# Recursive Least Squares Filtering Algorithms for On-Line Viscoelastic Characterization of Biosamples

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## Abstract

**:**

## 1. Introduction

## 2. The Experimental Device

## 3. The Mathematical Model

- l is the common length of the two links representing the arms, i.e., the distances $AB$ and $CD$;
- d is the distance between the hinges ($AD$), i.e., the frame length;
- ${k}_{2}$, ${k}_{4}$ and k are the torsional stiffness of the two arms hinges and the stiffness of the sample, respectively;
- ${c}_{2}$, ${c}_{4}$ and c are the viscous damping coefficients of the two arms and of the sample, respectively;
- ${I}_{2}$ and ${I}_{4}$ are the moments of inertia of the left and right arms around A and D, respectively;
- ${\tau}_{2}$ and ${\tau}_{4}$ are the input torques generated by the left and right comb drives, respectively.

## 4. Mechanical Characteristics Estimation

#### 4.1. Forgetting Factor Based RLS

**Remark**

**1.**

#### 4.2. Normalised Gradient Based RLS

## 5. Simulations

^{®}(MathWorks, Inc., Natick, MA, USA) and Simulink

^{®}(MathWorks, Inc., Natick, MA, USA) tools, were performed in order to show effectiveness, benefits and differences of the proposed estimation methods. Three different numerical cases were analysed, considering predominant elastic or dissipative behaviours.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Backman, D.E.; LeSavage, B.L.; Shah, S.B.; Wong, J.Y. A Robust Method to Generate Mechanically Anisotropic Vascular Smooth Muscle Cell Sheets for Vascular Tissue Engineering. Macromol. Biosci.
**2017**, 17. [Google Scholar] [CrossRef] [PubMed] - Mijailovic, A.S.; Qing, B.; Fortunato, D.; Van Vliet, K.J. Characterizing viscoelastic mechanical properties of highly compliant polymers and biological tissues using impact indentation. Acta Biomater.
**2018**, 71, 388–397. [Google Scholar] [CrossRef] [PubMed] - Qasaimeh, M.; Sokhanvar, S.; Dargahi, J.; Kahrizi, M. A micro-tactile sensor for in situ tissue characterization in minimally invasive surgery. Biomed. Microdevices
**2008**, 10, 823–837. [Google Scholar] [CrossRef] [PubMed] - Zhang, W.; Liu, L.F.; Xiong, Y.J.; Liu, Y.F.; Yu, S.B.; Wu, C.W.; Guo, W. Effect of in vitro storage duration on measured mechanical properties of brain tissue. Sci. Rep.
**2018**, 8, 1247. [Google Scholar] [CrossRef] [PubMed] - Laksari, K.; Assari, S.; Seibold, B.; Sadeghipour, K.; Darvish, K. Computational simulation of the mechanical response of brain tissue under blast loading. Biomech. Model. Mechanobiol.
**2015**, 14, 459–472. [Google Scholar] [CrossRef] [PubMed] - Vivanco, J.; Aiyangar, A.; Araneda, A.; Ploeg, H.L. Mechanical characterization of injection-molded macro porous bioceramic bone scaffolds. J. Mech. Behav. Biomed. Mater.
**2012**, 9, 137–152. [Google Scholar] [CrossRef] [PubMed] - Prasadh, S.; Wong, R.C.W. Unraveling the mechanical strength of biomaterials used as a bone scaffold in oral and maxillofacial defects. Oral Sci. Int.
**2018**, 15, 48–55. [Google Scholar] [CrossRef] - Edsberg, L.E.; Cutway, R.; Anain, S.; Natiella, J.R. Microstructural and mechanical characterization of human tissue at and adjacent to pressure ulcers. J. Rehabil. Res. Dev.
**2000**, 37, 463–471. [Google Scholar] [PubMed] - Hsu, C.K.; Lin, H.H.; Hans, I.; Harn, C.; Hughes, M.; Tang, M.J.; Yang, C.C. Mechanical forces in skin disorders. J. Dermatol. Sci.
**2018**, 90, 232–240. [Google Scholar] [CrossRef] [PubMed] - Whitford, C.; Movchan, N.V.; Studer, H.; Elsheikh, A. A viscoelastic anisotropic hyperelastic constitutive model of the human cornea. Biomech. Model. Mechanobiol.
**2018**, 17, 19–29. [Google Scholar] [CrossRef] [PubMed] - Erath, B.D.; Zañartu, M.; Peterson, S.D. Modeling viscous dissipation during vocal fold contact: the influence of tissue viscosity and thickness with implications for hydration. Biomech. Model. Mechanobiol.
**2017**, 16, 947–960. [Google Scholar] [CrossRef] [PubMed] - Garcés-Schröder, M.; Metz, D.; Hecht, L.; Iyer, R.; Leester-Schädel, M.; Böl, M.; Dietzel, A. Characterization of skeletal muscle passive mechanical properties by novel micro-force sensor and tissue micro-dissection by femtosecond laser ablation. Microelectron. Eng.
**2018**, 192, 70–76. [Google Scholar] [CrossRef] - Huveneers, S.; Daemen, M.J.; Hordijk, P.L. Between Rho (k) and a hard place: the relation between vessel wall stiffness, endothelial contractility, and cardiovascular disease. Circ. Res.
**2015**, 116, 895–908. [Google Scholar] [CrossRef] [PubMed] - Pogoda, K.; Chin, L.; Georges, P.C.; Byfield, F.J.; Bucki, R.; Kim, R.; Weaver, M.; Wells, R.G.; Marcinkiewicz, C.; Janmey, P.A. Compression stiffening of brain and its effect on mechanosensing by glioma cells. New J. Phys.
**2014**, 16, 075002. [Google Scholar] [CrossRef] [PubMed][Green Version] - Hu, S.; Yang, C.; Hu, D.; Lam, R.H. Microfluidic biosensing of viscoelastic properties of normal and cancerous human breast cells. In Proceedings of the 2017 IEEE 12th International Conference on Nano/Micro Engineered and Molecular Systems (NEMS), Los Angeles, CA, USA, 9–12 April 2017; pp. 90–95. [Google Scholar]
- Zouaoui, J.; Trunfio-Sfarghiu, A.; Brizuela, L.; Piednoir, A.; Maniti, O.; Munteanu, B.; Mebarek, S.; Girard-Egrot, A.; Landoulsi, A.; Granjon, T. Multi-scale mechanical characterization of prostate cancer cell lines: Relevant biological markers to evaluate the cell metastatic potential. Biochim. Biophys. Acta Gen. Subj.
**2017**, 1861, 3109–3119. [Google Scholar] [CrossRef] [PubMed] - Rubiano, A.; Delitto, D.; Han, S.; Gerber, M.; Galitz, C.; Trevino, J.; Thomas, R.M.; Hughes, S.J.; Simmons, C.S. Viscoelastic properties of human pancreatic tumors and in vitro constructs to mimic mechanical properties. Acta Biomater.
**2018**, 67, 331–340. [Google Scholar] [CrossRef] [PubMed] - Kauer, M.; Vuskovic, V.; Dual, J.; Székely, G.; Bajka, M. Inverse finite element characterization of soft tissues. Med. Image Anal.
**2002**, 6, 275–287. [Google Scholar] [CrossRef] - Dargahi, J.; Najarian, S. Advances in tactile sensors design/manufacturing and its impact on robotics applications—A review. Ind. Robot
**2005**, 32, 268–281. [Google Scholar] [CrossRef] - Dargahi, J.; Najarian, S.; Mirjalili, V.; Liu, B. Modelling and testing of a sensor capable of determining the stiffness of biological tissues. Can. J. Electr. Comput. Eng.
**2007**, 32, 45–51. [Google Scholar] [CrossRef] - Addae-Mensah, K.A.; Wikswo, J.P. Measurement techniques for cellular biomechanics in vitro. Exp. Biol. Med.
**2008**, 233, 792–809. [Google Scholar] [CrossRef] [PubMed] - Rodriguez, M.L.; McGarry, P.J.; Sniadecki, N.J. Review on cell mechanics: Experimental and modeling approaches. Appl. Mech. Rev.
**2013**, 65, 060801. [Google Scholar] [CrossRef] - Chen, K.; Yao, A.; Zheng, E.E.; Lin, J.; Zheng, Y. Shear wave dispersion ultrasound vibrometry based on a different mechanical model for soft tissue characterization. J. Ultrasound Med.
**2012**, 31, 2001–2011. [Google Scholar] [CrossRef] [PubMed] - Lim, C.; Zhou, E.; Quek, S. Mechanical models for living cells—A review. J. Biomech.
**2006**, 39, 195–216. [Google Scholar] [CrossRef] [PubMed] - Johnson, M.L.; Faunt, L.M. Parameter estimation by least-squares methods. In Methods in Enzymology; Elsevier: Amsterdam, The Netherlands, 1992; Volume 210, pp. 1–37. [Google Scholar]
- Xi, J.; Lamata, P.; Lee, J.; Moireau, P.; Chapelle, D.; Smith, N. Myocardial transversely isotropic material parameter estimation from in-silico measurements based on a reduced-order unscented Kalman filter. J. Mech. Behav. Biomed. Mater.
**2011**, 4, 1090–1102. [Google Scholar] [CrossRef] [PubMed] - Boonvisut, P.; Çavuşoǧlu, M.C. Estimation of soft tissue mechanical parameters from robotic manipulation data. IEEE/ASME Trans. Mechatron.
**2013**, 18, 1602–1611. [Google Scholar] [CrossRef] [PubMed] - Yang, S.; Xu, Q.; Nan, Z. Design and development of a dual-axis force sensing MEMS microgripper. J. Mech. Robot.
**2017**, 9, 061011. [Google Scholar] [CrossRef] - Verotti, M.; Dochshanov, A.; Belfiore, N.P. A comprehensive survey on microgrippers design: Mechanical structure. J. Mech. Des.
**2017**, 139, 060801. [Google Scholar] [CrossRef] - Dochshanov, A.; Verotti, M.; Belfiore, N.P. A comprehensive survey on microgrippers design: Operational strategy. J. Mech. Des.
**2017**, 139, 070801. [Google Scholar] [CrossRef] - Cauchi, M.; Grech, I.; Mallia, B.; Mollicone, P.; Sammut, N. Analytical, Numerical and Experimental Study of a Horizontal Electrothermal MEMS Microgripper for the Deformability Characterisation of Human Red Blood Cells. Micromachines
**2018**, 9, 108. [Google Scholar] [CrossRef] - Velosa-Moncada, L.A.; Aguilera-Cortés, L.A.; González-Palacios, M.A.; Raskin, J.P.; Herrera-May, A.L. Design of a Novel MEMS Microgripper with Rotatory Electrostatic Comb-Drive Actuators for Biomedical Applications. Sensors
**2018**, 18, 1664. [Google Scholar] [CrossRef] [PubMed] - Potrich, C.; Lunelli, L.; Bagolini, A.; Bellutti, P.; Pederzolli, C.; Verotti, M.; Belfiore, N.P. Innovative Silicon Microgrippers for Biomedical Applications: Design, Mechanical Simulation and Evaluation of Protein Fouling. Actuators
**2018**, 7, 12. [Google Scholar] [CrossRef] - Di Giamberardino, P.; Bagolini, A.; Bellutti, P.; Rudas, I.J.; Verotti, M.; Botta, F.; Belfiore, N.P. New MEMS Tweezers for the Viscoelastic Characterization of Soft Materials at the Microscale. Micromachines
**2018**, 9, 15. [Google Scholar] [CrossRef] - Verotti, M.; Crescenzi, R.; Balucani, M.; Belfiore, N.P. MEMS-based conjugate surfaces flexure hinge. J. Mech. Des.
**2015**, 137, 012301. [Google Scholar] [CrossRef] - Verotti, M.; Dochshanov, A.; Belfiore, N.P. Compliance synthesis of CSFH MEMS-based microgrippers. J. Mech. Des.
**2017**, 139, 022301. [Google Scholar] [CrossRef] - Cecchi, R.; Verotti, M.; Capata, R.; Dochshanov, A.; Broggiato, G.B.; Crescenzi, R.; Balucani, M.; Natali, S.; Razzano, G.; Lucchese, F.; et al. Development of micro-grippers for tissue and cell manipulation with direct morphological comparison. Micromachines
**2015**, 6, 1710–1728. [Google Scholar] [CrossRef] - Bagolini, A.; Ronchin, S.; Bellutti, P.; Chiste, M.; Verotti, M.; Belfiore, N. Fabrication of novel MEMS microgrippers by deep reactive ion etching with metal hard mask. IEEE J. Microelectromech. Syst.
**2017**, 26, 926–934. [Google Scholar] [CrossRef] - Bagolini, A.; Bellutti, P.; Di Giamberardino, P.; Rudas, I.J.; D’Andrea, V.; Verotti, M.; Dochshanov, A.; Belfiore, N. Stiffness characterization of biological tissues by means of MEMS-technology based micro grippers under position control. Mech. Mach. Sci.
**2018**, 49, 939–947. [Google Scholar] - Verotti, M. Analysis of the center of rotation in primitive flexures: Uniform cantilever beams with constant curvature. Mech. Mach. Theory
**2016**, 97, 29–50. [Google Scholar] [CrossRef] - Verotti, M. Effect of initial curvature in uniform flexures on position accuracy. Mech. Mach. Theory
**2018**, 119, 106–118. [Google Scholar] [CrossRef] - Sanò, P.; Verotti, M.; Bosetti, P.; Belfiore, N.P. Kinematic Synthesis of a D-Drive MEMS Device with Rigid-Body Replacement Method. J. Mech. Des.
**2018**, 140, 075001. [Google Scholar] [CrossRef] - Flacco, F.; De Luca, A.; Sardellitti, I.; Tsagarakis, N.G. Robust estimation of variable stiffness in flexible joints. In Proceedings of the 2011 IEEE/RSJ International Conference on Intelligent Robots and Systems, San Francisco, CA, USA, 25–30 September 2011. [Google Scholar]
- Lundquist, C.; Schön, T.B. Recursive Identification of Cornering Stiffness Parameters for an Enhanced Single Track Model. IFAC Proc. Vol.
**2009**, 42, 1726–1731. [Google Scholar] [CrossRef][Green Version] - Vahidi, A.; Stefanopoulou, A.; Peng, H. Recursive least squares with forgetting for online estimation of vehicle mass and road grade: Theory and experiments. Veh. Syst. Dyn.
**2005**, 43, 31–55. [Google Scholar] [CrossRef] - Lee, S.D.; Jung, S. A recursive least square approach to a disturbance observer design for balancing control of a single-wheel robot system. In Proceedings of the 2016 IEEE International Conference on Information and Automation (ICIA), Ningbo, China, 31 July–4 August 2016; pp. 1878–1881. [Google Scholar]
- Ljung, L. System Identification: Theory for the User, 2nd ed.; Prentice Hall: Upper Saddle River, NJ, USA, 1999. [Google Scholar]

**Figure 1.**Optical microscope images of the device. The whole microgripper (

**a**) and two overlapping frames showing the left arm in neutral (0 V) and actuated (28 V) configurations (

**b**).

**Figure 2.**Schematic representations of the microgripper: compliant mechanism (

**a**) and corresponding pseudo-rigid body model (

**b**).

**Figure 4.**Time evolution of the estimated parameters k (

**a**) and c (

**b**) for the first case: $c=8.4\times {10}^{-6}$ Nms/rad, $k=2.5\times {10}^{-3}$ Nm/rad.

**Figure 5.**Time evolution of the estimated parameters c (

**a**) and k (

**b**) for the second case: $c=8.4\times {10}^{-3}$ Nms/rad, $k=2.5\times {10}^{-6}$ Nm/rad.

**Figure 6.**Time evolution of the estimated parameters k (

**a**) and c (

**b**) for the third case: $c=8.4\times {10}^{-11}$ Nms/rad, $k=2.5\times {10}^{-5}$ Nm/rad.

Parameter | Value | Unit |
---|---|---|

${\widehat{\theta}}_{2}$ | $1.44$ | rad |

${\widehat{\theta}}_{4}$ | $1.70$ | rad |

$\widehat{u}$ | $150\times {10}^{-6}$ | m |

Parameter | Value | Unit |
---|---|---|

d | $5.47\times {10}^{-4}$ | m |

l | $1.50\times {10}^{-3}$ | m |

${I}_{2}$, ${I}_{4}$ | $1.25\times {10}^{-14}$ | kg m^{2} |

${k}_{2}$, ${k}_{4}$ | $0.30\times {10}^{-6}$ | N m^{2} |

${c}_{2}$, ${c}_{4}$ | $1.24\times {10}^{-12}$ | N m^{2} s^{−1} |

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**MDPI and ACS Style**

Di Giamberardino, P.; Aceto, M.L.; Giannini, O.; Verotti, M. Recursive Least Squares Filtering Algorithms for On-Line Viscoelastic Characterization of Biosamples. *Actuators* **2018**, *7*, 74.
https://doi.org/10.3390/act7040074

**AMA Style**

Di Giamberardino P, Aceto ML, Giannini O, Verotti M. Recursive Least Squares Filtering Algorithms for On-Line Viscoelastic Characterization of Biosamples. *Actuators*. 2018; 7(4):74.
https://doi.org/10.3390/act7040074

**Chicago/Turabian Style**

Di Giamberardino, Paolo, Maria Laura Aceto, Oliviero Giannini, and Matteo Verotti. 2018. "Recursive Least Squares Filtering Algorithms for On-Line Viscoelastic Characterization of Biosamples" *Actuators* 7, no. 4: 74.
https://doi.org/10.3390/act7040074