Magnetic microrheometry of tumor-relevant stiffness levels and probabilistic quantification of viscoelasticity differences inside 3D cell culture matrices

Abstract

The progression of breast cancer involves cancer-cell invasions of extracellular matrices. To investigate the progression, 3D cell cultures are widely used along with different types of matrices. Currently, the matrices are often characterized using parallel-plate rheometry for matrix viscoelasticity, or liquid-like viscous and stiffness-related elastic characteristics. The characterization reveals averaged information and sample-to-sample variation, yet, it neglects internal heterogeneity within matrices, experienced by cancer cells in 3D culture. Techniques using optical tweezers and magnetic microrheometry have measured heterogeneity in viscoelasticity in 3D culture. However, there is a lack of probabilistic heterogeneity quantification and cell-size-relevant, microscale-viscoelasticity measurements at breast-tumor tissue stiffness up to ≃10 kPa in Young’s modulus. Here, we have advanced methods, for the purpose, which use a magnetic microrheometer that applies forces on magnetic spheres within matrices, and detects the spheres displacements. We present probabilistic heterogeneity quantification using microscale-viscoelasticity measurements in 3D culture matrices at breast-tumor-relevant stiffness levels. Bayesian multilevel modeling was employed to distinguish heterogeneity in viscoelasticity from the effects of experimental design and measurement errors. We report about the heterogeneity of breast-tumor-relevant agarose, GrowDex, GrowDex–collagen and fibrin matrices. The degree of heterogeneity differs for stiffness, and phase angle (i.e. ratio between viscous and elastic characteristics). Concerning stiffness, agarose and GrowDex show the lowest and highest heterogeneity, respectively. Concerning phase angle, fibrin and GrowDex–collagen present the lowest and the highest heterogeneity, respectively. While this heterogeneity information involves softer matrices, probed by ≃30 μm magnetic spheres, we employ larger ≃100 μm spheres to increase magnetic forces and acquire a sufficient displacement signal-to-noise ratio in stiffer matrices. Thus, we show pointwise microscale viscoelasticity measurements within agarose matrices up to Young’s moduli of 10 kPa. These results establish methods that combine magnetic microrheometry and Bayesian multilevel modeling for enhanced heterogeneity analysis within 3D culture matrices.

Measuring mechanical cues for modeling the stromal matrix in 3D cell cultures

“…We used a Bayesian hierarchical model to analyze the mechanical data from microrheometry and rheometry, because the experimental data have systematic errors and uncertainties, complicating direct comparisons of investigated conditions. 40 Therefore, the reported results hereafter are based on the model output, unless stated otherwise. The hydrogel material's viscoelastic properties (| G *| or ϕ ) of interest may be hidden due to biased parameters, such as changing magnetic probe size, which necessitates a more principled approach to the analysis.…”

“…Logarithmic transformation was used to facilitate the modeling, 46 while the positive data values allow this. 47 The function f is a linear function with a saturation point, allowing for a change in the slope for stiffness (| G *|) ( e.g. , Fig.…”

“…S5, ESI†). The probe coatings ( μ probe coating ) and the effects ( μ experimental ) due to experimental data collection 40 are also unique parameters shifting the measured values independently of the calcium crosslinker concentration. Collagen concentration ( C collagen concentration ) and polymerization temperature ( T ) are assumed to have a linear correlation ( k collagen concentration and k polymerization temperature ) with the measured mechanical properties.…”

“…Furthermore, a key aspect explored by the magnetic microrheometer is the IPN material heterogeneity. We build on our previously reported model 40 to decouple the material's intrinsic variation in viscoelasticity from systematic biases. The heterogeneity in viscoelasticity is modeled as follows: y std = GP( C Ca 2+ ) + σ probe size + σ collagen concentration + σ polymerization temperature The heterogeneity was quantified as the standard deviation ( y std ) of the modeled data at the sample holder level, as outlined in our previous article.…”

“…The heterogeneity in viscoelasticity is modeled as follows: y std = GP( C Ca 2+ ) + σ probe size + σ collagen concentration + σ polymerization temperature The heterogeneity was quantified as the standard deviation ( y std ) of the modeled data at the sample holder level, as outlined in our previous article. 47 The trend is modelled with a non-linear function, specifically with a Gaussian process (GP), 48 which is assumed to be a function of the calcium crosslinker concentration ( C Ca 2+ ). Furthermore, the probe size ( σ probe size ), collagen concentration ( σ collagen concentration ) and polymerization temperature ( σ polymerization temperature ) may affect the heterogeneity independently of calcium concentration.…”