Non-identifiability of parameters for a class of shear-thinning rheological models, with implications for haematological fluid dynamics

Abstract

Choosing a suitable model and determining its associated parameters from fitting to experimental data is fundamental for many problems in biomechanics. Models of shear-thinning complex fluids, dating from the work of Bird, Carreau, Cross and Yasuda, have been applied in highly-cited computational studies of heamodynamics for several decades. In this manuscript we revisit these models, first to highlight a degree of uncertainty in the naming conventions in the literature, but more importantly to address the problem of inferring model parameters by fitting rheology experiments. By refitting published data, and also by simulation, we find large, flat regions in likelihood surfaces that yield families of parameter sets which fit the data equally well. Despite having almost indistinguishable fits to experimental data these varying parameter sets can predict very different flow profiles, and as such these parameters cannot be used to draw conclusions about physical properties of the fluids, such as zero-shear viscosity or relaxation time of the fluid, or indeed flow behaviours. We verify that these features are not a consequence of the experimental data sets through simulations; by sampling points from the rheological models and adding a small amount of noise we create a synthetic data set which reveals that the problem of parameter identifiability is intrinsic to these models.

An Iterative Approach for the Parameter Estimation of Shear-Rate and Temperature-Dependent Rheological Models for Polymeric Liquids

“…However, when the characteristic shear rate is high, then the difference in the predicted shear viscosity values from the two sets of the best fitting parameters in the relevant shear rate range is rather small (within 10%); for this reason, they yield similar results in flow simulations, as demonstrated in Figure 11. Our results for the local shear rate, as shown in Figures 10b and 11b, were also obtained from two approaches, one from solving a nonlinear Equation (by combining Equations (10) and (11)) directly (shown as solid lines) and the other from numerical evaluations of the velocity gradient (see Equation (12)) from our numerical solutions of the velocity distribution. The latter results are shown as dashed lines in Figures 10b and 11b.…”

“…As stated by Singh et al (2019) [9], people subjectively choose the fitting approach-thus, leading to nonunique inferences. Moreover, Gallagher et al (2019) [10] reported the non-identifiability in a family of shear-thinning rheological models. They found a class of parameter sets that fit the data evenly well.…”

Influence of Shear-Thinning Blood Rheology on the Laminar-Turbulent Transition over a Backward Facing Step

“…Curve fitting of the common shear-thinning rheology models; Cross, Bird, Carreau and Yasuda relies on determination of parameters. Gallagher et al [57] found that the most common shear-thinning rheology models for blood had wide spectrum of uncertainty when fitting parameters to experimental data and an existing problem of parameter identifiability. This can ultimately affect the flow profiles, particularly important in complex flow fields such as those experienced in VADs and stenotic flows.…”

Improving the Rheological Properties of Water Based Mud with Moringa oleifera Leaves

“…Four different temperatures (30ºC, 49ºC, 50ºC and 70ºC) and four concentrations (1%, 2%, 3% and 4%) were used to check their effect on the rheological properties of the mud. The Bingham model was used to explain the behavior of fluid which calculates the plastic viscosity and yield point as follows: [20].…”