The cardiac torsion that occurs during looping is one of the first morphological signs of left-right asymmetry in the vertebrate embryo. Looping abnormalities likely underlie some of the cardiac malformations that occur in as many as 1% of liveborn and 10% of stillborn human births [25
], and such defects may result in numerous spontaneous abortions during the first trimester [26
]. Thus, this problem has received considerable attention for decades. However, while rapid progress is being made in defining the genetic perturbations behind abnormal cardiac morphogenesis [25
], the biomechanical mechanisms that drive and regulate looping are not yet completely understood [1
The impetus for this study stems from our previous experiments that explored the behavior of the looping chick heart following removal of the SPL [5
]. Although the SPL normally plays a major role in the torsional component of c-looping [3
], those experiments suggest that the heart contains an intrinsic ability to adapt and undergo torsion when this membrane is removed [5
]. Such built-in redundancy ensures proper looping even under severely perturbed conditions. Because heart development is vital to the survival of embryo, it is not surprising that such redundancies exist.
In our earlier study of this problem, we found evidence that the observed adaptive torsion is driven by an abnormal cytoskeletal contraction that develops primarily on the right side of the heart and pulls the HT rightward [5
]. However, we did not investigate in detail the underlying mechanics to determine whether such a mechanism is fully consistent with physical principles and the observed 3D morphology. Moreover, we did not explore how this response is regulated. The purpose of the present study is to examine these issues using both experimental and computational models.
It is important to recognize that the present work is phenomenological in nature. The simulated mechanisms are based on data regarding cellular activity, but the molecular events that produce this activity are not considered. Moreover, developmental events occurring before and during heart tube formation likely influence looping directionality [25
A. Remarks on stiffness measurements
During the last decade, a number of papers have been published on the mechanical properties of embryonic chick hearts during post-looping stages 16 to 31 (2.5 to 7 days) [31
]. Since an early study of the compressive behavior of the stage-12 heart [36
], however, Zamir and colleagues [8
] have provided the only measurements of regional stiffness and material properties of the looping heart (also at stage 12). Notably, Zamir et al. [8
] found that the right side, left side, and outer curvature of the HT have similar stiffnesses, but the inner curvature is significantly stiffer, possibly due the seam left there by the fused edges of the DM. They speculated that the relatively stiff dorsal side plays a role in the bending component of c-looping.
Here, we measured the stiffness in several other regions of normal stage-11 hearts (developed with SPL intact), including the right and left sides of the conotruncus and the top and bottom of the primitive atria. Taken together, our data show that all regions of the heart at stages 11 and 12 have approximately the same stiffness, with the exception of the lower atrial region, which is significantly stiffer (). The lower atrial stiffness dropped on exposure to blebbistatin, indicating that the elevated stiffness in this region is at least partially due to cytoskeletal contraction, a finding consistent with previous data [3
]. The present 3D model supports the view that this contraction plays a role in the torsional component of c-looping; turning it off in the model results in a significant loss of rotation ().
The indentation problem can be treated as a plate or shallow shell (myocardium) on an elastic foundation (cardiac jelly). Hence, in relatively soft regions of the heart, the deformation is localized to the region near the indenter, and differences in stiffness primarily reflect changes in myocardial thickness or material properties [8
]. However, in the relatively stiff lower atrial region, boundary conditions on the heart may influence the measurements. Our finite element analysis of the indentation experiment accounts for these effects and shows that the relatively high stiffness in this region can be attributed to a combination of contraction, the attached SPL, and longitudinal curvature (). The material properties of the myocardium need not differ from those in other regions of the heart.
B. Remarks on the 3D model (no feedback)
In prior work, we used a 2D finite element model to show that contraction on the right side of the HT and conotruncus can cause the rightward rotation observed following SPL removal [5
]. That model, however, does not account for the complex 3D geometry of the embryonic heart and ignores the possible effects of the primitive atria. Here, we developed a more realistic model to test more thoroughly the feasibility of our hypothesis. The model includes circumferential contraction on the right side of the heart (HT, DM, and conotruncus), longitudinal contraction in the lower half of the atria, and longitudinal growth in the upper half of the atria (). It is important to note that the contraction along the right side of the heart occurs only when the SPL is removed [5
]. In this model, the growth stretch ratios for all of these processes are specified a priori.
The deformation of the model, including torsion, is similar qualitatively to that seen experimentally (compare and ). The general behavior is consistent with the hypothesis illustrated in , but the model provides more detail about the physical mechanism. First, the forces that the atria exert on the HT arise from two sources. One is the push of cells being added to the upper side of the atria, and the other is an additional push exerted by these same cells as they are pulled toward the heart by contraction of the lower side of the atria. In this model, the left atrium exerts a greater force because it is larger than the right atrium, as generally seen in vivo (e.g., see ). Second, geometric asymmetries convert the atrial forces into forces that cause the HT to rotate, rather than bend, rightward. In particular, the DM serves as a pivot for rotation, and the offset in the atrial cross-sections where they intersect the HT produce a moment that has a torsional component after the heart begins to rotate. The abnormal out-of-plane orientation of the atria when the SPL is removed () enhances the torsional moment (). Third, when the SPL is removed, all of the forces due to growth and contraction contribute to the observed torsion.
The results from the looping simulations following removal of one or both atria () seem to provide strong support for our general hypothesis for c-looping, as depicted in . In each simulation, all model parameters have the same values. The only difference is that the left atrium, right atrium, or both atria are removed. In each case, the model yields a shape that agrees reasonably well with that of the experimental heart.
C. Remarks on the 2D model (with feedback)
Feedback likely is involved in the regulation of morphogenesis. In traditional control systems, feedback is used to achieve two goals: (1) set point regulation, which involves taking the system response toward a desired value (e.g., cruise control in automobiles) and (2) disturbance rejection, which ensures that perturbations do not interfere with set point regulation (e.g., integral control in servo machines). In this paper, we investigate a control law for the rotational component of c-looping, which ensures that (1) the set point (i.e., a fully rotated heart) is reached and (2) rotation is completed even when external disturbances (i.e., SPL removal) are encountered.
The 2D model for looping employs a phenomenological control law based on the HR Hypothesis of Beloussov [23
]. The main idea of this hypothesis is that embryonic tissues attempt to maintain a homeostatic stress state by actively changing size and shape (e.g., by growth or contraction) whenever homeostasis is disturbed. The induced response generally overshoots the homeostatic target stress, however, leading to a new response, and so on, until the desired form is created. In this scheme, the perturbation that initiates a morphogenetic process may require genetic activity, but thereafter the cells are autonomous. Recently, we used computational models based on Eq. (5)
and Eq. (6)
in the present paper to explore the feasibility of the HR Hypothesis for some fundamental problems in morphogenesis [38
]. Comparing numerical and experimental results gave mixed agreement, and we concluded that genes must occasionally step in to alter or trigger further cellular activity.
The results from the 2D model show that the HR Hypothesis captures quite well the dynamics of the rotational component of c-looping. The behavior of the model is consistent with the results from three different experiments, where the SPL is removed at stages 10, 11, and 12 ( and ). We again emphasize that the same parameters are used for all three simulations. In this model, asymmetric atrial forces supply an initial perturbation that causes the HT to rotate and the DM to bend slightly rightward. The bending produces tension on the convex side and compression on the concave side of the DM, and Eq. (6)
makes the initially zero target stresses compressive on the convex side and tensile on the concave side. In response, the concave side contracts and the convex side grows, inducing more bending. This response continues until the HT contacts the wall of the foregut. Interestingly, DM bending stops after the initial perturbation if there is no overshoot, i.e., if b
is set to zero in Eq. (6)
. Thus, stress hyper-restoration is crucial for this model to work.
According to our models, contraction and growth in the DM is necessary for proper rightward rotation, as contraction and growth in the HT does little in itself. Recently, Linask et al. [39
] presented evidence supporting this idea. These authors found that the cell proliferation rate is normally higher on the left side than the right side of the DM (and associated splanchnic mesoderm adjacent to the foregut wall). Moreover, inducing hyperplasia on the right side is associated with left looping. They speculate that the added cells on the left side of the DM push the HT rightward, and vice versa, in agreement with the results from our model. Here, we suggest that the observed asymmetric cellular hyperplasia is an HR response to asymmetric stresses caused by external loads.
If the DM is truly needed for proper rotation, then rotation would be compromised if the DM were removed. Experiments confirm that rotation is severely disrupted when the DM is cut away (results not shown). Also, in an earlier study, we showed, using a 3D model and experimental perturbations, that the DM is critical for the rotational component during the early stages of c-looping [4
As mentioned above, element inversion causes convergence issues in the 3D model. The model has approximately 60,000 elements, and the simulation stops even if only one element inverts. This problem is especially acute at the intersection between the HT and the atria. For this reason, some simulations did not proceed to completion. This is the reason that the HT rotation angle (see schematic in ) at the end of c-looping given by the 3D model (29°, ) is smaller than the measured angle (54°, ). (Growth on the left side of the DM, which is not included in the 3D model, also may be a factor in the relatively small rotation.) Unfortunately, problems involving large deformation, such as the one considered here, are more susceptible to element inversion. Currently, the limited adaptive remeshing capability found in ABAQUS is insufficient to alleviate this problem.
Because of a lack of experimental data, the values of certain model parameters are just rough estimates. These include the growth law coefficients, growth and contraction stretch ratios, and changes to the material constant A due to contraction. Moreover, it is likely that cardiac jelly is highly viscoelastic. However, viscoelastic material properties for the heart during c-looping also are currently unavailable, and hence these effects are not included.
Our models appear to support the notion that c-looping is driven primarily by forces exerted on the heart tube by the SPL and the primitive atria. At the cell level these forces are generated by a combination of cytoskeletal contraction and growth. When these forces are perturbed, the models show that stress hyper-restoration can generate an adaptive mechanism that restores normal looping. Future work is needed to determine whether this concept applies to other developmental processes.