A narrative review of the measurement methods for biomechanical properties of plantar soft tissue in patients with diabetic foot

This article provides an overview of the development history and advantages and disadvantages of measurement methods for soft tissue properties of the plantar foot. The measurement of soft tissue properties is essential for understanding the biomechanical characteristics and function of the foot, as well as for designing and evaluating orthotic devices and footwear. Various methods have been developed to measure the properties of plantar soft tissues, including ultrasound imaging, indentation testing, magnetic resonance elastography, and shear wave elastography. Each method has its own strengths and limitations, and choosing the most appropriate method depends on the specific research or clinical objectives. This review aims to assist researchers and clinicians in selecting the most suitable measurement method for their specific needs.

Mechanical Behaviour of Plantar Adipose Tissue: From Experimental Tests to Constitutive Analysis

Plantar adipose tissue is a connective tissue whose structural configuration changes according to the foot region (rare or forefoot) and is related to its mechanical role, providing a damping system able to adsorb foot impact and bear the body weight. Considering this, the present work aims at fully describing the plantar adipose tissue's behaviour and developing a proper constitutive formulation. Unconfined compression tests and indentation tests have been performed on samples harvested from human donors and cadavers. Experimental results provided the initial/final elastic modulus for each specimen and assessed the non-linear and time-dependent behaviour of the tissue. The different foot regions were investigated, and the main differences were observed when comparing the elastic moduli, especially the final elastic ones. It resulted in a higher level for the medial region (89 ± 77 MPa) compared to the others (from 51 ± 29 MPa for the heel pad to 11 ± 7 for the metatarsal). Finally, results have been used to define a visco-hyperelastic constitutive model, whose hyperelastic component, which describes tissue non-linear behaviour, was described using an Ogden formulation. The identified and validated tissue constitutive parameters could serve, in the early future, for the computational model of the healthy foot.

Definition and evaluation of a finite element model of the human heel for diabetic foot ulcer prevention under shearing loads

Diabetic foot ulcers are triggered by mechanical loadings applied to the surface of the plantar skin. Strain is considered to play a crucial role in relation to ulcer etiology and can be assessed by Finite Element (FE) modeling. A difficulty in the generation of these models is the choice of the soft tissue material properties. In the literature, many studies attempt to model the behavior of the heel soft tissues by implementing constitutive laws that can differ significantly in terms of mechanical response. Moreover, current FE models lack of proper evaluation techniques that could estimate their ability to simulate realistic strains. In this article, we propose and evaluate a FE model of the human heel for diabetic foot ulcer prevention. Soft tissue constitutive laws are defined through the fitting of experimental stretch-stress curves published in the literature. The model is then evaluated through Digital Volume Correlation (DVC) based on non-rigid 3D Magnetic Resonance Image Registration. The results from FE analysis and DVC show similar strain locations in the fat pad and strain intensities according to the type of applied loads. For additional comparisons, different sets of constitutive models published in the literature are applied into the proposed FE mesh and simulated with the same boundary conditions. In this case, the results in terms of strains show great diversity in locations and intensities, suggesting that more research should be developed to gain insight into the mechanical properties of these tissues.

Experimental characterisation of porcine subcutaneous adipose tissue under blunt impact up to irreversible deformation

A deeper understanding of the mechanical characteristics of adipose tissue under large deformation is important for the analysis of blunt force trauma, as adipose tissue alters the stresses and strains that are transferred to subjacent tissues. Hence, results from drop tower tests of subcutaneous adipose tissue are presented (i) to characterise adipose tissue behaviour up to irreversible deformation, (ii) to relate this to the microstructural configuration, (iii) to quantify this deformation and (iv) to provide an analytical basis for computational modelling of adipose tissue under blunt impact. The drop tower experiments are performed exemplarily on porcine subcutaneous adipose tissue specimens for three different impact velocities and two impactor geometries. An approach based on photogrammetry is used to derive 3D representations of the deformation patterns directly after the impact. Median values for maximum impactor acceleration for tests with a flat cylindrical impactor geometry at impact velocities of 886 mm/s, 1253 mm/s and 2426 mm/s amount to 61.1 g, 121.6 g and 264.2 g, respectively, whereas thickness reduction of the specimens after impact amount to 16.7%, 30.5% and 39.3%, respectively. The according values for tests with a spherically shaped impactor at an impact velocity of 1253 mm/s are 184.2 g and 78.7%. Based on these results, it is hypothesised that, in the initial phase of a blunt impact, adipose tissue behaviour is mainly governed by the behaviour of the lipid inside the adipocytes, whereas for further loading, contribution of the extracellular collagen fibre network becomes more dominant.

MR-compatible loading device for assessment of heel pad internal tissue displacements under shearing load

In the last decade, the role of shearing loads has been increasingly suspected to play a determinant impact in the formation of deep pressure ulcers. In vivo observations of such deformations are complex to obtain. Previous studies only provide global measurements of such deformations without getting the quantitative values of the loads that generate these deformations. To study the role that shearing loads have in the etiology of heel pressure ulcers, an MR-compatible device for the application of shearing and normal loads was designed. Magnetic resonance imaging is a key feature that allows to monitor deformations of soft tissues after loading in a non-invasive way. Measuring applied forces in an MR-environment is challenging due to the impossibility to use magnetic materials. In our device, forces are applied through the compression of springs made of polylactide. Shearing and normal loads were applied on the plantar skin of the human heel through a flat plate while acquiring MR images. The device materials did not introduce any imaging artifact and allowed for high quality MR deformation measurements of the internal components of the heel. The obtained subject-specific results are an original data set that can be used in validations for Finite Element analysis and therefore contribute to a better understanding of the factors involved in pressure ulcer development.

Coupled experimental and computational approach to stomach biomechanics: Towards a validated characterization of gastric tissues mechanical properties

Gastric diseases are one of the most relevant healthcare problems worldwide. Interventions and therapies definition/design mainly derive from biomedical and clinical expertise. Computational biomechanics, with particular regard to the finite element method, provides hard-to-measure quantities during in-vivo tests, such as strain and stress distribution, leading to a more comprehensive and promising approach to improve the effectiveness of many different clinical activities. However, reliable finite element models of biological organs require appropriate constitutive formulations of building tissues, whose parameters identification needs an experimental campaign consisting in different tests on human tissues and organs. The aim of the reported here research activities was the identification of mechanical properties of human gastric tissues. Human gastric specimens were tested at tissue, sub-structural and structural levels, by tensile, membrane indentation and inflation tests, respectively. On the other hand, animal experimentations on tissue layers from literature pointed out the mechanical response at sub-tissue level during tensile loading conditions. In detail, the analysis of experimental results at sub-tissue and tissue levels led to a fibre-reinforced visco-hyperelastic constitutive formulation and to the identification of gastric layers mechanical behaviour. Results from experimentations on human samples were coupled with data derived from animal models. Data from sub-structural and structural experimentations were exploited to upgrade and validate the constitutive formulations and parameters. The developed investigations led to a reliable constitutive framework of human gastric tissues that both describe stomach mechanical functionality and allow computational investigations. Indeed, the comparisons among average computational data and experimental medians provided the following RMSEs (Root Mean Square Errors): 0.89 N, 0.15 N for corpus and fundus during membrane indentation test, respectively, and 0.44 kPa during inflation test. Accounting for the magnitude of experimental and computational data, the RMSEs can be considered low and acceptable because they concerned biological samples. In fact, biological tissues and structures are affected by a high inherent inter-samples' variability, which is detectable in both the geometrical configuration and the mechanical behaviour. The specific values of the here reported RMSEs ensured the reliability of the achieved parameters and the quality of the overall developed procedure. Reliable computational models of the gastric district could become efficient clinical tools to find out the main crucial aspects of bariatric procedures, such as the mechanical stimulation of gastric mechano-receptors. Moreover, the methods of computational biomechanics will permit to run the preliminary tests of new and innovative bariatric procedures, on one hand, predicting the successful rate and the effectiveness, and, on other hand, reducing the use of animal testing.

Computational methods for the investigation of ski boots ergonomics

Ski boots are known to cause vasoconstriction in the wearer’s lower limbs and, thus, cause a “cold leg” phenomenon. To address this problem, this work provides a computational framework for analysing interactions between the ski boot and the lower limb. The geometry of the lower limb was derived from magnetic resonance imaging and computed tomography techniques and anthropometric data. The geometry of the ski boot shell was obtained by means of three-dimensional computer aided design models from a manufacturer. Concerning the ski boot liner, laser scanning techniques were implemented to capture the geometry of each layer. The mechanical models of the ski boot and the lower limb were identified and validated by means of coupled experimental investigations and computational analyses. The computational models were exploited to simulate the buckling process and to investigate interaction phenomena between the boot and the lower limb. Similarly, experimental activities were performed to further analyse the buckling phenomena. The obtained computational and experimental results were compared regarding both interaction pressure and displacements between the buckle and the corresponding buckle hooks. These comparisons provided reasonable agreement (mean value of discrepancy between the model and mean experimental results in the tibial region: 20%), underlining the model’s capability to correctly interpret results from experimental measurements. Results identified the critical areas of the leg, such as the tibial region, the calcaneal region of the foot and the anterior sole, which may suffer the most due to the hydrostatic pressure and compressive strain exerted on them. The results highlight that computational methods allow investigation of the interaction phenomena between the lower leg and ski boot, potentially providing an effective framework for a more comfortable and ergonomic design of ski boots.

Feasibility Study of a Rapid Evaluate and Adjust Device (READ) for Custom Foot Orthoses Prescription

OBJECTIVE

Custom foot orthoses (CFOs) are typically used for the prevention and cure of lower extremity injuries (LEIs). Typically, CFOs are designed and prescribed based on iterative loops including: (1) follow-up loops between the patient and the physician, and (2) design feedback loops between the physician and the fabricator. The current prescription methodology has some deficiencies, i.e. excessive time to satisfactory treatment, and low repeatability in custom fabrication because of missing alignment, soft tissue considerations, and subjective feedback. There are significant opportunities to develop a new CFOs prescription procedure which can improve accuracy prior to the fabrication process by reducing time through minimizing iterations.

METHODS

First, a novel "rapid evaluate and adjust device" (READ) prescription methodology is proposed for CFO design by combining the follow-up loops with design feedback loop. To support the idea of the READ prescription method a novel 3D ergonomic measurement system (3DEMS) is developed. The 3DEMS is designed for the following key targets to: (1) improve the communication between the patient/physician and the fabricator, (2) reduce time to satisfactory treatment, (3) improve repeatability by considering the alignment and the soft tissue deformations, (4) archive digitally with minimal data, (5) reduce the system complexity, and (6) validate with plantar pressure measurements (i.e. Novel Pedar^®). The design process of the 3DEMS involved the following steps: (1) 3D data collection at the desired loading, (2) nested optimization to determine optimal segment design, and (3) system fabrication considering alignment and feedback control.

RESULTS

The results show that the READ prescription method with 3DEMS can be used to recreate the medial longitudinal arch for a range of arch height indices (AHI) by using a minimal number of parameters i.e. 6 parameters, and significant increases in mean peak pressure are observed between optimized and barefoot or flat segments.

CONCLUSION

This study establishes that the proposed READ prescription method with the 3DEMS may be used for CFOs prescription due to better communication between individuals in the follow-up and design loops, less time for satisfactory treatment, improved repeatability, archivable data, and low system complexity.

SIGNIFICANCE

The developed system may be used as measurement systems for CFOs, and in the future the proposed 3DEMS may prove highly important for the measurement of CFOs for flat feet.

A COUPLED EXPERIMENTAL AND NUMERICAL APPROACH TO CHARACTERIZE THE ANISOTROPIC MECHANICAL BEHAVIOR OF AORTIC TISSUES

Nowadays, the investigation of aortic wall biomechanics is a fundamental tool in clinical research and vascular prosthesis design. This study aims at analyzing the biomechanical behavior of aortic tissues using a coupled experimental and computational approach. Considering the typical fiber-reinforced configuration of aortic tissues, uni-axial tensile tests along six different loading directions were performed on specimens from pig aorta. Starting from the obtained experimental data, a suitable constitutive framework was defined and a methodology for the identification of the constitutive parameters was developed using the inverse analysis of mechanical tests. Transversal stretch versus loading stretch and nominal stress versus loading stretch curves were evaluated, showing the anisotropic and nonlinear mechanical behavior determined by tissue conformation with fibers distributed along preferential directions. In detail, experimental data showed different mechanical responses between longitudinal and circumferential directions, with a greater tissue stiffness along the longitudinal one. The reliability of the developed constitutive framework was evaluated by the comparison between experimental data and model results. The mentioned analysis can be considered as a useful tool for the development of reliable computational models, which allow a better understanding of the pathophysiology of cardiovascular diseases and can be applied for a proper planning of surgical procedures.

In vitro biomechanical properties of sole tissues: Comparison between healthy and ulcerated bovine claws

Sole ulcers are reportedly one of the most prevalent diseases associated with lameness in dairy cattle, significantly affecting animal welfare and farm profitability. The degree to which sole soft tissues, healthy or ulcerated, are able to maintain their structure and function when subjected to compressive forces remains unknown. Therefore, the aims of the present study were to assess sole tissue biomechanics in healthy and ulcerated claws and to describe correlated histology. Cylindrical samples were harvested from zones 4 and 6, as described by the international foot map, from hind lateral healthy (n = 12) and ulcerated bovine claws (n = 8; animals n = 12). Tissue biomechanics and morphology were evaluated via compressive tests and hematoxylin-eosin-phloxine-saffron staining, respectively. A 2-sample t-test was used to compare zones' mechanical properties between healthy and ulcerated tissues, and the Cochran-Mantel-Haenszel test was used to measure the effect of claw zone on histology. The fibril modulus (E_f) and permeability (k) respectively increased and decreased in ulcerated claws (E_f = 0.201 ± 0.104 MPa; k = 0.128 ± 0.069 mm²/MPa·s) compared with healthy claws (E_f = 0.105 ± 0.050 MPa; k = 0.452 ± 0.365 mm²/MPa·s) only for zone 6. Histology scores equal to or greater than 3 were associated with macroscopic presence of ulceration. A higher proportion of adipose tissue (30% or more) was associated with zone 6 compared with zone 4, but no difference was seen between healthy and ulcerated claws. Ulcerated claws had a higher prevalence of exostoses compared with healthy ones (33% vs. 8%). Sole soft tissues showed, as hypothesized, a viscoelastic behavior using unconfined compression testing, which, however, may not reflect in vivo loading conditions. Clinical and histological signs of sole ulceration were not associated with decreased strength of the supportive apparatus of the distal phalanx in zone 4 in this study.

In-vivo viscous properties of the heel pad by stress-relaxation experiment based on a spherical indentation

Identifying the viscous properties of the plantar soft tissue is crucial not only for understanding the dynamic interaction of the foot with the ground during locomotion, but also for development of improved footwear products and therapeutic footwear interventions. In the present study, the viscous and hyperelastic material properties of the plantar soft tissue were experimentally identified using a spherical indentation test and an analytical contact model of the spherical indentation test. Force-relaxation curves of the heel pads were obtained from the indentation experiment. The curves were fit to the contact model incorporating a five-element Maxwell model to identify the viscous material parameters. The finite element method with the experimentally identified viscoelastic parameters could successfully reproduce the measured force-relaxation curves, indicating the material parameters were correctly estimated using the proposed method. Although there are some methodological limitations, the proposed framework to identify the viscous material properties may facilitate the development of subject-specific finite element modeling of the foot and other biological materials.

Investigation of the mechanical behaviour of the plantar soft tissue during gait cycle: Experimental and numerical activities

The aim of this work is to investigate the mechanical response of the plantar soft tissue from the heel strike to the midstance, developing both experimental and numerical activities. Using force plates and motion tracking system, the dynamic and kinematic data of 10 subjects are evaluated. The average kinematics data obtained from the experimental tests are assumed as boundary and loading conditions for the computational analyses. A three-dimensional virtual solid model of the foot is developed from the analysis of Digital Imaging and Communications in Medicine images from computed tomography and magnetic resonance. Constitutive formulations that interpret the mechanical response of the biological tissues are defined. Because of the major role of plantar soft tissue in the proposed analysis, a specific visco-hyperelastic constitutive formulation is provided considering the typical features of the tissue mechanics. The three-dimensional numerical model permits to evaluate the capability of the plantar soft tissue to redistribute the deformations, especially during the midstance, and to define quantitative aspects related to the energy absorption. The numerical results highlight the stress distribution from the heel strike to the midstance. The values of stress and strain reached are more intensive during the midstance, when there is a single support of the foot.

Analysis of the structural behaviour of colonic segments by inflation tests: Experimental activity and physio-mechanical model

A coupled experimental and computational approach is provided for the identification of the structural behaviour of gastrointestinal regions, accounting for both elastic and visco-elastic properties. The developed procedure is applied to characterize the mechanics of gastrointestinal samples from pig colons. Experimental data about the structural behaviour of colonic segments are provided by inflation tests. Different inflation processes are performed according to progressively increasing top pressure conditions. Each inflation test consists of an air in-flow, according to an almost constant increasing pressure rate, such as 3.5 mmHg/s, up to a prescribed top pressure, which is held constant for about 300 s to allow the development of creep phenomena. Different tests are interspersed by 600 s of rest to allow the recovery of the tissues' mechanical condition. Data from structural tests are post-processed by a physio-mechanical model in order to identify the mechanical parameters that interpret both the non-linear elastic behaviour of the sample, as the instantaneous pressure-stretch trend, and the time-dependent response, as the stretch increase during the creep processes. The parameters are identified by minimizing the discrepancy between experimental and model results. Different sets of parameters are evaluated for different specimens from different pigs. A statistical analysis is performed to evaluate the distribution of the parameters and to assess the reliability of the experimental and computational activities.

Constitutive formulation and numerical analysis of the biomechanical behaviour of forefoot plantar soft tissue

The aim of this work is to provide a numerical approach for the investigation of the mechanical behaviour of the forefoot soft tissues. The development of reliable numerical models of biological structures requires the definition of constitutive formulations that actually interpret the mechanical response of the constituent biological tissues and their structural arrangement. A specific visco-hyperelastic constitutive model is provided to account for the typical features of soft plantar tissue mechanics, as geometric and material non-linearity, almost-incompressible behaviour and time-dependent phenomena. Constitutive parameters are evaluated by the analysis of experimental data from compression and stress relaxation tests on tissue samples. A three-dimensional finite element model of the forefoot region is developed starting from the analysis of biomedical images, leading to the evaluation of overall structural response. The reliability of model and analyses is assessed by the comparison of experimental and numerical results pertaining to indentation tests. The numerical model developed allows to evaluate the mechanical response of plantar soft tissue in terms of stress and strain distribution.

The in vivo plantar soft tissue mechanical property under the metatarsal head: implications of tissues׳ joint-angle dependent response in foot finite element modeling

Material properties of the plantar soft tissue have not been well quantified in vivo (i.e., from life subjects) nor for areas other than the heel pad. This study explored an in vivo investigation of the plantar soft tissue material behavior under the metatarsal head (MTH). We used a novel device collecting indentation data at controlled metatarsophalangeal joint angles. Combined with inverse analysis, tissues׳ joint-angle dependent material properties were identified. The results showed that the soft tissue under MTH exhibited joint-angle dependent material responses, and the computed parameters using the Ogden material model were 51.3% and 30.9% larger in the dorsiflexed than in the neutral positions, respectively. Using derived parameters in subject-specific foot finite element models revealed only those models that used tissues׳ joint-dependent responses could reproduce the known plantar pressure pattern under the MTH. It is suggested that, to further improve specificity of the personalized foot finite element models, quantitative mechanical properties of the tissue inclusive of the effects of metatarsophalangeal joint dorsiflexion are needed.

Characterization of the anisotropic mechanical behaviour of colonic tissues: experimental activity and constitutive formulation

The aim was to investigate the biomechanical behaviour of colonic tissues by a coupled experimental and numerical approach. The wall of the colon is composed of different tissue layers. Within each layer, different fibre families are distributed according to specific spatial orientations, which lead to a strongly anisotropic configuration. Accounting for the complex histology of the tissues, mechanical tests must be planned and designed to evaluate the behaviour of the colonic wall in different directions. Uni-axial tensile tests were performed on tissue specimens from 15 fresh pig colons, accounting for six different loading directions (five specimens for each loading direction). The next step of the investigation was to define an appropriate constitutive framework and develop a procedure for identification of the constitutive parameters. A specific hyperelastic formulation was developed that accounted for the multilayered conformation of the colonic wall and the fibre-reinforced configuration of the tissues. The parameters were identified by inverse analyses of the mechanical tests. The comparison of model results with experimental data, together with the evaluation of satisfaction of material thermomechanics principles, confirmed the reliability of the analysis developed. This work forms the basis for more comprehensive activities that aim to provide computational tools for the interpretation of surgical procedures that involve the gastrointestinal tract, considering the specific biomedical devices adopted.

Analysis of the biomechanical behaviour of gastrointestinal regions adopting an experimental and computational approach

An integrated experimental and computational procedure is provided for the evaluation of the biomechanical behaviour that characterizes the pressure-volume response of gastrointestinal regions. The experimental activity pertains to inflation tests performed on specific gastrointestinal conduct segments. Different inflation processes are performed according to progressively increasing volumes. Each inflation test is performed by a rapid liquid in-flaw, up to a prescribed volume, which is held constant for about 300s to allow the development of relaxation processes. The different tests are interspersed by 600s of rest to allow the recovery of the specimen mechanical condition. A physio-mechanical model is developed to interpret both the elastic behaviour of the sample, as the pressure-volume trend during the rapid liquid in-flaw, and the time-dependent response, as the pressure drop during the relaxation processes. The minimization of discrepancy between experimental data and model results entails the identification of the parameters that characterize the viscoelastic model adopted for the definition of the behaviour of the gastrointestinal regions. The reliability of the procedure is assessed by the characterization of the response of samples from rat small intestine.

Constitutive formulations for the mechanical investigation of colonic tissues

A constitutive framework is provided for the characterization of the mechanical behavior of colonic tissues, as a fundamental tool for the development of numerical models of the colonic structures. The constitutive analysis is performed by a multidisciplinary approach that requires the cooperation between experimental and computational competences. The preliminary investigation pertains to the review of the tissues histology. The complex structural configuration of the tissues and the specific distributions of fibrous elements entail the nonlinear mechanical behavior and the anisotropic response. The identification of the mechanical properties requires to perform mechanical tests according to different loading situations, as different loading directions. Because of the typical functionality of colon structures, the tissues mechanics is investigated by tensile tests, which are performed on taenia coli and haustra specimens from fresh pig colons. Accounting for the histological investigation and the results from the mechanical tests, a specific hyperelastic framework is provided within the theory of fiber-reinforced composite materials. Preliminary analytical formulations are defined to identify the constitutive parameters by the inverse analysis of the experimental tests. Finite element models of the specimens are developed accounting for the actual configuration of the colon structures to verify the quality of the results. The good agreement between experimental and numerical model results suggests the reliability of the constitutive formulations and parameters. Finally, the developed constitutive analysis makes it possible to identify the mechanical behavior and properties of the different colonic tissues.

On a staggered iFEM approach to account for friction in compression testing of soft materials

An inverse finite element method (iFEM) to estimate material parameters from compression tests of soft materials is presented, where alginate hydrogel was used as a phantom material. The method applies if the boundary conditions at the loaded surfaces are not ideal, i.e. neither free of friction nor fully constrained, as it may be the case in most realistic testing set-ups. Assuming a linear friction law, the friction coefficient μ was considered unknown and estimated in a first step by minimising the difference between the contours of the sample, obtained by optical measurements, and the simulated shape. Force-displacement data were used in a second step to determine the parameters of the constitutive law. Staggering these two steps, both friction and material parameters were identified by optimisation. Skipping the first step and predefining μ instead, a unique parameter set could only be clearly identified if the deviations of the contours were considered in addition to the deviations in the force-displacement data. Finally, forward FEM calculations with differently shaped specimens were used to verify the goodness of the obtained parameter sets.

Investigations on the viscoelastic behaviour of a human healthy heel pad: In vivo compression tests and numerical analysis

The aim of this study was to investigate the viscoelastic behaviour of the human heel pad by comparing the stress–relaxation curves obtained from a compression device used on an in vivo heel pad with those obtained from a three-dimensional computer-based subject-specific heel pad model subjected to external compression. The three-dimensional model was based on the anatomy revealed by magnetic resonance imaging of a 31-year-old healthy female. The calcaneal fat pad tissue was described with a viscohyperelastic model, while a fibre-reinforced hyperelastic model was formulated for the skin. All numerical analyses were performed to interpret the mechanical response of heel tissues, with loading conditions and displacement rate in agreement with experimental tests. The heel tissues showed a non-linear, viscoelastic behaviour described by characteristic hysteretic curves, stress–relaxation and viscous recovery phenomena. The reliability of the investigations was validated by the interpretation of the mechanical response of heel tissues under the application of three pistons with diameter of 15, 20 and 40 mm, at the same displacement rate of about 1.7 mm/s. The maximum and minimum relative errors were found to be less than 0.95 and 0.064, respectively.