A Bayesian approach to blood rheological uncertainties in aortic hemodynamics

Computational hemodynamics has received increasing attention recently. Patient‐specific simulations require questionable model assumptions, for example, for geometry, boundary conditions, and material parameters. Consequently, the credibility of these simulations is much doubted, and rightly so. Yet, the matter may be addressed by a rigorous uncertainty quantification. In this contribution, we investigated the impact of blood rheological models on wall shear stress uncertainties in aortic hemodynamics obtained in numerical simulations. Based on shear‐rheometric experiments, we compare the non‐Newtonian Carreau model to a simple Newtonian model and a Reynolds number‐equivalent Newtonian model. Bayesian Probability Theory treats uncertainties consistently and allows to include elusive assumptions such as the comparability of flow regimes. We overcome the prohibitively high computational cost for the simulation with a surrogate model, and account for the uncertainties of the surrogate model itself, too. We have two main findings: (1) The Newtonian models mostly underestimate the uncertainties as compared to the non‐Newtonian model. (2) The wall shear stresses of specific persons cannot be distinguished due to largely overlapping uncertainty bands, implying that a more precise determination of person‐specific blood rheological properties is necessary for person‐specific simulations. While we refrain from a general recommendation for one rheological model, we have quantified the error of the uncertainty quantification associated with these modeling choices.

particular interest since they cause half of all deaths worldwide, 1 a large portion of which is attributed to pathological conditions of the aorta, 2 for example, aortic dissection. In aortic dissection, blood pushes through a tear in the inner aortic wall and delaminates the layers, forming a new cavity. The hemodynamics exert a mechanical stress on the aortic wall, and a popular hypothesis is that these wall shear stresses (WSS) are linked to aortic dissection. 3 Particularly, a reduced oscillatory shear index of these WSS has been associated with an increased risk of endothelial cell disease, for example, tissue inflammation. 4 For blood flow modeling, blood is still widely assumed to be a Newtonian fluid also in large arteries, for example, for simulations of aortic dissections, 5 despite the fact that blood undoubtedly is a shear-thinning fluid. The importance of this shear-thinning behavior in large arteries is still disputed. Several studies [6][7][8][9][10] deemed the influence of the shear-thinning behavior on the hemodynamics negligible, justified by the prevailing shear rates being in a regime of nearly constant viscosity. In contrast, other studies stress the importance of shear-thinning in complex geometries, such as the aorta. [11][12][13][14] It has been argued that the shear-thinning behavior is associated to the break-up of red blood cell rouleaux formed in regions of high flow residence time, and a hybrid Newtonian/non-Newtonian rheological model has been proposed for aneurysmal flows accordingly. 15 All of these studies assumed a given set of parameters for their rheological models, but in a clinical scenario, these parameters are not given precisely. There may be a vague notion of the bounds on these parameters, or even a population statistic, yet the rheological parameters remain uncertain. These rheological uncertainties then enter the simulation, and subsequently any simulation result will be uncertain too. In this article, we shall quantify these rheological uncertainties.
The importance of uncertainty quantification (UQ) is accepted in computational fluid dynamics 16 and biomechanics. 17 The UQ problem is particularly challenging for turbulent flows due the presence of multiple spatial and temporal scales 18,19 and the large computational effort. Turbulence has not been considered in this work, as will be justified later. A comprehensive collection of reviews on UQ can be found from the engineering point of view 20 and a statistician's perspective. 21 Here, we will use Bayesian Probability Theory. 22,23 The uncertainties in hemodynamics simulations specifically are not completely understood yet. The main sources of uncertainty may be ascribed to (1) boundary conditions, (2) geometry and (3) material parameters. Scarce literature is available on quantification and/or propagation of (1) boundary condition uncertainties 24,25 and (2) geometry uncertainties. 26,27 Regarding (3) material parameters, 28 compared the uncertainties of different non-Newtonian viscosity models for small Reynolds numbers in an idealized bifurcation model of a vein, however surrogate uncertainties are neglected and ad-hoc assumptions on the parameter statistics made. The authors are aware of only one study 29 that considered all sources (1)(2)(3), however for a Newtonian fluid only.
In this contribution, we discuss a Bayesian perspective on the viscosity uncertainties and how the uncertainties propagate to the aortic WSS via a surrogate model. All in all, this comprises the inverse problem of viscosity parameter estimation from experimental rheological data, another inverse problem of constructing a surrogate model from finite simulation data (i.e., estimation of surrogate parameters), and, based on that, eventually the solution of the forward problem of uncertainty propagation. In contrast to previous work, we derive a coherent Bayesian theory and do not need ad-hoc assumptions. The uncertainty propagation demands a large number of simulations, that is, a prohibitively large computational effort. A common work-around is to run the simulation for a few selected parameter values, and use that data to "learn" a surrogate model. The simulation uncertainties are then inferred from this surrogate model instead of the original simulation model at a significantly reduced computational effort. Popular surrogate models are Polynomial Chaos Expansion [30][31][32] and Gaussian Process Regression, 33,34 the latter of which has had its renaissance recently from within the machine learning community. The reduction in computational effort however involves a trade-off for accuracy. The weak point of this procedure is the credibility of the surrogate model. Our Bayesian approach allows to include the uncertainties of this surrogate model itself, too.
We ultimately compare the uncertainties of the non-Newtonian model with the uncertainties of the Newtonian models, and how they affect the uncertainties of the hemodynamic shear stresses exerted on the aortic wall. We additionally compare personalized rheological models to population-averaged models.
The paper is organized as follows: Section 2 describes the methods and model assumptions regarding blood rheology and fluid dynamics. Section 3 describes the assumptions and statistical theory for the uncertainty quantification. Section 4 presents the results. In Section 5, the paper concludes.

| METHOD AND MODELING: BLOOD RHEOLOGY AND AORTIC HEMODYNAMICS
The following section describes governing equations, rheological model, flow domain, boundary conditions, and numerics. In the present work, we investigate aortic blood flow and the WSS on the aortic wall. The emphasis lies on uncertainties of WSS due to the uncertainty of the rheological model parameters. The flow is treated as laminar, that is, turbulence is not considered. This will be justified in Section 2.5 and supported by Appendix D.

| Fluid dynamics-Governing equations
The fluid is assumed incompressible, thus its density ρ is constant. Then the mass balance of a given control volume reduces to the requirement that the velocity field u must be solenoidal: To formulate the equation of motion for a fluid, the momentum of the same control volume is balanced. Neglecting body forces yields, The term on the right hand side is the divergence of the Cauchy stress tensor T, which is split into two parts as per, where p is the pressure, I the unit tensor, and τ the extra stress tensor. The extra stress in the fluid depends on the instantaneous local flow field and on the rheological material behavior. A constitutive model for the rheological properties needs to be defined in order to determine τ: Thermal energy is not balanced and the flow is assumed isothermal.

| Rheology
We introduce three different rheological models below: (1) the non-Newtonian Carreau model, (2) a Newtonian model that is Reynolds number-equivalent to the non-Newtonian Carreau model, and (3) a simple Newtonian model. We further document the shear-rheometric experiment. The three models' impact on the WSS uncertainty will later be compared.

| Non-Newtonian fluid
The choice of the rheological model is crucial because its behavior determines the extra stress in the flow due to the rates of fluid deformation. Experiments have shown that blood has a shear rate-dependent viscosity, which decreases as the shear rate increases. 7 An inelastic and shear rate-dependent, that is, shear-thinning, viscosity model is therefore needed in our application. The formulation from Bird et al. 35 for a generalized Newtonian model is chosen. The material model for the extra stress tensor τ describing the inelastic shear-thinning behavior is, determined by the rate-of-deformation tensor, given by the symmetrical part of the velocity gradient tensor ru and the dynamic viscosity η as a function of the local shear rate _ γ: In rheology, efforts have been made to find adequate mathematical model equations to represent such a rheological behavior. Roughly divided, three model approaches have been established for the phenomenological description of the viscosity as a function of the shear rate in hemodynamics. Namely, models of the type power-law, 36 Casson 37 and Carreau. 38 It is important that the model is valid for all shear rates between high and low. Abovementioned rheometry shows that blood is a shear-thinning liquid with a first Newtonian plateau at low shear rates. This behavior suggests representation of blood flow behavior by the Carreau model. This model proved capable of capturing of the shear-thinning behavior of blood by adapting four model parameters to the rheometric data. The model is not restricted to a specific range of shear rates and applies to liquids in a wide range of plateau viscosities. Thus, the Carreau model formulation is chosen, which reads, where, η 0 represents the asymptotic zero-shear viscosity (1st Newtonian plateau) and η ∞ equals the limiting value for high shear rates (2nd Newtonian plateau). λ and n determine shape and slope of the nonlinear regime. Parameters and uncertainties are estimated from shear-rheometric experiments described below. The shear rate _ γ is given as in Böhme, 39 where tr denotes the trace.

| Reynolds number-equivalent Newtonian fluid
We aim at a fair comparison of the rheological models through comparable flow regimes achieved by a dimensional analysis. In case of a Newtonian fluid, the viscosity is constant, that is, it is independent of the shear rate. This simplifies Equation (4). In order to characterize a flow, a dimensional analysis enables the identification of a set of nondimensional parameters. Therefore, all quantities characterizing the flow, including the fluid properties, need to be identified. The spatial average of a quantity : ð Þ is denoted by : ð Þ, and f : ð Þ is the temporal average. The given problem is governed by the following quantities: the cardiac output equivalent velocity u, the difference Δu between the peak systolic velocity and minimal diastolic velocity, the angular frequency ω of the pulsation (heart rate), the diameter D AA of the vessel, and the density ρ and a representative dynamic viscosity μ of blood. From these six quantities, three nondimensional numbers can be derived, 40 The Reynolds number Re represents the ratio of convective to diffusive rates of momentum transport. Due to the variable viscosity, the Reynolds number requires a special treatment. We adapted the concept of a generalized Reynolds number in non-Newtonian flows, proposed by Metzner and Reed 41 for a power-law fluid, to the rheological model in Equation (6). The result is, where, with the actual and the apparent wall shear rates, _ γ w ¼ _ γ w app 3m þ 1 ð Þ=4m and _ γ w app ¼ 8 e u=D AA , respectively. The generalized Reynolds number represents the dynamic behavior in internal flow of a non-Newtonian, shear-thinning liquid described by the Carreau model in Equation (6). The derivation of this Reynolds number, which we sketch in Appendix A, is oriented at equal dynamic behavior of the non-Newtonian and the Newtonian liquids with respect to resistance, that is, to pressure drop, in internal flow. Liquids with this generalized Reynolds number exhibit the same flow regimes as their Newtonian counterparts. The generalized Reynolds number is used to determine the dynamic viscosity, which a Newtonian fluid should have to be dynamically equivalent to the non-Newtonian one. That equivalent Newtonian liquid viscosity μ is given as The Reynolds number in Equation (8) represents the general definition, while Equation (9) is the one relevant for the flow of a non-Newtonian liquid with shear-thinning behavior according to the Carreau model. The Reynolds number in Equation (8) uses the representative dynamic viscosity calculated in Equation (11), leading to the value of the Reynolds number given by Equation (9).
The second dimensionless number is the Womersley number Wo. 42 It is a ratio of oscillatory to viscous time scales. Analogously to the Reynolds number, the non-Newtonian viscosity demands a generalized form. The following expression for the Womersley number results: The third dimensionless number is Gosling's pulsatility index (PI). 43

| Simple Newtonian fluid
We use an established model by Merrill 7 for determining the Newtonian blood viscosity as a function of the hematocrit H : The dynamic viscosities of blood and plasma are denoted by η bl and η pl , respectively, and hematocrit H is in units of [%]. For the plasma viscosity, η pl ¼ 1:2 AE 0:1 ð ÞmPas was reported in Reference 44. We chose a hematocrit of H ¼ 40% for all simulations. Note that this simple Newtonian model is not Reynolds number-equivalent, in contrast to the Newtonian model defined previously in Section 2.2.2. We will later compare the non-Newtonian model and the Reynolds number-equivalent Newtonian model in the context of this simple Newtonian model as the standard.

| Rheological experiments
Rheological properties of human blood were measured in simple shear at a temperature of 37 in a shear rheometer with a double cylindrical gap geometry for shear rates ranging from 1 to 1000s À1 (Anton Paar MCR 301 inner gap: 0.417 mm, outer gap: 0.462 mm, radii: 13.769 mm and 12.33 mm). The experimental data originate from five volunteers, each sample was adjusted to the corresponding hematocrit values of 30%, 40%, 50% and 60% by centrifugation.

| Flow domain
We focus on the rheological aspects and use a simplified aortic geometry, that is, a bent pipe with a circular cross section, to avoid uncertainties associated with the geometry, for example, MRT/CT, rendering, interpolation or meshing. Figure 1 shows the geometry. The circular cross section of the pipe has the constant diameter D AA over the entire flow domain. L AA is the ascending length of the aorta and the aortic arc has the radius R Arc with a curvature angle of 180 ending in the descending part of the aorta, which is of the length L DA . The various lengths are listed in Table 1. The carotid arteries and other branches were not taken into account in the modeling. At the points A to J, the simulation results will later be evaluated.

| Boundary conditions
The velocity profile at the inlet is assumed to be of parabolic shape. The velocity distribution over the cross section for a given volumetric flow rate Q t ð Þ reads, The radius R ¼ D AA =2 defines the inlet cross section. The temporal profile of the volumetric flow rate Q t ð Þ (Figure 2), represented by a power series, was adapted from Alastruey et al. 45 The boundary condition for the walls is the no-slip condition for the velocity, and the pressure is set to zero at the outlet. The spatio-temporal velocity profile here is due to the blood flow from the heart, which in the real case is very complicated-and unknown. Conversely, for example, a Womersley profile would be a profile that is formed in a fully developed laminar flow (i.e., in a very long tube or channel) under the influence of pulsating feed of blood. For the flow profile at the entrance only, such a Womersley profile is not more realistic than our parabolic profile. Both variants are model approximations to the flow field. The velocity profile develops under the geometric influence in the flow down the aorta. For that flow development to be studied, we consider both possible entrance flow profiles as equivalent.

| Investigated cases
The uncertainty analysis was carried out for three different cases corresponding to three different physiological states. The physiological state is characterized by the cardiac output e Q (CO), that is, the mean volumetric flow rate over one cardiac cycle, which is defined as the volume of blood pumped by the heart per unit of time. So it is the product of the stroke volume V stroke (SV), and the heart rate f heart (HR, given by ω=2πÞ : The first case corresponds to a person at rest with a relatively low stroke volume and low heart rate at the same time (LoRe_LoWo). The second case (HiRe_LoWo) represents a person at rest, that is, low heart rate, but with twice the stroke volume of the first case. The third case (HiRe_HiWo) corresponds to a person at moderate activity, that is, higher demand of cardiac output and increased stroke volume and heart rate. The average volumetric flow rate equals that of the second case. The ranges of the absolute parameters and nondimensional numbers were set according to data available from literature, [46][47][48] and the hematocrit was 40%. The parameters for the three cases are documented in Table 2. The uncertainties of the rheological models described above are compared for these three physiological states. The comparison of the uncertainties of these three cases allows an insight into how the uncertainties are affected by the nonlinear modeling itself.
The flow is treated as laminar. This can be justified by the fact that the values of the Reynolds number, which characterize our investigated cases, are significantly smaller than the critical value of 2300 known as the threshold value for a transition from laminar to turbulent pipe and channel flow, with overwhelming probability. This will become apparent from the analysis of the rheological data in the light of these investigated cases, see also Figure D1A in Appendix D. There is, therefore, no need for turbulence modeling in the simulations here.

| Numerics
Equations (1) and (2) were solved numerically using the finite volume method of the open source software OpenFOAM. The discretization of the geometry was achieved using Ansys ICEM 2019 R1 with a structured O-grid solely consisting of hexahedral cells for minimal geometric distortion and non-orthogonality. The mesh was gradually refined toward the wall to correctly resolve the gradients near the wall surface. A grid independence study of the velocity profiles was carried out with a steady volumetric flow rate at the inlet. The grid independence of the simulation results was ensured by steady-state simulations using the simpleFOAM solver. The simulations for verifying grid convergence were carried out at the highest value of the flow rate-equivalent velocity of all the flow situations studied, which is about 1.1 m s À1 . This puts the strongest requirements to the grid resolution close to the wall and is therefore significant for ensuring grid independence of the results. A mesh independent solution of the velocity profiles was obtained with $ 290, 000 cells. Accordingly, this grid was used for the simulations. A first order implicit scheme was chosen for time derivatives for which the time step was adaptively adjusted to the velocity field in order to keep the Courant number below a value of 0.9, ensuring numerical stability and decreasing the simulation time. Gradient and divergence schemes were calculated with second order central differences. In addition, gradients were limited with a multidimensional limiter to reduce numerical diffusion. The non-orthogonality of wall-normal gradients was taken into account with surface normal gradient scheme. A solution periodic in time was obtained for all the transient simulations.
The solver pimpleFOAM was used as the numerical solver for the transient simulations. That is, the pimple method of pressure-velocity coupling was used.
Five cycles were simulated to obtain the periodic solution. The results of the 6th cycle are presented here.

| METHOD AND MODELING: BAYESIAN UNCERTAINTY QUANTIFICATION
The goal is to quantify the uncertainties of the simulation results for the shear stress at different points on the inner aortic wall. The WSS are obtained from numerical simulations (finite volume method), based on the (non-) linear viscosity models discussed in the previous section. The latter depends on unknown parameters a, which shall be inferred from the data d exp obtained in rheological experiments. The uncertainty propagation will be made feasible with a surrogate model. The entire problem then constitutes a combination of two inverse problems with a forward problem. The first inverse problem is the viscosity parameter estimation from the experimental data. The forward problem is the propagation of the so found uncertainties through the CFD simulation to the WSS. The second inverse problem is the construction of a surrogate model from finite simulation data in order to eventually solve the forward problem. It is proven mathematically that Bayesian Probability Theory (BPT) is unique and the only consistent calculus for partial truths or rather uncertainty quantification. 49,50 The Bayesian approach to uncertainty quantification developed here is accordingly inspired by O 0 Hagan 51 and Jaynes. 23 For the readers unfamiliar with BPT, we provide a brief introduction in Appendix B. In Section 3.1, we apply BPT to analyze the experimental viscosity data (as described in Section 2.2) to find the probability density functions (pdf) and infer the uncertainty of the Carreau parameters. In Section 3.2, these uncertainties are then propagated through the subsequent computational fluid dynamics simulation of aortic hemodynamics to obtain the uncertainty of the WSS. Analogous discussions concern the Reynolds number-equivalent Newtonian model and the simple Newtonian model.

| Bayesian analysis of rheological experiments
Here, we quantify the uncertainties of the viscosity parameters based on shear-rheometric experiments (Section 2.2.4). In the Carreau model in Equation (6), the viscosity η is a function of the shear rate _ γ and parametrized by η ∞ , η 0 , λ, n f g : For notational convenience, we re-label the parameters to a a 1 ,a 2 , a 3 , a 4 f g : with The measured viscosity is corrupted by additive noise ε, that is, Applying Bayes' theorem to the rheology data set The product of likelihood, p ηja, _ γ ð Þ, and prior, p a ð Þ, is proportional to the so-called posterior. The so-called evi- Þp a ð ÞdV a , is a normalization constant that is independent of a: The evidence therefore does not affect the parameter estimation and forward uncertainty propagation envisaged in this work, that is, the posterior only needs to be known up to this proportionality constant. The other two factors, likelihood and prior will be discussed next.

| The likelihood function
We use experimental data of the blood viscosity η measured for a set of shear rates _ γ ¼ _ γ m f g N m m¼1 on N p volunteers. The experimental viscosities of volunteer n 1, …,N p È É for shear rate m 1, …, N m f gare denoted by η mn and combined in the vector η n for each volunteer and the matrix η ¼ η 1 ,…, η N m À Á for all volunteers. We note that in the actual experiment not all volunteers' samples have been measured at all N m shear rates _ γ m : To avoid notation overload, this is not accounted for in the following notation, but taken into account correctly in the evaluation. We assume a Gaussian measurement error ε as a safe default choice since no further knowledge is given. This defines the likelihood of a single data point η mn as, where σ 2 m is the scale of the experimental uncertainty. Note that σ 2 m includes all forms of uncertainty, for example, instrumental measurement error, physiological variability, and model inadequacy. A particular advantage of the Bayesian approach is that we need not treat those forms of uncertainty distinctly, since the nature of the uncertainty does not change the propositional probability calculus (see Appendix B). Note also that the viscosity η is physically bounded to be nonnegative, which could be ensured by choosing a Gamma-type likelihood. However, we will see in the data that the likelihood is centered far away from the line η 0: Then, a Gamma-type likelihood looks similar to a Gaussian likelihood, and the numerical difference to the analytically more convenient Gaussian likelihood is negligible here. The experimental error may depend on the shear rate _ γ m but not on the person. The data suggest that the standard deviation, σ m , is inversely proportional to the logarithm of the shear rate _ γ m , with scaling factor α. The measured shear rates range from 10 0 to 10 3 s À1 , hence þ1 is added to ensure b σ m is real at low shear rates. AssumptionsEquation (21a -21b) are motivated by naive estimates of the variance from the data, at given shear rates. The variance estimates are shown in Figure 3, which suggests a power-law relationship. In contrast, the assumption of a constant, shear rate-independent variance has led to unreasonable results and diverging uncertainties.
No instrumental properties are known to the authors that explain this behavior. A physiological-physical explanation for the shear rate-dependent variability might be, that fewer red blood cells align in flow direction at lower shear rates, and a minimal shear stress is required for this alignment. As we will see later, there is little data at high shear rates with small uncertainty, and comparatively lots of data at low shear rates with large uncertainty, due to the scale of the considered shear rates. Since the value of α defining the power-law variance model is not known, we will average all estimates over all possible values for α, that is, no particular value for α is assumed. This averaging mitigates the influence of choosing a particular value for α and is analytically tractable for our simple model for the variance. We note that more complex variance models, for example, with more hyperparameters in the denominator of Equation (21b), would significantly increase the computational effort. Assuming that the measurement errors at different shear rates are uncorrelated, the likelihood for a specific person is, where Z subsumes all constant factors independent of the parameters. Since the measurements for different persons are uncorrelated, the likelihood for the population is, F I G U R E 3 Naive estimate of the variance as a function of shear rate obtained from rheometric data. Note the double-log scale. The data suggests a power-law relationship as in Equations (21a)-(21b). Outliers are visible but effectively dealt with via averaging over all possible α's with N ¼ N m N p : From the normalization constant of the likelihood we have only mentioned explicitly the α dependence, as it will become important in the next step. Here α is an unknown hyper parameter, which has to be marginalized over with the appropriate Jeffreys' prior for scale variables, that is, p α ð Þ / 1=α: Then the resulting marginal likelihood is a Student-t distribution, where Z 0 and Z 0 0 are normalization constants that are independent of a: This model is robust to outliers, a property very favorable in the light of the large physiological variability. We note that discarding possible outliers, occurring particularly at low shear rates, here had the consequence that the low shear rate plateau η 0 could not be determined anymore to be within a finite range and the uncertainties diverge correspondingly. Thus an outlier-tolerant model is required also from this point of view.

| The prior
BPT allows to rigorously and consistently incorporate any prior knowledge about the experiment before taking a look at the data. This knowledge shall be elicited here. The viscosity is modeled by a generalized power-law (Equation 6). It is reasonable to assume that our inference must not depend on the exact parametrization. That is, if we re-label the parameters, the results must not change. This rescaling invariance is ensured by Jeffreys' generalized prior and is given by the Riemann metric R (or the determinant of the Fisher information matrix) 22 as follows, With the likelihood defined above, the result is (see Appendix C), where either N ¼ N m (person-specific likelihood) or N ¼ N m N p (population likelihood). The derivatives are available analytically by routine mathematics. This prior is the least-informative prior that adheres to the physical requirement that the viscosity does not depend on the units of the parameters η 0 ,η ∞ , λ, n ð Þ , that is, the units (scales) of the parameters can be set arbitrarily as long as they are consistent. A particular strength of this prior, in contrast to a flat prior, here is that it ensures a normalizeable posterior and results are not sensitive to the choice of any hyperparameters, for example, integration bounds. In addition to the reparametrization-invariance, we also have the prior knowledge that viscosities are positive. That is ensured by a 1 ≥ 0 ,a 2 ≥ 0 ,a 1 þ a 2 ≥ 0: Moreover, the viscosity decreases with increasing shear rate, that is, a 4 > 0: Finally, we can restrict the third parameter to a 3 ≥ 0 as it only enters quadratically. We note that incorporating the scale-invariance greatly increases the authority of the results. Prior information is important in the case of weak data information. This is the case here as far as the individual parameters a 2 and a 3 are concerned. The prior is further illustrated and argued for in Figure C1 in Appendix C.

| Parameter and viscosity estimation and associated uncertainties
With Bayes' theorem given in Equation (19) we can now compute the joint posterior probability density function (pdf) for all Carreau parameters, p ajd exp À Á : The marginal posterior pdf for one specific parameter is given by the integral with respect to the remaining parameters, for example, The probabilities for the other parameters are computed analogously. Similarly, joint pdf-s for any combination of parameters, such as p a 1 , a 2 jd exp À Á , are obtained by the volume integral with respect to the remaining parameters a 3 and a 4 (see Equation B1 in Appendix B).
Another interesting topic is regression, that is, we want to determine the pdf for the viscosity η _ γ for a specific shear rate _ γ, based on the measured rheological data, that is, p ηj_ γ, d exp À Á : The background information includes the fact that we use the Carreau model for the viscosity. The estimated viscosity η _ γ for a shear rate _ γ of interest is then given by the marginalization rule (see Equation B1 in Appendix B) Obviously, insignificant conditionals have been striked out. The second pdf has already been determined. The first pdf is simply a Dirac delta because the background information tells us to use the Carreau model f _ γja ð Þ (see Equation 18) to compute the viscosity and the required parameters are also given. Hence, Consequently, mean and second moment of η _ γ are simply, From that we obtain the uncertainty, It should be emphasized that the Bayesian approach has two important advantages over ad-hoc approaches. Firstly, common practise would be to determine the maximum likelihood estimate for the Carreau parameters a ML ð Þ based on the data d exp , which in the Gaussian case is equivalent to minimizing the misfit. This parameter vector a ML ð Þ is then used to determine the estimate for the viscosity by, This is not the same as the consistent use of the data and all available information, given by the Bayesian result. The latter yields only the same answer if the posterior is dominated by the likelihood and the prior is negligible (in the sense that the prior is comparatively nearly constant in regions where the likelihood has pronounced structure), and if the likelihood is sharply peak at its maximum. The first condition is met if we have very many data points. The second condition is rarely fulfilled if the data is sufficiently informative. Particularly in the present case, neither of the two conditions hold, as we will see in the discussion of the results. This finding is supported by Figure C1 in Appendix C, where the prior is shown and compared to the likelihood. Another very important advantage of the Bayesian approach is that it allows to determine the uncertainty of any estimate on the same probabilistic footing.

| Probability for Reynolds numbers
The posterior p ajd exp À Á also allows to infer the pdf for the Reynolds number Re gen and the equivalent Newtonian viscosity μ as, where we have used the marginalization rule (Equation B1) and the background information that Re gen a ð Þ and μ Re gen À Á are defined by Equations (9) and (11). The distribution of Womersley numbers can be computed analogously. The integrals can be evaluated numerically.

| Bayesian uncertainty propagation through the CFD simulation
The uncertainties of the blood rheological parameters have been calculated above, and are now propagated to the simulated WSS in the aorta. This is done for the three different rheological models defined in Section 2.2 (non-Newtonian Carreau model, Reynols-number equivalent Newtonian model, and simple Newtonian model).

| Non-Newtonian fluid
The uncertainty of the Carreau parameters a entails an uncertainty in the WSS. Let τ ¼ τ x ,τ y , τ z À Á T be the WSS vector.
Then let w ¼ τ be the absolute value of the WSS vector, or short absolute WSS, at location x and time instance t, denoted by w x,t ð Þ : The mean then is, The uncertainty of this estimate is given by the diagonals of, with, The numerical evaluation of the integral over the four Carreau parameters would imply the need for performing CFD simulations at least 10 5 times, which is way too CPU expensive, as a single CFD simulations requires $ 18 h on 8 CPUs in parallel. This can be avoided if the CFD simulations are replaced by a surrogate model that approximates the WSS w by a suitably parametrized surrogate function w sur ¼ g ajc ð Þ, where c are yet unknown parameters. Clearly, the parameters of the surrogate model will also depend on those positions and the time instance. So we actually have, where we have expanded the surrogate function in basis functions ϕ ν : In our work, we chose multivariate Legendre polynomials as basis without loss of generality. The unknown parameters c x,t ð Þ ν will be inferred from a suitable training data set. To this end, CFD simulations are performed for a moderately sized set of Carreau parameters A s ¼ a f g: Now we use the background information that we can take the WSS w entering the integral in Equation (40) from the surrogate model rather than from the expensive CFD simulation. Under these assumptions, we can use the formulas derived in Ranftl and von der Linden. 52 We emphasize that the underlying theory addresses the weak point of this approach by explicitly incorporating the surrogate uncertainties. The prior for the absolute value of the WSS was zero for negative values.
Note that the surrogate model in Equation (43) is very similar to a nonintrusive Polynomial Chaos Expansion, 30-32 however the basis functions ϕ ν chosen here do not have the defining orthonormality property. 52 Finding such an orthonormal basis is simple for standard distributions and conditionally independent parameters a, but elaborate for complex posteriors (see Section 3.1) as in this work. Although there are sophisticated schemes for constructing such an orthonormal basis also for more complex input pdfs, 53-55 the simulation's uncertainty must be independent of any orthonormality property of the surrogate's basis and we are free to choose a simpler basis, provided that the chosen basis can represent a sufficiently trustworthy surrogate. 52 In place of a truncation scheme for the expansion in Equation (43), the choice of N b was achieved via a Bayesian surrogate model comparison through the surrogates' evidences as described in detail by Ranftl and von der Linden. 52 In this procedure, the probability for each considered value of N b (or more generally for each considered set of basis functions) is computed, that is, the evidence for the surrogate model given the training data D sim : Then, the most probable value for N b (or most probable set of basis functions) is chosen to construct the surrogate.
An additional difference to traditional Polynomial Chaos Expansions is, that the Bayesian approach 52 here incorporates the uncertainty of the surrogate itself. This is achieved by recognizing that the expansion coefficients c x,t ð Þ ν in Equation (43) implicitly appear as unknown regression coefficients in the simulation uncertainty denoted by Equations (40)-(42) upon substitution of Equation (43) into Equation (40)- (42). One then needs to find the posterior pdf for the c's given the simulation samples, and average Equations (40)-(42) over the c's accordingly. The exact procedure was detailed by Ranftl and von der Linden. 52

| Reynolds number-equivalent Newtonian fluid
Analogous to the procedure leading to Equation (39) and departing from the previous subsection, we find for the average and uncertainties of the WSS of the Reynolds number-equivalent Newtonian model, The WSS w x,t ð Þ is to be taken from the corresponding simulations with the Reynolds number-equivalent Newtonian fluid. The uncertainty of this estimate is then given analogously by Equation (41). No surrogate model is needed here since the one-dimensional integrals can be computed directly.

| Simple Newtonian fluid
The literature references in Section 2.2.3 assume that the pdf of the Newtonian viscosity is a Gaussian with mean and standard deviation documented therein. The same formulas as in Section 3.2.2 apply, where p μjd exp À Á is substituted with said Gaussian from the literature, and the WSS w x,t ð Þ is to be taken from the corresponding simulations resulting with the simple Newtonian fluid.

| RESULTS
All numerical integrals for the uncertainty quantification were computed with Riemannian quadrature and convergence checked with successive grid refinement. Data and code are available in in a digital repository. 56

| Rheology
Here, we compare rheological results. We first compare the posterior probability density functions for the viscosity parameters, and its consequences for the viscosity (both non-Newtonian and Reynolds number-equivalent Newtonian) itself, for specific persons and the population average. We then compare the population-averaged posterior probability density functions for the viscosity parameters, and its consequences for the non-Newtonian viscosity, for different hematocrits. Figure 4A shows the posterior pdfs for the four Carreau parameters for two distinct volunteers and the population averages. Volunteer 1 shows two distinguished peaks in a 1 ≔ η ∞ , and all pdfs are skewed and asymmetric. Apparently, the statistics cannot be sufficiently described with mean, maximum likelihood or variance alone. This argument is supported by the deformed banana shapes of the joint pdfs in Figure 4B-D. The unusual shape of these pdfs poses no obstacle to our Bayesian uncertainty quantification. The inference is rather vague on the difference between the two plateaus of zero-shear limit viscosity to infiniteshear limit viscosity, a 2 ≔ η 0 À η ∞ , and the slope, a 3 ≔ λ, while the posterior for the infinite-shear limit viscosity alone, a 1 ≔ η ∞ , covers a smaller range. In conjunction with the shear rate-dependent variance of the viscosity (Equations 21a-21b), this also causes larger uncertainty bands for the viscosity ( Figure 5A) at low shear rates as compared to high shear rates. The tilt and "narrow" width of the deformed "ellipsis" of the panels "a 2 versus a 3 " in Figure 4B-D further mean, that the zero-shear plateau (offset) a 2 and the slope a 3 are strongly correlated. This in turn means, that better information on the zero-shear plateau, a 2 / η 0 , would consequently allow to improve the inference on the slope, a 3 ≔ λ, too, and vice versa. An analog statement can be made for the pair a 2 versus a 4 : The size of the ellipses however suggest that more or better data on either the slope or the zero-shear plateau (e.g., measurements at lower shear rates) are most promising for better estimates for the viscosity.

| Population-averaged versus person-specific probabilities
The population-averaged ( Figure 4B) and person-specific ( Figure 4C,D) pdfs live in the same region of parameter space and, albeit translated slightly, overlap significantly. This behavior translates further to the viscosities, Figure 5A. The estimated viscosities can be, naively, distinguished, however the uncertainty bands overlap considerably. The pdfs for the Reynolds number-equivalent Newtonian viscosity, Figure 5B, have a heavy tail, and are rather Student-t distributions than Gaussians. The person-specific results overlap considerably with population averages. Notably, the distribution for volunteer 3 in physiological states 2-3 overlaps considerably with the population average in physiological state one. This exemplifies the inadequacy of simple Newtonian models to account for state-dependent viscosity in the light of uncertainty, which is properly addressed in our approach via Reynolds number-equivalence. Figure 6A shows the pdf for the Carreau parameters for different hematocrit values at otherwise same conditions, and Figure 6B the corresponding viscosity models. The viscosities are different at low shear rates including the nonlinear regime, however uncertainty bands partially overlap at high shear rates. Each hematocrit's viscosity model can be easily distinguished as a whole though. Consequently, uncertainty bands derived from simulations based on these four viscosity models will not overlap, at least not due to the viscosity model. This implies that the hematocrit needs to be known within a small margin for person-specific simulations, albeit the quantification of this margin is not the matter of this work.

| Hemodynamics
We restrict our investigation to the case of hematocrit 40%. For the non-Newtonian WSS, we used N s ¼ 100 training points in parameter space, yielding a set of 100 simulations. An illustrative example of these primary results is  52 This data is then used to learn the surrogate models. It turned out that a total polynomial order of 2 is sufficient, corresponding to 15 basis functions in Equation (43) (1 zeroth order, 4 first order, 10 second order, that is, 10 distinct pairs a i , a j Þ: The amount of training data is thus roughly 3 times the amount typically recommended for the similar Polynomial Chaos Expansion. Uncertainties are then computed with a Riemannian integration scheme on an equi-spaced 50 Â 50 Â 50 Â 50 grid. No surrogate models were required for the Newtonian WSS, that is, the uncertainties were determined directly from Equation (40) with a Riemannian integration scheme with 40 equally spaced pivot points. The design of the computer experiments yielded an acceptable surrogate uncertainty, comprising <1%->50% in extreme cases of the total uncertainty, depending on location and time instance. Note that the surrogate's uncertainty is assessed according to Ranftl and von der Linden, 52 and that the surrogate's uncertainty is directly included in and propagated to the uncertainty of the WSS. This also means, that the corresponding standard paradigm of uncertainty quantification, for example, with Polynomial Chaos Expansions or Kriging, would neglect these surrogate uncertainties, and therefore naively underestimate the uncertainties precisely by the surrogate's contribution. Since all pdfs live in the same region of parameter space, the same surrogate can be used for all volunteers.
The results imply that a more precise determination of blood rheological properties is necessary for sensible personspecific simulations, for example, by repeated or more precise measurements. A measurement of the viscosity at lower shear rates < 10 0 s À1 ð Þwould be particularly promising, as this would allow to determine the low shear rate plateau η 0 , if it exists. If that is not possible, one might as well use population averages. Indeed, the measurement of blood viscosity (see Section 2.2.4) at shear rates lower than < 10 0 s À1 is very difficult. The reason is, that the red blood cells begin sedimentation at such low shear rates before a measurement can be completed, which measurement in turn takes longer at lower shear rates.  Table 1 and Equation (38). Color code as in left panel. Solid (dashed) line: Physiological State 1 (2 & 3) Further, the Newtonian models typically underestimate the uncertainty as compared to the non-Newtonian model. Although the authors are not aware of directly comparable studies with located probes, the resulting values generally agree with the ranges in the literature. 57,58 F I G U R E 6 (A) Population-averaged marginal probability densities (Equation 31) for the four model parameters a 1 , a 2 , a 3 , a 4 of the dynamic Carreau viscosity (Equation 6). The colors black/red/blue/green belong to hematocritcs of 30%/40%/50%/60%. All pdfs are similar in shape, however shifted in a 1 and a 2 . (B) shows viscosity over shear rate. The colors black/red/blue/green belong to hematocritcs of 30%/40%/50%/60%. + is the data, the continuous line is the estimate of the Carreau viscosity (Equation 6) and the shaded areas are corresponding uncertainties F I G U R E 7 Primary results. t is time and T is the heart cycle duration. Top (Bottom) shows 10 (100) examples of simulation results at measurement point C (J) (see Figure 1). Each set of connected dots (data points) corresponds to one simulation with one parameter set (see also Appendix E) 4.2.1 | Population-averaged versus person-specific & moderate activity versus at rest In Figure 8B we compare the WSS of the non-Newtonian model for two volunteers and the population average based on the posterior pdfs shown in Figure 4. Results and uncertainties overlap mostly. This raises two questions: (1) whether personalized viscosity measurements on this level of accuracy are necessary as opposed to population-averaged rheological parameters, and (2) whether personalized surrogate models (i.e., a surrogate as a function of rheological parameters) are really necessary, as opposed to population average-based surrogate models. The results for population averages may be deemed "good enough" here, although this depends on the application. More importantly, the answer could be quite different if viscosity measurements were more precise or available in a larger range of shear rates. The overlap in Figure 8B was found to hold throughout and was expected, based on the similarity of Figure 4B-D. We proceed to compare different rheological models with population averages only.

| Newtonian versus non-Newtonian & moderate activity versus at rest
We compare the population-averaged ( Figure 4B) uncertainties of Newtonian and non-Newtonian models at different measurement locations (Figure 1) for the three different physiological states ( Table 2). We select four examples for demonstration in Figure 9.
The WSS (not its uncertainty) of the simple Newtonian model mostly lies between non-Newtonian model and Reynolds number-equivalent Newtonian model, however this model does not capture flow regime or shear-thinning. For a large part, both Newtonian models underestimate the uncertainty as compared to the non-Newtonian model uncertainty. However, the non-Newtonian uncertainties are not necessarily larger than the Newtonian model uncertainties. At specific time-instances, they can occasionally also be in between the two Newtonian models, see for example,  Figure 9B, bottom panel at ca 0:7 t=T: At specific locations, they can also be smaller than both Newtonian models, see for example, row B of case 2 in Table E3 in Appendix F. The numbers demonstrate that both is possible, that a Newtonian model underestimates or overestimates the specific index or its uncertainty as compared to the non-Newtonian model. Whether the differences are "biologically relevant," for example, in terms of the response of the endothelial cells or arteriosclerosis, remains questionable.
Tables on time-averaged quantities that are sometimes used as "biologically relevant indices," such as timeaveraged WSS, oscillatory shear index and relative residence time, are provided for the population-averaged posterior ( Figure 4B) in Appendix F. While the average residence times of Newtonian and non-Newtonian models are arguably often close, it is evident that the uncertainty of residence times can be very different by up to a factor of two. This finding could have biological implications, for example, for the modeling of thrombus growth. 59 The uncertainty of the residence times could also have important implications particularly for the hybrid Newtonian/non-Newtonian rheological model put forth by Arzani et al., 15 in which the shear-thinning behavior of the blood is only activated in regions where the residence time is above a specified threshold. The emphasis of that study lies on the importance of residence times in comparison with the time scales of rouleaux formation. Consequently, the model by Arzani was applied to aneurysm flows, where blood residence times are indeed important for the resulting blood flow behavior. Conversely, the generic geometry in the flow of our study, in contrast, does not exhibit zones where residence times could be pronounced, so that this aspect, and the related influence of rouleaux formation on the blood rheology are of no relevance here.

| Further discussion and limitations
The adaption of this approach to other rheological models than the Carreau model (Equation 6) is in principle straightforward. If the measurement error is assumed to be Gaussian, then the likelihood is also Gaussian. If the noise level of the Gaussian is not known, then the likelihood would be a Student-t. The resulting prior will have the form of Equation (30), and depend only on the rheological model's derivatives. Other measurement errors, for example, of the form of a Gamma-or Beta-distribution or its variants, are usually not encountered in rheological experiments, and their corresponding Riemann prior might not necessarily be well defined or convenient.
In the case of the Carreau model, the Riemann prior turned out to be improper, that is, non-normalizable. However, a weakly informative likelihood ensured a proper posterior. Note that a flat prior, or a prior of the form / 1=λ (both are improper), here would have yielded an improper (non-normalizeable) posterior as the likelihood alone was not informative enough. This would thus refuse meaningful parameter estimates and viscosity estimates (due to divergent integrals) without a strong prior. In other words, the authors were not aware of other a priori knowledge, that would justify to cut off heavy tails of the likelihood. This also means, that the results, both for viscosity and also the WSS, would depend strongly on the choice for the hyperparameters (boundaries) of such a strong prior. The importance of the prior thus strongly depends on the information in the data. In our case, particularly information on the value of the zeroshear limit viscosity, η 0 , and the slope, λ, was not unambiguous. Although based on first principles, the choice for our Riemann prior is best argued for by sensible results, that is, reasonable and finite uncertainty estimates, which could not be obtained by naive priors.
The matter could also be addressed by measurements at low shear rates < 10 0 s À1 , this however is very difficult due to red blood cell sedimentation, and additionally limited by the yield stress. 60 In fact, our model does not consider (visco-)elastic properties below the yield stress at all. The existence of the plateau in the extrapolation in Figure 6B to lower shear rates is thus questionable, as it is only a result of the underlying assumptions of the Carreau model but not backed by experimental data. It is further unknown how in vitro conditions in the rheometer differ from in vivo conditions.
A further problem exists in the definition of blood. While the macroscopic view tempts to accept blood as a liquid, the microscopic view identifies blood as a suspension. Suspension properties affect flow properties in small conduits, 61 for example, through red blood cell arrangement, 62 and may be effective also when a blood portion flows through, for example, a dissected aortic wall. Flow skewness, unsteadiness or possibly turbulence disturb the homogeneity of blood further, [63][64][65] which makes blood a very dynamic material.
The simulations here are idealized and do not capture turbulence, fluid-structure interactions, or arterial branches or other complex features of physiological geometries. Shear-thinning effects are possibly more important in more complex simulations.
A different geometry of the flow domain, including the twisting of the aorta or an out-of-plane curvature, would lead to a more complex flow field with different vortical structures and zones of separated flow with recirculation areas. These flow structures, in particular with separated flow, are characterized by shear rates not larger than the ones encountered in the present flow field geometry, so that the more complex flow field, although it is closer to the realistic case, would not substantially affect the conclusions to be drawn. What matters is the generic behavior of the non-Newtonian fluid in the shear-dominated flow field.
The comparison of the WSS of different persons (Figure 8) depends, here, on the assumption that the probability density function for the viscosity parameters for different persons is located in roughly the same region of parameter space. This allows to use one surrogate model for all. For person-specific viscosity data that is far outside the uncertainty bands in Figure 5A, this assumption would not hold, and one would need to construct person-specific surrogate models. This would also affect the estimate for the Reynolds-equivalent Newtonian viscosity, and subsequently the WSS. The assumption was reasonable for the available data, however interpersonal variability of viscosity might in truth be much larger than what was found here. That would require either personalized surrogate models, or the construction of surrogate models in larger regions of the parameter space. We used roughly three times the amount of simulation data for surrogate construction than typically recommended for Polynomial Chaos Expansions, however the surrogate's uncertainty could in principle be reduced with even more data. Standard paradigms in uncertainty quantification usually do require some measure of "trustworthiness" for the surrogate, but usually neglect this uncertainty in the quantity of interest entirely. Unaffected by that, WSS uncertainty here is mostly rooted in rheological uncertainty.
Lastly, a clear, general recommendation for a Newtonian or a non-Newtonian model is difficult also from this perspective. We have shown that the uncertainties in the WSS can be significantly underestimated by Newtonian model and have quantified the associated trade-off for a given amount of simulation data of WSS. However, the required accuracy of this uncertainty quantification strongly depends on the specific use case, the quantity of interest, etc. The interplay of rheological uncertainties with other sources of uncertainties, for example, boundary conditions or geometry reconstructions, have not been considered in this study.

| CONCLUSIONS
We have derived a consistent and rigorous method to estimate (non-)linear rheological parameters of blood and associated uncertainties in aortic hemodynamics based on BPT. We achieve robust estimates despite the large interpersonal variability via consistent inclusion of prior knowledge such as rescaling invariance. The uncertainty quantification of the non-Newtonian model was made practically feasible via a surrogate model and surrogate uncertainties were incorporated. We consider three different comparisons: (1) Newtonian versus non-Newtonian rheology, (2) person-specific versus population averages and (3) moderate activity versus at rest, reflected by different parameters characterizing the inlet conditions.
Our quantities of interest were the WSS since these are believed to be linked to pathologies such as aortic dissection. However, the statistical theory can be applied without modification to, for example, pressure or flow field. Several representative measurement points were selected over a simplified aortic geometry.
The uncertainties of the non-Newtonian model were then compared to the uncertainties of two Newtonian models, a simple Newtonian model and a Reynolds number-equivalent Newtonian model. The estimation for the Reynoldsequivalent Newtonian model has been subjected to yield a flow regime comparable to the corresponding non-Newtonian flow, that is, equal Reynolds numbers and Womersley numbers at the aortic root. This assumption ensured a fair and sound comparison on the same footing. We note that a rejection of this assumption would either result in unreasonably high uncertainties for the Newtonian viscosity, demand an arbitrary preselection of data points or parameters, or require to use entirely different data in the first place (as was done for the simple Newtonian model, Section 2.2.3).
The results show that uncertainties can be considerably underestimated by the Newtonian models as compared to the non-Newtonian model. However, also the opposite can occur, that is, the Newtonian models overestimate the uncertainty. This seems counterintuitive at first since the non-Newtonian model has four uncertain parameters while the Newtonian models only have one uncertain parameter. However, the number of uncertain parameters is not necessarily an indicator for the propagated uncertainty in the quantity of interest. The probability distribution and range of the parameters might be more important. Since the probability density functions for the Newtonian and the non-Newtonian viscosity here correspond to each other based on the same experimental data and the same assumptions, the decisive factor here is how the models depend on the uncertain parameters. Note that we do not claim that one model is less uncertain than another in general. We merely quantify the error associated to certain model assumptions in the uncertainty quantification of the simulation. This is not necessarily the error of the simulation itself.
The effect is attributed to the larger uncertainties of the non-Newtonian viscosity at low shear rates, as compared to uncertainties at high shear rates or at the representative shear rate defining the Newtonian models. The larger uncertainties at low shear rates are attributed either to higher interpersonal variability or less informative data in that regime. This too implies that the Newtonian models underestimate the uncertainty. The issue may be settled by more precise determination of viscosity parameters by, for example, repeated or more precise measurements. Most promising are measurements at lower shear rates., this however faces experimental and physiological difficulties.
Our first conclusion is, based on the comparisons in Section 4.2.1, that a non-Newtonian model should be considered because the Newtonian models can significantly underestimate the uncertainties. However this is not the general rule, as in some cases it can be the other way around. Our second conclusion, based on the comparisons in Section 4.2.2 and independent of the first conclusion, is that either using population-averaged parameters for the viscosity is sufficient for the non-Newtonian model, or more precise or repeated person-specific measurements of blood viscosity, particularly at lower shear rates, are necessary if meaningful person-specific simulations are desired. Based on the overlap of the pdf's in Figure 5B, it is expected that an analogous behavior might be found for the Newtonian model. The corresponding verifying experiment has not been conducted due to the fact that such a finding would be based on wrongly estimated uncertainties.
What remains open is the interaction of the rheological uncertainties with inlet conditions' uncertainties, geometric uncertainties, and the dynamic motion of the aortic wall. These uncertainties are not necessarily additive, as was found by Reference 29 for a Newtonian viscosity, and can be understood on the same grounds as why a viscosity model with four uncertain parameters is not necessarily more uncertain than a viscosity model with one uncertain parameter. We hypothesize that the found discrepancy in the uncertainties might be more pronounced in more complex, realistic geometries since low shear rates will become more frequent. This further suggests to pay more attention to the elastic behavior below the yield stress.

ACKNOWLEDGMENTS
This work was funded by Graz University of Technology (TUG) through the LEAD Project "Mechanics, Modeling, and Simulation of Aortic Dissection" (https://www.tugraz.at/projekte/biomechaorta/home/), and supported by GCCE: Graz Center of Computational Engineering and HPC resources of TUG ZID. Open Access supported by TUG Library.

CONFLICT OF INTEREST
The authors declare that they have no conflicts of interest.

AUTHOR CONTRIBUTIONS Sascha Ranftl and Thomas
Stephan Müller conceived and designed the study. Ursula Windberger designed and performed the shear-rheometric experiments. Thomas Stephan Müller designed and performed the CFD simulations with support by Günter Brenn. Sascha Ranftl designed, derived the theory and implemented the computer code for the statistical analysis with support by Wolfgang von der Linden. Sascha Ranftl, Thomas Stephan Müller, Günter Brenn, and Wolfgang von der Linden participated in data analysis. Sascha Ranftl and Thomas Stephan Müller wrote the manuscript with editing by Günter Brenn, Wolfgang von der Linden and Ursula Windberger. All authors contributed to the discussions.

ETHICS STATEMENT
We followed a protocol approved by the local ethics committee, "Ethikkommission der Medizinischen Universität Wien," under ethical clearance number 2114/2019. The volunteers provided informed and written consent.

A P P END I X B: A BRIEF INTRODUCTION TO BPT
Probability is a measure of the truthfulness of a proposition, a statement which can either be true or false. BPT is thus a generalization of Boolean algebra to partial truths. It is a consistent and rigorous framework to treat uncertainties of any kind, be it due to noisy experimental data, missing information, unknown parameters or uncertain models. Like in Boolean algebra, propositions can be combined by logical AND, which will here be denoted by a comma ",." Let b and d be two such propositions, where b could, for example, be the proposition specifying a parameter and d the proposition specifying the values of a data set. We distinguish the probability p b ð Þ that b is true without further information, and the conditional probability p bjd ð Þ that b is true given further information denoted by d, for example, measurement data. Strictly speaking there are no un-conditional probabilities. Probabilities always depend on the so-called background information, ℐ, that specifies the problem, the meaning of the parameters and all additional information. For example, the proposition ℐ contains, in the context of this manuscript, the information that we use the Carreau model for the shear rate dependent nonlinear viscosity. It could also contain the information that the WSS z is computed by CFD simulations. Along with the necessary Carreau parameters a the quantity p zja, ℐ ð Þ is the probability density (pdf) for obtaining the result z in a CFD simulation, based on the parameters a: In this case, the distribution is extremely narrow, as fluctuations are merely due to numerical errors. Another example would be that e ℐ encodes the information that we use a surrogate model to compute the WSS corresponding to Carreau parameters a: The corresponding pdf is p zja, e ℐ : We see the background information is crucial. Nevertheless, we will omit it where it is easily possible, in order to keep the formulas transparent.
From the sum rule and the product rule, 22 we find the marginalization rule for continuous variables, and Bayes' theorem, which allows to solve inverse problems consistently. The meaning of the four terms in Bayes' theorem are: p b ð Þ is the prior probability for a before taking the experiment data into account. p djb ð Þ is the so-called likelihood, which is the probability for measuring d given the underlying true model parameters b. The likelihood is usually known as it quantifies the uncertainties of the experiment. Strictly speaking, an experiment is meaningless, if the inaccuracies cannot be quantified. We are interested in the posterior, p ajd ð Þ, that is, the probability that b is the underlying model parameter in the light of the measured data d. The last term in Equation (40), the so-called data evidence p d ð Þ, is independent of b, and not of concern for the inverse problems of parameter estimation and forward uncertainty propagation discussed in this manuscript. The v-th moment of b is defined by, The first moment is the mean value of b and is commonly used as an estimate for the unknown parameter b. The uncertainty of this estimate is given by the standard deviation, where, and h χ 2 ð Þ h ifor an arbitrary function h is independent of the parameters a. Both results are obtained by the substitution d n À f _ γ n ja ð Þ!t n in the integrals. Therefore, Again, based on the previous substitution, the average in the second line is independent of the parameters. Hence, Now we insert the original indices, resulting in, The remaining derivatives can easily be determined by routine mathematics. The prior is illustrated and compared to the likelihood (as defined in Section 3.1.1) in Figure C1. In the marginal pdfs in Figure C1A, in the panels for a 1 and a 4 the marginal prior is arguably flat as compared to the likelihood, but the marginal prior is clearly not nearly constant in the region of the likelihood in the panels for a 2 and a 4 : Therefore, the prior p a ð Þ is not approximately flat near the likelihood mass. This is further illustrated by comparison of the contour plots of the prior in Figure C1B to the contour plots of the likelihood in Figure C1C. Regarding parameter identifiability, 66 the contours of the likelihood in Figure C1C illustrate that some parameters suffer from practical nonidentifiability with a flat (i.e., non-identifying) prior, or conversely why the results based on such a flat prior would strongly depend on the hyperparameters (i.e., the bounds) specified for the flat prior, as has been mentioned in Section 3.1.2 and 4.3. This would also cause diverging uncertainties for the WSS. The non-identifiability in the likelihood is obviously rooted in the data. Accordingly, these findings could be different for different data sets. As suggested in Section 4.3, the non-identifiabiltiy could possibly be resolved by viscosity measurements at lower shear rates < 10 0 s À1 , but this is experimentally very difficult.
We note that the resulting Riemann prior is an improper prior, that is, non-normalizable. The prior does imply structural non-identifiability a priori in parameters a 1 and a 3 , but the likelihood suggests no such structural nonidentifiability. The inference however must be based on the posterior / likelihood Â prior: It is clear from the likelihood contours that parameters remain non-identifiable when choosing a naive constant prior. The choice of the here argued for Riemann prior is based on first principles, but best supported by the fact that the corresponding posterior in Figure 4 is normalizable, does not suffer from structural or practical parameter non-identifiability, and leads to sensible results, that is, finite uncertainty estimates for the WSS.

A P P END I X D: DISTRIBUTION OF REYNOLDS AND WOMERSLEY NUMBERS
The resulting distribution of Reynolds numbers and Womersley numbers as an intermediate result prior Figure 5B are shown in Figure D1A,B, both skewed and asymmetric with a heavier tail toward higher numbers. Particularly the Reynolds numbers vary over a large range, however the physiological states moderate activity/at rest (Low Reynolds number/High Reynolds number) clearly reside in their respective regimes.

A P P END I X E: DESIGN OF COMPUTER EXPERIMENT
From a regular grid of 50 Â 50 Â 50 Â 50, the 40,000 most probable population-averaged parameter combinations were preselected with the aim to "zoom in" on the posterior, or in other words; to avoid simulating parameter combinations that carry a negligible probability weight. However this may introduce a bias, and 40,000 simulations would take roughly 80 cpu-years. To tackle both issues, this preselection was then thinned out by randomly selecting 100 out of those 40,000, mitigating the bias, covering a larger volume and the outer regions of the posterior while avoiding utterly implausible parameter combinations and retaining computational feasibility. An illustration of this sample set is shown F I G U R E D 1 (A) Population-averaged posterior probability distribution for the generalized Reynolds number according to Table 1 and Equation (38). Black (Red): Physiological State 1 (2 & 3). (B) Population-averaged posterior probability distribution for the Womersley number according to Table 1 and Equation (38). Black/Red/Blue: Physiological State 1/2/3 F I G U R E E 1 (A) Family of viscosity models for the computer experiments. Each line corresponds to one cross (=parameter sample) in Figure E1B. The circles are measurement data from the shear-rheometric experiments. (B) Position of the samples for the computer experiments in parameter space relative to the population-averaged joint posterior. The semi-randomized selection procedure is described in the text. The samples roughly span roughly over the posterior volume Figure E1A,B. Remark: The spreading / mean distance discrepancy of the preselection samples can be controlled with the grid size and the number (here 40,000) of samples included in the "lottery." The number of samples drawn from the lottery usually is given by the computational budget. This procedure is heuristic but educated, and in contrast to classic latin hyper cube sampling allows to consider the posterior and improve the surrogate where it is important. An alternative procedure would be to draw samples directly from the posterior via Markov Chain Monte Carlo, where one sample is taken each after a specific Markov time has passed.
A P P END I X F : TIME-AVERAGED SUMMARIES Let τ ¼ τ x , τ y , τ z À Á T be the WSS vector. Then the time-average of the absolute value of the WSS (TAWSS), kτk, is defined as, kτkdt : Note that kτk ¼ w as used in Section 3. The mean is, where, p ajd exp À Á is the population-averaged posterior of Figure 4B. The oscillatory shear index (OSI) is defined as, The mean then is,

RE-Newton
Simple Newton Carreau

RE-Newton
Simple Newton Carreau

RE-Newton
Simple Newton  Table 2.