Abstract
Computational physiological models are promising tools to enhance the design of clinical trials and to assist in decision making. Organ-scale haemodynamic models are gaining popularity to evaluate perfusion in a virtual environment both in healthy and diseased patients. Recently, the principles of verification, validation, and uncertainty quantification of such physiological models have been laid down to ensure safe applications of engineering software in the medical device industry. The present study sets out to establish guidelines for the usage of a three-dimensional steady state porous cerebral perfusion model of the human brain following principles detailed in the verification and validation (V&V 40) standard of the American Society of Mechanical Engineers. The model relies on the finite element method and has been developed specifically to estimate how brain perfusion is altered in ischaemic stroke patients before, during, and after treatments. Simulations are compared with exact analytical solutions and a thorough sensitivity analysis is presented covering every numerical and physiological model parameter.
The results suggest that such porous models can approximate blood pressure and perfusion distributions reliably even on a coarse grid with first order elements. On the other hand, higher order elements are essential to mitigate errors in volumetric blood flow rate estimation through cortical surface regions. Matching the volumetric flow rate corresponding to major cerebral arteries is identified as a validation milestone. It is found that inlet velocity boundary conditions are hard to obtain and that constant pressure inlet boundary conditions are feasible alternatives. A one-dimensional model is presented which can serve as a computationally inexpensive replacement of the three-dimensional brain model to ease parameter optimisation, sensitivity analyses and uncertainty quantification.
The findings of the present study can be generalised to organ-scale porous perfusion models. The results increase the applicability of computational tools regarding treatment development for stroke and other cerebrovascular conditions.
1 Introduction
Cerebrovascular diseases including stroke impose a heavy burden on society [1]. The majority of stroke cases are caused by a thrombus blocking a major cerebral artery leading to severe blood shortage in the brain (ischaemic stroke). In recent decades, ischaemic stroke treatment has been revolutionised by thrombolysis (dissolving of blood clot with a thrombolytic agent) [2] and thrombectomy (mechanical thrombus removal with a stent retriever) [3, 4]. Advancing the related clinical procedures has great potential to benefit stroke patients but is tied entirely to resource intensive animal experiments and clinical trials [3–7]. To ease this task, the In Silico clinical trials for the treatment of acute Ischaemic STroke (INSIST) consortium aims to develop computational models of stroke and its treatments [8]. Once reliable models are available, in silico clinical trials could sharpen the focus of clinical trials and hence save time and money [9–11].
Estimating brain perfusion in both healthy and occluded scenarios is an important element of the INSIST pipeline and serves as a bridge between localised treatment effects and patient outcome [8]. To this end, a multi-scale haemodynamic model is under development which combines a one-dimensional (1D) network model of large arteries [12, 13] and a three-dimensional (3D) porous perfusion model [14] to simulate blood flow in the entire vasculature. The models will be coupled strongly through the cortical surface. The envisaged interface will facilitate communication between the boundary conditions of the models, namely pressure and volumetric flow rate values averaged over cortical territories [13]. Previously somewhat similar weak-coupling schemes have been established between a lumped parameter model and a porous continuum model to cover arteriole boundary conditions at the brain surface [15–19]. This approach preserves the two models as independent functional units and avoids the positioning of volumetric sources required for volumetric coupling [20–22]. The anatomical connection between large arteries and the microcirculation is crucial in ischaemic stroke modelling because it determines the location and extent of ischaemic regions. Another advantage of placing the interface at the cortical surface is that anatomical connections are preserved at the cortical surface so that subcortical vessels can be homogenised to lower computational costs. In the case of volumetric coupling, subcortical arterioles must be added to the one-dimensional network model to account for such connections which can drastically increase the number of discretised branches.
Computational porous and poroelastic models have been applied over a number of decades and have been proposed to describe aspects of cardiac [23–25] and cerebal biomechanics [15–19, 26]. Such organ-scale computational models have been applied to capture both healthy and pathophysiological states, such as brain injury [27–29], oedema [30], conditions associated with dementia [15, 18, 19], and ischaemic stroke [14]. The majority of these models utilise the Finite Element (FE) method [14–19, 27–30] with relatively few exceptions [20–22]. The present study focuses on a porous perfusion FE model developed specifically to capture blood flow changes in ischaemic stroke [14] as part of the in silico trial pipeline of INSIST [8]. Even though organ-scale porous and poroelastic brain models are becoming increasingly popular [14–19, 22, 26–32], efforts regarding their verification, validation [15], comprehensive sensitivity analysis [22, 33], and uncertainty quantification [31, 32] remain relatively rare and there is thus a need to fill this gap before the models are applied for clinical trial design and decision making.
Recently, the American Society of Mechanical Engineers (ASME) has introduced the Verification and Validation (V&V) 40 standard (titled “Assessing Credibility of Computational Modeling through Verification and Validation: Application to Medical Devices”) [34], which details the requirements that a credible engineering software needs to satisfy to ensure safe applicability in the medical device industry. The present study thus sets out to establish guidelines regarding the efficient usage of a porous cerebral perfusion Finite Element model [14] both in terms of accuracy and computational cost. To this end, simulations will be compared to exact analytical solutions and a thorough sensitivity analysis will be carried out covering every numerical and physiological model parameter. The results prepare the ground for uncertainty quantification, virtual patient generation, and validation, which are essential to carry out reliable in silico trials of ischaemic stroke treatments.
2 Methodology
Inspired by recent advances in the fields of heart [23–25] and brain [15–19] haemodynamic modelling, a multi-compartment porous continuum framework is utilised to describe time-averaged cerebral blood flow in the microcirculation embedded within brain tissue. The general partial differential equation set describing a steady-state multi-compartment Darcy system [15–19, 23–25] is written in a compact form1 as Here, pi denotes the Darcy pressure in the ith compartment and Ki is the corresponding permability symbolising flow conductance through pores with well-separated length scales. Considering n compartments, i = 1, 2,…n and βi,j is the intercompartment coupling coefficient matrix with n × n elements representing conductance between compartments i and j. It follows that βi,j = 0 if i = j. A volumetric source term in the ith compartment is defined by Si.
Considering a domain of interest denoted by Ω with a boundary ∂Ω = Γ = ΓD,i ∪ ΓN,i, a set of Dirichlet type boundary conditions (DBC) and Neumann type boundary conditions (NBC) are formulated as Here, n is the outward-pointing unit vector perpendicular to the boundary surface. Thereafter, the weak form can be derived by multiplying every component of Eq. (1) with a non-zero test function ti, integrating over Ω and applying the divergence theorem: Eq. (4) is discretised by the continuous Bubnov-Galerkin method [35]. Numerical solutions of pi are obtained with the open source FEniCS library [36–38] using first or second order Lagrange elements denoted by P1 and P2, respectively. The permeability tensors Ki and the coupling coefficients βi,j are captured by piecewise constant, zeroth order discontinuous elements (dP0) unless stated otherwise. For simplicity, we restrict ourselves to a model with arteriole (i = 1 = a), capillary (i = 2 = c), and venule (i = 3 = v) compartments, and then generalise our findings whenever it is possible.
Numerical solutions of linear equation systems originating from the FE discretisation are obtained iteratively based on the BiCojungate Gradient STABilised method (BiCGSTAB) [39]. Computations are accelerated with an Algebraic MultiGrid (AMG) preconditioner [40]. Simulations were run on a desktop with an Intel Xeon E-2146G processor and 32 GB RAM unless stated otherwise. In order to verify the resulting model and carry out a comprehensive sensitivity analysis, three test cases are considered as detailed in the following subsections.
2.1 Three-dimensional synthetic data
To derive manufactured solutions [41], we assume solution functions of Eq. (1) of the form Polynomial functions are chosen to ensure straightforward differentiation of the pressure terms in Eq. (1). The permeability tensors are set to whereas the non-zero elements of the coupling coefficient matrix are The source terms S1, S2 and S3 satisfying Eq. (1) can be obtained simply by substituting Eqs. (5c)–(7b) into Eq. (1). The source terms are set to zero in every other test case (details in Sections 2.3 and 2.2). Ki and βi,j are approximated by fourth order polynomials for the finite element computations.
The domain of interest chosen here is a unit cube Ω = [0, 1] × [0, 1] × [0, 1] with periodic boundary conditions applied on four sides: Eqs. (8) and (9) define four DBCs in every compartment and another one is added in the arteriole (i = 1) and the venule (i = 3) compartments based on the manufactured solutions such that Thereafter, the equation set is closed by prescribing homogeneous NBCs on the remaining surfaces (in Eq. (3) bi = 0).
2.2 Three-dimensional human brain
Details of the baseline human brain simulation are provided in [14] and therefore only the most important settings are covered here. The computational domain (Ω) is obtained by postprocessing a tetrahedral patient-specific head model utilised in multiple recent studies [14, 42–44]. The geometry depicted in Fig. 1a is remeshed with tetrahedral elements using Tetgen [45]. The volumetric region of interest includes both grey matter (ΩG) and white matter (ΩW) subdomains, so that Ω = ΩG ∪ ΩW as depicted in Fig. 1b. The bounding surface regions (∂Ω) include a transverse cut-plane of the brainstem ΓBS, the ventricles ΓV and the pial surface ΓP so that ∂Ω = ΓBS ∪ ΓV ∪ ΓP as depicted in Figs. 1a and c. The boundary region associated with the pial surface (Fig. 1a) is subdivided into eight perfusion territories corresponding to major feeding arteries as shown in Figs. 1d-f. Perfusion territories have been identified using a voxelised atlas created based on vessel-encoded arterial spin labelling (ASL) perfusion magnetic resonance imaging (MRI) [46–50]. Then, the surface region that is perfused, for instance, by the Right Middle Cerebral Artery (R-MCA) is denoted by ΓR-MCA. This approach ensures that blood arrives to the domain through cortical surface regions mirroring anatomical connections between large arteries and the microcirculation as shown in Figs. 1d-f.
The imposed baseline boundary conditions prescribe zero flow through the transverse cut-plane of the brainstem and the surface of the ventricles: Flow through the pial surface in the capillary compartment is zero: Because of the incompressible fluid flow model, the results depend solely on the cerebral perfusion pressure (CPP) defined as the arteriole-venule pressure difference at the pial surface: CPP = pa − pv. Setting the zero level of the pressure at the outlet of the venous compartment (pv) leads to Pressure values in the case of 3D brain simulations are presented relative to the venous outlet pressure pv. To account for totally occluded scenarios, blood flow through the perfusion territory of an occluded vessel is set to zero whereas surface pressure is assumed to remain constant in other regions. Accordingly, a R-MCA occlusion is modelled with zero flux through the corresponding cortical territory (ΓR-MCA) so that the resulting mixed boundary conditions become
The model is parametrised based on several simplifications as discussed in [14]. Considering that the arteriole-capillary and capillary-venule coupling coefficients are known in the grey matter, it is assumed that the ratio of grey and white matter coupling coefficients (βG/βW) is constant. It has been demonstrated based on microscale one-dimensional haemodynamic network simulations that the capillary permeability tensor is isotropic and characterised by a scalar k2 [51]. It is hypothesised that a reference coordinate system [ξ, η, ζ] can be found at every point so that ζ is parallel to the local axis of the descending arterioles and ascending venules. In this coordinate system, the anisotropic arteriole and venule permeability tensors are modelled as tensors with a single non-zero diagonal element (k1 and k3) because they represent vessel bundles: The [ξ, η, ζ] coordinate system is determined assuming that penetrating vessels grow from the pial surface to the ventricles following a vector field. This vector field is computed as a gradient of a scalar field governed by a single diffusion equation [14]. It is worth noting that no other methods have been proposed to obtain these anisotropic permeability fields even though they play a crucial role in predicting perfusion response to vessel occlusion.
2.3 One-dimensional brain tissue column
Eq. (1) can be simplified so that a one-dimensional problem with homogeneous permeabilities can be posed and defined by a set of ordinary differential equations in the form of Here, the permeability of each compartment (ki) is a scalar. Introducing the dimensionless variables x* = x/ls and p* = p/ps based on a length scale (ls) and a pressure scale (ps) leads to a dimensionless form of Eq. (16) as The dimensionless matrix group provides a similarity condition for Eq. (16).
Equation (17) describes n second order ordinary differential equations. This system can be replaced by 2n first order ordinary differential equations once the pressure gradient is introduced. Thereafter, the unknown vector contains the pressure and pressure gradient functions for each compartment. Finally, a matrix differential equation based on Equation (17) can be formulated as Here, the prime (′) denotes spatial differentiation and the A matrix is where every submatrix has a dimension of n × n, and I is the identity matrix. The connection between the coupling coefficient matrix α = αi,j and the submatrix B is given by Here, the n dimensional vector en is filled with ones, and the “diag” operator is used to turn a column vector into a diagonal square matrix.
The solution of Equation (19) is Here, λj and vj are the jth eigenvalue and eigenvector of A, respectively. is used to take the real part of the expression and to handle complex eigenvalue and eigenvector pairs.
The Cj coefficients are determined based on combinations of DBCs and NBCs at xb. A DBC in compartment i reads as Based on the unknown vector defined in Eq. (18) and the solution expression Eq. (22), a DBC can be satisfied by imposing Here, denotes the ith element of r and is the ith element of jth eigenvector. A NBC in compartment i is defined as The second half of the unknown vector defined in Eq. (18) includes the pressure gradients. Therefore, a NBC can be satisfied by imposing
Eqs. (24) and (26) provide a single equation for each boundary condition. Considering n compartments, the 2n boundary conditions lead to 2n linear equations based on the 2n coefficients of the solution expressions (Cj). The resulting linear equation system determines uniquely the coefficients of the solution expressions (Cj). Using the DBC (23) and the NBC (25), the procedure can be extended to handle multiple subdomains with different coupling coefficients. In such cases, interface conditions have to be applied to ensures that the solution is continuously differentiable. Therefore, this analytical method is suitable to obtain solutions for a column of brain tissue stretching between the cortical and ventricular surfaces. The method can handle a domain including grey and white matter with identical permeabilities but different coupling coefficient matrices.
3 Results and Discussion
3.1 Spatial resolution
3.1.1 Manufactured solutions
The suitability of the finite element method is evaluated based on the manufactured solution shown in Fig. 2. A uniform tetrahedral mesh is generated with two elements along each edge of the cube and uniformly refined four times. Then, a grid convergence study is carried out using first (P1) and second order (P2) elements as summarised in Table 1. The analytical solution and the finite element approximation along the sampling line depicted in Fig. 2 are shown in Fig. 3a, highlighting that a very good agreement is achievable. Fig. 3b presents the L2-norm defined over the entire domain based on the difference between the analytical and the numerical solutions. With increasing spatial resolution, the L2-norm decreases as expected. Superconvergence of the solution with first order elements can be observed which is a well-known feature of this FE approximation [52].
According to Fig. 3b and Table 1, solutions achieved using a second order scheme on a given mesh have about the same L2-norm as first order approximations after a single refinement. However, it is important to emphasise that the convergence of pointwise variables does not necessarily follow the convergence trend of integral quantities. Fig. 3c suggests that the second order FE scheme outperforms the first order scheme in terms of gradient estimation even if computations with the latter are carried out with a threefold refined mesh. Beyond accuracy, the computational cost of simulations as a function of the spatial resolution and the element order can be examined in Fig. 3d. In general, computations with second order elements take one order of magnitude longer and are comparable to the computational cost of simulations with first order elements on a refined mesh (see also Table 1).
3.1.2 Whole brain model
Next, simulations are carried out with the whole brain model to investigate how uncertainty of the gradient estimation near the boundaries impacts the results. Two scenarios are considered using previously optimised settings [14] summarised in Table 2: baseline (healthy) and an RMCA occlusion. The algorithm used for parameter optimisation is described in Appendix A. In a porous framework, the pressure gradient is directly connected to the Darcy velocity vector which can be introduced in compartment i as From here, the overall volumetric flow rate entering the arteriole compartment through the pial surface ΓP is simply defined as The Γn product corresponds to an outward-pointing vector with magnitude equal to the area of the Γ boundary surface. (The Darcy velocity has non-zero components perpendicularly to the boundary surface only at ΓP and therefore in Eq. (28) the integration domain ΓP is replaced with Γ for a more concise notation.)
Therefore, any uncertainty in the pressure gradient propagates directly to the volumetric flow rate. Accurate flow rate computation is important for two reasons: (i) superficial arteriole pressure and flow rates are cornerstones of a coupling scheme between the present continuum model and a network model of large arteries as detailed in Section 1; (ii) surface fluxes associated with major cerebral arteries can be directly compared with values obtained from phase-contrast MRI [55] for validation purposes.
The multi-compartment formulation offers an alternative method to compute volumetric flow rate through the brain. Introducing perfusion F = β1,2(p1 − p2), and integrating it over the brain volume Ω leads to According to mass conservation QΩ = QΓ so the imbalance between the two formulations is present because the FE method is not conservative by default, unlike the finite volume method [56]. Two strategies are employed to mitigate the associated error: (i) grid refinement; and (ii) increasing FE orders of both the pressure and the velocity. For P1 and P2 pressure elements, the natural velocity pair is dP0 and dP1 but higher order velocity approximations can be obtained by projection. The results in Table 3 summarise the tested cases and show that the volumetric flow rate from Eq. (29) depends on the grid size and the element order only negligibly. The error in QΩ should be relatively low in the investigated cases because the computed values overlap independently from the spatial resolution. For this reason, and because of the mass conservation principle, the relative difference, is used for error measurement.
Figs. 4a and b show the imbalance between QΩ and QΓ highlighting that without improved ui approximation the surface flux is always underestimated by QΓ. Both refinement and increased FE order can mitigate the error of QΓ. However, higher order pressure and velocity approximations are required to keep |ΔrQΓ| < 5%. Fig. 4c displays the wall time of the simulations emphasising the increased computational cost required for more accurate flux estimations. In summary, second order velocity and pressure elements on a relatively coarse mesh (about 106 elements) can provide a reasonable compromise between accuracy and performance if computing QΓ is important. Otherwise, perfusion and pressure distributions can be estimated reasonably well on coarse meshes with about 1 million elements.
Once it is established under what conditions the present 3D model can provide accurate estimation of volumetric flow rate through cortical surface regions, it is possible to compute volumetric flow rate of the major cerebral arteries. Simulation results are compared with values from a clinical study in Tab. 4. Whereas the model is optimised for overall brain perfusion, blood flow rate through these major arteries provide a chance for independent validation. It is promising that volumetric flow rate in both the ACA and the MCA is predicted within one standard deviation of the experimental values. Volumetric flow rate in the PCA is somewhat overestimated. This simple comparison is an important milestone in the quantitative validation of the 3D brain model and provides guidance about development directions for the future.
The results emphasise the importance of accurate mapping between voxelised vascular atlases and the triangulated brain surface and draw attention to geometrical flaws. The brain model does not capture the lateral sulcus and the insula. Because of these neglected regions, the brain surface is underestimated whereas the brain volume is overestimated (see Tab. 4). It is anticipated that the 3D porous model will provide more reliable volumetric flow rate estimations once these geometrical issues are overcome. However, obtaining accurate discretised brain geometries suitable for FE computations remains a challenging task, partially because of labelling the boundary regions, and therefore such efforts are left to a future study.
3.2 Boundary conditions
Results from the previous section draw attention to issues related to flux computation in a porous FE framework. Instead of constant pressure, uniform velocity has been used as an inlet BC [15–19], where the velocity is calculated as the ratio of a given volumetric flow rate and the inlet surface area. This NBC at the inlet might distort the results because of errors related to gradient estimation so we next investigate the difference between pressure and velocity inlets using the present model. To this end, a single simulation with constant inlet velocity is run with the same settings as the case in Tab. 3 marked with *.
The overall volumetric flow rate QΩ = 600 [ml/min], and hence brain perfusion, is insensitive to the inlet boundary conditions. QΓ is impacted only slightly: pressure inlet leads to 559 [ml/min] whereas velocity inlet results in 572 [ml/min] (both values should be 600 [ml/min] based on mass conservation). The effects of the inlet boundary condition on other statistics regarding the pressure and the perfusion fields are shown in Fig. 5. Most of the volume averaged pressure (〈pi〉) and perfusion (〈pi〉) values are altered only slightly by the inlet boundary conditions. However, extrema of both the pressure and the perfusion fields change substantially when the inlet velocity BC is used. Minimum values are underestimated so that the relative difference of the minimum arteriole pressure and perfusion values are approximately −100% because min p1 and min F) are close to zero when uniform inlet velocity BCs are used. By comparison, the inlet velocity BCs lead to overshot maximum values nearly by a factor of 10 compared to the case with inlet pressure BCs (see max p1 and max F in Fig. 5).
In Figs. 6a-d, and b-e, the pressure and perfusion fields corresponding to inlet pressure and velocity BCs, respectively, emphasise further the difference between the two solutions. The arteriole velocity magnitude displayed over the cortical surface (Fig. 6c) shows that in the case of pressure inlet BC the velocity is strongly inhomogeneous. It is a remarkable feature of this scenario that high velocity values are predicted along the major cerebral arteries branching from the circle of Willis because the whole brain model does not include any information about the location of these vessels. The solution obtained with constant pressure BC suggests that descending arterioles branching from the major cerebral arteries play a key role in providing well-balanced grey and white matter perfusion. The velocity distribution over the cortical surface is not known a priori and therefore it is challenging to impose feasible velocity distributions with velocity inlet BC.
The volumetric flow rates corresponing to boundary regions of the major cerebral arteries are summarised in Tab. 4. Based on previous clinical investigations [55, 57], it is clear that each major cerebral artery contributes differently to volumetric blood flow rate to the brain. Even if we assume that superficial perfusion territories of cerebral arteries are proportional to the associated blood flow rate, it is important to recognise that the volumetric flow rate should be heterogeneous as dictated by underlying grey and white matter volumes. Although the validity of homogeneous inlet velocity is physiologically questionable, prescribing constant pressure inlets in healthy scenarios is supported indirectly by clinical data. Based on ASL MRI [47], cerebral perfusion regions have been found to be well-separated indicating that the pressure gradient between these territories is small. Furthermore, dynamic computed tomography angiograms indicate that flow through leptomeningeal collaterals is activated as a result of cerebral artery occlusion [60]. We hypothesise that in healthy cases, leptomeningeal collaterals [2, 50, 60–63] as well as pial arterioles [64–66] act to equalise blood pressure in descending arterioles near the cortical surface. This hypothesis could be tested once the present perfusion model and the 1D model incorporating the larger arteries [13] are coupled, and it is supported by pressure data obtained in rodents which suggest small pressure drop in large arteries compared to the microcirculation [67–69].
3.3 Sensitivity analysis
Determining the actual values of parameters in physiological models is a challenging task because of limited experimental data, high patient-specific variability, and the typically high dimensionality of the parameter spaces. Considering porous cerebral haemodynamics, this issue is far from being settled as indicated by the broad range of reported parameters summarised in Tab. 2. With increasing computational costs, mapping a high dimensional parameter space quickly becomes unfeasible. Beyond parameter optimisation, sensitivity analysis and the associated uncertainty quantification are other tasks which require large batches of simulations. Here we aim to utilise the one-dimensional model described in Section 2.3 in order to ease these tasks.
3.3.1 One-dimensional brain tissue column
The one-dimensional multi-compartment porous model enables the investigation of tissue columns including both grey and white matters so that the corresponding model parameters are identical with the ones used for the three-dimensional brain model (Tab. 2). In the 1D case, x = 0 and x = l correspond to the pial and the ventricualar surfaces, respectively. Therefore, the same boundary conditions can be imposed in the 1D tissue column and the 3D brain. (Similarly to the 3D brain, pressure values in the 1D case are presented relative to the venous outlet pressure pv.) Thereafter, the one-dimensional model has only two free parameters: the lengths of the grey (lG) and the white (lW) matter domains defined so that l = lG + lW. The equivalent lG = 13.55 and lW = 7.99 [mm] is uniquely determined by the human brain model used here and can be obtained by optimisation. The corresponding cost function is zero if the one-dimensional model returns 〈F〉 = 43, 〈F〉G = 56, and 〈F〉W = 21 [ml/min/100ml] which overlap exactly with volume-averaged perfusion values of the three-dimensional brain model.
Fig. 7 displays the analytical pressure distributions in the arteriole, capillary and venule compartments. A one-dimensional FE approximation with 100 P1 elements is also shown which is in excellent agreement with the analytical results. The average pressure values provided by the 1D model are within ±4% of those computed using the 3D brain in every case. One-dimensional simulations are approximately 3000 times faster compared to their 3D counterparts.
3.3.2 Mapping the parameter space
The 1D model is a good candidate for parameter space mapping because of its low computational cost. A disadvantage of the analytical formulation is that with the parameters presented in Tab. 2, the linear equation system governing the boundary conditions is very stiff probably because of the parameters with different. The condition number depends nonlinearly on the reference length scale used for the nondimensionalisation. To avoid the issue of suitable reference length scale selection, we will use the 1D finite element model verified in Fig. 7 for a detailed sensitivity analysis.
Next, each parameter is perturbed independently from others. The one-at-a-time approach is applicable and the corresponding sensitivity analysis is representative of the system behaviour becayse the governing equation set (1) is linear [33]. The parameter range is [10%;1000%] compared to the values in Tab. 2 with 101 samples along each dimension. In addition, a geometrical scaling factor is introduced based on the ratio of the original and the perturbed volume of the computational domain. In order to evaluate whether the 1D model can capture the system behaviour, 11 samples along each dimension of the parameter space are tested with the 3D brain model using P1 pressure elements.
Results are summarised in Fig. 8 showing brain perfusion as a function of the model parameters in the healthy scenario. Based on the 3D simulations, RMCA occlusion is also considered so that the perfusion in the occluded scenario and infarcted volume change are also displayed. Infarcted volume is estimated solely based on the perfusion distribution. A region with more than 70% perfusion drop compared to the baseline scenario is identified as an infarcted zone inspired by perfusion imaging-based methods [70, 71]. The obtained value estimating infarcted volume is used here solely as an indicator of perfusion drop severity because this method overestimates the ischaemic region compared to diffusion-weighted magnetic resonance imaging [72].
According to Figs. 8a-h, the 1D solutions follow the trend of the 3D simulations in a wide parameter range in every case except the geometrical scaling factor. Considering that the present 1D model does not account for curvature effects, the results could potentially be improved with a cylindrical or spherical formulation. The sensitivity of each parameter is estimated as the normalised partial derivative of brain perfusion and infarcted volume at 100% of the abscissa (around baseline values presented in Tab. 2). The derivatives computed based on the 3D simulations in Fig. 8a-h are listed above the abscissa and indicate the percentage change of brain perfusion and infarcted volume as a response to unit percentage change in model parameters.
In terms of haemodynamics, it should be noted that the present model captures solely the resistive part of the microcirculation and neglects conductance and inductance associated with vessel wall stiffness and pulsatility. The linear relationship between brain perfusion and perfusion pressure is a direct manifestation of the “hydraulic Ohm’s law” (Fig. 8a). As other parameters are increased, brain perfusion tends to exhibit a saturating behaviour even if it is not obvious within the displayed parameter range (Fig. 8b-g) Considering the infarcted volume fraction, the insensitivity to CPP change originates from its definition restricted to relative perfusion change (as CPP is increased perfusion in the healthy and the occluded cases increases by the same amount).
Perturbations in the capillary and venous permeabilities have a relatively weak impact on both Fb and IV. In terms of the capillaries, this is a misleading behaviour which is a direct consequence of the structure of the brain model. Fluid flow within the capillary compartment is relatively small. This result can be interpreted as flow “avoiding” the capillaries because of their high resistance. Considering an indefinite domain with solely grey or white matter and constant but different arteriole and venous pressures, blood flow is non-zero between the compartments but it is zero inside each compartment. In the microcirculation, capillaries cannot be bypassed so it is important to recognise that the coupling coefficients must account not only for pre- and post-capillaries but also for the smallest capillary vessels. Therefore, the results show strong sensitivity to changes in coupling coefficients, which are responsible for the majority of the system resistance and consequently most of the pressure drop.
The geometry scaling factor has a strong influence on the infarcted volume fraction. If one of the major feeding arteries is occluded, it seems logical that for a given purely resistive vasculature a larger brain suffers more compared to a smaller brain. Such effects might be observable in clinical data but they are unlikely to be statistically significant compared to other factors, such as the extent of collateralisation. Nevertheless, the results suggest that a representative brain geometry is essential to keep the numerical uncertainty of the simulations low. In addition, the data presented in Fig. 8, and the one-dimensional model particularly, can help to accelerate virtual patient generation, where parameter sets must be found so that the resulting perfusion distribution is representative of a patient cohort.
3.4 Limitations
This subsection summarises some of the most important overlooked factors in the present study. Investigations are limited thus far to a single patient-specific brain geometry because of challenges associated with mesh generation. The porous model relies on multiple scale separation even though the vessel diameter in the vasculature changes continuously [20]. The present study demonstrated promising quantitative validation based on blood flow rate in major cerebral arteries but a more comprehensive validation is essential to ensure wider, multi-purpose applicability of the model.
The study is restricted to a porous model with three compartments (arteriole, capillary, venule) which are connected by homogeneous coupling coefficients both in grey and white matter. The model cannot represent arterioles which branch relatively far away from the cortical surface and hence feed deeper tissue regions. For this reason, low perfusion regions caused by occluded cerebral arteries are always connected to the brain surface so that isolated white matter infarction near the ventricles cannot be captured. On the one hand, such effects could be resolved by model expansion, for instance, by introducing two arteriole compartments which perfuse grey and white matter separately and rely on different boundary conditions. On the other hand, models with a single or two [22] compartments might be found more suitable in certain cases. Considering that the present study focuses on the sensitivity analysis of a three-compartment brain model for acute ischaemic stroke, exploring alternative porous frameworks for other specific (patho)physiological problems is left to future investigations.
From the haemodynamics point of view, the present model is purely resistive and aims to capture statistically steady state (time-averaged) brain perfusion. Therefore, the following physiological phenomena have been neglected: vessel wall stiffness (conductance), unsteady flow phenomena (inductance) [73], and cerebral autoregulation [74]. The associated processes and mechanisms are strongly time-dependent and often rely on nonlinear processes leading to significant complexity which is beyond the limits of the present study. Nevertheless, the authors hope that the present work will contribute to the creation of a comprehensive cerebral blood flow model which can incorporate these effects, in addition to pathophysiological processes, such as cerebral oedema [30], emboli advection and blockage of the microcirculation [53, 75, 76], and spreading of ischaemic tissue damage [77–79].
4 Conclusions
The present study set out to explore the sensitivity of a porous brain perfusion model and to prepare the ground for validation and uncertainty quantification. To this end, simulations were carried out using the finite element method and three test cases: a unit cube with manufactured solutions, a human brain geometry, and a one-dimensional brain tissue column. For the unit cube and the brain tissue column, analytical solutions were presented and used for verification. This approach allowed direct comparison between analytical and numerical results and enabled the identification of certain settings which were found crucial to minimise numerical errors.
The key findings of the study can be itemised as
Blood pressure and perfusion can be estimated reasonably well on a relatively coarse grid and first order pressure elements.
In order to approximate volumetric flow rate through superficial territories accurately with minimal computational effort, higher order finite elements are essential. Volumetric flow rate estimation can be further improved with local grid refinement in the vicinity of the superficial territories.
Porous models can provide a good estimation of blood flow rate through major cerebral arteries when higher order elements and constant inlet pressure boundary conditions are used.
Using uniform velocity inlet boundary conditions does not have solid physiological foundations because blood flow is likely not distributed proportionally to the cortical surface area. However, when considering healthy scenarios, constant pressure inlets can be prescribed with higher confidence.
A multi-compartmental porous system can be solved analytically in one-dimensional cases. After determining the equivalent thickness of the subdomains, the one-dimensional system behaviour is representative of the three-dimensional organ-scale model. The one-dimensional case is suitable for parameter mapping and sensitivity analysis at a computational cost three orders of magnitude smaller compared to three-dimensional models.
The results lay down the foundations for validation, uncertainty quantification, and for the development of a multiscale model which incorporates large arteries as well as the microcirculation. The present work contributes to the creation of a reliable computational model which can assist in the design of clinical trials and in clinical decision making related to cerebrovascular diseases.
Authors’ contributions
TIJ carried out the research and prepared a draft of the manuscript under the supervision of SJP. RMP developed the brain territory clustering algorithm under the supervision of AGH. TIJ, WKE-B, and SJP designed the research and every author contributed to the revision of the paper.
Conflict of interest declaration
The authors declare that they have no conflict of interest.
Acknowledgements
This work was funded by the European Union‘s Horizon 2020 research and innovation programme, the INSIST project, under grant agreement No 777072. Thanks are due to the members of the INSIST consortium for useful discussions and their sustained support.
Appendix A Whole brain model parametrisation
The baseline model parameters listed in Tab. 2 are determined using the method established in [14] as summarised below.
The capillary permeability k2 and the venule-arteriole permeability ratio k3/k1 are assumed to be given based on previous studies [22, 51];
The ratio of the grey matter coupling coefficients ( and ) can be calculated based on the spatially averaged pressure values in the grey matter (〈pi〉G) using [14] The ratio of the pressure drops is inferred from rodent experiments [67, 68]. Thereafter, both and can be calculated once the cerebral perfusion pressure is estimated [54] and grey matter perfusion is selected.
The remaining two parameters, namely k1 and , are obtained using optimisation [80, 81] relying on the minimalisation of the cost function. The target perfuions in grey and white matter are set to physiologicall realistic values [82]: and [(ml blood)/min/(100 ml tissue)]. Furthermore, a penalty term (Jpenalty) is applied to restrict the minimum and maximum perfusion values: Here, H is the Heaviside function resulting in a non-zero Jpenalty only if the extrema are out of the Fmin,target = 10 and the Fmax,target = 80 [(ml blood)/min/(100 ml tissue)] range.
Footnotes
↵1 Einstein summation convention is not utilised in this study.
References
- [1].↵
- [2].↵
- [3].↵
- [4].↵
- [5].
- [6].
- [7].↵
- [8].↵
- [9].↵
- [10].
- [11].↵
- [12].↵
- [13].↵
- [14].↵
- [15].↵
- [16].
- [17].
- [18].↵
- [19].↵
- [20].↵
- [21].
- [22].↵
- [23].↵
- [24].
- [25].↵
- [26].↵
- [27].↵
- [28].
- [29].↵
- [30].↵
- [31].↵
- [32].↵
- [33].↵
- [34].↵
- [35].↵
- [36].↵
- [37].
- [38].↵
- [39].↵
- [40].↵
- [41].↵
- [42].↵
- [43].
- [44].↵
- [45].↵
- [46].↵
- [47].↵
- [48].
- [49].
- [50].↵
- [51].↵
- [52].↵
- [53].↵
- [54].↵
- [55].↵
- [56].↵
- [57].↵
- [58].
- [59].
- [60].↵
- [61].
- [62].
- [63].↵
- [64].↵
- [65].
- [66].↵
- [67].↵
- [68].↵
- [69].↵
- [70].↵
- [71].↵
- [72].↵
- [73].↵
- [74].↵
- [75].↵
- [76].↵
- [77].↵
- [78].
- [79].↵
- [80].↵
- [81].↵
- [82].↵