Fig 1.
TFM-UQ method improves traction stress inference and provides uncertainty bounds.
(A) Traction Force Microscopy experiment. Reference image of the substrate markers is obtained at a stress-free state. Session image(s)
of the substrate marker are obtained at the time point of interest. (B) Existing TFM workflow provides a point estimate of forces but does not consider the variability in the traction stress due to the microscopy images and TFM implementation. (C) The proposed TFM-UQ method considers the local image quality, ambiguity in regularization parameter selection, and the numerical implementation to provide locally adaptive smoothing and error bars.
Fig 2.
Particle Image Velocimetry with Uncertainty Quantification (PIV-UQ) method.
Top: Schematic of PIV and PIV-UQ. (A) The image data is sectioned into overlapping square sub-windows
of length WL (n2 pixels), their centers separated by WS. PIV computes the most likely displacement for each sub-window
that corresponds to the maximum of the cross-correlation metric. Each pixel of
contributes to the correlation value, hence the maximization procedure. (B) PIV-UQ method bootstraps the contribution of individual pixels to the cross-correlation metric. Pixel indices are randomly sampled with replacement and the unsampled indices are set to 0 (Black pixels). Backslash denotes the set difference operation and
represents uniformly distributed n2 samples in the interval [1,n2]. The bootstrapped sample of size nB is analyzed for multiple possible clusters and outliers, subsequently providing a metric of variability for each sub-window (i.e., for each discrete vector of the deformation field). Bottom: PIV-UQ bootstrap algorithm.
Fig 3.
Hierarchical Bayesian TFM formulation for adaptive and self-consistent regularization.
The problem of choosing a regularization parameter is treated in a Bayesian formulation. Substrate deformation () is the observed quantity, and it is modeled as the additive contribution of traction stresses (
), locally resolved PIV measurement uncertainty
and a global model error
that is unknown. The hierarchical formulation allows to express unknown hyper-parameters (
for the prior on traction stress field and the model error term as random variables to be inferred. Therefore, a non-informative hyper-prior (
) is specified to the hyper-parameters. The priors can encode reasonable assumptions such as smoothness and global force balance. Markov Chain Monte Carlo (MCMC), specifically an hybrid Gibbs sampling, is used for inference from the marginal posterior distribution,
.
Fig 4.
Hybrid Gibbs sampler for TFM UQ inference.
Fig 5.
(A, D) Gaussian profile synthetic beads are randomly generated at varying signal-to-noise ratio (SNR) of image pixel noise. A uniform (U) displacement field (B, E) or a shear (S) displacement field (C, F) is applied to the synthetic beads and PIV-UQ is performed as described in § 2.2. In (B,C,E and F), left columns show ground-truth data consisting of applied displacement field vectors overlaid on contour maps of ensemble standard deviation ( and
). The right columns represent the same data estimated from PIV-UQ,
and
. Units are in pixels (px). (G) Joint distribution of
for uniform (U) and shear (S) deformation fields for WL of 32 and 64 px. Analysis procedure is described in Fig A in S1 Text (H) RMS of ensemble standard deviation and PIV UQ bootstrap estimates are compared for varying SNR values. (I) PIV Window size as a function of SNR with
isolines derived from data presented in (H). N = 50 realizations of
pixels images were generated.
Fig 6.
Synthetic pipeline to simulate experimentally relevant, spatially heterogeneous noise levels in TFM.
(A) Synthetic traction stress generated from four traction islands of Gaussian profile, , one in each quadrant with a maximum stress of
. (B) Synthetic displacement field resulting from the traction profile in (A). Substrate of
KPa and Poisson’s ratio of 0.45 is used here. (C) Schematic depicting spatially heterogeneous noise addition directly to the synthetic image by varying fluorescent bead density in X-direction. The synthetic image also has image pixel noise equivalent to SNR = 50 (D) Synthetic fluorescent bead image encoding image pixel noise and bead density variations. (E) PIV-UQ deformation measurement
of the simulated bead images (D). (F) PIV-UQ estimation of uncertainty (standard deviation
) showing higher uncertainty increasing with x-direction in agreement with noise addition process (C).
Fig 7.
TFM-UQ adaptively regularizes based on the local displacement uncertainty.
(A) Mean marginal posterior traction stress distribution () approximated from Hybrid-Gibbs sampling. (Based on synthetic data described in Fig 6A.) (B) Pointwise marginal posterior uncertainty (standard deviation
) of posterior traction stress (C) L-curve to determine Tikhonov regularization parameter
in traditional TFM methods.
and
are regularization parameters obtained from L-curve corners of homoskedastic synthetic simulations with spatially uniform low noise (L) or high noise (H). (D, E, F) Tikhonov regularized traction stress field corresponding to L-curve corner
, low noise
and high noise
respectively.
is the discretized Laplacian operator. (G, H, I) Tikhonov regularized traction stress field, with
identity matrix.
,
and
were determined to match 95th percentile of traction stress magnitude with the respective fields in (D-F).
Table 1.
Performance comparison of locally regularized TFM-UQ with classical (global) Tikhonov regularization (discrete Laplacian Prior).
The best and worst performing methods are highlighted in bold and italic respectively for each metric.
Fig 8.
TFM-UQ captures variability associated with microscopy image quality.
(A) Brightfield image of C3H/10T1/2 cell cultured on fibronectin micropatterned island of length (Red dashed outline). (B) “Raw” wide-field fluorescent bead image (beads of size
). (C) “BGS” Background subtracted image processed from (B). (D, G, J)
(arrows) overlaid on PIV-UQ uncertainty field (
) corresponding to raw WL-128, BGS WL-128 and BGS WL-64 respectively. Here, WL denotes the PIV interrogation window size WL. (E, H, K) Mean marginal posterior traction stress field (
) corresponding to Raw WL-128, BGS WL-128 and BGS WL-64 respectively. (F, I, L) Traction stress signal-to-noise ratio (
) plotted as a heatmap corresponding to Raw WL-128, BGS WL-128 and BGS WL-64 respectively. Overlaid uncertainty arrows denote the pointwise angular uncertainty corresponding to 1 circular std. dev. of marginal posterior
. Scale bar:
.
Fig 9.
TFM-UQ applied to endothelial monolayer experiment demonstrates uncertainty propagation.
(A) Membrane labeling of HUVEC monolayers with CellMask. (B) Corresponding fluorescent bead image (beads of size ). Arrows indicate bead-related image artifacts (C) PIV-UQ displacement field
(D) PIV-UQ uncertainty map,
. White regions indicate “bad” PIV windows that were deleted and replaced as described in § 2.2. Uncertainty arrows denote the pointwise angular uncertainty corresponding to 1 circular std. dev. of bootstrapped PIV-UQ distribution. (E) Inferred mean marginal posterior traction stress,
(F) Marginal posterior traction stress uncertainty field (
). Uncertainty arrows denote the pointwise angular uncertainty corresponding to 1 circular std. dev. of marginal posterior
. Scale bar :