Phase changes of the flow rate in the vertebral artery caused by debranching thoracic endovascular aortic repair: effects of flow path and local vessel stiffness on vertebral arterial pulsation
Naoki Takeishia Li Jialongb Naoto Yokoyamac Hisashi Tanakad Takasumi Gotoe Shigeo Wada

1
.001
Phase changes of the flow rate in the VA caused by dTEVAR
N. Takeishi & N. Yokoyama et al.


mode = title]Phase changes of the flow rate in the vertebral artery caused by debranching thoracic endovascular aortic repair: effects of flow path and local vessel stiffness on vertebral arterial pulsation


1]Naoki Takeishi[
   type=editor,
   auid=000, bioid=1,
   
   
   orcid=0000-0002-9568-8711
   ]
[1]

takeishi.naoki.008@m.kyushu-u.ac.jp


2]Li Jialong[
   
   ]

3]Naoto Yokoyama[
   
   
   orcid=0000-0003-1460-1002
   ]
[1]

n.yokoyama@mail.dendai.ac.jp


4]Hisashi Tanaka
5]Takasumi Goto
2]Shigeo Wada









[1]Department of Mechanical Engineering, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395,
Japan
[2]Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, 560-8531, Osaka, Japan
[3]Department of Mechanical Engineering, Tokyo Denki University, 5 Senju-Asahi, Adachi, 120-8551, Tokyo, Japan
[4]Graduate School of Medicine, Division of Health Science, Osaka University, 2-2 Yamadaoka, Suita, Osaka 565-0871, Japan
[5]Department of Cardiovascular Surgery, Osaka University Graduate School of Medicine, Suita, Osaka, 565-0871, Japan

[cor1]Corresponding author










§ ABSTRACT 
Despite numerous studies on cerebral arterial blood flow,
there has not yet been a comprehensive description of hemodynamics in patients undergoing debranching thoracic endovascular aortic repair (dTEVAR),
a promising surgical option for aortic arch aneurysms.

A phase delay of the flow rate in the left vertebral artery (LVA) in patients after dTEVAR compared to those before was experimentally observed, 
while the phase in the right vertebral artery (RVA) remained almost the same before and after surgery.
Since this surgical intervention included stent graft implantation and extra-anatomical bypass,
it was expected that the intracranial hemodynamic changes due to dTEVAR were coupled with fluid flow and pulse waves in cerebral arteries.
To clarify this issue,
A one-dimensional model (1D) was used to numerically investigate the relative contribution (i.e., local vessel stiffness and flow path changes) of the VA flow rate to the phase difference.

The numerical results demonstrated a phase delay of flow rate in the LVA but not the RVA in postoperative patients undergoing dTEVAR relative to preoperative patients.
The results further showed that the primary factor affecting the phase delay of the flow rate in the LVA after surgery compared to that before was the bypass, i.e., alteration of flow path,
rather than stent grafting, i.e., the change in local vessel stiffness.
The numerical results provide insights into hemodynamics in postoperative patients undergoing dTEVAR,
as well as knowledge about therapeutic decisions.













Intracranial blood flow, vertebral artery, 1D analysis, nonlinear wave dynamics, Riemann invariants, debranching TEVAR, computational biomechanics


[
    [
    3 September 2024
====================





§ INTRODUCTION


Aortic aneurysm, which is morphologically defined as the focal dilation and structural degradation of the aorta,
is an asymptomatic disease,
and its rupture is a significant cause of death worldwide.
For instance, in the United States of America, aortic aneurysms and dissections cause over 10,000 deaths each year <cit.>.
To date, thoracic endovascular aortic repair (TEVAR) is the only effective treatment for aortic aneurysm.
TEVAR has been a particularly attractive surgical intervention because it is less invasive than conventional open surgical repair <cit.>.

The aorta is the largest conduit artery in the body, and due to its extraordinary ability to expand and contract, it serves as a reservoir that transforms the highly pressured and pulsatile heart output into a flow with moderate fluctuations <cit.>.
Deterioration of this buffering effect due to stiffening of the arterial wall is the main cause of hypertension <cit.>.
These dynamics of the aorta are also important for propagation of the pulse wave,
which has been recognized as an early indicator of the health status of the cardiovascular system <cit.>.
Thus, understanding the hemodynamic differences between pre- and postoperative TEVAR patients based on pulse-wave dynamics is of paramount importance, not only for surgical decision-making to achieve optimal clinical outcomes, but also for evaluating postoperative hemodynamics.

In TEVAR, the aortic arch is mapped according to the segmentation of the vertebral column or landing zone <cit.>,
where zone 0 ranges from the ascending aorta (AA) to the origin of the innominate artery (IA),
zone 1 ranges from the IA to the origin of the left common carotid artery (LCCA),
zone 2 ranges from the LCCA to the left subclavian artery (LSA),
and zones 3 and 4 follow in the longitudinal direction.
TEVAR for partial arch debranching,
known as debranching TEVAR (dTEVAR),
includes both stent graft implantation and extra-anatomical bypass,
where the stent graft is placed from zone 1 or 2 with a length of several millimeters depending on the patient.
This operation is most often performed in high-risk patients with thoracic aortic aneurysms (TAA) <cit.>.
Currently, expanded polytetrafluoroethylene (ePTFE) and polyethylene therephthalate (PET) are the two most popular graft materials in abdominal aortic aneurysm repair due to their remarkable biocompatibility and durability.
Additional supra-arch vessel reconstruction in dTEVAR is known to prevent cerebral infarction <cit.> and ischemia in the cerebral circulation <cit.>.




A previous experimental study has shown that total intracranial blood flow was preserved after dTEVAR,
with significant decrease of flow in the LVA and significant increase of flow in the RVA <cit.>.
The hemodynamic mechanism of such changes in posterior cerebral circulation after dTEVAR remains uncertain yet due the technical difficulty of measurements.
Thus, our primary concern here is to evaluate bilateral vertebral artery blood flows under pulsation between the patients before and after 1dTVEAR with axillo-axillary artery (AxA-AxA) bypass,
and to clarify the hemodynamic mechanism associated with postoperative significant changes of posterior cerebral circulation. 
Since dTEVAR includes extra-anatomical bypass,
it was expected that the intracranial hemodynamic changes due to dTEVAR would be coupled with fluid flow and pulse waves in the VA.




Therefore, the objective of this study was to numerically clarify whether there exist phase differences of the flow rate in the VA,
as a representative intracranial vessel,
between patients before and after dTEVAR,
especially in cases with one bypass from the right axillary artery (RAxA) to the left axillary artery (LAxA) and embolization at the branch point between a stent-grafted aortic arch and the LSA (see Figure <ref>).
If phase differences of the flow rate exist,
we clarify how surgery contributes to it.
Hereafter, we refer to this operation as single-debranching TEVAR (1dTEVAR).

A one-dimensional (1D) model analysis was used to numerically investigate blood flow rates in the L/RVA in pre- and postoperative patients after 1dTEVAR.

In a human aorta with ∼10 mm radius and ∼0.3 mm thickness,
the wave speed, quantified by the traditional foot-to-foot wave velocity method, e.g., by <cit.>, is over 5 m/s <cit.>.
Such conventional experimental methodology, however, is still difficult to quantify the effect of pulse waves in the VA on the intracranial blood supply.


Furthermore, assuming that one pulse has a duration of 1 second,
the ratio between the wave length and radius is 500, i.e., long wave.


Numerical analysis for blood flow with long-wave pulses still involves a heavy computational load even with a two-dimensional model,
although several attempts for this issue were reported, e.g., in <cit.>.

1D model analysis is one of the most effective and practical non-invasive solutions to investigate hemodynamics with long-wave pulses <cit.>.
Thus, for practical purposes, 1D modeling has been widely used in circulatory systems,
such as the systemic arteries <cit.>,
coronary circulation <cit.>,
the circle of Willis <cit.>,
and large vasculature, including venous systems coupled with contributions of the microcirculation modeled by lumped parameters <cit.>.
Recently, 1D modeling was applied to a problem involving both blood flow rate and blood oxygenation <cit.>.
<cit.> used patient-specific medical imaging data on the anatomy of the circle of Willis to perform a 1D model analysis of intracranial blood flow assuming a steady state,
coupled with three-dimensional tissue perfusion.



In this study, 
the experimental evidence on differences in the flow rate in the L/RVA between patients before and after 1dTEVAR was reported.
Next, using 1D numerical model, the phase difference of the flow rate in the L/RVA in pre- and postoperative patients was evaluated.
Simulations were also performed for different local vessel stiffnesses and lengths of vessels that had stiffened due to stent grafting.



§ MATERIALS AND METHODS




 §.§ Subjects and measurements

This retrospective study was conducted in accordance with the guidelines of the Declaration of Helsinki.
All experimental protocols were approved by the institutional review board of Osaka University.
All subjects provided oral and written informed consent to participate in this study.
A prospective analysis of blood flow in the left and right internal carotid arteries (L/RICA) and the L/RVA in 9 patients before and after dTEVAR was performed between January 2015 and January 2020 at Osaka University Hospital,
where RAxA-LAxA bypass was performed (see also Figure <ref>B).
A ringed 8-mm ePTFE graft (FUSION, MAQUET Getinge Group, Japan) was used in all the debranching procedures (see also <cit.>).
Two-dimensional (2D) cine phase-contrast data was acquired with 3T magnetic resonance imaging (MRI) (MR750-3T, GE Healthcare, Waukesha, WI, USA).
Validations of the resolution of 3T MRI have been performed in in vivo studies, for instance, by <cit.> and <cit.>.
In all patients, MRI was performed within 6 months prior to the dTEVAR procedure and within 1 month afterward <cit.>.






The vascular tree models of both pre- and postoperative patients were reconstructed based on geometrical parameters of vessel diameter and length between branch points, as shown in Figure <ref>.
Arterial vascular geometries, including reference radius (r_0) and length (L) in a representative patient, were measured both pre- and postoperatively at Osaka University Hospital,
and the data are summarized in Table <ref>.
The diameters of the end terminals,
which are denoted by the vessel ID = χ^' in Figure <ref>(A),
were determined so that the time-average flow rate became similar to that obtained with experimental measurements by <cit.> or <cit.> (see Figure <ref>).
The lengths of the end terminals were uniformly set to 50 mm.
For simplicity, branch angles were uniformly set to 30 deg.
The effect of branch angle is mentioned in  <ref>.
To represent arterial trees in patients after 1dTEVAR,
The Young's modulus was set to be almost 100 times larger in the stent-grafted area (vessel ID = 4; see Table <ref>) than in preoperative patients,
and also created a bypass from the RAxA to the LAxA (vessel ID = 7 and 16) with embolization at the branch point from the stent-grafted thoracic aorta to the LSA (see Figure <ref>(B)).
In this study,
the bypass was considered to have a radius of 4.5 mm and a length of 262 mm,
and the stent-grafted artery was considered to have a radius of 12.5 mm and a length of 40 mm.
The diameters and lengths of end terminals were also the same as those in the preoperative vascular tree model.



 §.§ Mathematical model and simulation

The 1D governing equations describe the conservation of mass and momentum:


    ∂ A/∂ t + ∂ Q/∂ x = 0, 
       ∂ Q/∂ t + ∂/∂ x( αQ^2/A) + A/ρ∂ p/∂ x + K_R Q/A = 0,



where x is the axial direction,
A = A (x, t) is the area of a cross-section at x and at time t,
Q (= UA) is the mean volumetric flow rate across a section,
U = U (x, t) is the velocity of the fluid averaged across the section,
p = p (x, t) is the pressure,
ρ (= 1.05 × 10^3 kg/m^3) is the blood density,
α is the coefficient of the velocity profile,
and K_R (= 22 πV = 22 πμ/ρ) is the drag coefficient for the blood viscosity μ (= 4.5 × 10^-3 Pa·s).
A flat velocity profile was assumed, and set to be α = 1 <cit.>.
Assuming static equilibrium in the radial direction of a cylindrical tube or thin elastic shell,
one can derive a pressure relationship of the form <cit.>

    p - p_ext = β( √(A) - √(A_0)),

and

    β   = √(π) h_0 E(x)/( 1 - ν^2 ) A_0, 
    
      E(x)    = r_0/h_0[ k_1 exp( k_2 r_0 ) + k_3 ],

where h_0 = h_0 (x), r_0 = r_0 (x), and A_0 = π r_0^2 are the vessel thickness, vessel radius, and sectional area, respectively, at the equilibrium state (p, Q) = (p_ext, 0),
E(x) is the Young's modulus,
p_ext (= 0) is the external pressure, assumed as a constant,
ν is the Poisson ratio, which is set to be ν = 0.5 for practical incompressibility, and
k_i (i = 1–3) are the coefficients set to k_1 = 2 × 10^6 kg/(s^2·m), k_2 = -2.253 × 10^3 1/m, and k_3 = 8.65 × 10^4 kg/(s^2·m) <cit.>.
These parameter values were also used in previous 1D blood flow analyses, e.g., by <cit.> and <cit.>.
Since the effect of h_0 does not appear in β owing to equations (<ref>) and (<ref>),
and since the vessel stiffness is simply characterized by β, which is determined mainly by the reference radius r_0 and parameters k_i,
the value of h_0 in each artery was simply assumed as one-tenth of the diameter 2 r_0, i.e., h_0 = 2 r_0/10.
Thus, the order of magnitude of the calculated Young's modulus, referring to the values of r_0 in Table <ref>, was O(E) = 10^-1 MPa,
which is consistent with that of aortic elasticity in conscious dogs <cit.>.
The Young's modulus of the bypass was set to be 100 times larger than that obtained with the reference radius (r_0 = 4.5 mm),
i.e., E_bypass = 43.3 MPa.
Given that a previous numerical analysis of ePTFE stent grafts used a Young's modulus of 55.2 MPa <cit.>,
we set the same order of magnitude of the Young's modulus in the stent grafted region, i.e., E_s = 10 MPa.

The governing equations (<ref>) can be written as an advection equation:

    ∂_t v̌ + 𝐉(v̌) ·∂_x v̌ = b̌ →∂_t v̌ + ∂_x F̌ = b̌,

where the variable v̌,
advection term F̌,
source term b̌,
and Jacobian J̌ are written as


    v̌ = 
      [ Q; A ],   b̌ = 
      [ -K_R Q/A  
      0 ],   F̌ = 
      [ Q^2/A + β/3 ρ A^3/2  
      Q ],   𝐉(v̌) =
      [                  2 Q/A -( Q/A)^2 + β/2 ρA^1/2;                      1                      0 ].


The flow and pulse wave are integrated over time by the Lax-Wendroff method with Δ t = 10^-5 s from t^n to t^n+1 = t^n + Δ t.
The precise descriptions are given in the Appendix.



To capture the internal pressure profile in the heart during a cardiac cycle,
wherein vascular pumping is accelerated in the systolic phase and attenuated in the diastolic phase <cit.>,
the inlet pressure p_in(t) is given as <cit.>

    p_in(t) = p_0 +
      p_a2[1
       + 11-ϵ_s(
            sin(πT̃_s)
         + ϵ_ssin(3 πT̃_s)
          )
      ]   for  
      0 ≤ t  < T4
    p_a2[1
      - 11-ϵ_d(
          sin(πT̃_d)
       + ϵ_dsin(3πT̃_d)
        )
      ]   for  T4≤ t < T
      
    ,

and

    T̃_s = t - T/8T/4,   T̃_d = t - 5T/83T/4,

where p_a (= 4333 Pa) is the amplitude from the base pressure p_0 (= 10666 Pa),
T (= 1 s) is the wave period,
and ϵ_s or ϵ_d is the waveform parameter (black line in Figure <ref>).
Hereafter, the subscripts s and d denote the systolic and diastolic phases, respectively.
In this study, ϵ_s = 0.1 and ϵ_d = 0 <cit.> were used,
so that the derivatives up to the fourth of p(t) are continuous.
The waveform reflects a fast expansion during the systolic phase 0 ≤ t < T/4 and a slow contraction during the diastolic phase T/4 ≤ t < T.
The parameters in our simulations and their values are shown in Table <ref>.

Figure <ref> shows pressures and calculated flow rates during a period T (= 1 s).
The model pressure given in equations equation (<ref>) and (<ref>) well captures the profile of human aortic pressure (Figure 3a in the work by <cit.>).
A no-reflection condition was applied for each outlet by a backward Riemann invariant W_- = 0.
A more precise description of the methodology is presented in the Appendix.







§ RESULTS




 §.§ Measurements of flow rate in pre-and postoperative patients

The flow rates in both the L/RVA in 9 patients before and after 1dTEVAR during a single heartbeat were measured by 2D cine phase-contrast MRI.
The mean flow rates are shown in Figure <ref>,
where the heartbeat was divided into 12 parts and the flow rate at the i-th phase Q_i was normalized by the total flow rate Q_all (= ∑_i^12 Q_i) in both the L/RVA during a period T.
The first phase (i.e., 1/12) was defined when the pulse wave was detected by an accelerometer (GE Healthcare) with the tip of a forefinger.
Compared to the peak in the second phase of the cardiac cycle (i.e., 2/12) in the LVA of preoperative patients,
the peak in postoperative patients was delayed to the third phase (i.e., 3/12).
Thus, a 1/12-period (∼8%) phase delay in the LVA was observed between before and after surgery (Figure <ref>(A)).
On the other hand, 
the peak of the flow rate ratio in the RVA (the first phase, 1/12) remained almost the same before and after surgery (Figure <ref>(B)).
Since the difference of flow rate ratio in the RVA between the first and second phases of the cardiac cycle was very small,
we concluded that the first phase of the cardiac cycle was the time point of the peak of the flow rate ratio.





 §.§ Model validation

Using a preoperative vascular tree model (Figure <ref>),
the mean flow rates Q_mean in eight different arteries were investigated,
specifically AA, VA (= LVA + RVA), L/RICA, and L/RSA.
The calculated mean flow rates were normalized by the mean inlet flow rate Q̅_in = (1/T)∫_0^T Q_in dt,
and the flow rate ratio Q_RATIO = Q_mean/Q̅_in in each vessel was compared with data in a previous experimental study by <cit.> (Figure <ref>A).
The outlet diameters were determined so that the errors in the flow rate ratio in each vessel were uniformly less than 5%, i.e. | Q_RATIO^sim/Q_RATIO^exp - 1| < 0.05,
where the superscripts “sim” and “exp” denote the simulation and experiment, respectively. 

Using the same preoperative vascular tree model,
the flow rate ratio between the L/RICA and L/RVA was calculated.
The flow rate ratio in patients after 1dTEVAR was also calculated with the postoperative vascular tree model (Figure <ref>B).
The diameters of the end terminals were the same as those in the preoperative vascular tree model.
The calculated flow rate ratio in each artery was compared with previous experimental measurements by <cit.> (Figures <ref>B and <ref>C).
In the experiments,
although there were no large differences before and after surgery in the flow rate ratio in the L/RICA or the LVA,
the flow rate ratio in the RVA increased after surgery.
The numerical results qualitatively agree with these experimental measurements.







 §.§ 1D model analysis of the flow rate in the ICA in pre-and postoperative patients

To gain insight into the mechanism of the phase difference in the LVA and the lack of such a difference in the RVA (Figure <ref>),
1D numerical simulations were performed using arterial vascular tree models before and after surgery (Figure <ref>).
Model verifications are shown in the Appendix (see Figures <ref>).
The time history of the pressure and flow rates in the L/RVA during a cardiac cycle T are shown in Figure <ref>(A) and <ref>(B), respectively,
where the results before and after surgery are superposed.
Data are shown after the pressure and flow rates have reached the stable periodic phase (t ≥ 2 s).
As described regarding the experimental measurements of flow rates (Figure <ref>(A)),
the numerical results of the flow rate waveform in the LVA were delayed, by 6.8% of a single period,
which is similar to the experimental measurements (Figure <ref>(A)).
As in experimental measurements, the phase difference was also quantified by the time point observed at the maximum flow rate.

On the other hand,
the numerical results of the flow rate waveform in the RVA were only slightly delayed, by 0.84% of a single period (Figure <ref>(B)).
We concluded that our numerical results of the phase difference in the L/RVA are consistent with the experimental measurements (Figure <ref>(B)).






 §.§ Effects of local vessel stiffness and flow path

To clarify whether the flow rate waveforms in the LVA (Figure <ref>(A)) can be caused by the local vessel stiffness, flow path changes, or both,
the effect of both of these factors on the flow rate in the L/RVA was investigated with two different additional vascular tree models.
The Young's modulus in the stent-grafted region was increased by 100 times postoperatively compared to the preoperative state without the bypass or embolization, the so-called “stent model",
while the flow path change was the same as that in the postoperative vascular tree model (i.e., new bypass and embolization) and the Young's modulus was the same as that in the preoperative model, the so-called “flow path model".
The flow rate and pressure waveforms during a cardiac cycle T are shown in Figure <ref>(A) and <ref>(B).
The flow rate waveforms obtained with the stent model were different from those in the postoperative vascular tree model in both the L/RVA.
On the other hand,
the flow rate waveforms obtained with the flow path model collapsed with those obtained with the preoperative vascular tree model.
These agreements or disagreements were also commonly observed in the pressure waveforms.
The phase difference δ of the maximum flow rate in both the L/RVA between the preoperative vascular tree model and the three aforementioned models (stent, flow path, and postoperative) are summarized in Figure <ref>(C),
where the results were normalized by T.
Changing the flow path, as in the flow path model, decreased the ratio of the phase difference of the flow rate δ/T by only 7.1% in the LVA and 1.3% in the RVA relative to the ratios obtained in the postoperative model.
On the other hand, changing the local Young's modulus, as in the stent model, caused quite small phase differences, i.e., O(δ/T) = 10^-2%.





 §.§ Effects of stent grafting and bypass angle

The effects of the local vessel stiffness (Young's modulus) and stent-grafted length on the phase difference in the L/RVA were further investigated using the postoperative vascular tree model.


Simulations were performed for different Young's modulus valuses E/E_s (= 0.01, 0.1, 1, 10) in the stent-grafted artery with standard length L_0 = 40 mm,
where E_s (= 10 MPa) was the original Young's modulus in the stent-grafted region.
The ratio of the phase difference δ/T remained almost the same (less than 0.2%) in both the L/RVA.
Even when the length of the stent-grafted region increased to L/L_0 = 1, 2.25, 3.5, 4.75, 6,
the results of δ/T remained the same (less than 0.1%; data not shown),
where L_0 (= 40 mm) was the original length of the stent-grafted region.

The phase delay in the LVA was also insensitive to the bypass angle θ,
defined at the branch point between the vessel and bypass,
especially for θ≥ 60 deg as shown in Figure <ref>.
Note that the relative difference of the mean flow rate in the RVA obtained with the reference angle (30 deg) and that obtained with 120 deg at the branch angle of the bypass was smaller than 1% (data not shown).
The results were obtained with the postoperative vascular tree model with a standard stent length of L_0 = 40 mm and a Young's modulus of E_s = 10 MPa.
These results, including those shown in Figure <ref>(C), suggest that the phase delay of the flow rate in the VA between before and after surgery (Figure <ref>) arises mostly from the alteration of the flow path, i.e., by the new bypass and embolization rather than by local vessel stiffness due to stent grafting.

Note that the effect of the stiffness (or Young's modulus E) of the straight tube on the fluid velocity is described in the Appendix,
and the results showed that the pulse-wave speed c increased with E with the ratio of velocity O(c/U) > 10^2 (Figure <ref>D).







§ DISCUSSION

A previous clinical study by <cit.> showed hemodynamic changes in blood flow through the L/RVA after 1dTEVAR, while postoperative total intracranial blood flow was almost the same as that measured preoperatively.
This might be due to systemic hemodynamic compensation causing cerebral blood flow to be strictly maintained through cerebral autoregulation <cit.>.
Despite the maintained flow mass in patients after 1dTEVAR,
experimental measurements demonstrated phase differences of the flow rate in the LVA but not in the RVA between pre- and postoperative patients undergoing 1dTEVAR (Figure <ref>).
It is expected that the aforementioned hemodynamic compensation and the phase differences in the VA in postoperative patients might be due to two mechanical factors: structural alterations in flow paths due to the new bypass and embolization, and an increase of local vessel stiffness due to stent grafting.
However, much is still unknown about this matter, in particular about the relative contribution of these two factors to the phase of the flow rate in the VA between pre- and postoperative patients undergoing 1dTEVAR.
To explore this issue,
A 1D model was used to numerically investigate flow rates in the VA, both before and after 1dTEVAR.
Using a postoperative arterial tree model (Figure <ref>(B)),
The numerical results demonstrated a phase delay in the LVA compared to preoperative models, as shown in Figure <ref>(A).
Stent grafting,
which is characterized by a locally increasing Young's modulus,
contributed negligibly to the phase delay in the LVA (see <ref>).
Thus, the experimentally observed phase delay (Figure <ref>(A)) was mainly caused by alteration of the flow path, i.e., by the new bypass and embolization, rather by local vessel stiffness due to stent grafting (Figure <ref>).









In this simulation,
branch angles were uniformly set to 30 deg for simplicity.
A previous experimental study showed that the energy loss at bifurcations was very small as reported in <cit.>.
This is because the branch angle only affects the total pressure continuity (equations <ref>(b, c) and <ref>(b, c), see the Appendix),
and the order of magnitude of the static pressure term p is much greater than those of the other terms.
Indeed,
the relative difference of the mean flow rate in the RVA obtained with the reference angle (30 deg) and that obtained with 120 deg at the branch angle of the bypass is smaller than 1% (data not shown).
Thus, at least in this model, the phase delay in the LVA was insensitive to the bypass angle θ.
We did not consider postoperative shape deformation and diameter changes due to device placement.
In the future we will perform systematic analyses of the effect of postoperative vascular configurations on the phase change.


In this study, the Young's modulus in model vasculatures was also approximated using equation (<ref>).
Referring to previous numerical assessments of the mechanical behavior of ePTEF,
we set the same order of magnitude of the Young's modulus for the stent-grafted area (i.e., E_s =10 MPa),
which was approximately 100 times larger than that of vessels with similar diameter.
Considering different Young's moduli of PET stent from O(10^0) MPa <cit.> to O(10^3) MPa <cit.> in previous numerical studies,
the simulations were performed for different orders of magnitude of Young's modulus in an ePTFE-stent-grafted area (E/E_s = 0.01-10).
However, the ratio of the phase difference δ/T remained almost the same (less than 0.2%) in both the L/RVA.
Since model parameters in the bifurcation model (γ_1 and γ_2 in equations <ref>(b, c) and <ref>(b, c), see the Appendix) and in Young's modulus (k_1, k_2, and k_3 in equation (<ref>)) were fixed in this study,
further precise analysis of the effects of bifurcation angles and Young's modulus of the stent should be performed experimentally to confirm whether those mechanical factors have less impact on the phase differences.



The velocity profile was implicitly expressed as flat,
derived from Newtonian and laminar fluid flow.
However, it is expected that the blood flow profile, especially in a real artery, is much more complex,
and therefore it is usually modeled as plug flow due to its turbulent nature and cellular dynamics.
Since the Reynolds number in the aorta,
estimated based on the mean flux between systolic and diastolic phases,
is over 4 × 10^3,
the flow should be turbulent.

Furthermore, the blood velocity profile is also affected by frequency-dependent inertia.
The ratio between the transient inertia force and the viscous force can be estimated by the Womersley number Wo = r_0 (ω/V)^1/2≈ 1.2-12,
where r_0 (= 1-10 mm) is the radius of the artery (see Table <ref>),
ω (= 2 π f) is the angular frequency,
f (= 1/T) is the cardiac frequency that is roughly estimated as 1 Hz,
and V (= μ/ρ≈ 4.3 × 10^-6 m^2/s) the kinematic viscosity of the blood.

Thus, it is expected that a more rigorous velocity flow profile can reproduce unsteady hemodynamics in patients before and after dTEVAR.
In this study, however, we focused on explaining the experimentally observed phase delay of the mean volumetric flow rate in the LVA seen in postoperative 1dTEVAR patients relative to that before surgery,
and thus, the flow profile predominantly affecting the unit mean volumetric flow rate should not change the calculated phase delay.












In this simulation,
we simply modeled arterial trees from the heart (or LV) to eight end terminals,
whose diameters and lengths were determined so that the time average flow rate became similar to that obtained in experimental measurements by <cit.> or <cit.> (Figure <ref>).
Even with the simplest boundary condition at the end terminals (no-reflection condition, i.e., W_- = 0),
our numerical model reproduced the phase delay of the flow rate in the LVA without the phase difference in the RVA in postoperative patients,
and also clarified that the phase delay arose mostly from alteration of the flow path (Figure <ref>).

Further precise boundary conditions for both the inlet (heart) and outlets will leads to patient-specific analyses preoperatively and to the evaluation of hemodynamics postoperatively, topics that we will address in the future.





Depending on the areas of TAA,
different numbers of bypasses (1d, 2d, and 3d) can be selected.
If phase delays of flow rate are present in the LVA of postoperative patients who have undergone different surgical operations,
it is expected that the phase differences between before and after surgery are caused by the new bypasses.
Quantitative analyses of hemodynamic changes caused by these surgeries should be performed in future studies.
Modification of the aforementioned model factors, e.g., boundary conditions, will potentially reproduce hemodynamics in patients who undergo these various surgeries.
The numerical results based on pulse-wave dynamics provide fundamental knowledge regarding hemodynamic changes between pre- and postoperative patients undergoing dTEVAR,
and will be helpful not only in surgical decision-making for optimal clinical outcomes but also in evaluating hemodynamics after surgery.




§ CONCLUSION

A 1D model was used to numerically investigate blood flow rates, with the goal of explaining experimental evidence on the phase delay of the flow rate in the LVA but not in the RVA after 1dTEVAR relative to before.
The numerical model can reproduce the flow distribution in the major arteries from the heart,
and can capture the flow rate ratio in the L/RVA in both pre- and postoperative patients.
The numerical results showed that the phase delay was mainly caused by the bypass, i.e., by alteration of the flow path, rather than by stent grafting, i.e., the change of local vessel stiffness.
Bypass angles and the effects of the length and Young's modulus of the stent were also investigated,
but all were insensitive to the phase delay.

We hope that our numerical results will provide fundamental knowledge about therapeutic decisions for dTEVAR.











§ ETHICAL APPROVAL

Not required.



§ DECLARATION OF COMPETING INTEREST

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.



§ ACKNOWLEDGEMENTS

The presented study was partially funded by Daicel Corporation.
Last but not least, N.T. and N.Y. thank Mr. Tatsuki Shimada for his assistance in the preparation of this work.







§ APPENDIX



 §.§ A1. Methodology

The variables of the ith-vessel segment at specific time t^n (v̌_i^n and F̌_i^n) in equation (<ref>) are described as:

    v̌_i^n = 
      [ Q_i^n; A_i^n ],   F̌_i^n =
      [ (Q_i^n)^2A_i^n + β3 ρ (A_i^n)^3/2;                             Q_i^n ].

The discretized form of equation (<ref>) is obtained with the Lax-Wendroff method:


    v̌_i^n+1   = v̌_i^n
      - 1/2 Δ c( F̌_i+1^n - F̌_i-1^n ) 
        + 1/2 (Δ c)^2[ 𝐉( v̌_i^n + v̌_i+1^n/2) ( F̌_i+1^n - F̌_i^n )
      - 𝐉( v̌_i^n + v̌_i-1^n/2) ( F̌_i^n - F̌_i-1^n ) ], 
       = v̌_i + 1/2 Δ c[ F̌_i-1 + F̌_i^n
      + 1/Δ c𝐉( v̌_i^n + v̌_i-1^n/2) ( F̌_i-1^n - F̌_i^n ) . 
       . - {F̌_i^n + F̌_i+1^n + 1/Δ c𝐉( v̌_i^n + v̌_i+1^n/2) ( F̌_i^n - F̌_i+1^n ) }],


where Δ c = Δ x/Δ t, and Δ x is the segment length of the vessel.

The boundary values of Q and A are determined by the Riemann invariants (W_+ and W_-),
which represent a forward- and backward-traveling wave at speeds λ_+ and λ_- as eigen values of the Jacobian 𝐉.
Riemann invariants are the characteristic variables of the following hyperbolic system transformed from equation (<ref>):

    ∂_t W̌ + λ̌·∂_x W̌ = 0,

where W̌ = (W_+, W_-)^T,
and λ̌ = (λ_+, λ_-)^T.
By choosing the reference conditions (A = A_0, U = 0),
we obtain the solutions to system (<ref>):

    W_±   = Q/A± 4 √(β/2 ρ)( A^1/4 - A_0^1/4), 
       = U ± 4 ( c - c_0), 
    and  λ_±   = U ± c,

where c = √(β/(2 ρ)) A^1/4 is the wave speed,
and c_0 is the wave speed at A = A_0.









Flow conditions along the lines (1 → 2) (see Figure <ref>(A)) are described by mass conservation and total pressure continuity <cit.>:


    Q_1 - Q_2 = 0, 
       p_1 + ρ U_1^2/2 - ( p_2 + ρ U_2^2/2 ) = 0,


and W_1^n+1(L) and W_2^n+1(0) are derived from values at the previous time step at the distal end (denoted by L) and proximal end (denoted by 0) of an artery by extrapolating the outgoing Riemann invariants along the characteristic lines <cit.>:


    W_1, +^n+1 (L) = W_1, +^n( L - λ_1, +Δ t ), 
       W_2, -^n+1 (0) = W_2, +^n( 0 - λ_2, -Δ t ).


Flow conditions at the bifurcations (3 → 1 + 2) (Figure <ref>(B)) are described as <cit.>:


    -Q_1 - Q_2 + Q_3 = 0, 
       p_3 + ρ U_3^2/2 - sign(Q_3) f_m (Q_3/A_3) - [ p_1  + ρ U_1^2/2 + sign(Q_1) f_t( Q_1/A_1 + A_3/2) ] = 0, 
       p_3 + ρ U_3^2/2 - sign(Q_3) f_m (Q_3/A_3) - [ p_2  + ρ U_2^2/2 + sign(Q_2) f_t( Q_2/A_2 + A_3/2) ] = 0, 
       W_3, +^n+1 (L) - W_3, +^n (L - λ_3, +Δ t) = 0, 
       W_1, -^n+1 (0) - W_1, -^n (0 - λ_1, -Δ t) = 0, 
       W_2, -^n+1 (0) - W_2, -^n (0 - λ_2, -Δ t) = 0,



where


    f_m (Q/A) = γ_1 ρ (Q/A)^2, 
       f_t (Q/A) = γ_2 ρ (Q/A)^2 √(2 ( 1 - cosα)).


In this study, we set γ_1 = 0, γ_2 = 2, and α = π/6.
At the confluences (1 + 2 → 3) (Figure <ref>(C)), we have:


    Q_1 + Q_2 - Q_3 = 0, 
       p_1 + ρ U_1^2/2 - sign(Q_1) f_t( Q_1/A_1 + A_3/2) - [ p_3  + ρ U_3^2/2 + sign(Q_3) f_m (Q_3/A_3) ] = 0, 
       p_2 + ρ U_2^2/2 - sign(Q_2) f_t( Q_2/A_2 + A_3/2) - [ p_3  + ρ U_3^2/2 + sign(Q_3) f_m (Q_3/A_3) ] = 0, 
       W_1, +^n+1 (L) - W_1, +^n (L - λ_1, +Δ t) = 0, 
       W_2, +^n+1 (L) - W_2, +^n (L - λ_2, +Δ t) = 0, 
       W_3, -^n+1 (0) - W_3, +^n (0 - λ_3, -Δ t) = 0.









 §.§ A2. Weakly non-linear form

To verify the calculated solutions,
the numerical results are compared with those obtained with a weak non-linear equation instead of the theoretical formula.
When Q = Q_0 + dQ and A = A_0 + dA,
Equation (<ref>) can be written as:

    W_±   = Q_0 + dQ/A_0 + dA± 4 c_0 ( ( 1 + dA/A_0)^1/4 - 1 ), 
       ≈Q_0/A_0( 1 - dA/A_0) + dQ/A_0± c_0 dA/A_0.

When Q_0 = 0, we finally have a weak non-linear form of the Riemann invariants (<ref>)

    W_±   ≈dQ/A_0± c_0 dA/A_0.

Equation (<ref>) predicts that wave propagations, including reflective wave W_-, are negligible for the small flow rate dQ ≈ 0 and small vessel contraction A ≈ A_0.
Indeed, the maximum values of |W_±| in the middle point of the straight tube almost collapse on those obtained with p_a = 0 when the maximum inlet flow rate Q_in^max decreases (Figure <ref>(B)),
where the tube length and reference radius are set to L = 1 m and r_0 = 10^-3 m, respectively,
and the Young's modulus is E_0 = 0.1 MPa,
referring to the value of aortic elasticity in conscious dogs <cit.>.
The inlet pressure wave form (<ref>) and no-reflection condition for the outlet W_- = 0 are also considered.
The small maximum inlet inflow rate Q_in^max is controlled by the inlet pressure amplitude p_a

(see Figure <ref>(A)). 

Furthermore, the aforementioned weakly non-linear formalization simultaneously leads equation (<ref>) to be approximated as λ_±≈ c_0±,
and the results are shown in Figure <ref>(C).
We also confirmed that the average fluid velocity increased with tube stiffness, as shown in Figure <ref>(D).







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