# Young’s modulus computation¶

The computation of the Young’s modulus uses a look-up table (LUT) that was derived from simulations based on the finite elements method (FEM) [MMM+17] and the analytical solution [MOG+15]. The LUT was generated with an incompressible (Poisson’s ratio of 0.5) linear elastic sphere model (an artificial viscosity was added to avoid division-by-zero errors) in an axis-symmetric channel (2D). Although the simulations were carried out in this cylindrical symmetry, they can be mapped onto a square cross-sectional channel by adjusting the channel radius to approximately match the desired flow profile. This was done with the spatial scaling factor 1.094 (see also supplement S3 in [MOG+15]). The original data used to generate the LUT are available on figshare [WMM+20].

All computations take into account:

• scaling laws: The original LUT was computed for a specific channel width $$L$$, flow rate $$Q$$, and viscosity $$\eta$$. If the experimental values of these parameters differ from those in the simulation, then they must be scaled before interpolating the Young’s modulus. The scale conversion rules can be derived from the characteristic length $$L$$ and stress $$\sigma=\eta \cdot Q/L^3$$ [MOG+15]. For instance, the event area scales with $$(L_\text{exp}/L_\text{LUT})^2$$, the Young’s modulus scales with $$\sigma_\text{exp}/\sigma_\text{LUT}$$, and the deformation is not scaled as it has no units. Please note that the scaling laws were derived for linear elastic materials and may not be accurate for other materials (e.g. hyperelastic). The scaling laws are implemented in the submodule dclab.features.emodulus.scale_linear.

• pixelation effects: All features (including deformation and area) are computed from a pixelated contour. This has the effect that deformation is overestimated and area is underestimated (compared to features computed from a “smooth” contour). While a slight change in area does not have a significant effect on the interpolated Young’s modulus, a systematic error in deformation may lead to a strong underestimation of the Young’s modulus. A deeper analysis is visualized in the plot pixelation_correction.png which was created with pixelation_correction.py. Thus, before interpolation, the measured deformation must be corrected using a hard-coded correction function [Her17]. The pixelation correction is implemented in the submodule dclab.features.emodulus.pxcorr.

• shear-thinning and temperature-dependence: The viscosity of a medium usually is a function of temperature. In addition, complex media, such as 0.6% methyl cellulose (CellCarrier B), may also exhibit shear-thinning. The viscosity of such media decreases with increasing flow rates. Since the viscosity is required to apply the scaling laws (above), it must be corrected which is done using hard-coded correction functions [Her17]. The computation of viscosity is implemented in the submodule dclab.features.emodulus.viscosity.

Fig. 1 Visualizations of the support and the values of the look-up table (LUT) used for determining the Young’s modulus from deformation and cell area. The values of the Young’s moduli in the regions shown depend on the channel size, the flow rate, the temperature, and the viscosity of the medium [MOG+15]. Here, they are computed for a 20 µm wide channel at 23°C with an effective pixel size of 0.34 µm. The data are corrected for pixelation effects according to [Her17].

Since the Young’s modulus is model-dependent, it is not made available right away as an ancillary feature (in contrast to e.g. event volume or average event brightness).

In [1]: import dclab

In [2]: ds = dclab.new_dataset("data/example.rtdc")

# "False", because we have not set any additional information.
In [3]: "emodulus" in ds
Out[3]: False


Additional information is required. There are three scenarios:

1. The viscosity/Young’s modulus is computed individually from the chip temperature for each event. Required information:

• The temp feature which holds the chip temperature of each event

• The configuration key [calculation]: ‘emodulus medium’

• The configuration key [calculation]: ‘emodulus model’

1. Set a global viscosity. Use this if you have measured the viscosity of your medium (and know all there is to know about shear thinning [Her17]). Required information:

• The configuration key [calculation]: ‘emodulus model’

• The configuration key [calculation]: ‘emodulus viscosity’

1. Compute the Young’s modulus using the viscosities of known media.

• The configuration key [calculation]: ‘emodulus medium’

• The configuration key [calculation]: ‘emodulus model’

• The configuration key [calculation]: ‘emodulus temperature’

Note that if ‘emodulus temperature’ is given, then this temperature is used, even if the temp feature exists (scenario A).

The key ‘emodulus model’ currently (2019) only supports the value ‘elastic sphere’. The key ‘emodulus medium’ must be one of the supported media defined in dclab.features.emodulus.viscosity.KNOWN_MEDIA and can be taken from [setup]: ‘medium’. The key ‘emodulus temperature’ is the mean chip temperature and could possibly be available in [setup]: ‘temperature’.

import matplotlib.pylab as plt

import dclab

ds = dclab.new_dataset("data/example.rtdc")

# Add additional information. We cannot go for (A), because this example
# does not have the temperature feature ("temp" not in ds). We go for
# (C), because the beads were measured in a known medium.
ds.config["calculation"]["emodulus medium"] = ds.config["setup"]["medium"]
ds.config["calculation"]["emodulus model"] = "elastic sphere"
ds.config["calculation"]["emodulus temperature"] = 23.0  # a guess

# Plot a few features
ax1 = plt.subplot(121)
ax1.plot(ds["deform"], ds["emodulus"], ".", color="k", markersize=1, alpha=.3)
ax1.set_ylim(0.1, 5)
ax1.set_xlim(0.005, 0.145)
ax1.set_xlabel(dclab.dfn.get_feature_label("deform"))
ax1.set_ylabel(dclab.dfn.get_feature_label("emodulus"))

ax2 = plt.subplot(122)
ax2.plot(ds["area_um"], ds["emodulus"], ".", color="k", markersize=1, alpha=.3)
ax2.set_ylim(0.1, 5)
ax2.set_xlim(30, 120)
ax2.set_xlabel(dclab.dfn.get_feature_label("area_um"))

plt.show()