Mining data from hemodynamic simulations via Bayesian emulation
- Vijaya B Kolachalama^{1}Email author,
- Neil W Bressloff^{2} and
- Prasanth B Nair^{2}
https://doi.org/10.1186/1475-925X-6-47
© Kolachalama et al; licensee BioMed Central Ltd. 2007
Received: 27 June 2007
Accepted: 13 December 2007
Published: 13 December 2007
Abstract
Background:
Arterial geometry variability is inevitable both within and across individuals. To ensure realistic prediction of cardiovascular flows, there is a need for efficient numerical methods that can systematically account for geometric uncertainty.
Methods and results:
A statistical framework based on Bayesian Gaussian process modeling was proposed for mining data generated from computer simulations. The proposed approach was applied to analyze the influence of geometric parameters on hemodynamics in the human carotid artery bifurcation. A parametric model in conjunction with a design of computer experiments strategy was used for generating a set of observational data that contains the maximum wall shear stress values for a range of probable arterial geometries. The dataset was mined via a Bayesian Gaussian process emulator to estimate: (a) the influence of key parameters on the output via sensitivity analysis, (b) uncertainty in output as a function of uncertainty in input, and (c) which settings of the input parameters result in maximum and minimum values of the output. Finally, potential diagnostic indicators were proposed that can be used to aid the assessment of stroke risk for a given patient's geometry.
Introduction
Vascular diseases such as atherosclerosis and thrombosis are known to cause fluid mechanic derangements that affect blood flow in arteries [1]. Previous research involving in vitro studies [2–5] aimed at understanding these disorders have found that the main factors contributing to changes in blood flow patterns are the pulsatile behavior, viscosity of blood, the geometry of arteries and the arterial wall distensibility [1, 6]. Of these, geometry of the vessel was found to be the most important factor influencing the flow behavior [6]. In recent years, in silico blood flow simulations have gained lot of popularity. In particular, three dimensional computational fluid dynamics (CFD) studies at the sites of curvature, bifurcations, and junctions have facilitated the identification of vulnerable atherogenic sites [1, 7, 8]. Simulations were performed to analyze blood flow patterns at several sites such as carotid [9–13], coronary [14–17], aortic [18, 19] and iliac [20] arteries. Many of these have highlighted the disturbed flow patterns caused due to alterations in arterial geometry. Furthermore, fluid mechanical forces such as wall shear stress (WSS) have been identified to play a major role in the pathogenesis and pathophysiology of atherosclerosis [21].
Most of the work in the literature on computational hemodynamics assume that arterial geometry definition is precisely known. In practice, however, it is known to vary both with and within individuals [22, 23]. Hence, there is a need for efficient numerical methods that can systematically account for geometric uncertainty and predict true flow behavior in real time. This problem is particularly challenging because it is difficult to characterize the observed geometric variability using a small number of variables – straightforward univariate or bivariate parameter studies that involve varying a subset of the geometric variables tend to be of limited use. A statistical assessment hence becomes beneficial to gain insights into the relationship between flow patterns and geometric attributes. The basic idea of this approach is to construct probabilistic models for the input uncertainties and subsequently propagate this through the computer model to assess the impact of variability in inputs on the outputs of interest. A standard approach for statistical analysis is the Monte Carlo simulation technique [24], where the computer model is run repeatedly for randomly generated values of the inputs, and subsequently, the resulting data is postprocessed to estimate the output statistics. However, due to the requirement of a large sample-size, this approach becomes computationally prohibitive, particularly when high-fidelity models are used.
In the present work, we proposed the application of Bayesian Gaussian process modeling [25–27] to study the relationship between geometric factors and hemodynamic metrics. The present approach can be viewed as a computer-based data mining strategy which extracts useful information and synthesizes interesting relationships from datasets generated during multivariable parameter studies. The input and output observational data was generated by running computer simulations on selected cases and emulators constructed via Bayesian Gaussian process modeling were used to analyze and summarize the data in novel ways that are understandable and can potentially be of clinical benefit.
To demonstrate the applicability of the proposed approach, the human carotid artery bifurcation was chosen as the anatomical site for analysis since it is a common site for arterial disease to occur [6]. A novel three dimensional parametric computer aided design (CAD) representation of the human carotid artery was defined. The objective behind using a parametric geometry model for statistical analysis is so that a range of probable geometries can be automatically generated. However, comparison between the data obtained for different simulations requires a method for ranking individual performance. Whilst it is possible to visually differentiate surface contours and/or velocity profiles between any two cases, a numerical value (metric or indicator) is preferable because it eases the level of comparison. A range of metrics are available in the literature, and most of these evaluate shear stress related expressions on a cell by cell basis [3, 5, 28–34]. Following some of this work, we considered the maximal wall shear stress (MWSS) as the metric for statistical analysis. It has already been shown that large changes in the magnitude of MWSS can play a role in the embolic mechanism by which carotid lesions can induce stroke [30]. Hence, it is important to understand the correlation between the geometric variability and MWSS in the human carotid artery.
A design of experiments (DOE) approach [35] was employed to generate a set of candidate geometries for steady state three dimensional flow analysis. The data generated from these runs was then used to construct a Bayesian Gaussian process emulator which approximated the MWSS as a function of the geometric variables. This model was subsequently employed as a computationally cheap emulator to compute the statistics of the output when the inputs were modeled as random variables and to estimate the degree of influence of each geometry parameter on the MWSS. Later, the application of the Bayesian emulator to identify geometries that have the highest and lowest MWSS values was demonstrated. Subsequently, pulsatile simulations on the same candidate geometries were performed to compare these results with the steady state case. Finally, we proposed potential diagnostic indicators that are capable of estimating the degree of severity with respect to the MWSS metric for a given patient's geometry thereby aiding the assessment of stroke risk.
Methods
Parametric model of the carotid bifurcation
Parametric computer aided design (CAD) definition of the carotid bifurcation enables the automatic generation of typical geometries by varying the parameters in the baseline CAD model. In so doing, not only is it possible to explore the impact of alternative definitions on the flow and its associated shear stress parameters but also, more importantly, the relationships between hemodynamics and a wide range of geometric parameters can be investigated in detail. The power of parametric geometry representation lies in its ability to simply generate a range of alternative geometries using the baseline shape as a template that is then reconfigured according to new values of the variable parameters. Note that it is also possible to morph the parametric model in order to reproduce a patient-specific geometry obtained from magnetic resonance imaging (MRI) [36–38] or contrast-enhanced x-ray computer tomography [39] scans. Image reconstruction techniques have reached a new level in the recent past where realistic geometry shapes were extracted from these scans in real time and subsequently, CFD studies were performed. In the context of our work, for example, the CAD model parameters can be estimated from a scan by solving an optimization problem involving minimization of an appropriate distance metric characterizing how well the CAD model reproduces the scanned geometry. In combination with regularization methods, the parametric CAD approach can be potentially useful in smoothening geometries obtained from imaging techniques.
Careful consideration of a number of geometry descriptions including [6] and [40] shows that, from a CAD perspective, insufficient information is available in all of them to construct a parametric CAD model from scratch, without making a number of assumptions, and/or unrealistic constraints are imposed that limit the overall flexibility of the models. Consequently, new parametric CAD definitions of the carotid geometry were presented in this paper. There are some similarities with older models but a number of important innovations were introduced that yield complete and reusable definitions and provide sufficient flexibility to facilitate parametric studies. This CAD representation was mainly based on the Y-shaped model developed in [6].
Dimensions of the human carotid artery bifurcation (Locations in Figure 1). Note that the split ratio values are not mentioned here and units of all the parameters shown here (excluding the angles) are in mm
Description | Location | Mean value |
---|---|---|
Angles | 1 | 25.1° |
2 | 25.4° | |
External carotid | 3 | 5.6 |
4 | 15.0 | |
5 | 5.5 | |
6 | 15.0 | |
7 | 4.6 | |
8 | 15.0 | |
9 | 4.6 | |
Internal carotid | 10 | 8.3 |
11 | 7.28 | |
12 | 8.9 | |
13 | 7.2 | |
14 | 8.2 | |
15 | 9.52 | |
16 | 6.0 | |
17 | 12.0 | |
18 | 5.7 | |
19 | 9.0 | |
20 | 5.7 | |
Common carotid | 21 | 8.0 |
22 | 24.0 | |
23 | 8.0 | |
24 | 24.0 | |
25 | 8.0 | |
26 | 24.0 | |
27 | 8.0 | |
Other | 28 | 8.0 |
Bayesian Gaussian process modeling
In this section, we describe the theoretical and computational aspects of Bayesian Gaussian process modeling and describe its application to mine data obtained from computational simulations. To illustrate, consider a computer code which takes as input the vector x ∈ ℝ^{ p }and returns a scalar output y(x). Let a design of computer experiments (DOE) strategy be applied to decide the settings of the inputs at which the computer code must be run [35]. In the context of hemodynamics, this step essentially involves generating a set of candidate geometries at which the flow solver is run to evaluate a hemodynamic metric of interest. The observational dataset thus created can be compactly written as $\mathcal{D}$ := {X, y}, where X = {x^{(1)}, x^{(2)}, ..., x^{(l)}} ∈ ℝ^{ p × l }and y = {y^{(1)}, y^{(2)}, ..., y^{(l)}} ∈ ℝ^{ l }.
The objective of Gaussian process modeling is to construct a computationally cheap emulator that can be used in lieu of the original computer code. The basic assumption made here is that the observed outputs {y^{(1)}, y^{(2)}, ..., y^{(l)}} are realizations of a Gaussian random field with parameterized mean and covariance functions. The assumed model structure used for the emulator can hence be written as
Y(x) = β + Z(x), (1)
where θ _{ j }≥ 0, 0 <p _{ j }≤ 2, j = 1, 2, ..., p are the hyperparameters. We shall hence use the symbol θ to denote the vector of hyperparameters. Furthermore, we have chosen p _{ j }= 2 to reflect the belief that the underlying function being modeled is smooth and infinitely differentiable. The hyperparameters θ which control the nonlinearity of the emulator are estimated from the data. For example, small values of θ _{ j }indicate that the output is a smooth function of the j th variable while large values indicate highly nonlinear behavior. It is also possible to tune the parameters p _{ j }to the data which allows for the possibility of modeling functions which may be discontinuous. In theory, the choice of an optimal covariance function is data-dependent. However, in practice it has been found that the parameterized covariance function in Equation (2) offers sufficient flexibility for modeling smooth and highly nonlinear functions [45].
where P(θ, β, ${\sigma}_{z}^{2}$|$\mathcal{D}$) is the posterior probability of the hyperparameters,
P($\mathcal{D}$|θ, β, ${\sigma}_{z}^{2}$) is the likelihood, P(θ, β, ${\sigma}_{z}^{2}$) is the assumed prior for the hyperparameters and P($\mathcal{D}$) is a normalizing constant called the evidence. To ensure computational efficiency, we adopt an empirical Bayesian approach, wherein the posterior distribution of the hyperparameters is approximated by point values obtained by maximizing the likelihood function – see the subsection that follows for details.
where R ∈ ℝ^{ l × l }is the correlation matrix computed using the training points whose ij th element is computed as R _{ ij }= R(x^{(i)}, x^{(j)}). r(x) = {R(x, x^{(1)}), R(x, x^{(2)}), ..., R(x, x^{(l)})} ∈ ℝ^{ l }is the correlation between the new point x and the training points, and 1 = {1, 1, ..., 1} ∈ ℝ^{ l }. It can be observed from Equations (5) and (6) that the Bayesian inferencing approach ultimately leads to an approximation of the computer code as a multidimensional Gaussian random field. The posterior variance computed using Equation (6), i.e., C(x, x), can be interpreted as an estimate of the uncertainty involved in making predictions at a new point x. Note that this uncertainty arises from the fact that only a finite set of points are used to construct the emulator. The quantification of uncertainty in the output due to variability in the inputs will be dealt with in the next section.
In practice, for the sake of computational efficiency, we compute the Cholesky decomposition of R. This allows the posterior mean to be computed at any point of interest using a vector-vector product, i.e., $\widehat{Y}$(x) = β + r(x)^{ T } w, where w = R^{-1}(y - 1 $\widehat{\beta}$). However, the computation of the variance (or error bar) of the posterior process (i.e., C(x, x)) requires a forward and back substitution.
Maximum likelihood estimation
Numerical optimization techniques are required for the maximization of Equation (8) to estimate the unknown parameters; see reference [27] for a detailed discussion of the computational and implementation aspects. Since computing L(θ, β, ${\sigma}_{z}^{2}$) and its gradients involves computing and decomposing the dense l × l covariance matrix R (requiring $\mathcal{O}$(l^{3}) resources) at each iteration, maximum likelihood estimation can be prohibitively expensive even for moderately sized data, e.g., say a few thousand points. Recent work to address this issue has led to data parallel techniques [47] which can cope with large datasets. In the context of the present study, due to the high computational cost involved in solving the Navier-Stokes equations, only a modestly sized training dataset is available for model building. As a consequence, the computational cost associated with maximization of L(θ, β, ${\sigma}_{z}^{2}$) is negligible.
It is worth noting here that for some datasets the correlation matrix R may be ill-conditioned, particularly when two or more training points lie close to each other. To circumvent this difficulty, we add a small term (say 10^{-6}) to the diagonal elements of R. A more robust strategy would be to employ a singular value decomposition of R – this would however be computationally expensive for large datasets.
Model validation
After emulator construction, validation studies will help in assessing how well the approximate model agrees with the true model. A brute force approach would be to run the numerical simulation for a number of additional geometries (testing points) and check how well the approximate model correlates at these points. In most practical applications, generation of additional testing data is computationally expensive. This motivates the development of alternative model diagnostic measures that can be evaluated more cheaply.
where $\widehat{Y}$ _{-i }(x) and C _{-i }(x, x) denotes the mean and variance of the prediction at a point x without using the i^{ th }training point. SCVR_{ i }can be computed for all the training points by removing the contribution of the corresponding point from the correlation matrix R.
In the cross validation procedure, it is generally assumed that the maximum likelihood estimates for the hyperparameters do not change when one training point is removed from the training set. If the Gaussian process prior is appropriate for the problem under consideration, SCVR_{ i }will roughly lie in the interval [-3, +3]. This implies that given the posterior predictions of the emulator at a new design point x, the actual output value lies in the interval $[\widehat{\text{Y}}(x)-3\sqrt{C(x,x)},\widehat{\text{Y}}(x)+3\sqrt{C(x,x)}]$ with a high level of confidence. More details and additional validation procedures such as leave-one-out and leave-two-out validation can be found in [27].
Post processing of the emulator
Once the Bayesian Gaussian process emulator has been constructed and validated, it can be mined in the post-processing phase. In this section, we describe how the emulator can be applied to: (i) compute statistics of the output when the inputs are modeled as random variables with a specified joint probability distribution, (ii) identify the relative importance of each input variable and (iii) identify which settings of the inputs lead to maximum and minimum values of the output.
Uncertainty analysis
where y(x) is a function calculated by running an expensive computer model.
In the MCS technique, y(x) is evaluated at numerous points by drawing samples from the distribution P(x) and the integral is subsequently approximated as $I\approx (1/m){\displaystyle {\sum}_{i=1}^{m}y({x}^{(i)})}$. The convergence rate of the Monte Carlo estimate is $\mathcal{O}\left(\frac{1}{\sqrt{m}}\right)$. Hence, to ensure accurate approximations, a large sample size becomes necessary; for example, to improve accuracy by one decimal place, around 100 times more samples will be required. Another drawback of the MCS technique is that it wastes information since the realizations of the inputs x^{(i)}, are not used in the final estimate [49]. It is worth noting here that it is possible to employ quasi-MCS techniques that enjoy faster convergence rates [27]. However, even so, the resulting reductions in computational cost are often not very significant. The emulator-assisted uncertainty analysis approach discussed below presents a computationally more efficient alternative to simulation techniques. Further, the emulator based approach uses both the input realizations x^{(i)}and the corresponding output values to estimate ⟨y(x)⟩.
and χ denotes the support of the distribution function P(x).
It is to be noted here that the multidimensional integrals in the preceding equations can be analytically computed if the elements of the input vector x are uncorrelated Gaussian random variables [50]. For more general distributions, simulation techniques can be applied to compute various statistics of interest including the probability distribution functions. This can be done efficiently since the prediction of the output at a new point x using the emulator is computationally very cheap. Furthermore, it is also possible to exploit the emulator sensitivities which are readily computable in order to accelerate the convergence of simulation schemes [51].
Sensitivity analysis
The above integral can be numerically computed by approximating it by a sum over a grid of m points x _{ h } ^{(1)}, ..., x _{ h } ^{(m)}. These points can be generated using the Latin hypercube sampling (LHS) technique [52]. Note that since the Bayesian emulator also provides the posterior variance, error bars on the main effects can be readily computed. Similarly, the effect of two or more covariates (joint effects) can be investigated by integrating out all the other covariates or by fixing the other covariates at some values [44, 53].
The values of S _{ i }, i = 1, 2, ..., p can subsequently be plotted in a pie chart to indicate the sensitivity of each variable. Similarly, it is also possible to define sensitivity factors whose magnitudes indicate the importance of interaction effects [54, 55].
Estimating worst-case settings
This section briefly discusses how the Bayesian emulator can be used to estimate settings of the inputs that give rise to maximum and minimum values of the output being modeled. A straightforward way to do this would be to directly maximize/minimize the posterior mean predicted by the emulator as a function of the input variables. This naive approach would work well provided the approximation quality of the emulator is sufficiently high. More general-purpose statistical criteria that employ the posterior variance along with the mean to enable the solution of complex optimization problems can be found in the engineering design literature [27].
In the present study, the probability of improvement (PoI) criterion was used to identify input settings that result in worst-case values of the output. The PoI criterion encapsulates the posterior mean as well as the variance (error bar) predicted by the emulator. For example, to find a geometry with maximum MWSS, the following optimization problem is solved.
Maximise : P[Y(x) > y^{+}], (18)
where y^{+} denotes the highest value of MWSS among all the geometries in the training dataset used to construct the emulator. It can be noted that the PoI criterion indicates the probability that the output at a given point is greater than y^{+}. Hence, by maximizing P[Y(x) > y^{+}], it becomes possible to identify geometries that are likely to have higher values of MWSS compared to those in the training dataset.
Since the emulator prediction Y(x) at a given point x is a Gaussian random variable, the PoI can be computed exactly in terms of the standard normal distribution function. Note that for the case when it is desired to find geometries which have lowest MWSS, the criterion P[Y(x) <y^{-}] can be maximized, where y^{-} denotes the lowest MWSS among all the geometries in the training dataset.
Results and discussion
Model geometry and problem description
and density, ρ = 1035 kg.m^{-3}, viscosity, μ = 0.0035 kg.m^{-1}.s^{-1}, the CCA diameter, w = 0.008 m and $\overline{U}$ is the mean velocity. The mass fluxes through the ICA and ECA were fixed in the ratio 70 : 30 (as used in the experimental studies conducted by [2]). The semi-implicit method for pressure-linked equations (SIMPLE) pressure correction algorithm was used with second order spatial accuracy. Other solver settings included a second order upwind scheme for momentum discretization; a standard scheme for the discretization of the pressure equation. The principal assumptions in the present study were to consider the flow as steady, blood as a Newtonian fluid and the walls to be rigid. After obtaining a converged solution, the magnitude of MWSS was extracted for each geometry in order to create the training data for constructing the emulator.
Mesh dependence and flow setup
Mesh dependence configurations for a candidate geometry in the training data
Interval Size | Cell Count | Magnitude of MWSS(Pa) |
---|---|---|
0.96 | 18432 | 4.29743 |
0.84 | 23466 | 4.30954 |
0.72 | 35362 | 4.37759 |
0.60 | 53156 | 4.45912 |
0.48 | 115980 | 4.40611 |
0.36 | 241795 | 4.40634 |
0.24 | 835996 | 4.38985 |
Influence of geometry
Distribution of maximal wall shear stress
Main effects
Probability of improvement
Unsteady simulations
So far, the applicability of Bayesian modeling for statistical analysis has been demonstrated by performing steady state simulations on the candidate geometries. In this section, unsteady simulations were performed on the same candidate geometries and the PoI criterion was applied. As an inflow boundary condition at the inlet of the CCA, we used a mean inflow pulse taken from [56]. A time dependence study was performed on the mesh interval size 0.48 (mesh size taken for the steady state flow analysis) with time steps 0.001, 0.0001 and 0.00001 seconds. Since the flow is pulsatile, we computed the maximum value of the average wall shear stress in a time period (T = 0.917 sec) for each case. Henceforth, this new metric is denoted as maximal average shear stress (MASS). The MASS values for these cases were found to be 4.1405, 4.1347 and 4.1057 Pa, respectively. Consequently, 0.0001 sec was selected providing a reasonable balance between accuracy and computational cost. All unsteady simulations employed the pressure implicit with splitting of operators (PISO) pressure-velocity coupling scheme in FLUENT [43]. For the purpose of validation, the average wall shear stress for the case with mesh interval size 0.48 mm and time step 0.0001 sec was computed and found to be 0.692 Pa. This value lies within the accepted range of values reported in [57]. Recall that the dimensions and boundary conditions of the baseline case closely match with the data described in [2, 6, 40, 56].
Potential diagnostic indicators
Alternatively, the geometry of the patient can be compared with those predicted in the previous section. A similar approach can be applied for evaluating the bounds with respect to any indicator which synthesizes different flow behavior from which $\mathcal{D}$ _{ s }can be estimated. It should be noted that Γ_{g} and Γ_{b} have been estimated in a laminar flow regime with fixed outflow boundary conditions and by assuming the walls to be rigid. To extend this concept of assessment on the severity of disease for a more physiologically realistic case, we can incorporate realistic boundary conditions and the arterial wall distensibility to estimate $\mathcal{D}$ _{ s }. Furthermore, offline simulations on an extended set of geometries can be carried out to improve the predictivity of the emulator.
Conclusion
This paper presented a computer-based Bayesian modeling approach for data mining. The methodology was applied on a hemodynamic problem in which the geometric parameters affecting maximal wall shear stress (MWSS) in the human carotid bifurcation were analyzed. A design of experiments approach was employed to generate a set of geometries for numerical simulations using a Navier-Stokes solver. The results obtained from these runs were then used as training data to construct a Bayesian Gaussian process model. Applications of the resulting emulator include (i) uncertainty quantification when the inputs were modeled as random variables, (ii) the quantification of the sensitivity of an output quantity of interest with respect to the input parameters, and (iii) the ability of the model to predict geometries having low and high values of maximal wall shear stress using the probability of improvement criterion which can be subsequently used for estimating degree of severity with respected to any patient.
The work in this paper seeked to directly compare the effect of varying the geometric parameters on the maximal wall shear stress for the human carotid bifurcation. However, this metric only highlights the role of elevated shear regions. Using the same principle, new metrics can be used to correlate the regions of disturbed flow and sites of arterial disease [31–34, 59]. The main focus here was to demonstrate the applicability of the Bayesian modeling approach to mine data generated from hemodynamic simulations. Physiologically more relevant results can be obtained by considering the fluid-wall interactions and non-Newtonian nature of blood along with more realistic boundary conditions. More importantly, this analysis can be carried out on any arterial location that is vulnerable to hemodynamic derangements. Furthermore, new parameters can be introduced into the CAD model that can simulate the presence of stents or grafts. Subsequently, optimization studies can be performed on these cases that can improve the design and control of these artificially implanted devices [60–66]. We hope that the Bayesian Gaussian process modeling methodology can be efficiently integrated with well known image reconstruction techniques to generate a powerful paradigm of designing, optimizing and controling patient-specific blood flow behavior.
Appendix A
Construction of the parameterized CAD model
Declarations
Acknowledgements
The authors would like to thank the reviewers for their useful comments and suggestions. This research was performed during V. B. Kolachalama's doctoral study and he would like to thank the School of Engineering Sciences at the University of Southampton, UK for providing a studentship.
Authors’ Affiliations
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