One assumption of DSC-MRI is that the injected contrast agent is kept totally intravascular and the arterial wall is impermeable to contrast agent. The assumption is unreal for such small contrast agent as Gd-DTPA can leak into the arterial wall. To investigate whether the unreal assumption is valid for the estimation of the delay and dispersion of the contrast agent bolus, we simulated flow and Gd-DTPA transport in a model with multilayer arterial wall and analyzed the bolus delay and dispersion qualified by mean vascular transit time (MVTT) and the variance of the vascular transport function. Factors that may affect Gd-DTPA transport hence the delay and dispersion were further investigated, such as integrity of endothelium and disturbed flow. The results revealed that arterial transmural transport would slightly affect MVTT and moderately increase the variance. In addition, although the integrity of endothelium can significantly affect the accumulation of contrast agent in the arterial wall, it had small effects on the bolus delay and dispersion. However, the disturbed flow would significantly increase both MVTT and the variance. In conclusion, arterial transmural transport may have a small effect on the bolus delay and dispersion when compared to the flow pattern in the artery.
Dynamic susceptibility contrast magnetic resonance imaging (DSC-MRI) has been shown to be a powerful technique to qualify cerebral blood flow and is playing an increasing role in diagnosis of acute ischemic stroke [
Cerebral blood flow quantification requires knowledge of the arterial input function that is the concentration of contrast agent in the feeding vessel to the tissue of interest. In theory the AIF should be measured at the feeding vessel close to the tissue of interest [
Another implicit assumption of the technique of DSC-MRI related to the estimation of AIF is that no or a negligible amount contrast agent penetrates the arterial wall, due to the fact that, in general, the vascular wall is considered irrelevant for the contrast mechanism in DSC-MRI. The assumption is tenable for the intact blood brain barrier (BBB) [
To investigate the delay and dispersion effects of a bolus of contrast agent, two approaches were usually applied. One that has been commonly used was convolving the estimated AIF with a vascular transport function (VTF). The VTFs were quite differently assumed, ranging from the simple model of a single-exponential to the more sophisticated model of a feeding artery in series with small parallel vessels [
In the present study, to investigate whether the unreal basic assumption of DSC-MRI is valid for the estimation of the delay and dispersion of the contrast agent bolus, we formulated the arterial wall as a five-layer model and numerically simulated the flow and the transport of contrast agent in the model using CFD. This five-layer model included the endothelial glycocalyx layer (EGL), the endothelium, the intima, the internal elastic lamina (IEL), and the media, which were all treated as macroscopically homogeneous porous media. The effects of different factors that may affect contrast agent transport such as the integrity of endothelium and the disturbed flow after the stenosis of artery on the delay and dispersion of contrast agent were further analyzed.
As a basic geometrical model, the arterial segment concerned was simplified as a straight axisymmetric cylinder. The inner diameter of model
Schematic illustration of the computational geometry and boundary conditions. (a) The computational geometry of the stenosed model; (b) five-layer arterial wall with the thickness of each layer illustrated in the parentheses; (c) flow waveform at the inlet. EGL: endothelial glycocalyx layer; IEL: internal elastic lamina;
For the stenosed model, the variation of the vessel radius along the stenosis was described using a cosine function and the reduction in the cross-sectional area of the lumen was set as 75% [
The flow simulation in the lumen of the arterial segment was based on the incompressible Navier-Stokes and continuity equations:
The mass transport of contrast agent Gd-DTPA in the flowing blood can be described by
The transmural flow across the arterial wall can be described by the Brinkman equation as follows [
The transport of Gd-DTPA across the arterial layers was modeled by the following equation [
In order to solve (
Model parameters for each layer.
EGL | Endothelium | Intima | IEL | Media | |
---|---|---|---|---|---|
Hydraulic permeability ( |
6.0383 × 10−18 | 1.7383 × 10−20 | 4.2 × 10−17 | 8.4 × 10−20 | 6.09 × 10−19 |
Effective diffusivity ( |
1.0128 × 10−10 | 3.0276 × 10−13 | 1.2219 × 10−10 | 2.4094 × 10−13 | 9.3392 × 10−14 |
Reflection coefficient ( |
0.0555 | 0.1212 | 0.2514 | 0.2514 | 0.3617 |
Porosity ( |
0.6735 | 0.0005 | 0.8025 | 0.002 | 0.258 |
As shown in Figure
The boundary conditions for the mass transport equations (( where
The numerical simulations were carried out using a validated finite element algorithm COMSOL Multiphysics. Three full cardiac cycles (3 s) simulation of the pulsatile flow with a time step of 4 ms were carried out to achieve a periodic flow independent of the initialization. Based on the initial velocity field, the mass transport equations were solved coupling with flow transport equations for 37 s with time step from 1 ms to 4 ms depending on the temporal variations of contrast concentration.
The effect of the dispersion of the contrast agent bolus on the AIF can be described by convolving the estimated AIF with a vascular transport function (VTF) [
To demonstrate the deformation of the concentration-time curve along the flow direction in the unstenosed model, the area weighted average of the concentration of contrast agent on cross-sections at axial direction of
The concentration-time curves at inlet,
The concentration-time curves across the arterial wall at the axial direction of
The development of the bolus delay and dispersion in the unstenosed model are quantified by the mean vascular transit time (MVTT) and the variance of the VTF from (
Comparison of the quantitative parameters of the bolus delay (MVTT) and dispersion (variance) in the unstenosed artery between the model with arterial transmural transport and without. (a) The mean vascular transit time (MVTT); (b) the variance of vascular transport function (variance); (c) the difference and percentage difference in variance.
When the endothelium is damaged in diseased condition, the transport of the contrast agent may be affected and hence the bolus dispersion. To investigate the role of endothelium in the transport of the contrast agent, the endothelium was assumed to be totally damaged and the four transport parameters of the damaged endothelium and EGL were simplified to be similar to that of the intima. In addition, the transport parameters of other layers were consistent with that of the model with intact endothelium.
Figures
Effects of endothelium on contrast agent transport and the bolus dispersion in the unstenosed model. (a–d) Comparison of the concentration-time curve across the arterial wall between the damaged endothelium model and the intact one at the axial direction of
As illustrated in Figures
Until now, the numerical simulations have been carried out only for a simplified straight axisymmetric blood vessel. However, the geometry of the physiological blood vessel is much more complex with such characteristics as branching, twisting, taper, and curvature, which would lead to far more than parabolic flow profile in the simplified model but very complicated flow patterns [
Figure
Effects of disturbed flow on contrast agent transport. (a) Concentration distribution in the stenosed region at three time points of the contrast injection at the inlet.
Figures
In the stenosed model the influence of the disturbed flow on the bolus delay and dispersion was observed immediately behind the stenosis (Figure
Comparison of the quantitative parameters of the bolus delay (MVTT) and dispersion (variance) in the stenosed artery between the model with arterial transmural transport and without. (a) The mean vascular transit time (MVTT); (b) the variance of vascular transport function (variance).
One unreal assumption of DSC-MRI is that the arteries are impermeable to contrast agent. In the present study, we numerically coupled contrast agent transport in the arterial wall with that in arterial lumen to investigate whether the unreal assumption is valid for the estimation of the delay and dispersion of the contrast agent bolus. The results obtained reveal that the arterial transmural transport would slightly affect the bolus delay qualified by MVTT and moderately increase the bolus delay qualified by the variance. The MVTT and the variance induced by the arterial transmural transport are much less than that by the disturbed flow after the stenosis. In addition, although the integrity of endothelium can significantly affect the accumulation of contrast agent in the arterial wall, it has small effects on the bolus delay and dispersion.
The bolus dispersion would lead to a systematic blood flow underestimation. Graafen et al. simulated the dispersion in coronary arteries using a computational fluid dynamics approach and demonstrated that the variance between 1.0 and 2.5 s2 would lead to the myocardial blood flow underestimation between about 6% and 10% [
As the transport of contrast agent is mainly governed by the convective transport in the lumen and the diffusion transport in lumen and the arterial wall, the local flow pattern, the flow rate, and the arterial transmural transport would affect the degree of bolus dispersion. Among the three factors, the arterial transmural transport may be the least important one, since the present simulation indicated that the disturbed flow produced much more dispersion of the bolus than the arterial transmural transport. However, the contribution of local flow pattern and the flow rate to the dispersion is comparable. Our simulations demonstrated the disturbed flow induced by the stenosis would significantly increase the bolus dispersion. In contrast, Graafen et al. indicated that stenosis in the coronary arterial model leads to a reduction of dispersion [
This study revealed that local flow pattern caused by the stenosis would significantly affect the local contrast agent transport and hence the local bolus dispersion, which is constant with mass transport simulations based on the patient-specific model. For instance, Liu et al. demonstrated that the disturbed flow would significantly hinder the mass transport in the human aorta and computational studies indicated that the oxygen transport was significantly low in the outer wall of carotid artery where disturbed flow developed [
In this pilot study, the accuracy of the results may be reduced by the simplifications of the transport parameters, the blood rheological properties, and the geometries.
Due to the scarcity of the transport parameters, most of the parameters were obtained from theoretical models. The estimated parameters, especially the diffusivity of the contrast agent, would affect the simulation results. The diffusivity used in the present study (6.5847 × 10−11 m2 s−1) is lower than the estimated diffusion coefficient of contrast agent in the previous studies (5.5 × 10−10 m2 s−1 and 1.5 × 10−10 m2 s−1) [
The blood in the present study was simplified as Newtonian fluid and the non-Newtonian rheological properties such as shear thinning nature of the blood were neglected. It is reported that the shear thinning non-Newtonian nature of blood could slightly reduce oxygen flux (similar micromolecule as contrast agent) in most regions of the arteries, and this effect became much more apparent in areas with disturbed flow [
Another limitation of the present study is that all simulations were performed on simplified geometries. The geometry of the physiological blood vessel is very complex with such characteristics as branching, twisting, taper, and curvature, which would lead to complicated flow patterns. Therefore, it is necessary to use the realistic geometries to simulate the exact dispersion of the contrast agent bolus. As the main aim of the present study is not to estimate the exact dispersion errors in an individual patient, the results obtained from the typical parabolic flow and disturbed flow can still shed some light on the effects of arterial transmural transport of the contrast agent on the bolus delay and dispersion.
The arterial transmural transport would slightly affect the bolus delay and small increase the bolus dispersion, which may have small effects on the estimation of the blood flow estimation. In comparison with disturbed flow induced by the presence of stenosis, the arterial transmural transport plays a less important role in the bolus delay. Although the assumption of impermeable arterial wall of DSC-MRI is unreal, it may still be a good simplification for the estimation of the delay and dispersion of the contrast agent bolus.
The EGL layer of the arterial wall was assumed to be composed of symmetrical EGL fibers and leaky junctions of cells resulting from either dying or being in mitosis. According to the results by Liu et al. [
The total filtration reflection coefficient of the EGL layer is determined by
In (
The effective diffusivity (
Filtration flow can move through the endothelium via both normal junctions and leaky junctions of the endothelial cells. Therefore, the hydraulic permeability
The reflection coefficient of the normal junction is defined as
Using the results from Dabagh et al., the porosity and hydraulic permeability
The IEL is assumed to have a constant thickness with fenestral pores. It seems that the fenestral pores were filled with the same fiber matrix as intima [
The media are modeled as a porous medium composed of smooth muscle cells with a hydraulic permeability of
The effective diffusivity of the media is obtained from the following equation:
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is supported by Grants-in-Aid from the National Natural Science Research Foundation of China (nos. 31200703, 11228205, 10772019, 11332003, 11421202, and 61190123) and Specialized Research Fund for the Doctoral Program of Higher Education (20121102120038).