Biological materials exhibit time and history-dependent stress-strain viscoelastic behaviour (Fung, 1967; Nasseri et al., 2002). In the frequency domain, soft tissues exhibit a power law behaviour which suggests a broad range of relaxation processes that are likely present (Bonfanti et al., 2020). Fractional calculus can be used to model the power-law behaviour of linear viscoelastic models (Torvik and Bagley, 1984; Bagley and Torvik, 1986; Suki et al., 1994; Pritz, 1996; Fabry et al., 2001; Rossikhin and Shitikova, 2001) using elements known as springpots or Scott-Blair elements (Craiem and Armentano, 2006). Fractional viscoeleastic models have been applied to materials such as rubber, polymers and gels (Alcoutlabi and Martinez-Vega, 1998; Chen et al., 2004; Kobayashi et al., 2017; Aime et al., 2018; Bouzid et al., 2018; Kaplan et al., 2019; Katsourinis and Kontou, 2019; Bonfanti et al., 2020) and tissue such as the brain (Sack et al., 2009), liver (Kobayashi et al., 2005; Kobayashi et al., 2009; Nicolle et al., 2010; Kobayashi et al., 2012a; Kobayashi et al., 2012b), kidney (Nicolle et al., 2010), red blood cells (Craiem and Magin, 2010), and arteries (Craiem and Armentano, 2007; Craiem et al., 2008).

In this paper, viscoelastic properties of various porcine organs were measured and fit to a number of three-parameter biomechanical models. It was found that a semi-fractional Kelvin Voight (S-FKV) biomechanical model was the best fit to the power law behaviour of the organs up to 10% axial strain. Fractional viscoelastic models have previously been used in various studies (Kobayashi et al., 2005; Craiem and Armentano, 2007; Craiem et al., 2008; Kobayashi et al., 2009; Sack et al., 2009; Craiem and Magin, 2010; Nicolle et al., 2010; Kobayashi et al., 2012a; Kobayashi et al., 2012b), and the results reported here adds to the quantity of data available to researchers. Further in this study, the S-FKV model was fit to frequency domain measurements, and the model was then used to predict the time domain relaxation behaviour of the organs which were compared with the time domain experimental data. The parameters obtained from the S-FKV model were also extrapolated to higher frequencies and were comparable to elastography measurements.

The time scales (0.1–100 s) and strains (0.01%–10%) reported in these measurements are relevant to a number of clinical applications, in particular to provide input data for simulations. For example, elastography techniques which have been employed for diagnostic applications such as diagnosing liver fibrosis (Yin et al., 2007), characterising brain lesions (Kruse et al., 2008), and characterising cardiac diseases such as myocardial infarction (Kolipaka et al., 2009). A second example is minimally invasive surgery where needles (and other instruments) are inserted into tissue, such as: needle insertion into liver for radiofrequency ablation or percutaneous ethanol injection therapy (Kobayashi et al., 2009). In neurosurgery (Yundt et al., 1997), a biomechanical model could be used to optimise retractor-applied pressure and retractor position for reducing injury to tissue from retractor strains (Kyriacou et al., 2002). A final example is kidney stone treatment, both percutaneous nephrolithotomy done as key hole-surgery (Kallidonis et al., 2020; Barua et al., 2022) and shock wave lithotripsy (Pace et al., 2005) where the pulse repetition frequency of 1–2 Hz are typically employed.

Measurements of the mechanical properties of porcine heart, kidney, liver and brain are reported here. Storage modulus and loss modulus remained steady during strain sweep tests from 0.01% to 0.1% shear strain suggesting that all organs exhibit linear viscoelastic behaviour in this region. Frequency sweep tests were performed at 0.1% shear strain, and a S-FKV model fitted to the frequency sweep data at different axial strains for all the porcine organs with a correlation coefficient better than 0.99. The S-FKV model had a better fit than both the SLS, which couldn’t capture the storage modulus and loss modulus, and the FKV which couldn’t capture the frequency dependence of the loss modulus. We note that including more viscoelastic elements in the SLS, i.e., generating a Prony series (Mishra and Cleveland, 2024), could result in a better fit but at the cost of needing to fit more parameters. Here all the models we compared only had three fitting parameters.

The coefficient of consistence (K _α) increased monotonically with the axial strain for all the organs and there are two potential reasons behind this increase. Firstly, there is a close contact between connective tissue layers during the pre-compression forcing the interstitial fluid out of the tissue matrix and leading to a more solid like behaviour. Secondly, a higher stress is accumulated at small strains in soft tissues due to their hyperelastic nature thereby requiring a higher torque to deform these samples and hence, a higher storage and loss modulus. The stiffness in shear modulus due to compression observed in our study is similar to the result obtained in the literature (Hrapko et al., 2008; Engstrom et al., 2019). The shear modulus in the study (Hrapko et al., 2008) performed on ex-vivo porcine brains increased by 20% as the precompression force was increased from 5 to 10 mN. A compressional study (Engstrom et al., 2019) performed on mammalian brain tissue observed a linear increase in the shear storage modulus with the uniaxial prestress.

The predictive power of the S-FKV model was tested by comparing the predicted response to a step shear strain with the measured response. The stress relaxation amplitudes of the predictions and measurements were within 30%. One potential reason for the discrepancy is that the S-FKV model was fit to data between 0.1 Hz and ∼10 Hz and so may not capture the response outside this range. We note that the decay of the predicted and measured curves in Figure 8 are very similar and the cause of the discrepancy appears to be associated with the initial values, which will be dependent on the high frequency response of the tissue. A second potential reason is that measurements in the frequency and time domains were done on different tissues samples. The standard deviation in the fitted values for K_α values were 32% in heart, 28.4% in kidney, 22% in liver and 12% in brain which are similar to the differences between the S-FKV model and the relaxation modulus and so biological variability could also contribute to the differences.

The data reported here were all taken at 20 ℃ , however living tissue would be at body temperature, normally 37 ℃ . The effect of this change in temperature was assessed by performing frequency sweep tests on kidney samples immersed in PBS solution at 37 ℃ by adjusting the peltier plate temperature in rheometer. The shear modulus of kidney samples at 1.2 Hz varied less than 10% as the temperature was increased from 20 ℃ to 37 ℃ (see Supplementary Material) and assuming other organs are similar this difference is within 10%.

The fitted S-FKV model was used to predict the shear modulus of organs at frequencies reported in the literature and comparison with literature values are shown in Table 6. It can be seen that in many cases the S-FKV model prediction of shear modulus is consistent with literature values but in some cases there are large discrepancies: up to a factor of 10 is observed in animal organs and up to 27 in human organs. One potential reason is that S-FKV data was measured at frequencies <10 Hz and literature values included measurements up to 1,000 Hz; and the largest discrepancies were typically for data extrapolated beyond 10 Hz. Also, the power law exponent reported here, ∼0.15, is less than reported in the literature 0.3–0.9 (Hoffman et al., 2006; Sack et al., 2009; Grosz et al., 2019; Bonfanti et al., 2020) but again the literature values were obtained in the frequency range of 0.1 Hz–2000 Hz well above the bandwidth measured here. This gives further support to the high-frequency response being responsible for the discrepancies observed in comparison with the relaxation modulus. Other reasons could include variability in the tissue samples and variations in the methods employed.

In this paper, the rheological behaviour of four porcine organs has been studied from 0.1 Hz up to 9.5 Hz for a range of strain amplitudes and compression. Strain sweep results show that 0.1% strain is within the linear viscoelastic region for all organs. Frequency sweep results showed a monotonic increase in both the storage modulus and loss modulus as a function of frequency and axial strain. This behaviour was better captured with a S-FKV model than a SLS or FKV model. The stress relaxation behaviour of different porcine organs was compared with the stress predicted by S-FKV model and the amplitudes were within 30%. The S-FKV model was extrapolated to predict the shear storage modulus at higher frequencies, and the predictions were consistent with values reported in the literature using other non-invasive methods.