Applied Science, Faculty of
Mechanical Engineering, Department of
DSpace
UBCV
Helisaz, Hamed
Doctor of Philosophy - PhD
University of British Columbia
Mechanical properties of biological tissues have been shown to vary in the presence of pathological disorders. This dissertation aims to investigate the effect of cancer on the viscoelastic properties of prostate gland. We use the quasi-linear viscoelastic model to characterize the elastic and viscous characteristics of thirty-five fresh prostate glands, removed within two hours of radical prostatectomy surgery. The viscoelastic properties of normal and cancerous prostates are compared to reveal the influence of cancer grade and tumor volume on the elasticity and viscosity of prostate tissue. We first present a novel method for describing the time-dependent behavior of soft materials using one-dimensional quasi-linear viscoelastic model. Our model derives the elastic (Young’s modulus) and viscous (shear relaxation modulus) properties by analyzing the stress response after a sudden uniaxial compression. The model is validated using experimental data on the phantoms that mimic the elasticity of normal and cancerous prostate tissues. The applicability of the model for material characterization is demonstrated by introducing a procedure for obtaining the mechanical properties via indentation test. Using three-dimensional equations and pre-calculated finite element analysis, the procedure introduces a fast and straightforward approach that is superior to inverse finite element methods. We evaluate our procedure by comparing the derived properties with those obtained from uniaxial compression test, where the results show high precision (standard deviation less than 10%) and good accuracy (error less than 20%). Finally, the procedure introduced for material characterization is used to characterize the viscoelastic properties of prostate gland. We test four different locations on fresh, unfixed prostate samples and compare the properties of locations where cancer spreads with those where cancer is not present. Our data-driven results demonstrate that tumor volume increases elastic and viscous properties of prostate with high statistical significance (p-value < 2%), which reflects the potential of using the mechanical properties of prostate as a cancer biomarker. However, the author admits the challenges for realizing an in vivo apparatus to measure the viscoelastic properties of prostate. In fact, in vivo application of the findings requires other considerations that are not discussed here.
An Investigation of Cancer Effect on ViscoelasticProperties of Prostate Gland via Quasi-Linear ViscoelasticModelbyHamed HelisazB.Sc., Sharif University of Technology, 2014M.Sc., Sharif University of Technology, 2016A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Mechanical Engineering)The University of British Columbia(Vancouver)September 2022© Hamed Helisaz, 2022The following individuals certify that they have read, and recommend to the Fac-ulty of Graduate and Postdoctoral Studies for acceptance, the thesis entitled:An Investigation of Cancer Effect on Viscoelastic Properties of ProstateGland via Quasi-Linear Viscoelastic Modelsubmitted by Hamed Helisaz in partial fulfillment of the requirements for the de-gree of Doctor of Philosophy in Mechanical Engineering.Examining Committee:Mu Chiao, Professor, Mechanical Engineering, UBCSupervisorMattia Bacca, Assistant Professor, Mechanical Engineering, UBCSupervisory Committee MemberAbbas Samani, Professor, Electrical and Computer Engineering, Western Univer-sityExternal ExaminerPurang Abolmaesumi, Professor, Electrical and Computer Engineering, UBCUniversity ExaminerMauricio Ponga de la Torre, Assistant Professor, Mechanical Engineering, UBCUniversity ExaminerAdditional Supervisory Committee Members:Boris Stoeber, Professor, Electrical and Computer Engineering and MechanicalEngineering, UBCSupervisory Committee MemberClarence de Silva, Professor, Mechanical Engineering, UBCSupervisory Committee MemberRyozo Nagamune, Professor, Mechanical Engineering, UBCSupervisory Committee MemberiiAbstractMechanical properties of biological tissues have been shown to vary in the pres-ence of pathological disorders. This dissertation aims to investigate the effect ofcancer on the viscoelastic properties of prostate gland. We use the quasi-linear vis-coelastic model to characterize the elastic and viscous characteristics of thirty-fivefresh prostate glands, removed within two hours of radical prostatectomy surgery.The viscoelastic properties of normal and cancerous prostates are compared to re-veal the influence of cancer grade and tumor volume on the elasticity and viscosityof prostate tissue.We first present a novel method for describing the time-dependent behavior ofsoft materials using one-dimensional quasi-linear viscoelastic model. Our modelderives the elastic (Young’s modulus) and viscous (shear relaxation modulus) prop-erties by analyzing the stress response after a sudden uniaxial compression. Themodel is validated using experimental data on the phantoms that mimic the elastic-ity of normal and cancerous prostate tissues.The applicability of the model for material characterization is demonstratedby introducing a procedure for obtaining the mechanical properties via indentationtest. Using three-dimensional equations and pre-calculated finite element analysis,the procedure introduces a fast and straightforward approach that is superior to in-verse finite element methods. We evaluate our procedure by comparing the derivedproperties with those obtained from uniaxial compression test, where the resultsshow high precision (standard deviation less than 10%) and good accuracy (errorless than 20%).Finally, the procedure introduced for material characterization is used to char-acterize the viscoelastic properties of prostate gland. We test four different loca-iiitions on fresh, unfixed prostate samples and compare the properties of locationswhere cancer spreads with those where cancer is not present. Our data-driven re-sults demonstrate that tumor volume increases elastic and viscous properties ofprostate with high statistical significance (p-value< 2%), which reflects the poten-tial of using the mechanical properties of prostate as a cancer biomarker. However,the author admits the challenges for realizing an in vivo apparatus to measure theviscoelastic properties of prostate. In fact, in vivo application of the findings re-quires other considerations that are not discussed here.ivLay SummaryProstate cancer is the second most common type of cancer in the World and one ofthe leading causes of cancer deaths among men. When diagnosed at early stages,prostate cancer can be cured with minimal side effects. One of the primary meth-ods for detecting prostate cancer is monitoring the viscoelastic properties of theprostate, as cancer affects the elastic and viscous properties of the tissue. This dis-sertation introduces a model for finding the viscoelastic properties of the prostateand investigates the influence of cancer on prostate properties.First, we introduce a model to describe prostate tissue in one dimension andthen present a method to extend the model to three-dimension. Finally, we useour model to study the behavior of normal and cancerous prostate tissue and com-pare the findings to understand the effect of cancer. Our results show that cancerincreases the stiffness and viscosity of prostate tissue.vPrefaceThis dissertation entitled ”An Investigation of Cancer Effect on Viscoelastic Prop-erties of Prostate Gland via Quasi-Linear Viscoelastic Model” describes the re-search projects that I carried out during my PhD. The research projects that in-troduced in this dissertation were mainly designed and supervised by Mu Chiao(Supervisor). In this preface, the contributions and collaborations to the projectsand published papers are briefly explained.A version of Chapter 2 has been published as: ”H. Helisaz, M. Bacca, andM. Chiao, “Quasi-Linear Viscoelastic Characterization of Soft Tissue-MimickingMaterials,” ASME J. Biomech. Eng., vol. 143, no. 6, Jun. 2021, doi: 10.1115/1.4050036.” This research project was designed by Hamed Helisaz, Mattia Bacca,and Mu Chiao. Hamed Helisaz (author) derived the model formulation, and con-ducted numerical simulations and data collection. Mu Chiao and Mattia Baccasupervised the content and formula and contributed to the editing of the paper.A version of Chapter 3 has been published as: ”H. Helisaz, M. Bacca, andM. Chiao, “A New Characterization Procedure for Quasi-Linear Viscoelastic Ma-terials Using Indentation Test: Validation with Finite Element and ExperimentalResults,” Exp. Mech., no. 0123456789, 2022, doi: 10.1007/s11340-022-00837-7.” This research project was designed by Hamed Helisaz, Mattia Bacca, and MuChiao. Hamed Helisaz (author) conducted the finite element analysis, designedexperiments, and derived the results. Mattia Bacca supervised the formula and MuChiao and Mattia Bacca supervised content and contributed to the editing of thepaper.Hamed Helisaz, Mu Chiao, Peter Black, Eric Belanger designed the researchproject in chapter 4. Hamed Helisaz designed and coordinated the clinical study,viand performed the data and statistical analysis on the results. Ethical and opera-tional approval for the clinical study was obtained from the Research Ethics Boardat the University of British Columbia (CREB Number: H19-03592) and the Van-couver Coastal Health Authority (VCH Number: V19-03592). Peter Black per-formed the surgeries for removing prostate glands, Eric Belanger conducted his-tology analysis on the prostate glands, and Mu Chiao supervised the process. Thesensor design and fabrication in the last section was also performed with significantcontribution from Anoush Sepehri, who also provided the data for this section.viiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiNomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxviiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Measurement of Tissue Viscoelasticity . . . . . . . . . . . . . . . 21.1.1 Tissue Actuation . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Response Detection . . . . . . . . . . . . . . . . . . . . . 21.1.3 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Material Characterization with the Quasi-linear Viscoelastic Model 91.3 Effect of Cancer on Mechanical Properties of Prostate Tissue . . . 111.4 Dissertation Organization . . . . . . . . . . . . . . . . . . . . . . 13viii2 Quasi-linear Viscoelastic Calibration of Soft Materials via UniaxialTest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.1 Methods and Materials . . . . . . . . . . . . . . . . . . . . . . . 162.1.1 QLV Model in One Dimension . . . . . . . . . . . . . . . 162.1.2 Data-Fitting Algorithm . . . . . . . . . . . . . . . . . . . 192.1.3 Tissue-Mimicking Phantoms . . . . . . . . . . . . . . . . 212.1.4 Method Validation and Confidence Interval . . . . . . . . 222.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . 252.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 Quasi-linear Viscoelastic Characterization of Soft Materials via In-dentation Test and Finite Element Analysis . . . . . . . . . . . . . . 313.1 Methods and Materials . . . . . . . . . . . . . . . . . . . . . . . 323.1.1 Three-dimensional QLV Constitutive Equation . . . . . . 323.1.2 Finite Element Implementation . . . . . . . . . . . . . . 353.1.3 Model Validation . . . . . . . . . . . . . . . . . . . . . . 363.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . 393.2.1 The Elastic Part of Indentation Behavior . . . . . . . . . . 393.2.2 The Viscous Part of Indentation Behavior . . . . . . . . . 413.3 The New Material Characterization Procedure . . . . . . . . . . . 463.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524 Effect of Cancer on the Viscoelastic Properties of Prostate Tissue . . 564.1 Experimental Design for Ex Vivo Testing . . . . . . . . . . . . . 574.1.1 Model Validation with Prostate-mimicking Phantoms . . . 574.1.2 Experiment with Fresh Prostate Tissue . . . . . . . . . . . 604.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . 624.2.1 Application of the QLV model to Prostate Tissue . . . . . 624.2.2 Effect of Cancer on Mechanical Properties of Prostate . . 644.2.3 Effect of Cancer Grade on the Prostate Gland . . . . . . . 704.3 Application in Prostate Cancer Diagnosis . . . . . . . . . . . . . 744.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . 78ix5.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785.1.1 Calibrating the QLV Model via the Uniaxial Test . . . . . 785.1.2 QLV Characterization via the Indentation Test . . . . . . . 795.1.3 Effect of Cancer on Prostate Viscoelasticity . . . . . . . . 805.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83A A Possible Approach for In Vivo Diagnosis of Prostate Cancer . . . . 96A.1 Sensor Design and Fabrication . . . . . . . . . . . . . . . . . . . 96A.2 Validation on Prostate-mimicking Phantoms . . . . . . . . . . . . 98B Prostate Glands for Clinical Study . . . . . . . . . . . . . . . . . . . 103xList of TablesTable 2.1 The composition of tissue-mimicking phantoms . . . . . . . . 22Table 2.2 The QLV parameters for tissue-mimicking phantoms; 95% con-fidence intervals are reported . . . . . . . . . . . . . . . . . . 23Table 2.3 95% confidence intervals for relaxation parameters of ElastomerIII under different strain levels (see Fig. 2.3) . . . . . . . . . . 25Table 3.1 The QLV parameters of the fabricated silicon-based polymer;the parameters were obtained from uniaxial compression tests(see Chapter 2) and were used as the input for the ABAQUS®simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Table 3.2 Comparing the relaxation parameters obtained from uniaxialcompression and indentation: the parameters are first obtainedfrom the compression test using the method described in Sec-tion 2.1 and compared with those calculated from an indenta-tion test using our proposed methods (Eqs. (3.12) and (3.13)).Compression and indentation tests were repeated three times foreach material and the mean values are reported along with thestandard deviations. The table demonstrates that Eq. (3.13) in-troduces parameters with a high precision (standard deviationwithin 10%) and in a good agreement with uniaxial parameters(maximum error within 20%). . . . . . . . . . . . . . . . . . . 52xiTable 4.1 95% confidence intervals for the ratio between the propertiesof the cancerous and normal sides of the posterior prostate.As seen in Fig. 4.10, the properties for the cancerous side aregreater than those for the normal side. . . . . . . . . . . . . . . 69Table 4.2 Descriptive and inferential statistics for the cancer effect onthe viscoelastic properties of the prostate gland when cancerpresents in the posterior prostate. The table also includes p-values for the alternative hypothesis with gcanceri /gnormali > 1and σ cancerE /σnormalE > 1. From the table, cancer appears to havea greater statistically significant effect on g1 and σE (lower p-values). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75Table 4.3 95% confidence interval for non-cancerous prostate samples.Since cancer is absent from the prostate sections, the proper-ties for the left and right sides are expected to be similar. Thetable shows that g1 and g3 are relatively biased towards higherratios and g2 and E are balanced around a ratio of one. . . . . 76Table B.1 Characteristics of fresh prostate samples used in the clinicalstudy: Thirty-five fresh prostate samples, identified as P1-P35were collected within two hours after removal by prostatectomysurgery. The table includes the location(s) of the cancer in midand apical sections, cancer grade, Gleason score, prostate vol-ume, prostate mass, and the patient’s age. The results for sam-ples P1-P3 are used in Section 3.1 to check the suitability of theQLV model for the prostate gland, and the results for samplesP4 - P35 are used in Sections 3.1 and 3.2 for studying the effectof cancer on the properties of the prostate gland. Some of theprostate characteristics were not available and appear empty inthe table. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104xiiList of FiguresFigure 1.1 Comparison of the actual mechanical response of the rabbitmesentery (solid line) with the corresponding incremental lines(dashed lines). Reproduced with permission from [1]. . . . . 5Figure 1.2 Time-dependent characteristic of tissue behavior: a) changein loading/unloading behavior of an anterior cruciate ligamentthrough time; b) pre-conditioning of a dog’s carotid artery. Re-produced with permission from [1]. . . . . . . . . . . . . . . 7Figure 1.3 Different forms of spring-dashpot models; (a) Maxwell model,(b) Kelvin-Voight model, (c) Standard Linear Solid model. Re-produced with permission from [2] . . . . . . . . . . . . . . . 8Figure 2.1 Experimental procedure: (a) a series of low-range compres-sions: each is a 2.5% strain and kept long enough until thestress reaches the steady-state; (b) stress response versus timeunder the prescribed displacement at (a): elastic propertiesare obtained by plotting the steady-state stress (orange dots)against the compression; (c) ramp-and-hold compression with50% final strain: strain is kept long enough until the stressreaches the steady-state; (d) stress response versus time underthe prescribed displacement at (c): the stress amplification fac-tors g1 and g2, and g3 measure the stress relaxation in the timeintervals (τ0, τ1), (τ1, τ2), and (τ2, τ3) . . . . . . . . . . . . . 21xiiiFigure 2.2 Tissue-mimicking phantoms: (a) Elastomer I (67% Ecoflex™00-10 and 33% NOVOCS™); (b) Elastomer II (80% Ecoflex™00-30 and 20% NOVOCS™); c) Elastomer III (100% Ecoflex™00-10): Elastomer I mimics normal tissue in terms of Young’smodulus while elastomers II and III mimic cancerous tissues. . 23Figure 2.3 Model validation: a) ramp-and-hold compression with differ-ent final strains; b) stress response of Elastomer III versus timeunder the prescribed displacement at (a): lines are experimen-tal results while symbols denote predictions of the QLV model(Eq. (2.9)). The experiments are repeated three times for eachsample, and the mean values for the results are presented withthe error bars. . . . . . . . . . . . . . . . . . . . . . . . . . 24Figure 2.4 Testing machine used to obtain stress response of materials: acustom compression has slightly larger diameter than samplesto guarantee a uniaxial compression force. . . . . . . . . . . . 26Figure 2.5 Material characterization: lines are experimental results whilesymbols denote predictions of the QLV model (Eq. (2.9)).QLV parameters, including α , E, and gi, can well distinguishthe phantoms. The experiments are repeated three times foreach sample, and the mean values for the results are presentedwith the error bars. . . . . . . . . . . . . . . . . . . . . . . . 29Figure 3.1 ABAQUS® simulation: a) axisymmetric model with a deformablepart, for the soft material (shown in green), a rigid analyticalpart, for the indenter (shown in blue), and a rigid analyticalpart for the bottom support (shown in red); b) the indentationstress is expressed as von Mises stress; the indenter size ϕ andindentation depth δ are also shown in the figure. The resultsare depicted for ϕ = 7.60 mm, H = 18.88 mm, and D = 25 mm 37xivFigure 3.2 Model validation: The ABAQUS® simulation results are com-pared with those obtained from experiments on a silicon-basedpolymer. Lines denote the QLV model prediction, and thesymbols are the experimental results. The experiment includescompressing an indenter into the specimen at a rate of 10 mm/sand recording the force response (F ). The force data is thenconverted to stress (σ ) by dividing force by the indenter sur-face area (π4ϕ2). Each experiment is repeated three times, andthe mean values for the results are depicted along with the errorbars. The results are depicted for specimen height H = 18.88mm, specimen diameter D = 25 mm, and indenter diameterϕ = 7.60 mm . . . . . . . . . . . . . . . . . . . . . . . . . . 38Figure 3.3 Comparing the stress feedback with and without using WD-40silicone lubricant (WD-40 Company Inc., California, USA).The figure demonstrates no difference between the results, whichconfirms the frictionless contacts in the experiment. The sym-bols indicate the experimental results for a silicon-based poly-mer with height H=18.88 mm and diameter D = 25 mm. . . . 40Figure 3.4 Model validation for indenter diameter ϕ = 4.95 mm: the ABAQUS®simulation results are compared with those obtained from ex-periments on a silicon-based polymer. Lines denote the QLVmodel prediction, and the symbols are the experimental results.The results are depicted for specimen height H = 18.88 mmand specimen diameter D = 25mm. . . . . . . . . . . . . . . 41Figure 3.5 Model validation for indenter diameter ϕ = 6.33 mm: the ABAQUS®simulation results are compared with those obtained from ex-periments on a silicon-based polymer. Lines denote the QLVmodel prediction, and the symbols are the experimental results.The results are depicted for specimen height H = 18.88 mmand specimen diameter D = 25mm. . . . . . . . . . . . . . . 42xvFigure 3.6 Model validation for indenter diameter ϕ = 8.88 mm: the ABAQUS®simulation results are compared with those obtained from ex-periments on a silicon-based polymer. Lines denote the QLVmodel prediction, and the symbols are the experimental results.The results are depicted for specimen height H = 18.88 mmand specimen diameter D = 25mm. . . . . . . . . . . . . . . 43Figure 3.7 Effect of elastic parameters (µ and α) on indentation behavior:effect of a) shear modulus µ and b) strain-stiffen parameter αon the stress response of the material. The behavior shows apeak stress followed by stress relaxation until the results reacha steady-state. The steady-state stress occurs when the viscousproperties of a material vanish, and only the elastic propertiesremain. The results demonstrate that the indentation stress issensitive to both µ and α: it has a linear relationship with µand a non-linear relationship with α . . . . . . . . . . . . . . . 44Figure 3.8 Effect of indenter size and indentation depth on the elasticstress of the material. The results are nondimensionalized toclarify the impact of each parameter; elastic stress is nondi-mensionalized with shear modulus µ and indentation depth δwith indenter diameter ϕ . Unlike the prediction of linear elas-ticity, the curves for different indenter diameters do not inter-sect, and the data suggests a non-linear relationship betweenstress and indentation depth. The results are depicted for spec-imen height H = 18.88 mm and specimen diameter D = 25mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Figure 3.9 Nondimensionalized stress behavior for different indenter di-ameters and indentation depths: nondimensionalized stress re-sponse at indentation diameters of a) ϕ = 4.95 mm, b) ϕ =6.33 mm, c) ϕ = 7.60 mm, and d) ϕ = 8.88 mm. The resultsdemonstrate that relaxation behavior does not depend on theindentation size and depth. . . . . . . . . . . . . . . . . . . . 46xviFigure 3.10 Effect of material properties on the viscous behavior: evolutionof nondimensionalized stress relaxation over time for differentvalues of a) g1, b) g2, and c) g3: the results indicate that thestress relaxation is proportional to the value of gis. The re-laxation caused by gi = 2 is two-times that of gi = 1; d) thenondimensionalized stress relaxation associated with multiplegis; the figure shows that the stress relaxation caused by sev-eral gis is obtained by adding the stress relaxations made byeach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47Figure 3.11 Stress relaxation in different time intervals: four time inter-vals are determined by τ0, τ1, τ2, and τ3, where τ0 measuresthe indentation time and τ1, τ2, and τ3 denote the relaxationtimes. The stress relaxation in the time interval (τi−1,τi) is ob-tained by subtracting the stress at t = τi(στi) from the stress att = τi−1(στi−1). The data is nondimensionalized by the elasticstress σE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Figure 3.12 Effect of indentation rate on the viscous behavior: evolution ofnondimensionalized stress relaxation over time when a) g1 = 1,b) g2 = 1, and c) g3 = 1. The indentation rate is measured byindentation time τ0, which is measured against τi. Stress relax-ation increases as τi/τ0 increases; in particular, when τi/τ0 ≥10, the indentation rate is sufficiently fast to fully capture thestress relaxation (σ −σE/σE ≈ 1). In addition, the stress re-laxation is shown to depend on τi/τ0 and not on the individualvalues of τi and τ0. While τi differs in different plots, the samestress relaxation is observed when τi/τ0 is the same. . . . . . 49xviiFigure 3.13 The stress relaxation of Fig. 3.12 in different time intervals;the data is nondimensionalized by elastic stress, and expressedas a percentage. The coefficient gi affects the stress relaxationmainly between (τi−1,τi+1); in particular, the stress relaxationover the last time interval is only influenced by g3 since ∆στ3−∞is zero when g1 = 1 or g2 = 1. Likewise, stress relaxation in(τ2,τ3) depends on g2 and g3 (∆στ2−τ3) is zero when g1 = 1and the stress relaxation between (τ1,τ2) depends on all threerelaxation coefficients. . . . . . . . . . . . . . . . . . . . . . 50Figure 3.14 The suggested characterization procedure; a) stress relaxationfrom a ramp-and-hold indentation experiment. The figure showsstress relaxation created by b) g1 = 1, c) g2 = 1, and d) g3 =1. Note that the values for ∆σˆg1=1τ2−τ3 ,∆σˆg1=1τ3−∞, and ∆σˆg2=1τ3−∞ areall zero. Eq. (3.12) compares the stress relaxation at threepoints (t = τ1, t = τ2, and t = τ3 in figure a) with those ofpre-calculated simulation results (figures b-d) and derives thematerial’s parameters. The results are depicted for indentationtime τ0 = 0.5 s. . . . . . . . . . . . . . . . . . . . . . . . . . 51Figure 3.15 Effect of indentation rate (measured by τ0) on the stress re-laxations created by g1 = 1,g2 = 1, and g3 = 1. The resultsdemonstrate the stress relaxation between a) (τ1−τ2), b) (τ2−τ3), and c) (τ3−∞). The stress relaxation decreases as the in-dentation time (τ0) increases (slower indentation). No stressrelaxation is created by g1 after t = τ2 (∆σˆg1=1τ2−τ3 and ∆σˆg1=1τ3−∞)are zero in figures b and c) and no stress relaxation is made byg2 after t = τ3 (∆σˆg2=1τ3−∞ is zero in figure c)). Since the valuesof ∆σˆg1=1,∆σˆg2=1, and ∆σˆg3=1 are dependent on the indenta-tion rate, the values substituted in Eq. (3.12) and (3.13) shouldcorrespond to τ0 used in the experiment. . . . . . . . . . . . . 53xviiiFigure 3.16 Normalized stress for indentation tests with different indenta-tion tilt angles, defined as the angle between the indenter andcontact surface normal (θ ). The results demonstrate that tiltangle (misalignment) up to 20◦ does not affect the stress re-laxation. Note that the figure represents the experimental datasince the indentation test with 0 cannot be modelled with anaxisymmetric simulation. . . . . . . . . . . . . . . . . . . . . 54Figure 3.17 Algorithm for writing the UMAT software in Fortran; the UMATreceives the deformation gradient tensor (from ABAQUS cal-culations) and material properties (from user inputs) and pro-vides the Cauchy stress and elasticity tensor. . . . . . . . . . 55Figure 4.1 Procedure for mechanical testing: a) the sample is compressedat 10 mm/s and kept at an indentation depth (here, δ = 5mm for45 seconds), b) stress response after the compression shownin (a). Stress peaks occurred at t = τ0 and then they relaxedover time until reaching the steady-state stress σE . Relaxationshear moduli gis were determined by the stress relaxation thatoccurred in different time periods, as specified by relaxationtimes τks. Three relaxation times τ1, τ2, and τ3 were chosenthat were 0.1, 1, and 10 seconds, respectively, after t = τ0. . . 58xixFigure 4.2 Tissue-mimicking phantoms with embedded nodules to resem-ble cancerous prostate tissue: a) phantoms were fabricated byEcoflex™ 00-10 with elasticity close to the Young’s modu-lus of normal prostate gland. The anatomical model was usedto ensure that the phantom’s shape was similar to that of theposterior prostate, b) a hard nodule was embedded on the leftside of the phantom to mimic the presence of cancer. Nod-ules were made by Ecoflex™ 00-50 with a Young’s modulusclose to that of prostate cancer. Three phantoms were fabri-cated with different-sized nodules; i.e., 5 mm, 7 mm, and 10mm, representing the presence of cancer in early, intermediate,and advanced stages, respectively, c) mechanical tester (Mach-1 V500CS) equipped with a multiple-axis load cell to recordthe stress response of tissue at a sampling rate of 100 Hz. . . . 59Figure 4.3 Model validation with tissue-mimicking phantoms: compari-son of relaxation shear moduli between the side with the hardnodule (i.e., the “cancerous” side) and the side without a nod-ule (i.e., the “normal” side), for nodule sizes of a) 10 mm, b) 7mm, and c) 5 mm. The figure shows that larger nodules led tohigher viscous properties. . . . . . . . . . . . . . . . . . . . 61Figure 4.4 Model validation with tissue-mimicking phantoms: compari-son of elastic properties (steady-state stress) between the sidewith the hard nodule (i.e., the “cancerous” side), and the sidewithout a hard nodule (i.e., the “normal” side), for nodule sizesof 10 mm, 7 mm, and 5 mm. The figure shows that larger nod-ules led to higher elastic properties. . . . . . . . . . . . . . . 61Figure 4.5 Fresh prostate gland: tissues were collected within two hoursafter being removed by radical prostatectomy surgery. Testswere performed on mid and apex sections of the posterior prostateand the mechanical properties were compared to those at theleft and right sides of the posterior prostate. . . . . . . . . . . 62xxFigure 4.6 The prostate was placed into a container filled with phosphatebuffered saline (PBS) to avoid dehydration. Samples were sup-ported from beneath by a plastic holder and oriented so that theanterior prostate was at the bottom (touching the holder) andthe posterior prostate was at the top (touching the indenter). . 63Figure 4.7 Applicability of the QLV model for prostate tissue: the QLVmodel’s applicability for predicting the viscoelastic behaviorof prostate tissue was checked by testing the separation be-tween the viscous and elastic parts of the stress response. Test-ing was performed with different indentation depths; i.e., 4mm, 5 mm, and 6 mm, and the stress was normalized by theelastic stress to obtain the viscous part of the stress. The resultsshow that the elastic and viscous parts of the prostate stress re-sponse are separable since while the indentation depth varies,and hence the elastic part of stress, the viscous part remainedthe same. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65Figure 4.8 Applicability of the QLV model for the prostate gland: me-chanical testing was performed with different indentation depths;i.e., δ = 4 mm, 5 mm, and 6 mm, and the relaxation shearmoduli gis were obtained for each test. Each test was repeatedeight times and the average and standard deviation of gis forthe last six tests were reported in the figure (blue, orange, andgreen boxes). The properties are normalized by gavgi s; i.e., theaverage of the properties for the three indentation depths. Thefigure shows that a) g1, b) g2, and c) g3 from tests at differ-ent indentation depths are close. The greater variation of g1 ismainly due to overshoot; i.e., when the tester cannot preciselycontrol the final deformation at a high indentation rate. Thefigure also shows that the variation within gis for the differ-ent tests is within 10%, which indicates the precision of themeasurements. . . . . . . . . . . . . . . . . . . . . . . . . . 66xxiFigure 4.9 Precision of the QLV model: each mechanical test was re-peated eight times and the results of last six tests were usedfor the figure. σE from each test was compared to the averageelastic stress σavgE to determine the precision of the measure-ments. From this figure (and Fig. 4.8) the QLV model has ahigh precision for reporting the viscoelastic properties of theprostate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Figure 4.10 Effect of cancer on the mechanical properties when cancer is inposterior prostate (a-d); and when cancer is in anterior prostate(a-h). Samples that had cancer on both sides were excluded.The properties were normalized by dividing the properties ofthe cancerous side by the properties of the normal side for eachsection. Tests were repeated eight times for each quadrant andthe average of the last six tests were used to calculate the prop-erties. The figures show higher values for the properties onthe cancerous side, especially for a) g1, and d) σE as the ra-tio for the properties on the cancerous and normal sides aremostly above one. The effect of cancer on the properties isless significant when cancer occurred in the anterior prostatee-h), mainly because the tests were performed on the posteriorprostate, which limits the accuracy in capturing the anteriorprostate properties. . . . . . . . . . . . . . . . . . . . . . . . 68Figure 4.11 Mechanical properties of non-cancerous prostate gland. Theproperties are normalized by dividing the properties for theright side by the properties for the left side in each section.Since no cancerous nodules are in the prostate, the proper-ties for the left and right sides are expected to be similar. Theshaded area shows the 10% variation in the properties, whichcan be considered as insignificant (see Fig. 4.8 and Fig. 4.9).The figures also show more similarity between the propertiesfor the left and right sides for a) g1 and d) σE . The higherdifference in g2, c) g3 may be explained by the presence of anodule in a nearby section. . . . . . . . . . . . . . . . . . . 69xxiiFigure 4.12 Correlation between different properties of the prostate gland:a) correlation between cancer grade, prostate mass, and prostatevolume, and b) correlation between cancer grade, prostate vol-ume, and patient’s age. a) shows a strong correlation betweenthe prostate volume and mass, but the cancer grade does notappear to be correlated to the mass or volume. For example,cancer grade 5 could be assigned to either a small or a largeprostate gland. b) shows a positive correlation between pa-tient’s age and prostate volume. While a clear relationship be-tween cancer grade and age is not seen, high-grade cancer (i.e.,grades 4 and 5) are more likely to occur in older patients. . . . 71Figure 4.13 Gleason scores for prostate cancer cells: a) patterns for Glea-son score 3 have well-formed glands without fusion, b) pat-terns for Gleason score 4 have back-to-back fused glands anda sieve-like structure formed by the cells, and c) patterns forGleason score 5 show cell invasion in the glands. . . . . . . . 72Figure 4.14 Correlation between tumor volume and properties when cancerpresents in the posterior prostate. The properties were normal-ized by dividing the properties for the cancerous side to theproperties for the normal side. Viscous properties are affectedby tumor volume (see a, b, and c) while the elastic propertiesdo not seem to vary with tumor volume (see d). g1 and g3 havethe highest statistical significance (lowest p-value). The cancergrade does not appear to be correlated with tumor volume orthe viscoelastic properties. International Society of UrologicalPathology (ISUP) Cancer Grade was reported, and calculatedbased on the Gleason score. Cancer grades 2, 3, 4, and 5 are as-sociated with Gleason Score 3+4, 4+3, 4+4, and 4+5 (or 5+4),respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . 73xxiiiFigure 4.15 Correlation between tumor volume and mechanical propertieswhen cancer presents in the anterior prostate. The propertiesare normalized by dividing the properties on the cancerous sideby the properties on the normal side. Unlike Fig. 4.14, noclear correlation can be seen between tumor volume and theviscoelastic properties. This may be due to our testing thatwas performed on the posterior prostate, which fails to accu-rately capture the properties in the anterior prostate. Interna-tional Society of Urological Pathology (ISUP) Cancer Gradewas reported, and calculated based on the Gleason score. Can-cer grades 2, 3, 4, and 5 are associated with Gleason Score3+4, 4+3, 4+4, and 4+5 (or 5+4), respectively. . . . . . . . . . 74Figure 4.16 Correlation between cancer grade and relative tumor volume(tumor volume divided by prostate volume). No clear correla-tion is seen between cancer grade, patient’s age, and the rela-tive tumor volume. International Society of Urological Pathol-ogy (ISUP) Cancer Grade was reported, and calculated basedon the Gleason score. Cancer grades 2, 3, 4, and 5 are asso-ciated with Gleason Score 3+4, 4+3, 4+4, and 4+5 (or 5+4),respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . 75Figure A.1 Design of the proposed FBG sensor: The sensor is composedof an FBG sensor bonded to a elastomer (polyurethane) rodand it is supported by a 3D printed enclosure. The rod is de-signed to contact the material through a sensor head with adiameter of 6 mm. The sensor is compressed into a materialby using an actuator and the force reaction of the material isrecorded over time. The size of the sensor is designed to allowfor in vivo applications. . . . . . . . . . . . . . . . . . . . . . 98xxivFigure A.2 Sensor calibration: the mechanical tester described in Section2.1 was used to calibrate the sensor. The FGB sensor wasplaced beneath the tester’s indenter and a range of strains from0 to 2.25% were applied in intervals of 0.225%. The sensor in-formation was read through the interrogator and calibrated bythe forces recorded for the load cell. The calibration showed alinear relationship between the sensor’s measurements (wave-length shift in the FBG sensor) and Force, with a maximumnon-linearity error of 22 mN. . . . . . . . . . . . . . . . . . . 99Figure A.3 Testing the FBG sensor on the prostate-mimicking phantoms:phantoms were fabricated as described in Section 4.1.1 andthe nodule was located in L1; c) the FBG sensor was used tomeasure the force response of the prostate phantoms after eachtest. The sensor was connected to the actuator with an adaptorand a custom-built rig. Experiments were repeated three timesfor each quadrant (L1-L4) and the average QLV parameterswere recorded for each quadrant. . . . . . . . . . . . . . . . . 100Figure A.4 Validation of the FBG sensor applicability by testing on prostate-mimicking phantoms: viscous properties (gi s) are reported fora) L1, b) L2, c) L3, and d) L4 for hard nodules with diametersof 5 mm, 7 mm, and 10 mm. The figure shows that the mostvariation occurs for the properties at L1, where the nodules arelocated. The properties at L4 also vary slightly for the differ-ent nodule sizes, which is likely due to the effect of a hardnodule in a nearby quadrant. L2 and L3, however, have similarproperties for the different nodule sizes. . . . . . . . . . . . . 101xxvFigure A.5 Validation of the FBG sensor applicability by testing the prostate-mimicking phantoms: elastic stress is shown for quadrantswith different nodule sizes. The figure shows that the great-est variation in elastic stress occurs with the nodule sizes atL1. The elastic stress at L4 also varies slightly with the nodulesize, which is likely due to the effect of a nearby quadrant. L2and L3, however, have similar amounts of elastic stress for thedifferent nodule sizes. . . . . . . . . . . . . . . . . . . . . . 102xxviNomenclatureC Right Cauchy-Green Deformation TensorD Specimen DiameterE Young’s ModulusG(t) Reduced FunctionH Specimen HeightI1 First Invariant (Trace)J Volumetric StretchNi EigenvectorP Hydrostatic Pressureα Ogden’s Model ConstantF Deformation Gradient TensorS 2nd Piola-Kirchhoff Tensorℓ Elasticity Tensorδ Indentation Depthε Strainκ Bulk Modulusxxviiλ Axial Compressionλi Eigenvalueµ Ogden’s Model Constantφ Material Constitutive Parameterψ Strain Energy Density Functionσ Cauchy StressσE Steady-state (Elastic) Cauchy Stressτ Relaxation Timeϕ Indenter Sizegi Shear Relaxation ModulusxxviiiAcknowledgmentsI would first like to thank my advisor, Prof. Mu Chiao. He has been very support-ive in the course of my Ph.D. program and provided me with the resources, withoutwhich the work could not be imagined. His generosity, guidance, sharpness, andapproach to science have all contributed to shape who I am in my research career.I also want to thank my lab mates, Minghua Wu, Hiroshan Gunawardane, ShayanJannati, Anoush Sepehri, Ali Shademani, and Ali Moalemi for creating an intel-lectually stimulating environment where I could better develop my research ideas.Furthermore, I very much appreciate the supports I received from John Jackson,who helped me to shape my ideas at the beginning of my study.I want to thank Dr. Mattia Bacca, Dr. Peter Black, and Dr. Eric Belanger, whohave worked closely with me through the years and given me invaluable advice.Dr. Bacca supported me in the mathematical side of the project and Dr. Black andDr. Belanger made the clinical study possible and facilitated the collaboration withVancouver General Hospital. I would also like to thank my supervisory committee,Prof. Boris Stoeber, Prof. Clarence de Silva, and Prof. Ryozo Nagamune, for theirinput and guidance in turning this into a stronger dissertation. I would also liketo show my appreciation to Dr. Danmei Liu and Vivian Chung who provided mewith excellent support in The Robert HN Ho Research Centre. In addition, I wantto thank Ali Hussein and Mathieu Roumiguie for facilitating the clinical study.To my friends: thank you so much, Vahid Karamzadeh, MohammadhosseinAkhavanfar, Abdollah Safari, Amin Arefi, Amirhossein Afshinfard, and HamidrezaEslami, for all your assistance and for giving me strong encouragement when, attimes, I had to grapple with deep frustration and self-doubt. My warm gratitudealso goes to Alireza Zakeri, Kasra Ghorbaninejad, and Ali Karimi for all their helpxxixand emotional support.Last but not least, I would like to thank my lovely wife, parents, parents-in-law,sister, brother, and brothers-in-law, for their moral support and encouragement inthe pursuit of education. Without foundation they gave me, I would never havereached this stage.xxxChapter 1IntroductionProstate cancer (PCa) is the second most common type of cancer in the world andthe fifth leading cause of cancer death among men in 2020 [3]. Being diagnosed inearly stages, PCa can be cured with manageable side effects. The initial screeningmethod for PCa includes digital rectal examination (DRE) [4], where a physicianexamines the stiffness of the posterior prostate gland by finger. Higher stiffness inthe prostate gland marks the presence of cancer [5]. The results of DRE though arenot accurate as they are subjective and dependent on the sensitivity of the humanfinger. Naji et al. [6], Soeterik et al. [7], and Jones et al. [8], in different stud-ies, showed that more than 50% of the people with abnormal DRE results do nothave prostate cancer. Alternative methods have been introduced for measuring themechanical properties of the prostate gland in an objective manner, among whichultrasonic elastography (USE) [9], magnetic resonance elastography (MRE) [10],and direct elasticity measurement [5] are more common. Despite their differences,all of these methods have the same basis: measuring the response of tissue uponapplying a specific actuation [11]. This common basis requires all viscoelasticitymeasurement techniques to follow three main steps: actuating tissue, detecting theresponse, and extracting the properties based on a model.11.1 Measurement of Tissue Viscoelasticity1.1.1 Tissue ActuationTissue can be excited by applying either a displacement or a force. In the former,tissue is actuated by static or dynamic deformation. Taka´cs et al. [12] measuredthe viscoelastic properties of liver by conducting a relaxation test, where the livertissues were compressed at a rate of 750 mm/min, and then kept compressed at4 mm for 60 seconds. The stress relaxation of tissues during this procedure wasrecorded to measure the viscous characteristics. In another work, dynamic defor-mation was utilized for measuring the stiffness of synthetic and natural sponges[13]. In this article, the sponges were excited by a sinusoidal deflection with anamplitude between 0.5-1.0 mm and frequency between 5-40 Hz. Alternatively,static and dynamic forces can be used in the actuation mode. Magnetic, piezo-electric, and pneumatic drivers are among the conventional methods for generatingactuation force [14]. Mechanical drivers, however, are only appropriate when theproperties of superficial organs are of interest. For deep organs, another techniquecalled acoustic radiation force impulse (ARFI) should be considered [15, 16]. InARFI, force is generated by focalizing an ultrasound beam to a small spot (a fewsquare centimeters) [11, 17]. Since the actuation is produced by ultrasonic waves,this method can work without physical contact and so it is suitable for deep tis-sues. ARFI technique actuates tissue with high frequency (i.e., few MHz) [18].High-frequency actuation, however, does not allow tissues to exhibit their entireviscoelastic characteristics. For example, it has been shown that stress relaxationin biological tissues can last for several minutes [19–21]. That is to say, usingmechanical drivers with low actuation frequency is preferable to better capture theviscoelastic properties of materials, but the application is limited to superficial or-gans.1.1.2 Response DetectionThe second step is detecting tissue response upon the applied actuation. Whentissue is excited by displacement, its force response is recorded, and vice versa[12]. In the literature, it is more common to use force as actuation and then mea-2sure the deformation. One of the methods for measuring tissue deformation ismedical imaging, in which the images before and after actuation are compared toestimate displacement. Various imaging techniques, including ultrasound (US) andmagnetic resonance (MR) can be used for this purpose. Knowing the tissue dis-placement over an applied force, the viscoelasticity of tissue can then be calculated.Despite providing a straightforward method for measuring biomechanical proper-ties, this method suffers from that fact that the exact value of applied stress is notalways known, especially when it comes to deep organs. To address this problem,researchers have developed another approach, called wave imaging, whereby tissueproperties are determined without the stress field. In this approach, the waves cre-ated inside tissue as the result of actuation are considered for calculating stiffness.The waves induced after an actuation can move either in the same direction as theactuation (longitudinal waves) or normal to that (shear waves). It can be shownthat the propagation speed of longitudinal and shear waves are dependent on thebulk modulus and shear modulus of the material, respectively [9]. Bulk modulusis less commonly used for characterizing biological tissues as they are almost in-compressible and so highly resistant to change of volume. On the contrary, shearwaves can well characterize tissues, and so the shear wave imaging is more com-mon in the literate [9, 11]. Capturing shear wave velocity though is challenging.Shear waves propagate in opposite directions away from the actuation source, andso they are difficult to be monitored. Moreover, because of their high attenuationrate, shear waves can be detected only in a small region.Alternatively, tissue can be actuated by a deformation and the stress responseis measured by using electronic [22, 23] or optical pressure sensors [24, 25]. Theelectronic sensors, like piezoresistive and capacitive sensors, have a simple work-ing mechanism and inexpensive fabrication cost [23]. Meanwhile, they have somedrawbacks that compromise their applicability, especially when it comes to bio-medicine. For example, they need to be in direct contact with the tissue whilekeeping the sensor fixed on tissue is challenging [26]. Also, miniaturizing theelectronic sensors decreases their mechanical strength; that is why brittleness isaccounted for one of the main problems of small (i.e., few millimeters) electronicsensors [22]. This is especially important in biomedical applications where unre-liability of sensors can risk the patient’s health. Optical sensing is an alternative3sensing scheme for biomedical tactile sensors. In these sensors, the force-induceddeflection is correlated by the optical properties of light in a fiber. The non-contactnature of optical sensors facilitates their application in biomedical systems. Also,optical sensors show good mechanical strength in small sizes [27]. These advan-tages have been encouraging an increasing number of researchers to select opticalsensors for the biomedical applications. In particular, the Fibre-Bragg grating sen-sors (FBG) are more popular due to the simplicity in packaging process and lowfabrication cost [25]. FBG sensors are made by inscribing diffraction grating on anoptical fiber. The grating reflects a specific wavelength of light dependent on thegrating period [28]. Reading the reflected wavelength, therefore, one can measurethe strain in the fiber and hence the corresponding stress. FBG sensors, however,are subject to chirp failure; that is when the stress is not distributed uniformly in thegrating zone, and multiple wavelengths are reflected [25]. To avoid the ambiguitycaused by chirp failure, the sensor’s design should guarantee uniform distributionof stress through fiber. Also, FBG sensors are susceptible to temperature so thattemperature cross-sensitivity can exceed force sensitivity [24]. This problem canalso be addressed by considering another optical fiber for temperature compensa-tion.Hitherto, techniques for actuating tissue and measuring the tissue feedbackbehavior are discussed. For discovering the tissue properties, a viscoelastic modelis needed to translate the tissue behavior into quantifiable parameters reflecting itscharacteristics. This point brings us to the third and last step of viscoelasticitymeasurement which is discussed in the next subsection .1.1.3 ModelingThe last step in determining materials’ viscoelastic properties is extracting the in-trinsic characteristics from the experimental data. No matter how accurate the firsttwo steps (actuation and response measurement) are carried out, using an inap-propriate model prevents obtaining accurate results. Modeling biological tissues,however, is difficult since their stress-strain behavior is non-linear and it can bedifferent in loading and unloading cycles ([1]). To relive this difficulty, some tech-niques have been proposed in the literature to approximate the viscoelastic behavior4Figure 1.1: Comparison of the actual mechanical response of the rabbitmesentery (solid line) with the corresponding incremental lines (dashedlines). Reproduced with permission from [1].of tissue with that of elastic materials, which is easier to be simulated.Incremental LawViscoelastic substances demonstrate non-linear behavior when an actuation is ap-plied. This non-linear behavior decays to linear form when perturbation is in-finitesimal [1]. Therefore, dividing the actual actuation of tissue into a set of smalldisturbances, the stress-strain curve of tissue can be approximated by a set of lines,called incremental lines, each reflects a small disturbance (Fig. 1.1) compares theactual behavior of the rabbit mesentery with the corresponding incremental lines[1]. As it is seen in the figure, the actual behavior of tissue is estimated with sev-eral lines (dashed lines), each can be described by an elastic model. It should benoted that the incremental lines are not necessarily tangent to the actual curve ofmaterials. In fact, these lines are not mathematically related to the actual behaviorand should be obtained experimentally.5Pseudo-Elasticity and HyperelasticityThe hysteresis loop shown in (Fig. 1.1) implies the fact that biological tissues havedifferent behavior in loading and unloading cycles. Therefore, one can suggest twoseparate elastic models for describing the loading and unloading curves. Doingthis, however, requires tissue to show consistent behavior over time, which is notnecessarily true for soft tissues [1]: for example, Fig. 1.2a demonstrates the changein behavior of a ligament through time. Fortunately though, the time dependency oftissue’s reaction can be removed by a process called pre-conditioning (Fig. 1.2b).In this process, the tissue completes several cycles before conducting the actualtest. Accordingly, the discrepancy between the behavior of tissue in different cyclesdecreases, which results in a same loading/unloading curves through time. Afterpre-conditioning, tissue can be treated as a pseudo-elastic material exhibiting twodifferent elastic behavior in loading and unloading cycles, each can be modeledwith a separate model.The techniques above enable us to consider elastic models for representing vis-coelastic materials. Hyperelastic models are a classification of elastic models usedto represent soft tissues. In Hyperelastic models, an analytical function, calledstrain energy density (SED), is proposed to relate the stress tensor of materials totheir strain rate. Examples of SED functions can be found in the comprehensivereview paper of Chagnon et al. [29]. Hyperelastic models are widely used formodeling soft tissues [30–32]; however, they can only provide an approximateddescription for the behavior of biological materials as they ignore the viscous char-acteristics. In fact, these models, like other elastic models, cannot accurately pre-dict some behavior of viscoelastic materials, such as creep and stress relaxation.Therefore, when the accurate characterization of tissue is of interest, elastic mod-els are not appropriate, and instead viscoelastic models should be considered. Inthis document, we review two viscoelastic models that are widely-used for describ-ing biological tissues.Spring-dashpot modelsSpring-dashpot models introduce a set of springs and dashpots for describing a ma-terial; the spring part is supposed to explain the elastic behavior and the dashpots6Figure 1.2: Time-dependent characteristic of tissue behavior: a) change inloading/unloading behavior of an anterior cruciate ligament throughtime; b) pre-conditioning of a dog’s carotid artery. Reproduced withpermission from [1].represent viscous characteristics. The relation between force and strain in springsand dashpots can be either linear [2] or non-linear [12]. In their simplest forms,spring-dashpot models are comprised of a spring and dashpot which are placedeither in series (Maxwell model) or parallel (Kelvin-Voight model). Figs. 1.3aand b display Maxwell and Kelvin-Voight models, each suggests two parame-ters, E and η , for describing the stress-strain relation. Complete agreement withexperiments, however, is not usually achievable by tuning only two parameters.Therefore, multi-parameter models are often employed. In particular, the three-parameter model of Standard Linear Solid (SLS) is commonly used in the litera-ture (Fig. 1.3c) [2]. Despite providing an appropriates relation for capturing theviscoelastic properties, spring-dashpot models are empirical and therefore lackinga physical basis [2]. This fact indicates that the parameters introduced for springand dashpot coefficients are not necessarily meaningful. That is why the same tis-sue can be represented well by using two different spring-dashpot models [12, 33],each introduces different values for E and η . This problem has been discussed byParker et al. [34] with the goal of reaching a consensus on the appropriate modelfor soft tissues. They concluded that, among linear viscoelastic models, fractionalstandard linear model and Kelvin–Voigt fractional derivative (KVFD) model pro-7Figure 1.3: Different forms of spring-dashpot models; (a) Maxwell model,(b) Kelvin-Voight model, (c) Standard Linear Solid model. Reproducedwith permission from [2]vide the most meaningful description for viscoelastic behavior. Meanwhile, theproperties obtained by these models are still dependent on the frequency by whichthe tissue is actuated; for example, the loss modulus given by Kelvin-Voigt modelincreases with frequency, which compromises the model’s generality. It shouldbe noted that having a physically interpretable model is essential when investi-gating actual properties of tissue. Therefore, it is necessary to introduce anotherviscoelastic model which has better biophysical background.Quasi-linear Viscoelastic ModelTo better describe the viscoelastic properties of materials, Fung introduced Quasi-linear Viscoelastic (QLV) in 1981 [1]. Unlike spring-dashpot models, the QLVmodel characterizes tissues, as opposed to describing them. In other words, spring-dashpot models describe the observed behavior of tissue by an arrangement ofsprings and dashpots while there is no guarantee that this description keeps itsvalidity for other behavior of tissue. However, the QLV explores the relaxationcharacteristics, which are inherent properties of tissue, and so it provides a morevalid model. This feature enables the QLV model to predict the non-linear behav-ior of viscoelastic materials, and to provide better estimations than spring-dashpotmodels. The QLV model calculates the stress by the convolution of two functions:one describes the elastic response, defined by Fung as the stress generated instan-taneously after the deformation, and another accounts for the stress relaxation [1].Fung’s QLV model is shown to predict the non-linear characteristics of inelas-tic materials and provide an accurate representation for tissue. In this dissertation,8we aim to use the QLV model for characterizing the prostate tissues. Nevertheless,correct implementation of the QLV model requires delicate considerations; disre-garding those considerations caused some authors to mistakenly doubt the accuracyof the model [35]. For example, in order to preserve objectivity, the stress in theQLV model should be expressed by the second Piola–Kirchhoff stress form. Also,the determination of the functions for describing elastic stress and stress relaxationrequires experimental elicitation so that the functions are derived independentlyfrom each other. The characterization of viscoelastic materials via the QLV modelis discussed in the next section.1.2 Material Characterization with the Quasi-linearViscoelastic ModelAfter being introduced in 1981, the QLV model has been reappraised in later re-search works [35–37]. For example, the relaxation function that is originally sug-gested by Fung in 1981 is shown to results in viscoelastic parameters with largeconfidence intervals, which complicate material identification [37]. To avoid thisproblem, it was suggested to use a discrete spectrum approximation [38], whichidentifies a limited number of relaxation timescales over which a discrete spectrumrepresents the relaxation behavior. The QLV model with a discrete spectrum re-laxation function has been widely used for modeling soft tissues ([37–40]). Usinga discrete approach though cannot guarantee the reliability of the QLV parame-ters, especially when the model parameters increase and cause overfitting. Thismay explain why Calvo-Gallego et al. [40] concluded that the stress relaxation inadipose tissue is maximal after 0.01 seconds, which contradicts the experimentalfindings. There have been attempts in the literature to provide enhanced techniquesfor calibrating the QLV parameters. Yang et al. [39] normalized the data by usingthe differences between the peak and post-relaxation stresses and checked if theQLV parameters fit the normalized and denormalized stresses. A similar approachwas followed in the works of Xu et al.[41] and Babaei et al. [37]. Alternatively,Troyer et al. [42] separated the elastic and relaxation terms of the QLV model byassuming a Heaviside strain form and derived the elastic and relaxation parametersindependently. While they addressed the overfitting problem by reducing the pa-9rameters being fitted at a time, the deviation of the actual strain from the Heavisideform challenges their parameters’ validity.Material characterization by the QLV model requires experimental data thatexpress the force feedback under specific deformation. Different deformation pro-tocols, including unconfined compression, tension, torsion, and indentation canbe considered for this purpose, but because of its simplicity, indentation is of-ten the preferred method [43]. In an indentation test, the tissue is pressed by asmall indenter (i.e., a flat punch) and the mechanical properties are derived fromthe force-displacement relationship. The test is simple and appropriate for in vivoassessments of tissues with various sizes and geometries [43]. In addition, its non-invasive nature preserves the tissue’s physiological structure while showing theactual properties [44].QLV Characterization by Indentation TestResearchers have tried to incorporate analytical indentation models into the QLVanalysis, and for example, Kermani et al. [45] substituted Sneddon’s model for theelastic part of the QLV formulation. The authors predicted the behavior of porcineaorta using nanoindentation tests with a conical indenter. Other researchers haveused spherical indenters in either nanoindentation [46] or traditional (macro) inden-tation tests [47]. Flat-ended tips, however, are more commonly used in indentationtests as they 1) can keep a constant contact area during the indentation, and 2) leadto higher forces compared to those obtained with spherical or conical tips at thesame depth [48]. Ling et al. [43] used a flat-ended indentation test to characterizefoot plantar tissue by substituting the QLV relaxation modulus in Hayes’ solution.Similarly, Qiu et al. [49] derived the properties of breast cancer tissue with a flatpunch indentation test. In this work, the QLV shear modulus was replaced with theshear modulus in Fischer-Cripps’s indentation model. The same method was alsoadopted by Zheng Mak [50] and Huang et al. [51] to calculate the properties ofbiological tissues by applying the QLV model in indentation tests. In any case, theliterature suggests that analytical indentation models cannot accurately model bio-logical tissues as the reported properties differ from those obtained from uniaxialcompression tests [52–54]10Finite element (FE) methods are more accurate for studying indentation testsas they can model the three-dimensional deformation that occurs during the inden-tation. Moreover, ABAQUS software has been widely used as an FE tool for thispurpose. Seyf et al. [55] derived elastic and viscous properties of meniscus tissueusing the inverse FE method and the QLV model in ABAQUS. In inverse FE meth-ods, FE analyses are conducted iteratively with different parameters until the re-sults match the experiment. Nevertheless, inverse FE methods are time-consuming,especially when it comes to 3D modeling. Consequently, attempts have been madeto improve the speed of the inverse FE method when QLV is considered. For exam-ple, Kim and Srinivasan [56] separated the characterization of elastic and viscousparameters to enhance the model’s speed, which could be completed in a few hours.In another study, Pierrat et al. [57] found elastic parameters from pre-computed FEsimulation data, and then used a one-dimensional QLV model to derive the viscousparameters. By using pre-computed data, the authors introduced a method that wassufficiently rapid for real-time characterizations.In this dissertation, we aim to use the QLV model to characterize the elasticand viscous characteristics of prostate tissue and study the effect of cancer on themechanical properties. The effect of cancer on the viscoelasticity of prostate hasbeen the subject of many research works, which some of them are reviewed in thenext section.1.3 Effect of Cancer on Mechanical Properties ofProstate TissueAs mentioned earlier, imaging shear waves after an actuation is a common methodfor mapping the mechanical properties of soft tissues (i.e., shear wave elastography(SWE)). Using ultrasound SWE, Boehm et al. [58] recorded the elasticity of theprostate gland for 60 patients and concluded that cancerous tissues are two timesstiffer than normal tissues. Their study included the samples with cancer grades1ranging from one (Gleason Score 3+3) to four (Gleason Score 4+4). In anotherwork, Fu et al. [59] concluded that prostate tissue with a Gleason Score of 10 isfour times stiffer than the tissue with a Gleason score of 6. A similar trend of works1International Society of Urological Pathology (ISUP) Cancer Grade11on elasticity of prostate tissue can be found in the recent review paper by Anbarasanet al. [60]. Like elastic properties, viscous properties affect the propagation ofshear waves, and so neglecting them leads to inaccurate estimations [61]. Whilethere are multiple studies on applying SWE to map viscoelastic properties of softtissues [15, 62, 63], to our best knowledge, there is only one targeting prostatetissue, in which Mitri et al. [64] incorporated Kelvin-Voigt models with shear waveequations to obtain the shear elasticity and viscosity for non-cancerous prostates.Characterizing three freshly excised prostates from cadavers, they demonstratedthat the elasticity and viscosity of the prostates for different people are not similar.Alternatively, MRE can be used to capture the waves propagation speed afteran actuation. Hu et al. [65], for instance, applied MRE to derive the stiffnessof prostate using a 3 Tesla scanner and SWE techniques. Their results showedthat PCa stiffness increases over 30% when cancer metastasis occurs. A similarapproach has been followed by Asbach et al. [66] to differentiate PCa with can-cer grade 1 and above (Gleason score ≥ 3+3) from benign prostate hyperplasia(BPH). Estimating shear wave speed as a measure of stiffness, they concluded thatprostates show 10% more stiffness under PCa than BPH. Some other works on ex-ploring elastic properties of the prostate with MRE have been reviewed by Pepin etal. [67]. The MRE research works that include viscoelastic properties are also fre-quent. As an example, Li et al. [68] derived viscoelasticity of prostate, and showedthat cancerous prostate tissue is tree times stiffer and more viscous than prostatitistissue. Another example is the work of Reiter et al. [69], who compared storageand loss moduli between normal and cancerous prostate. Their results showed thatthe complex shear modulus for the cancerous prostates with Gleason scores ≥ 3is two times more than healthy tissues. Some other similar works include those ofReiss-zimmermann et al. [70] and Asbach et al. [66], who reported complex shearmodulus to study prostate viscoelasticity. In line with other similar works, theirresults concluded that the stiffness and viscosity of prostate tissue increase in thepresence of cancer.The research works above describe the viscoelastic properties by using spring-dashpot models that, as discussed earlier, are not reliable since the reported proper-ties are dependent on the arrangement of spring and dashpot elements and also thefrequency that the tissue is actuated. That is to say, the tissue properties obtained12from different actuation frequencies are not comparable. For example, the resultsof Reiter et al. [69] and Li et al. [68] cannot be compared as the former uses 500Hz actuation while the latter used 100 Hz actuation. In this dissertation, for the firsttime, we propose a novel method for calibrating the QLV model and characterizethe mechanical properties of soft materials. This method is different from previousmethods that presented for QLV characterization in two aspects. First, we considerthe elastic stress as the one that emerges after a sufficiently long time, when thetime-dependent viscous part of stress vanishes, and the steady-state elastic part re-mains. This is different from Fung’s traditional definition for elastic response: thestress generated instantaneously after the deformation [1]. Our definition for theelastic response can reduce the undesirable effect of overshooting on the parameterestimation as it removes the focus on the immediate stress response [71]. Second,the elastic and relaxation parameters are being derived separately using two differ-ent sets of experiments: one for representing the elastic response and another forthe relaxation response. As a result, the number of fitting parameters decreases,and hence overfitting does not occur. The current method is shown to derive theQLV parameters with narrow confidence intervals. We validate our method usingtissue-mimicking phantoms, with elastic properties that resemble normal and can-cerous prostate tissues. The application of the method for material characterizationvia indentation test is also discussed by introducing a straightforward procedurefor characterizing viscoelastic materials. We then apply our method for derivingthe elastic and viscous properties of cancerous and normal prostate tissue and studythe effect of cancer of the QLV parameters. Our results demonstrate that the cancerhas statistically significant effect on the stiffness and shear relaxation modulus ofthe prostate.1.4 Dissertation OrganizationThe rest of the dissertation is organized as follows: in Chapter 2, we provide a newmethod for calibrating the QLV model using uniaxial compression experiment. Themethod is shown to obtain the parameters with small confidence interval, indepen-dently from the loading conditions. We applied the method on tissue-mimickingphantoms to validate its applicability for biological tissues. Chapter 3 presents13a procedure for characterizing the QLV parameters from indentation experiment.Using pre-calculated finite element results, we show the material parameters areobtained in a few seconds, with good accuracy (error withing 20%) and precision(standard error within 10%). The findings of Chapters 2 and 1 are then applied forfinding the viscoelastic properties of normal and cancerous prostate tissue in Chap-ter 4. Having tested thirty-five fresh, unfixed prostate samples, we investigated theeffect of cancer on the elastic and viscous properties of cancer and performed hy-pothesis testing to find the significance of findings. Finally, Chapter 5 concludesthe findings and discuss future work.14Chapter 2Quasi-linear ViscoelasticCalibration of Soft Materials viaUniaxial TestThis chapter 1 provides a new method for calibrating the QLV parameters fromthe uniaxial compression test (one-dimensional stress-strain state). The findingswill be used as a base in the following chapter for calibrating the parameters fromindentation test (three-dimensional stress-strain state).This chapter is organized as follows: first, we provide the constitutive equationsfor one-dimensional QLV model and provides an approximated analytical solution.The QLV equations are then solved numerically to provide a better descriptionfor the viscoelastic characteristics of materials. Finally, based on the numericalsolution provided, we present a characterization method for deriving the Young’smodulus and shear relaxation modulus from uniaxial compression test, and validateour method by testing on phantoms with similar Young’s modulus as normal andcancerous prostate tissue.1The results of this chapter were published in: ”H. Helisaz, M. Bacca, and M. Chiao, “Quasi-Linear Viscoelastic Characterization of Soft Tissue-Mimicking Materials,” ASME J. Biomech. Eng.,vol. 143, no. 6, Jun. 2021, doi: 10.1115/1.4050036.”152.1 Methods and Materials2.1.1 QLV Model in One DimensionThe parameters of the QLV model can be obtained via calibration from uniaxialstress relaxation experiments, where the sample is compressed at high rate and thenwait for stress to be fully relaxed. We apply a compression strain ε and measurethe evolution of the stress σ over time t. The stress state is uniaxial and σ can bedescribed as:σ(t) =∫ t0φ(t− τ)∂ε(τ)∂τdτ (2.1)where φ identifies the constitutive behavior of materials. The QLV model proposes[1]:φ(t) = G(t)∂σE(ε)∂ε(2.2)with G, the relaxation function, is a function related to the viscous (rate-dependent)response of the material and σE is a function related to its elastic behavior. Thelatter corresponds to the true (Cauchy) stress of the material under quasi-static de-formation, and it can be obtained from hyperelastic constitutive models. We usea one-term Ogden model to determine σE , which has been shown to predict theelastic response of human tissue accurately [72]. Note that the one-term Ogdenmodel can describe the behavior of materials under fully compressive or fully ten-sile experiments while characterizing the whole compression-tension behavior re-quires at least two terms [72]. Assuming incompressibility, the axial compressionλ = 1−|ε| generates an elastic Cauchy stress σE of [40]:σE(λ ) =−µ(λα − 1λα/2)(2.3)where µ and α are the materials’ constants related to the elastic modulus E asfollows:E =32µα (2.4)In compression tests, the applied deformation follows a ramp-and-hold function(see Fig. 2.1c). Hence, the deformation increases from t = 0 to t = τ0, and re-mains constant for t ≥ t = τ0, where t = τ0 is determined by the compression rate.16This implies that for t ≥ t = τ0 the applied strain rate is zero, reducing the inte-gral bounds on the right-hand side of Eq.(2.1). Also, the rate of change of ε perunit time in Eq. (2.1) is equivalent to that of the axial compression λ ; hence, thesubstitution of Eq. (2.2) into Eq. (2.1) gives:σ(t) =−∫ τ00G(t− τ)∂σE(λ )∂λ∂λ (τ)∂τdτ (2.5)G describes the time-dependent behavior of the material as a stiffness decay. Fungproposed the following relaxation function to describe the relaxation behavior ofbiological soft tissues [1]:G(t) =1+κ [Z1 (t/v2)−Z1 (t/v1)]1+κ ln(v2/v1)(2.6)where v1, v2, and κ are viscoelastic material coefficients and Z1 is defined as fol-lows:Z1(t) =∫ ∞te−ttdt (2.7)Eq. (2.6), however, prohibits the numerical integration of Eq. (2.5) [38] and willlead to viscoelastic parameters with large confidence intervals [37]. To avoid theseproblems, we consider the Prony’s series below as a discrete spectrum approxima-tion for the relaxation function [38]G(t) = 1+∑igie−t/τi (2.8)where gis are shear relaxation moduli created by the viscous response of the mate-rial decaying to zero at the rate defined by the relaxation time τi. Eq. (2.8) has beenshown to well represent Fung’s relaxation function when relaxation times spreadone decade apart [38]. Substituting Eqs. (2.8), (2.4), and (2.8) into Eq. (2.5), wefinally have:σ(t) =−2E3∫ τ00[1+∑igie−(t−τ)/τi](λα−1+12λα/2+1)∂λ (τ)∂τdτ (2.9)17This integro-differential equation is solved numerically, as explained in the nextsection.We now try to find the analytical solution of Eq. (2.9) for a specific conditionof super-fast compression. By use of the mean value theorem for integrals, a timeτm exists with 0≤ τm ≤ τ0, for which Eq. (2.5) can be rewritten as:σ(t) =−τ0G(t− τm) ∂σE (λ (τm))∂λ∂λ (τm)∂ t(2.10)For fast compression, we have τ0 ≪ τi ∀i; that is, the ”compression time” τ0 ismuch smaller than the relaxation timescales. This also implies τm ≪ τi ∀i and soG(t− τm) from Eq. (2.8) can be approximated by G(t). Rewriting Eq. (2.10), wehave:σ(t)≈−τ0(1+∑igie−t/τi)∂σE (λ (τm))∂λ∂λ (τm)∂ t(2.11)The stress in Eq. (2.11) is maximal at the beginning of the stress relaxation process,at t = τ0, giving:σmax = σ (τ0)≈−τ0(1+∑igi)∂σE (λ (τm))∂λ∂λ (τm)∂ t(2.12)and it reaches the steady-state when stress relaxation is completed at t →∞, giving:σsteady = σ(∞) =−τ0 ∂σE (λ (τm))∂λ∂λ (τm)∂ t(2.13)Equating Eqs. (2.12) and (2.13), we obtain the nominal amount of stress relaxationas: ∣∣∣∣σmax−σsteadyσsteady∣∣∣∣=∑igi (2.14)while at any jth relaxation time τ j we have:∣∣∣∣σ (τ j)−σsteadyσsteady∣∣∣∣=∑igie−τ j/τi (2.15)Eqs. (2.14) and (2.15) correlate the shear relaxation moduli to the amount of stressrelaxation occurring in different time intervals; specifically, gis measure the de-crease in stress response between τi−1 and τi+1. In our approach, the choice of18relaxation times τi is arbitrary. As mentioned earlier, Eq. (2.9) has shown to beaccurate for a choice of relaxation times as decades [40, 42, 73], that is when τi+1is ten times more than τi. We adopt the same testing protocol and try to find the ma-terial parameters (E, α and gis) by fitting the experimental results with Eq. (2.9).2.1.2 Data-Fitting AlgorithmThis section presents our novel method for deriving the elastic and viscous param-eters of materials by invoking the QLV model. We characterize the elastic behaviorby prescribing displacements following the protocol shown in Fig. 2.1a. The com-pression at each step is then matched with the steady-state stress developed afterrelaxation (evidenced by the solid orange circles in Fig. 2.1b). The material is com-pressed by a displacement of 0.5 mm (∼% 2.5 engineering strain) in each step andthen maintained until the stress reaches a steady-state. The steady-state stress rep-resents the elastostatic response as it excludes the time-dependent (viscous) part.In the following of this manuscript, we use elastic stress instead of elastostaticstress for the sake of simplicity. The figure has an inset showing the steady-statestresses that are fit to σE(λ ) from Eq. (2.3) to obtain µ and α and hence E (seeEq. (2.4)). Notably, the literature suggests deriving elastic properties, E and α , byfitting Eq. (2.9) with stress relaxation tests (Fig. 2.1d) [40, 73, 74]. Nevertheless,a stress relaxation test cannot characterize the elastic properties of the materials.In fact, including E and α in the curve fitting causes overfitting, which leads toincorrect parameters. Babaei et al. [37] reported that the QLV model can fit relax-ation data by parameters that fail to predict the behavior of the material under otherdynamic loadings.We then substitute E and α in Eq. (2.9) and solve by deriving σ(t) in termsof gi, given τi as a priori. The experiments reveal that relaxation does not startbefore 0.01 seconds from the peak stress, at t = τ0, and terminates after 10 seconds.Hence, three relaxation times, τ1, τ2, and τ3 are chosen so that τ1 − τ0 = 0.1s,τ2− τ0 = 1s, and τ3− τ0 = 10s (see Fig. 2.1d). G(t) is derived as:G(t) = 1+g1e−t/τ1 +g2e−t/τ2 +g3e−t/τ3 (2.16)The integral in Eq. (2.9) is solved numerically by invoking the Right Riemann19sum formula with time intervals of 0.01 seconds. The solution expresses the stressin terms of shear relaxation moduli gis. We then derive g1,g2, and g3 by fittingthe solution with the experimental results (Fig. 2.1d). gis characterize the viscousproperties by determining the stress relaxation in specific time scales (see Eqs.(2.14, 2.15)). High-rate compression (30 mm⁄s) is considered in the experiment toenhance the accuracy of the QLV estimation [75]. The obtained gi along with Eand α define the QLV parameters required for material characteristics.We fit the experimental results by minimizing the weighted least square errorfunction as follows:e =∑iWi(σQLVi (t)−σExpi (t))2(2.17)where σQLVi (t) is the stress predicted by the QLV model (obtained by Eq. (2.9))and σExpi (t) is obtained by experiments; experiments provide stress in the time in-tervals of 0.01 seconds. We repeat the experiments three times for each sample andsubstitute the mean values as σExpi (t). Wi is also the assigned weight for each datapoint. Larger weights are considered for the data points neighboring the stress peakdue to smaller points in that region. In the first five points nearing the stress peak,Wi = 50 for the three middle points and Wi = 20 for the remaining two; Wi = 1 forother points. We use the interior-point algorithm performed by MATLAB functionfmincon to minimize the error. However, the interior-point algorithm cannot guar-antee a global minimum because of the non-convexity in the objective function[76]. The outcome of the interior-point algorithm is the local minimum aroundthe initial guess, and so, it is highly sensitive to the first guess [40]. To addressthis issue, we adopt a two-step minimization. In the first step, we use a geneticalgorithm (GA) by MATLAB function ga to estimate the global minimum in thespace of the variables gis. GA starts from a set of random points distributed inthe entire domain of gis and finds a set of values for gis where the error functionis minimal. Nevertheless, the solution of the genetic algorithm is not optimal asit uses a heuristic method. Therefore, in the second step, we input the solution ofthe genetic algorithm as the initial guess for the interior-point algorithm to find theoptimal values for gis. To confirm that the minimization algorithm is consistent fora given choice of relaxation times, we repeat the algorithm ten times with the sameτi and compare the derived values for gis. While the estimation of GA is different20Figure 2.1: Experimental procedure: (a) a series of low-range compressions:each is a 2.5% strain and kept long enough until the stress reaches thesteady-state; (b) stress response versus time under the prescribed dis-placement at (a): elastic properties are obtained by plotting the steady-state stress (orange dots) against the compression; (c) ramp-and-holdcompression with 50% final strain: strain is kept long enough until thestress reaches the steady-state; (d) stress response versus time under theprescribed displacement at (c): the stress amplification factors g1 andg2, and g3 measure the stress relaxation in the time intervals (τ0, τ1),(τ1, τ2), and (τ2, τ3)each time, the final solution provided by the interior-point algorithm is same andinsensitive to the initial guess. A similar procedure is followed for finding E and αby fitting Eq. (2.3) to the elastic response from the experiments (see Fig. 2.1b).2.1.3 Tissue-Mimicking PhantomsWe evaluated the ability of the QLV model to characterize the phantoms shown in(Fig. 2.2). The phantoms were fabricated by mixing a silicon-based polymer andcross-linker in the presence of platinum as a catalyst. We used the commerciallyavailable elastomer kits Ecoflex™ 00-10 and Ecoflex™ 00-30 (Smooth-On Inc.,Pennsylvania, U.S.). The mechanical properties of the elastomers are tailored by21changing the concentrations of cross-linker and polymer. Having less cross-linkerthan polymer results in a smaller number of polymer chains and a softer elastomer[77]. A non-reactive silicone fluid NOVOCS™ (Smooth-On Inc., Pennsylvania,U.S.) was also added to further adjust the properties. The silicone fluid decreasesthe friction between the polymer chains and leads to a lower viscosity. It also re-duces the relative concentration of cross-linker with respect to polymer and hencethe ultimate stiffness. The elastomers were fabricated so that Young’s modulus ofElastomer I matches with that suggested by the literature for normal prostate tissue(15-20 kPa) [78], and Elastomers II and III mimic that of cancerous prostate tissue(24-40 kPa) [78]; the composition and the properties of the elastomers are listedin Table 2.1 and Table 2.2. Unlike the elastic properties, the viscous propertiesof normal and cancerous prostate tissues have not been well established [79, 80].The findings of the few available studies on the viscosity of prostate tissues arealso not comparable as they reported different parameters for representing viscos-ity [79, 81]. Therefore, we cannot confirm whether the viscous properties of ourphantoms resemble those of normal and cancerous prostate tissues. To fabricate thephantoms, we mixed the polymer, cross-linker, and silicone fluid for one minute,degassed the mixture for 10 minutes in a 30 kPa vacuum, and waited overnight forthe solution to cure at room temperature. All samples had a cylindrical shape with adiameter of 25.2±0.24 mm and a height of 19.9±0.30 mm (Fig. 2.2). The color pig-ment (Silc Pig™, Smooth-On Inc., Pennsylvania, U.S.) was added to distinguishthe elastomers.Table 2.1: The composition of tissue-mimicking phantomsEcoflex™ 00-10 Ecoflex™ 00-30 NOVOCS™Elastomer I 0% 67% 33%Elastomer II 0% 80% 20%Elastomer III 100% 0% 0%2.1.4 Method Validation and Confidence IntervalWe compare the relaxation behavior for one of the tissue-mimicking phantoms(Elastomer III) with that predicted by the QLV model. The phantom is subject to22Figure 2.2: Tissue-mimicking phantoms: (a) Elastomer I (67% Ecoflex™ 00-10 and 33% NOVOCS™); (b) Elastomer II (80% Ecoflex™ 00-30 and20% NOVOCS™); c) Elastomer III (100% Ecoflex™ 00-10): Elas-tomer I mimics normal tissue in terms of Young’s modulus while elas-tomers II and III mimic cancerous tissues.Table 2.2: The QLV parameters for tissue-mimicking phantoms; 95% confi-dence intervals are reportedα Young’s Modulus g1 g2 g3Elastomer I 2.63 ± 0.03 16.0 ± 0.1kPa 0.31 ± 0.03 0.08 ± 0.01 0.01 ± 0.00Elastomer II 2.53 ± 0.03 33.0 ± 0.1 kPa 0.15 ± 0.02 0.18 ± 0.00 0.00 ± 0.00Elastomer III 3.70 ± 0.01 28.5 ± 0.0 kPa 0.87 ± 0.03 0.46 ± 0.01 0.00 ± 0.00four different compression strains, i.e. ε = 30%,40%,50% and 60% (Fig. 2.3a).The QLV parameters are obtained by fitting the theory to the experimental resultsfor 50% compression strain. We then use the same parameters for describing thebehavior of the same material under 30%, 40%, and 60% strain levels and compareour prediction with the experimental results (Fig. 2.3b). The experiments were per-formed by the mechanical tester Mach-1 V500CS (Biomomentum Inc., Laval, QC,Canada). The tester has the multiple-axis load cell MA234 (Biomomentum Inc.)with a range of 35 N, a resolution of 1.75 mN, and a recording rate of 100 Hz. Theload cell is connected to the phantom by a compression plate with a diameter of 25mm (Fig. 2.4). The compression plate has slightly larger size than the samples toguarantee a uniaxial stress state. Also, the lateral force of the loadcell is monitoredto confirm a friction-less boundary condition between tissue and the fixture plates.As shown in Fig. 2.3, QLV can well predict the phantom’s relaxation behavior un-der different loading conditions. The observed discrepancy between the theoretical23Figure 2.3: Model validation: a) ramp-and-hold compression with differentfinal strains; b) stress response of Elastomer III versus time under theprescribed displacement at (a): lines are experimental results while sym-bols denote predictions of the QLV model (Eq. (2.9)). The experimentsare repeated three times for each sample, and the mean values for theresults are presented with the error bars.and experimental results is partly because of considering a limited number of relax-ation times (see Eq. (2.16)). One can enhance the model’s accuracy by consideringmore relaxation times but at the expense of increasing the number of calibrationparameters. It should be noted that, however, considering more relaxation timesdoes not guarantee a perfect fit to the experiments, because, like any other model,the QLV model has its own limitations. In particular, the model assumes that theeffects of strain and time on the viscoelasticity are separable (see Eq. (2.2)), whichdoes not perfectly represent the materials’ characteristics.To measure the reliability of the QLV model, the confidence intervals of theparameters in Fig. 2.3 are determined by using the method described by Yin etal. [82]. Using this method, we first create sixty-four new data sets (like that ofFig. 2.1d) that could be observed in the experiments. Then, we calculate gis foreach of the new data to obtain a population for gis values, from which confidenceintervals are measured. For finding the new set of data, we calculate the differencesbetween the experimental stress response with that predicted by the QLV model to24draw a residual plot. It is assumed that the residual stress is the sum of a smootherror representing systematic error and a rough error rendering experimental noise.Assuming that the systematic errors can be derived by fitting a polynomial func-tion to residual plot [82] , we obtain the noise as the difference between the resid-ual stress and systematic error. We use eight polynomial functions to representthe systematic error, and for each, we obtain a random noise distribution. Then,the systematic error functions and noise distributions are paired and added to thestress predicted by the QLV model. This creates sixty-four new data whose sys-tematic error and noise distributions could be observed in the experiments. Finally,we implement our data fitting algorithm to calculate the relaxation parameters forthe new data set and find the confidence intervals. The 95% confidence intervalsfor the relaxation parameters are obtained for Elastomer III under different strainlevels (Table 2.3). The table shows the relaxation parameters for different strainlevels have a good agreement proving our method’s ability to predict the behaviorof materials under different loading conditions. As expected, more difference isobserved between the parameters of the lowest and highest strain levels. This canalso be seen in Fig. 2.3 where the parameters for ε = 50% predict the behaviorof ε = 40% better than ε = 30% and ε = 60%. (Table 2.3) confirms a narrowconfidence interval for all reported parameters.Table 2.3: 95% confidence intervals for relaxation parameters of ElastomerIII under different strain levels (see Fig. 2.3)g1 g2 g3ε = 30% 0.74 ± 0.06 0.51 ± 0.01 0.05 ± 0.00ε = 40% 0.84 ± 0.05 0.46 ± 0.01 0.00 ± 0.00ε = 50% 0.87 ± 0.03 0.46 ± 0.01 0.00 ± 0.002.2 Results and DiscussionWe used the process described in the previous section to characterize the tissue-mimicking phantoms. The identification of viscoelastic materials requires detect-ing both elastic and viscous properties. In particular, distinguishing between nor-mal and cancerous prostate tissue involves extracting both properties as cancer can25Figure 2.4: Testing machine used to obtain stress response of materials: acustom compression has slightly larger diameter than samples to guar-antee a uniaxial compression force.change either. The uncontrolled proliferation of cancer cells generates compres-sive stress in the tissue, which causes an increase in the stiffness [83]. On theother hand, the depletion of collagen in the presence of cancer changes the time-dependent behavior of tissue and hence its viscous properties [84]. Cancer effecton the mechanical properties of prostate tissue will be discussed more in Chapter4.Mixing a non-stoichiometric ratio of cross-linker and polymer leaves somemolecules unattached to the network (unreacted monomers) or attached only at oneend (dangling chains). The movement of monomers and dangling chains under de-formation causes a time-dependent behavior [77]. The ramp-and-hold compression(shown in Fig. 2.1c) compels the monomer molecules to move between the poly-mer chains. Nevertheless, the movement is resisted by viscosity, which is high atthe beginning and diminishes over time until the stress reaches a steady state. Thatis why we observe a peak stress in Fig. 2.1d, which is followed by a gradual stressrelaxation over time. The amount and duration of stress relaxation depend on the26concentration of the unreacted monomers and the structure of the dangling chains[77]. We used the QLV model to calculate the amount and rate of stress relaxation,based on which a material can be identified.Fig. 2.5 illustrates the relaxation behavior of the elastomers. We identified thephantoms based on the shear relaxation modulus gi, which reflect the stress relax-ation in (τi−1, τi+1) intervals (see Eqs. (2.14) and (2.15)). The 95% confidenceintervals for QLV parameters are listed in Table 2.2. The table demonstrates thatour method derives the QLV parameters with narrow confidence intervals. In par-ticular, our confidence interval for Young’s modulus is a third of that reported byAbramowitch and Woo [75], who used Fung’s original continuous spectrum forQLV analysis. The parameters are also more reliable as we derive E and α basedon the steady-state stress (see Fig. 2.1b), as opposed to instantaneous stress thatis associated with overshooting errors [71]. Gimbel et al. [71] have shown thata 10% overshoot in the strain input results in a more than 30% error in the QLVparameters.The QLV parameters for the elastomers are listed in Table 2.2 and Fig. 2.5.Table 2.2 demonstrates that the shear relaxation moduli (gis) can well characterizeElastomers II and III that have similar Young’s moduli but different viscous proper-ties. The higher stress relaxation in Elastomer III is reflected in a larger g1, g2. Thetable also compares the rate of stress relaxation in the elastomers: while ElastomerII relaxes the same amount of stress in intervals (τ0, τ1) and (τ1, τ2), ElastomerIII does twice as much in the first interval. Both elastomers have negligible stressrelaxation in the third interval (τ2, τ3). These results reflect the potential of ourmethod for distinguishing materials with similar elasticity (Young’s modulus) butdifferent viscous properties.The stress relaxation behavior of Elastomer I is also demonstrated in Fig. 2.5.Elastomers I and II are fabricated with the same polymer and cross-linker; how-ever, the larger amount of silicon fluid in Elastomer I decreases the viscosity andleads to a higher stress relaxation rate in the early time interval. Also, Elastomer Ishows a smaller Young’s modulus than Elastomer II, reflecting the lower concen-tration of polymer and cross-linker (see Table 2.2). We compare Young’s modulusderived from the QLV model with that represented by the linear regression of thesteady-state stress-strain data (orange dots in the inset of Fig. 2.1b). Linear regres-27sion is implemented by invoking the MATLAB function fitlm. Young’s modulus iscalculated from the experimental data of up to 40% strain, where the stress-strainbehavior of phantoms is still in the linear region [85]. The measured Young’s mod-uli from linear regression match with those predicted by the QLV model.As explained in the previous section, the relaxation times τis are selected know-ing (from experiments) that no stress relaxation occurs in the first 0.01 secondsand after 10 seconds. To culminate this section, we examine the outputs of theQLV model without considering this a priori information. For this purpose, weshould add two more relaxation times to consider stress relaxation before 0.01 sec-onds and after 10 seconds. Thus, τ ′ and τ ′′ are added where τ ′− τ0 = 0.01s andτ ′′−τ0 = 100s. Having these two relaxation times in Eqs. (2.8) and (2.9), we re-fitthe experimental data to Eq. (2.9) and re-obtain the stress amplification factors.The results show that the amplification factors corresponding to the time intervals(τ ′, τ0) and (τ3, τ ′′) are negligible and other gis match the previous results. Thatis to say, the QLV model works well even with no a priori information about therelaxation behavior of materials.While the proposed method demonstrates the potential for characterizing softmaterials, some limitations exist. First, the derived parameters and the accuracy ofthe data fit depend on the choice of relaxation times. One can improve the accuracyof data fit by considering more relaxation times but at the expense of increasingcalibration parameters. Also, the QLV model assumes that the effects of time andstrain on the material’s viscoelasticity are separable, which is not necessarily truefor all the materials. That is to say, when the proposed model does not demonstratean acceptable agreement with the actual behavior of a material, one should considerusing another model.2.3 SummaryThis chapter presents a method for characterizing the QLV properties of soft mate-rials using uniaxial compression test. Unlike the previous approaches for calibrat-ing the QLV model, we use two different experimental data for deriving elastic andviscous parameters separately. Our method can reduce the negative effect of over-shooting on the estimation of the QLV parameter. The proposed model is shown28Figure 2.5: Material characterization: lines are experimental results whilesymbols denote predictions of the QLV model (Eq. (2.9)). QLV param-eters, including α , E, and gi, can well distinguish the phantoms. Theexperiments are repeated three times for each sample, and the mean val-ues for the results are presented with the error bars.to characterizes the material by parameters with narrow confidence intervals (highaccuracy), independently from the loading conditions. Hence, it represents an im-provement with respect to other viscoelastic models (e.g., spring-dashpot models)that introduce different parameters for each loading condition [12, 33]. To test ourmethod, we developed low-cost tissue-mimicking materials that match Young’smodulus of healthy and cancerous prostate tissues. Characterization of these sam-ples with our method allows for a precise distinction between materials havingsimilar elastic (viscous) properties but different viscous (elastic) properties. How-ever, the uniaxial compression test is not always applicable for in-vivo applications.Also, compressing the whole tissue may cause some irreversible damages. For clin-ical applications, indentation test is recommended as 1) it can be applied on tissueswith various size and geometries [43], and 2) it is non-invasive and so preservesthe tissue’s structure [44]. Unlike uniaxial compression test, the indentation test29causes three-dimensional stress-strain fields and so Eq. (2.1) is not valid. There-fore, a procedure is required to extend the proposed method to a three-dimensionalfield, which is discussed in the next chapter.30Chapter 3Quasi-linear ViscoelasticCharacterization of SoftMaterials via Indentation Testand Finite Element AnalysisIn this chapter1 we present a new method for characterizing the QLV parametersof viscoelastic materials via indentation test. Using pre-calculated finite elementanalysis on three-dimensional stress-strain field, our method is able to obtain theparameters in a few seconds that suggest its application in real-time.This chapter is organized as follows: first, we provide the equation for three-dimensional QLV model and a finite element approach for solving the equation.The finite element method is implemented in ABAQUS® by invoking a user ma-terial subroutine (UMAT). We validate our finite element simulation with exper-imental results and then use it for studying the indentation stress response of amaterial under different QLV parameters, indentation depths, indenter sizes, andindentation rates. Our results, for the first time, demonstrate that1A form of this chapter were published in: ”H. Helisaz, M. Bacca, and M. Chiao, “A NewCharacterization Procedure for Quasi - Linear Viscoelastic Materials Using Indentation Test: Val-idation with Finite Element and Experimental Results,” Exp. Mech., no. 0123456789, 2022, doi:10.1007/s11340-022-00837-7.”31• Indentation depth and indenter size do not have effect on the stress relaxationbehavior of materials in indentation test• Each QLV parameters has a specific (and quantifiable) effect on the stressrelaxations over different time intervals• Stress relaxation is linearly correlated with the shear relaxation moduliBased on these findings, we propose a procedure for material characterizationusing an indentation test with a flat-ended tip. The procedure provides a straight-forward approach for material characterization that is significantly faster than in-verse finite element methods. Finally, we evaluate the accuracy of the procedureby characterizing the same material once with the procedure and once again withthe method described in Section 2 and compare the parameters. Our procedure isshown to have high precision (standard deviation less than 10%) and good accuracy(error less than 20%).3.1 Methods and Materials3.1.1 Three-dimensional QLV Constitutive EquationThe equation presented in Eq. (2.1) is only valid for one-dimensional stress-strainfield and it cannot be used for indentation test. In three dimensions, QLV-basedconstitutive models describe the stress tensor S(t) in terms of 2nd Piola-Kirchhoff:[35, 86]S(t) = SE(t) : G(0+)+∫ t0SE(τ) :∂G(t− τ)∂ (t− τ) dτ (3.1)where SE is the elastic stress response, emerging at steady-state, which is expressedin the form of 2nd Piola-Kirchhoff [35]. The colon operator ( : ) also represents thedouble dot product. In the general case, the Cauchy stress tensor is obtained from:σ (t) = J−1F (t)S(t)F (t)T (3.2)where F is the deformation gradient tensor (bold letter indicates a tensor) andJ = det(F ) is volumetric stretch and S is obtained from Eq. (3.1). Considering32isotropic relaxation behavior, i.e., stress relaxation behavior is the same in all di-mensions, the reduced relaxation tensor in Eq. (3.1) becomes the scalar functionG(t), and the equation is rewritten as:S(t) = SEG(0+)+∫ t0SE(τ)∂G(t− τ)∂ (t− τ) dτ (3.3)Assuming a hyperelastic constitutive model for predicting elastic stress, Se is de-rived as follows:SE = 2∂ψ∂C−PC−1 (3.4)where ψ is the strain energy density function, and P is the hydrostatic pressure.C is also the right Cauchy-Green deformation tensor, defined as C = F T F =∑3i=1λ 2i C i ⊗N i with λ 2i and Ni the ith eigenvalue and eigenvector of C, respec-tively, and⊗ the outer product symbol. Note that λi represents the principal stretchapplied in the ith direction. Linear elastic compressibility gives P as a function ofbulk modulus κ:p = κ(J−1) (3.5)Therefore, one can enforce incompressibility (J = 1) by selecting a large value forbulk modulus since, as κ → ∞, J should approach one to provide a finite valuefor P. Because large κ can cause instability in the simulation, the remedy is toselect an appropriate finite element mesh. Here, we assume incompressibility byadopting κ ≥ 104µ . A one-term Ogden model expresses the strain energy densityas:ψ =2µα2(I1(Cα/2)−3)(3.6)where I1 represents the first invariant (trace) of the tensor. Substituting Eq. (3.6)into Eq. (3.4) we have:SE =(2µαCα/2− p)C−1 (3.7)Note that C is a symmetric matrix, and so ∂∂C I1(Cα/2)=Cα/2−1. The relaxationfunction in Eq. (3.3) can be represented by a Prony’s series presented in Eq. (2.8).One can substitute Eqs. (3.7) and Eq. (2.8) into Eq. (3.3), and solve the integral33with the trapezoid method as follows:S(t) =SE(1+n∑k=1gk)− ∆t2n−1∑j=1N∑k=1[SE (t j)gktke−(t−t j)tk +SE (t j+1)gktke−(t−t j+1)tk] (3.8)where the subscript j specifies the variable calculated at t = t j. SubstitutingEq. (3.8) in Eq. (3.2), one can derive the expression for three-dimensional Cauchystress for an incompressible, isotropic material. The finite element implementationof Eq. (3.8) will be discussed in the following section. To incorporate Eq. (3.8)into the commercial software ABAQUS®, we also need to calculate the elasticitytensor ℓ= ∂σ /∂ε where ε is the strain tensor. This can be derived by transformingL = 2∂S/∂C using the gradient tensor. Assuming incompressibility, L has thefollowing index form [87]:ℓi jhk = FipFjqFhrFksLpqrs (3.9)where ℓi jhk is the component i, j,h,k of l; Fi j is the component i, j of ; Li jhk is thecomponent i, j,h,k ofL ; and repeated indices indicate a sum. Differentiating S inEq. (3.8) with respect to C results in the following expression for tensorL :L = 2∂SE∂C(1+n∑k=1gk− ∆t2N∑k=1gkτk)(3.10)Note that all of the terms in the sum over the time steps (∑n−1j=1∑Nk=1[. . .] in thesecond term) are eliminated except the last term since the derivation is calculatedat time t = tn [88]. The derivative of SE in Eq. (2.7) with respect to C results in thefollowing expression:∂SE∂C= µ(1− 2α)Cα/2−2+PC−2− ∂P∂CC−1 (3.11)where, from Eq. (3.4), ∂ p∂C = κJ2C−1. Further calculations for ∂Se∂C are availablein the works of Connolly et al. [89]. For the sake of brevity, we do not repeat34the calculations here and refer the reader to these references. To incorporate Eq.(3.3) and Eq. (3.10) into ABAQUS®, the FE approach needs to be implementedfor these equations, which is discussed in the next section.3.1.2 Finite Element ImplementationABAQUS® is able to model finite-strain viscoelasticity by splitting stress into volu-metric and deviatoric parts (see Abaqus Theory Manual Version 6.11, section 4.8.2and [90]). In particular, the QLV model can be implemented in ABAQUS® bycombining built-in hyperelastic models (for elastic properties) and Prony’s series(for viscous properties). Meanwhile, the expression for Prony’s series in ABAQUSis different from Eq. (2.8) (see ABAQUS Analysis User’s Guide Version 6.12, sec-tion 22.7.1). Using the ABAQUS’s expression, the elastic part in the QLV formulais defined as the instantaneous stress, which appears immediately after the defor-mation (see ABAQUS Analysis User’s Guide Version 6.12, section 22.7.1). Thisdiffers from the current formula that defines elastic stress as that which emergesat steady-state (see Eq. (3.1) and Section 2.1). The steady-state stress happenswhen viscous part of the stress vanishes, and the purely elastic part remains, andso we believe it is a better representative of the elastic stress. Also, unlike instan-taneous stress, steady-state stress only depends on the strain and the parameters ofthe material, and not on the indentation speed. So, we translate Eqs. (3.2) and (3.9)into ABAQUS® using a UMAT subroutine in Fortran. The UMAT was writtenbased on the program by Connolly et al. [87] for simulating hyperelastic materi-als. We modified their code so that it can model quasi-linear viscoelastic materials.Fig. 3.17 demonstrates the flow chart for the algorithm we used in the UMAT. Thesoftware receives material properties and deformation gradients as inputs and cal-culates Cauchy stress (Eqs. (3.2) and (3.8)) and the elasticity tensor (Eqs. (3.9) and(3.10)) as outputs. The material properties, including the coefficients for the hy-perelastic model and Prony’s series, are inputted by the user, and the deformationgradient is calculated at each timestep by ABAQUS. Then, the UMAT calculatesthe right Cauchy-Green tensor C and its eigenvalues α2 and eigenvectors N . TheOgden strain energy density ψ is also calculated and its first-order derivative iscomputed. Having the parameters for Prony’s series (g and τ), the code estimates352nd Piola-Kirchhoff stress from Eq. (3.8) and converts it to Cauchy stress using Eq.(3.2). Note that the summation term in Eq. (3.8) requires storing the informationfrom previous steps. For this purpose, we define state variables (STATEV) to storethe normal and shear components of stress in three dimensions. Finally, the soft-ware finds L by substituting ∂Se∂C from Eq. (3.11) and obtains the elasticity tensorℓ from Eq. (3.9).The UMAT described in Fig. 3.17 is translated into ABAQUS 2018 by Intel’sParallel Studio (XE 2018) compiler. The axisymmetric model shown in Fig. 3.1ais created in ABAQUS by including a deformable part, as the soft material, anda rigid analytical part, as the indenter. The deformable part is supported fromthe bottom by a rigid analytical layer with the encastre boundary condition. Allcontacts in the model are assumed to be frictionless and hard contacts. We uselinear quadrilateral mesh elements (CAX4H) for the deformable part and performa mesh independence analysis to ensure that the solution is independent of meshsize. The model includes two steps: first, the indenter compresses the material byimposing a displacement boundary condition. The indentation rate is controlled bytailoring the step’s time period. In the second step, the indenter remains at the samedisplacement and waits 10 seconds for stress relaxation. A fixed time incrementsize of 10−4 s is chosen for both steps to guarantee simulation stability. Each runtakes around 10 hours, after which we derive the indenter’s normal stress in they-direction by averaging the stress of elements beneath the indenter.3.1.3 Model ValidationFig. 3.2 validates the results of the FE analysis with those obtained from the in-dentation experiment on a silicon-based polymer. We fabricated the polymer usingthe commercially available elastomer kit Ecoflex™ 00-10 (Smooth-On Inc., Penn-sylvania, USA), following the procedure mentioned in Section 2.1. The fabricatedmaterial had diameter D = 25 mm and height H = 18.88 mm. The material’s QLVparameters were then obtained by following the process described in the previ-ous section. Table 3.1 lists the QLV parameters for the material, which was usedin ABAQUS®. For the relaxation function, we selected three relaxation times:τ1 = 0.1 s, τ2 = 1 s, and τ3 = 10 s, same as Section 2.1.36Figure 3.1: ABAQUS® simulation: a) axisymmetric model with a de-formable part, for the soft material (shown in green), a rigid analyticalpart, for the indenter (shown in blue), and a rigid analytical part for thebottom support (shown in red); b) the indentation stress is expressed asvon Mises stress; the indenter size ϕ and indentation depth δ are alsoshown in the figure. The results are depicted for ϕ = 7.60 mm, H =18.88 mm, and D = 25 mmTable 3.1: The QLV parameters of the fabricated silicon-based polymer; theparameters were obtained from uniaxial compression tests (see Chapter2) and were used as the input for the ABAQUS® simulation.α µ g1 τ15.39 ±0.07 3.98±0.02 kPa 0.33 ±0.10 0.1 sg2 τ2 g3 τ30.48 ±0.02 1 s kPa 0.13±0.00 10 sThe mechanical tester Mach-1 V500CS (Biomomentum Inc., Laval, QC, Canada)was used to perform the indentation experiments. The tester was equipped with themultiple-axis load cell MA234 (Biomomentum Inc.) with a range of 35 N, a reso-lution of 1.75 mN, and a recording rate of 100 Hz. The load cell was connected tothe material by an indenter. Multiple indenters were made with different diametersϕ; i.e., ϕ = 4.95 mm, 6.33 mm, 7.60 mm, and 8.88 mm . The indenters were madeof 316 stainless steel, which is considered rigid when contacting a soft material.37Figure 3.2: Model validation: The ABAQUS® simulation results are com-pared with those obtained from experiments on a silicon-based poly-mer. Lines denote the QLV model prediction, and the symbols are theexperimental results. The experiment includes compressing an indenterinto the specimen at a rate of 10 mm/s and recording the force response(F ). The force data is then converted to stress (σ ) by dividing forceby the indenter surface area (π4ϕ2). Each experiment is repeated threetimes, and the mean values for the results are depicted along with theerror bars. The results are depicted for specimen height H = 18.88 mm,specimen diameter D = 25 mm, and indenter diameter ϕ = 7.60 mmThe specimen was supported by a petri dish from beneath. The indentation testsincluded fast compression of the sample at a rate of 10 mm/s for several indenta-tion depths δ , and then waiting for 10 seconds for stress relaxation. The indentationforce F was recorded by the load cell as the normal force component. Note thatindentation force is not uniform and F represents the average stress beneath thepunch. To ensure a frictionless contact, we compared F with and without lubri-cant, where no difference was observed (see Fig. 3.3). The ramp-and-hold indenta-tion protocol compels the monomer molecules of the sample to move through thepolymer chains. The movement, however, is resisted by the viscosity of monomersthat causes a peak stress feedback at the beginning. As viscosity diminishes over38time, the resistance against movement decreases, and stress relaxation starts untilthe stress reaches a steady-state. Fig. 3.2 compares the indentation stress obtainedduring the experiment with those predicted by ABAQUS® from simulation. Theexperimental stress was obtained by dividing the indentation force (F ) by the in-denter surface area π4ϕ2. As shown in the figure, a good agreement is seen betweenthe experimental and simulation results for different indentation depths δ . Fig. 3.2is depicted for an indenter size of ϕ = 7.60 mm; the results for other indenter sizesalso show a good agreement with the experiments (Figs. 3.4- 3.6). These resultsvalidate the applicability of the FE analysis that we developed in this section.3.2 Results and DiscussionAfter the QLV finite element model was validated by experiments, we focused onsimulating indentation experiments under different conditions. We changed theelastic (µ and α) and relaxation parameters (gis) and investigated the indentationbehavior over time. We also changed the indenter size, indentation depth, inden-tation tilt angle, and indentation rate to review the effects of loading condition onthe indentation stress. We then used the findings to construct a general indenta-tion procedure for characterizing viscoelastic materials without using inverse FEmethods.3.2.1 The Elastic Part of Indentation BehaviorFigs. 3.7a and b demonstrate the effects of µ and α on the indentation behavior,when ϕ = 7.6 mm and δ = 5 mm. The relaxation coefficients are chosen to beg1 = 1 and g2 = g3 = 0 for both figures. As expected for viscoelastic materials, weobserve a peak stress followed by a stress relaxation over time until stress reaches asteady-state, which as mentioned earlier, represents elastic part of the stress. Fromthis point forward in this manuscript, the steady-state stress is referred to as elasticstress and denoted by σE . The figure suggests a proportional effect for µ on theelastic stress, which is consistent with Eq. (3.6), where µ serves as a multiplierfor the elastic stress. Unlike µ , the effect of α on elastic stress is nonlinear, whichagain, is expected from Eq. (3.6). Fig. 3.7 demonstrates that indentation stressesare sensitive to both µ and α . This result differs from those of Fellay et al. [91]39Figure 3.3: Comparing the stress feedback with and without using WD-40silicone lubricant (WD-40 Company Inc., California, USA). The fig-ure demonstrates no difference between the results, which confirms thefrictionless contacts in the experiment. The symbols indicate the experi-mental results for a silicon-based polymer with height H=18.88 mm anddiameter D = 25 mm.who reported that the indentation stress remains unchanged for α ≥ 2. Consideringthe same stress for different values of α , the authors concluded that the indentationtest is not able to characterize the materials with α ≥ 2. Our data in Fig. 3.7,however, show that the stress response is changing with α even when α ≥ 2, andso the indentation test can characterize the materials.The results in Fig. 3.7 are depicted for a specific indentation diameter ϕ andindentation depth δ . In Fig. 3.8, we investigate the effects of ϕ and δ on the elas-tic stress. The figure displays the elastic stress for different sizes of indenter atdifferent indentation depths. We nondimensionalize δ by ϕ and σ by µ . The re-sults show that the relation between indentation depth and stress is non-linear. Inaddition, the curves for different indenter sizes do not collide after nondimension-alization, which deviates from the prediction of the analytical indentation models40Figure 3.4: Model validation for indenter diameter ϕ = 4.95 mm: theABAQUS® simulation results are compared with those obtained fromexperiments on a silicon-based polymer. Lines denote the QLV modelprediction, and the symbols are the experimental results. The results aredepicted for specimen height H = 18.88 mm and specimen diameterD = 25mm.[54]. Attempts have been made to modify the analytical models for biological tis-sues, mainly by introducing a correction factor. For the sake of brevity, we do notrepeat the same process here, and readers are referred to previous works [54, 92].3.2.2 The Viscous Part of Indentation BehaviorIn the QLV model, the effects of elastic and viscous parameters on the material’sstress are assumed to be separable [35]. That is to say, SE(t) and G(t) in Eq. (3.1)are independent. To reflect this separability, we nondimensionalize stress by σEin the figures to exclude the effect of elastic parameters, so the figures becomeindependent of the values for µ and α . Fig. 3.9 demonstrates the evolution ofnondimensionalized stress for different values of δ and ϕ , revealing that the relax-ation behavior is the same for different values of indentation sizes and depths. Our41Figure 3.5: Model validation for indenter diameter ϕ = 6.33 mm: theABAQUS® simulation results are compared with those obtained fromexperiments on a silicon-based polymer. Lines denote the QLV modelprediction, and the symbols are the experimental results. The results aredepicted for specimen height H = 18.88 mm and specimen diameterD = 25mm.results demonstrate that the same is also true for tilted indentation, when the inden-ter is not aligned with contact surface normal (see Fig. 3.16). These observationsare consistent with our reduced relaxation function (see Eq. (2.4)), where G(t) isonly dependent on time and scalar coefficients gis and τis. The same relaxationbehavior is hence expected in Figs. 3.9a-d since gis and τis are the same.Having the same relaxation behavior for different δ and ϕ is not necessarilytrue for materials that do not comply with the QLV model. In particular, differentbehaviors might be seen for materials whose elastic and viscous characteristicsare not separable, such as collagenous soft tissues [93]. Comparing Fig. 3.8 andFig. 3.9, one can conclude that relaxation behavior can serve as a better tool formaterial characterization than elastic behavior. That is because, unlike the former,the latter is dependent on the indentation size and depth, and so the effect of δ and42Figure 3.6: Model validation for indenter diameter ϕ = 8.88 mm: theABAQUS® simulation results are compared with those obtained fromexperiments on a silicon-based polymer. Lines denote the QLV modelprediction, and the symbols are the experimental results. The results aredepicted for specimen height H = 18.88 mm and specimen diameterD = 25mm.ϕ should be excluded before characterization.Fig. 3.10 examines the effect of material properties on relaxation behavior.Fig. 3.10a, b, and c review the impact of g1, g2, and g3, respectively. The resultsare expressed in the nondimensionalized form of σˆ = (σ −σE)/σE . As seen inthe figure, the stress relaxation is directly proportional to the value for gi; the stressrelaxation for gi = 2 is twice that of gi = 1. This observation is expected since gisserve as scalar multipliers in the relaxation function (see Eq. (2.8)). Fig. 3.10devaluates the relaxation behavior created by more than one relaxation coefficient.The results demonstrate that the stress relaxation associated with multiple gis isderived by adding the stress relaxation made by each. For instance, the solid linein Fig. 3.10d is obtained by adding the red lines in Figs. 3.10a-c. Therefore, nocross-link effects occur between the stress relaxation coefficients gis. A similar43Figure 3.7: Effect of elastic parameters (µ and α) on indentation behavior:effect of a) shear modulus µ and b) strain-stiffen parameter α on thestress response of the material. The behavior shows a peak stress fol-lowed by stress relaxation until the results reach a steady-state. Thesteady-state stress occurs when the viscous properties of a material van-ish, and only the elastic properties remain. The results demonstrate thatthe indentation stress is sensitive to both µ and α: it has a linear rela-tionship with µ and a non-linear relationship with α .conclusion was also reached by Amabili et al. [86], who introduced independenteffects for relaxation coefficients under different directions.The relaxation behavior of a material is influenced by the indentation time,which determines the time when stress relaxation begins (τ0 in Fig. 3.11), andmore stress relaxation is expected for lower indentation time. In particular, as it isshown in Section 2.1, when τ0 → 0, stress relaxation reaches its maximum value∑i gi. Fig. 3.12 plots the stress relaxation at different indentation rates. The resultsare obtained by keeping the indentation depth constant (δ = 5mm) and changingthe indentation time. τ0 is nondimensionalized by τi to compare the indentationtime against the relaxation times. As expected, we observe more stress relaxationfor small values of τ0, when the indentation rate is high. Interestingly, the figureshows that stress relaxation depends on the ratio of τi/τ0 and not on individualvalues for τi and τ0.The relaxation behavior in Fig. 3.12 approaches one as the ratio of τi/τ0 in-creases. This is consistent with our expectation for stress relaxation to reach ∑i giin super-fast indentation (when τi/τ0 →∞). Our results reveal that when τi/τ0 ≥ 10,the indentation can be assumed to be super-fast, and the stress relaxation can be44Figure 3.8: Effect of indenter size and indentation depth on the elastic stressof the material. The results are nondimensionalized to clarify the im-pact of each parameter; elastic stress is nondimensionalized with shearmodulus µ and indentation depth δ with indenter diameter ϕ . Unlikethe prediction of linear elasticity, the curves for different indenter di-ameters do not intersect, and the data suggests a non-linear relationshipbetween stress and indentation depth. The results are depicted for spec-imen height H = 18.88 mm and specimen diameter D = 25 mm.approximated as ∑i gi. That is to say, one can capture the full relaxation behaviorwhen the indentation time is ten times smaller than the relaxation time. On theother hand, when the indentation rate is low (τi/τ0 ≤ 10), more than 80% of thestress relaxation is not perceived. Based on the data in Fig. 3.12, we recommendthe indentation time to be 0.1τm ≤ τ0 ≤ τm, where τm is the smallest relaxation timeof the material, to ensure that at least 60% of the stress relaxation is captured.Fig. 3.13 investigates the stress relaxation occurring in Fig. 3.12 during dif-ferent time intervals. As described in Fig. 3.11, ∆σˆτi−τi+1), the normalized stressrelaxation in the time interval (τi− τi+1), is obtained by subtracting the normal-ized stress at τi from that of τi+1(∆σˆτi−τi+1 =(στi −στi+1)/σE). According toFig. 3.13, the stress relaxation caused by gi mostly occurs in (τi−1,τi+1), and for45Figure 3.9: Nondimensionalized stress behavior for different indenter diam-eters and indentation depths: nondimensionalized stress response at in-dentation diameters of a) ϕ = 4.95 mm, b) ϕ = 6.33 mm, c) ϕ = 7.60mm, and d) ϕ = 8.88 mm. The results demonstrate that relaxation be-havior does not depend on the indentation size and depth.example, the stress relaxation made by g1 = 1 mainly occurs between τ0 and τ2.Furthermore, the stress relaxation in the last time interval (∆σˆτ3−∞) is affected onlyby the last coefficient g3 because ∆σˆτ3−∞ is zero when g1 = 1 or g2 = 1. Likewise,(∆σˆτ2−τ3) is dependent on g2 and g3, and (∆σˆτ1−τ2) is changing with all three gis.As expected, the stress relaxation in different time intervals depends on the in-dentation rate, where a faster indentation rate (higher τi/τ0) leads to more stressrelaxation.3.3 The New Material Characterization ProcedureOur results in Section 3.2 demonstrate that each gi has specific effect on the stressrelaxation which is proportionally related to the value for gi (see Fig. 3.10). Ithas been also shown that the indentation depth and indenter size do not affect thestress relaxation (see Fig. 3.9). Based on these findings, we present a new method46Figure 3.10: Effect of material properties on the viscous behavior: evolutionof nondimensionalized stress relaxation over time for different valuesof a) g1, b) g2, and c) g3: the results indicate that the stress relaxationis proportional to the value of gis. The relaxation caused by gi = 2 istwo-times that of gi = 1; d) the nondimensionalized stress relaxationassociated with multiple gis; the figure shows that the stress relaxationcaused by several gis is obtained by adding the stress relaxations madeby each.for deriving gi from stress relaxation response of a material. Fig. 3.14 explainsour suggested indentation procedure for material characterization. After record-ing the stress response of material from the indentation test (see Fig. 3.14a), wecalculate the stress relaxation in different time intervals; i.e., ∆σˆτ1−τ2 , ∆σˆτ2−τ3 ,and ∆σˆτ3−∞. The results of Fig. 3.13 prove that stress relaxation in each time in-terval can be separated into three parts, each associated with one of the gis; thatis to say, ∆σˆτi−τi+1 = ∑k∆σˆgkτi−τi+1 where the superscript gk shows that the corre-sponding part is associated with gk. Moreover, our findings in the previous sec-tion show that stress relaxation is proportionally related to the value for gi, and so∆σˆgkτi−τi+1 = gk∆σˆgk=1τi−τi+1 where ∆σˆgk=1τi−τi+1 is stress relaxation created by gk = 1 (seeFigs. 3.14b-d). Therefore, we have:47Figure 3.11: Stress relaxation in different time intervals: four time intervalsare determined by τ0, τ1, τ2, and τ3, where τ0 measures the indentationtime and τ1, τ2, and τ3 denote the relaxation times. The stress relax-ation in the time interval (τi−1,τi) is obtained by subtracting the stressat t = τi(στi) from the stress at t = τi−1(στi−1). The data is nondimen-sionalized by the elastic stress σE .∆σˆτi−τi+1 =3∑k=1gk∆σˆgk=1τi−τi+1 (3.12)where ∆σˆτi−τi+1 is obtained from the indentation experiment (Figs. 3.14a) and∆σˆgk=1τi−τi+1 is calculated from simulation (Figs. 3.14b-d). Eq. (3.11) should be solvedfor three unknowns, gis, which characterize the relaxation coefficients, gis, withoutneeding to invoke time-consuming inverse methods. It should be mentioned thatthe values given in Figs. 3.14b-d are valid for a specific indentation rate (τ0 =0.5 s). Figs. 3.15 shows the values for ∆σˆg1=1,∆σˆg2=1, and ∆σˆg3=1 at differentindentation rates. Based on the indentation rate used in the experiment, one canfind the corresponding values for stress relaxation and substitute them in Eq. (3.12)to derive g1, g2, and g3.We evaluated the accuracy of our method by comparing the parameters de-48Figure 3.12: Effect of indentation rate on the viscous behavior: evolutionof nondimensionalized stress relaxation over time when a) g1 = 1, b)g2 = 1, and c) g3 = 1. The indentation rate is measured by indentationtime τ0, which is measured against τi. Stress relaxation increases asτi/τ0 increases; in particular, when τi/τ0 ≥ 10, the indentation rate issufficiently fast to fully capture the stress relaxation (σ −σE/σE ≈ 1).In addition, the stress relaxation is shown to depend on τi/τ0 and noton the individual values of τi and τ0. While τi differs in different plots,the same stress relaxation is observed when τi/τ0 is the same.rived by Eq. (3.12) from the indentation test with those obtained from the 1Duniaxial compression test. We used the method described in Section 2.1 to obtainthe QLV parameters from uniaxial compression. The indentation and compressiontests were done on three different elastomers and the parameters are comparedin Table 3.2. Elastomers were made using commercially available elastomer kitsEcoflex™ 00-10, Ecoflex™ 00-30, and Ecoflex™ 00-50 along with NOVOCS™solvents (Smooth-On Inc., Pennsylvania, USA). The elastomers’ fabrication andthe experimental apparatus are similar to what is described before in Section 2.1.Each experiment was repeated three times and the mean values are reported with49Figure 3.13: The stress relaxation of Fig. 3.12 in different time intervals; thedata is nondimensionalized by elastic stress, and expressed as a per-centage. The coefficient gi affects the stress relaxation mainly between(τi−1,τi+1); in particular, the stress relaxation over the last time inter-val is only influenced by g3 since ∆στ3−∞ is zero when g1 = 1 or g2 = 1.Likewise, stress relaxation in (τ2,τ3) depends on g2 and g3 (∆στ2−τ3)is zero when g1 = 1 and the stress relaxation between (τ1,τ2) dependson all three relaxation coefficients.the standard deviation. Table 3.2 demonstrates a good precision (small standarddeviation), though the accuracy is not appropriate for all cases; in particular, thederived values for g1 from the indentation test are different from those obtainedfrom the compression test. To improve the method’s accuracy, we increased thenumber of points in Eq. (3.12) and minimize the difference E:E =∞∑τi=τ0(∆σˆτi−∞−3∑k=1gk(∆σˆgk=1τi−∞))2(3.13)We used the 0.01s time steps in Eq. (3.13) and derived the materials’ propertiesagain (see Table 3.2). The results show a high precision (standard deviation lessthan 10%) and good accuracy (error within 20%) when compared to the parameters50Figure 3.14: The suggested characterization procedure; a) stress relaxationfrom a ramp-and-hold indentation experiment. The figure shows stressrelaxation created by b) g1 = 1, c) g2 = 1, and d) g3 = 1. Note thatthe values for ∆σˆg1=1τ2−τ3 ,∆σˆg1=1τ3−∞, and ∆σˆg2=1τ3−∞ are all zero. Eq. (3.12)compares the stress relaxation at three points (t = τ1, t = τ2, and t = τ3in figure a) with those of pre-calculated simulation results (figures b-d) and derives the material’s parameters. The results are depicted forindentation time τ0 = 0.5 s.from the uniaxial test.Eq. (3.13) derives the materials’ parameters within a few seconds (∼ 5 sec-onds), making it appropriate for real-time applications. Also, as mentioned earlier,the same method can be applied for indentation tests with different indenter diam-eters, indentation depths, and indentation tilt angles (misalignment) (see Figs. 3.9and 3.16). Nevertheless, the accuracy of the results is compromised when the in-denter size increases and the stress state approaches the uniaxial compression case(ϕ ∼ D); in particular, the normalized stress relaxation for uniaxial compression(1D QLV model) is different from those shown in Fig. 3.12 by 40%. Our results donot suggest that the method should be used when the indenter is larger than half ofthe specimen diameter. Furthermore, the new characterization procedure is validfor materials that follow the QLV constitutive equations and the results are not nec-51essarily true for materials whose elastic and viscous behavior cannot be separated.Finally, this chapter describes the viscous behavior by identifying the relaxationparameters, and it does not analyze other viscous aspects, like the loss modulus.The loss modulus describes the short-term viscoelastic behavior while our focus ison long-term viscoelastic response characterization. Studying the long-term vis-coelastic behavior is more effective for detecting subtle microstructural changesin the material. Finally, the proposed method is more focused on viscous proper-ties, and it is useful when the material can be characterized by viscous parameters.When elastic and viscous parameters are both needed for material characterization,inverse FE methods can be used, even with the longer processing time.Table 3.2: Comparing the relaxation parameters obtained from uniaxial com-pression and indentation: the parameters are first obtained from the com-pression test using the method described in Section 2.1 and comparedwith those calculated from an indentation test using our proposed meth-ods (Eqs. (3.12) and (3.13)). Compression and indentation tests wererepeated three times for each material and the mean values are reportedalong with the standard deviations. The table demonstrates that Eq.(3.13) introduces parameters with a high precision (standard deviationwithin 10%) and in a good agreement with uniaxial parameters (maxi-mum error within 20%).ElastomerUniaxial Compression(using the method inSection 2.1 )Indentation(Eq. (3.12))Indentation(Eq. (3.13))80% Ecoflex™ 00-10+20% NOVOCS™0.57±0.010.16±0.000.09±0.000.63±0.000.13±0.000.09±0.000.51±0.010.13±0.000.11±0.00Ecoflex™ 00-30 0.13±0.020.03±0.010.03±0.010.18±0.050.03±0.010.03±0.000.11±0.010.03±0.000.03±0.00Ecoflex™ 00-50 0.15±0.010.05±0.000.04±0.000.25±0.030.05±0.010.04±0.000.18±0.000.05±0.000.04±0.003.4 SummaryIn this chapter, we studied the viscoelastic behavior of a material under differentindenter sizes and indentation depths, and revealed that the stress relaxation is de-pendent on neither. The stress relaxation was also investigated for different shear52Figure 3.15: Effect of indentation rate (measured by τ0) on the stress relax-ations created by g1 = 1,g2 = 1, and g3 = 1. The results demonstratethe stress relaxation between a) (τ1−τ2), b) (τ2−τ3), and c) (τ3−∞).The stress relaxation decreases as the indentation time (τ0) increases(slower indentation). No stress relaxation is created by g1 after t = τ2(∆σˆg1=1τ2−τ3 and ∆σˆg1=1τ3−∞) are zero in figures b and c) and no stress relax-ation is made by g2 after t = τ3 (∆σˆg2=1τ3−∞ is zero in figure c)). Sincethe values of ∆σˆg1=1,∆σˆg2=1, and ∆σˆg3=1 are dependent on the in-dentation rate, the values substituted in Eq. (3.12) and (3.13) shouldcorrespond to τ0 used in the experiment.relaxation moduli and it was shown that each relaxation modulus governs the stressrelaxation in a specific time interval, so that the stress relaxation is proportionallyrelated to the relaxation modulus. Based on these findings, we introduced a proce-dure for characterizing materials via indentation test and showed that our methodcan derive the material properties in a few seconds. Moreover, we compared theparameters derived with our method with those obtained from uniaxial compres-sion test (Section 2.1) and demonstrated that our method has high accuracy (errorwithin 20%) and precision (standard deviation within 10%). Having a method tocharacterize soft tissue, we are now ready to derive the viscoelastic properties ofprostate tissue with and without cancer and study the effect of cancer on the prop-53Figure 3.16: Normalized stress for indentation tests with different indenta-tion tilt angles, defined as the angle between the indenter and contactsurface normal (θ ). The results demonstrate that tilt angle (misalign-ment) up to 20◦ does not affect the stress relaxation. Note that thefigure represents the experimental data since the indentation test with0 cannot be modelled with an axisymmetric simulation.erties, which is the subject of next chapter.54Figure 3.17: Algorithm for writing the UMAT software in Fortran; theUMAT receives the deformation gradient tensor (from ABAQUS cal-culations) and material properties (from user inputs) and provides theCauchy stress and elasticity tensor.55Chapter 4Effect of Cancer on theViscoelastic Properties ofProstate TissueIn this chapter, we aim to use the characterization method described in Chapters2 and 3 for studying the effect of cancer on the stiffness and viscous properties ofprostate tissue. Thirty-five fresh prostate samples were collected within two hoursafter their removal by radical prostatectomy surgery and characterized by the pre-viously described process. We performed tests for four locations of the posteriorprostate, and then transferred the samples to the pathology lab for histological anal-ysis to reveal the presence or absence of cancer at each tissue site. The character-ization results were then correlated with the pathology reports to study the effectof cancer on the viscoelastic properties of the prostate. Our results demonstratethat the presence of cancer has statistically meaningful effects on the stiffness andrelaxation shear modulus of the prostate gland. In particular, we show that the firstshear relaxation modulus and stiffness increase by 14% and 30%, respectively dueto the presence of cancer.This chapter is organized as follows: first, we provide a brief review on ourcharacterization procedure and the design of a stress relaxation experiment forprostate tissue. We tested our method on prostate-mimicking phantoms with anembedded hard nodule representing a tumor. The phantoms were made using an56anatomical model that represents the shape of the prostate. After validating themethod, it was applied to real prostate tissues and the results are provided in thesecond section. The results include the effect of cancer and cancer grade on themechanical properties of the prostate. We also performed statistical hypothesistesting on the data to reveal the significance of our conclusions. The applicationof the results for PCa diagnosis is discussed in the third section where we suggesta possible approach for in vivo detection of PCa. Finally, the conclusion of thechapter is provided in the fourth section.4.1 Experimental Design for Ex Vivo TestingAs described in Chapter 3, material characterization was performed by feedingCauchy stress data from a ramp-and-hold experiment into Eqs. (3.2), (3.3), and(2.8) and solving for σE and gis. The stress relaxation experiment we used tocharacterize the prostate included a ramp-and-hold compression (Fig. 4.1a) wherethe sample was compressed at a rate of 10 mm/s from t = 0 to t = τ0 and kept at anindentation depth (δ = 5 mm in the figure) for 45 seconds. Note that we increasedthe waiting period after the compression to 45 seconds to ensure that the stress hadfully relaxed. Fig. 4.1b illustrates how the QLV parameters were derived from therelaxation stress response. Having σE and gis, the viscoelastic properties of thesoft tissues can be characterized.4.1.1 Model Validation with Prostate-mimicking PhantomsWe validated the applicability of this procedure by testing it on tissue-mimickingphantoms. Phantoms (Figs. 4.2a and b) were made to represent the posteriorprostate with a cancerous nodule on the left side (Prostate Model, 3B Scientific,Georgia, USA). Approximately 80% of the PCa has been shown clinically to stemfrom the posterior prostate [94]. After consulting with a pathologist expert, weconsidered three spherical nodules with diameters 5 mm, 7 mm, and 10 mm todemonstrate PCa in early, intermediate, and advanced stages, respectively. Eachnodule was embedded near the surface (see Fig. 4.2). Previous prostate-mimickingphantom constructions for elastography studies used similar approaches [95, 96].Phantoms were made using a commercial elastomer kit Ecoflex™ 00-10 (Smooth-57Figure 4.1: Procedure for mechanical testing: a) the sample is compressedat 10 mm/s and kept at an indentation depth (here, δ = 5mm for 45seconds), b) stress response after the compression shown in (a). Stresspeaks occurred at t = τ0 and then they relaxed over time until reachingthe steady-state stress σE . Relaxation shear moduli gis were determinedby the stress relaxation that occurred in different time periods, as spec-ified by relaxation times τks. Three relaxation times τ1, τ2, and τ3 werechosen that were 0.1, 1, and 10 seconds, respectively, after t = τ0.On Inc., Pennsylvania, USA), following the instructions given in Section 2.1. Wetailored the mechanical properties by adding NOVOCS™ solvents (Smooth-OnInc., Pennsylvania, USA) so that the Young’s modulus of our phantoms matchedthat of normal prostate tissue (15-20 kPa) [78]. The hard nodules were made withEcoflex™ 00-50 (Smooth-On Inc., Pennsylvania, USA) with a Young’s modulussimilar to that of cancerous prostate tissue (40-60 kPa) [78]. As mentioned earlier,the viscous properties of the phantoms did not necessarily resemble normal andcancerous prostate since the viscous properties of prostate tissue have not been wellestablished [79, 80]. To ensure that the phantoms had the same shape as the poste-rior prostate, we cured the polymer solution inside molds that had the same shapeas our prostate anatomy model. The molds were fabricated using Mold Start™ 15(Smooth-On Inc., Pennsylvania, USA).The testing apparatus was similar to that described in Sections 2.1 and 3.1: thespecimen was placed on a custom-built holder and an indenter with diameter ofϕ = 7.60 mm compressed the surface at δ = 5 mm and then waited for 45 secondsfor the stress to fully relax. Compression tests were repeated six times for eachside and the σE and gi were derived from each test. The values for σE and gis were58Figure 4.2: Tissue-mimicking phantoms with embedded nodules to resemblecancerous prostate tissue: a) phantoms were fabricated by Ecoflex™00-10 with elasticity close to the Young’s modulus of normal prostategland. The anatomical model was used to ensure that the phantom’sshape was similar to that of the posterior prostate, b) a hard nodule wasembedded on the left side of the phantom to mimic the presence of can-cer. Nodules were made by Ecoflex™ 00-50 with a Young’s modulusclose to that of prostate cancer. Three phantoms were fabricated withdifferent-sized nodules; i.e., 5 mm, 7 mm, and 10 mm, representing thepresence of cancer in early, intermediate, and advanced stages, respec-tively, c) mechanical tester (Mach-1 V500CS) equipped with a multiple-axis load cell to record the stress response of tissue at a sampling rate of100 Hz.averaged and then reported in Figs. 4.3 and 4.4 to compare the properties of the“cancerous” side (the side with the nodule) to the “normal” side (the side withouta nodule) under different nodule sizes. The relaxation shear moduli gis for thenodules were four-times smaller than the substrate material, and hence, gis at thecancerous side were expected to be smaller than those at the normal side (Fig. 4.3).Interestingly, the larger nodules were associated with more significant differencesbetween the properties. Also, variations of g1 were more significant than those ofg2 and g3. Fig. 4.4 illustrates that the cancerous side had more stiffness than thenormal side and the larger nodules led to more significant differences in elastic59stress σE . Figs. 4.3 and 4.4 confirm our procedure’s potential for detecting nodulesinside the phantoms.4.1.2 Experiment with Fresh Prostate TissueAfter testing the tissue-mimicking phantoms, we used the same procedure to ob-tain the viscoelastic properties of fresh prostate tissue. Ethical and operationalapproval was obtained from the Research Ethics Board at the University of BritishColumbia (H19-03592) and the Vancouver Coastal Health Authority. Thirty-fiveprostate glands were collected within two hours of their removal by radical prosta-tectomy surgery. Previously, it has been shown that the mechanical properties ofexcised prostate gland do not change within two hours after removal [97]. Weexcluded any cases that had received prior therapy, including radiation, high inten-sity focused ultrasound (HIFU), and androgen deprivation therapy (ADT) due topossible effects to the prostate properties. The prostatectomy surgeries were per-formed at Vancouver General Hospital (Vancouver, British Columbia, Canada) andthe prostate samples were delivered to the pathology laboratory immediately aftertheir removal. After being registered at the pathology laboratory, the fresh, unfixedsamples were transported to a nearby facility with biosafety level 2 for the mechan-ical testing. Tissues were kept on ice during the transportation for 10 minutes. Ithas been shown that short-term contact with ice does not adversely affect the me-chanical properties of biological tissues [98]. The mechanical tests were performedby compressing the tissue at a rate of 10 mm/s and then waiting 45 seconds for thestress relaxation to occur. The indentation strain did not exceed 15% to ensure thatthe test did not damage the tissue [5].The mechanical test was repeated eight times at each site on the prostate (i.e.,mid, apex, and at the left and right sides) (see Fig. 4.5). The first two tests wereconsidered as tissue preconditioning (see Section 1.1) and the remaining six testswere used for the analysis. Before each test was begun, sites on the prostate wereidentified with help from an expert. Mechanical testing took approximately onehour, while the prostate was kept in a container filled with phosphate-bufferedsaline (PBS) to avoid dehydration. Samples were placed onto a custom-built plasticholder, so that the posterior prostate was at the top (in contact with the indenter),60Figure 4.3: Model validation with tissue-mimicking phantoms: comparisonof relaxation shear moduli between the side with the hard nodule (i.e.,the “cancerous” side) and the side without a nodule (i.e., the “normal”side), for nodule sizes of a) 10 mm, b) 7 mm, and c) 5 mm. The figureshows that larger nodules led to higher viscous properties.Figure 4.4: Model validation with tissue-mimicking phantoms: comparisonof elastic properties (steady-state stress) between the side with the hardnodule (i.e., the “cancerous” side), and the side without a hard nodule(i.e., the “normal” side), for nodule sizes of 10 mm, 7 mm, and 5 mm.The figure shows that larger nodules led to higher elastic properties.and the anterior prostate was at the bottom (in contact with the holder) (shown inFig. 4.6). We performed the tests on the posterior side of the prostate to mimicthe DRE procedure. The stress response of the prostate after compression wasrecorded, and the procedures described in Chapter 3 were followed to calculate σEand gis for each tissue site. After testing, the samples were brought back to thepathology laboratory for histological examination. The histological analysis re-61Figure 4.5: Fresh prostate gland: tissues were collected within two hours af-ter being removed by radical prostatectomy surgery. Tests were per-formed on mid and apex sections of the posterior prostate and the me-chanical properties were compared to those at the left and right sides ofthe posterior prostate.ported: i) the location(s) of predominant tumor(s), ii) the cancer grade and Gleasonscore, iii) prostate volume, iv) tumor volume, v) prostate mass, and vi) the patient’sage (Table B). Features that were not available for the report are left empty in thetable. Prostate samples P1-P3 were used to verify the applicability of the QLVmodel for prostate tissue (Section 4.2.1) and samples P4-P35 were used to studythe effect of cancer on prostate properties (Sections 4.2.2 and 4.2.3).4.2 Results and Discussion4.2.1 Application of the QLV model to Prostate TissueAccording to Eq. (3.3), the QLV model assumes that the elastic and viscous partsof a material’s behavior are separable, which may not necessarily be true for all soft62Figure 4.6: The prostate was placed into a container filled with phosphatebuffered saline (PBS) to avoid dehydration. Samples were supportedfrom beneath by a plastic holder and oriented so that the anteriorprostate was at the bottom (touching the holder) and the posteriorprostate was at the top (touching the indenter).tissues [93]. To verify this assumption for prostate tissue, we changed the elasticcomponent of the stress by performing the indentation test at different indentationdepths and examined if the viscous behavior remained the same. Fig. 4.7 shows theviscous behavior of the prostate gland (P1, mid-section) under different indentationdepths. We normalized the stress with σE to remove the effect of elastic stress. Thenormalized stress showed similar behaviors under different indentation depths (δ ),which confirms that the elastic and viscous properties are separable in prostatetissue. We repeated the same procedure for the P2 and P3 sections, which ledto the same conclusion. For further analysis, we derived the gis for experimentswith different indentation depths and compared the results (Fig. 4.8). The figureincludes the results for the P1-P3 samples, and shows the variation of gis with63respect to indentation depth δ . The results are normalized with gavgi , which is theaverage gi obtained from different indentation depths. The figure shows that g2(Fig. 4.8b) and g3 (Fig. 4.8c) have smaller between variation than g1 (Fig. 4.8a),which may be due to the undesirable effect of overshoot on g1. Overshoot occurswhen the tester cannot precisely control the indentation depth at a high rate, whichmay compromise the measurement accuracy for peak stress. This may also accountfor the larger error bars seen in Fig. 4.7 for recording stress when it is close tot = τ0. Overshoot has been shown to cause up to 30% error in deriving the QLVvalues [71].Fig. 4.8 also shows the model’s precision by the variation within gis. The testwith δ = 6 mm indicates a higher precision as it had the lowest variation withingis (green boxes in Fig. 4.8). To ensure that the indentation strain remained below15%, the test with δ = 5 mm (orange boxes in Fig. 4.8) was performed and italso showed a variation within gis that was below 10%. The same is true for theprecision of σE (Fig. 4.9).4.2.2 Effect of Cancer on Mechanical Properties of ProstateThe prostate gland is a complex tissue with different mechanical properties at dif-ferent locations; that is to say, the properties of the mid and apical areas are notsimilar. In a normal prostate gland, each section has symmetrical properties onthe left and right sides (see Fig. 4.5). Consequently. the properties on the left andright sides of each section can be compared to study the effect of cancer on theviscoelastic properties. Fig. 4.10 shows the effect of cancer on the prostate prop-erties, including the mid and apical sites. In cases where the predominant tumorsare in both sides (mid site in sample P11, apical site in sample P14, and mid site insample P21) the tests were excluded since the information on tumor size was notavailable. The properties were normalized by dividing the properties for the can-cerous side (the side with a predominant tumor) by the properties for the normalside. Figs. 4.10a-d show the viscoelastic properties when the cancer was present inthe posterior prostate and Figs. 4.10e-h show the properties when the cancer wasin the anterior prostate. In most cases, gis and σE are greater for the canceroussides, especially in the posterior prostate. This observation is consistent with pre-64Figure 4.7: Applicability of the QLV model for prostate tissue: the QLVmodel’s applicability for predicting the viscoelastic behavior of prostatetissue was checked by testing the separation between the viscous andelastic parts of the stress response. Testing was performed with dif-ferent indentation depths; i.e., 4 mm, 5 mm, and 6 mm, and the stresswas normalized by the elastic stress to obtain the viscous part of thestress. The results show that the elastic and viscous parts of the prostatestress response are separable since while the indentation depth varies,and hence the elastic part of stress, the viscous part remained the same.vious studies that used MRE to derive the viscoelastic properties of prostate cancer[66, 68–70]. Figs. 4.10e-h show less significant differences, possibly because wemeasured the posterior prostate and so, the properties of the anterior prostate wereless well captured. The 95% confidence intervals for the results in Fig. 4.10a-d areshown in Table 4.1.The increase in gis and σE due to cancer can be explained by changes in the tis-sue’s cell structure. Tissue stress relaxation depends on the ability of cells to diffusewater (water diffusivity) and exchange their position with that of their neighbors(tissue fluidity). High water diffusivity and fluidity leads to smaller stress relax-ation as the cells can rapidly relax the stress either by removing the water from thecells or rearranging their positions. PCa has been shown to decrease water diffusiv-ity [66], which decreases the water movement between cells and hence increasesthe stress relaxation. This can explain why gis increase in the presence of cancer.65Figure 4.8: Applicability of the QLV model for the prostate gland: mechani-cal testing was performed with different indentation depths; i.e., δ = 4mm, 5 mm, and 6 mm, and the relaxation shear moduli gis were ob-tained for each test. Each test was repeated eight times and the aver-age and standard deviation of gis for the last six tests were reported inthe figure (blue, orange, and green boxes). The properties are normal-ized by gavgi s; i.e., the average of the properties for the three indentationdepths. The figure shows that a) g1, b) g2, and c) g3 from tests at differ-ent indentation depths are close. The greater variation of g1 is mainlydue to overshoot; i.e., when the tester cannot precisely control the finaldeformation at a high indentation rate. The figure also shows that thevariation within gis for the different tests is within 10%, which indicatesthe precision of the measurements.The increase in elastic stress σE can also be explained by the uncontrolled prolifer-ation of cancerous cells, so that the increased number of cells causes more stiffness.To further analyze the effect of cancer on the prostate properties, we performed astatistical analysis on the results. Table 4.2 shows the statistics for gcanceri /gnormaliand σ cancerE /σnormalE in the posterior prostate (Fig. 4.10a-d). The table also in-cludes the one-tailed p-values for the alternative hypotheses: gcanceri /gnormali > 1and σ cancerE /σnormalE > 1 from a t-Student distribution. The effect of cancer is sta-tistically significant for g1 and σE with p-values less than 2%.Table B shows that in some samples, both sides of a section were non-cancerous(like the mid-section in P4). In these samples, the properties at the left and rightsides would be expected to be similar. Fig. 4.11 shows the properties for thesecases, after the properties for the right side are normalized by the properties forthe left side, without losing generality. The shaded area in the figure specifies the66Figure 4.9: Precision of the QLV model: each mechanical test was repeatedeight times and the results of last six tests were used for the figure. σEfrom each test was compared to the average elastic stress σavgE to deter-mine the precision of the measurements. From this figure (and Fig. 4.8)the QLV model has a high precision for reporting the viscoelastic prop-erties of the prostate.10% variation (as indicated in Section 3.1) that can be regarded as insignificant.The figure shows that some of the properties do not always fall within the 10%variation, especially in regards to g3. While more data would be needed to con-firm the observations, the different properties for the left and right sides can beexplained by the presence of a tumor in a nearby section. Table 4.3 shows the 95%confidence interval (CI) for the results from Fig. 4.11, and both gis and σE haveacceptable confidence intervals. Moreover, while the results for g2 and σE appearto be balanced around one, g1 and g3 appear to be slightly biased toward higherratios. By comparing Tables 4.1 and 4.3, the properties of the prostate gland withand without the presence of cancer can be seen. In particular, the tables show thatthe presence of cancer increases g1s and σE by 14%, and 30%, respectively.67Figure 4.10: Effect of cancer on the mechanical properties when cancer is inposterior prostate (a-d); and when cancer is in anterior prostate (a-h).Samples that had cancer on both sides were excluded. The propertieswere normalized by dividing the properties of the cancerous side bythe properties of the normal side for each section. Tests were repeatedeight times for each quadrant and the average of the last six tests wereused to calculate the properties. The figures show higher values for theproperties on the cancerous side, especially for a) g1, and d) σE as theratio for the properties on the cancerous and normal sides are mostlyabove one. The effect of cancer on the properties is less significantwhen cancer occurred in the anterior prostate e-h), mainly because thetests were performed on the posterior prostate, which limits the accu-racy in capturing the anterior prostate properties.68Table 4.1: 95% confidence intervals for the ratio between the properties of thecancerous and normal sides of the posterior prostate. As seen in Fig. 4.10,the properties for the cancerous side are greater than those for the normalside.Count gcancer1 /gnormal1 gcancer2 /gnormal2 gcancer3 /gnormal3 σcancerE /σnormalE95% CI 46 (1.03 – 1.50) (0.96 – 1.15) (0.90 – 1.69) (1.16 – 1.50)Figure 4.11: Mechanical properties of non-cancerous prostate gland. Theproperties are normalized by dividing the properties for the right sideby the properties for the left side in each section. Since no cancerousnodules are in the prostate, the properties for the left and right sides areexpected to be similar. The shaded area shows the 10% variation in theproperties, which can be considered as insignificant (see Fig. 4.8 andFig. 4.9). The figures also show more similarity between the propertiesfor the left and right sides for a) g1 and d) σE . The higher differencein g2, c) g3 may be explained by the presence of a nodule in a nearbysection.694.2.3 Effect of Cancer Grade on the Prostate GlandIn Section 3.2, the effect of cancer on the mechanical properties of the prostatewas investigated. PCa is classified into different grades, depending on how fast itspreads and whether or not it affects other organs (metastatic cancer). Higher can-cer grades are associated with fast-growing cancers that can invade nearby organs.In this section, we investigate the possible correlation between cancer grade andviscoelastic properties of the prostate. The relationships between cancer grade andother properties of the prostate are also discussed.Fig. 4.12 illustrates the correlation between cancer grade, prostate volume,prostate mass, and patient’s age. Fig. 4.12a shows that, as expected, the prostatemass and volume are highly correlated. Interestingly, our results show that theprostate volume (in mL) is close to the prostate mass (in grams). Nevertheless,these factors are not correlated with cancer grade and, for example, a cancer gradeof five (red dots in Fig. 4.12a) can appear in small, intermediate, or large-sizedprostate glands. Here, we reported the International Society of Urological Pathol-ogy (ISUP) Cancer Grade, calculated based on the Gleason patterns observed in theprostate (Table B). Cancer grade 2, shows that the top two patterns in the prostateare Gleason scores of 3 and 4 (shown as 3+4); Similarly, cancer grades 3,4, and5 are associated with Gleason Score 4+3, 4+4, and 4+5 (or 5+4), respectively.Fig. 4.13 shows the patterns that correspond to Gleason scores of 3, 4, and 5, whichrepresent cancerous cells in early, intermediate, and advanced stages, respectively.Since the cancer grade mostly depends on the cell patterns and not on the prostatesize, a clear correlation between cancer grade and prostate mass/volume would notbe expected.The correlations between cancer grade, patient’s age, and cancer volume areshown in Fig. 4.12b. Prostate gland enlargement is a common condition that occursas men age and so a relationship between prostate volume and patient’s age wouldbe expected. Moreover, while no apparent correlation is observed between age andcancer grade, high-grade cancers (i.e., grades 4 and 5) are more likely to appear inolder patients. Similarly, Alibhai et al. [99] reported a weak correlation betweencancer grade and patient’s age. The presence of high-grade cancer in older patientscan be explained by a lower number of screening tests, which can lead to more70Figure 4.12: Correlation between different properties of the prostate gland: a)correlation between cancer grade, prostate mass, and prostate volume,and b) correlation between cancer grade, prostate volume, and patient’sage. a) shows a strong correlation between the prostate volume andmass, but the cancer grade does not appear to be correlated to the massor volume. For example, cancer grade 5 could be assigned to either asmall or a large prostate gland. b) shows a positive correlation betweenpatient’s age and prostate volume. While a clear relationship betweencancer grade and age is not seen, high-grade cancer (i.e., grades 4 and5) are more likely to occur in older patients.advanced PCa levels [99]. Changes in cancer biology can also explain the highgrades of cancer in older patients [100].Besides cancer grade, the tumor volume is another important factor that can beused to evaluate the cancer progression. Tumor volume is the total volume of theprostate that is occupied by a tumor nodule(s) and higher tumor volumes are asso-ciated with more progressive cancers. Figs. 4.14 and 4.15 show the effect of tumorvolume on the mechanical properties of prostate when cancer presents in the pos-terior and anterior prostates. A higher variation in the viscoelastic properties (i.e.,larger gcanceri /gnormali and σ cancerE /σnormalE ) would be expected with an increase intumor volume. Fig. 4.14 shows that an increase in tumor volume in the poste-rior prostate increases the viscous properties (Figs. 4.14a-c) but does not affect theelastic properties (Figs. 4.14d). Among the viscous properties, g1 and g3 are cor-71Figure 4.13: Gleason scores for prostate cancer cells: a) patterns for Gleasonscore 3 have well-formed glands without fusion, b) patterns for Glea-son score 4 have back-to-back fused glands and a sieve-like structureformed by the cells, and c) patterns for Gleason score 5 show cell in-vasion in the glands.related with the highest statistical significance (lowest p-values). That is to say, g1and g3 can provide a good approximation of the tumor volume. As expected, lesssignificant effects are observed when cancer is in the anterior prostate since theproperties of the anterior prostate are not well captured by our testing. In addition,no correlation appears to exist between cancer grade and tumor volume. The sameis also true for cancer grade and relative tumor volume (Fig. 4.16). The literature isgenerally controversial about whether or not tumor volume should be an indicatorfor PCa progression (e.g., [101, 102]). In 2007, Nature published work by Mayet al. [103] who concluded that relative tumor volume below 25% cannot serve asan independent factor for determining PCa progression. This would explain whywe did not see an independent influence of tumor volume in most of our prostatesamples, where the relative tumor volume was below 25% (Fig. 4.16). A largersample of prostate glands with high tumor volumes would be needed to providemore insight into the relationship between tumor volume and PCa progression.72Figure 4.14: Correlation between tumor volume and properties when cancerpresents in the posterior prostate. The properties were normalized bydividing the properties for the cancerous side to the properties for thenormal side. Viscous properties are affected by tumor volume (see a,b, and c) while the elastic properties do not seem to vary with tumorvolume (see d). g1 and g3 have the highest statistical significance (low-est p-value). The cancer grade does not appear to be correlated withtumor volume or the viscoelastic properties. International Society ofUrological Pathology (ISUP) Cancer Grade was reported, and calcu-lated based on the Gleason score. Cancer grades 2, 3, 4, and 5 areassociated with Gleason Score 3+4, 4+3, 4+4, and 4+5 (or 5+4), re-spectively.73Figure 4.15: Correlation between tumor volume and mechanical propertieswhen cancer presents in the anterior prostate. The properties are nor-malized by dividing the properties on the cancerous side by the prop-erties on the normal side. Unlike Fig. 4.14, no clear correlation canbe seen between tumor volume and the viscoelastic properties. Thismay be due to our testing that was performed on the posterior prostate,which fails to accurately capture the properties in the anterior prostate.International Society of Urological Pathology (ISUP) Cancer Gradewas reported, and calculated based on the Gleason score. Cancergrades 2, 3, 4, and 5 are associated with Gleason Score 3+4, 4+3, 4+4,and 4+5 (or 5+4), respectively.4.3 Application in Prostate Cancer DiagnosisThis thesis, which is focused on the possible effects of cancer on the properties ofthe prostate gland does not intend to introduce a diagnostic model for PCa. In fact,any diagnostic model for PCa would require a larger number of prostate samples (∼50) for classification algorithms to be applied [104]. Nevertheless, the initial resultsof this study suggest that stiffness σE and the relaxation shear modulus g1 couldbe used together to detect PCa with a high sensitivity (Table 4.2) and specificity74Figure 4.16: Correlation between cancer grade and relative tumor volume(tumor volume divided by prostate volume). No clear correlation isseen between cancer grade, patient’s age, and the relative tumor vol-ume. International Society of Urological Pathology (ISUP) CancerGrade was reported, and calculated based on the Gleason score. Can-cer grades 2, 3, 4, and 5 are associated with Gleason Score 3+4, 4+3,4+4, and 4+5 (or 5+4), respectively.Table 4.2: Descriptive and inferential statistics for the cancer effect on theviscoelastic properties of the prostate gland when cancer presents in theposterior prostate. The table also includes p-values for the alternativehypothesis with gcanceri /gnormali > 1 and σ cancerE /σnormalE > 1. From thetable, cancer appears to have a greater statistically significant effect ong1 and σE (lower p-values).gcancer1 /gnormal1 gcancer2 /gnormal2 gcancer3 /gnormal3 σcancerE /σnormalECount 24 24 24 24Average 1.24 1.06 1.29 1.33StandardDeviation 0.49 0.22 0.94 0.40p-values 0.012 0.12 0.07 0.0075Table 4.3: 95% confidence interval for non-cancerous prostate samples.Since cancer is absent from the prostate sections, the properties for theleft and right sides are expected to be similar. The table shows that g1 andg3 are relatively biased towards higher ratios and g2 and E are balancedaround a ratio of one.Count gright1 /gleft1 gright2 /gleft2 gright3 /gleft3 σrightE /σleftE95% CI 9 (0.99 – 1.20) (0.91 – 1.09) (0.92 – 1.29) (0.88 – 1.14)(Table 4.3). σE has been widely used for detecting prostate cancer but to the bestof our knowledge, this is the first time that the relaxation shear modulus is reportedto serve as a possible diagnostic tool. We also report that the method can estimatetumor volume (Fig. 4.14). Nevertheless, our current results do not suggest that themethod would be appropriate for detecting cancer in the anterior prostate, whichaccounts for approximately 20% of all PCa cases [94]. The same problem has alsobeen reported for detecting PCa via ultrasound elastography [105] or by biopsy[94]. Furthermore, the current results are only valid for ex vivo examination andtheir applicability for in vivo tests remains for future studies. For instance, sincein vivo examination involves the indenter approaching the prostate gland throughthe rectum, the effect of the rectal wall on force measurements should be analyzed.That is to say, possible clinical applications of this method in vivo would requireother considerations that are beyond the scope of this study. A brief discuss onthe possible approach for in vivo applications of our results for PCa diagnosis isincluded in the Appendix, though we leave the actual implementation to futurework.4.4 SummaryIn this chapter, we derived the QLV properties for fresh prostate tissue and inves-tigated the effect of cancer on the viscoelastic characteristics. Thirty-five prostateglands were collected within two hours following prostatectomy surgery and theirQLV parameters were obtained by performing stress relaxation experiments. Theproperties were then correlated with the presence and grade of cancer to studythe effect of cancer on the stiffness and shear relaxation moduli of the prostate.76The results show that cancer increases both elastic and viscous properties of theprostate with a high statistical significance (p-value < 2%). For example, the firstshear relaxation modulus and stiffness increase by 14% and 30%, respectively inthe presence of cancer. In addition, the cancer grade is not directly related to themechanical properties of the prostate or prostate volume and mass. Furthermore,tumor volume is highly correlated with the viscous properties, so that a highertumor volume appears to lead to a higher shear relaxation moduli. No similar cor-relation was found for the elastic properties. In this chapter, the application of theresults to the in-vivo diagnosis of PCa was also discussed. A FBG sensor was de-signed and fabricated for possible use as a diagnostic tool in-vivo and the sensor’sapplicability for detecting a hard nodule within prostate-mimicking phantoms wasdemonstrated.77Chapter 5Conclusions and Future Work5.1 ConclusionThe mechanical properties of biological soft tissues have attracted much attentiondue to their importance in diagnosing diseases. In particular, assessing the me-chanical properties of the prostate gland is suggested to be a primary method fordetecting prostate cancer. In this thesis, we presented a new method for calibrat-ing the quasi-linear viscoelastic (QLV) model by using uniaxial stress relaxationexperiments (Chapter 2) and extended the method for a possible indentation test(Chapter 3). Finally, we characterized the viscoelastic properties of the prostategland and studied the effect of cancer and cancer grade on shear relaxation moduliand stiffness of prostate tissue (Chapter 4). In the following sections, we presentconcluding remarks for each chapter.5.1.1 Calibrating the QLV Model via the Uniaxial TestA new method for calibrating the QLV model was presented that derives the shearrelaxation moduli and Young’s modulus for soft materials using uniaxial compres-sion tests. The method differs from previously reported methods that are based onthe QLV model in two aspects: 1) we define elastic stress as the stress that emergesafter a sufficiently long time, where the time-dependent part vanishes and only theelastic part remains; and 2) we designed two different experiments to calculate theelastic and viscous characteristics independently. Considering these two modifi-78cations, we derived the viscoelastic properties with confidence intervals that wereone-third of those previous reported.To ensure the applicability of our method, we tested it on tissue-mimickingphantoms; i.e., soft materials that represent the mechanical properties of biologi-cal tissue. For this purpose, we performed stress relaxation tests at specific com-pression depths and used the results to characterize the material. The derived pa-rameters were then used to predict the behavior of materials at other compressiondepths, and the predicted results were in good agreement with the experimentalresults. Thus, the ability of the model to derive the properties independently fromthe loading conditions was demonstrated. These results represent an improvementwith respect to other viscoelastic models (e.g., spring-dashpot models) that intro-duce different parameters for each loading condition. We tested the applicabilityof the method to distinguish prostate phantoms with the same Young’s modulusas normal and cancerous prostate tissue, and the results allowed for the precisedistinction between the materials. We also showed that our model can distinguishmaterials with similar elastic properties but different viscous properties, and vice-versa. On the limitation side however, we admit the fact that a better fit with exper-imental data could be presented by increasing the number of relaxation times, andhence the relaxation moduli, but at the risk of over-fitting. Another way to improvethe modeling could be optimizing relaxation times instead of pre-selecting them,but this may further complicate the fitting algorithm.5.1.2 QLV Characterization via the Indentation TestBecause the method for QLV calibration by a uniaxial compression test may notnecessarily be applicable to in vivo testing, we presented a procedure for QLVcharacterization by an indentation test. For this purpose, we studied the inden-tation behavior of viscoelastic materials with an indentation test under differentloading conditions. The effect of the QLV parameters on stress relaxation behaviorwas also discussed. The results show that stress relaxation is not dependent on theindenter size or indentation depth, and it is only governed by the material proper-ties and the indentation rate. The shear relaxation moduli were also shown to affectthe stress relaxation at specific time intervals and thus, they are proportional to the79amount of stress relaxation. We also calculated the effect of different relaxationmoduli on stress relaxation by superimposing the effect of each modulus. Whiledemonstrating a good performance for characterizing soft materials, it should benoted that our model (and hence the reported parameters) is affected by the se-lected relaxation times. We argue that our selection was limited by experimentalparameters (e.g. sensors/actuators specifications). In the future, one can derive acontinuous spectrum of relaxation moduli by invoking Eq. (2.6) for the relaxationfunction and hence report a better description of the relaxation behaviour; however,it might be challenging to combine with FEA.Based on the findings, we present a new procedure for fast and straightforwardcharacterization of soft materials. Using a finite element analysis, we can calcu-late the stress relaxation for different time intervals and estimate the value for therelaxation moduli. To validate the accuracy of our procedure, we compared theparameters obtained from an indentation test with those calculated from the uniax-ial compression tests (Chapter 2), where a good accuracy (error within 20%) andhigh precision (standard deviation within 10%) have been reported. Our proce-dure can also characterize materials within a few seconds, which distinguishes itfrom inverse finite element methods, making it appropriate for real-time materialcharacterization. It is important to mention that the applicability of the calibra-tion method for a more complex phantom was not discussed in the section as ourphantoms were all homogeneous with a simple cylindrical shape. We acknowledgethe possibility of less accurate performance for the model when it comes to morecomplex phantoms.5.1.3 Effect of Cancer on Prostate ViscoelasticityWe used the models presented in Chapters 2 and 3 to characterize the mechani-cal properties of the prostate gland and study the effect of cancer on the proper-ties. First, we used the model to detect a hard nodule within a prostate-mimickingphantom. The hard nodule and the phantom have the same Young’s modulus asnormal and cancerous prostate tissue, respectively. We performed stress relaxationtests on the sides with and without the hard nodule and compared the propertiesto determine whether or not the model can capture differences in the mechanical80properties. The results confirm that the model can recognize different mechanicalproperties for the side with a small nodule (5 mm in diameter) compared to the sidewithout a nodule.For clinical assessments, we collected 35 fresh prostate glands within two hoursof their removal and tested their viscoelastic properties. First, we validated theapplicability of the QLV model for prostate tissue by checking whether or not theelastic and viscous aspects can be separated. We repeated the stress relaxation testswith different indentation depths (to change the elastic aspect) and found that theviscous part of stress relaxation remained the same. After validating the applicabil-ity of the QLV model for prostate tissue, we compared the properties of cancerousand non-cancerous prostate tissues to assess the effect of cancer and cancer gradeon the relaxation moduli and stiffness. We found that cancer can change prostategland properties with a high statistical significance (p-value < 2%). Cancer in-creases the first relaxation modulus by 14% and the stiffness by 30%. The relation-ship between cancer grade and the prostate properties were also assessed thoughno apparent correlation was observed. Nevertheless, cancer grade was found to becorrelated with the patient’s age and high-grade cancers were more likely to occuramong older patients. A similar trend was not observed for low-grade cancers.Furthermore, tumor volume was also shown to be highly correlated with relaxationmoduli though the same was not true for the elastic properties.5.2 Future WorkThe clinical applications of the findings for detecting prostate cancer would requirethe following future work:1. The QLV model in Chapters 2 and 3 are derived for a homogeneous andisotropic material, which is not the case for complex biological tissues, likeprostate gland. While the current model seems to provide accurate results,the accuracy can be improved by considering anisotropic relaxation functionin the QLV model to reflect the complex composition of the tissue. This stepwould be required if future results show that the model’s accuracy diminishesby including more data.812. More prostate samples should be tested ex vivo and analyzed with classi-fication algorithms to obtain a model predicting the presence and grade ofprostate cancer based on the mechanical properties. For this purpose, theimportant features in detecting prostate cancer should be specified and thenimportance of each is determined by employing ensemble classification al-gorithms.3. In vivo testing should be performed to confirm if the conclusion from ex vivotesting is also valid for in vivo setting. The testing can be first performed ona life-like prostate examination simulator and then on human participants.For this purpose, a mechanism should be designed to navigate the sensor(described in the Appendix) through the rectum so that it smoothly accessesthe prostate gland and stays fixed during the force measurement. In vivo test-ing can investigate whether or not the rectum wall affects the force readings.It can also show if the prostate position variation from a person to anothercan make difference in the results. That being said, it should be mentionedthat the current results cannot be extended to in vivo setting. For example,the guidance of the sensor through the rectum needs to be considered beforeany in vivo experiments. Appropriate guidance should ensure the sensor canreach the prostate quadrants, so the indentation test is performed the sameas ex vivo. Also, in vivo applications might require using a smaller indenterthan what we used in this work, which may alter the results. Last but notleast, it should be mentioned that the sample we used is biased toward thepeople with confirmed prostate cancer, which is different from our actualsample, i.e., all people who are suspicious of having prostate cancer. Whilewe can address this issue by doing in vivo testing on a more diverse sample,the difference in the sample sizes might lead to different outcomes.4. Finally, the results of in vivo testing should be compared with those obtainedfrom digital rectal examination by practicing both on the same participants.The comparison can demonstrate how much the method is effective in im-proving the prostate cancer diagnosis.82Bibliography[1] Yuan-Cheng Fung. Biomechanics. 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(October 2021), 2022. → page7695Appendix AA Possible Approach for In VivoDiagnosis of Prostate CancerA.1 Sensor Design and FabricationAs discussed in Chapter 1, fiber Bragg grating (FBG) sensors are appropriate forforce measurements in biomedical applications. A fiber Bragg grating is a mi-crostructure, generally a few millimeters in length, that is photo inscribed in thecore of a single mode optical fiber. When the full spectrum of light passes throughthe fiber Bragg grating, a specific wavelength is reflected back and measured byan interrogator. When the fiber Bragg grating is subject to strain, the reflectedwavelength changes accordingly. We proposed using a FBG sensor to test the invivo stress relaxation in the prostate gland. The design for the sensor is shownin Fig. A.1. The sensor is composed of a 3D printed enclosure, sensor head, andpolyurethane rod. A fiber Bragg grating sensor (FBGS Technologies Company,Jena, Germany) was first bonded onto a 5 mm polyurethane rod using NorlandUV-curable Adhesive NOA 61 (Norland Products, New Jersey, USA). The rod wassanded on one side to provide a suitable flat surface for bonding. Once the sensorwas bonded to the rod, it was glued into the sensor head and the bottom of thesensor enclosure. The rod was suspended in the middle of the enclosure to allowfor unconfined compression when an axial load is applied. The sensor head hasan indentation face of 6 mm (dia.) and is manufactured using Master Mold Resin96(MiiCraft, Vaughan, Canada) via stereolithography 3D printing. The sensor headalso contains two grooves to prevent torsional strain from affecting the axial load-ing during compression. The assembled sensor has an outer diameter of 15 mmand a total length of 26 mm. The enclosure was fabricated from polylactide acid(PLA) using the Monoprice Mini Select V2 3D Printer.The fabricated sensor was attached to a linear actuator for stress relaxationexperiments. To ensure proper compression and reliability in capturing the stressrelaxation, the actuator requires a pull-in force of 2 N, linear speed of 10 mm/s,and accuracy of at least 0.5 mm. The specifications are chosen so that the sensorcan provide enough force to compress the prostate tissue by 5 mm at a rate that isappropriate for capturing the stress relaxation. To fulfill the criteria, an L12 linearactuator supplied by Actuonix (12 V, 100:1 gearing, with position feedback) wasused. An adaptor was also 3D printed to connect the sensor to the actuator (theadapter is not shown in Fig. A.1).The sensor is calibrated using the mechanical tester described in Section 2.1(see Fig. A.2). The sensor was placed directly under the indenter tip and forcecalibration experiments were conducted over a range of 0 to 2.25% strain using thefiber Bragg grating at strain intervals of 0.225%. The FBG sensor was connected toan FBG-Scan 908 interrogator (FBGS Technologies Company, Jena, Germany) andsensor readings were recorded at 100 Hz. For each increment, the static force fromthe load cell and the central wavelength shift from the interrogator were recorded.The calibration procedure was repeated three times to ensure consistent results.The sensor calibration results showed the sensitivity to be 4 mN in a range of 0-4.0N, corresponding to a strain of 2.5%. The calibration shows linearity between themeasurements (wavelength shift in the FBG sensor) and force, which is reflectedby the R-squared value of 0.9998 and a maximum non-linearity error of 22 mN.The current sensor design is 15 mm in diameter, which is sufficiently small to besuitable for in-vivo prostate tissue examinations. The sensor design greatly simpli-fies the fabrication procedure as no complex machining is required, and the numberof different components in the assembled sensor are kept at a minimum. This helpsto reduce sources of error, improves reliability, and minimizes the cost.97Figure A.1: Design of the proposed FBG sensor: The sensor is composedof an FBG sensor bonded to a elastomer (polyurethane) rod and it issupported by a 3D printed enclosure. The rod is designed to contact thematerial through a sensor head with a diameter of 6 mm. The sensor iscompressed into a material by using an actuator and the force reactionof the material is recorded over time. The size of the sensor is designedto allow for in vivo applications.A.2 Validation on Prostate-mimicking PhantomsThe applicability of the sensor was demonstrated by testing the prostate-mimickingphantoms (Fig. A.3). The phantoms were fabricated following the procedure de-scribed in Section 4.1.1. As shown in Fig. A.3, a test rig was fabricated to rigidlyhold the actuator and sensor for the stress relaxation experiments. The sensor com-presses the prostate phantom in the four quadrants (as shown in Fig. A.3a) follow-ing the protocol described in Fig. 4.1. Each test was repeated three times. For eachphantom that was tested, a cancerous nodule was inserted in quadrant L1. TheQLV parameters for each quadrant (gis and σE) were then calculated according to98Figure A.2: Sensor calibration: the mechanical tester described in Section 2.1was used to calibrate the sensor. The FGB sensor was placed beneaththe tester’s indenter and a range of strains from 0 to 2.25% were appliedin intervals of 0.225%. The sensor information was read through the in-terrogator and calibrated by the forces recorded for the load cell. Thecalibration showed a linear relationship between the sensor’s measure-ments (wavelength shift in the FBG sensor) and Force, with a maximumnon-linearity error of 22 mN.the procedure outlined in Chapter 3, and compared to Figs. A.4 and A.5.The QLV parameters for each location in the prostate phantom are shown inFig. A.4. In all tests, the gi parameters from the quasi-linear viscoelastic modelwere smaller in the cancerous region of the phantom compared to the non-cancerousregions. As mentioned earlier, this result is expected since the shear relaxationmoduli for the hard nodules are four-times smaller than those of the phantom.Based on the results, the sensor was able to identify both the elastic and viscousproperties of the phantoms. Furthermore, in all tests, the elastic stress was highestin quadrant L1 where the cancerous nodule had been implanted (Fig. A.5). Theelastic and viscous parameters are directly related to the size of the nodules, withlarger nodules leading to higher elastic stresses and shear relaxation moduli. Theresults shown in Figs. A.4 and A.5 demonstrate that the FBG sensor, equippedwith the QLV model, can identify the hard nodules in the phantoms. Nevertheless,99Figure A.3: Testing the FBG sensor on the prostate-mimicking phantoms:phantoms were fabricated as described in Section 4.1.1 and the nod-ule was located in L1; c) the FBG sensor was used to measure theforce response of the prostate phantoms after each test. The sensor wasconnected to the actuator with an adaptor and a custom-built rig. Ex-periments were repeated three times for each quadrant (L1-L4) and theaverage QLV parameters were recorded for each quadrant.the applicability of the sensor for in vivo diagnosis of PCa requires further clini-cal testing. In particular, the current design for connecting the actuator and sensorwould require modification to accommodate in vivo testing. Specific guidance ofthe sensor in the rectum would also need to be considered so that movement of thesensor does not cause injury. While we did not assess whether or not the currentdesign would satisfy the in vivo requirements, we believe that it is a step forwardin providing a better tool for detecting PCa cancer in the early screening stages.100Figure A.4: Validation of the FBG sensor applicability by testing on prostate-mimicking phantoms: viscous properties (gi s) are reported for a) L1,b) L2, c) L3, and d) L4 for hard nodules with diameters of 5 mm, 7mm, and 10 mm. The figure shows that the most variation occurs forthe properties at L1, where the nodules are located. The properties atL4 also vary slightly for the different nodule sizes, which is likely dueto the effect of a hard nodule in a nearby quadrant. L2 and L3, however,have similar properties for the different nodule sizes.101Figure A.5: Validation of the FBG sensor applicability by testing theprostate-mimicking phantoms: elastic stress is shown for quadrantswith different nodule sizes. The figure shows that the greatest varia-tion in elastic stress occurs with the nodule sizes at L1. The elasticstress at L4 also varies slightly with the nodule size, which is likely dueto the effect of a nearby quadrant. L2 and L3, however, have similaramounts of elastic stress for the different nodule sizes.102Appendix BProstate Glands for ClinicalStudyTable B includes the prostate samples we used for the clinical study as well as thecancer location(s), cancer grade, prostate volume, tumor volume, prostate mass,and patient’s age.103Table B.1: Characteristics of fresh prostate samples used in the clinical study: Thirty-five fresh prostate samples, iden-tified as P1-P35 were collected within two hours after removal by prostatectomy surgery. The table includes thelocation(s) of the cancer in mid and apical sections, cancer grade, Gleason score, prostate volume, prostate mass,and the patient’s age. The results for samples P1-P3 are used in Section 3.1 to check the suitability of the QLVmodel for the prostate gland, and the results for samples P4 - P35 are used in Sections 3.1 and 3.2 for studying theeffect of cancer on the properties of the prostate gland. Some of the prostate characteristics were not available andappear empty in the table.Cancer Location (s)Cancer Grade(Gleason Score)Prostate Volume(ml)Tumor Volume(ml)Prostate Mass(g)Patient’s AgeMid Section Apex SectionP1Right(Posterior)Right(Posterior)3 (4+3) 50 5 50 68P2Right and Left(Anterior)None 2 (3+4) 40 2 46 54P3Left(Anterior)None 5 70 5 86 61P4 NoneLeft(Posterior)3 (4+3) 178 5 165 72P5Left(Anterior)Left(Anterior)2 (3+4) 35 5 43 70P6Left(Posterior)Left(Posterior)3 (4+3) 37 46 68104P7Right(Anterior)Right(Anterior)2 (3+4) 42 6 47 73P9Right(Posterior)Right(Posterior)4 87 13 90 70P11Right and Left(Anterior)Left(Anterior)2 (3+4) 63 46 66P12Right(Posterior)Right(Posterior)3 (4+3) 49 55 79P13Left(Anterior)Left(Anterior)2 (3+4) 51 62P14Right(Anterior)Right and Left(Anterior)2 (3+4) 63 79P15Left(Posterior)Left(Posterior)2 (3+4) 48 12 63 64P16Right(Posterior)None 5 94 104 79P17 NoneRight(Anterior)5 41 6 45 70P18Right(Anterior)None 3 (4+3) 45 9 51 69105P19Right(Anterior)Left(Anterior)4 44 2 47 67P20Left(Posterior)Left(Posterior)3 (4+3) 70 11 68P21Right and Left(Anterior)Left(Anterior)2 (3+4) 67 23 70 71P22Right(Posterior)Right(Posterior)2 (3+4) 40 6 44 61P23Left(Anterior)Left(Anterior)3 (4+3) 43 3 38 72P24Left(Posterior)Right(Posterior)5 53 8 60 73P25Left(Posterior)Left(Posterior)2 (3+4) 42 6 48 59P26Left(Posterior)None 3 (4+3) 42 49 60P27Left(Posterior)None 3 (4+3) 78 5 87 69P28Left(Posterior)None 2 (3+4) 77 61106P29Left(Posterior)Left(Posterior)3 (4+3) 37 6 57P30Left(Posterior)Left(Posterior)2 (3+4) 38 11 47 69P31 NoneLeft(Anterior)3 (4+3) 65 7 69 70P32 NoneRight(Anterior)2 (3+4) 81 4 80 77P33Left(Anterior)Right and Left(Anterior)2 (3+4) 53 11 48 71P34Left(Anterior)Left(Anterior)3 (4+3) 58 6 67 71P35Right(Anterior)Right(Anterior)3 (4+3) 35 35 69107
Thesis/Dissertation
eng
Mechanical Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Chiao, M. (Mu)
An investigation of cancer effect on viscoelastic properties of prostate gland via quasi-linear viscoelastic model
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