Computational Associate I Broad Institute of MIT and Harvard, United States
Introduction: AFM has served as a powerful tool for researchers to quantitatively characterize cancer tissue mechanics. Overall, studies have demonstrated distinct mechanical characteristics for healthy and cancerous tissue samples [1,2,3].
Importantly, the lateral resolution of AFM is on the order of just a few nanometers, resulting in highly precise measurements across the surface of a tissue sample. Nonetheless, AFM is a costly and technically-sophisticated method; moreover, AFM does not allow for the characterization of tissue in situ. These limitations prevent AFM from being a viable technique for clinical use. Additionally, tissue mechanics is contingent on the tissue’s environment—for example, blood circulation and perfusion affects tissue stiffness [4]—so measuring stiffness in situ is essential to accurately assess the mechanical properties of cancerous lesions.
To this end, devices have been developed for measuring tissue elasticity in patients [5]. Handheld devices for clinical use are unlikely to collect measurements with as high a lateral resolution as AFM; however, signal processing techniques may allow for the interpolation of lower-resolution device readings to generate elastic modulus “maps” with improved resolution.
The aim of this work was to understand whether elastic modulus “maps” obtained via AFM can be down-sampled and accurately reconstructed through various interpolation methods. Elastic moduli were obtained from a public dataset of AFM data, corresponding to measurements taken for both healthy and tumor tissue. The findings presented draw preliminary conclusions about whether signal processing techniques—in the time or frequency domains—can be applied to data obtained from lower-resolution devices.
Materials and
Methods: The analyzed dataset, made available by the EPSRC Centre for Interventional and Surgical Sciences at the University College London, includes samples from seven patients with metastatic pancreatic or colorectal cancer. 10-20 sites across each tissue sample were measured using AFM to obtain 8x8 maps of Young’s modulus values [6].
AFM maps were visualized as grayscale heatmaps. Maps were down-sampled and then interpolated using the following methods: ideal, Fourier domain, nearest neighbor, linear, or cubic interpolation (Figure 1). Sampled maps were padded with edge values to allow for interpolation, and cubic interpolation was only applied to 4x4 sampled maps. Ideal interpolation was manually implemented using the scheme in Figure 2.
To evaluate the accuracy of reconstruction with interpolation, the normalized root mean squared error (NRMSE) between each original map and the estimated map was computed (Table 1). To evaluate the extent to which the overall distribution of values in the true AFM map is retained after interpolation, histograms of the measurements in original and reconstructed maps were assessed. The RMSE between the respective density curves (RMSE_density) was computed (Table 1).
Both metrics for reconstruction error were modeled as a linear combination of the experimental covariates within the data and image processing parameters (xi) applied. The coefficients of the model (βi) were assessed to evaluate the influence of each variable upon reconstruction accuracy (Table 2).
Packages used: Python (Version 3.12.1) NumPy, SciPy, and Matplotlib; R (Version 4.1.1) stats and ggplot2
Results, Conclusions, and Discussions: Heterogeneity within AFM maps presented a challenge in interpolating down-sampled maps and highlighted a significant barrier to mathematical recovery of high resolution tissue mechanics data. Among attempted interpolation methods, nearest neighbor interpolation was most unsuccessful, demonstrating the need for interpolation in the first place. Fourier interpolation appeared more inaccurate than the ideal method at boundaries of reconstructed maps because the original signal was treated as periodic, which is inconsistent with trends that display heterogeneity across soft tissue. However, trends in reconstruction error indicated that Fourier interpolation performed better than cubic and ideal methods, possibly because the incorporation of random high frequencies during reconstruction reduces error metrics (Table 1). According to the results of the linear models fitted to predict NRMSE and RMSE_density, linear interpolation performed best according to the former and worst according to the latter. This is reasonable because this method best retains the range of measurements after interpolation while significantly altering (smoothening) the distribution. Ideal interpolation performed second worst and proved less robust to increases in sampling rate and prefiltering, likely due to high frequencies. Error distributions were skewed left, indicating that outliers were reconstructed less accurately than the majority of samples (Figure 3).
Results demonstrated using a larger sampling rate and prefiltering with an averaging filter reduced reconstruction accuracy significantly. The former indicates, naturally, that lateral resolution of tissue mechanical measurement devices should be maximized to optimize the accuracy of measurement maps that can be interpolated from the raw data produced. The latter is expected considering the high frequencies within the data and suggests it is better for lower resolution instruments to minimize the size of the discrete points at which they measure tissue mechanical properties, rather than measure the bulk properties of areas that are set closer together.
Looking ahead, it will be necessary to obtain larger AFM maps; perform down-sampling and interpolation for more sampling rates; and assess the NRMSE/RMSE_density as a function of various sampling rates (ex. with an elbow plot of RMSE vs. sampling rate). These results can help understand the minimum distance between stiffness measurements required for clinical devices.
Acknowledgements (Optional): 1. Brás MM, Sousa A, Cruz TB, et al. Microrheological comparison of melanoma cells by atomic force microscopy. J Biol Phys. 2024;50(1):55-69. doi:10.1007/s10867-023-09648-w 2. Lekka M. Discrimination Between Normal and Cancerous Cells Using AFM. Bionanoscience. 2016;6:65-80. doi:10.1007/s12668-016-0191-3 3. Prasad S, Rankine A, Prasad T, et al. Atomic Force Microscopy Detects the Difference in Cancer Cells of Different Neoplastic Aggressiveness via Machine Learning. Adv. NanoBiomed Res. 2021;1:2000116. doi:10.1002/anbr.202000116 4. Wei JC, Cartmill ID, Kendall MA, Crichton ML. In vivo, in situ and ex vivo comparison of porcine skin for microprojection array penetration depth, delivery efficiency and elastic modulus assessment. J Mech Behav Biomed Mater. 2022;130:105187. doi:10.1016/j.jmbbm.2022.105187 5. Bartsch K, Brandl A, Weber P, et al. Assessing reliability and validity of different stiffness measurement tools on a multi-layered phantom tissue model. Scientific reports. 2023;13(1):815. doi: 10.1038/s41598-023-27742-w. 6. Zajiczek LN. AFM liver tissue data. 2024. doi:10.5522/04/25020907.v1
