm deep) of living biological samples with high spatial resolution (10 μm or less) in three dimensions and high sensitivity (>110 dB; Fujimoto, 2003). Hence, this method is ideal for dynamic imaging of the embryonic heart.
Cardiac looping is a crucial process during early heart development. During looping, the initially straight heart tube transforms into a curved tube normally directed rightward, thereby creating one of the first visible signs of left–right asymmetry in the embryo. Because abnormal looping is a major source of congenital heart defects (Hoffman and Kaplan, 2002), this problem has received considerable attention for many decades. Nevertheless, the biophysical mechanisms of looping have remained poorly understood.
Here, we show how OCT can be used to define the deformation of the embryonic chick heart during the first phase of cardiac looping (c-looping), as the heart tube bends and twists into a c-shaped tube (Männer, 2000). In particular, we examine the morphomechanical effects of the splanchnopleure (SPL), which is a membrane that compresses the ventral surface of the heart tube. Recent work has shown that the splanchnopleure supplies much of the mechanical force that drives the torsional component of looping (Voronov et al., 2004; Nerurkar et al., 2006). Removing the SPL before looping alters subsequent looping morphogenesis, as little torsion is observed for several hours. Later, however, an asymmetric cytoskeletal contraction apparently develops in the myocardium to restore normal looping (Nerurkar et al., 2006). The present OCT-based analysis provides new insight into this adaptive response.
Here, OCT was used to obtain two types of data. First, cross-sectional images were derived from three-dimensional (3D) OCT data sets of the looping heart. These images show that SPL forces are large enough to significantly affect the spatial relationship between the heart tube and the primitive atria as well as to alter the distribution of cardiac jelly (CJ). (Primitive atria also are referred to as omphalomesenteric veins in the literature; Patten, 1951; Romanoff, 1960). Second, morphogenetic strain maps were computed by using time-lapse OCT to track the 3D positions of markers placed on the myocardium. The measured strains are consistent with the hypothesis that abnormal contraction on the right side of the heart tube causes the delayed torsion when the SPL is removed.
MATERIALS AND METHODS
White Leghorn chicken eggs were incubated in a forced draft incubator for approximately 35 hr to stage 10 of embryonic development (Hamburger and Hamilton, 1951). A paper ring (Whatman #2 filter paper) was used to remove the embryo and the surrounding vitelline membrane from the egg. The technique for culturing embryos has been described previously (Voronov and Taber, 2002). Briefly, the embryo on the paper ring was placed ventral side up, sandwiched with another paper ring, and submerged in culture media by placing a stainless steel ring on top of the paper surrounding the embryo. The culture medium was composed of 89% Dulbecco's Modified Eagle's Medium (Sigma), 10% chick serum (Sigma), and 1% penicillin/streptomycin/neomycin from 5,000 U/ml stock (Invitrogen).
For time-lapse studies, the embryo was cultured in a 0.17-mm-diameter Delta T Dish (Bioptechs, Butler, PA) containing 1.2 ml of culture medium and 50 μM verapamil (Sigma). Verapamil, a calcium chelator, was used to stop the heartbeat so that the strain measurements would include only deformation caused by morphogenesis. Consistent with previous studies, we found that c-looping proceeded normally in the absence of a heartbeat (Manasek and Monroe, 1972; Porter et al., 2003; Rémond et al., 2006). The dish was covered with a glass lid that was maintained at 37°C using a Delta T4 Culture Dish Controller (Bioptechs). A heated and humidified 95% oxygen, 5% carbon dioxide gas was supplied to the embryo with a Mini-Pump Variable Flow device (Fisher Scientific). Time-lapse images of the developing embryo were captured with a Retiga 1300 camera mounted on a fluorescent microscope (Leica DMLB). Images were recorded at 10-min intervals over durations of 24 and 36 hr to verify proper embryo development (data not shown).
In general, embryos were staged according to the system of Hamburger and Hamilton (1951), which is based on somite counts and morphological characteristics of the developing embryo. However, observing somites with OCT would have required moving the x–y stage, which sometimes knocks the microspheres (used for strain measurements) loose from the heart. Additionally, when the sample was focused under OCT, the focus was centered on the heart tube, and structures at different z-heights (such as the somites) often could not be clearly resolved. For OCT studies, therefore, the shape of the heart alone was used for staging. Normal hearts with SPL intact were staged by correlation with the Hamburger–Hamilton based images of Männer (2000). For SPL-lacking hearts, we conducted a separate study to correlate the abnormal heart morphology with stage as defined by characteristic features of the embryo (Hamburger and Hamilton, 1951). Labeling the ventral midline of the heart with DiI (1,1′, di-octa-decyl-3,3,3′,3′,-tetramethylindo-carbocyanine perchlorate - Molecular Probes, D-282) indicated the amount of rotation, which helped us stage these hearts (Ramasubramanian et al., 2006).
Optical Coherence Tomography
The 3D OCT imaging was performed with a laser output of 15 mW at a wavelength of 1,310 nm using a LDC-37448 Laser Diode Controller (ILX Lightwave). This system was custom built at Case Western Reserve University in the laboratory of Dr. Andrew Rollins as described previously (Jenkins et al., 2006). Images were obtained by measuring the delay time for the light traveling to the sample versus a known reference path (Fig. 1A). OCT images were acquired by means of cross-sectional (x-z) scans along the y axis, with 400 images acquired over a distance of 2 mm (5-μm intervals between scans). Image analysis software (Volocity, Improvision) was then used to reconstruct the image files into 3D volumes. Alternatively, Caret (Computerized Anatomical Reconstruction and Editing Tool Kit developed in the laboratory of Dr. David Van Essen) was used to produce some of the volumes (Fig. 1C) to show the anatomical detail of the 3D heart more clearly in all orientations (Van Essen et al., 2001). For time-lapse studies, stacks of OCT images were collected every 60 min on average.
Morphological observations using OCT were performed for three embryos with the SPL intact and seven embryos with the SPL removed. In SPL removal experiments, images were first acquired at stage 10 with the membrane intact. More images were obtained from the same embryo after the SPL was removed at stage 10 and after the embryo developed to stage 12. These embryos were compared with other normal stage-12 embryos (with intact SPL).
Surface strains were measured in seven SPL-lacking hearts. The SPL was dissected away from the heart tube at stage 10, and a syringe was used to place 8.6-μm-diameter polystyrene microspheres (Bangs Laboratories) on the myocardium as described previously (Alford and Taber, 2003). (Fluorescent labels cannot be detected using traditional OCT.) Microspheres were diluted 10:1 in phosphate buffered saline (Gibco) from a stock solution of 10% solids. The microspheres remained adhered to the myocardium during development, and little sliding was observed of beads not being firmly attached. Volocity software was used to determine the x, y, z coordinates of bead centroids at each time point.
Myocardial Lagrangian strain distributions were calculated by fitting 2D surfaces to the 3D coordinates of the beads in the initial (stage 10) state of the heart and to subsequent deformed states (stages 11, 11+, 12−, and 12). Three strain components (longitudinal, E11; circumferential, E22; and shear, E12) were calculated with the longitudinal direction defined as being parallel to the embryonic heart tube in the reference configuration at stage 10 (see dashed line in Fig. 7), and the circumferential direction was perpendicular to the longitudinal direction. A MATLAB program was used to compute strain fields relative to stage 10 as described in the Appendix.
All data are reported as mean ± SD. Statistical analyses were done using SigmaStat (SPSS Science). Inter-region strain data at stage 12 was compared usingone-way analysis of variance with post hoc pairwise comparison made using the Bonferroni / Dunn Test (P < 0.05 for statistical significance).
Development of the embryonic chick heart was investigated during stages 10–12 (35–48 hr incubation). OCT cross-sections of the heart tube were reconstructed into volumes, allowing for the clear resolution of embryonic structure in three dimensions (Fig. 1).
The effects of the forces exerted by the SPL were readily apparent. For example, in embryos with an intact SPL, the heart tube and the primitive atria remained relatively planar through stage 11 (Fig. 2). When the SPL was removed at stage 10, however, the heart tube “popped up” out of the ventral side of the embryonic plane, while the distal ends of the outflow tract and atria maintained their attachments within the plane (Fig. 3). As the embryo approached stage 11, the atria rotated dorsally until they made a relatively steep angle with the embryonic plane (Fig. 3C′,C″). In the ventral view, this rotation made the atria appear more horizontal (Fig. 3A–C). Hence, the flattened appearance of the normal heart and atria apparently is due in large part to the SPL, which is under considerable tension (Voronov and Taber, 2002), pressing against the ventral surfaces of these structures.
To gain a better understanding of the load imposed by the SPL on the developing heart, cross-sections of stage 12 embryos with the SPL intact (Fig. 4) and with the SPL removed (which had been previously removed at stage 10; Fig. 5) were investigated. In normal embryos, as shown previously (Voronov and Taber, 2002), the SPL compresses the heart tube only slightly (Fig. 4A), with the layer of CJ being somewhat thinner immediately under the SPL. However, the compressive force on the primitive left atrium near the heart tube is large enough to squeeze CJ away from the region of contact, and the presumptive myocardium takes an elliptical shape (dashed line in Fig. 4B). More distal to the heart tube, the left atrium appears more circular (Fig. 4C). The right atrium also appears to be compressed by the SPL (Fig. 4A′,B′).
In SPL-lacking embryos, the CJ is more evenly distributed in the heart tube and proximal left atrium, which are more circular in shape than in embryos with intact SPL (Fig. 5A,B). The cross-sections of the primitive right atrium also are more circular in appearance (Fig. 5A′,B′).
Myocardial strain distributions were measured for seven hearts developing from stage 10 to stage 12 with the SPL removed. Three surface strain components (E11, longitudinal; E22, circumferential; E12, shear) were calculated relative to the configuration at stage 10. As a representative example, Figure 6 shows longitudinal strain distributions, as well as the locations of the microspheres used in the analysis, at stages 11, 11+, 12−, and 12 for one embryo. Spatially, the strain generally increased toward the caudal end of the heart, as well as the original ventral surface, which becomes the outer curvature of the looped tube.
To characterize temporal–spatial variations in strain, the strain components for the seven hearts were averaged in each of seven regions (Fig. 7). Average strains in each region are plotted as functions of stage relative to the reference stage-10 configuration. Longitudinal and circumferential strains were uniformly positive, indicating an overall expansion of the myocardium. During development, longitudinal strains increased steadily in all regions from stage 11 to 12− and appeared to level out in some regions approaching stage 12. Circumferential strains approached maximum values at stages 11+ and 12− and then dropped in all regions by stage 12. Shear strains remained relatively small everywhere except in region 7, where the largest circumferential and longitudinal strains also occurred.
Statistical analysis revealed that the average longitudinal strain in region 7 was significantly larger than the longitudinal strains in regions 1–5 at stage 12. All other differences between regions (including circumferential strains) were not statistically significant at this stage.
Beads adhered poorly to the primitive right atrium and the caudal half of the primitive left atrium, where the SPL remains attached. Thus, these regions were excluded from the present study. In addition, some beads became dislodged during development and were excluded. This latter problem became acute when we tried to measure strains in embryos with the SPL intact by placing beads on the heart through small holes in the SPL. In this case, beads moved as the rotating heart rubbed against the SPL. Future work should consider injecting nonfluorescent labels directly into the myocardium.
To correlate regional variations in strains with global looping patterns, alternative heart regions were defined (Fig. 8A). By stage 12, the middle region (MHT) bends and rotates toward the outer curvature of the bent heart tube, while the left and right sides (LHT and RHT) become located near the inner curvature. In this analysis only longitudinal and circumferential strains were considered. At stage 12, the longitudinal strains in MHT and RHT were similar but significantly greater than those in LHT (Fig. 8B). Circumferential strains in all three regions remained similar through stage 11+, but from that point forward the average circumferential strains on the right side of the heart tube were smaller than those of the left side of the heart tube (Fig. 8C). These differences were not statistically significant due to the variance in the data, but a trend of decreased circumferential strain on the right side compared with the left side was observed after stage 11+.
To understand the role of mechanical forces in shaping the embryo, accurate descriptions of morphological changes are of paramount importance. As highlighted in a recent study (Männer, 2004), the looping heart poses an especially challenging problem in this regard, as the complex changes in shape that occur during looping are difficult to visualize in three dimensions. This finding has caused some misunderstandings of looping morphology (Männer, 2000), and consequently, experimental results sometimes have been misinterpreted (Taber, 2006). Partly for these reasons, the biophysical mechanisms of looping have remained poorly understood, despite many decades of study. Here, we have shown how OCT can help overcome these problems to provide new information on looping morphogenesis.
C-looping consists of combined ventral bending and dextral torsion of the heart tube (Männer, 2000). While studies have shown that cardiac bending is driven primarily by intrinsic forces in the heart (Butler, 1952; Latacha et al., 2005), torsion is largely a result of loads imposed by the SPL, which is under considerable tension (Voronov and Taber, 2002; Voronov et al., 2004). Recent work, however, has shown that there may be interaction between the internal and external forces. In particular, Nerurkar et al. (2006) found that removing the SPL from the region over the embryonic chick heart disrupts torsion for approximately 6 hr (from stage 10 to stage 11). However, full rotation of the heart is restored after an additional 6 hr in culture (stage 11 to stage 12).
Experiments by Nerurkar et al. (2006) suggested that this late-onset torsion is caused by abnormal cytoskeletal contraction that develops on the right side of the heart tube. This adaptive response likely is triggered by removing the compressive loads normally applied by the SPL. In the present study, we used this experiment to illustrate the utility of using OCT to explore a specific problem of cardiac morphogenesis, as well as to gain new insight into this unexpected behavior.
The ability of OCT to resolve embryonic structures at micron-scale resolution has been well documented (Boppart et al., 1996, 1997; Yelbuz et al., 2002; Luo et al., 2006), and some investigators have applied this technology to the embryonic chick heart (Yelbuz et al., 2002; Jenkins et al., 2006). While Yelbuz and colleagues showed that histological cross-sections bear a strong correlation with OCT images, Jenkins and colleagues used a gated OCT imaging technique to measure heart parameters such as cardiac volume, ejection fraction, and wall thickness. This gating technique allowed structures to be digitally sectioned and visualized in three dimensions at any time during the cardiac cycle. While these investigations have shown that OCT can be used to study morphogenesis and function of the developing heart, to our knowledge OCT has not previously been used to study the mechanics of morphogenesis. Here, we have used time-lapse OCT imaging to examine 3D, dynamic, morphomechanical changes of the embryonic heart.
Effects of SPL on Heart Morphology
Recent studies have shown that the SPL exerts considerable compressive force on the heart tube during c-looping (Voronov and Taber, 2002; Voronov et al., 2004). This force is key to proper torsion, as it pushes the bent tube into its proper orientation, appearing as a “c” when viewed from the ventral side of the embryo (Voronov et al., 2004). This effect is confirmed by the present study (see Figs. 1E, 2). However, our OCT images also show that the forces supplied by the SPL influence the morphology of the looping heart to a greater extent than previously realized. Without the SPL, the heart tube protrudes prominently outward from the plane of the embryo, altering its spatial relationships with the primitive atria and outflow tract, which remain attached at the distal ends to the embryo (Fig. 3). As discussed next, this additional deformation may have important implications for how the heart adapts to this perturbation.
To repeat, Nerurkar et al. (2006) speculated that late-onset torsion after SPL removal is driven by cytoskeletal contraction on the right side of the heart tube. However, Voronov et al. (2004) have shown that the primitive atria (omphalomesenteric veins) also affect torsion by pushing against the caudal end of the heart. Computational modeling has shown that these forces, combined with geometric asymmetry of the atria, generate a slight rightward torsion, which is then magnified in normal embryos by the pressure applied by the SPL (Ramasubramanian et al., 2006). Recent modeling studies in our lab have shown that the out-of-plane orientation of the atria following SPL removal can enhance the torsional effects of the atria (results not shown), and we speculate that this effect plays a role in the delayed torsion.
In addition to this influence on global looping behavior, our OCT images show that the force applied by the SPL strongly affects regional morphology. As shown previously (Voronov and Taber, 2002), the SPL compresses the cross section of the heart tube (Fig. 4A), but perhaps more significant are the effects on the primitive atria. By the end of c-looping at stage 12, for example, the compressive load on the left atrium close to the heart is high enough to squeeze the CJ almost completely out of the region next to the SPL (Fig. 4B). Of interest, when the SPL is removed at stage 10 and the heart is allowed to develop to stage 12, both the heart tube and left atrium have more homogeneous distributions of CJ and more circular cross-sections (Fig. 5A,B).
These observations show that CJ can flow from one region to another under the action of mechanical forces. However, isolated CJ maintains its shape (Nakamura and Manasek, 1978), and it exhibits pseudoelastic behavior when indented on a time scale of minutes (Zamir et al., 2003; Zamir and Taber, 2004). Taken together, these results suggest that CJ behaves as a viscoelastic fluid with a relatively long time constant. (Under loading, a viscoelastic fluid behaves as an elastic solid for short times and a fluid for long times; Flügge, 1975). Hence, CJ likely offers little resistance to the relatively slow process of looping, as it flows to follow the deformation of the myocardium.
Our results also suggest that the effects of the SPL on CJ localization are greater in the primitive atria than in the heart tube. This finding could be due to softer CJ in the atria, stiffer myocardium in the heart tube, greater SPL tension in the atrial regions, or a combination of these factors. In addition, the compression of the atria may affect blood flow, which is just beginning at stage 12.
Myocardial Strain Distributions
Strains quantify local changes in tissue size and shape, which can be caused by both mechanical stress and volumetric growth. In the embryonic heart, strains are associated with the heartbeat and morphogenesis. Because the heart loops normally in the absence of sarcomeric activity (Manasek and Monroe, 1972; Porter et al., 2003; Rémond et al., 2006), we were able to isolate morphogenetic strains by stopping the heartbeat. Myocardial strains were measured in the myocardium by fitting 2D surfaces to the coordinates of multiple surface markers, similar to the technique used by Hashima et al. (1993) to measure epicardial strains in beating dog hearts. We computed longitudinal, circumferential, and shear strains relative to the local coordinate directions defined in the reference configuration at stage 10 (Fig. 7).1
Most previous measurements of myocardial strain in the embryo have focused on the deformation that occurs during a heartbeat (Taber et al., 1994; Tobita and Keller, 2000a, b; Alford and Taber, 2003). To our knowledge only two studies have investigated morphogenetic strains in the heart. Lacktis and Manasek (1978) tracked the motions of carbon particles on the surface of the chick heart during c-looping (stages 10–12). Using a strictly 2D analysis, they computed regional changes in surface area and some deformation patterns for a single heart. More recently, Ramasubramanian et al., (2006) used fluorescent labels and image stacks to obtain 3D coordinates of injected markers. However, their technique suffers from decreased resolution in the z-direction and only partially describes the mechanics of specific regions. Extending these prior analyses, we used OCT imaging of developing hearts to obtain accurate 3D marker coordinates. From these coordinates, we mapped strain distributions over the surface of the heart.
Some investigators have used digital image correlation techniques to map morphogenetic strains during other developmental processes (Brodland et al., 1996; Zamir et al., 2005). Currently, however, these methods are limited to 2D deformations.
Because the present study focuses on an experiment in which looping is perturbed, it is difficult to make direct comparisons with previous strain measurements for normal looping. However, some observations are appropriate. First, the longitudinal and circumferential strains were positive in all of the studied regions (Fig. 7), consistent with the data of Ramasubramanian et al., (2006) for these regions. This finding likely reflects a combination of growth (Soufan et al., 2006) and general inflation of the heart due to accumulation and swelling of CJ (Nakamura and Manasek, 1978; Manasek et al., 1984; see Fig. 6). Second, the largest longitudinal strains occurred on the cranial side of the primitive left atrium (Fig. 7, region 7), also in agreement with the results of Ramasubramanian et al. (2006). Changes in cell size, high rates of cell proliferation, and bending of the atrium all may contribute to the high strain observed in this region (Soufan et al., 2006). These large atrial strains are consistent with speculation that the primitive left atrium plays a role in determining left–right looping directionality by pushing the caudal end of the heart toward the right side of the embryo (Voronov et al., 2004; Ramasubramanian et al., 2006).
The measured strain patterns in SPL-lacking hearts appear to be consistent with observed morphological changes. All strain components remained relatively small from stages 10 to 11, a period when only some ventral bending (and little or no torsion) occurs. The heart subsequently bends, twists, and inflates to stage 12−, whereupon the heart tube appears to deflate somewhat, but ventral bending continues through stage 12 (Fig. 6). These trends are clearly reflected in the strain data of Figure 8. Specifically, longitudinal strain (bending strain) on the outer curvature (region MHT) significantly increases from stage 11 through 12 (Fig. 8B). It also increases in the RHT, consistent with the rightward bending described by Voronov et al. (2004). The relatively small increase in longitudinal strain on the left side (region LHT) likely is due to CJ swelling. Circumferential strains generally increase with observed heart tube inflation toward stage 12 and subsequently decrease due to the diminution of the CJ (Figs. 6, 8C).
Finally, we note that the right-sided cytoskeletal contraction found after SPL removal by Nerurkar et al. (2006) should be associated with a decreased circumferential strain on the right side compared with the left as the heart rotates. Our results showed this trend after stage 11+ (Fig. 8C), but the differences were not statistically significant.
In regions of the heart that were accessible, the use of polystyrene microspheres provided a reliable method for tracking developmental patterns in chick hearts. Past studies have used microspheres to track migrating cells (Bronner-Fraser, 1982) and to measure strain during the cardiac cycle (Alford and Taber, 2003). The microspheres were clearly visualized and tracked using volume reconstructions of image cross-sections. It is likely that the relatively large diameter of the beads (8.6 μm) and semisolid composition, combined with placement on the ventral surface of the heart, led to the strong contrast signals for these markers. The strong signals caused the beads to appear larger using OCT compared with images from traditional light microscopy (Fig. 6). However, this should not significantly affect the coordinates of the bead centroids.
Beads were useful for measuring strains in SPL-lacking hearts. In studies of normal looping, however, beads present some serious problems, as they tend to dislodge when they come into contact with the overlying SPL. Future work should explore injectable contrast agents that are appropriate for OCT.
The present analysis is limited to surfaces that are single-valued functions of X and Y. In the future, the analysis can be extended to general surfaces by fitting multiple regions independently and then combining the results.
The morphology of even normal hearts can be quite variable, as reflected in the strain measurements (Figs. 7, 8). This finding makes statistical comparisons a challenge, but meaningful trends in the data could be discerned.
In conclusion, OCT can be an extremely useful tool for obtaining qualitative and quantitative descriptions of morphogenesis of living systems in three dimensions. The ability to image the entire heart in any plane at high resolution makes OCT-based strain analysis an ideal method for studies of the mechanics of cardiac morphogenesis.
L.A.T. was funded by the National Institutes of Health.
Longitudinal and circumferential strains characterize the fractional change in length along the corresponding directions, with positive strains indicating stretching and negative strains indicating shortening. Shear strains characterize changes in the angles at the corners of a small, originally rectangular, surface element.
Consider the deformation of a surface S into the surface s (Fig. 9). A point on S with Cartesian coordinates (X,Y,Z) moves to the coordinates (x,y,z) on s. The surface S is described by the relation Z = Z(X,Y), and the coordinates of s are given by x = x(X,Y,Z) = x(X,Y,Z(X,Y)) = x(X,Y), etc. Hence, we have x = x(X,Y), y = y(X,Y), and z = z(X,Y).
The position vectors of a point on S and its deformed image on s are, respectively,
where the ei are Cartesian base (unit) vectors of the global coordinate system. The base vectors in S and the convected base vectors in s are given by the respective relations (Taber, 2004)
where I = 1, 2 and XI = (X,Y). Substituting Equations (1) into (2) yields
To compute the functions Z(X,Y), x(X,Y), y(X,Y), and z(X,Y), each of these coordinates was plotted individually against X and Y. Then the MATLAB routine GRIDFIT (www.mathworks.com/matlabcentral/fileexchange) was used to fit a surface to these points, and derivatives with respect to the X and Y coordinates were calculated.
Lagrangian surface strains are defined by Eij = ½(gi · gj − Gi · Gj)/[(gi · gi)(Gj · Gj,)]1/2. Inserting Eqs. (3) yields the surface strain components
These components were transformed into strain components relative to the local longitudinal (X1) and circumferential (X2) directions (see Fig. 7 for X1 direction). This was done using the standard strain transformations
where θ is the local rotation angle from the X-Y system to the X1-X2 system. Further details can be found in Filas (2006).