Sacrificial layer technique for axial force post assay of immature cardiomyocytes

Abstract

Immature primary and stem cell-derived cardiomyocytes provide useful models for fundamental studies of heart development and cardiac disease, and offer potential for patient specific drug testing and differentiation protocols aimed at cardiac grafts. To assess their potential for augmenting heart function, and to gain insight into cardiac growth and disease, tissue engineers must quantify the contractile forces of these single cells. Currently, axial contractile forces of isolated adult heart cells can only be measured by two-point methods such as carbon fiber techniques, which cannot be applied to neonatal and stem cell-derived heart cells because they are more difficult to handle and lack a persistent shape. Here we present a novel axial technique for measuring the contractile forces of isolated immature cardiomyocytes. We overcome cell manipulation and patterning challenges by using a thermoresponsive sacrificial support layer in conjunction with arrays of widely separated elastomeric microposts. Our approach has the potential to be high-throughput, is functionally analogous to current gold-standard axial force assays for adult heart cells, and prescribes elongated cell shapes without protein patterning. Finally, we calibrate these force posts with piezoresistive cantilevers to dramatically reduce measurement error typical for soft polymer-based force assays. We report quantitative measurements of peak contractile forces up to 146 nN with post stiffness standard error (26 nN) far better than that based on geometry and stiffness estimates alone. The addition of sacrificial layers to future 2D and 3D cell culture platforms will enable improved cell placement and the complex suspension of cells across 3D constructs.

Appendix

The natural frequency of the cantilever is related to its stiffness and mass using the classical equation for a translational mechanical harmonic oscillator, where k _c is the bending stiffness of the cantilever for a point load applied to its tip and m _eff is the mass of the cantilever:

$$ {\omega_n} = \frac{1}{{2\pi }}\sqrt {{\frac{{{k_c}}}{{{m_{{eff}}}}}}} $$

(2)

We substitute cantilever mass with density times volume. In the following equation, ρ is the density of silicon and L, W, and T are the cantilever length, width, and thickness, respectively:

$$ {\omega_n} = \frac{1}{{2\pi }}\sqrt {{\frac{{{k_c}}}{{0.24\rho LWT}}}} $$

(3)

The effective mass is only 0.24 times the overall mass of the cantilever beam due to the mode shape of the first eigenmode. If stiffness is calibrated from a higher-order eigenmode, then the effective mass should be calculated accordingly. Laser Doppler vibrometry was used to determine the natural frequency, ω _n , 2.37 kHz, and when we rearrange the previous equation we obtain the following expression for k_c:

$$ {k_c} = 0.24\rho LWT{\left( {2\pi {\omega_n}} \right)^2} $$

(4)

The force F applied by the cantilever is calculated from the amplifier output-referred differential voltage, ΔV, and the cantilever force sensitivity, S _f :

$$ {F_c} = \frac{{\varDelta V}}{{G{S_f}}} $$

(5)

where G is the overall gain of the signal-conditioning circuit. The force sensitivity scales linearly with the Wheatstone bridge bias voltage and depends on the cantilever design parameters. The force sensitivity is calculated from the product of the cantilever stiffness and displacement sensitivity.

The micropost stiffness, k _p , is the quotient of the applied force, F, and the optical displacement of the micropost. Since we measured micropost displacement in pixels, we multiply our resolution [microns/pixel], R, and our pixel deflection, Δx, to obtain the deflection in microns:

$$ {k_p} = \frac{{\varDelta V}}{{G{S_f}R\varDelta x}} $$

(6)

Finally, we must express force sensitivity, S _f , in terms of the displacement sensitivity, S _d , since we do not measure force sensitivity directly during cantilever calibration. The displacement sensitivity is determined by driving the cantilever on resonance and measuring the ratio of the circuit voltage output to the tip deflection, again using laser Doppler vibrometry (Park et al. 2007). The micropost stiffness can finally be written in terms of experimentally measured parameters as follows:

$$ {k_p} = \frac{{{k_c}\varDelta V}}{{{V_b}G{S_d}R\varDelta x}} $$

(7)

$$ {k_p} = \frac{{0.24\rho LWT{{\left( {2\pi {\omega_n}} \right)}^2}\varDelta V}}{{{V_b}G{S_d}R\varDelta x}} $$

(8)

As we have detailed previously (Kim et al. 2011), uncertainty in the cantilever stiffness contributes to the accuracy or systematic error in our reported forces and is calculated using a root mean sum of squares analysis of the measurement uncertainties (Holman 1988). Lithographic variations and uncertainty in displacement sensitivity contribute to this device uncertainty (Park et al. 2007). We convolve this error with the error in optical measurements, which is limited by pixel resolution.