## Abstract

The calculation of tensile mechanical properties from stress–strain curves is a fundamental step in characterizing material behavior, yet no standardized method exists to perform these calculations for soft tissue. To address this deficiency, we developed a free web application called Dots-on-Plots^{2} that fully automates the calculation of tensile mechanical properties from stress–strain curves. The analyzed mechanical properties include the strength, strain, and energy at four points of interest (transition, yield, ultimate, and rupture), and the linear modulus. Users of Dots-on-Plots can upload multiple files, view and download results, and adjust threshold settings. This study determined a threshold setting that minimized error when calculating the transition point, where the stress–strain curve “transitions” from a nonlinear “toe” region to a linear region. Using the optimal threshold (2% stress deviation from a linear region fit), Dots-on-Plots calculated the transition strains from twenty tensile experiments of human meniscus to be 0.049 ± 0.007, which nearly matched the known transition strain values of 0.050 ± 0.006 (determined using finite element parameter optimization). The sensitivity of the calculated transition strain to the shape of various stress–strain curves was analyzed using sets of model-generated synthetic data. This free web application offers a convenient and reliable tool to systematically enhance the speed, transparency, and consistency of mechanical analysis across biomedical research groups.

## Introduction

Uniaxial tensile tests are conventional experiments to characterize the mechanical behavior of engineered and biological materials. Stress-strain curves generated from tensile tests provide a graphical representation of a tissue's normalized load response to axial stretch. Several key mechanical properties can be quantified from these curves, including ultimate tensile strength (UTS), yield strength, energy to failure, and the transition strain [1–3] (Fig. 1). International testing standards for many engineered materials (e.g., plastics, polymer matrix composites) [4,5] provide guidelines for identifying and calculating relevant properties from stress–strain curves. These standards reduce the probability that the reported properties are biased by their testing environment or calculation method, thus improving reproducibility between different research groups. These standards have also led to the development of software packages [6,7] to automate the calculation of mechanical properties, further reducing the subjectivity and burden of data analysis on research groups. Unfortunately, no testing standards exist for tensile testing of soft biological tissues, and existing software packages are unable to account for soft tissue's nonlinear stress–strain behavior (Fig. 1).

The lack of standards for tensile testing of soft fibrous tissues has resulted in biomedical research groups using different methods to calculate mechanical properties. One such mechanical property, the transition strain, physically represents the straightening of collagen fibers [1]. At the transition point, soft fibrous tissue “transitions” from an exponential stress–strain response (toe region), to an approximately linear response (Fig. 1). This point on the stress–strain curve is important, as the normal physiological functions of tendon and ligament occur mostly in the toe region [8] where the tissue is highly resistant to fatigue damage [9]. Methods to calculate the transition point vary widely across research groups, and include using inflection points of a polynomial fit [10,11], bimodal linear fitting algorithms [12,13], piecewise fitting algorithms [14], or deviation of either stress or strain from a linear fit to the linear region [15,16]. While descriptions of these methods are published, the custom programs written to do the analysis are not publicly available. This lack of standardization and transparency could be contributing to the wide variance of reported transition strains that exist between research groups (Table 1) [2,10–12,16,17]. It's possible to reduce cross-lab variability and accelerate the pace of discovery by giving research groups access to a standardized computational tool that automates the calculation of transition strain and other tensile properties in soft tissue. A type of computational tool that provides exceptional accessibility is a web-based software application, which permits instant access to anyone with an internet connection. The objectives of this work are to 1) develop a free, web-based software application for the calculation of soft-tissue tensile properties, and 2) identify optimal program settings to minimize transition strain error when evaluating experimental stress–strain curves.

Study | Species | Method | Transition strain |
---|---|---|---|

Nesbitt et al. [11] | Human | 10% stress deviation from linear fit | 3.6 ± 0.4% |

Tissakht and Ahmed [2] | Human | Intersection of non-linear and linear curve fits | 7.4 ± 2.6% |

He et al. [17] | Porcine | Bimodal linear fitting | 4.6 ± 0.7% |

Abdelgaied et al. [12] | Porcine | Bimodal linear fitting | 5.5 ± 0.9% |

Wale et al. [10] | Bovine | 5% stress deviation from linear fit | 5.2 ± 1.2% |

Danso et al. [16] | Bovine | 0.6% strain deviation from linear fit | 5.9 ± 1.9% |

Study | Species | Method | Transition strain |
---|---|---|---|

Nesbitt et al. [11] | Human | 10% stress deviation from linear fit | 3.6 ± 0.4% |

Tissakht and Ahmed [2] | Human | Intersection of non-linear and linear curve fits | 7.4 ± 2.6% |

He et al. [17] | Porcine | Bimodal linear fitting | 4.6 ± 0.7% |

Abdelgaied et al. [12] | Porcine | Bimodal linear fitting | 5.5 ± 0.9% |

Wale et al. [10] | Bovine | 5% stress deviation from linear fit | 5.2 ± 1.2% |

Danso et al. [16] | Bovine | 0.6% strain deviation from linear fit | 5.9 ± 1.9% |

## Materials and Methods

### Overview.

A web-based application called Dots-on-Plots was built to automatically calculate and output mechanical properties from tensile stress–strain curves. The optimal threshold setting for computing transition strain was determined by finding what threshold value resulted in the most accurate transition strains when analyzing stress–strain curves from twenty tensile tests of human meniscus. The “gold standard” transition strain used to measure accuracy was determined by curve fitting a hyperelastic-damage model, which included transition strain as a material parameter, to the experimental data using finite element parameter optimization. In addition, 27 unique stress–strain curves were synthetically generated (using a finite element solver) to determine whether the calculation of transition strain was sensitive to curve shape.

### Automated Calculation of Mechanical Properties.

*σ*(

*ε*) deviated from the stress predicted from the linear fit $\sigma linear(\epsilon )$ by a set percentage of the stress at the point of maximum positive slope

*σ*

_{ms}(Fig. 2). This can be mathematically expressed as follows, where the largest value of strain

*ɛ*that satisfies this equation is marked as the transition strain.

Stress and strain at each of these four points of interest are output as mechanical properties. In addition, the strain energy density at each point of interest is calculated. Strain energy density represents the potential energy absorbed by the tissue, normalized by volume, and was calculated as the area under the stress–strain curve using trapezoidal integration up to the point of interest (integration domain defined by the user-inputted strain data). For example, the strain energy density that accumulates from no strain to ultimate strain would be the ultimate energy, and from no strain to the rupture strain would be the rupture energy (Fig. 1). All calculations are independent of units and will display the stress and strain magnitude as provided in the input file. For example, if the stress column is calculated in MPa, the yield strength will be in MPa. Importantly, the user-input stress and strain measures must be energy conjugates to properly calculate the strain energy density (area under the stress–strain curve) [18].

### Web Application Development.

A web application was developed to satisfy four primary design criteria: efficiency, reliability, flexibility, and convenience (Table 2). These design criteria were addressed by incorporating specific design features into a program that was initially developed in Python using the aforementioned algorithms for calculating mechanical properties. This original Python code (available on GitHub^{3}) was installed on a server and a web interface was written in standard HTML, Javascript, and CSS. The server processes data files in parallel, writing out the output images and data to the client.

Design criteria | Design feature |
---|---|

1) Efficient | Fast, automated batch processing of multiple input files. |

2) Reliable | Transparent calculations using graphical displays. |

3) Flexible | Thresholds that define properties are adjustable. |

4) Convenient | Accessible via the web. Properties and plots exported as CSV and PDF. |

Design criteria | Design feature |
---|---|

1) Efficient | Fast, automated batch processing of multiple input files. |

2) Reliable | Transparent calculations using graphical displays. |

3) Flexible | Thresholds that define properties are adjustable. |

4) Convenient | Accessible via the web. Properties and plots exported as CSV and PDF. |

### Determining an Optimal Threshold to Calculate Transition Strain.

The calculation of transition strain in Dots-on-Plots is based on a user-specified threshold for percent stress deviation (Eq. (1)). To determine the optimal threshold setting that minimized error, stress–strain curves were analyzed that had a known answer for transition strain (*ɛ*_{trans_actual}). This known answer was then compared to the transition strain output by Dots-on-Plots for a given threshold (*ɛ*_{trans_dots}). For this analysis, stress–strain curves were acquired from 20 monotonic uniaxial tensile tests (loading rate = 1% strain/s) taken from five young human menisci (age = 33 ± 5 years) and five older human menisci (age = 72 ± 7 years) [11]. The two age groups have significant differences in mechanical properties [11], and therefore we included both age groups to have a more diverse set of stress–strain curves. The twenty experiments were analyzed in Dots-on-Plots using five different threshold settings of percent stress deviation (1, 2, 3, 5, and 10%) to calculate transition strain (*ɛ*_{trans_dots}). We selected these settings to span the range of threshold values applied in previous studies to calculate transition strain (Table 1).

*ɛ*

_{actual}), the experimental stress–strain curves were curve fit to a hyperelastic-damage model using the free finite element solver, FEBio [19]. For this simulation, a single hexahedral element was pulled in tension using a sliding elastic contact to allow displacement along the axial load direction. A transversely isotropic Veronda-Westmann model was selected for the hyperelastic material, where the model's strain energy density

*Ψ*is represented by a ground substance

*F*

_{1}, a fiber matrix

*F*

_{2}

*,*and a hydrostatic term

*I*

_{2}are the first and second invariants of the deviatoric portion of the right Cauchy Green deformation tensor, $\lambda \u0303$ is deviatoric fiber stretch,

*J*is the volume change ratio, and

*K*is bulk modulus, which was set to 1000 MPa to enforce near-incompressibility [20]. The functions for

*F*

_{1}and

*F*

_{2}can be written

where the strain energy of the ground substance *F*_{1} is dependent on two user-specified material coefficients (*C*_{1}, *C*_{2}), and the strain energy of the fiber matrix *F*_{2} is dependent on four user-specified material coefficients (*C*_{3}, *C*_{4}, *C*_{5} and *λ _{m}*). Of particular significance to this study is the coefficient

*λ*, which defines the transition stretch between the exponential and linear regions [14]. The material coefficient

_{m}*C*

_{6}(not user-specified) ensures that stress is continuous at the transition stretch. The strain energy density can be converted to an effective Cauchy stress

**σ**[19].

_{0}_{}*D*can range from values of 0 to 1, and is a function of the first principle Lagrange strain (

*Ξ*). The limits of damage evolution (

*μ*

_{min},

*μ*

_{max}) determine the strain values where damage initiates and reaches a maximum (

*D*

_{max}), respectively. Finally, Cauchy stress

**σ**was calculated by scaling the effective undamaged stress

**with damage**

*σ*_{0}*D*

### Parameter Optimization.

The material parameters were curve fit to experimental data (axial force versus time) using the Levenberg–Marquardt parameter optimization module in FEBio [19,21]. Parameter optimization was conducted in two steps. In the first step, only the hyperelastic model was run (Eqs. (2)–(4)), and the elastic material parameters (*C _{1}*,

*C*,

_{2}*C*,

_{3}*C*,

_{4}*C*,

_{5}*λ*) were fit to the toe and linear region of the stress–strain curve (Fig. 1). Initial guesses for

_{m}*C*,

_{1}*C*, and

_{5}*λ*were based on previously measured mechanical properties [11] with optimization limits of more than ± two standard deviations to help ensure convergence to a global minima. Initial guesses for the remaining elastic parameters were determined by trial and error, with maximum and minimum limits of approximately double and one-half the initial guess, respectively. In the second step, the full hyperelastic-damage model was run (Eq. (6)), and the damage parameters were fit to the strain-softening region of the stress–strain curve (Fig. 1) with initial guesses set as the values of Lagrange strain at yield and rupture (70% of UTS) from our previous experiments [11]. When optimization returned a parameter extremum, the extrema was expanded by 30% and optimization was reperformed.

_{m}The axial component of stress from the model and experiment were plotted together as a function of engineering tensile strains (*ε* = *λ − *1). The hyperelastic-damage model resulted in excellent fits for all experimental data (Fig. 3), with an average *R*^{2} value of 0.998 ± 0.002, and an NRMSE of 2.9 ± 1.2% (normalized to mean stress). The average model coefficients used to fit these twenty experiments are given in Table 3. The transition stretch *λ _{m}* from the optimized model fit was converted to the known or “actual” transition strain (

*ɛ*

_{actual}=

*λ*− 1). This known transition strain was then used to estimate the accuracy of Dots-on-Plots by calculating the error (

_{m}*ɛ*

_{trans_dots}−

*ɛ*

_{trans_actual}) and absolute error (|

*ɛ*

_{trans_dots}−

*ɛ*

_{trans_actual}|) of the measured transition strain. A similar approach to estimate accuracy has been used in previous studies [22,23].

C_{1} (MPa) | C_{2} | C_{3} (MPa) | C_{4} | C_{5} (MPa) | λ_{m} | μ_{min} | μ_{max} | D_{max} | NRMSE (%) | |
---|---|---|---|---|---|---|---|---|---|---|

Young | 0.75 ± 0.53 | 1.39 ± 1.00 | 0.42 ± 0.10 | 37.9 ± 6.8 | 115.0 ± 38.7 | 1.052 ± 0.006 | 0.14 ± 0.08 | 0.29 ± 0.10 | 1 | 2.42 ± 0.99 |

Older | 0.71 ± 0.78 | 1.24 ± 1.18 | 0.47 ± 0.17 | 37.0 ± 8.1 | 100.2 ± 36.0 | 1.048 ± 0.007 | 0.11 ± 0.05 | 0.26 ± 0.13 | 1 | 3.29 ± 1.23 |

Total | 0.73 ± 0.65 | 1.32 ± 1.07 | 0.45 ± 0.14 | 37.5 ± 7.3 | 107.6 ± 37.2 | 1.050 ± 0.006 | 0.12 ± 0.07 | 0.28 ± 0.11 | 1 | 2.86 ± 1.18 |

C_{1} (MPa) | C_{2} | C_{3} (MPa) | C_{4} | C_{5} (MPa) | λ_{m} | μ_{min} | μ_{max} | D_{max} | NRMSE (%) | |
---|---|---|---|---|---|---|---|---|---|---|

Young | 0.75 ± 0.53 | 1.39 ± 1.00 | 0.42 ± 0.10 | 37.9 ± 6.8 | 115.0 ± 38.7 | 1.052 ± 0.006 | 0.14 ± 0.08 | 0.29 ± 0.10 | 1 | 2.42 ± 0.99 |

Older | 0.71 ± 0.78 | 1.24 ± 1.18 | 0.47 ± 0.17 | 37.0 ± 8.1 | 100.2 ± 36.0 | 1.048 ± 0.007 | 0.11 ± 0.05 | 0.26 ± 0.13 | 1 | 3.29 ± 1.23 |

Total | 0.73 ± 0.65 | 1.32 ± 1.07 | 0.45 ± 0.14 | 37.5 ± 7.3 | 107.6 ± 37.2 | 1.050 ± 0.006 | 0.12 ± 0.07 | 0.28 ± 0.11 | 1 | 2.86 ± 1.18 |

### Sensitivity of Transition Strain to the Shape of Stress-Strain Curves.

A sensitivity analysis was conducted to determine if factors that affect the shape of the stress–strain curve will significantly influence the calculation of transition strain. We used the aforementioned hyperelastic-damage model (Eqs. (2)–(6)) to artificially generate a set of 27 unique stress–strain curves by adjusting the toe region, linear modulus, and damage onset (Fig. 4). The toe region was adjusted by inputting *λ _{m}* values into the model that corresponded to transition strains of 3, 4, and 5% (small, moderate, and large, respectively); linear modulus was adjusted by inputting modulus values into the model (

*C*) of 30, 110, and 190 MPa (low, medium, and high, respectively); and damage onset was adjusted by inputting different

_{5}*μ*

_{min}values to have yield strain occur at 5 ± 1%, 12 ± 2%, and 19 ± 1% (early, average, and late, respectively). The three values for each tested group were based on the average ± two standard deviations from our previous mechanical study on human meniscus [11]. Since the transition stretch

*λ*was a model input (Eq. (4)), the actual transition strain was known (

_{m}*ɛ*

_{trans_actual}) for all generated stress–strain data.

### Statistics.

All statistical analysis was performed using spss software (IBM; Armonk, NY, v25). For analyzed data that were not normally distributed (determined using a Kolmogorov-Smirnov normality test), nonparametric tests were used. For experimental stress–strain curves, the effect of stress deviation threshold (within-subject) and age group (between-subject) on the absolute error of the transition strain calculated by Dots-on-Plots was measured using a repeated measures anova, with Tukey HSD posthoc testing. For synthetically generated stress–strain curves, the effect of input parameters (toe region, linear modulus, damage onset) on the absolute error of the transition strain calculated by Dots-on-Plots was measured using a nonparametric Kruskal-Wallis test. For all statistical tests, significance was set at *p *<* *0.05. All means are reported with one standard deviation.

## Results

### Web Application.

A free web application called Dots-on-Plots was developed (Fig. 5), tested, and is now available online^{2}. This application automatically calculates and exports mechanical properties for soft tissue tensile tests, and allows for multiple files to be input and run simultaneously. Users can upload .xlsx, .csv, or .txt files containing two columns of equal length data, with strain in the first column and stress in the second column. The program generates a stress–strain curve and a table of calculated mechanical properties for each uploaded file (Fig. 5(a)). The threshold settings can be adjusted by the user, including the stress deviation threshold used to calculate the transition point. Results from all analyzed files can be downloaded as one .pdf report and one .csv summary spreadsheet (Fig. 5(b)). The report has graphical displays of the results, including derivative plots used to determine the maximum slope of the stress–strain curve. The summary spreadsheet allows for convenient plotting or statistical analysis of mechanical properties (Fig. 5(b)).

### Optimal Threshold to Calculate Transition Strain.

The error of the transition strain calculated by Dots-on-Plots was influenced by the threshold setting (Fig. 6; *p *<* *0.001), where a stress deviation threshold of 2% was most accurate. For all twenty experiments, the mean transition strain calculated using a 2% threshold in Dots-on-Plots (0.049 ± 0.007) was within 0.0007 (Fig. 6) of the known mean transition strain (0.050 ± 0.006), corresponding to a mean percent error of 1.4%. For each individual experiment, the mean absolute error when using a 2% threshold in Dots-on-Plots was 0.004 ± 0.004, corresponding to a mean absolute percent error of 8.2 ± 8.0%. This absolute error was significantly less than the 10% deviation setting (*p *<* *0.001), but not significantly different than the 1%, 3%, or 5% threshold settings (*p *=* *0.23; *p *=* *0.98; *p *=* *0.18, respectively) (Fig. 6). There were no significant differences in the absolute error of the calculated transition strain due to age group (*p *=* *0.88).

### Sensitivity of Transition Strain to the Shape of Stress-Strain Curves.

The calculation of transition strain was most sensitive to damage onset (Fig. 7; *p *<* *0.001). Stress-strain curves with a late damage onset (Fig. 4, orange curves) had significantly greater mean transition strains that were 2.5 times greater than the known mean (Fig. 7, dashed line). The calculation of transition strain was insensitive to changes in linear modulus (*p *=* *0.95), and to changes in the size of the toe region (*p *=* *0.99) (Fig. 7).

### Automated Calculation of Mechanical Properties.

The tensile mechanical properties of the twenty meniscus specimens (Table 4) were computed in Dots-on-Plots using the optimized threshold setting for the transition point (2% stress deviation from the linear fit), and a rupture point setting at 15% of the ultimate stress (based on our previous experimental work) [11]. The linear fit used to calculate linear modulus (Fig. 2) had an average NRMSE of 0.30 ± 0.12% (*R*^{2} = 0.998 ± 0.001). The mean and standard deviation values in Table 4 were calculated from the .csv output file generated by Dots-on-Plots (Fig. 5(b)). The total runtime for Dots-on-Plots to analyze all twenty stress–strain curves was approximately 15 s.

Transition | Yield | Ultimate | Rupture | |
---|---|---|---|---|

Strain (%) | 4.94 ± 0.68 | 7.08 ± 1.18 | 15.49 ± 5.23 | 24.37 ± 8.46 |

Stress (MPa) | 2.45 ± 1.08 | 4.66 ± 2.21 | 11.57 ± 6.63 | 4.02 ± 5.40 |

Energy (J/ml) | 0.04 ± 0.02 | 0.13 ± 0.08 | 0.97 ± 0.75 | 1.32 ± 0.87 |

Modulus (MPa) | 103.9 ± 37.2 |

Transition | Yield | Ultimate | Rupture | |
---|---|---|---|---|

Strain (%) | 4.94 ± 0.68 | 7.08 ± 1.18 | 15.49 ± 5.23 | 24.37 ± 8.46 |

Stress (MPa) | 2.45 ± 1.08 | 4.66 ± 2.21 | 11.57 ± 6.63 | 4.02 ± 5.40 |

Energy (J/ml) | 0.04 ± 0.02 | 0.13 ± 0.08 | 0.97 ± 0.75 | 1.32 ± 0.87 |

Modulus (MPa) | 103.9 ± 37.2 |

## Discussion

The objectives of this work were to provide a free, web-based computational tool to calculate the tensile mechanical properties of soft tissue, and to identify the optimal settings to minimize error when calculating the transition strain, a mechanical property with important implications for soft collagenous tissues. We met our first objective by developing a web application called Dots-on-Plots that is now freely available on the internet^{2} (Fig. 5). We met our second objective by determining that a 2% stress deviation was an optimal threshold for calculating transition strain from a set of uniaxial tensile experiments of human meniscus (Fig. 6). We further identified characteristics of stress–strain curves that affect the calculation of the transition strain (Fig. 7).

The development of Dots-on-Plots represents an important advance for the standardization of material characterization in biomedical engineering and beyond. To our knowledge, Dots-on-Plots will be the first web-based application that allows users to upload tensile test data to calculate mechanical properties. Existing software for analyzing stress–strain curves includes downloadable software packages that are tailored toward more common engineering materials, like metals or semiconductors [24,25]. While these programs are capable of calculating a wide variety of mechanical and thermodynamic properties, they would not be appropriate for handling the analysis of tissues that exhibit nonlinear behavior. A recently released downloadable software, mechanalyze [26], has sought to help fill this gap by automating the analysis of compressive force–displacement curves. mechanalyze calculates the ultimate stresses and strains, as well as compressive moduli for hydrogels or tissues, but is not currently capable of calculating transition, yield, strain energy, or analyzing tensile data. Dots-on-Plots is unique in automating the analysis of tensile data for soft tissue, and importantly, is unique in being a web application. A striking advantage of web applications is they provide on-demand access to a scalable software platform. By eliminating barriers related to downloading and installing a software package on a particular operating system, web-based software becomes easily accessible to a worldwide research community. In the future, Dots-on-Plots can be expanded to analyze other types of stress–strain curves and other types of materials.

The automated calculation of mechanical properties with Dots-on-Plots can help researchers conduct an objective and comprehensive analysis of mechanical behavior. For example, our automated calculation of linear modulus can eliminate the subjectivity inherent in previous methods that defined modulus as the slope between two user-defined points [27,28]. Our program also automatically calculates strain energy density, which is a single scalar measure of material resilience and toughness [29] that can be calculated at yield [30], ultimate [31], and rupture points [32]. While strain energy density is often reported in biomechanical studies using computational models [33,34], it is not a commonly reported property in experimental studies of soft tissue. Automating this calculation could encourage research groups to consistently report this property, thus improving our understanding of tissue material behavior. Our study determined yield strength by finding the point on the stress–strain curve with a maximum positive slope, as this point indicates the beginning of strain-softening (Fig. 1). For collagenous tissues, strain softening correlates with the onset of tissue damage in the form of unfolding collagens [9], though the precise location of this point along the stress–strain curve is debatable. To account for this uncertainty and support program flexibility (Table 2), Dots-on-Plots allows users to adjust the positioning of the yield point from the point of maximum slope. The downloadable files from Dots-on-Plots (PDF and CSV) provide an archivable record of all results and settings that can be readily included as supplementary data for journal articles.

An innovation of this project was the use of a constitutive model to determine the known (actual) transition strain from sets of stress–strain data. This allowed us to quantify the accuracy of Dots-on-Plots relative to a known answer. We selected a hyperelastic-damage model that 1) gave excellent fits to experimental stress–strain curves (Fig. 3), and 2) included a material coefficient (λ* _{m}*) equivalent to the transition stretch, defined as the intersection of the toe and linear region (Fig. 1). This model curve fitting served as a “gold standard” to quantify the error of our algorithm for calculating transition strains. We found that a 2% stress deviation threshold gave the best results, with a mean absolute error of 0.004 engineering strain for each individual experiment, corresponding to a mean absolute percent error of 8%. This level of accuracy was independent of stress–strain curves from young and older groups that exhibited different mechanical properties [11]. Moreover, if we examine the average transition strain for all twenty experiments using a 2% threshold in Dots-on-Plots (0.049) and compare it to the average known transition strain using finite element parameter optimization (0.050), we see little overall difference between the methods (1.4% mean percent error). This gives us confidence that our algorithm for calculating transition strain can provide a fast and reliable alternative to time-intensive parameter optimization.

To determine whether the calculation of transition strain by Dots-on-Plots was sensitive to the shape of the input stress–strain curve, we analyzed a manufactured set of curves with unique shapes (Fig. 4). We found that our algorithm was generally robust when analyzing data with different lengths of toe region and magnitudes of linear modulus, but was quite sensitive to damage onset (Fig. 7). The late damage onset group gave the largest magnitude of transition strain error due the linear region of the stress–strain curve having a steady curvature that prematurely triggered the stress deviation threshold and resulted in large overpredictions of the transition strain. For this reason, we recommend increasing the threshold to calculate the transition strain when analyzing stress–strain curves with high-yield strains.

The mechanical properties calculated in this study using Dots-on-Plots can be compared to previous biomechanical meniscus studies. Our calculated transition strain of 4.9% (Table 4; using the optimized 2% stress deviation setting) is close to the overall mean transition strain of approximately 5.4% that was computed in six previous meniscus studies using various methods to compute transition strain (Table 1). Our mean linear modulus value of 104 MPa is similar to the mean modulus values of 90–140 MPa reported for human meniscus by Lechner et al. and Tissakht & Ahmed [2,28]. Importantly, our method for calculating the linear modulus, where we applied a linear fit to stress–strain data near the maximum slope of the stress–strain curve (Fig. 2), gave excellent fits to our experimental data in this region, with an average NRMSE of 0.30 ± 0.12%. The goodness of fit persisted across the entirety of the linear region between the transition and yield point, maintaining an average NRMSE of 0.99 ± 0.53%. The close comparison of our calculated linear modulus to prior studies, and the quality and consistency of our linear fits, give us confidence that our algorithm is accurately determining the linear modulus. It's also worth noting that the tensile properties computed by Dots-on-Plots are nearly identical to our prior analysis of this same set of tensile stress–strain curves [11]. This was expected since the custom matlab script used in our prior analysis was eventually converted into the Dots-on-Plots web application. The one exception is that the mean transition strain in the current study (4.9%) is larger than we previously reported (3.6%). The reason for this difference is that we used a 10% stress deviation threshold in our prior study, which we now know is too large a threshold to accurately estimate the transition strain (Fig. 6).

This study has several notable limitations. First, Dots-on-Plots was designed to evaluate tensile pull-to-failure data of soft tissue, and may not be appropriate for analyzing other materials and test configurations (e.g., compression, shear). However, in practice, any stress–strain curve with a toe and linear region could be analyzed with Dots-on-Plots. For example, compression tests of intervertebral disk exhibit stress–strain profiles with a toe and linear region [35], and therefore could be analyzed using our software. Second, the optimal threshold value to calculate transition strain was determined from a single material (human lateral meniscus), which may not represent all variations in stress–strain behavior the program may encounter. To account for this, we (1) conducted sensitivity tests to better understand stress–strain shapes that could pose problems to our algorithm, and (2) designed the software to allow users to adjust threshold settings. Third, this study focused on a single set of algorithms to automate the calculation of transition and yield points, and the accuracy of other techniques to detect these points of interest was not quantified [2,12,16,17,36]. Nevertheless, the algorithms used in this study were shown to be accurate (Fig. 6), and they have proven to be robust in previous work [3,10,11]. Finally, the program performs unitless calculations. While this provides program simplicity and user flexibility, it does mean that the accurate calculation of strain energy density is dependent on the user inputting stress and strain measures that are energy conjugates (e.g., 2nd Piola-Kirchhoff stress vs. Green-Lagrange strain) [18].

In conclusion, this study has developed and evaluated a free, web-based program for the calculation of tensile mechanical properties. This program can provide researchers with a fast, convenient, and reliable tool to analyze mechanical data, and along with other recent work from our group [10,37], can support the broad adoption of standard test methods for tensile testing of biological tissue.

## Funding Data

National Science Foundation (Grant No. 1554353; Funder ID: 10.13039/100000147).

National Institute of Arthritis and Musculoskeletal and Skin Diseases (Grant No. 1R15AR075314-01; Funder ID: 10.13039/100000069).

## Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.