#  Hemodynamic Effects of Entry and Exit Tear Size in Aortic Dissection
Evaluated with In Vitro Magnetic Resonance Imaging and Fluid-Structure
Interaction Simulation

Judith Zimmermann  Stanford University, Department of Radiology, Stanford, CA
University of California, San Francisco, Department of Radiology and
Biomedical Imaging, San Francisco, CA  Kathrin Bäumler  Stanford University,
Department of Radiology, Stanford, CA  Corresponding author, email:
<EMAIL_ADDRESS>Michael Loecher  Stanford University, Department of
Radiology, Stanford, CA  Veterans Affairs Health Care System, Division of
Radiology, Palo Alto, CA  Tyler E. Cork  Stanford University, Department of
Radiology, Stanford, CA  Stanford University, Department of Bioengineering,
Stanford, CA  Veterans Affairs Health Care System, Division of Radiology, Palo
Alto, CA  Alison L. Marsden  Stanford University, Department of
Bioengineering, Stanford, CA  Stanford University, Department of Pediatrics,
Stanford, CA  Daniel B. Ennis  Stanford University, Department of Radiology,
Stanford, CA  Veterans Affairs Health Care System, Division of Radiology, Palo
Alto, CA  Dominik Fleischmann  Stanford University, Department of Radiology,
Stanford, CA

###### Abstract

Understanding the complex interplay between morphologic and hemodynamic
features in aortic dissection is critical for risk stratification and for the
development of individualized therapy. This work evaluates the effects of
entry and exit tear size on the hemodynamics in type B aortic dissection by
comparing fluid-structure interaction (FSI) simulations with in vitro 4D-flow
magnetic resonance imaging (MRI). A baseline patient-specific 3D-printed model
and two variants with modified tear size (smaller entry tear, smaller exit
tear) were embedded into a flow- and pressure-controlled setup to perform MRI
as well as 12-point catheter-based pressure measurements. The same models
defined the wall and fluid domains for FSI simulations, for which boundary
conditions were matched with measured data. Results showed exceptionally well
matched complex flow patterns between 4D-flow MRI and FSI simulations.
Compared to the baseline model, false lumen flow volume decreased with either
a smaller entry tear ($-17.8-18.5\text{\,}\mathrm{\char 37\relax}$, for FSI
simulation and 4D-flow MRI, respectively) or smaller exit tear
($-16.0-17.3\text{\,}\mathrm{\char 37\relax}$). True to false lumen pressure
difference (initially $11.07.9\text{\,}\mathrm{mmHg}$, for FSI simulation and
catheter-based pressure measurements, respectively) increased with a smaller
entry tear ($28.914.6\text{\,}\mathrm{mmHg}$), and became negative with a
smaller exit tear ($-20.6-13.2\text{\,}\mathrm{mmHg}$). This work establishes
quantitative and qualitative effects of entry or exit tear size on
hemodynamics in aortic dissection, with particularly notable impact observed
on FL pressurization. FSI simulations demonstrate acceptable qualitative and
quantitative agreement with flow imaging, supporting its deployment in
clinical studies.

###### keywords:

Aortic Dissection, 4D-Flow MRI, Fluid Structure Interaction, 3D-Printing

## Introduction

Aortic dissection is a life-threatening cardiovascular disease, typically
presenting with an acute and dramatic onset, followed by a chronic life-long
phase of increased risk for late adverse events. An estimate of up to
$138\,000$ individuals in the United States alone live with chronic aortic
dissection [1]. Its pathological substrate and hallmark is the formation of a
secondary flow channel within the aorta, the false lumen (FL), caused by
abrupt delamination of the inner aortic wall layers. The delaminated tissue
that seperates the FL from the original true lumen (TL) is referred to as the
dissection flap. Blood flow enters the FL through the entry tear upstream, and
returns to the TL downstream through one or more exit tears [2]. The outer
boundary of the FL is mechanically weak and prone to fatal aortic rupture,
particularly in the first 24 to 48 hours [3], which presents the most common
cause of death in the acute phase. Other major concerns are branch vessel
ischemia resulting in organ malperfusion, and aneurysmal dilation of the FL.
For all patients reaching the chronic phase, life-long clinical monitoring is
mandatory, and surgical or endovascular intervention is required in
$4060\text{\,}\mathrm{\char 37\relax}$ of patients within the first five years
[4, 5].

Anatomical characteristics of aortic dissections vary substantially between
patients. Specifically, an aortic dissection without the involvement of the
ascending aorta (Stanford type B aortic dissection, TBAD) is managed medically
for the majority of patients ($6070\text{\,}\mathrm{\char 37\relax}$) [1].
However, studies suggest that late adverse events occur in
$3438\text{\,}\mathrm{\char 37\relax}$ of initially uncomplicated TBADs [1].
According to latest guidelines [6], preventive thoracic endovascular aortic
repair (TEVAR) may be considered for TBAD; nevertheless, the role and timing
of TEVAR remains an evolving matter of debate.

Surveillance imaging is critical to identify potentially vulnerable patients
and initiate intervention informed by measurable predictive biomarkers. A
number of previous works investigated morphological features that were mostly
derived from computed tomography angiography (CTA) image data [7, 8, 9, 10,
11, 12, 13, 14]. Adding to morphological measures, recent studies increasingly
investigated hemodynamics to further advance prediction models. For example,
decreased outflow through FL branch vessels [11], increased FL ejection
fraction [15, 16, 17], and FL pressurization [18, 19, 20] were determined to
promote late adverse events. Further, these hemodynamic features have been
associated with patient-specific morphology. Cuellar-Calabria et al. [21]
presented entry tear dominance (i.e. entry tear much larger than exit tear) as
a multivariable predictor, while Fleischmann and Burris [22] hypothesize a
linkage between tear dominance, increased outflow resistance, and resulting FL
pressurization. That is, to optimize risk stratification, predictive models
must comprise both morphological and hemodynamic indices.

Hemodynamic parameters can be obtained via computational fluid dynamics (CFD)
and through flow imaging. CFD simulates patient-specific flow fields and
pressures at high spatio-temporal resolution, and fluid-structure interaction
(FSI) additionally incorporates the effects of the deformable vessel wall [23,
24]. If simulations were able to reliably replicate _in vivo_ hemodynamics, it
would tremendously expand non-invasive means to predict risk related to
pathological changes and response to interventions. Numerous studies have
deployed CFD frameworks (with and without FSI) to study hemodynamics in
patient-specific TBAD geometries [25, 26, 19, 18, 27, 20]. But, all face three
key technical challenges: 1) it is difficult to obtain accurate patient-
specific material parameters such as wall stiffness and fluid viscosity; 2)
sparse clinical data, particularly of luminal pressure, often prevents a
complete prescription of the boundary conditions; and 3) lacking _in vivo_
measured flow and pressure data hinders rigorous validation.

4D-flow magnetic resonance imaging (MRI) measures time-resolved 3D vector-
valued velocity maps within a 3D volume of interest [28], and has shown
promising potential for measuring flow dynamics in dissections [29, 30, 31].
Owing to lengthy scan times, however, clinical in vivo 4D-flow MRI must be
performed using high acceleration rates and ultra-fast sequences, but trades-
off in spatio-temporal image resolution and signal-to-noise-ratio (SNR) are
inevetiable [32, 33, 34, 35, 36, 37]. Sub-optimal data quality ultimately
hampers the retrieval of quantitative metrics, which in turn challenges the
applicability of in vivo 4D-flow MRI for validating simulations [23, 38].

We recently reported a flow- and pressure-controlled MRI-compatible setup
which embeds subject-specific 3D-printed aortas with compliant walls that is
appropriate for comparison with FSI simulations [39, 40]. The setup enables
direct measurement of all boundary conditions and material parameters, and
yields prolonged acquisition times for high-quality 4D-flow MRI data, thus
overcoming the technical challenges of studies that deploy a CFD-only
approach. Herein, we build upon this work and generate in vitro 4D-flow MRI
from three TBAD geometries to establish a comprehensive comparison with FSI
simulations (Fig. 1). The objectives of this study were: (1) to evaluate the
impact on hemodynamic metrics in TBAD amid variations in entry and exit tear
area; and (2) to compare these hemodynamic metrics as obtained from FSI
simulations with in vitro 4D-flow MRI.

Figure 1:  Pipeline showing the key methodological steps: (1) image-based
patient-specific model generation from CTA data; (2) compliant 3D-printing;
(3) experiments (MRI + catheter-based pressure mapping) with in vitro setup;
(4) FSI simulations with boundary conditions informed by measured data (dashed
lines); and (5) data analysis, i.e. quantitative and qualitative comparison of
hemodynamics between measured and simulated data. CTA: computed tomography
angiogram, $Q(t)$: 2D-PC measured net flow, $P_{sys}$: systolic pressure,
$P_{dias}$: diastolic pressure, $P_{MAP}$: mean arterial pressure, $E_{y,t}$:
tangent Young’s modulus wall, $\rho_{s}$: density wall, $\rho_{f}$: density
fluid, $\mu_{f}$: dynamic viscosity fluid, RCR: Windkessel components, ETS:
external tissue support, TL: true lumen, FL: false lumen.

## Methods

### Patient-specific model generation

#### Anatomy and imaging data

A 3D CTA dataset of a 25 year old woman with an uncomplicated TBAD was
selected retrospectively from our institutional database. CTA and all other
methods were performed in accordance with standard of care procedures. Data
retrieval was approved by Stanford University Institutional Review Board
(#39377, “Image Registry For Computer Simulations and Image-Based Modeling in
Congenital and Acquired Cardiovascular Disease”, PI: Alison L. Marsden). The
requirement for written consent was waived due to the retrospective nature of
the study. Contrast enhanced CTA images were acquired at
$1\text{\,}\mathrm{mm}$ section thickness and
$0.7\text{\times}0.7\text{\,}{\mathrm{mm}}^{2}$ pixel size from the thoracic
inlet to the common femoral arteries. The imaging findings revealed an acute
TBAD in the descending thoracic aorta with a dissection flap extending from
the left subclavian artery origin down to the distal thoracic aorta, ending
above the level of the diaphragm. The proximal entry tear into the false lumen
was located immediately distal to the left subclavian artery, and a distal
exit tear was located above the diaphragm, superior to the celiac artery
origin. The cross-sectional area of entry and exit tear was
$228\text{\,}{\mathrm{mm}}^{2}$ and $227\text{\,}{\mathrm{mm}}^{2}$,
respectively. No branch vessels except for small intercostal arteries branched
off of the TL or FL.

#### Segmentation and digital wall model

A digital model of the patient’s thoracic aorta was generated from the CTA
dataset. The model included the ascending aorta from above the sinotubular
junction, the aortic arch, and the descending thoracic aorta to the level of
the diaphragm. The three aortic arch branches were included in the model:
brachiocephalic trunk (BCT), left common carotid artery (LCC), and left
subclavian artery (LSA). The intercostal arteries were excluded. The model
domain was extended to $35\text{\,}\mathrm{mm}$ distal to the re-entry tear.
Two mesh domains were generated: The ‘fluid domain’ – representing the aortic
lumen – was segmented using active contours with manual refinements (itk-SNAP
v3.4.0). The ‘wall domain’ – representing the outer aortic wall and the
dissection flap – was extruded from the fluid domain (Autodesk Meshmixer v3.5)
with uniform thickness $h=$2\text{\,}\mathrm{mm}$$ (Fig. 2a). A detailed
description of the TBAD model generation is provided in Bäumler et al. [23].

Figure 2:  TBAD models. (a) Original digital wall model with definition of 18
landmarks that were defined for tuning and analysis purposes. Note that
${DAo}_{prox}$ cuts through the entry tear, while ${DAo}_{dist}$ is positioned
just below the exit tear. Inset images show 2D-cine MRI frames for cross-
sections along the dissected region. Close-up view of entry and exit tear
regions for the (b) original model $\mathrm{TBAD}_{OR}$ , (c) smaller entry
tear model $\mathrm{TBAD}_{EN}$ , and (d) smaller exit tear model
$\mathrm{TBAD}_{EX}$ .

#### Digital Modification of Entry and Exit Tear Size

The original patient-specific model $\mathrm{TBAD}_{OR}$ was digitally
modified to generate two additional models with reduced entry or exit tear
size: $\mathrm{TBAD}_{EN}$ with reduced entry tear and original exit tear
size, and $\mathrm{TBAD}_{EX}$ with unchanged entry tear but reduced exit tear
size (Fig. 2b, c, d). The smaller size tears were
$\approx$27\text{\,}\mathrm{\char 37\relax}$$ of their original size,
measuring $62\text{\,}{\mathrm{mm}}^{2}$ (entry tear) and
$61\text{\,}{\mathrm{mm}}^{2}$ (exit tear), respectively. All models were
augmented with visual landmarks to enable a precise definition of cross-
sectional planes for data analysis and comparison. Additionally, the
3D-printed models were augmented with cylindrical caps to facilitate tubing
connections. Meshing and model modification tasks were performed using
SimVascular [41] and Autodesk Meshmixer.

### Compliant 3D-printing

The wall models of $\mathrm{TBAD}_{OR}$ , $\mathrm{TBAD}_{EN}$ ,
$\mathrm{TBAD}_{EX}$ were 3D-printed using a stereolithography technique
(PolyJet J735, Stratasys Inc.) and a photopolymer print resin (Agilus30,
Stratasys Inc.). Printed models were finished with a thin coating film
(Bectron, Elantas) to prevent fluid absorption and maintain the material
characteristics during the flow experiments. The compliant print material
underwent uniaxial tensile testing [39] and proved to be comparable to in vivo
aortic wall compliance (tangent Young’s modulus
$E_{y,t}=$1.2\text{\,}\mathrm{MPa}$$).

### Experiments

#### Flow setup

An MRI-compatible flow setup (Supplementary Fig. S1) was assembled, driven by
a programmable pump (CardioFlow 5000 MR, Shelley Medical Imaging Technologies)
capable of providing highly-controlled flow and pressure conditions within a
physiologic range [39]. In brief, each model was connected to the circuitry
tubing via custom designed 3D-printed barbed connectors with tapered
transitions (Form 3B, Formlabs Inc.). Each model was then embedded into a
solid ballistics gelatin block (ClearBallistics) for stabilization and to
provide static MRI signal regions for velocity (phase) offset corrections. The
model and gelatin mold were placed inside an enclosed box and connected
through box-mounted flow fittings to the periphery. Two capacitance elements
controlled the system’s downstream compliance, and in turn, pressure waveform
amplitude. The net flow splits were adjusted through pinch valves distal to
the capacitance elements; an ultrasonic flow sensor (ME-PXL14, Transonic) was
used to fine tune the flow splits of the system. The same pinch valves were
used to control the outlet resistance and to tune the mean pressure conditions
in the system. A pressure transducer (SPR-350S, Millar) was inserted through
two ports to record pressure measurements at 12 locations along the model.

Glycerol-water (ratio = 40/60) with added MR contrast medium (ferumoxytol,
concentration: $0.75\text{\,}\mathrm{mL}\text{/}\mathrm{L}$) was used as a
blood-mimicking fluid with strong $T_{1}$ MRI signal enhancement. Fluid
density ($\rho_{f}$) and dynamic viscosity ($\mu_{f}$) measurements confirmed
$\rho_{f}=$1.1\text{\,}\mathrm{g}\text{\,}{\mathrm{mm}}^{-3}$$ and
$\mu_{f}=$0.0042\text{\,}\mathrm{Pa}\text{\,}\mathrm{s}$$. A typical aortic
flow waveform was applied (heart rate = $60\text{\,}\text{/}\mathrm{min}$,
stroke volume = $74.1\text{\,}\mathrm{mL}\text{/}\mathrm{s}$, peak flow rate =
$300\text{\,}\mathrm{mL}\text{/}\mathrm{s}$, total flow =
$4.45\text{\,}\mathrm{L}\text{/}\mathrm{min}$, Supplementary Fig. S1b).

#### System tuning and pressure mapping

For each of the three models, pressure and flow split conditions were tuned on
the MRI scanner table prior to image acquisition. Tuning targets were defined
as follows: flow split (outlet vs. combined arch branches) 70/30, inlet
systolic pressure ($P_{s}$) $120\text{\,}\mathrm{mmHg}$ or greater, and inlet
diastolic pressure ($P_{d}$) $70\text{\,}\mathrm{mmHg}$ or greater. The
downstream resistance was reduced in the models with smaller entry or exit
tear sizes ($\mathrm{TBAD}_{EN}$ and $\mathrm{TBAD}_{EX}$ compared to
$\mathrm{TBAD}_{OR}$ ) to maintain pressure targets. Once tuning targets were
met, luminal pressure data were recorded at twelve locations: one proximal to
the dissection, five locations within the TL, five within the FL, and one
distal to the dissection (Fig. 2a); three recordings of five consecutive pump
cycles (five seconds) were obtained per landmark.

#### MR imaging protocol

MRI acquisitions were performed on a $3\text{\,}\mathrm{T}$ MRI scanner
(Skyra, Siemens Healthineers) with a 32-channel spine and 18-channel body
matrix coil. The pump provided an external trigger signal to retrospectively
gate time-resolved sequences. Image data were acquired immediately after
system tuning and pressure mapping.

The protocol included sequences with parameters as follows: (1) 3D spoiled
gradient echo (3D-SPGR): voxel size =
$1.1\text{\times}1.1\text{\times}1.1\text{\,}{\mathrm{mm}}^{3}$, FoV =
$220\text{\times}123\text{\,}{\mathrm{mm}}^{2}$, TE = $3\text{\,}\mathrm{ms}$,
TR = $5.25\text{\,}\mathrm{ms}$, flip angle = $None$. 3D-SPGR datasets were
acquired under the following conditions: (i) no flow (“flow off”) and (ii)
steady flow, thus no cardiac gating was applied. (2) 2D phase contrast (2D-PC)
at eleven cross-sections (Fig. 2a): pixel size =
$1.1\text{\times}1.1\text{\,}{\mathrm{mm}}^{2}$, slice thickness =
$6\text{\,}\mathrm{mm}$, FoV =
$220\text{\times}123\text{\,}{\mathrm{mm}}^{2}$, TE = $3\text{\,}\mathrm{ms}$,
TR = $5.25\text{\,}\mathrm{ms}$, flip angle = $None$, averages = 2, $V_{enc}$
= 100–170 $\mathrm{cm}\text{\,}{\mathrm{s}}^{-1}$, retrospective gating (40
frames, temporal resolution = $25\text{\,}\mathrm{ms}$). (3) 2D cine gradient
echo (2D-cine) at eleven cross-sections (Fig. 2a): pixel size =
$0.9\text{\times}0.9\text{\,}{\mathrm{mm}}^{2}$, slice thickness =
$6\text{\,}\mathrm{mm}$, FoV = $240\text{\times}150\text{\,}{\mathrm{mm}}^{2}$
TE = $3\text{\,}\mathrm{ms}$, TR = $4.75\text{\,}\mathrm{ms}$, flip angle =
$None$, averages = 2, retrospective gating (40 frames, temporal resolution =
$25\text{\,}\mathrm{ms}$). (4) 4D-flow with Cartesian k-space sampling: FoV =
$340\text{\times}236\text{\times}84\text{\,}{\mathrm{mm}}^{3}$, matrix =
$220\text{\times}156\text{\times}56\text{\,}\mathrm{,}$ voxel size =
$1.5\text{\times}1.5\text{\times}1.5\text{\,}{\mathrm{mm}}^{3}$, TE = $2.7$
$\mathrm{ms}$, TR = $5.6$ $\mathrm{ms}$, flip angle = $None$, parallel imaging
(GRAPPA, R=2), $V_{enc}$ = 120–180 $\mathrm{cm}\text{\,}{\mathrm{s}}^{-1}$,
lines/segment = 2, retrospective gating (20 frames, temporal resolution =
$50\text{\,}\mathrm{ms}$). Scan time per model was 1 hour and 30 minutes, and
total end-to-end experiment time for all models was 12 hours.

### FSI simulations

This work’s simulation approach was built on previously presented work on FSI
simulations in a patient-specific aortic dissection case [23]. To match the
experimental conditions for the present study, the following adaptations were
made: (i) we prescribed homogeneous material properties throughout the
structural domain, i.e. no distinction between intimal flap and outer material
wall properties; (ii) prestress of the structural domain was set to zero.

#### Mesh generation

For each geometric model ($\mathrm{TBAD}_{OR}$ , $\mathrm{TBAD}_{EN}$ , and
$\mathrm{TBAD}_{EX}$ ) we created refined unstructured tetrahedral meshes
with the TetGen mesh generator [42] (embedded in SimVascular [41]). Based on
previous simulations, a mesh size of $1.3\text{\,}\mathrm{mm}$ was found to be
sufficiently small [23]. Each mesh consisted of approximately
$1.1\text{\times}{10}^{6}$ tetrahedral elements for the fluid domain and
approximately $0.5\text{\times}{10}^{6}$ tetrahedral elements for wall domain.

#### Governing equations

The fluid flow was governed by the Navier–Stokes equations with viscosity
$\mu_{f}=$0.0042\text{\,}\mathrm{Pa}\text{\,}\mathrm{s}$$ and density
$\varrho_{f}=$1100\text{\,}\mathrm{kg}\text{\,}{\mathrm{m}}^{-3}$$. The FSI
equations were formulated in arbitrary Lagrangian-Eulerian (ALE) coordinates
[43, 23], which allowed us to capture the coupled motion of the fluid and
structural domain using a finite element method (described below). The
glycerol-water mixture used in the experimental setup was modeled as
incompressible and Newtonian. The 3D printed material representing the
arterial wall was modeled as a homogeneous, isotropic, nonlinear, hyperelastic
material, and described by a Neo-Hookean material model [43]. Structural
density (per manufacturer’s spec sheet) and elasticity (as measured) were
$\varrho_{s}=$1450\text{\,}\mathrm{kg}\text{\,}{\mathrm{m}}^{-3}$$ and
$E_{y,t}=$1.2\text{\,}\mathrm{MPa}$$, respectively. To account for the
presence of the gelatin mold surrounding the printed models we applied
external tissue support (ETS) to the outer arterial wall in the form of a
Robin boundary condition [44]:

$\bm{\sigma_{S}n}=-k_{S}\bm{u_{S}}-c_{S}\partial_{t}\bm{u_{S}}-p_{0}\bm{n},$
(1)

where $\bm{\sigma_{S}}$ denotes the Cauchy stress tensor of the structural
domain, $\bm{n}$ the unit outer normal vector, $\bm{u_{S}}$ the displacement
field, and $p_{0}$ the external (or intrathoracic) pressure. The user-defined
scalar parameters $k_{S}$ and $c_{S}$ denote the elastic and viscoelastic
response of the external tissue, respectively.

#### Boundary data and tuning

At the model inlet we prescribed a pulsatile flow waveform that was informed
from in vitro 2D-PC MRI measurements; a parabolic velocity profile was used.
At the four model outlets, we applied three-element Windkessel boundary
conditions which account for downstream vascular effects [45]. The total
resistance $R_{T}$ and capacitance $C_{T}$ were tuned and distributed across
all outlets such that every outlet $i$ was described by a proximal resistance
$R_{P,i}$, distal resistance $R_{D,i}$, and capacitance $C_{i}$. The ratio of
distal to proximal resistance $k_{d}$ was fixed for each model, and the
distribution of total resistance and capacitance were governed by the
experimentally determined flow splits. Details of the tuning process have been
described in Bäumler et al. [23]. The target pressure values comprised
systolic pressure ($P_{sys}$), diastolic pressure ($P_{dias}$), and mean
pressure ($P_{MAP}$) values — informed by experimental pressure mapping.
Target flow ratios across outlets BCT, LCC, LSA, and outlet were informed by
2D-PC net flow measurements. Tuning targets (Table 1) were matched with
relative residual errors of $\leq 1.4\text{\,}\mathrm{\char 37\relax}\leq
4.1\text{\,}\mathrm{\char 37\relax}\leq 4.6\text{\,}\mathrm{\char 37\relax}$
for pressure and $\leq 3.4\text{\,}\mathrm{\char 37\relax}\leq
7.4\text{\,}\mathrm{\char 37\relax}\leq 7.3\text{\,}\mathrm{\char 37\relax}$
for flow splits (corresponding to $\mathrm{TBAD}_{OR}$ , $\mathrm{TBAD}_{EN}$
, and $\mathrm{TBAD}_{EX}$ , respectively)

|  |  pressure tuning at inlet |  | flow split tuning across outlets
---|---|---|---|---
|  | $P_{sys}$ | $P_{dias}$ | $P_{MAP}$ |  | BCT | LCC | LSA | outlet
$\mathrm{TBAD}_{OR}$ | simulated (mmHg) | 127.5 | 59.1 | 86.1 | simulated (%) | 15.5 | 3.2 | 5.6 | 75.7
| measured (mmHg) | 126.4 | 60.0 | 86.1 | measured (%) | 15.0 | 3.1 | 5.4 | 76.5
| residual error (%) | 0.82 | 1.42 | 0.08 | residual error (%) | 3.37 | 3.22 | 3.08 | 1.02
$\mathrm{TBAD}_{EN}$ | simulated (mmHg) | 141.9 | 72.7 | 95.7 | simulated (%) | 12.3 | 3.0 | 6.1 | 78.6
| measured (mmHg) | 137.7 | 69.9 | 96.2 | measured (%) | 11.4 | 2.8 | 5.7 | 80.1
| residual error (%) | 3.02 | 4.11 | 0.57 | residual error (%) | 7.43 | 7.21 | 7.00 | 1.84
$\mathrm{TBAD}_{EX}$ | simulated (mmHg) | 148.6 | 75.1 | 100.4 | simulated (%) | 11.7 | 3.3 | 5.0 | 80.1
| measured (mmHg) | 142.1 | 73.5 | 98.9 | measured (%) | 10.9 | 3.1 | 4.6 | 81.4
| residual error (%) | 4.58 | 2.24 | 1.5 | residual error (%) | 7.26 | 7.06 | 6.93 | 1.63
Table 1:  Inlet pressure ($\mathrm{mmHg}$) and flow splits ($\mathrm{\char
37\relax}$ of total outflow) for the three models. The FSI simulations matched
target pressure values (systolic ($P_{sys}$), diastolic ($P_{dias}$), and mean
($P_{MAP}$)) with a relative error of $\leq$4.58\text{\,}\mathrm{\char
37\relax}$$ and absolute error of $\leq$6.5\text{\,}\mathrm{mmHg}$$. Flow
split ratios between outlets were matched in the FSI simulations with a
relative error of $\leq$7.43\text{\,}\mathrm{\char 37\relax}$$ and an absolute
error or $\leq$1.5\text{\,}\mathrm{\char 37\relax}$$.

ETS parameters $k_{S}$ and $c_{S}$ were chosen such that the FSI-simulated
minimum-to-maximum lumen dilation at AAo matched the MRI-measured minimum-to-
maximum dilation in the 3D-printed models. ETS tuning yielded a relative
dilation of $5.0\text{\,}\mathrm{\char 37\relax}5.0\text{\,}\mathrm{\char
37\relax}5.3\text{\,}\mathrm{\char 37\relax}$ in FSI compared to
$4.4\text{\,}\mathrm{\char 37\relax}5.0\text{\,}\mathrm{\char
37\relax}5.4\text{\,}\mathrm{\char 37\relax}$ in 2D-cine MRI (for
$\mathrm{TBAD}_{OR}$ , $\mathrm{TBAD}_{EN}$ , and $\mathrm{TBAD}_{EX}$ ,
respectively). Tuning parameter settings and pressure traces are provided in
Supplementary Table S2 and Supplementary Fig. S3, respectively.

#### Numerical solver

The governing equations were solved with the open source finite element solver
svFSI (SimVascular [41, 46]). svFSI features a second order generalized
$\alpha$-time stepping scheme, linear tetrahedral elements for pressure and
velocity (with pressure and momentum stabilization), two-way coupling of the
fluid and structural domain, pre-conditioning of the resulting linear systems,
and backflow stabilization at the fluid outlets [47, 48, 49]. To assure
periodicity of the simulation, at least 6 cardiac cycles were simulated before
results were extracted for the final cycle. The simulations were carried out
on two nodes (Intel(R) Xeon(R) Gold 5118, 2.30 GHz) with a total of 48 CPUs;
approximate runtime was ten hours per cardiac cycle.

### Data analysis

For MRI data, TBAD lumen segmentation was performed on high-resolution 3D-SPGR
steady flow data (using watershed-based region-growing) and facilitated image-
based flow visualization. 2D-cine datasets were used to automatically track
through-cycle aortic wall and flap deformation [50] and to compute cross-
sectional area change. 2D-PC datasets were corrected for velocity (phase)
offsets and processed to retrieve the inlet flow rate waveform as well as
percentage net flow splits across outlets. 4D-flow datasets were corrected for
the following conditions: (i) Maxwell terms (during MR reconstruction), (ii)
gradient non-linearity distortion [51], and (iii) phase offsets. 4D-flow
datasets were reformatted at each analysis location (Fig. 2a) to generate
cross-sectional phase and magnitude images. Reformatted magnitude images were
used to track lumen boundaries which served to calculate quantitative flow
parameters based on reformatted phase images. Post-processing of the MRI data
was completed using MevisLab modules (v3.5a, Fraunhofer Institute for Digital
Medicine) and ParaView (v5.7, Kitware). Pressure measurements were analyzed
using dedicated data acquisition system software (LabChart 8, ADInstruments)
and pressure signals were averaged over five cardiac cycles. FSI simulation
results were exported as VTK unstructured grid files (.vtu) from svFSI at 80
time-steps over a single cardiac cycle (i.e. temporal frame length:
$12.5\text{\,}\mathrm{ms}$). Qualitative visualization and quantitative
analysis of simulated datasets were performed using ParaView (v.5.7, Kitware).

## Results

### Flow patterns

Each of the three models exhibited a unique flow pattern and velocity
distributions, with excellent qualitative agreement between 4D-flow MRI and
FSI simulations (Figs. 3, 4, and Supplementary Videos S4-S8). Highly complex
and distinct flow patterns were observed in the proximal false lumen adjacent
to the entry tear region. A high-velocity flow jet through the entry tear with
impingement on the opposite FL wall was only observed in the
$\mathrm{TBAD}_{EN}$ model. The impingement zone was also associated with a
steep local pressure gradient affecting the FL wall (Fig. 4c). Helical flow
patterns were also visible in the proximal TL close to the entry tear and were
well captured by both techniques (Fig. 3b). In the distal part of the
dissected descending aorta, the modified models, $\mathrm{TBAD}_{EN}$ and
$\mathrm{TBAD}_{EX}$ , showed increased flow velocities in the TL (Fig. 3a,
arrows), which was consistent with the higher TL net flow volumes in these
models, as reported below. In addition, $\mathrm{TBAD}_{EX}$ showed increased
velocity through the exit tear compared to the $\mathrm{TBAD}_{OR}$ and
$\mathrm{TBAD}_{EN}$  models. The smaller exit tear in the
$\mathrm{TBAD}_{EX}$ model also caused reversed flow distal to the exit tear
(Fig. 3c, Supplementary Video S8), whereas $\mathrm{TBAD}_{OR}$ and
$\mathrm{TBAD}_{EN}$ showed unidirectional laminar flow in this region. During
diastole, additional flow oscillations through the entry tear (in particular
for $\mathrm{TBAD}_{EN}$ , Fig. 4b), and persistent helical flow in FL
(Supplementary Videos S6, S7) were observed.

### Flowrate measurements

#### FL flow ratio

The FL flow ratio ($\mathrm{FR}_{\mathrm{FL}}$, in $\mathrm{\char 37\relax}$)
was defined based on the TL and FL net flow volumes
($\overline{\mathrm{Q}}_{\mathrm{TL}}$, $\overline{\mathrm{Q}}_{\mathrm{FL}}$)
as:

$\mathrm{FR}_{\mathrm{FL}}\coloneqq\frac{\overline{\mathrm{Q}}_{\mathrm{FL}}}{\overline{\mathrm{Q}}_{\mathrm{FL}}+\overline{\mathrm{Q}}_{\mathrm{TL}}}*100.$
(2)

$\mathrm{FR}_{\mathrm{FL}}$ was substantially decreased in the modified tear
models, $\mathrm{TBAD}_{EN}$ and $\mathrm{TBAD}_{EX}$ , compared to the
original $\mathrm{TBAD}_{OR}$  model. Specifically,
$\mathrm{FR}_{\mathrm{FL}}$ was $73.7\text{\,}\mathrm{\char
37\relax}55.2\text{\,}\mathrm{\char 37\relax}56.4\text{\,}\mathrm{\char
37\relax}$ for 4D-flow MRI and $77.6\text{\,}\mathrm{\char
37\relax}59.8\text{\,}\mathrm{\char 37\relax}61.6\text{\,}\mathrm{\char
37\relax}$ for FSI simulations (values given for $\mathrm{TBAD}_{OR}$ ,
$\mathrm{TBAD}_{EN}$ , and $\mathrm{TBAD}_{EX}$ , respectively).

Figure 3:  (a) Streamlines at peak systole ($t=$200\text{\,}\mathrm{ms}$$)
rendered from 4D-flow MRI (top) and FSI-simulated (bottom) data. Each model
exhibits unique local flow characteristics that are in agreement between
techniques. Key observations include: (i) increased flow velocities through
the entry tear region, particularly in $\mathrm{TBAD}_{EN}$ , and local
helical flow in the proximal TL and FL in the vicinity of the entry tear (blue
box, close-up view in (b)); (ii) increased TL flow velocity for the modified
$\mathrm{TBAD}_{EN}$ and $\mathrm{TBAD}_{EX}$  models (arrows); (iii) flow
jet through small size exit tear in $\mathrm{TBAD}_{EX}$ , with recirculating
TL flow distal to the exit tear (orange box, close-up view in (c)). Graphics
created using ParaView (v5.7, https://www.paraview.org).  Figure 4:  Flow and
pressure dynamics in the arch region. Velocity vectors in the aortic arch at
(a) peak systole ($t=$200\text{\,}\mathrm{ms}$$) (b) and mid-diastole
($t=$600\text{\,}\mathrm{ms}$$) based on 4D-flow MRI data and FSI simulations.
During systole, flow patterns are in agreement between both modalities and for
each model, but velocities are higher in FSI simulations. Mid-diastole
renderings reveal a secondary push through the entry tear, that is most
pronounced in $\mathrm{TBAD}_{EN}$ . For full-cycle animations refer to
Supplementary videos 1-5. (c) Absolute pressure at the lumen boundary
($t=$250\text{\,}\mathrm{ms}$$) from the FSI simulations. $\mathrm{TBAD}_{EN}$
shows a local pressure difference of $\approx 35\text{\,}\mathrm{mmHg}$ in the
FL impingement zone. Graphic created using ParaView (v5.7,
https://www.paraview.org).

#### Dynamic flowrates over the cardiac cycle

Transient flow rates at multiple cross-sectional landmarks showed higher
systolic peaks in FSI simulations, whereas higher diastolic flow was observed
in 4D-flow MRI (Fig. 5a). Damping of the flow waveform from
$\mathrm{DAo}_{\mathrm{prox}}$ to $\mathrm{DAo}_{\mathrm{dist}}$ as well as
along the individual lumina was observed in all three models, with stronger
damping in 4D-flow and 2D-PC data than in FSI simulations. Diastolic flow
oscillations were most pronounced in FSI simulations, with short periods of
reverse (i.e. negative) flow at $t\approx$500\text{\,}\mathrm{ms}$$. To
further assess potential underestimation of systolic flow in 4D-flow MRI,
2D-PC measurements at identical locations were added to Fig. 5a.

Figure 5:  (a) Flow rates over the cardiac cycle along the dissected
descending aorta based on 4D-flow MRI (green), FSI simulations (blue), and
2D-PC MRI (black, dashed). (b) Cross-sectional area over the cardiac cycle
based on 2D-cine MRI (light blue) and FSI simulations (purple). Relative area
change was defined as absolute area divided by area in the first cycle frame.
For landmark label definition see Fig. 2a.

### Cross-sectional area measurements

The cross-sectional lumen area was evaluated as absolute area, and as relative
area changes over the cardiac cycle. For the latter, the first frame was used
as the end-diastolic reference for normalization. In addition to area
measurements during pulsatile flow, we also obtained absolute area values from
the STL model file, from both “flow-off” and steady flow MRI imaging datasets,
and from steady flow FSI simulations. These additional measurements were all
based on the $\mathrm{TBAD}_{OR}$  model.

#### Absolute area measurements

FSI simulations, compared to 2D-cine MRI based wall contour tracking, resulted
in smaller absolute cross-sectional area at the majority of landmarks (Fig.
6b). Specifically, FSI-based area measurements in peak-systole and end-
diastole (mean $\pm$ SD) differed from MRI-based values by $-16.6\pm
10.5\text{\,}\mathrm{\char 37\relax}-7.0\pm 11.2\text{\,}\mathrm{\char
37\relax}-3.1\pm 12.2\text{\,}\mathrm{\char 37\relax}$ for
$\mathrm{TBAD}_{OR}$ , $\mathrm{TBAD}_{EN}$ , and $\mathrm{TBAD}_{EX}$
respectively. Additional “flow-off” experiments showed acceptable agreement
with STL-based values ($-4.7\pm 16.1\text{\,}\mathrm{\char 37\relax}$), but
measurements during steady flow affirmed smaller absolute cross-sectional
areas of the simulated deformable wall domain than in the printed model
($-0.6\pm 10.4\text{\,}\mathrm{\char 37\relax}$). Numerical results for all
area measurements are listed in Supplementary Table S9 and plotted in
Supplementary Fig. 10.

#### Relative area changes over cardiac cycle

Figure 5b displays transient curves of relative cross-sectional areas for
$\mathrm{TBAD}_{OR}$ , $\mathrm{TBAD}_{EN}$ , and $\mathrm{TBAD}_{EX}$ ,
respectively: TL areas increased by a factor of up to $1.021.041.03$ in the
experiments and up to $1.031.061.03$ in FSI simulations over the cardiac
cycle. FL areas increased by a factor of up to $1.041.041.05$ in experiments
and up to $1.051.031.05$ in FSI simulations for $\mathrm{TBAD}_{OR}$ ,
$\mathrm{TBAD}_{EN}$ , and $\mathrm{TBAD}_{EX}$ , respectively. TL in
$\mathrm{TBAD}_{EX}$ showed a slight collapse at the distal landmarks TL4
($\approx$-1\text{\,}\mathrm{\char 37\relax}$$) and TL5
($\approx$-3\text{\,}\mathrm{\char 37\relax}$$), as captured best in FSI
simulations; 2D-cine MRI data did not fully reveal this behavior.

### Pressure

#### Pressure drops along centerline

Peak-systolic pressure ($P_{sys}$) at twelve pressure mapping landmarks was
normalized with respect to the peak-systolic pressure at inlet (Fig. 6a).
Overall, inlet to outlet pressure drops were smallest in the original
$\mathrm{TBAD}_{OR}$ model ($2\text{\,}\mathrm{\char
37\relax}7\text{\,}\mathrm{\char 37\relax}$), and larger for the modified
$\mathrm{TBAD}_{EN}$ ($9\text{\,}\mathrm{\char
37\relax}20\text{\,}\mathrm{\char 37\relax}$) and $\mathrm{TBAD}_{EX}$
($18\text{\,}\mathrm{\char 37\relax}19\text{\,}\mathrm{\char 37\relax}$);
given value pairs correspond to experiments and FSI simulations.

Pressure drop across the entry tear (i.e. TL entering FL, see inlet to FL1)
was largest in $\mathrm{TBAD}_{EN}$ , with a decrease of
$17\text{\,}\mathrm{\char 37\relax}20\text{\,}\mathrm{\char 37\relax}$ in
experiments and FSI simulations, respectively. Pressure drop across the exit
tear (i.e. FL merging back into the TL, see FL5 to outlet) was largest in
$\mathrm{TBAD}_{EX}$ , with a pressure decrease by $11\text{\,}\mathrm{\char
37\relax}14\text{\,}\mathrm{\char 37\relax}$ in experiments and FSI
simulations, respectively.

For all models, peak-systolic pressure did not drop noticeably along the FL
centerline (FL1 to FL5). However, pressure values steadily decreased along the
TL (TL2 to TL5), with the strongest TL pressure decline observed in
$\mathrm{TBAD}_{EX}$  ($15\text{\,}\mathrm{\char
37\relax}20\text{\,}\mathrm{\char 37\relax}$ in the experiments and FSI
simulations, respectively). The largest discrepancy between experimental
pressure measurements and FSI simulations was found in the
$\mathrm{TBAD}_{EN}$  model, particularly in the FL. Here, inlet to FL
pressure (at FL1) dropped by $\geq$20\text{\,}\mathrm{\char 37\relax}$$ in FSI
simulations, but $\leq$10\text{\,}\mathrm{\char 37\relax}$$ in the
experiments.

#### TL-FL pressure differences

Inter-luminal pressure differences were calculated as

$\Delta P_{TL-FL}\coloneqq P_{TL}-P_{FL}$ (3)

at five locations along the dissected aorta. $\Delta P_{TL-FL}$was
substantially affected by tear size and steadily decreased from proximal to
distal locations (Fig. 7). $\Delta P_{TL-FL}$was positive at all landmarks and
throughout the cardiac cycle for $\mathrm{TBAD}_{OR}$ and $\mathrm{TBAD}_{EN}$
, with maximum systolic differences of ($7.9$, $11.0$) $\mathrm{mmHg}$ for
$\mathrm{TBAD}_{OR}$ and ($14.6$, $28.9$) $\mathrm{mmHg}$ for
$\mathrm{TBAD}_{EN}$ . $\mathrm{TBAD}_{EX}$ FL pressure exceeded TL pressure
by up to ($13.2$, $20.6$) $\mathrm{mmHg}$ during systole; values are given for
catheter-based and simulated data, respectively.

At end-diastole, FSI-simulated TL and FL pressures were in approximate
equilibrium ($$-0.5\text{\,}\mathrm{mmHg}$<\Delta P_{TL-
FL}<$1\text{\,}\mathrm{mmHg}$$), whereas experimental pressure measurements
resulted in $\Delta P_{TL-FL}\neq 0$. Inter-luminal diastolic pressure
differences measured slightly positive for the $\mathrm{TBAD}_{OR}$ and
$\mathrm{TBAD}_{EN}$ models ($$0.9\text{\,}\mathrm{mmHg}$<\Delta P_{TL-
FL}<$3.2\text{\,}\mathrm{mmHg}$$), and negative for the $\mathrm{TBAD}_{EX}$
model ($$-7.6\text{\,}\mathrm{mmHg}$<\Delta P_{TL-
FL}<$-5.5\text{\,}\mathrm{mmHg}$$).

Figure 6:  (a) Relative pressure (i.e. normalized to peak pressure at inlet)
drops along the aortic centerline including inlet), outlet), as well five
landmarks in both the TL and FL. (b) Maximum cross-sectional area measurements
at the identical landmarks used for pressure mapping, except for AAo, which
was chosen over ’inlet’, since inlet is fixed both in the 3D-printed model and
in the simulation setup. Dashed bars denote area values obtained from the STL
model file, squares denote size of entry tear, and triangles denote size of
exit tear. For landmark label definition see Fig. 2a. Figure 7:  Inter-luminal
pressure difference $\Delta P_{TL-FL}$ between TL and FL according to
pressure transducer measurements (top) and FSI simulations (bottom). $\Delta
P_{TL-FL}$ increased when the entry tear was made smaller
($\mathrm{TBAD}_{EN}$ ), and decreased — including negative pressure
difference at distal landmarks — when exit tear made smaller
($\mathrm{TBAD}_{EX}$ ).

## Discussion

Hemodynamic factors — together with biomechanical, mechanobiological, genetic,
and morphological factors — underlie the development and progression of aortic
dissection and are being investigated as biomarkers for prognosis and
treatment decisions. The complex interplay between these factors is
incompletely understood, and evolving computational methods to obtain patient-
specific hemodynamic must be rigorously validated. The present study assessed
the hemodynamic effects of entry and exit tear size variations in a patient-
specific TBAD model using two modalities. We directly compared hemodynamic
features of FSI simulations against in vitro MRI and catheter-based pressure
data in compliant 3D-printed TBAD models. The in vitro approach enabled the
study of quantitative hemodynamics in a highly controlled environment and
without a scan time limitation. Moreover, the experiments informed the
boundary conditions and material parameter specifications in the FSI
simulations with catheter-based pressure data and high fidelity material
property estimates, which are not typically available in any clinical setting
[23].

Local and global flow helices, recirculation zones, and flow jets through
tears were visualized in fine detail via streamlines, and were well captured
in both 4D-flow MRI and FSI simulations. In particular, local helices in the
vicinity of the entry tear (Fig. 3b, $\mathrm{TBAD}_{EN}$ ) and distal to the
exit tear (Fig. 3c, $\mathrm{TBAD}_{EX}$ ) are strikingly similar using both
techniques. The emergent entry tear flow jet in $\mathrm{TBAD}_{EN}$ impinging
on the opposite FL wall, creating a local pressure gradient was equally
demonstrated by 4D-flow MRI and FSI (Fig. 4a). This impingement zone may
promote tissue degradation and destructive remodeling via mechanobiological
pathways and potentially lead to aneurysmal degeneration [52, 53].

Comparing results between both modalities (FSI and MRI), in systole, FSI
simulations displayed overall higher flow velocities, whereas in diastole,
4D-flow measured higher velocities. Similarly, simulated flow rates were
higher in systole and lower in diastole when compared to MRI data,
specifically in 4D-flow MRI (Fig. 5a). Two confounding effects may explain
this result: First, there is general consensus that 4D-flow MRI underestimates
peak velocities when compared to 2D-PC MRI. Our results confirm that 2D-PC MRI
consistently resulted in higher peak velocities, and therefore showed smaller
discrepancies with FSI simulations. Second, less dampening seems to occur in
FSI simulations. This is nicely reflected by the initially well matched flow
rates (between 2D-PC and FSI) at landmark inlet, but then higher systolic and
lower diastolic flow (for FSI simulations) at all downstream landmarks.

Entry and exit tear size considerably affected inter-luminal pressure
differences and true and false lumen flow splits. Peak-systolic $\Delta P_{TL-
FL}$ was positive in the $\mathrm{TBAD}_{OR}$ model (up to
$11.0\text{\,}\mathrm{mmHg}$), then further increased in the reduced entry
tear model (up to $28.9\text{\,}\mathrm{mmHg}$), but flipped to negative in
the reduced exit tear model (up to $-20.6\text{\,}\mathrm{mmHg}$).
Additionally, both of the tear-modified models significantly reduced
$\mathrm{FR}_{\mathrm{FL}}$ down to $75\text{\,}\mathrm{\char 37\relax}$ of
the initial $\mathrm{TBAD}_{OR}$ value. This drastic
$\mathrm{FR}_{\mathrm{FL}}$ reduction indicates that the flow throughput is
dictated by the total resistance and independent of the location of the
narrowing. Consequently, it appears that despite the decrease in
$\mathrm{FR}_{\mathrm{FL}}$ an increase in outflow resistance — which only
occurred in the $\mathrm{TBAD}_{EX}$ model — contributes to FL pressurization,
which is thought promote FL degeneration and aneurysm formation.

Inter-modality comparison showed overall acceptable agreement for $\Delta
P_{TL-FL}$. Both FSI simulations and catheter-based measurements link a small
exit tear to false lumen pressurization, which is hypothesized to increase the
risk for late adverse events in TBAD patients. Moreover, our results align
with findings by Cuellar-Calabria et al. [21], who identified “entry tear
dominance” (here: $\mathrm{TBAD}_{EX}$ model) as a predictor of late adverse
events. Absolute numbers for FSI-derived $\Delta P_{TL-FL}$ deviated from
catheter-based measurements and showed (1) greater systolic inter-luminal
pressure gradients, and (2) diastolic inter-luminal pressure equilibrium at
all landmarks, rather than $\Delta P_{TL-FL}$$\neq 0$ in catheter-based
measurements. To explain these effects, we revisited potential causes as
follows: differences of actual tear size between the 3D-printed model and the
digital wall model, wall and fluid mesh coarseness distal to the exit tear,
and catheter-based measurement inaccuracies relative to measured $\Delta
P_{TL-FL}$. But, none of these additional analyses led to a sound explanation
as to why these discrepancies occured.

Tear size alterations also affected peak-systolic pressure drops along the
luminal centerline. First, simulated and measured data agreed on substantially
greater inlet to outlet pressure drops in both of the tear-modified models.
Similar trends that describe increased pressure drops have been reported for
aortic coarctation [54, 55], which refers to a focal narrowing of the proximal
descending aorta. Second, the modification of tear size led to different
locations of the steepest pressure drop along the centerline (Fig. 6).
$\mathrm{TBAD}_{EN}$ exhibited the largest gradient across the entry tear;
whereas $\mathrm{TBAD}_{EX}$ exhibited the largest gradient across the exit
tear. Although simulated data presented larger pressure drops between
centerline locations, relative trends again were well-matched between
modalities.

In addition to the effect of entry and exit tear sizes, our results also
showcase the relationship between the cross-sectional luminal area and the
$P_{sys}$ drops along the respective luminal centerlines. Along the TL
centerline, $P_{sys}$ decreased incrementally, while the cross-sectional area
narrowed considerably further downstream — that is, the area at TL5 was
$24\text{\,}\mathrm{\char 37\relax}$ of the area at TL1. In comparison, our
results suggest no considerable drop of $P_{sys}$ along the FL centerline,
along which cross-sectional area values remained approximately constant.

The combined results for $\Delta P_{TL-FL}$ and relative area change over the
cardiac cycle also revealed the well known clinical phenomenon of TL
compression and collapse. If only FSI simulation are considered, one can
deduce: $\Delta P_{TL-FL}$$>0$ caused an increase in TL area, $\Delta P_{TL-
FL}$$\approx 0$ was reflected by negligible area change (that is, constant
area through the cycle), and $\Delta P_{TL-FL}$$<0$ (as observed in the
$\mathrm{TBAD}_{EX}$ model only) caused a decrease in TL area, corresponding
to a compression of the TL. Area measurements from 2D-cine MRI also hint at
potential TL collapse in the $\mathrm{TBAD}_{EX}$  model.

Three limitations of the present study should be addressed. First, several
assumptions simplified the actual TBAD in vivo scenario: Both experiments and
simulations utilized a uniform thickness and elasticity for the outer aortic
wall and the dissection flap. Thus, we are unable to analyze effects of
spatially varying or non-isotropic wall characteristics on hemodynamics. The
FSI simulations also did not pre-stress the wall domain. While this matches
the in vitro measurements for a 3D-printed models, any study that considers
inherently pre-stressed in vivo tissue and data may require incorporation pre-
stress effects [23]. In addition, the ETS modeling parameters in the FSI
simulations were applied uniformly. Both the presented in vitro embedding of
the models, and likely also the surrounding of the thoracic aorta in vivo, is
expected to non-uniformly restrict the movement of the outer aortic wall.

Second, retrieving precise wall deformation measurements of the 3D-printed
models was limited by 2D-cine MRI spatial resolution as well as the inherent
error of the registration-based wall tracking algorithm. Specifically, this
affected area measurements in the relatively small-sized TL. While obtaining
structural motion from FSI simulations can be considered error-free, measuring
area based on the given 2D-cine MRI is not. Future works must refine methods
for measuring wall and flap motion in the experimental setup.

Lastly, this study investigated only a single TBAD case with its unique
patient-specific features; specifically, no fenestration points (that is,
additional small communications between the TL and FL along dissection flap)
were present, the dissection did not extend distal to the celiac trunk, and —
for feasibility purposes — we excluded intercoastal arteries when building the
model. Novel advances in 3D printing, specifically the integration of tissue
mimicking materials, provide excellent versatility in model manufacturing
regarding global and local geometry, thickness and elasticity — a technical
tool that could be leveraged in future works.

In conclusion, this work describes changes in the TBAD hemodynamics due to
tear size alterations while comparing results from in vitro MRI and pressure
mapping experiments with FSI simulations. In particular, the results
demonstrate FL pressurization owing to a decreased exit tear size — with well-
matched observations between measurements and simulation. The present study
contributes to a better understanding of the interplay between TBAD morphology
and associated quantitative hemodynamics.

## Acknowledgements

We thank Shannon Walters, Chris LeCastillo, and Kyle Gifford (Stanford 3DQ
Lab) for their 3D printing support and services, and Nicole Schiavone
(Stanford Mechanical Engineering) for advice about the technical setup. We
acknowledge the computational infrastructure (Sherlock HPC cluster) provided
by the Stanford Research Computing Center. Funding was received through DAAD
doctoral candidate scholarship (to J.Z.), NIH R01 LM013120 (to A.L.M.) and NIH
R01 HL131823 (to D.B.E.)

## Author contributions

J.Z. and K.B. jointly conceptualized the study, performed cross-modality data
analysis, prepared figures, and wrote the manuscript. J.Z. engineered the flow
loop setup and led experimental MRI studies. K.B. developed the computational
simulations framework and performed FSI simulations. M.L. and T.E.C. supported
experimental data acquisition and reconstruction. A.L.M. conceptualized the
study and supervised computational simulation work. D.B.E. conceptualized the
study and supervised experimental data acquisition. D.F. conceptualized the
study and provided overall advice to the research objective. All authors
critically revised and approved the manuscript.

## Additional Information

Data availability: TBAD model files (.stl), MRI data (.dcm), pressure traces
(.csv), and FSI-simulated data (.vtu) will be made publicly available upon
publication.

## References

  * [1] Fleischmann, D. _et al._ Imaging and Surveillance of Chronic Aortic Dissection: A Scientific Statement from the American Heart Association.  _Circ Cardiovasc Imaging_ 15, 200–221, DOI: 10.1161/HCI.0000000000000075 (2022).
  * [2] Nienaber, C. A. _et al._ Aortic dissection.  _Nat. Rev. Dis. Primers_ 2, 16053, DOI: 10.1038/nrdp.2016.53 (2016).
  * [3] Chiu, P. & Miller, D. C.  Evolution of surgical therapy for Stanford acute type A aortic dissection.  _Ann Cardiothor Surg_ 5, 275–295, DOI: 10.21037/ACS.2016.05.05 (2016).
  * [4] Pape, L. A. _et al._ Presentation, Diagnosis, and Outcomes of Acute Aortic Dissection: 17-Year Trends From the International Registry of Acute Aortic Dissection.  _JACC_ 66, 350–358, DOI: 10.1016/J.JACC.2015.05.029 (2015).
  * [5] Afifi, R. O. _et al._ Outcomes of Patients With Acute Type B (DeBakey III) Aortic Dissection: A 13-Year, Single-Center Experience.  _Circulation_ 132, 748–754, DOI: 10.1161/CIRCULATIONAHA.115.015302 (2015).
  * [6] Isselbacher, E. M. _et al._ 2022 ACC/AHA Guideline for the Diagnosis and Management of Aortic Disease: A Report of the American Heart Association/American College of Cardiology Joint Committee on Clinical Practice Guidelines.  _Circulation_ 146, 334–482, DOI: 10.1161/cir.0000000000001106 (2022).
  * [7] Schwartz, S. I. _et al._ Predictors of late aortic intervention in patients with medically treated type B aortic dissection.  _J Vasc Surg_ 67, 78–84, DOI: 10.1016/J.JVS.2017.05.128 (2018).
  * [8] Spinelli, D. _et al._ Current evidence in predictors of aortic growth and events in acute type B aortic dissection.  _J. Vasc. Surg._ 68, 1925–35, DOI: 10.1016/j.jvs.2018.05.232 (2018).
  * [9] Kunishige, H. _et al._ Predictors of surgical indications for acute type B aortic dissection based on enlargement of aortic diameter during the chronic phase.  _Jpn J Thorac Cardiovasc Surg_ 54, 477–482, DOI: 10.1007/S11748-006-0039-9 (2006).
  * [10] Tsai, T. T. _et al._ Partial Thrombosis of the False Lumen in Patients with Acute Type B Aortic Dissection.  _NEJM_ 357, 349–359, DOI: 10.1056/NEJMoa063232 (2007).
  * [11] Sailer, A. M. _et al._ Computed Tomography Imaging Features in Acute Uncomplicated Stanford Type-B Aortic Dissection Predict Late Adverse Events.  _Circ. Cardiovasc. Imaging_ 10, e005709, DOI: 10.1161/CIRCIMAGING.116.005709 (2017).
  * [12] Lavingia, K. S. _et al._ Volumetric analysis of the initial index computed tomography scan can predict the natural history of acute uncomplicated type B dissections.  _J Vasc Surg_ 62, 893–899, DOI: 10.1016/J.JVS.2015.04.449 (2015).
  * [13] Evangelista, A. _et al._ Long-term outcome of aortic dissection with patent false lumen: predictive role of entry tear size and location.  _Circulation_ 125, 3133–3141, DOI: 10.1161/CIRCULATIONAHA.111.090266 (2012).
  * [14] Tolenaar, J. L. _et al._ Morphologic predictors of aortic dilatation in type B aortic dissection.  _J Vasc Surg_ 58, 1220–1225, DOI: 10.1016/J.JVS.2013.05.031 (2013).
  * [15] Burris, N. S. _et al._ False lumen ejection fraction predicts growth in type B aortic dissection: Preliminary results.  _Eur. J. Cardio-thoracic Surg._ 57, 896–903, DOI: 10.1093/ejcts/ezz343 (2020).
  * [16] Marlevi, D. _et al._ False lumen pressure estimation in type B aortic dissection using 4D flow cardiovascular magnetic resonance: comparisons with aortic growth.  _J Cardiovasc Magn Reson_ 23, 51, DOI: 10.1186/S12968-021-00741-4 (2021).
  * [17] Zadrazil, I., Corzo, C., Voulgaropoulos, V., Markides, C. N. & Xu, X. Y.  A combined experimental and computational study of the flow characteristics in a Type B aortic dissection: Effect of primary and secondary tear size.  _Chem Eng Res Design_ 160, 240–253, DOI: 10.1016/J.CHERD.2020.05.025 (2020).
  * [18] Tse, K. M., Chiu, P., Lee, H. P. & Ho, P.  Investigation of hemodynamics in the development of dissecting aneurysm within patient-specific dissecting aneurismal aortas using computational fluid dynamics (CFD) simulations.  _J Biomech_ 44, 827–836, DOI: 10.1016/J.JBIOMECH.2010.12.014 (2011).
  * [19] Cheng, Z., Wood, N. B., Gibbs, R. G. & Xu, X. Y.  Geometric and flow features of type B aortic dissection: initial findings and comparison of medically treated and stented cases.  _Ann Biomed Eng_ 43, 177–189, DOI: 10.1007/s10439-014-1075-8 (2014).
  * [20] Xu, H. _et al._ Computed tomography-based hemodynamic index for aortic dissection.  _J Thorac Cardiovasc Surg_ 162, e165–e176, DOI: 10.1016/J.JTCVS.2020.02.034 (2020).
  * [21] Cuellar-Calabria, H. _et al._ Differences in the Area of Proximal and Distal Entry Tears at CT Angiography Predict Long-term Clinical Outcomes in Aortic Dissection.  _Radiol Cardiothorac Imaging_ 3, e210029, DOI: 10.1148/RYCT.2021210029 (2021).
  * [22] Fleischmann, D. & Burris, N.  Entry Tear Dominance at CT Angiography Predicts Long-term Clinical Outcomes in Aortic Dissection: Another Piece of the Puzzle.  _Radiol Cardithorac Imaging_ 3, e210271, DOI: 10.1148/RYCT.2021210271 (2021).
  * [23] Bäumler, K. _et al._ Fluid–structure interaction simulations of patient-specific aortic dissection.  _Biomech Model Mechanobiol_ 19, 1607–1628, DOI: 10.1007/s10237-020-01294-8 (2020).
  * [24] Bäumler, K., Zimmermann, J., Ennis, D. B., Marsden, A. L. & Fleischmann, D.  Hemodynamic Effects of Entry Versus Exit Tear Size and Tissue Stiffness in Simulations of Aortic Dissection.  In _CMBBE. Lect Notes Comput Vision Biomech_ , 143–152, DOI: 10.1007/978-3-031-10015-4_13 (Springer, Cham, 2022).
  * [25] Karmonik, C. _et al._ Computational study of haemodynamic effects of entry- and exit-tear coverage in a DeBakey type III aortic dissection: Technical report.  _Eur J Vasc Endovasc Surg_ 42, 172–177, DOI: 10.1016/j.ejvs.2011.04.008 (2011).
  * [26] Shang, E. K. _et al._ Use of computational fluid dynamics studies in predicting aneurysmal degeneration of acute type B aortic dissections.  _J. Vasc. Surg._ 62, 279–284, DOI: 10.1016/j.jvs.2015.02.048 (2015).
  * [27] Osswald, A. _et al._ Elevated Wall Shear Stress in Aortic Type B Dissection May Relate to Retrograde Aortic Type A Dissection: A Computational Fluid Dynamics Pilot Study.  _Eur. J. Vasc. Endovasc. Surg._ 54, 324–330, DOI: 10.1016/j.ejvs.2017.06.012 (2017).
  * [28] Markl, M. _et al._ Time-resolved three-dimensional phase-contrast MRI.  _J. Magn. Reson. Imaging_ 17, 499–506, DOI: 10.1002/jmri.10272 (2003).
  * [29] de Beaufort, H. W. _et al._ Four-dimensional flow cardiovascular magnetic resonance in aortic dissection: Assessment in an ex vivo model and preliminary clinical experience.  _J Thorac Cardiovasc Surg_ 157, 467–476, DOI: 10.1016/j.jtcvs.2018.06.022 (2019).
  * [30] Dillon-Murphy, D., Noorani, A., Nordsletten, D. & Figueroa, C. A.  Multi-modality image-based computational analysis of haemodynamics in aortic dissection.  _Biomech. Model. Mechanobiol._ 15, 857–76, DOI: 10.1007/s10237-015-0729-2 (2016).
  * [31] Ruiz-Muñoz, A. _et al._ False lumen rotational flow and aortic stiffness are associated with aortic growth rate in patients with chronic aortic dissection of the descending aorta: a 4D flow cardiovascular magnetic resonance study.  _J Cardiovasc Magn Reson_ 24, 20, DOI: 10.1186/S12968-022-00852-6 (2022).
  * [32] Valvano, G. _et al._ Accelerating 4D flow MRI by exploiting low-rank matrix structure and hadamard sparsity.  _Magn. Reson. Med._ 78, 1330–1341, DOI: 10.1002/mrm.26508 (2017).
  * [33] Ma, L. E. _et al._ Aortic 4D flow MRI in 2 minutes using compressed sensing, respiratory controlled adaptive k-space reordering, and inline reconstruction.  _Magn. Reson. Med._ 81, 3675–3690, DOI: 10.1002/mrm.27684 (2019).
  * [34] Schnell, S. _et al._ K-t GRAPPA accelerated four-dimensional flow MRI in the aorta: Effect on scan time, image quality, and quantification of flow and wall shear stress.  _Magn. Reson. Med._ 72, 522–533, DOI: 10.1002/mrm.24925 (2014).
  * [35] Garg, P. _et al._ Comparison of fast acquisition strategies in whole-heart four-dimensional flow cardiac MR: Two-center, 1.5 Tesla, phantom and in vivo validation study.  _J. Magn. Reson. Imaging_ 47, 272–281, DOI: 10.1002/jmri.25746 (2018).
  * [36] Loecher, M., Magrath, P., Aliotta, E. & Ennis, D. B.  Time-optimized 4D phase contrast MRI with real-time convex optimization of gradient waveforms and fast excitation methods.  _Magn. Reson. Med._ 82, 213–2024, DOI: 10.1002/mrm.27716 (2019).
  * [37] Dillinger, H., Walheim, J. & Kozerke, S.  On the limitations of echo planar 4D flow MRI.  _Magn. Reson. Med._ 84, 1806–16, DOI: 10.1002/mrm.28236 (2020).
  * [38] Pirola, S. _et al._ 4D Flow MRI-Based Computational Analysis of Blood Flow in Patient-Specific Aortic Dissection.  _IEEE Trans. Biomed. Eng._ 66, 3411–19, DOI: 10.1109/TBME.2019.2904885 (2019).
  * [39] Zimmermann, J. _et al._ On the impact of vessel wall stiffness on quantitative flow dynamics in a synthetic model of the thoracic aorta.  _Scientific Reports_ 11, 6703, DOI: 10.1038/s41598-021-86174-6 (2021).
  * [40] Lan, I. S. _et al._ Validation of the Reduced Unified Continuum Formulation Against In Vitro 4D-Flow MRI.  _Ann Biomed Eng_ DOI: 10.1007/S10439-022-03038-4 (2022).
  * [41] Updegrove, A. _et al._ SimVascular: An Open Source Pipeline for Cardiovascular Simulation.  _Ann. Biomed. Eng._ 45, 525–541, DOI: 10.1007/s10439-016-1762-8 (2017).
  * [42] Si, H.  Adaptive tetrahedral mesh generation by constrained Delaunay refinement.  _Int J Numerical Methods Eng_ 75, 856–880, DOI: 10.1002/nme.2318 (2008).
  * [43] Bazilevs, Y. _et al._ Computational vascular fluid-structure interaction: methodology and application to cerebral aneurysms.  _Biomech Model Mechanobiol_ 9, 481–498, DOI: 10.1007/S10237-010-0189-7 (2010).
  * [44] Moireau, P. _et al._ External tissue support and fluid-structure simulation in blood flows.  _Biomech Model Mechanobiol_ 11, 1–18, DOI: 10.1007/s10237-011-0289-z (2012).
  * [45] Vignon-Clementel, I. E., Alberto Figueroa, C., Jansen, K. E. & Taylor, C. A.  Outflow boundary conditions for three-dimensional finite element modeling of blood flow and pressure in arteries.  _Computer Methods in Applied Mechanics and Engineering_ 195, 3776–3796, DOI: 10.1016/J.CMA.2005.04.014 (2006).
  * [46] Zhu, C. _et al._ svFSI: A Multiphysics Package for Integrated Cardiac Modeling.  _J Open Source Software_ 7, 4118, DOI: 10.21105/JOSS.04118 (2022).
  * [47] Marsden, A. L. & Esmaily-Moghadam, M.  Multiscale modeling of cardiovascular flows for clinical decision support.  _Appl Mech Rev_ 67, 030804, DOI: 10.1115/1.4029909 (2015).
  * [48] Esmaily-Moghadam, M., Bazilevs, Y. & Marsden, A. L.  A bi-partitioned iterative algorithm for solving linear systems arising from incompressible flow problems.  _Comput. Meth. Appl. Mech. Eng._ 286, 40–62, DOI: 10.1016/j.cma.2014.11.033 (2015).
  * [49] Esmaily-Moghadam, M., Bazilevs, Y., Hsia, T.-Y., Vignon-Clementel, I. E. & Marsden, A. L.  A comparison of outlet boundary treatments for prevention of backflow divergence with relevance to blood flow simulations.  _Comp Mech_ 48, 277–291, DOI: 10.1007/s00466-011-0599-0 (2011).
  * [50] Tautz, L., Hennemuth, A. & Peitgen, H.-O.  Motion Analysis with Quadrature Filter Based Registration of Tagged MRI Sequences.  In _Stat. Atlases Comput. Model. Heart. STACOM 2011. Lect. Notes Comput. Science_ , vol. 7085, 78–87, DOI: 10.1007/978-3-642-28326-0_8 (Springer, Berlin, Heidelberg, 2012).
  * [51] Markl, M. _et al._ Generalized reconstruction of phase contrast MRI: Analysis and correction of the effect of gradient field distortions.  _Magn Reson Med_ 50, 791–801, DOI: 10.1002/MRM.10582 (2003).
  * [52] Meng, H., Tutino, V. M., Xiang, J. & Siddiqui, A.  High WSS or Low WSS? Complex Interactions of Hemodynamics with Intracranial Aneurysm Initiation, Growth, and Rupture: Toward a Unifying Hypothesis.  _American Journal of Neuroradiology_ 35, 1254–1262, DOI: 10.3174/AJNR.A3558 (2014).
  * [53] Wang, H., Uhlmann, K., Vedula, V., Balzani, D. & Varnik, F.  Fluid-structure interaction simulation of tissue degradation and its effects on intra-aneurysm hemodynamics.  _Biomech Model Mechanobiol_ 21, 671–683, DOI: 10.1007/S10237-022-01556-7 (2022).
  * [54] Riesenkampff, E. _et al._ Pressure fields by flow-sensitive, 4D, velocity-encoded CMR in patients with aortic coarctation.  _JACC Cardiovasc. Imaging_ 7, 920–926, DOI: 10.1016/j.jcmg.2014.03.017 (2014).
  * [55] Urbina, J. _et al._ Realistic aortic phantom to study hemodynamics using MRI and cardiac catheterization in normal and aortic coarctation conditions.  _J. Magn. Reson. Imaging_ 44, 683–697, DOI: 10.1002/jmri.25208 (2016).

## Supplementary Material

See below for supplementary figures and tables S1-S3, S9, S10. Upon
publication we will make supplementary videos (.mp4) available. These are
numbered S4-S8, with captions printed at the end of this document.

S1 Experimental setup. (a) Schematic drawing of the flow loop setup; fluid lines are displayed in blue. (b) Programmed flow rate waveform with stroke volume of $74.1\text{\,}\mathrm{mL}\text{/}\mathrm{s}$. (c) Photograph of the setup (with gel top and inner box lid both removed). All shown parts were inside the scanner bore during image acquisition. All other equipment (not shown in photograph) was positioned either outside the bore (pump unit), or outside the scanner room (pump control, data acquisition system (DAQ), bridge amplifier, flow meter). |  $R_{T}$ |  $C_{T}$ |  $k_{d}$ |  $k_{s}$ |  $c_{s}$ |  $\varrho_{f}$ |  $\mu_{f}$ |  $\varrho_{s}$ |  $E_{y,t}$
---|---|---|---|---|---|---|---|---|---
|  ($\mathrm{MPa}\text{\,}\mathrm{s}\text{\,}{\mathrm{m}}^{-3}$) |  (${\mathrm{m}}^{3}\text{\,}{\mathrm{Pa}}^{-1}$) |  (-) |  ($\mathrm{MN}\text{\,}{\mathrm{m}}^{-3}$) |  ($\mathrm{kN}\text{\,}\mathrm{s}\text{\,}{\mathrm{m}}^{-3}$) |  ($\mathrm{kg}\text{\,}{\mathrm{m}}^{-3}$) |  ($\mathrm{Pa}\text{\,}\mathrm{s}$) |  ($\mathrm{kg}\text{\,}{\mathrm{m}}^{-3}$) |  ($\mathrm{MPa}$)
$\mathrm{TBAD}_{OR}$ |  150 |  $6.74\text{\times}{10}^{-9}$ |  0.84 |  $-18$ |  $-30$ |  1100 |  0.0042 |  1450 |  1.2
$\mathrm{TBAD}_{EN}$ |  161 |  $1.23\text{\times}{10}^{-8}$ |  0.87 |  $-18$ |  $-30$ |  1100 |  0.0042 |  1450 |  1.2
$\mathrm{TBAD}_{EX}$ |  170 |  $1.02\text{\times}{10}^{-8}$ |  0.86 |  $-18$ |  $-30$ |  1100 |  0.0042 |  1450 |  1.2
S2. FSI simulations parameters. The three-element Windkessel boundary
parameters include the total resistance $R_{T}$, total capacitance $C_{T}$,
and ratio of distal to proximal resistance $k_{d}$. $R_{T}$ and $C_{T}$ are
distributed across outlets according to measured flow splits (see Table 2 in
main article) and the respective value for $k_{d}$. ETS scalar parameters
$k_{s}$ (elastic response) and $c_{s}$ (viscoelastic response) were chosen to
match the minimum-to-maximum dilation of the simulation to MRI-measured
values. Fluid and structural density ($\varrho_{f}$ and $\varrho_{s}$), fluid
viscosity $\mu_{f}$, and elastic modulus $E_{y,t}$ were prescribed according
to benchtop measurements or manufacturer’s information.

S3. Pressure waveforms at inlet (solid) and outlets (dotted) for catheter-based measurements (green) and FSI simulations (blue): (a) $\mathrm{TBAD}_{OR}$ , (b) $\mathrm{TBAD}_{EN}$ , and (c) $\mathrm{TBAD}_{EX}$ . Numbers in plotting areas report $P_{sys}$/$P_{dias}$ ($P_{\textrm{MAP}}$) in $\mathrm{mmHg}$.  | AAo | TL1 | TL2 | TL3 | TL4 | TL5 | FL1 | FL2 | FL3 | FL4 | FL5 | $DAo_{dist}$
---|---|---|---|---|---|---|---|---|---|---|---|---
MRI (pulsatile flow, retrieved from 2D-cine data) |  |  |  |  |
$\mathrm{TBAD}_{OR}$ |  |  |  |  |  |  |  |
$A_{0}$ |  $612.4$ |  $339.1$ |  $182.0$ |  $118.8$ |  $73.5$ |  $62.3$ |  $517.4$ |  $587.7$ |  $540.3$ |  $551.5$ |  $569.5$ |  $444.7$
$A_{max}$ |  $633.9$ |  $345.9$ |  $185.7$ |  $121.7$ |  $74.5$ |  $63.2$ |  $530.9$ |  $606.2$ |  $553.5$ |  $572.1$ |  $590.9$ |  $453.9$
$A_{mean}$ |  $620.2$ |  $340.9$ |  $183.0$ |  $120.3$ |  $73.8$ |  $62.2$ |  $521.3$ |  $592.8$ |  $543.4$ |  $558.5$ |  $577.0$ |  $447.2$
$\mathrm{TBAD}_{EN}$ |  |  |  |  |  |  |  |
$A_{0}$ |  $630.3$ |  $253.7$ |  $148.9$ |  $97.3$ |  $83.7$ |  $57.2$ |  $490.5$ |  $489.6$ |  $475.1$ |  $477.0$ |  $450.2$ |  $398.5$
$A_{max}$ |  $656.9$ |  $262.6$ |  $155.0$ |  $99.8$ |  $85.7$ |  $57.5$ |  $496.6$ |  $499.8$ |  $486.4$ |  $493.6$ |  $466.1$ |  $411.5$
$A_{mean}$ |  $640.1$ |  $254.4$ |  $150.0$ |  $98.0$ |  $84.8$ |  $57.0$ |  $490.4$ |  $492.0$ |  $476.7$ |  $480.8$ |  $454.6$ |  $403.5$
$\mathrm{TBAD}_{EX}$ |  |  |  |  |  |  |  |
$A_{0}$ |  $559.0$ |  $272.2$ |  $151.9$ |  $95.0$ |  $77.6$ |  $54.3$ |  $428.5$ |  $466.2$ |  $456.2$ |  $477.1$ |  $464.8$ |  $382.9$
$A_{max}$ |  $581.6$ |  $274.8$ |  $152.9$ |  $95.9$ |  $78.0$ |  $55.5$ |  $444.1$ |  $481.6$ |  $470.3$ |  $500.1$ |  $482.9$ |  $389.1$
$A_{mean}$ |  $566.8$ |  $271.7$ |  $151.6$ |  $95.3$ |  $77.3$ |  $54.7$ |  $433.1$ |  $470.3$ |  $459.4$ |  $484.8$ |  $470.3$ |  $384.9$
MRI (steady flow, retrieved from 3D-SPGR data) |  |  |  |  |
$\mathrm{TBAD}_{OR}$ |  $527.0$ |  $315.0$ |  $154.0$ |  $85.0$ |  $81.0$ |  $65.0$ |  $547.0$ |  $587.0$ |  $553.0$ |  $564.0$ |  $457.0$ |  $457.0$
MRI (“flow-off”, retrieved from 3D-SPGR data) |  |  |  |  |
$\mathrm{TBAD}_{OR}$ |  $435.0$ |  $268.0$ |  $152.0$ |  $85.0$ |  $83.0$ |  $60.0$ |  $418.0$ |  $431.0$ |  $411.0$ |  $438.0$ |  $445.0$ |  $373.0$
FSI (pulsatile flow) |  |  |  |  |
$\mathrm{TBAD}_{OR}$ |  |  |  |  |  |  |  |
$A_{0}$ |  $513.5$ |  $239.6$ |  $150.7$ |  $68.4$ |  $58.9$ |  $56.9$ |  $497.2$ |  $457.4$ |  $494.8$ |  $479.0$ |  $448.0$ |  $379.6$
$A_{max}$ |  $538.7$ |  $247.4$ |  $155.7$ |  $70.7$ |  $59.8$ |  $57.1$ |  $516.8$ |  $477.7$ |  $515.4$ |  $500.8$ |  $468.3$ |  $394.6$
$A_{mean}$ |  $523.2$ |  $242.2$ |  $152.1$ |  $69.1$ |  $59.1$ |  $56.9$ |  $505.2$ |  $465.5$ |  $502.9$ |  $487.5$ |  $455.9$ |  $385.5$
$\mathrm{TBAD}_{EN}$ |  |  |  |  |  |  |  |
$A_{0}$ |  $518.5$ |  $238.3$ |  $151.2$ |  $68.7$ |  $59.0$ |  $56.9$ |  $499.6$ |  $462.2$ |  $499.3$ |  $483.7$ |  $452.2$ |  $382.9$
$A_{max}$ |  $544.2$ |  $250.2$ |  $160.6$ |  $73.0$ |  $60.5$ |  $57.1$ |  $509.9$ |  $472.6$ |  $511.2$ |  $497.9$ |  $466.2$ |  $393.2$
$A_{mean}$ |  $526.7$ |  $241.5$ |  $153.5$ |  $69.8$ |  $59.4$ |  $57.0$ |  $503.4$ |  $466.1$ |  $503.9$ |  $489.0$ |  $457.4$ |  $386.8$
$\mathrm{TBAD}_{EX}$ |  |  |  |  |  |  |  |
$A_{0}$ |  $519.3$ |  $240.8$ |  $151.1$ |  $68.6$ |  $59.0$ |  $56.9$ |  $502.9$ |  $463.3$ |  $500.3$ |  $484.7$ |  $453.0$ |  $355.8$
$A_{max}$ |  $546.8$ |  $247.8$ |  $155.2$ |  $69.9$ |  $59.1$ |  $57.1$ |  $525.5$ |  $486.2$ |  $524.0$ |  $510.0$ |  $476.9$ |  $366.1$
$A_{mean}$ |  $528.5$ |  $243.1$ |  $152.2$ |  $68.9$ |  $58.9$ |  $56.5$ |  $510.6$ |  $471.2$ |  $508.3$ |  $493.2$ |  $461.0$ |  $359.7$
FSI (steady flow) |  |  |  |  |
$\mathrm{TBAD}_{OR}$ |  $522.1$ |  $241.6$ |  $151.6$ |  $68.9$ |  $59.1$ |  $57.0$ |  $505.2$ |  $465.4$ |  $502.6$ |  $487.0$ |  $455.4$ |  $385.2$
STL model (“flow-off”) |  |  |  |  |
$\mathrm{TBAD}_{OR}$ |  $492.6$ |  $235.6$ |  $149.2$ |  $67.6$ |  $58.4$ |  $56.6$ |  $476.3$ |  $436.3$ |  $475.0$ |  $458.7$ |  $429.3$ |  $365.5$
S9. Cross-sectional area (in $\mathrm{m}\mathrm{m}^{2}$) at twelve landmarks
for pulsatile flow, steady flow, and “flow-off” modes. $A_{0}$ (area at first
frame of the cardiac cycle) and $A_{max}$ (maximum area of cardiac cycle) were
considered to define end-diastolic and peak-systolic cross-sectional area,
respectively. $A_{mean}$ reports the area averaged over the cardiac cycle. For
landmark label definition see Fig. 2a of main article. Refer to Supplementary
Fig. S10 for data plots.

S10. Absolute cross-sectional area evaluated at twelve landmarks with 3D-SPGR
or 2D-cine MRI of the 3D-printed model (horizontal axis) and based on the
deformable structural domain, i.e. aortic wall, in FSI simulations (vertical
axis): (a) “flow-off” and steady flow measurements; (b, c, d) first frame
($A_{0}$, end-diastolic) and maximum ($A_{max}$, peak-systolic) area
measurements in pulsatile mode for each model. Refer to Supplementary Table S9
for underlying data.

Supplementary video file description: In all animated vector visualizations,
cycle length was stretched from $1\text{\,}\mathrm{s}$ to
$2\text{\,}\mathrm{s}$ to better display complex patterns, and the video file
was exported with 24 fps.

S4 4D-flow MRI velocity vector visualizations in three TBAD models. Cycle
length was slowed down to $2\text{\,}\mathrm{s}$.

S5 CFD-FSI simulations displaying velocity vectors of fluid domain in three
TBAD models.

S6 Entry tear close-up view of velocity vector visualizations of 4D-flow MRI
and CFD-FSI simulations.

S7 Entry tear close-up view of velocity vector visualizations of 4D-flow MRI
and CFD-FSI simulations. Identical data as in video S6, but with camera view
rotated around aorta long axis.

S8 Exit tear close-up view of velocity vector visualizations of 4D-flow MRI
and CFD-FSI simulations.


# The Spectrum of Triangle-free Graphs

József Balogh111Department of Mathematics, University of Illinois at Urbana-
Champaign, Urbana, Illinois 61801, USA. E-mail<EMAIL_ADDRESS>Research
is partially supported by NSF Grants DMS-1764123, RTG DMS 1937241 and FRG
DMS-2152488, Arnold O. Beckman Research Award (UIUC Campus Research Board RB
22000), the Langan Scholar Fund (UIUC), and the Simons Fellowship. Felix
Christian Clemen 222Department of Mathematics, University of Illinois at
Urbana-Champaign, Urbana, Illinois 61801, USA, E-mail<EMAIL_ADDRESS>Bernard Lidický 333Iowa State University, Department of Mathematics, Iowa
State University, Ames, IA., E-mail: lidicky@ iastate.edu. Research of this
author is partially supported by NSF grants DMS-1855653 and FRG DMS-2152498
and Scott Hanna fellowship. Sergey Norin444Department of Mathematics and
Statistics, McGill University, Montreal, QC, Canada<EMAIL_ADDRESS>Research of this author was supported by an NSERC Discovery Grant. Jan
Volec555 Department of Mathematics, Czech Technical University in Prague,
Prague, Czech Republic<EMAIL_ADDRESS>Research of this author is partially
supported by Czech Science Foundation GAČR, grant 23-06815M.

###### Abstract

Denote by $q_{n}(G)$ the smallest eigenvalue of the signless Laplacian matrix
of an $n$-vertex graph $G$. Brandt conjectured in 1997 that for regular
triangle-free graphs $q_{n}(G)\leq\frac{4n}{25}$. We prove a stronger result:
If $G$ is a triangle-free graph then
$q_{n}(G)\leq\frac{15n}{94}<\frac{4n}{25}$. Brandt’s conjecture is a
subproblem of two famous conjectures of Erdős:

(1) Sparse-Half-Conjecture: Every $n$-vertex triangle-free graph has a subset
of vertices of size $\lceil\frac{n}{2}\rceil$ spanning at most $n^{2}/50$
edges.

(2) Every $n$-vertex triangle-free graph can be made bipartite by removing at
most $n^{2}/25$ edges.

In our proof we use linear algebraic methods to upper bound $q_{n}(G)$ by the
ratio between the number of induced paths with 3 and 4 vertices. We give an
upper bound on this ratio via the method of flag algebras.

##  1 Introduction

We prove a result on eigenvalues of triangle-free graphs which is motivated by
the following two famous conjectures of Erdős.

######  Conjecture 1.1 (Erdős’ _Sparse Half Conjecture_ [9, 10]).

Every triangle-free graph on $n$ vertices has a subset of vertices of size
$\lceil\frac{n}{2}\rceil$ vertices spanning at most $n^{2}/50$ edges.

Erdős offered a $250 reward for proving this conjecture. There has been
progress on this conjecture in various directions [12, 14, 17, 15, 4]. Most
recently, Razborov [17] proved that every triangle-free graph on $n$ vertices
has an induced subgraph on $n/2$ vertices with at most $(27/1024)n^{2}$ edges.

For a graph $G$, denote by $D_{2}(G)$ the minimum number of edges which have
to be removed to make $G$ bipartite.

######  Conjecture 1.2 (Erdős [9]).

Let $G$ be a triangle-free graph on $n$ vertices. Then $D_{2}(G)\leq
n^{2}/25$.

There also has been work on this conjecture [11, 13, 3, 1, 18], most recently,
Balogh, Clemen and Lidický [3] proved $D_{2}(G)\leq n^{2}/23.5$.

Brandt [5] found a surprising connection between these two conjectures and the
eigenvalues of triangle-free graphs. Denote by
$\lambda_{n}(G)\leq\ldots\leq\lambda_{1}(G)$ the eigenvalues of the adjacency
matrix of an $n$-vertex graph $G$. Brandt [5] proved that

$\displaystyle D_{2}(G)\geq\frac{\lambda_{1}(G)+\lambda_{n}(G)}{4}\cdot n$ (1)

for regular graphs and conjectured the following.

######  Conjecture 1.3 (Brandt [5]).

Let $G$ be a triangle-free regular $n$-vertex graph. Then

$\lambda_{1}(G)+\lambda_{n}(G)\leq\frac{4}{25}\cdot n.$

Towards this conjecture, Brandt [5] proved a bound
$\lambda_{1}(G)+\lambda_{n}(G)\leq(3-2\sqrt{2})n\approx 0.1715n$ for regular
triangle-free graphs, which was very recently shown to hold also in the non-
regular setting by Csikvári [7]. Brandt also noted that
$\lambda_{1}(G_{HS})+\lambda_{n}(G_{HS})=0.14n$ for the so-called Higman-Sims
graph $G_{HS}$, which is the unique strongly regular graph with parameters
$(n,d,t,k)=(100,22,0,6)$. Recall that an $(n,d,t,k)$-_strongly regular graph_
is an $n$-vertex $d$-regular graph, where the number of common neighbors of
every pair of adjacent vertices is $t$ and the number of common neighbors of a
non-adjacent pair of vertices is $k$.

The value $4/25$ is motivated by the fact that if either of Conjectures 1.1 or
1.2 were true, it would imply Conjecture 1.3. As observed by Brandt [5],
Conjecture 1.1 implies Conjecture 1.3 by applying the following version of the
Expander Mixing Lemma for a set $S\subset V(G)$ of size $n/2$ with $e(S)\leq
n^{2}/50$.

######  Lemma 1.4 (Bussemaker-Cvetković-Seidel [6], Alon-Chung [2]).

Let $G$ be an $n$-vertex $d$-regular graph. Then, for every $S\subseteq V(G)$,
we have

$\displaystyle e(S)\geq|S|\cdot\frac{|S|d+(n-|S|)\lambda_{n}(G)}{2n}.$

Given a graph $G$, denote by $Q=A+D$ the _signless Laplacian matrix_ of $G$,
where $D$ is the diagonal matrix of the degrees of $G$ and $A$ is the
adjacency matrix of $G$. Let $q_{n}(G)\leq\ldots\leq q_{1}(G)$ be the
eigenvalues of $Q$. By considering the signless Laplacian matrix, De Lima,
Nikiforov and Olivera [8] extended (1) beyond regular graphs as follows.

######  Theorem 1.5 (De Lima, Nikiforov and Olivera[8]).

For every $n$-vertex graph $G$ we have

$D_{2}(G)\geq\frac{q_{n}(G)}{4}\cdot n.$

By Theorem 1.5, if Conjecture 1.2 holds then $q_{n}(G)\leq\frac{4n}{25}$ for
every triangle-free $n$-vertex graph $G$. Motivated by this observation De
Lima, Nikiforov and Olivera [8] proposed investigating upper bounds on
$q_{n}(G)$, and proved $q_{n}(G)\leq\frac{2n}{9}$ for $n$-vertex triangle-free
graphs $G$. Our main result is an improvement of this bound, which solves
Conjecture 1.3.

######  Theorem 1.6.

If $G$ is a triangle-free $n$-vertex graph, then

$q_{n}(G)\leq{\frac{15}{94}}\cdot n<0.1596n.$

Note that, if $G$ is $d$-regular, then $\lambda_{1}(G)=d$ and
$q_{n}(G)=\lambda_{n}(G)+d=\lambda_{n}(G)+\lambda_{1}(G)$. Thus Theorem 1.6
implies that $\lambda_{1}(G)+\lambda_{n}(G)<0.1596n<\frac{4n}{25}$ for every
regular triangle-free $n$-vertex graph $G$, confirming Conjecture 1.3 in
strong form.

It remains open to determine a sharp upper bound for $q_{n}(G)/n$ for
triangle-free $n$-vertex graph $G$. While we only prove Theorem 1.6 with the
constant $\frac{15}{94}\approx 0.1596$, a larger flag algebra computation
yields $q_{n}(G)<{0.15467}n$. Also, one can additionally assume that $G$ is
regular and use flag algebras to show a slightly stronger bound
$q_{n}(G)=\lambda_{1}(G)+\lambda_{n}(G)<{0.15442}n$. As we believe neither of
these two bounds are sharp (see Section 3), we omit presenting their proofs.

##  2 Proof of Theorem 1.6

Our proof is based on bounding the ratio between the number of induced paths
with $3$ and $4$ vertices in triangle-free graphs. On one hand, we upper bound
$q_{n}(G)$ in terms of this ratio in Lemma 2.1 and Corollary 2.2. On the other
hand, Lemma 2.3, which is proved using flag algebras, gives a sufficiently
good bound on the ratio.

For an edge $e=xy$ of a graph $G$, let $m_{xy}$ be the number of edges $uv\in
E(G)$ such that $ux,vy\in E(G)$. For a vertex $x\in V(G)$, let $w_{x}$ to be
the number of walks of length two starting in $x$, i.e. $w_{x}$ is the number
of edges $uv\in E(G)$ such that $xu\in E(G)$.

######  Lemma 2.1.

If $G$ is an $n$-vertex triangle-free graph and $xy\in E(G)$, then

$\left(\deg(x)+\deg(y)\right)\cdot q_{n}(G)\leq w_{x}+w_{y}-2m_{xy}\,.$ (2)

###### Proof.

Define a vector $z=(z_{v})_{v\in V(G)}\in\mathbb{R}^{V(G)}$ by

$z_{v}=\begin{cases}+1,&\text{if }xv\in E(G),\\\ -1,&\text{if }yv\in E(G),\\\
0,&\text{otherwise.}\end{cases}$

The vector $z$ is well-defined since $G$ is triangle-free. Also note that
$\|z\|^{2}=\deg(x)+\deg(y)$. Let $Q$ be the signless Laplacian matrix of $G$.
We have

$\displaystyle z^{T}Qz$ $\displaystyle=\sum_{u,v\in
V(G)}Q_{uv}z_{u}z_{v}=\sum_{u\in V(G)}(z_{u})^{2}\deg(u)+2\cdot\sum_{uv\in
E(G)}z_{u}z_{v}$ $\displaystyle=w_{x}+w_{y}+2\cdot\sum_{uv\in
E(G)}z_{u}z_{v}=w_{x}+w_{y}-2m_{xy},$

where in the last equality we used that $G$ is triangle-free. Since $Q$ is
symmetric, $q_{n}(G)$ is upper bounded by the Rayleigh-Ritz quotient of $z$,
i.e.

$q_{n}(G)\leq\frac{z^{T}Qz}{\|z\|^{2}}=\frac{w_{x}+w_{y}-2m_{xy}}{\deg(x)+\deg(y)}\,,$

as desired. ∎

A map $\varphi:V(H)\to V(G)$ is _a strong homomorphism_ from a graph $H$ to a
graph $G$ if for every pair of vertices $u,v\in V(H)$ we have $uv\in E(H)$ if
and only if $\varphi(u)\varphi(v)\in E(G)$. Let $\operatorname{hom}_{s}(H,G)$
denote the number of strong homomorphisms from $H$ to $G$. Let $P_{k}$ denote
the $k$-vertex path. Summing the bound from Lemma 2.1 over all the edges of
$G$ yields the following.

######  Corollary 2.2.

If $G$ is an $n$-vertex triangle-free graph, then

$\operatorname{hom}_{s}(P_{3},G)\cdot
q_{n}(G)\leq\operatorname{hom}_{s}(P_{4},G)\,.$ (3)

###### Proof.

First, note that

$\sum_{xy\in E(G)}(\deg(x)+\deg(y))=\sum_{x\in
V(G)}\deg^{2}(x)=\operatorname{hom}_{s}(P_{3},G),$ (4)

where in the last equality we used that $G$ is triangle-free. Meanwhile,
$\sum_{xy\in E(G)}(w_{x}+w_{y})$ is equal to the number of walks of length
three in $G$, i.e. the number of maps $\phi:\\{1,2,3,4\\}\to V(G)$ such that
$\\{\phi(1)\phi(2),\phi(2)\phi(3),\phi(3)\phi(4)\\}\subset E(G)$. Similarly,
the expression $2\sum_{xy\in E(G)}m_{xy}$ is equal to the number of maps
$\phi:\\{1,2,3,4\\}\to V(G)$ such that
$\\{\phi(1)\phi(2),\phi(2)\phi(3),\phi(3)\phi(4),\phi(4)\phi(1)\\}\subset
E(G)$. It follows that $\sum_{xy\in E(G)}(w_{x}+w_{y}-2m_{xy})$ counts the
maps $\psi:\\{1,2,3,4\\}\to V(G)$ such that
$\\{\psi(1)\psi(2),\psi(2)\psi(3),\psi(3)\psi(4)\\}\subset E(G)$ and
$\psi(4)\psi(1)\not\in E(G)$, i.e.,

$\sum_{xy\in E(G)}(w_{x}+w_{y}-2m_{xy})=\operatorname{hom}_{s}(P_{4},G).$ (5)

Summing (2) over all $xy\in E(G)$ and using (4) and (5), we obtain (3). ∎

Theorem 1.6 is an immediate consequence of the above corollary and the
following lemma which is proved using standard, albeit computer-assisted flag-
algebra calculation.

######  Lemma 2.3.

If $G$ is an $n$-vertex triangle-free graph, then

$\operatorname{hom}_{s}(P_{4},G)\leq\frac{15n}{94}\cdot\operatorname{hom}_{s}(P_{3},G).$
(6)

###### Proof.

Suppose the lemma is false, and let $G$ be an $n$-vertex triangle-free graph
such that

$\operatorname{hom}_{s}(P_{4},G)=\frac{15n}{94}\cdot\operatorname{hom}_{s}(P_{3},G)+\varepsilon
n^{4}\,,$ (7)

for some $\varepsilon>0$. Let $G^{(b)}$ be the $b$-blowup of $G$, obtained by
replacing every vertex of $G$ by $b$ pairwise non-adjacent vertices. Then
$\operatorname{hom}_{s}(P_{k},G^{(b)})=\operatorname{hom}_{s}(P_{k},G)\cdot
b^{k}$ for $k=3,4$. In particular, for every $b\in\mathbb{N}$, the graph
$G^{(b)}$ satisfies the analogue of (7) as well.

Let us now reformulate (7) in the flag algebra language [16]. Given a graph
$H$, let
$p\left({\vbox{\hbox{\includegraphics[scale={0.34},page=15]{flags.pdf}}}},H\right)$
be the probability that a $3$-vertex subset of $V(H)$ chosen uniformly at
random induces exactly two edges. Analogously, let
$p\left({\vbox{\hbox{\includegraphics[scale={0.34},page=38]{flags.pdf}}}},H\right)$
be the probability that a randomly chosen $4$-vertex subset induces a path of
length $3$.

For every fixed $\ell$-vertex graph $F$ and a $k$-vertex graph $H$, only
$O(k^{\ell-1})$ maps $V(F)\to V(H)$ are non-injective. Therefore,
$\operatorname{hom}_{s}(F,H)=|\operatorname{Aut}(F)|\cdot
p(F,H)\cdot{k\choose\ell}+O(k^{\ell-1})$, so in particular every $k$-vertex
triangle-free graph $H$ satisfies

$\operatorname{hom}_{s}(P_{4},H)=\frac{k^{4}}{12}\cdot
p\left({\vbox{\hbox{\includegraphics[scale={0.34},page=38]{flags.pdf}}}},H\right)+O\left(k^{3}\right)\quad\hbox{and}\quad\operatorname{hom}_{s}(P_{3},H)=\frac{k^{3}}{3}\cdot
p\left({\vbox{\hbox{\includegraphics[scale={0.34},page=15]{flags.pdf}}}},H\right)+O\left(k^{2}\right)\,.$

Therefore, the fact $G^{(b)}$ satisfies (7), after multiplying by
$564/(bn)^{4}$ and rearranging, translates to

$\lim_{b\to\infty}\;30\cdot
p\left({\vbox{\hbox{\includegraphics[scale={0.34},page=15]{flags.pdf}}}},G^{(b)}\right)-47\cdot
p\left({\vbox{\hbox{\includegraphics[scale={0.34},page=38]{flags.pdf}}}},G^{(b)}\right)=-564\varepsilon^{4}.$

In order to derive a contradiction, we present a flag algebra computation
proving that an inequality

$30\cdot{\vbox{\hbox{\includegraphics[scale={0.34},page=15]{flags.pdf}}}}-{47}\cdot{\vbox{\hbox{\includegraphics[scale={0.34},page=38]{flags.pdf}}}}\geq
0$ (8)

asymptotically holds in the theory of triangle-free graphs. To see that,
consider the following $6$ flag-algebra expressions, which are all non-
negative:

1)
$\displaystyle{\left(13\cdot\left({\vbox{\hbox{\includegraphics[scale={0.34},page=24]{flags.pdf}}}}+{\vbox{\hbox{\includegraphics[scale={0.34},page=25]{flags.pdf}}}}+{\vbox{\hbox{\includegraphics[scale={0.34},page=26]{flags.pdf}}}}\right)-52\cdot\left({\vbox{\hbox{\includegraphics[scale={0.34},page=27]{flags.pdf}}}}+{\vbox{\hbox{\includegraphics[scale={0.34},page=28]{flags.pdf}}}}+{\vbox{\hbox{\includegraphics[scale={0.34},page=29]{flags.pdf}}}}\right)+84\cdot{\vbox{\hbox{\includegraphics[scale={0.34},page=30]{flags.pdf}}}}\right)^{2}}$
2)
$\displaystyle{\left(31\cdot\left({\vbox{\hbox{\includegraphics[scale={0.34},page=31]{flags.pdf}}}}+{\vbox{\hbox{\includegraphics[scale={0.34},page=32]{flags.pdf}}}}\right)-63\cdot\left({\vbox{\hbox{\includegraphics[scale={0.34},page=34]{flags.pdf}}}}+{\vbox{\hbox{\includegraphics[scale={0.34},page=35]{flags.pdf}}}}\right)+3\cdot{\vbox{\hbox{\includegraphics[scale={0.34},page=33]{flags.pdf}}}}\right)^{2}}$
3)
$\displaystyle{\left(94\cdot{\vbox{\hbox{\includegraphics[scale={0.34},page=17]{flags.pdf}}}}-55\cdot{\vbox{\hbox{\includegraphics[scale={0.34},page=18]{flags.pdf}}}}-14\cdot{\vbox{\hbox{\includegraphics[scale={0.34},page=19]{flags.pdf}}}}+58\cdot{\vbox{\hbox{\includegraphics[scale={0.34},page=20]{flags.pdf}}}}\right)^{2}}$
4)
$\displaystyle{{\vbox{\hbox{\includegraphics[scale={0.34},page=22]{flags.pdf}}}}\times\left(2\cdot{\vbox{\hbox{\includegraphics[scale={0.34},page=22]{flags.pdf}}}}+10\cdot{\vbox{\hbox{\includegraphics[scale={0.34},page=21]{flags.pdf}}}}-24\cdot{\vbox{\hbox{\includegraphics[scale={0.34},page=23]{flags.pdf}}}}\right)^{2}}$
5)
$\displaystyle{{\vbox{\hbox{\includegraphics[scale={0.34},page=41]{flags.pdf}}}}\times\left(14\cdot{\vbox{\hbox{\includegraphics[scale={0.34},page=39]{flags.pdf}}}}+19\cdot{\vbox{\hbox{\includegraphics[scale={0.34},page=41]{flags.pdf}}}}-44\cdot{\vbox{\hbox{\includegraphics[scale={0.34},page=40]{flags.pdf}}}}\right)^{2}}$
6)
$\displaystyle{{\vbox{\hbox{\includegraphics[scale={0.34},page=41]{flags.pdf}}}}\times\left(9\cdot{\vbox{\hbox{\includegraphics[scale={0.34},page=39]{flags.pdf}}}}-14\cdot{\vbox{\hbox{\includegraphics[scale={0.34},page=41]{flags.pdf}}}}-3\cdot{\vbox{\hbox{\includegraphics[scale={0.34},page=40]{flags.pdf}}}}\right)^{2}}$

Figure 1: The set $\mathcal{F}$ of $5$-vertex triangle-free graphs with at
least $2$ edges.

Let $\mathcal{F}$ be the set of all the $5$-vertex triangle-free graphs with
at least $2$ edges. A case analysis yields $|\mathcal{F}|=12$; see Figure 1.
Now observe that averaging over all choices of the labelled vertices in each
of the $6$ expressions yields a linear combination of subgraph densities,
where every term has $5$ vertices and at least $2$ edges. Thus a flag algebra
argument yields that the average of the $i$-th expression is equal to the
$i$-th coordinate of $M\cdot\left(v_{\mathcal{F}}\right)^{T}$, where
$v_{\mathcal{F}}=\left(F_{1},\dots,F_{12}\right)$ and

$\tiny
M=\frac{1}{30}\times\blockarray{cccccccccccc}\block{(cccccccccccc)}507&20280-4056-354901248811216224-13104021168\\\
0028833819610-3906-4098384463198450\\\
12100-23688-19140-236201217220184-372481748647664295686730-7392\\\
00614000-48-1000358-12000\\\ 1960798196-4202166762-1036-2464-702-3080792\\\
810-37881541176-16527-108-87-135279\\\ \,.$

On the other hand, another flag algebra argument yields that the left-hand
side of (8) is equal to

$3\cdot F_{1}+9\cdot F_{3}+3\cdot F_{4}-\frac{17}{5}\cdot F_{5}+18\cdot
F_{6}-\frac{34}{5}\cdot F_{7}-\frac{49}{5}\cdot F_{8}+12\cdot
F_{9}-\frac{4}{5}\cdot F_{10}-32\cdot F_{11}+27\cdot F_{12}.$

A tedious yet straightforward calculation reveals the following coordinate-
wise inequality

$\left(\frac{1}{33},\frac{12}{209},\frac{3}{1147},\frac{231}{163},\frac{17}{84},\frac{12}{293}\right)\cdot
M<\left(3,0,9,3,-\frac{17}{5},18,-\frac{34}{5},-\frac{49}{5},12,-\frac{4}{5},-32,27\right),$

which in turn shows that (8) asymptotically holds in the theory of triangle-
free graphs. ∎

The flag algebra calculations used in the proof of Lemma 2.3 can be
independently verified by a SAGE script, which is available as an ancillary
file of the arXiv version of this manuscript.

##  3 Concluding remarks

As we have already mentioned in the introduction, a significantly larger flag
algebra computation than the one used in our proof yields that
$q_{n}(G)<{0.15467}n$ for every triangle-free $n$-vertex graph. Similarly,
assuming that $G$ is regular allows us to show
$\lambda_{1}(G)+\lambda_{n}(G)<{0.15442}n$. On the other hand, our method will
be able to get neither of the coefficients below $42/275=0.15\overline{{}27}$.

Indeed, consider the Higman-Sims graph $G_{HS}$. It is edge-transitive so
$m_{xy}=21\cdot 6+22$ for every $xy\in E(G_{HS})$, and $w_{x}=22^{2}$ for
every $x\in V(G)$, where $m_{xy}$ and $w_{x}$ are defined as before Lemma 2.1.
Therefore,

$\frac{w_{x}+w_{y}-2m_{xy}}{(\deg(x)+\deg(y))\cdot|V(G_{HS})|}=\frac{2(22^{2}-21\cdot
6-22)}{2\cdot 22\cdot 100}=\frac{42}{275},$

for every $xy\in E(G_{HS})$, and so Lemma 2.1 only yields
$q_{n}(G_{HS})\leq\frac{42}{275}\cdot|V(G_{HS})|$. However, we have
$q_{n}(G_{HS})=\lambda_{1}(G_{HS})+\lambda_{n}(G_{HS})=0.14\cdot|V(G_{HS})|$.
It might be that $q_{n}(G)\leq 0.14n$ holds for every triangle-free graph $G$
on $n$ vertices.

## Acknowledgements

We thank anonymous referees for carefully reading the manuscript.

## References

  * [1] N. Alon.  Bipartite subgraphs.  Combinatorica, 16(3):301–311, 1996.  doi:10.1007/BF01261315.
  * [2] N. Alon and F. R. K. Chung.  Explicit construction of linear sized tolerant networks.  In Proceedings of the First Japan Conference on Graph Theory and Applications (Hakone, 1986), volume 72, pages 15–19, 1988.  doi:10.1016/0012-365X(88)90189-6.
  * [3] J. Balogh, F. C. Clemen, and B. Lidický.  Max cuts in triangle-free graphs.  Trends in Mathematics (Extended Abstracts EuroComb 2021), 14:509–514, 2021.  doi:10.1007/978-3-030-83823-2_82.
  * [4] W. Bedenknecht, G. O. Mota, C. Reiher, and M. Schacht.  On the local density problem for graphs of given odd-girth.  J. Graph Theory, 90(2):137–149, 2019.  doi:10.1002/jgt.22372.
  * [5] S. Brandt.  The local density of triangle-free graphs.  Discrete Math., 183(1-3):17–25, 1998.  doi:10.1016/S0012-365X(97)00074-5.
  * [6] F. C. Bussemaker, D. M. Cvetković, and J. J. Seidel.  Graphs related to exceptional root systems.  In Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Vol. I, volume 18 of Colloq. Math. Soc. János Bolyai, pages 185–191. North-Holland, Amsterdam-New York, 1978.
  * [7] P. Csikvári.  Note on the sum of the smallest and largest eigenvalues of a triangle-free graph, 2022.  arXiv:2205.08873.
  * [8] L. de Lima, V. Nikiforov, and C. Oliveira.  The clique number and the smallest $Q$-eigenvalue of graphs.  Discrete Math., 339(6):1744–1752, 2016.  doi:10.1016/j.disc.2016.02.002.
  * [9] P. Erdős.  Problems and results in graph theory and combinatorial analysis.  In Proceedings of the Fifth British Combinatorial Conference (Univ. Aberdeen, Aberdeen, 1975), pages 169–192. Congressus Numerantium, No. XV, 1976.
  * [10] P. Erdős.  Some old and new problems in various branches of combinatorics.  Discrete Math., 165/166:227–231, 1997.  doi:10.1016/S0012-365X(96)00173-2.
  * [11] P. Erdős, R. Faudree, J. Pach, and J. Spencer.  How to make a graph bipartite.  J. Combin. Theory Ser. B, 45(1):86–98, 1988.  doi:10.1016/0095-8956(88)90057-3.
  * [12] P. Erdős, R. J. Faudree, C. C. Rousseau, and R. H. Schelp.  A local density condition for triangles.  Discrete Math., 127:153–161, 1994.  Graph theory and applications (Hakone, 1990).  doi:10.1016/0012-365X(92)00474-6.
  * [13] P. Erdős, E. Győri, and M. Simonovits.  How many edges should be deleted to make a triangle-free graph bipartite?  In Sets, graphs and numbers (Budapest, 1991), volume 60 of Colloq. Math. Soc. János Bolyai, pages 239–263. North-Holland, Amsterdam, 1992.
  * [14] P. Keevash and B. Sudakov.  Sparse halves in triangle-free graphs.  J. Combin. Theory Ser. B, 96(4):614–620, 2006.  doi:10.1016/j.jctb.2005.11.003.
  * [15] S. Norin and L. Yepremyan.  Sparse halves in dense triangle-free graphs.  J. Combin. Theory Ser. B, 115:1–25, 2015.  doi:10.1016/j.jctb.2015.04.006.
  * [16] A. A. Razborov.  Flag algebras.  J. Symbolic Logic, 72(4):1239–1282, 2007.  doi:10.2178/jsl/1203350785.
  * [17] A. A. Razborov.  More about sparse halves in triangle-free graphs.  Sbornik: Mathematics, 213(1):109–128, 2022.  doi:10.1070/sm9615.
  * [18] J. B. Shearer.  A note on bipartite subgraphs of triangle-free graphs.  Random Structures & Algorithms, 3(2):223–226, 1992.  doi:10.1002/rsa.3240030211.