Introduction

The brain receives 15–20% of the body’s total cardiac output to maintain its function.6 Each part of the brain is perfused by the Circle of Willis (CoW), which works as the main center to distribute blood flow in the brain. Since hemodynamics in the CoW is related to various cerebrovascular diseases, such as the aneurysms, and stenosis,5,15,18 obtaining detailed CoW flow information can facilitate better understanding of disease progression and contribute to decisions about appropriate treatment. For example, blood flow distribution and pulse waveforms can offer useful references, as well as physiological and pathological information, for disease diagnosis and surgical planning.23,30,37,38 However, acquiring quantitative measurements of detailed blood flow information in the CoW is difficult due to current limitations in the spatial and temporal resolution of noninvasive medical imaging techniques.

To overcome this difficulty, computational fluid dynamics (CFD) is combined with medical imaging data to create a powerful tool for studying hemodynamics in the CoW of an individual patient.4,20,21

A three-dimensional (3D) simulation can provide detailed information about blood flow in the CoW.4,26 However, as the construction of geometry and generation of grids for 3D simulations is both time-consuming and resource intensive, 3D simulation tends to be performed only for a locally diseased area. However, if simulations are performed for case studies to examine postoperative effects on blood flow, the entire cardiovascular system must be considered in order to emulate the alteration of blood flow after surgery. 1D–0D simulations can estimate not only flow distribution but also pulse-wave propagation in the artery network with lower calculation costs than 3D simulations.

Many researchers have studied 1D–0D in the past few decades. Anliker et al. 3 constructed a 1D model of the canine cardiovascular system in 1971. Stergiopulos et al. 35 developed a model of the human cardiovascular system, and used the Windkessel model to represent the cumulative effects of distal vessels. Olufsen et al. 25 proposed an alternative for the Windkessel model by considering the distal vessels as a fractal network. Formaggia et al. 8 and Reymond et al. 29 developed a lumped model for the left ventricle of the heart, and then coupled it with the rest of the cardiovascular system. Alastruey et al. 1 constructed a model to investigate the effects of anatomical variations and occlusions on blood flow in cerebral circulation. Liang et al. 17 modeled the complete heart, including the right atrium, ventricle and left atrium, ventricle with varying elastance, so as to examine the entire cardiovascular system as a closed-loop system. Most of these studies used literature data to investigate general hemodynamics in the cardiovascular system. However, in order to obtain hemodynamic information on an individual patient using a 1D–0D simulation of the CoW, it is necessary to take patient-specific information into account.

We have been developing a 1D–0D simulation based on the method described in Liang et al. 16,17 Liang’s 1D–0D model uses the collected literature data for the different parts of the cardiovascular geometry1,2,6,24,27,29,35,36 and associated physiological parameters,6,22,25,29,35,36,39 such as the resistance and compliance of the blood vessels. In this paper, we introduced the patient-specific arterial geometry of the CoW into the original Liang’s 1D–0D model. When patient-specific geometry is applied to a part of the cardiovascular system, deviations from the literature data occur in the physiological parameters of the 0D simulation. Instead of literature data, some methods, such as the two-area method31 and pulse-pressure method,34 have been proposed calculating compliance and peripheral resistance for 0D simulations. In general, instantaneous pressure and flow rate over cardiac cycles for an individual patient can be used to calculate the physiological parameters. For our study of blood circulation in the CoW, however, it is difficult to measure those values. Thus, these methods cannot be applied in the present study.

In order to find appropriate physiological parameters and adjust deviations caused by literature data, we are developing a new method using SPECT data. Since SPECT data represent a map of flow rates in the peripheral regions, they can be used to determine the physiological parameters in those regions, which are represented by the 0D model in our study. In this paper, deviations in the physiological parameters due to literature data are corrected to minimize differences between the resulting flow rate and the SPECT data in each efferent artery in the CoW; this is accomplished by adjusting the peripheral cerebral resistance in the 0D simulation. The present method is applied to the cases of three patients after surgery to investigate hemodynamics in the CoW.

Governing Equations and Numerical Methods

As illustrated in Fig. 1, the large arteries in the cardiovascular system mainly consist of the ascending aorta, and the upper (cerebral, ECA and upper limb) and lower body (abdominal and lower limb) circulations. The blood flow in large arteries is assumed to be axisymmetric and to be governed by the 1D model. The effects of the capillaries, veins, and heart are realized by the 0D model. The 1D–0D model is formed as a closed-loop system to simulate the blood flow in the entire cardiovascular system.

Figure 1
figure 1

Arterial tree composed of a set of 83 vessels described by the 1D model.

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Figure 2a shows schematic views of the 1D–0D simulation for the entire cardiovascular system. The 1D simulation calculates blood flow passing through the large arteries in the upper and lower body; blood flow in the capillaries and veins are calculated in the 0D part, which is connected to the terminal arteries of the 1D cardiovascular system. Finally, the veins converge at the right atrium and ventricle. After passing through the pulmonary circulation, blood flow is pumped back into the ascending aorta by the left atrium and ventricle and continues to the next cycle. Figure 2b is a detailed description of the peripheral vessel network of the cerebral circulation. The variables R ip, R id and C i0 represent the proximal resistance, distal resistance and compliance of the peripheral arteries, respectively. The peripheral resistance R i  = R ip + R id has a dominant effect on the flow rates in the terminal arteries. In our 1D–0D simulation system, the peripheral arteries of the upper body converge at the capillary level. A similar structure is implemented in the lower body portion.

Figure 2
figure 2

(a) Schematic illustration of the 1D–0D multi-scale models for the entire cardiovascular system (b) A detailed description of the peripheral vessel network of the cerebral circulation.

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Governing Equations

1D Simulation

A large artery can be considered a straight, long, tapered segment with an elastic wall. There are three governing equations in the 1D simulation. The first two are the continuity and longitudinal momentum equations, as follows:

$$\frac{\partial A}{\partial t} + \frac{\partial Q}{\partial z} = 0,$$
(1)
$$\frac{\partial Q}{\partial t} + \frac{\partial }{\partial z}\left( {\frac{{Q^{2} }}{A}} \right) + \frac{A}{\rho }\frac{\partial P}{\partial z} + K_{\rm R} \frac{Q}{A} = 0,$$
(2)

where A, Q and P denote the cross-sectional area, the volumetric flow rate and the pressure, respectively. For the working fluid, blood is assumed to be a Newtonian fluid with a constant density ρ = 1.06 g cm−3 and a kinematic viscosity ν = 4.43 s−1 cm2. The parameter K R is a resistance parameter related to the viscosity of the blood. It is taken as 22πν corresponding to the fixed heart rate of 60 beats per minute in this study.33

The third governing equation describes the relationship between the pressure P and area A, and is given by8,32:

$$P - P_{0} = \frac{{Eh_{0} }}{{r_{0} \left( {1 - \sigma^{2} } \right)}}\left( {\sqrt {\frac{\textit{A}}{{{\textit{A}}_{0} }}} - 1} \right),$$
(3)

where A 0 and h 0 represent the cross-sectional area and wall thickness at the reference configuration, respectively. P 0 means the reference pressure state, E is the Young’s modulus, r 0 is the radius corresponding to A 0, and the Poisson ratio is set as σ = 0.5.

The pulse wave velocity c follows from Eqs. (1) and (2) as8,32:

$$c = \sqrt {\frac{\beta }{{2\rho {\text{A}}_{0} }}} \cdot {\text{A}}^{{\frac{1}{4}}} ,\quad \beta = \frac{{\sqrt \pi Eh_{0} }}{{1 - \sigma^{2} }}.$$
(4)

0D Simulation

As described in Fig. 2b, the 0D model is comprised of an electric circuit series with resistance, compliance, and inductance. The governing equations for the 0D simulation consist of mass and momentum conservation equations based on the electrical circuit analogies as follows:

$$C\frac{{{\textit{{\rm d}P}}_{1} }}{\textit{{\rm d}t}} = Q_{1} - Q_{2} ,$$
(5)
$$L\frac{{{\textit{{\rm d}Q}}_{2} }}{\textit{{\rm d}t}} + P_{2} + {\textit{{\rm R}Q}}_{2} = P_{1} ,$$
(6)

where R, C and L represent arterial and peripheral resistances, volume compliance of the vessels and inertia of the blood, respectively. P 1 represents upstream pressure and Q 1 inflow; P 2 and Q 2 are downstream pressure and outflow, respectively.

Coupling of 1D and 0D Simulations

The Riemann invariants W 1 and W 2 are the characteristic variables of the hyperbolic system for the 1D simulation, defined by8:

$$\frac{\partial }{\partial t}\left( {\begin{array}{*{20}c} {W_{1} } \\ {W_{2} } \\ \end{array} } \right) + \left( {\begin{array}{*{20}c} {\lambda_{1} } & 0 \\ 0 & {\lambda_{2} } \\ \end{array} } \right)\frac{\partial }{\partial z}\left( {\begin{array}{*{20}c} {W_{1} } \\ {W_{2} } \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} 0 \\ 0 \\ \end{array} } \right).$$
(7)

They represent the forward- and backward-traveling waves that contain the flux and pressure information at speed \(\lambda_{1,2} = \frac{Q}{\textit{A}} \pm c\). The solutions for this equation can be obtained by choosing a reference state A = A 0, Q = 0:

$$W_{1,2} = \frac{Q}{\textit{A}} \pm 4(c - c_{0} ),$$
(8)

where c 0 is the wave speed when A = A 0. At interface l between the 1D and 0D simulations, the Riemann invariant W 1,l can be extrapolated through:

$$W_{1,l}^{n + 1} = W_{1} \left( {t^{n + 1} ,l} \right) = W_{1} \left( {t^{n} ,l - \lambda_{1}^{n} \Delta t} \right).$$
(9)

The backward wave of W 2,l at interface l is decided by:

$$W_{2,l}^{n + 1} = W_{1,l}^{n + 1} - 8\sqrt {\frac{\beta }{2\rho }} \left( {\left( {\frac{{A_{l}^{n + 1} }}{{A_{l,0} }}} \right)^{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}}} - 1} \right),$$
(10)
$$A_{l}^{n + 1} = A_{l,0} \left( {\frac{{P^{n} }}{\beta } + 1} \right)^{2} ,$$
(11)

where P n is the pressure at the interface updated by calculation of the 0D simulation. The flow rate boundary condition at the outlet of 1D can be expressed as the following equation:

$$Q_{1D}^{n + 1} = A_{l}^{n + 1} \frac{{W_{1,l}^{n + 1} + W_{2,l}^{n + 1} }}{2}.$$
(12)

Numerical Method

The 1D governing Eq. (3) was substituted into Eq. (2) to eliminate P. Then Eqs. (1) and (2) were solved for Q and A using the two-step Lax–Wendroff method. The calculated A was substituted back into Eq. (3) to obtain P. The 0D governing equations were solved with a fourth-order Runge-Kutta method. At the interface between the 1D and 0D domains, the numerical iteration was performed for Eq. (7) at each time step to obtain the solutions of the Riemann invariants.17

Acquisition of Patient-Specific Data

In this paper, MRA, PC-MRA and SPECT data were utilized in the simulations for patient-specific cerebral circulation. Arterial geometry information, such as the proximal and distal radius, and the vessel length of the CoW were measured with 3D time-of-flight (3D TOF) MRA. The flow rates \(\overline{Q}_{{{\text{PC}} - {\text{MRA}}_i}}\) in the afferent arteries were measured with PC-MRA. MRA was implemented on a 1.5 Tesla MRI system (MAGNETOM Avanto, Siemens, Munich, Germany), with a field of view (FOV) of 200 mm, repetition time/echo time (Tr/Te) = 28/6 ms, and image resolution = 0.5 × 0.5 × 0.8 mm. The same system was used for PC-MRA flow rate measurement (slice thickness = 3 mm, pixel size = 256 × 192, FOV = 140 mm, Tr/Te = 24/5 ms, flip angle= 15°).

SPECT was originally applied to qualitatively assess the physiologic function of the brain.11 By scanning intravenously injected 123I-N-isopropyl-p-iodoamphetamine (123I-IMP) with an autoradiography (ARG) technique, the brain perfusion map of patients can be obtained. After a series of quantitative calculation and mathematical transformations, these data can be used to derive the regional cerebral blood flow (rCBF) and cerebral vascular reactivity (CVR) in response to an acetazolamide challenge.12 In recent years, a new quantitative approach for using the SPECT data has been proposed by Yamada et al.40 The flow rate \(\overline{Q}_{{{\text{SPECT}}_i}}\) in the efferent arteries of the CoW can be calculated as:

$$\overline{Q}_{{{\text{SPECT}}_i}} = rCBF_{i} \times V_{i} \times \rho_{b} ,\quad i = 58, 61, 63, 65, 67, 70.$$
(13)

where i denotes the efferent arterial segment number listed in Table 1. V i represents the vascular territory perfused by each efferent artery, and ρ b = 1.04 g mL−1 is the density of human brain tissue. The SPECT measurement was performed with a Siemens E-Cam rotating gamma camera (Siemens, Munich, Germany). The resolution was 64 × 64 matrix, 9-mm full-width at half-maximum.

Table 1 Arterial geometry of literature data and patient-specific measurement data, and flow rate measurement data in three cases.
Full size table

Application of Patient-Specific Data to 1D–0D Simulation

The 1D simulation required vessel radius and length as the patient-specific geometrical parameters. These parameters were obtained from MRA measurement data using the in-house software “V-modeler”, developed by Kobayashi et al.14

When patient-specific geometry data were applied to the current 1D–0D simulation based on the literature data, some deviations were generated in the physiological parameters (i.e., the peripheral cerebral resistance from the initial literature data1,29). The SPECT data were utilized to correct these deviations. Since the SPECT data give the brain perfusion map of an individual patient, they can represent a map of the peripheral flow rates in the 0D simulation regions, which connect to the corresponding efferent arteries in the 1D simulation regions. From this viewpoint, the SPECT data are considered to contain the nonlinear effects of the cerebral autoregulation.

In the 0D simulation, flow rate in the peripheral artery is mainly affected by peripheral resistance.10,28 In the present method, the peripheral cerebral resistance of each efferent artery of the CoW was adjusted in every cardiac cycle to match the corresponding reference flow rate \(\overline{Q}_{i}^{R}\) given by the SPECT data \(\overline{Q}_{{{\text{SPECT}}_i}}\) as follows:

$$\left. {\begin{array}{*{20}c} {R_{i}^{n + 1} = R_{i}^{n} \cdot \left( {1 - \alpha \cdot \frac{{\overline{Q}_{i}^{R} - \overline{Q}_{i}^{n} }}{{\overline{Q}_{i}^{R} }}} \right)} \\ {R_{ip}^{n + 1} = \frac{1}{5} \cdot R_{i}^{n + 1} } \\ {R_{id}^{n + 1} = \frac{4}{5} \cdot R_{i}^{n + 1} } \\ \end{array} } \right\},$$
(14)
$$\overline{Q}_{i}^{R} = \overline{Q}_{{{\text{SPECT}}_i}} ,\quad i = 58, 61, 63, 65, 67, 70$$
(15)

where i denotes the efferent arterial segment number listed in Table 1. R n+1 i and R n i are peripheral resistances calculated at the n+1th and the nth cycle, respectively; R n+1 ip and R n+1 id are proximal and distal peripheral resistance, respectively, as shown in Fig. 2b. \(\overline{Q}_{i}^{n}\) denotes the calculated averaged flow rate of the i artery at the nth cardiac cycle. The parameter α is a relaxation coefficient which is equal to 0.9 in the present simulation.

Figure 3 summarizes the procedure for optimization of peripheral cerebral resistances in the 1D–0D simulation. The resulting flow rate \(\overline{Q}_{i}^{n}\), based on the peripheral resistance from literature data and the patient-specific arterial geometry, was compared to the reference flow rate \(\overline{Q}_{i}^{R}\) at every cardiac cycle. The peripheral cerebral resistance of the i artery was adjusted to minimize the difference between \(\overline{Q}_{i}^{n}\) and \(\overline{Q}_{i}^{R}\). The peripheral cerebral resistance was determined when the difference between \(\overline{Q}_{i}^{n}\) and \(\overline{Q}_{i}^{R}\) was smaller than a threshold value ε.

Figure 3
figure 3

Procedure for optimization of peripheral cerebral resistance in the 1D–0D simulations.

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The present 1D–0D simulation method determines the total flow rate by the patient’s blood pressure and the total physiological impedance of the entire cardiovascular system. The blood pressure for an individual patient can be measured while the total physiological impedance such as the resistance and compliance can not be measured individually. When the SPECT data are used to adjust the peripheral cerebral resistance so as to minimize the deviation from the original literature data, the method can determine the flow distributions among the arteries in the CoW within the total flow rate governed by the SPECT data. The SPECT data represent the total flow rate in the cerebral peripheral circulation not the total flow rate into the CoW. On the other hand, the PC-MRA data can provide the total flow rate into the CoW by summing up all flow rates in the afferent arteries. Thus, we proposed a method to combine SPECT data with PC-MRA data in order to determine the flow distribution in the CoW by a more patient-specific manner.

First, the flow rate ratios FR i in the efferent arteries were obtained from the SPECT data \(\overline{Q}_{SPECT_i}\):

$${\text{FR}}_{i} = \frac{{\overline{Q}_{{{\text{SPECT}}_i}} }}{{\sum\limits_{i} {\overline{Q}_{{{\text{SPECT}}_i}} } }},\quad i = 58, 61, 63, 65, 67, 70$$
(16)

where the subscript i denotes the efferent arterial segment number.

The time-averaged total inflow rate \(\overline{Q}_{{{\text{CoW}}_{\text{total}}}}\) in CoW for the patient was calculated from the summation of the PC-MRA data in ICAs (i= 40, 47) and BA (i = 56) as follows:

$$\overline{Q}_{{{\text{CoW}}_{\text{total}}}} = \sum\limits_{i} {\overline{Q}_{{{\text{PC}} - {\text{MRA}}_i}} } ,\quad i = 40,47,56$$
(17)

where i denotes the afferent arterial segment number listed in Table 1.

The reference flow rate \(\overline{Q}_{i}^{R}\) was then obtained based on the flow rate ratios FR i and the total inflow rate \(\overline{Q}_{{{\text{CoW}}_{\text{total}}}}\) according to:

$$\overline{Q}_{i}^{R} = \overline{Q}_{{{\text{CoW}}_{\text{total}}}} \cdot FR_{i} .$$
(18)

The peripheral cerebral resistance was adjusted every cardiac cycle according to Eq. (14).

The peripheral resistances of the entire cardiovascular system excluding the cerebral circulation, which are initially referenced from the literature data,35 are adjusted to match the averaged pressure at the upper arm with the measurement data (1/3P_systolic + 2/3P_diastolic).

Patient Selection and Numerical Conditions

Research subjects were two male (Case 1, Case 2) and one female (Case 3) patients. The patient in Case 1 was 70 years old with 73% stenosis at Lt.ICA and carotid artery stenting (CAS). The patient in Case 2 was 79 years old with occlusion at the Lt.MCA; he had undergone a double bypass. The patient in Case 3 was 79 years old with 98% stenosis at Lt.ICA and carotid endarterectomy (CEA).

Since the simulations in this paper focused on behavior of blood flow, particularly in the CoW after the surgery, patient-specific simulations were conducted using measurement data collected after surgery. Table 1 summarizes the patient-specific arterial geometry information of the CoW from MRA using the V-modeler, and the mean flow rate measured by SPECT and PC-MRA. The extracted arterial geometries of the CoW were used for 1D simulation, while the geometries of other arteries in the cardiovascular system were based on literature data from Stergiopulos et al.35 and Olufsen et al.24. In order to compare with the patient-specific measurement of the arterial geometry of the CoW, the literature data from Fahrig et al.7 Devault et al.6 and Reymond et al.29 were also summarized as Lit as shown in Table 1. Although radius and length in the distal regions of the ICAs could be obtained from the MRA data, those in the proximal region of ICAs were not available from the MRA data. Thus, they were from the literature data.24,35 Note that Cases 1 and 2 have the entire CoW, but Case 3 had hyperplasia in both PCoAs, which are missing. Systolic/diastolic blood pressure in the three cases was 107/70, 106/90, and 125/60 mmHg, respectively. All the patient-specific data were provided by the Hamamatsu Rosai Hospital (Shizuoka, Japan). They were acquired with the patient consent as well as with the approval by the ethical committee for human research. The anonymized patient medical image data were used in the simulation after the personal information of the patients were removed from the medial image data by the hospital.

In order to investigate the effects of the present method utilizing SPECT data, as well as SPECT data combined with PC-MRA data, the following three types of simulations were performed: (1) Sim-Orig, (2) Sim-SPECT and (3) Sim-PS.

In Sim-Orig, the peripheral cerebral resistance were not tuned but the ones of the original Liang’s 1D–0D model16 were assigned for the 0D simulation. In Sim-SPECT, peripheral cerebral resistance was adjusted to minimize the difference between the resulting flow rate and the reference flow rate given by the SPECT data in each efferent artery of the CoW. In Sim-PS, the reference flow rate was obtained by combining the SPECT data with the PC-MRA data (the flow ratios in the efferent arteries were from the SPECT data, and the total flow rate was from the PC-MRA data).

The calculations were performed on the cluster nodes with Xeon® 3.3 GHz, 4 GB RAM. The 1D–0D simulation converged after about 10 cardiac cycles’ calculation regardless of the simulation methods. Sim-SPECT and Sim-PS did not show any increase in the calculation time when comparing to Sim-Orig. The averaged computation time for each case is about 50 min.

Results

The flow distribution results of simulations for three cases are summarized in Fig. 4. For all patients, flow distribution results in the CoW were the same (with less than 0.05% difference) between Sim-SPECT and Sim-PS. Since the same flow rate ratios based on SPECT data in the efferent arteries were used in the Sim-SPECT and Sim-PS, there was little difference in the flow distribution of the CoW for these two simulations. On the other hand, since the flow rates in the efferent arteries were mainly determined by the peripheral cerebral resistance from the literature data in Sim-Orig, 0.1–3.8% difference can be observed between the results of the flow rate ratios of Sim-SPECT, Sim-PS and those of Sim-Orig.

Figure 4
figure 4

Comparison of the flow distribution in the CoW between the simulation results of Case 1/2/3.

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The calculated flow rates in the afferent arteries of the three types of simulation were compared to those of the PC-MRA data, as shown in Fig. 5. In all three cases, Sim-PS showed better agreement with the PC-MRA data than did Sim-Orig and Sim-SPECT. In particular, for hyperplasia like that in Case 3, Sim-PS showed significant improvement over Sim-Orig and Sim-SPECT.

Figure 5
figure 5

Comparison of flow rates in the afferent arteries of CoW between the simulation results and the PC-MRA measurements: (a) Case 1; (b) Case 2; (c) Case 3.

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In order to examine how the peripheral cerebral resistances of the efferent arteries of the CoW changed in association with changes in the geometries, the ones adjusted in Sim-SPECT and Sim-PS are summarized in Table 2a, together with the ones from the literature data in Sim-Orig. In Table 2b, the differences between the adjusted peripheral cerebral resistances and the peripheral cerebral resistances from literature data were calculated as ((APR-LitPR)/LitPR), where APR and LitPR stand for the adjusted peripheral cerebral resistances and peripheral cerebral resistances from literature data, respectively.

Table 2 (a): Peripheral resistance from literature data and adjustment in three cases; (b): Difference between the adjusted peripheral resistances and the peripheral resistances from literature data.
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Since 1D–0D simulation can provide information on the transient behavior of pressure and flow rate, pressure and flow rate waveforms were compared among Sim-Orig, Sim-SPECT and Sim-PS for Case 1, as illustrated in Figs. 6 and 7. Note that only Case 1 is discussed because all three cases showed similar results. In order to investigate the pressure and flow rate waveforms in the afferent and efferent arteries, those in the Rt.MCA and Lt.ICA were representatively depicted over one cardiac cycle of 1.0 s. Similar pressure waveforms were found among the three simulations (Sim-Orig, Sim-SPECT and Sim-PS) except that the peaks of the waveforms varied slightly, as shown in Figs. 6a and 6b. With the instantaneous flow rates shown in Figs. 7a and 7b, the shapes of the flow rate remained similar among the three simulations, while the magnitudes varied.

Figure 6
figure 6

Comparison of the pressure waveforms at various artery locations in Case 1: (a) Rt.MCA; (b) Lt.ICA.

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Figure 7
figure 7

Comparison of the flow rate waveforms at various artery locations in Case 1: (a) Rt.MCA; (b) Lt.ICA.

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Discussion

The comparison of flow rates among Sim-Orig, Sim-SPECT and Sim-PS in Cases 1–3 (Fig. 5) demonstrates that the flow rates in the afferent arteries calculated by the method with combined SPECT and PC-MRA data (Sim-PS) showed better agreement with the PC-MRA data than the others, even for the patient with hyperplasia. If flow distribution in the CoW is the main interest, the simulation with SPECT data alone gives sufficient information, since the same flow distribution can be obtained for both SPECT data only (Sim-SPECT) and SPECT data combined with PC-MRA data (Sim-PS), as indicated in Fig. 4. The PC-MRA data can be used as the reference data to help determine the total flow rate of the blood flow into the cerebral circulation. Thus, Sim-PS showed better agreement in the distribution of blood flow as well as the magnitude of flow rates in the CoW comparing to Sim-Orig and Sim-SPECT as shown in Fig. 5.

If the arterial geometry of the patient’s CoW is similar to the literature data, the deviations in the peripheral cerebral resistance of Sim-PS from Sim-Orig will be small. In Case 2, the APR of Sim-PS show little difference from the LitPR, as indicated in Table 2b. Thus, the flow rates in BA and ICAs showed similar values between Sim-PS and Sim-Orig, as described in Fig. 5b. As may be observed in Table 1, the arterial geometries in Case 2 are close to the literature data, particularly for BA and ICAs. A linear least squares regression for the vessel radius of the CoW between Case 2 and the literature data gave a slope of 0.91, an intercept of 0.06 mm, and a correlation coefficient r = 0.77. On the other hand, in Cases 1 and 3, the slopes were 0.72 and 0.65, the intercepts were 0.42 and 0.35, and the correlation coefficients were r = 0.79 and 0.81, respectively. Since resistance is more sensitive to radius than length, the similarity in radius between the patient arterial geometry and literature data led to less difference between APR and LitPR in Case 2 as shown in Table 2b. In Case 2, the comparison of the peripheral resistances between Sim-PS and Sim-Orig showed that the deviation in Lt.MCA was 53.42%, and ones at other places were less than 13.26%. In Case 1, the deviations were more than 43.89% while in Case 3 the deviations were more than 38.16% in all locations. When the patient arterial geometries are very different in size and configuration from the literature data such as Case 1 and Case 3 with hyperplasia, the present method using the PC-MRA data combining with the SPECT data (Sim-PS) is effective to provide the flow distribution as described in Figs. 5a and 5c comparing to Sim-Orig.

As for the pressure waveforms, the results from Sim-Orig, Sim-SPECT and Sim-PS show similar patterns over one cardiac cycle (Fig. 6). However, the wave reflection caused by the difference in peripheral cerebral resistance leads to a slight difference near the peak systole. The same parameters of compliance and inductance were utilized in the 1D–0D system for the three different simulations; therefore a similar shape was observed for the pressure waveforms. The decrease in peripheral cerebral resistance brought a smaller reflection.19 As summarized in Table 2a, the APR for both Sim-SPECT and Sim-PS in Case 1 were lower than the LitPR. Therefore, a relatively smaller reflection in Sim-PS and Sim-SPECT was observed from the lower peaks of the pressure waveforms compared with those in Sim-Orig.

The shape of the flow rate waveforms was not affected by the adjustment of peripheral cerebral resistance. The 1D–0D simulation in the paper calculated the total volume of blood flow from blood pressure and entire resistance. Thus, adjustment of peripheral cerebral resistance affected the magnitude of the flow rate but not the shape of the waveforms, as shown in Fig. 7.

The detailed information on the flow rate and pressure from our simulations can provide useful information to the clinicians to assist their diagnosis and to avoid the surgical complications such as the cerebral hyperperfusion syndrome (CHS). The CHS is one of the major causes of the significant morbidity and mortality following the surgeries for the stenosis. According to the study reports, CHS can occur several hours to several days after the surgery. Some researchers regarded the impaired cerebral autoregulation due to the increase of the total cerebral blood flow (>100% increase from baseline) as the main cause of CHS.16,37 But some other researchers found CHS can also happen to the patients with moderate increase in the total cerebral blood flow.9,13 Thus, it is important to obtain the information of the flow distribution in the CoW. The present 1D–0D model enables us to perform the case study for an individual patient to provide the flow distribution in the CoW within a reasonable computational time.

The SPECT data were originally used to qualitatively assess CVR by comparing rest and acetazolamide-challenged states. Thus, utilization of the SPECT data is effective in determining the ratio of flow distribution in the CoW, but may have a limit to determining the flow rate. This can be improved by incorporating the PC-MRA data into the SPECT data. However, the further investigation is needed. In order to advance the application of our method to clinical studies, it will be important to investigate the error propagation by using the SPECT data to adjust peripheral cerebral resistance. It will also be important to develop a method for estimating the peripheral cerebral resistance of an individual patient without SPECT data, since the SPECT data are not easy to obtain.

Since our methods require different kinds of measurement data such as MRA, PC-MRA and SPECT data, the patient information with all these measurement data are difficult to be obtained. All three cases in our current research are patients who were treated with the stenosis surgeries. And the simulations were based on the measurement data collected after the surgeries. It would be necessary to validate our method with subjects who have more diverse characteristics, such as with regard to their pre/postoperative situations, different stenosis locations and different diseases.

Conclusions

In order to obtain hemodynamic information in the CoW for individual patients, we developed a 1D–0D simulation of the entire cardiovascular system utilizing patient-specific arterial geometry and flow-rate measurement data. Vessel radius and length from the MRA data were prescribed as the geometrical parameters of the 1D simulation. A new numerical method was proposed to optimize the peripheral cerebral resistance in the 0D simulation using the SPECT data. The present method was applied to three cases.

The simulation results showed that application of the SPECT data alone could provide the flow distribution in the CoW for an individual patient. Combining the SPECT data with PC-MRA data, we could further obtain flow distribution with the magnitude of flow rate in the afferent arteries of the CoW. We also investigated the effects of peripheral cerebral resistance adjustments on the waveforms. The wave reflection caused by the different peripheral cerebral resistances led to slight differences at the peak of the pressure waveforms. A change in a peripheral cerebral resistance affected the magnitude but not the shape of the flow rate waveform.

This paper demonstrates that SPECT data can be used independently or in combination with other measurement data to provide hemodynamic information for individual patients. The simulation results show that the proposed numerical method is useful for the optimization of peripheral cerebral resistance.