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980 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 70, NO. 3, MARCH 2023
Fluid Mechanics Approach to Perfusion
Quantification: Vasculature Computational Fluid
Dynamics Simulation, Quantitative Transport
Mapping (QTM) Analysis of Dynamics Contrast
Enhanced MRI, and Application in Nonalcoholic
Fatty Liver Disease Classification
Qihao Zhang , Xianfu Luo, Liangdong Zhou, Thanh D. Nguyen , Martin R. Prince,
Pascal Spincemaille , and Yi Wang
Abstract—Objective: We quantify liver perfusion using
quantitative transport mapping (QTM) method that is free
of arterial input function (AIF). QTM method is validated in a
vasculature computational fluid dynamics (CFD) simulation
and is applied for processing dynamic contrast enhanced
(DCE) MRI images in differentiating liver with nonalcoholic
fatty liver disease (NAFLD) from healthy controls using
pathology reference in a preclinical rabbit model. Methods:
QTM method was validated on a liver perfusion simulation
based on fluid dynamics using a rat liver vasculature model
and the mass transport equation. In the NAFLD grading
task, DCE MRI images of 7 adult rabbits with methionine
choline-deficient diet-induced nonalcoholic steatohepatitis
(NASH), 8 adult rabbits with simple steatosis (SS) were
acquired and processed using QTM method and dual-input
two compartment Kety’s method respectively. Statistical
analysis was performed on six perfusion parameters: ve-
locity magnitude |u|derived from QTM, liver arterial blood
flow LBF a, liver venous blood flow LBF v, permeability
Ktrans, blood volume Vpand extravascular space volume
Veaveraged in liver ROI. Results: In the simulation, QTM
method successfully reconstructed blood flow, reduced er-
ror by 48% compared to Kety’s method. In the preclinical
study, only QTM |u|showed significant difference between
high grade NAFLD group and low grade NAFLD group.
Conclusion: QTM postprocesses DCE-MRI automatically
through deconvolution in space and time to solve the in-
verse problem of the transport equation. Comparing with
Kety’s method, QTM method showed higher accuracy and
Manuscript received 6 June 2022; revised 2 August 2022; accepted
7 September 2022. Date of publication 15 September 2022; date of
current version 20 February 2023. (Qihao Zhang and Xianfu Luo con-
tributed equally to this work.) (Corresponding author: Yi Wang.)
Qihao Zhang, Liangdong Zhou, Thanh D. Nguyen, Martin R. Prince,
and Pascal Spincemaille are with the Department of Radiology, Weill
Medical College of Cornell University, USA.
Xianfu Luo is with the Northern Jiangsu People’s Hospital, Yangzhou,
China.
Yi Wang is with the Department of Radiology, Weill Medical Col-
lege of Cornell University, New York, NY 10065 USA (e-mail: yw233@
cornell.edu).
Digital Object Identifier 10.1109/TBME.2022.3207057
better differentiation in NAFLD classification task. Signifi-
cance: We propose to apply QTM method in liver DCE MRI
perfusion quantification.
Index Terms—Kinetic modeling, liver perfusion quantifi-
cation, magnetic resonance imaging, quantitative transport
mapping (QTM).
I. INTRODUCTION
DYNAMIC contrast enhanced MRI (DCE-MRI) captures
the passage of contrast agent in tissues, which can reflect
perfusion, vessel volume, and permeability information [1],
[2]. Traditional quantitative analysis of DCE-MRI is based on
Kety’s equation [3], also referred to as Toft’s model in MRI
[4], where blood flow F and blood volume Vpcan be calculated
if arterial input function (AIF) is known [5]. Although various
studies have revealed it’s promising to use this traditional tracer
kinetic modeling method in lesion classification, tumor grading
and treatment response [6], [7], [8], Kety’s model hasn’t been
commonly applied in clinical practice yet [9], [10], [11], largely
due to variation in kinetic modeling [4], [5] and dependency
of kinetic parameters on the choice of AIF [12]. Particularly,
AIF for a voxel is not well defined in principle, as there may be
several vessels input into that voxel, and it is nearly impossible
to estimate an AIF for a voxel from DCE-MRI data in principle
[13], [14], [15]. In practice, a single global AIF is often assumed
to supply all voxels, ignoring voxel-level deviations from the
global AIF in terms of dispersions; the AIF estimation from
DCE-MRI suffers from partial volume effects and varies highly
with the choice of location for AIF estimation [12].
The regional tissue blood flow can be determined by inverting
the transport equation in a fully automated manner without any
AIF input which is termed as quantitative transport mapping
(QTM) [15], [16], [17]. Using a vascular tree CFD simulation
to validate quantitative tissue perfusion, QTM is shown to be
substantially more accurate than Kety’s approach for kidney
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ZHANG et al.: FLUID MECHANICS APPROACH TO PERFUSION QUANTIFICATION 981
perfusion quantification [15]. Correlating with immunohisto-
chemistry, QTM processing of DCE-MRI provides more signif-
icant resolutions of pathological markers than Kety’s approach
in nasopharyngeal carcinoma [18] and in breast cancer [19].
In this study, we intend to establish QTM processing of DCE
MRI for liver perfusion quantification. To validate QTM method,
we extend the vascular tree model for contrast agent transport
simulation to a liver vasculature-tissue model [15]. To access
potential clinical value of voxel average velocity |u|derived
from QTM, we compare both QTM and Kety’s method with
pathological classification of nonalcoholic fatty liver disease
(NAFLD), the leading cause of chronic liver disease [20], [21],
[22], [23], [24].
II. MATERIALS AND METHODS
We organize this section into the following subsections: A.
review both Kety’s and QTM postprocessing methods. B. Vali-
dation of Kety’s and QTM method against CFD simulated flow
as ground truth using CFD simulated tracer concentration as
input. C. Application in NAFLD grading task.
A. Postprocessing Methods For DCE MRI
Two postprocessing methods were implemented, traditional
Kety’s method using a global arterial input function, and quan-
titative transport mapping (QTM).
1) Traditional Kety’s Method: One-compartment kinetic
modeling method was implemented in simulation:
∂c(ξ,t)
∂t =LBF(ξ)ca(t)−1
V(ξ)c(ξ,t)(1)
where t∈{12,...N
t−1}the time index with Ntas the number
of time frames, ξ=(ξx,ξy,ξz)is voxel index in a volume
of (Nx, ,N
y,N
z)voxels along (x, y, z)axis, ∂tis the time
derivative, ca(t)is the tracer concentration of feeding artery
(global AIF), LBF is liver blood flow, V(ξ)is the volume fraction
of vascular space, and c(ξ,t)is the tracer concentration scalar
field. Eq. (1) is a linear equation system for LBF and LBF
V, and
LBF can be solved using linear least squared method [25]. All the
reconstruction is performed using MATLAB R2018a (Natick,
Massachusetts: The MathWorks Inc.).
A two-compartment exchange model with dual inputs was
also implemented for the liver DCE-MRI:
∂Vpcp(t)
∂t =LBFaca(t)+LBFvcv(t)−Kepcp(t)
−Ktranscp(t)+Ktr ansce(t)(2)
∂Vece(t)
∂t =Ktrans cp(t)−Ktrans ce(t)(3)
Here ca(t)and cv(t)are arterial and portal vein input function,
LBFaand LBFvare arterial and portal venous blood flow,
Ktrans is exchange rate between vascular and extravascular
space, Vpand Veare volume fraction of vascular and extravascu-
lar space, c(t)=Vpcp(t)+Vece(t)is the tracer concentration
in the voxel, and Kep =LBFa+LBFvis the outflux rate. Eqs.
(2) and (3) was solved using linear least squared method [25].
2) Quantitative Transport Mapping: QTM method is used
to reconstruct the flow velocity from the 4D contrast agent
concentration profile. Given a time resolved 4D DCE-MRI data,
its contrast agent concentration can be calculated by assuming a
linear relationship between signal intensity change and contrast
agent concentration [26]. In quantitative transport mapping,
tracer concentration profile is modeled by transport equation
[15], [27]:
−∇ · c(ξ,t)u(ξ)+∇·D(ξ)∇c(ξ,t)=∂tc(ξ,t).(4)
Here ∇=(∂x,∂
y,∂
z)the gradient operator, u(ξ)=
(ux(ξ),u
y(ξ),u
z(ξ)) is an average velocity vector field with
magnitude |u(ξ)|=ux(ξ)2+uy(ξ)2+uz(ξ)2, and D(ξ)
the diffusion coefficient scalar field [26]. Both time derivative
and gradient operator are difference operations in the discretized
4D spacetime-resolved image space. For DCE MRI scans that
the images are acquired within 5 minutes after injection, the
diffusion effect is much smaller than convection effect and
cannot be accurately reconstructed from Eq. (4), therefore can be
neglected [27]. Eq. (4) is a linear partial differential equation.
The velocity is solved from an optimization problem with L1
total variation regularization as in a recent QTM study with the
regularization parameters λ=10
−3chosen according to the
L-curve method [15] :
u=argmin
u
Nt−1
t=1
∂tc+∇·cu2
2+λ∇u1.(5)
Comparing with traditional kinetic modeling method, QTM
method doesn’t require AIF as input. In numerical simulation,
uis converted to flow by multiplying with voxel cross section
area and vascular space volume and compared with ground truth
flow. In DCE MRI processing, we report nonalcoholic fatty liver
disease and simple steatosis classification accuracy of uand
traditional kinetic modeling output.
B. Validation Against CFD Simulation Ground Truth of
Contrast Agent Transport in Liver
1) Liver Perfusion Simulation Based on Fluid Dynamics:
Our liver perfusion model contains three parts: supplying vascu-
lar system part (hepatic artery and portal vein, denoted by SVS),
capillary system and homogenized hepatic space part (denoted
by HHS) and draining vein system part (hepatic vein, denoted
by DVS). For SVS and DVS, each artery and vein is assumed to
a cylinder with parabolic flow running through, and flow rate
is determined by Poiseuille’s law. For HHS, capillary space
and extravascular space are modeled as homogeneous porous
media, and the flow velocity is determined by Darcy’s law.
Contrast agent concentration can then be simulated using CFD
model given SVS&DVS structure, HHS volume and boundary
concentration at artery inlet. In this study, we used vasculature
and tissue volume from a 35um3resolution rat liver micro-CT
scan, consisting of SVS and DVS geometry from the root of the
vessel to its 11th branches [28]. Rat liver vasculature has similar
branching pattern and perfusion value with human liver [29],
[30], [31], therefore is commonly used for liver perfusion and
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982 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 70, NO. 3, MARCH 2023
Fig. 1. Vessel segments for simulated liver blood flow.
drug delivery study. Details of the CFD simulation are described
as follows, and all the computations below were performed on
a computer with an Intel i7-8700K 6-core CPU and 64 GB
memory:
Assuming a cylinder shape for each vessel segment and a
parabolic flow through the SVS and DVS vascular network,
velocity and flow in each vessel segment and pressure at each
connecting node can be calculated based on Poiseuille’s law and
boundary condition [32]:
j connected to i
πd4
ij
128μlij
(Pj−Pi)=0, i is branching point
(6)
πd4
ij
128μlij
(Pj−Pi)=F0, i is terminal point (7)
Here irepresents one specific node, jis the node connected to
i,dij and lij are the diameter and length of the segment between
iand j, and μ=3×10−3Pa·sis blood viscosity. If iis
a branching point, the net flow through the node should be 0
(flow conservation). And if iis a terminal point, a constant flow
F0as boundary condition is implied. In this study, we assumed
a constant flow rate of 0.4 mL/s at the root of SVS and DVS,
and assumed the flow is evenly distributed to the terminals of
SVS and DVS. The simulated average perfusion in liver is 62
mL/100g/min. The vasculature and flow in each segment are
shown in Fig. 1.
After pressure at each connecting node is calculated, the
flow of the segment between node iand jcan be calculated
as Fij =πd4
ij
128μlij (Pi−Pj), and the velocity at each point in
the segment can be expressed as uij (r)= 2Fij
πR2
ij
(1 −r2
R2
ij
).
Here, Rij is the radii of the segment and ris the distance of
the point to the axis of the cylinder. Tracer propagation in the
vascular network can be simulated based on transport equation
using finite element method [33]. To decrease the memory cost
because of saving and computing the tracer concertation in
all elements at each time step, 1D plug flow assumption is
commonly used to simplify the computation [34]. However,
plug flow assumption may introduce error when solving tracer
propagation. We propose a tracer propagation approximation
based on parabolic flow as follows:
Fig. 2. (a) liver blood flow in hepatic space. (b) AIF used in CFD sim-
ulation (red solid line) interpreted from measured AIF in rabbit hepatic
artery (red stars); simulated hepatic tissue enhancement (green line)
comparing with measured hepatic tissue enhancement (green stars) in
a rabbit DCE MRI.
Assuming parabolic flow inside each vessel segment and
ignoring diffusion effect because it’s small comparing with
convection effect in vessel network [35], the concentration at
each point can be expressed as c(x, r, t)= c(0,r,t−x
u(r)).
Here xis the distance of the point to the start of the cylinder
assuming the velocity direction is positive direction. At the
bifurcation point, we assume the radial waveform is preserved:
cdaughter (0,r,t)=cf ather lf ather ,Rf ather
Rdaughter
r, t(8)
At the meeting point, we assume the concentration of fa-
ther branch is the average of concentration in daughter branch
weighted by its flow:
cfather (0,r,t)=
all daughter branch
Fdaughteri
Ffather
×cdaughterildaughteri,Rdaughteri
Rfather
r, t
(9)
In arterial side, father branch means the branch in upstream
direction, while in venous side father branch means the branch
in downstream direction. As flow is conserved at the connecting
point, the concentration is also conserved. A concentration input
c(t)= 0.28te−t
45s+9.1(1 −exp(−t
15s)) was applied at the
arterial input for the simulation, which is acquired by fitting
relative enhancement of hepatic artery (average value in a 9
voxel hepatic artery ROI) in a NAFLD rabbit DCE MRI scan
to 4-parameter AIF c(t)=k
1te−t
k2+k
3(1 −exp(−t
k4)) (red
line in Fig. 2(b)). Eqs. (8) & (9) allow us to calculate the tracer
concentration at each segment independently instead of updating
all the segments at the same time and were solved at 1um
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ZHANG et al.: FLUID MECHANICS APPROACH TO PERFUSION QUANTIFICATION 983
spatial resolution and 1ms temporal resolution. The accuracy
of this simulation in SVS and DVS is validated on a three-level
vessel network where accurate tracer propagation simulation
based on finite element method is performed using COMSOL
Multiphysics (COMSOL AB, Stockholm, Sweden), which is
shown in appendix 1.
For the tracer transport in HHS, we employ a one compartment
porous media model. Velocity and pressure in HHS (referred as
volume V) are calculated based on Darcy’s law [28]:
−∇ · (α∇p)=gflow inside V (10)
∂vp=0on the boundary of V (11)
u=α
φ∇p(12)
Here gflow is the flow source from SVS and flow sink form
DVS. ∂vpis the gradient of pressure pvertical to liver surface.
αis effective permeability defined as permeability divided by
viscosity. αis set to 1 as velocity field udoesn’t depend on
α.φ=0.15 is the vascular space volume [36]. uis velocity
vector and can be converted to flow by multiplying with φ
and integrating on any surface area. Tracer concentration cin
HHS can be simulated using transport equation based on the
calculated velocity:
∂c
∂t =−∇ · (cu)+gc(13)
Here gcis the concentration source from SVS and concentra-
tion sink form DVS. Eqs. (10)-(13) were discretized and solved
using finite volume method [33] implemented on MATLAB
R2018a (Natick, Massachusetts: The MathWorks Inc.) at spatial
resolution 0.5 mm and temporal resolution 1ms. This simulation
resolution was determined by computer memory, and the accu-
racy under this resolution is validated against a high-resolution
tracer propagation simulation of porous media in a small volume,
which is shown in appendix 2.
After tracer propagation in SVS, HHS and DVS was cal-
culated, it was integrated in 0.5 mm3cubical voxels at 5s
temporal resolution and was used as input for |u| and LBF recon-
struction using QTM and Kety’s method. For Kety’s method,
the concentration input at artery inlet (c(t)= 0.28te−t
45s+
9.1[1 −exp(−t
15s)]) was used as AIF. The reconstruction ac-
curacy was evaluated using relative root mean squared error
(rRMSE), which is the L2 norm of the difference between
prediction and ground truth divided by L2 norm of ground truth.
2) Application of QTM in Pathological Nonalcoholic
Fatty Liver Disease (NAFLD) Classification: Basedonthe
spectrum of steatosis, lobular inflammation, and hepatocyte
ballooning, NAFLD can be divided into two categories: simple
steatosis (SS) and nonalcoholic steatohepatitis (NASH). While
patients with SS are at no higher risk of death than the general
population, patients with NASH are at increased risk of death,
as NASH may cause severe liver disease such as liver cirrhosis
and hepatocellular carcinoma [37]. NASH is reversible after
proper therapies to inhibit steatosis, inflammation [38], [39].
Therefore, detecting and staging of NAFLD is crucial to its
treatment. Current golden standard for NAFLD classification is
percutaneous liver biopsy [40]. However, it’s not widely used
in clinical assessments because of its invasiveness and variation
because of the sampling position.
A preclinical animal model was used to test the clinical value
of QTM in NAFLD grading task. Comparing with rats, rabbits
are easier to feed and scan because of the size. Moreover, rabbit
liver also has similar anatomical structure, vascular system, and
perfusion value with human liver, thus is widely used in studying
liver disease, especially when it’s difficult to acquire pathology
section in patients [41], [42], [43], [44]. This study was approved
by the animal care and use committee. 18 adult male New
Zealand rabbits weighing from 2 kg to 2.5 kg were included. All
the rabbits were maintained at 21°C with a 12-hour light-dark
cycle. The rabbits were randomly divided into 4 groups and were
fed with a high fat/cholesterol diet (standard diet with additional
2% cholesterol and 10% triglycerides, Product No. TP2R144;
Trophic Animal Feed High-tech Co. Ltd., China) for 46,8 and 10
weeks, respectively, to establish NAFLD. 3 rabbits died because
of infection or anesthesia before measurement.
After the nurturing cycle, DCE-MRI scans of the remaining 15
rabbits were acquired on a GE Discovery 750 3T scanner with 3D
spoiled gradient echo sequence (Liver Acquisition with Volume
Acquisition, LAVA) before and after the injection of gadolinium
contrast agent (Gd-DTPA-BMA, Omniscan, GE Healthcare,
Ireland) using the following parameters: TR/TE 5.71/1.51 ms,
voxel size 0.5 ×0.5 ×0.75 mm3, matrix size 256 ×256×36,
flip angle 12°, temporal resolution 5 s. 60 time points in total
are acquired. The rabbits were anaesthetized to avoid motion.
Registration between different time points was performed using
FSL toolbox (FMRIB Software Library v6.0, Analysis Group,
FMRIB, Oxford, U.K.) to further remove respiratory motion.
Region of interest (ROI) was drawn by an eight-year experienced
radiologist in the middle of liver where the tissue was enhanced,
and no obvious vessel can be observed. For two compartment
exchange model perfusion parameter reconstruction, dual input
function was applied. Arterial input was calculated as an average
of 9 voxels in rabbit hepatic artery, and venous input was
calculated as an average of 9 voxels in rabbit portal vein.
Liver samples were firstly immersed in 10% phosphate-
buffered formalin, then embedded in paraffin embedded and
sectioned. All the sample slices were stained using hematoxylin,
eosin, and Masson’s trichrome and were reviewed by a patholo-
gist with 15 years of experience blinded to the diet and imaging
information. Each rabbit was scored by summing up semi-
quantitatively determined indicators including steatosis (0–3
points), acinar inflammation (0–3 points) and hepatocellular
ballooning (0–2 points) based on the NASH clinical research net-
work’s (NASH CRN) NAFLD Activity Score (NAS) system. In
this study, 15 rabbits are divided into simple steatosis (SS) group
(N =8, NAS =12) and nonalcoholic steatohepatitis (NAFH)
group (N =7, NAS≥3). One representative pathology slice of
NASH and SS cases are shown in Fig. 3, which correspond to
the case shown in Figs. 5 and 6.
Statistical analysis was performed on the perfusion param-
eters averaged in liver ROI. Using the R Statistical Software
(Foundation for Statistical Computing, Vienna, Austria), a
Mann-Whitney U test was performed comparing ROI values
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984 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 70, NO. 3, MARCH 2023
Fig. 3. Pathology image of a NASH case (a) and a SS case (b).
for |u|,LBFaLBFv,Ktrans ,Vpand Vebetween the two
groups. Receiver operation curve (ROC) analysis was performed
to test the discrimination accuracy of these perfusion parameters.
P-values at or below 0.05 were considered to indicate statistical
significance.
III. RESULTS
A. CFD Simulation
CFD simulated flow in vessels and hepatic tissue are shown
in Figs. 1 and 2:Fig. 1 shows the flow in each vessel segment,
and Fig. 2(a) shows the flow magnitude in HHS at three cross
sections of the liver. Tracer transport was then solved based
on transport equation Eqs. (8), (9) & (13) using arterial input
function approximated from the tracer concentration measured
in hepatic artery, and the simulated tracer concentration profile
in hepatic tissue was shown in Fig. 2(b).
uand LBF ware then reconstructed based on Eqs. (1) and
(5) using the simulated 4D tracer concentration image. The
output uwas converted to flow by multiplying the voxel surface
area divided by plasma volume φ. Comparing with ground
truth (Fig. 4(a)), QTM method showed 60% reduction in error
(Fig. 4(b) and (d),rRMSE=0.24) than Kety’s method (Fig. 4(c)
and (e),rRMSE=0.46), indicating the feasibility of applying
QTM method into liver perfusion quantification.
B. Rabbit DCE-MRI Experiment
T1 weighted DCE-MRI images were acquired on all 15 rabbits
and were used to calculate the perfusion parameters using both
QTM method and two-compartment Kety’s parameters (Eqs 2,3
& 5). Perfusion parameter map of a representative NASH case
and a SS case are shown in Figs. 5 and 6, respectively. NASH
case showed a higher |u|, Ktrans and Vpcomparing with SS
case.
Statistically, a significant difference is observed between
SS group and NAFH group in QTM |u| (0.18±0.08
mm/s vs 0.08±0.05 mm/s, p =0.04, AUC =0.82), but
not in LBFa(78.50±57.38 mL/100g/min vs 77.67±61.53
Fig. 4. Liver blood flow reconstruction results: (a) ground truth, re-
constructed with (b) QTM method and (c) Kety’s method. Against the
ground truth flow (a), QTM method showed a lower error (rRMSE 0.24
vs 0.46). (d) absolute error of QTM method and (e) absolute error of
Kety’s method.
Fig. 5. Perfusion parameters of a NASH case. a) QTM |u|map,
b) LBFamap, c) LBFvmap, d) Ktrans map, e) Vpmap and f) Ve
map.
Fig. 6. Perfusion parameters of a SS case. (a) QTM |u|map, (b) LBFa
map, (c) LBFvmap, (d) Ktrans map, (e) Vpmap and (f) Vemap.
mL/100g/min, p =0.61, AUC =0.59), LBFv(71.47±27.04
mL/100g/min vs 137.95±85.91 mL/100g/min, p =0.09, AUC
=0.76), Ktrans(0.025±0.014 /min vs 0.023±0.012/min, p =
0.86, AUC =0.54), Vp(0.017±0.009 vs 0.013±0.006, p =1,
AUC =0.50), and Ve(0.24±0.16 vs 0.16±0.10,p =0.28, AUC
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ZHANG et al.: FLUID MECHANICS APPROACH TO PERFUSION QUANTIFICATION 985
Fig. 7. (a - f) NASH vs SS statistical comparison of QTM |u|,LBFa,
LBFv,Ktrans ,Vpand Verespectively. A significant difference is ob-
served on QTM |u|between NASH group and SS group.
=0.68). Details of the ROC analysis is shown in Fig. 5 and
appendix 9.
IV. DISCUSSION
Our results demonstrate the feasibility to quantify liver per-
fusion using QTM method. In the numerical simulation, QTM
method showed higher accuracy in flow quantification compar-
ing with traditional Kety’s kinetic modeling method. In NAFLD
grading task, significant difference of QTM velocity is found
between SS group and NASH group, and no difference of LBFa
,LBFv,Ktrans ,Vpand Veis found between these groups,
suggesting that velocity magnitude derived from QTM method
has the potential to differentiate NASH from SS, and may
perform better than kinetic parameters derived from traditional
dual input two compartment exchange model [45]. This finding
is consistent with liver vascular changes caused by steatosis
and inflammation in NAFLD; this finding warrants clinical
translational study in patients to address current lack of accurate
noninvasive classification methods [46].
Perfusion quantification is highly promising to capture the
changes in liver vascular morphology and flow during the pro-
gression of NAFLD. Steatosis may decrease blood flow by
increasing the resistance in vascular network [47], while liver
fibrosis may increase the portal venous flow [48]. This change
may be captured by dynamic imaging of tracer transport through
liver and quantitative perfusion modeling. However, the output
of traditional Kety’s kinetic modeling method largely depends
on the choice of AIF [49]. QTM method doesn’t require arterial
input function as an input, thus provides an automatic flow ve-
locity evaluation method and is proven promising in the NAFLD
grading task. The classification accuracy of QTM method may
be further improved by a better estimation of contrast agent
concentration and a higher resolution DCE MRI image [50].
Moreover, texture analysis may be used to improve the NAFLD
detection [51]. Logistic regression may be used to combine these
image measurements as well as laboratory test results to archive
a better accuracy and specificity. This QTM development would
be valuable for various noninvasive imaging techniques, such
as ultrasound, MRI and CT, used in first-line investigation of
NAFLD by detecting liver steatosis [52], [53], [54], particularly
for establishing an accurate noninvasive alternative to biopsy for
NAFLD detection and grading [55].
There are substantial technical advancements in the liver flow
and tracer transport simulation used in this study. Compared
to the vascular tree model used in the recent QTM validation
in the kidney, the liver vasculature here based on experimental
data is more complex and realistic with both artery part and
venous part. This suggests that the fluid mechanics approach to
tissue perfusion may be scaled to any organ tissue. There have
been CFD studies for liver perfusion and drug delivery (16-18)
using microvascular network (15,17,19), which can serve as
a ground truth for validating tissue perfusion quantification
(15,20) through integration of microvasculature over a voxel for
interpreting the contrast agent concentration changes in space
and time (21). However, these studies use plug flow for vascular
branches, possibly due to limitations in memory and computa-
tional power [28], [56]. If the ground truth flow is parabolic, this
plug flow assumption may cause around 10% error in tracer flux
calculation, as first shown in this study.
Although liver perfusion quantification has been widely stud-
ied using Kety’s kinetic modeling [57], [58], the calculated
perfusion parameters has not yet been compared with ground
truth in experimental measurements or simulation that are very
difficult to perform. In this study, we present microvasculature
CFD based simulation of tissue transport as ground truth. The
accuracies of various DCE-MRI postprocessing methods can be
validated on the correspondingly simulated 4D tracer propaga-
tion images against the ground truth. Here, we showed that there
was substantial error in flow estimation using traditional Kety’s
kinetic modeling method. One possible reason is that traditional
kinetic modeling method ignores the dispersion caused by tracer
transport in arterial system. Transport equation is a more proper
way to model the tracer propagation process [59].
In this study, we ignored the diffusion term in the vascular
space (Eq. (8)). The diffusion coefficient of Gd in blood is around
3×10−4mm2/s [35], which is very small during observation
time compared to the flow speed in vasculature network varies
from 0.5 mm/s to 100 cm/s. Therefore, the diffusive flux is much
smaller than convective flux, therefore can be neglected in flow
simulation of vasculature networks. The analytical solution of
parabolic flow transport process can be derived by assuming the
tracer waveform is preserved when it travels from father branch
to daughter branches, which allows a tracer transport simulation
with error smaller than 1%. Moreover, no extra memory is
needed, because global update of tracer concentration is vascular
system is not needed.
There are several limitations of this study. First, constrained
by the resolution of micro-CT, capillaries are not included in the
vasculature model, therefore perfusion in hepatic space is mod-
eled as flow in porous media. Liver vasculature acquired with
higher resolution imaging techniques may be used to improve
the accuracy of simulation. Second, vessels are considered as
straight cylinders in this study, and blood is considered as in-
compressible fluid with low Reynolds number. In this situation,
the flow in vessel network can be approximated as parabolic
flow. Whether this assumption holds in curved vessels with
varying radii and blood with high Reynolds number still need
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986 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 70, NO. 3, MARCH 2023
to be validated. Third, QTM model used in this study only
contains one compartment. Hepatic sinusoidal capillaries are
permeable to macromolecules such as gadolinium because of the
large fenestrae on the vessel wall [60]. After gadolinium leaves
capillary system, it will travel with interstitial fluid flow and
may be collected by capillary system again [61]. The interstitial
fluid flow speed is much smaller than capillary flow speed, and
therefore a two-compartment exchange QTM model should be
developed to better map the transport of contrast agent and com-
pared with Kety’s method. Fourth, the reconstruction accuracy
of QTM method and Kety’s method depend on spatial and tem-
poral resolution, which should be discussed in the future work
to determine the optimal acquisition resolution for perfusion
parameter estimation. Fifth, gadolinium concentration ([Gd])
was estimated from DCE-MRI signal magnitude according to
a linear model. The relationship between [Gd] and MRI signal
magnitude is highly complex and affected by inflow effects [62].
Quantitative susceptibility mapping based on simple static mag-
netism for MRI signal phase processing offers a robust accurate
alternative [63], [64], [65]. Sixth, high spacetime resolution 4D
DCE-MRI is desired for QTM input to capture the full transport
through the liver. Fast imaging can be employed for adequate
spatial and temporal resolution [66], [67], [68]. Respiratory
motion during imaging may be substantial and require effective
motion compensation [69], [70], [71]. Seventh, the data size of
the animal experiment is small (N =15). A larger dataset will
be helpful for a precise evaluation about whether QTM can be
used in early detection of NAFLD, which is important in clinical
practice.
V. C ONCLUSION
Both CFD simulation and preclinical imaging demonstrate
that QTM is superior to Kety for liver perfusion quantification
from DCE MRI data. |u| derived from QTM is promising in
NAFLD grading task. Clinical study with a larger data size is
warranted to compare the performance of QTM method against
traditional Kety’s kinetic modeling method and evaluate the
possibility of coupling QTM in current non-invasive NAFLD
grading assessment.
APPENDIX
A) Validation of Transport Simulation in SVS and DVS
A three-level vasculature model was constructed to validate
our flow and tracer concentration simulation method (shown in
Fig. 8(a)). The length and radii of the vessel segment for each
level are 1 mm, 1 mm, 0.7 mm and 0.1 mm, 0.0794 mm, 0.0630
mm, respectively. The included angle of level2 and level3 vessels
are 60◦and 80◦. Parabolic flow with maximum velocity 2mm/s
is set as inlet boundary condition and zero pressure is set as
outlet boundary condition. The diffusion coefficient is set to 3×
10−4mm2/s [35]. Flow and velocity inside the vasculature were
solved based on Navier-Stokes equation and transport equation
using finite element method. Velocity in radial direction and
tracer flux rate change with time were sampled at the middle
point of the segment at each level.
Fig. 8. (a) structure of 3 level tube used to validate the parabolic
flow simulation method. (b) the corresponding finite element mesh. (c),
(e) and (g) ground truth velocity profiles (solid line) comparing with
velocity profile simulated using plug flow (green dots) and parabolic flow
assumption (blue dots) in radii direction at the middle cross section of
level 1, 2 and 3 respectively. (d), (f) and (h) ground truth tracer fluxes
(solid line) comparing with tracer flux simulated using plug flow (green
dots) and parabolic flow assumption (blue dots) at the middle cross
section of level 1, 2 and 3, respectively.
Fig. 9. Perfusion simulation in 10 mm3volume (red box in a) using 0.5
mm grid size (figure b) and 0.25 mm grid size (figure c). The difference
in concentrations between the two resolutions is 4%.
We observed a high accuracy for flow simulation using
parabolic flow assumption. Comparing with plug flow assump-
tion, parabolic flow assumption provides a more accurate veloc-
ity estimation (shown in Fig. 8(c),(e) and (g)), and decreased the
tracer flux estimation error from 11% with plug flow simulation
to 0.5% (shown in Fig. 8(d),(f) and (h)).
B) Validation of Transport Simulation in HHS
For transport simulation in HHS, we used spatial resolution
0.5 mm, which is determined by computer memory. We tested
the simulation accuracy under this resolution by simulating a
small volume with higher spatial resolution: 10 mm cubical
volume was chopped from the liver volume (shown in Fig. 9)
and tracer propagation was simulated with 0.25 mm grid size.
The difference between simulated tracer concentrations at the
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ZHANG et al.: FLUID MECHANICS APPROACH TO PERFUSION QUANTIFICATION 987
Fig. 10. Liver blood flow reconstruction results with SNR=40 image:
(a) ground truth, reconstructed with (b) QTM method and (c) Kety’s
method. Against the ground truth flow (a), QTM method showed a lower
error (rRMSE 0.24 vs 0.46). (d) absolute error of QTM method and
(e) absolute error of Kety’s method.
two resolutions was 4%, indicating the spatial resolution used is
enough for accurate simulation.
C) QTM and Kety’s Reconstruction in Simulation With
Noise
We tested QTM and Kety’s reconstruction in simulation data
described in main test with Gaussian noise (SNR =40, while
the SNR of acquired DCE MRI is around 80). Reconstructed
perfusion parameter map is shown in Fig. 10.TherRMSEof
QTM method is 24%, while the rRMSE of Kety’s method is
46%.
D) Kety’s Reconstruction in Simulation With AIF
Sampled at Different Temporal Resolution
We tested Kety’s reconstruction in simulation data with AIF
sampled at different temporal resolution. AIF sampled at 1s and
15s resolution is shown in Fig. 11(d). Kety’s flow reconstruction
error is 42% using AIF sampled at 1s resolution, and Kety’s flow
reconstruction error is 48% using AIF sampled at 15s resolution.
E) Reconstruction of Liver DCE MRI Using Two
Compartment Exchange Model (2CXM) With
Population-Based AIF
We tested Kety’s reconstruction using population based AIF
by fitting measured AIF to the following 4-parameter model
using Levenberg-Marquardt method:
ca(t)=k1te−t
k2+(1−k3)e−t
k4(A1)
Parameter map estimated from a NAFH group is shown in
Fig. 12:
Statistically, there is no difference between NASH and
SS group for LBFa(96.51±85.42 vs 80.84±78.92, p =
0.86), LBFv(112.82±72.18 vs 80.57±57.86, p =0.39),
PS (0.020/min±0.014/min vs 0.006±0.005, p =0.28), Vp
Fig. 11. Liver blood flow reconstruction using Kety’s method with AIF
sampled at different temporal resolution: (a) ground truth flow, (b) Kety’s
method with 1s temporal resolution and (c) Kety’s method with 15s tem-
poral resolution. (d) AIF sampled at 1s and 15s resolution, (e) absolute
error of Kety’s method at 1s resolution and (f) absolute error of Kety’s
method at 15s resolution.
Fig. 12. Perfusion parameters of a NASH case reconstructed using
Kety’s method with population based AIF. a) LB Famap, b) LBFvmap,
c) Ktrans map, d) Vpmap and e) Vemap.
(0.012±0.010 vs 0.005±0.004, p =0.23) and Ve(0.20±0.19
vs 0.08±0.05, p =0.23).
F) Reconstruction of Liver DCE MRI Using Brix Model
In Brix model, tracer propagation is modeled using the fol-
lowing equation [72], [73]:
C(t)= −AH
Kep −Kel e−Kep (t−TA)−e−Kel(t−TA)(A2)
Here C(t)is tracer concentration of tissue, TAis tracer arrival
time, AH is magnitude scaling factor. Kel is the elimination
constant of plasma and Kep is the exchange rate of plasma
and extravascular extracellular space. Eq. (A2) was fitted using
Levenberg-Marquardt method, and parameter map estimated
from a NAFH group is shown in Fig. 13.
Statistically, there is no difference between NASH and
SS group for AH (0.04±0.01 vs 0.04±0.02, p =0.39),
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988 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 70, NO. 3, MARCH 2023
Fig. 13. Perfusion parameters of a NASH case reconstructed using
Brix model. a) AH map, b) Kep map, c) Kel map, d) TAmap.
Fig. 14. Perfusion parameters of a NASH case reconstructed using
tissue homogeneity model. a) Vai map, b) Vvi map, c) EFamap, d) EFv
map and e) Vemap.
Kep (2.14±0.51 vs 2.17±0.30, p =1), Kel (0.18±0.06 vs
0.16±0.09, p =0.34) and TA(0±0.01 vs 0.99±0.01, p =0.24).
G) Reconstruction of Liver DCE MRI Using Adiabatic
Approximation of Tissue Homogeneity (TH) Model
In adiabatic approximation of tissue homogeneity model,
tracer propagation is modeled using the following equation [74]:
C(t)=Vai Ca(t)+VviCv(t)+EFaCa(t)⊗e−EFa
Vet
+EFvCv(t)⊗e−EFv
Vet(A3)
Here C(t)is tracer concentration of tissue, Vai and Vvi are
product of arterial and venous blood flow with tracer transit time,
EFaand EFvare product of arterial and venous blood flow with
tracer extraction factor to extravascular extracellular space, and
Veis extravascular extracellular space volume. Eq. (A3) was
fitted using Levenberg-Marquardt method, and parameter map
estimated from a NAFH group is shown in Fig. 14:
Statistically, there is no difference between NASH and SS
group for Vai (0.10±0.05 vs 0.09±0.04, p =1), Vvi (0.36±0.09
Fig. 15. Perfusion parameters of a NASH case reconstructed using
reference tissue model. a) Ktr ans
Ktrans
RR
map, b) kep.Map,c) Ve
Ve,RR map.
vs 0.39±0.12, p =0.77), EFa(0.002±0.001 vs 0.004±0.002,
p=0.77), EFv(0.002±0.001 vs 0.004±0.002, p =0.69) and
Ve(0.06±0.02 vs 0.06±0.04, p =0.86).
H) Reconstruction of Liver DCE MRI Using AIF-Free
Reference Tissue Model
In reference tissue model, tracer propagation is modeled using
the following equation [75], [76]:
C(t)=Ktrans
Ktrans
RR
CRR (t)+Ktrans
Ve,RR t
0
CRR (τ)dτ
−Kep t
0
C(τ)dτ (A4)
Here C(t)is tracer concentration of tissue, CRR(t)is tracer
concentration of reference region. Eq. (S4) was solved using
a linear inversion method to estimate relative volume trans-
fer constant Ktrans
Ktrans
RR
, relative extravascular extracellular space
volume Ve
Ve,RR , and reflux time constant Kep. Averaged tracer
concentration of 9 voxels in muscle tissue near liver was used as
reference tissue region. Parameter map estimated from a NAFH
group is shown in Fig. 15.
Statistically, there is no difference between NASH and SS
group for relative volume transfer constant (0.77±0.35 vs
0.51±0.31, p =0.12), relative extravascular extracellular space
volume (0.02±0.01 vs 0.02±0.01, p =0.54), and reflux time
constant (0.58±0.20 /min vs 0.44±0.23/min, p =0.23).
I) Details of the ROC Analysis of NAFLD Differentiation
Task
In ROC analysis of NAFLD differentiation task, QTM |u|
showed AUC =0.82 (95% confidence interval C 0.46 to 1)
with sensitivity 0.86 (CI 0.33 to 1), specificity 0.75 (CI 0.33
to 1), optimal threshold 0.27mm/s (calculated by maximizing
sensitivity plus specificity); LBFashowed AUC =0.59 (CI
0.24 to 0.89), sensitivity 0.87 (CI 0.43 to 1), specificity 0.29
(CI 0 to 0.75), optimal threshold 35.97mL/100g/min; LBFv
showed AUC =0.76 (CI 0.35 to 0.96), sensitivity 0.57 (CI
0.16 to 1), specificity 0.87 (CI 0.42 to 1), optimal threshold
112.84mL/100g/min; Ktransshowed AUC =0.54 (CI 0.18 to
0.84), sensitivity 0.85 (CI 0.28 to 1), specificity 0.25 (CI 0 to
0.67), optimal threshold 0.04/min; Vpshowed AUC =0.50 (CI
0.19 to 0.80), sensitivity 0.85 (CI 0 to 1), specificity 0.25 (CI 0
to 0.67), optimal threshold 0.003; Veshowed AUC =0.68 (CI
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ZHANG et al.: FLUID MECHANICS APPROACH TO PERFUSION QUANTIFICATION 989
0.30 to 0.91), sensitivity 0.75 (CI 0.26 to 1), specificity 0.71 (CI
0.25 to 1), optimal threshold 0.20.
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