Quantification of embryonic atrioventricular valve biomechanics during morphogenesis

Abstract

Tissue assembly in the developing embryo is a rapid and complex process. While much research has focused on genetic regulatory machinery, understanding tissue level changes such as biomechanical remodeling remains a challenging experimental enigma. In the particular case of embryonic atrioventricular valves, micro-scale, amorphous cushions rapidly remodel into fibrous leaflets while simultaneously interacting with a demanding mechanical environment. In this study we employ two microscale mechanical measurement systems in conjunction with finite element analysis to quantify valve stiffening during valvulogenesis. The pipette aspiration technique is compared to a uniaxial load deformation, and the analytic expression for a uniaxially loaded bar is used to estimate the nonlinear material parameters of the experimental data. Effective modulus and strain energy density are analyzed as potential metrics for comparing mechanical stiffness. Avian atrioventricular valves from globular Hamburger-Hamilton stages HH25-HH34 were tested via the pipette method, while the planar HH36 leaflets were tested using the deformable post technique. Strain energy density between HH25 and HH34 septal leaflets increased 4.6±1.8 fold (±SD). The strain energy density of the HH36 septal leaflet was four orders of magnitude greater than the HH34 pipette result. Our results establish morphological thresholds for employing the micropipette aspiration and deformable post techniques for measuring uniaxial mechanical properties of embryonic tissues. Quantitative biomechanical analysis is an important and underserved complement to molecular and genetic experimentation of embryonic morphogenesis.

Fig. 1

Axisymmetric schematic of pipette aspirated tissue disk. The reference tissue is denoted by the dashed box and has dimensions of radius, R, and thickness, D. The region inside the pipette with radius, r_p, is loaded with aspiration pressure, ΔP, resulting in an aspirated length, L.

Fig. 2

Pipette stiffness measurements consistent for D > 4 r_p. (A) Plot of measured tissue stiffness normalized to Young’s Modulus, E, of a NeoHookean material versus tissue thickness, $\overline{D}$ Inset: FE generated ΔP versus λ curves from which stiffness was calculated from the slope. (B) Plot of radial stress normalized to stress at applied load surface as a function of depth along the axis of the symmetry. For $\overline{D}<4$, the radial component (likewise meridianal) experiences a compressive stress at the bottom edge, evidence of the tissue undergoing bending.

Fig. 3

Pipette aspiration approximated as a uniaxial load. (A) Along the axis of symmetry, the deformation transitions from equibiaxial (gray region) extension toward uniaxial extension with depth into the tissue. (B) The ΔP versus λ curve of an incompressible, NeoHookean material closely aligns with the Cauchy stress of a uniaxially loaded bar, not the 1st Piola-Kirchhoff stress. (C) The ΔP versus λ curve of an exponential material differs by a scale factor, λ, from the axial stress of a uniaxial load compared at same stretch ratio. (D) The scale factor, λ, is a function of only α, which is approximated with the cubic polynomial shown for α in [0, 2]. A=−0.052, B=0.252, C=0.053, and λ₀=1.09.

Fig. 4

Strain energy density is a meaningful parameter for nonlinear material comparison. Plot of axial Cauchy stress for an exponential material with (1) fixed effective modulus, E_Eff=αC (circles), and (2) fixed strain energy density from λ_{1 - 2} (solid lines) evaluated for α=0.5, 1, 1.5, and 2. Though both parameters are non-unique, the strain energy density generates a tighter range of material response curves over the same set of α values, while E_Eff has a larger spread of material response curves and no physical meaning for large α.

Fig. 5

AV cushion geometry transition necessitates alternative mechanical testing devices. (A) Cushion thickness, t_c, relative to pipette placement decreases during development. HH36 violates the D = 4r_p testing criteria for 70 μm pipette diameter. (B) Ratio of cushion thickness over cushion apex to base length, L_c, shows a transition to a more planar configuration at the completion of valvulogenesis (HH36). n=8-10 mean±SD, all pairs of different letters are statistically significant, ANOVA, p<0.001.

Fig. 6

Deformable post testing device for planar tissues. (A) Schematic of deformable cantilevers of height, h_p, and diameter, ø_p, in reference (dashed) and deformed configurations (solid). DP deflection equals v=(x_b x_t)/2, where x_t is the distance between the top of the posts, and x_b is the distance between the bases. (B) Side view of DP device, scale bar=1 mm (C) Top view of device with attached HH36 septal leaflet, and tip of posts highlighted for reference (dashed circles). Scale bar=500 μm. (D) Euler beam theory (dashed line) predicts a linear increase in bending stiffness with diameter for a fixed h_p/ø_p. Measured bending stiffness for three diameters with h_p/_p=3 showed agreement with theory, n=6 (3 silicone batches) mean±SD.

Fig. 7

Monotonic increase in AV cushion stiffness during development. (A) Representative PA data for septal leaflets from stages HH25–HH34, (n=4) (B) Calculated strain energy densities for septal (black) and mural (gray) cushions. HH25 inferior denoted as septal (black) and superior (white) bar. n=8-11 mean±SD 1-way ANOVA or t-test *p<0.05, # p<0.001 w.r.t. HH25 septal (C) Plot of applied force over initial cross-sectional area (~ 1st PK) for HH36 septal leaflet deformable post data. Average strain energy density denoted on figure as mean±SD.