# Physics-informed neural networks for improving cerebral hemodynamics
predictions

Mohammad Sarabian  Hessam Babaee  Kaveh Laksari The work was supported by the
National Institutes of Health (NIH) National Institute of Neurological
Disorders and Stroke (NINDS) grant number R03NS108167.M. S. is with the
Department of Biomedical Engineering, University of Arizona, Tucson, Az,
United States (email: msarabian@email.arizona.edu).H. B. is with the
Department of Mechanical Engineering, University of Pittsburgh, Pittsburgh,
PA, United States (email: h.babaee@pitt.edu).K. L. is with the Department of
Biomedical Engineering and Department of Aerospace and Mechanical Engineering,
University of Arizona, Tucson, Az, United States (email:
klaksari@arizona.edu).

###### Abstract

Determining brain hemodynamics plays a critical role in the diagnosis and
treatment of various cerebrovascular diseases. In this work, we put forth a
physics-informed deep learning framework that augments sparse clinical
measurements with fast computational fluid dynamics (CFD) simulations to
generate physically consistent and high spatiotemporal resolution of brain
hemodynamic parameters. Transcranial Doppler (TCD) ultrasound is one of the
most common techniques in the current clinical workflow that enables
noninvasive and instantaneous evaluation of blood flow velocity within the
cerebral arteries. However, it is spatially limited to only a handful of
locations across the cerebrovasculature due to the constrained accessibility
through the skull’s acoustic windows. Our deep learning framework employs in-
vivo real-time TCD velocity measurements at several locations in the brain and
the baseline vessel cross-sectional areas acquired from 3D angiography images,
and provides high-resolution maps of velocity, area, and pressure in the
entire vasculature. We validated the predictions of our model against in vivo
velocity measurements obtained via 4D flow MRI scans. We then showcased the
clinical significance of this technique in diagnosing the cerebral vasospasm
(CVS) by successfully predicting the changes in vasospastic local vessel
diameters based on corresponding sparse velocities measurements. The key
finding here is that the combined effects of uncertainties in outlet boundary
condition subscription and modeling physics deficiencies render the
conventional purely physics-based computational models unsuccessful in
recovering accurate brain hemodynamics. Nonetheless, fusing these models with
clinical measurements through a data-driven approach ameliorates predictions
of brain hemodynamic variables.

Neural network, Hemodynamics, TCD

##  1 Introduction

Advances in medical imaging technologies have brought about quantifiable
improvements in clinical diagnosis, patient management, and clinical outcomes
[1]. Novel imaging modalities are integral to our pathophysiological knowledge
from the molecular level to organ systems, and have led to excitement about
possibilities of personalized and predictive medicine [2]. However, solely
relying on medical imaging has manifested significant limitations [3, 4, 5]
since most diseases are characterized by complex and dynamic interactions that
require high spatiotemporal resolution, which is a noted advantage of
computational models. On the other hand, although computational modeling has
made significant progress in providing a quantitative understanding of complex
biological processes and their function in disease, their clinical adoption
requires sufficient evidence to convince regulatory agencies to approve
clinical trials. This has led to a search towards “the perfect model”, which
has proven elusive so far, and might remain so for years to come, due to the
simplifying assumptions and physics deficiencies inherent in modeling. A
natural question then arises: Can the current modeling approaches, despite all
their shortcomings, provide useful information to augment the clinical
measurements or should we do away with imperfect computational models for the
time being? The fundamental challenge in the task of combining computational
models and clinical measurements is the varying accuracy and uncertainty in
each source of prediction. In this work, we utilize recent advances in the
field of deep learning to go beyond the single-source clinical diagnosis to a
multi-source paradigm, where computational simulations and clinical
measurements are assimilated with the goal of rectifying the paucity of
clinical measurements by utilizing physics-based models. This methodology
could enable lab-to-bedside deployment of a vast array of existing and future
computational models and ultimately lead to a paradigm shift in clinical
workflow.

We will develop our framework for cerebral blood flow hemodynamics, which
constitutes a significant health marker for clinical diagnoses. As an example
for other clinical applications, we will test our approach for CVS, which
afflicts 70% of subarachnoid hemorrhage (SAH) patients. SAH is a type of
stroke typically due to intracranial aneurysm rupture with an annual incidence
of 8-10 per 100,000 adults [6, 7, 8]. CVS, defined as focal or diffuse
transient narrowing of the large intracranial arteries, typically occurs
between day 4 to 14 after hemorrhage and accounts for neurologic deterioration
in a third of SAH survivors. To improve the likelihood of early detection of
CVS, the patients stay in the intensive care unit (ICU) for a period of two
weeks and undergo frequent neurologic, physiologic, and neuroimaging
examinations. Computed tomography angiography (CTA) is commonly obtained at
presentation of the patient to accurately assess their vascular architecture
and screen for cerebral aneurysms. However, high irradiation dose and
administration of exogenous contrast agents prompt health concerns and limit
the application of repeat CTA studies. Instead, TCD ultrasonography is
performed at bedside on a daily basis during the CVS risk window. Despite its
ease of use and while it provides reasonable temporal resolution,
applicability of TCD as a stand-alone neuromonitoring tool is limited, since
it can only provide velocity time series at a limited number of location
across the Circle of Willis (CoW), accessible through the skull’s acoustic
windows. Furthermore, contrary to CTA, which gives geometrical features and
branching patterns of the entire cerebrovasculature, TCD provides quantitative
blood flow velocity measurements at several locations in the vasculature.
Changes in blood flow velocities measured through TCD are used as a surrogate
for change in vessel lumen diameters. However, in terms of the flow
hemodynamics, elevated mean flow velocities are not always linked only to
vessel narrowing, and could for example be due to systemic changes, such as
high blood pressure, which might lead to higher false-positive decisions [9].
As a result, the sensitivity and specificity of TCD for diagnosing CVS, while
reasonable, are significantly lower than CTA (sensitivity: 67% vs. 75.6%, and
specificity: 69% vs 95.3%, for TCD and CTA, respectively [10, 11, 12]). There
is a notable need to find an alternative and effective tool for diagnosing
this devastating disease that is as accurate as CTA scans without its
limitations.

Challenges in reliably measuring cerebrovascular structure and hemodynamic
properties in the clinic, have motivated the use of CFD modeling. However,
while computing power and numerical schemes have significantly improved and
resulted in high-resolution, subject-specific blood flow simulations of
complex vascular topologies [17], clinical adoption of such models has been
hampered by the lack of rigorous clinical validation, high computational cost,
tedious mesh generation, parameter calibration, and most importantly
uncertainty in boundary condition subscriptions. Recent advances in the field
of deep learning have overcome the challenge of combining two separate
datasets with different levels of accuracy and uncertainty [18, 19, 20, 21,
22, 23], which can lead to a paradigm shift in quantitative approaches towards
clinical diagnosis, wherein by augmenting the highly accurate (yet sparse)
clinical measurements with the more-readily available computational models in
order to increase the spatiotemporal resolution while maintaining the same
level of accuracy of the clinical measurements. In particular, physics-
informed neural networks (PINN) can resolve the above pitfalls which has
plagued the pure physics-based CFD models and improve the prediction of
hemodynamic variables significantly by combining the clinical measurements
with the physics-based models [20, 22, 24, 25]. Recently, this method was
implemented for a rather simple geometry at the carotid bifurcation and
trained using in vivo flow velocity and wall displacement measurements
obtained via 4D flow MRI and segmenting 2D cine images, respectively [22]. 4D
flow MRI is a non-invasive technique that provides time-dependent, three-
dimensional blood velocity measures within the vasculature, from which a
comprehensive representation of the blood flow hemodynamics could be extracted
[13, 14, 15]. However, this technique is not common in the current clinical
settings and is mainly limited to fundamental research endeavors as scan times
can become prohibitively long, especially when the high spatial resolution or
large volumetric coverage are required [13, 15]. Furthermore, the wall
displacements of intracranial arteries cannot be accurately measured due to
the small variation of brain lumen diameters in a cardiac cycle and limited
spatial resolution of the MRI data [26, 27]. As a result, extending this
method to the brain vasculature while strictly using clinically available
measurements poses significant challenges given the highly more complex
cerebrovascular architecture, including the number of branches and
significantly more tortuous nature of the vessels, compared to the carotid
bifurcation.

In the current study, we first present a deep learning framework that can
predict blood flow hemodynamics in the entire brain vasculature by utilizing
subject-specific 3D angiography and TCD measurements, and validate our
modeling predictions against 4D flow MRI measurements (Fig. 1). We extend the
current state of the art by designing the neural network, called “area
surrogate physics-informed neural network (ASPINN)”, without relying on vessel
wall displacements or spatially-resolved 4D flow MRI data. Once trained, the
model can predict local blood flow velocity, vessel cross-sectional area, and
blood pressure across the entire brain vasculature in a matter of seconds. We
further demonstrate the clinical efficacy of our approach in diagnosing CVS,
based on synthetically simulated blood flow data within healthy baseline and
spastic CoW vasculature.

Figure 1: Procedure to generate and validate the cerebral hemodynamics in the
CoW: a) 3D rendering of brain vasculature with several major arteries; MCA:
middle cerebral artery, VA: vertebral artery, BA: basilar artery, ICA:
internal carotid artery, ACA: anterior cerebral artery. b) Vessel segmentation
of brain vasculature to extract branching patterns and vessel cross-sectional
areas [28] in addition to the TCD velocity time-series. c) The structure of
the PINN model taking the velocities from TCD and the baseline areas from
segmentation, and generating brain hemodynamics in the arterial tree. d)
Representative example of 3D blood flow streamlines resulting from 4D flow MRI
that we employ to validate the PINN predictions. From left to right, the
streamlines are shown for left ICA and MCA, A1 and A2-segments of left ACA,
BA, and right ICA, MCA, and ACA at peak systolic, respectively.

##  2 Methods

###  2.1 Physics-informed neural networks

In this framework, the solution of partial differential equations is
parametrized by a neural network that is trained to fit the clinical
measurements, while being constrained to follow physical laws. Specifically,
one neural network (NN) is trained for each CoW arterial segment to generate
unknown blood flow variables, i.e., blood velocity, wall displacement, and
blood pressure, such that they satisfy three constraints simultaneously [22].
First, the NNs are trained to fit the measured clinical data. Second, they are
penalized to satisfy the underlying physical conservation laws governed by a
reduced-order one-dimensional (1D) model of pulsatile blood flow [29]. Third,
they are trained to satisfy the conservation of mass and momentum across all
interface points in the arterial tree. The total loss function is a sum of
three individual contributions: measurement loss (corresponds to fitting the
clinical measurements), residual loss (to penalize the 1D equations not being
satisfied on random points in the space-time domain, known as the “collocation
points”), and interface loss (to satisfy the conservation of mass and momentum
on the junctions and bifurcations) (see Fig. 1c). The mathematical expression
of the total loss function is as follows:

$\operatorname{\mathcal{L}_{t}}=\operatorname{\sum_{k=1}^{M_{m}}}\operatorname{\mathcal{L}_{m}^{k}}+\operatorname{\sum_{k=1}^{M}}\operatorname{\mathcal{L}_{r}^{k}}+\operatorname{\sum_{k=1}^{M_{I}}}\operatorname{\mathcal{L}_{I}^{k}},$
(1)

where $\operatorname{\mathcal{L}_{t}}$ is the total loss function,
$\operatorname{\mathcal{L}_{m}^{k}}$ is the measurement loss of vessel $k$ at
which we have collected the clinical data,
$\operatorname{\mathcal{L}_{r}^{k}}$ is the residual loss of $k^{th}$ vessel,
$\operatorname{\mathcal{L}_{I}^{k}}$ is the interface loss function at the
$k^{th}$ interface point, M is the total number of vessels in the arterial
network (M = 23), $\operatorname{M_{m}}$ is total number of vessels in which
we have measurements ($\operatorname{M_{m}}=9$), and $\operatorname{M_{I}}$ is
the total number of interface points ($\operatorname{M_{m}}=11$).

The measurement loss is computed via:
$\operatorname{\mathcal{L}_{m}}=\operatorname{\mathcal{L}_{init}^{V}}+\operatorname{\mathcal{L}_{init}^{A}}+\operatorname{\mathcal{L}_{b}^{V}}+\operatorname{\mathcal{L}_{b}^{A}},$
where $\operatorname{\mathcal{L}_{init}^{V}}$,
$\operatorname{\mathcal{L}_{b}^{V}}$, $\operatorname{\mathcal{L}_{init}^{A}}$,
and $\operatorname{\mathcal{L}_{b}^{A}}$ denote the measurement losses
corresponding to the initial and boundary conditions of velocity and area,
respectively. Note that here by “boundary conditions” we mean the spatial
positions in the CoW where we have access to TCD data and these points need
not necessarily be located on the boundaries of the network, which is one of
the main advantages of the PINN approach as compared to conventional pure CFD
models. To compute the positions of TCD data in each vessel, first, we compute
the velocity at equilibrium position (diastolic phase) in each segment of the
CoW by integrating the mass conservation law, i.e., equation 4 below, using
the TCD velocity and ToF vessel area data. Then, we compare the computed
equilibrium velocity against the same quantity obtained from 4D flow MRI.

The initial and boundary measurement losses are the mean square error (MSE)
loss functions and defined as:

$\operatorname{\mathcal{L}_{init}^{S}}=\frac{1}{N_{x}}\sum_{i=1}^{N_{x}}\left(|S^{*}\left(x_{i},0\right)-S\left(x_{i},0\right)|^{2}\right),$
(2)

$\operatorname{\mathcal{L}_{b}^{S}}=\frac{1}{N_{t}}\sum_{i=1}^{N_{t}}\left(|S^{*}\left(x_{0},t_{i}\right)-S\left(x_{0},t_{i}\right)|^{2}\right),$
(3)

where $N_{x}$ and $N_{t}$ denote the total number of spatial and temporal
measurement points, $S$ and $S^{*}$ represent measurement and predicted blood
velocities $V$ or vessel cross-sectional areas $A$ respectively. Finally,
$x_{i}$, $t_{i}$ represent the spatial and temporal positions, and $x_{0}$ is
the spatial position in the vessel at which we have collected clinical data.
To compute area boundary loss $\operatorname{\mathcal{L}_{b}^{A}}$, we require
area time-series and due to the lack of clinical data, we have developed an
ASPINN framework to recover this information, i.e.,
$A\left(x_{0},t_{i}\right)$ (see section 2.2). Further, we smoothed out the
velocity waveforms employing Gaussian Processes (GP) regression [42] with a
periodic kernel function before feeding them to the PINNs framework.

The second contribution to the total loss function is residual loss
$\operatorname{\mathcal{L}_{r}}$. We have selected 2000 points using the
Latin-hypercube sampling strategy [43]. Over these points, the physics of
pulsatile blood flow is enforced by satisfying the governing equations of a
one-dimensional model of blood flow dynamics coupled to a pressure-area
relationship characterizing the elasticity of the vessel wall. These equations
take the following form [38, 29]:

$\frac{\partial A}{\partial t}+\frac{\partial Q}{\partial x}=0,$ (4)

$\frac{\partial Q}{\partial t}+\frac{\partial}{\partial
x}\left(\alpha\frac{Q^{2}}{A}\right)+\frac{A}{\rho}\frac{\partial P}{\partial
x}=\frac{f}{\rho},$ (5)

$P=P_{ext}+\frac{\beta}{A_{0}\left(x\right)}\left(\sqrt{A}-\sqrt{A_{0}\left(x\right)}\right),$
(6)

where $Q(x,t)$ is the mass flux, $P(x,t)$ is the average internal pressure
over the cross section, $\alpha$ is a non-dimensional momentum flux correction
factor, and $\rho$ is the constant mass density of blood. Here, $f$ represents
the friction force per unit length and is given by $f=-22\mu\pi V$, where
$\mu$ is is the constant blood dynamic viscosity and it provides a good
compromise fit to experimental findings [44]. Here, we set $\alpha=1$ in the
convective inertia term of equation (5) [45]. In equation 6,
$\beta=\frac{4}{3}\sqrt{\pi}Eh$, where $h$ and $E$ denote elastic arterial
wall thickness and Young’s modulus respectively, with the values adopted from
literature [26]. Each cross section deforms axisymmetrically independently of
the others from a reference state
$\left(P,A\right)=\left(P_{ext},A_{0}\right)$, with an external pressure
$P_{ext}$ and diastolic cross-sectional area $A_{0}$. Here we assume
$P_{ext}=0$ when $A=A_{0}$.

Since we enforce equations 4–6 over all collocation points, the corresponding
residuals are defined as: $r_{A}:=\frac{\partial A}{\partial t}+\frac{\partial
Q}{\partial x},$ $r_{v}:=\frac{\partial Q}{\partial
t}+\frac{\partial}{\partial
x}\left(\alpha\frac{Q^{2}}{A}\right)+\frac{A}{\rho}\frac{\partial P}{\partial
x}-\frac{f}{\rho},$
$r_{p}:=P-P_{ext}-\frac{\beta}{A_{0}\left(x\right)}\left(\sqrt{A}-\sqrt{A_{0}\left(x\right)}\right),$
where $r_{A}$, $r_{v}$, and $r_{p}$ represent mass, momentum, and pressure
residuals, respectively, and are functions of spatial $x$ and temporal $t$
positions. The partial derivatives in the residual expression are computed
using automatic differentiation [46]. These residuals enforce the neural
networks to predict velocity, area, and pressure so as to satisfy the
underlying differential equations 4–6. Eventually, the residual loss function
for each vessel in the CoW arterial tree is computed via:

$\operatorname{\mathcal{L}_{r}}=\frac{1}{N_{f}}\sum_{i=1}^{N_{f}}\left(r_{A}\left(x_{i},t_{i}\right)\right)^{2}+\left(r_{v}\left(x_{i},t_{i}\right)\right)^{2}+\left(r_{p}\left(x_{i},t_{i}\right)\right)^{2},$
(7)

where $N_{f}$ is the total number of collocation points in each vessel.

The final contribution to the total loss function, i.e., interface loss,
corresponds to satisfying the conservation of mass and momentum across all
interface points and is computed at each bifurcation point via:

$\operatorname{\mathcal{L}_{I}}=\frac{1}{N_{b}}\sum_{i=1}^{N_{b}}\left[Q_{in}\left(x_{b},t_{i}\right)-\sum_{j=1}^{N_{outlets}}Q_{j}\left(x_{b},t_{i}\right)\right]^{2}+\frac{1}{N_{b}}\sum_{i=1}^{N_{b}}\left[\zeta_{in}\left(x_{b},t_{i}\right)-\sum_{j=1}^{N_{outlets}}\zeta_{j}\left(x_{b},t_{i}\right)\right]^{2},$
(8)

where $N_{b}$ represents the number of collocation points on the interface
boundaries, $Q_{in}$ is the inlet volumetric flow rate to each bifurcation,
$x_{b}$ is the spatial position of the interface boundaries, $N_{outlets}$ is
the number of outlets at each bifurcation, and $\zeta$ is the total momentum
which is the sum of average internal pressure $P$ over the cross-section at
each bifurcation and the dynamic pressure. The interface loss function ensures
the flow information measurements in one vascular domain can be propagated
throughout the neighboring domains [22].

Here, each neural network has 7 hidden layers and 100 hidden units per hidden
layer, and an exponential linear unit (elu) activation function is chosen for
all the hidden layers. Each neural network’s weights and biases
$\left\\{W^{l},b^{l}\right\\}$ of layer $l$ are learned by minimizing the
total loss function via stochastic gradient descent using Adam optimizer with
default settings [48]. The training is performed for a total number of
$500,000$ epochs with the learning rate equal to $10^{-3}$. Moreover, to
increase the computational efficiency, we train the neural networks by feeding
them mini-batches of size 1024 collocation points. The predictions of neural
networks are robust with respect to the chosen batch size.

###  2.2 Area surrogate physics-informed neural networks

As discussed previously, due to the lack of in vivo clinical data for the
vessel wall displacements in the brain vasculature, we are not able to compute
area boundary loss $\operatorname{\mathcal{L}_{b}^{A}}$. Hence, in the current
study, we have developed an area surrogate physics-informed neural network
(ASPINN) model to generate area time-series at the spatial positions, for
which we have collected TCD data. Once the vessel wall displacements are
determined, we use them along with the velocity data obtained from TCD
ultrasound as training data to the PINN model explained in section 2.1 to
generate physically consistent outputs in the entire vasculature.

The step-by-step procedure of the ASPINN model is demonstrated in Fig. 2. From
the 3D ToF scan, we first performed image segmentation and extract the vessel
centerlines using the pathline-based model construction workflow of
SimVascular [30]. Then, we perform one-dimensional CFD modeling of blood flow
through the vasculature using SimVascular (Step $1$). We set the inlet
boundary conditions from the processing of our 4D flow MRI dataset, and for
the outlet boundary conditions we use the three-element Windkessel model [31],
where we adopted the total arterial compliance and systemic vascular
resistance from the literature [26]. Next, we build our ASPINN model to learn
subject-specific velocity-area correlations for each segment within the CoW
network via training two fully-connected, feed-forward PINNs, i.e., Networks
(a) and (b) in series (Step $2$). Henceforth, we utilize the ASPINN model
(Network (b)) to predict area time series using the in vivo TCD velocity time-
series $v(t)$ and the spatial position of velocity measurement $x$ as inputs
to the neural Network (b) (Step 3). In addition, our ASPINN model generates
internal blood pressure time-series averaged over those cross-sections.
Finally, we validate the area predictions against segmenting 2D cine scans
(Step 4). Note that since the wall displacement of brain arteries cannot be
recovered from segmentation of cine images, we have validated our developed
ASPINN model predictions against the area time-series resulting from the
segmentation of 2D cine scan of aorta/common carotid bifurcation of a healthy
subject with the clinical data adopted from [22] (See appendix S 0.1 Fig.
S1).The velocity-area correlations in each vessel predicted by the optimized
ASPINN model along with the comparison against one-dimensional CFD modeling as
well as the estimated area time series using the TCD velocity data as input
are shown in the appendix S 0.1 Fig. S2.

Figure 2: Step-by-step procedure to build the subject-specific ASPINN model
for each vessel in the CoW vasculature to generate and validate physics-based
area time-series at TCD velocity measurement points.

Network (a) of the ASPINN model (step 2 in Fig. 2) is employed to approximate
the velocity, area, and pressure at each $x$ and $t$. Network (b) is designed
for approximating the correlation between blood flow velocity and vessel
cross-sectional area at each $x$ and $t$. Both Networks (a) and (b) are
trained based on the velocity and area information generated by performing
one-dimensional CFD modeling of blood flow through the CoW. It is noteworthy
to mention that during the training of Network (b) the velocity derivatives
with respect to input features, i.e., $\partial{v}/\partial{x}$,
$\partial{v}/\partial{t}$ are obtained from Network (a) in each iteration.

First, we perform one-dimensional CFD simulation of blood flow to generate
velocity, area, and pressure in the entire vasculature. Next, we train Network
(a) by employing velocity and area time series at similar spatial positions as
TCD measurements (as shown in Fig. 4a) derived from the CFD modeling (see
section 2.1). Note that since we train Network (a) based on the outputs of CFD
simulation, we have access to the area boundary conditions and can compute
area boundary loss $\operatorname{\mathcal{L}_{b}^{A}}$. Then, we train
Network (b) for each vessel in the CoW arterial tree using the outputs of
Network (a) to learn subject-specific velocity-area correlations. The
structure of hidden layers of Network (b), the choice of activation function,
the learning rate, the type of optimizer, the number of iterations, the mini-
batch size, and the number of collocation points are the same as Network (a)
and discussed in section 2.1.

Network (b) is trained by minimizing the same total loss function (equation 1)
as Network (a). The measurement loss for each vessel of Network (b) is
computed via:
$\operatorname{\mathcal{L}_{m}^{b}}=\operatorname{\mathcal{L}_{init}^{A^{b}}}+\operatorname{\mathcal{L}_{b}^{A^{b}}},$
where $\operatorname{\mathcal{L}_{m}^{b}}$ is the measurement loss of Network
(b), and $\operatorname{\mathcal{L}_{init}^{A^{b}}}$ and
$\operatorname{\mathcal{L}_{b}^{A^{b}}}$ are the measurement losses of Network
(b) corresponding to the are initial and boundary conditions respectively,
computed as described below:

$\operatorname{\mathcal{L}_{init}^{A^{b}}}=\frac{1}{N_{x}}\sum_{i=1}^{N_{x}}\left(|A^{*}\left(x_{i},0,V\left(x_{i},0\right)\right)-A\left(x_{i},0,V\left(x_{i},0\right)\right)|^{2}\right),$
(9)

$\operatorname{\mathcal{L}_{b}^{A^{b}}}=\frac{1}{N_{t}}\sum_{i=1}^{N_{t}}\left(|A^{*}\left(x_{0},t_{i},V\left(x_{0},t_{i}\right)\right)-A\left(x_{0},t_{i},V\left(x_{0},t_{i}\right)\right)|^{2}\right).$
(10)

Finally, the vessel cross-sectional area time-series predicted by the trained
ASPINN model at the spatial positions in the CoW architecture along with the
velocity data obtained from TCD ultrasound are served as training data and
used to train the PINN model discussed in section 2.1 to generate physically
consistent brain hemodynamics in the vasculature.

###  2.3 Cerebral vasospasm diagnosis using PINN

The general overview of our procedure to update vessel cross-sectional areas,
and cerebral hemodynamics in the entire vasculature after CVS is illustrated
in Fig. 3.

Figure 3: Step-by-step procedure to update the vessel cross-sectional areas
and cerebral hemodynamics in the CoW arterial network after CVS employing
PINNs for solving an inverse problem.

First, we perform image segmentation of the initial angiography scan of a SAH
patient to find initial cross-sectional areas, branching pattern, and vessel
lengths using SimVascular [30]. Note that as we don’t have access to the SAH
patient data we develop our model based on the healthy subject and
synthetically create a CVS subject as explained below. Next, we generate
synthetic data for training and validating the PINNs by performing 1D CFD
simulation of the baseline condition (before CVS) using SimVascular [30]. The
spatial positions of velocity and area training data are marked as red cross-
markers on the centerlines and are selected to be the same as shown in Fig.
4a. We then train a PINN model for each vessel for the baseline condition
using the method discussed in section 2.1. So far, we have estimated the
baseline brain hemodynamic features in the vasculature.

Next, we use post CVS data to re-evaluate local cross-sectional areas.
However, in this study we only synthetically construct CVS cases by shrinking
the M1-segment of the left MCA uniformly across the lumen, with no change in
vessel shape and perform one-dimensional CFD modeling for all the six CVS
cases. Nevertheless, we only use the updated velocity time-series at the same
spatial positions as the baseline TCD measurements, along with the pressure
distribution predicted by the PINNs at the baseline condition to solve an
inverse problem. We retrain the PINN model and update the vessel cross-
sectional areas, velocity, and blood pressure in the entire vasculature.
Finally, we validate the neural network predictions against the outputs of
one-dimensional CFD simulations, which serves as our ground-truth data. Here,
we assume that the pressure distributions in the affected vessel do not
significantly change compared to the baseline condition (see appendix S 0.1
Fig. S3).

The second PINN model (after CVS) is trained by minimizing the same total loss
function presented in equation 1. The residual and interface loss functions
remain the same as equations 7 and 8. However, the measurement loss is
computed via:
$\operatorname{\mathcal{L}_{m}^{CVS}}=\operatorname{\mathcal{L}_{init}^{V}}+\operatorname{\mathcal{L}_{b}^{V}}+\operatorname{\mathcal{L}^{P}},$
where $\operatorname{\mathcal{L}_{m}^{CVS}}$ is the measurement loss for each
vessel after CVS, $\operatorname{\mathcal{L}^{P}}$ denotes the pressure
measurement loss and is computed via:
$\operatorname{\mathcal{L}^{P}}=\frac{1}{N_{p}}\sum_{i=1}^{N_{p}}\left(|P^{*}\left(x_{i},t_{i}\right)-P\left(x_{i},t_{i}\right)|^{2}\right),$
where $N_{p}$ is the total number of pressure training data points in the
$x-t$ plane. Here, $P^{*}$ is the predicted pressure by neural networks at the
training data points. The second neural network that is trained after CVS has
the same architecture as the first one trained before CVS (see section 2.1),
but with more variables to get trained. Concretely, the local equilibrium
cross-sectional areas $A_{0}$ of each vessel are now defined as trainable
variables in addition to weights and biases of each neural network. Thus the
set of trainable parameters $\hat{\Theta}$ consists of
$\left\\{W^{l},b^{l}\right\\}_{l=1}^{D}$ and
$\left\\{A_{0}^{j}\right\\}_{j=1}^{N_{x}}$, where $D$ is the number of hidden
layers of each neural network. These variables are updated during training by
minimizing the total loss function (equation 1) using the Adam optimizer [48].
All the algorithms presented in sections 2.1, 2.2, and 2.3 are implemented in
Tensorflow v2.0 [49], and computations were performed in double precision
arithmetic on a single NVIDIA Tesla P200 GPU card. The MRI clinical data
acquisition and the pre-and-post processing steps of 4D flow MRI scans, which
are based on the procedure presented in [50, 13] are explained in the appendix
S 0.1.

##  3 Results

###  3.1 Brain hemodynamic predictions

We acquired in vivo structural properties as well as blood flow velocity data
across the CoW of a healthy volunteer as follows. First, using a 2MHz TCD
ultrasound probe (DigiLite, Rimed, Inc), we collected flow velocity time-
series at nine spatial locations in the CoW, namely, left and right VAs (LVA,
RVA), BA, left and right M1-segment of MCA, left and right A1 and A2 segments
of ACA (as depicted in Fig. 4a). Comparable to clinical measurements, this
data was collected at 14.7 ms temporal resolution and over 40 cardiac cycles.
We use the TCD velocity measurements for training the neural networks as
described below. We also performed MRI scans from the same subject, including
a ToF angiography scan, which is routinely performed in the clinic and,
similar to CTA, is used to build the geometrical features of the CoW for
training. In addition, we obtained a 4D flow MRI scan, which provides three-
dimensional velocity time series maps in the entire CoW vasculature, but is
not a part of routine clinical workflow and will only be used for validation
purposes (Fig. 4b). The spatial resolution of 4D flow MRI was $1.5\times
1.26\times 2$ $mm^{3}$, and the temporal resolution is 42 msec over 14 cardiac
phases. All the MRI scans were performed on a 3T Siemens Skyra at University
of Arizona’s Translational Bioimaging Resource facilities.

Figure 4: Clinical data of a healthy human volunteer (30-year old male) used
for training and validating of the PINNs. a) TCD ultrasound velocity
measurements at nine spatial points shown on the centerline of the CoW
arteries. b) 3D ToF MRI scan and 4D flow MRI data, representing the CoW
architecture (top-left panel) and the three-dimensional peak systolic blood
flow distributions (bottom-left panel) in the arterial tree. The middle panel
shows 3D velocity streamlines at peak systolic through the left ICA and MCA.
The dashed blue box represents the cross-sectional blood flow velocity at the
plane located at the inlet of the left ICA and the corresponding velocity
profile.

We compared blood flow velocity time series predicted by the PINN model at ten
different spatial positions in the CoW against the flow velocities measured
through 4D flow MRI, which provides more spatial information (Fig. 5). We
found good agreement in velocity time evolution between the predictions of
neural networks and the in-vivo velocity data resulting from 4D flow MRI
during each cardiac cycle at every test point considering the uncertainty
range of MRI measurements (Fig. 5b). We observed that despite the lack of
knowledge about the inlet and outlet boundary conditions, which is a major
setback in traditional CFD simulations, and the relatively few velocity
training data, our PINN model is capable of closely approximating the
physiologically correct blood flow velocities in the entire CoW network. Note
that to predict the cerebral hemodynamic parameters using the PINN model, we
first utilized our developed ASPINN model to estimate physically-consistent
area wave propagation at TCD velocity measurement points, which are shown in
Fig. S2b of the appendix S 0.1 for few locations across the vasculature. Each
component of the total loss function of the PINN model, i.e., equation 1, as a
function of the number of training epoch is shown in the appendix S 0.1 Fig.
S4.

In order to further demonstrate this advantage, we simulated the blood flow
velocities in the same CoW network using a purely physics-based axisymmetric
CFD model (SimVascular, Stanford, CA) [30]. We used the physiological velocity
values, extracted from the 4D flow MRI data, to prescribe boundary conditions
at the four inlets (LICA, RICA, LVA, and RVA). In addition, we used the
commonly-used three-element Windkessel model [31] for the outlet boundary
conditions, and adopted the total arterial compliance and systemic vascular
resistance from the literature [26]. As can be seen, there are substantial
discrepancies between the CFD predictions and the ground-truth MRI
measurements. Specifically, although the CFD model can predict the general
trend of the blood velocities, it either underestimates or overestimates the
velocity range or introduces a time delay in the peak velocity values. This is
despite the fact that we used the accurate inlet boundary conditions (from 4D
flow MRI measurements), which are normally not available for a subject-
specific simulation. This constitutes a significant advantage of our machine
learning approach over traditional CFD modeling approaches.

In addition, the probability density functions (PDF) of blood velocity over
one cardiac cycle resulting from 4D flow MRI measurements, predictions of
PINNs, and CFD model outputs are illustrated for a point on each of the CoW
branches in Fig. 5c. The distributions highlight the velocity field agreement
between the PINNs and the in-vivo MRI data and the substantial discrepancies
between those and the CFD model. The discrepancies become more apparent as we
move further away from the inlets of the CoW arterial network, which indicates
the effect of physics-deficiencies and more importantly the uncertainties in
the outflow boundary conditions (Fig. 5b). Hence, uncertainties in the outlet
boundary condition subscription (due to the lack of knowledge in pressure
distributions in the efferent vessels) lead to erroneous predictions of brain
hemodynamics by the one-dimensional computational model as can be observed in
Fig. 5. Note that the results of the CFD simulations are not shown at the four
inlets (LICA, RICA, LVA, RVA) as the flow velocities there are prescribed as
inlet boundary conditions directly from the 4D flow MRI measurements.

Figure 5: Comparison of predicted results: a) CoW centrelines, marked by
color-coded labels for each of the ten test point. b) Comparison of blood flow
velocity time-series between the PINN, 4D flow MRI, and CFD modeling. The
error bars in the 4D flow MRI measurements denote the standard deviation of
cross-sectional velocities in that particular time frame. c) Velocity
distributions in one cardiac cycle at ten test spatial points resulting from
PINN, 4D flow MRI, and CFD modeling. The color of each panel corresponds to
its spatial location in the arterial tree. The width of the probability
density function shows relative distribution density. In panels b) and c), the
color of each box corresponds to its spatial location in the arterial tree as
marked in a).

###  3.2 Cerebral vasospasm diagnosis

In this section, we utilize our PINN framework developed above to address a
specific clinical cerebrovascular disease, i.e., CVS, which requires precise
knowledge of brain hemodynamics and vessel cross-sectional areas for
diagnosis. As mentioned in the introduction, the primary parameter in
diagnosing CVS is the local cross-sectional areas and how they evolve during
the length of stay in the ICU compared to the equilibrium cross-sectional
areas, extracted from baseline CTA on the day of admission. However, high-dose
irradiation and administration of contrast agents limit the application of
repeat CTA studies. Consequently, in order to approximate updated vessel local
diameters, after CVS, instead of CTA, TCD is performed at the bedside daily
during the 14-day CVS window, which provides the velocity information at only
several spatial positions in the vasculature. Below, we present a data-driven
approach to automatically compute and update the local cross-sectional areas
in the entire CoW vasculature based on new blood flow velocities obtain from
TCD ultrasonography.

Figure 6: Application to CVS: a) Location of training data, and their
corresponding velocity time-series at baseline and six levels of vasospasm
(Box colors correspond to the marker colors on the CoW centerline). The
training data are velocity and area for the PINN model that is trained for
baseline condition and are velocity and pressure for the PINN model that is
trained for CVS cases. b-c) Spatial comparison of 3D systolic area and
velocity maps between predictions of PINNs and synthetic ground truth (GT)
data in the entire CoW for a CVS case with 30% uniform reduction in diameter.
d-e) Spatiotemporal comparison of the area and velocity maps at the left MCA
after CVS between PINN predictions and GT data. f) Non-dimensional systolic
cross-sectional area comparison at three locations ($x/L=$0, 0.5, 1) in the
MCA, for seven scenarios; baseline (no vasospasm), and six levels of vasospasm
starting from $10\%$ lumen diameter reduction (mild vasospasm) to $60\%$
(severe vasospasm) diameter reduction in $10\%$ increments. g) Comparison of
PINN predictions of CVS cross-sectional areas and the clinically utilized
hemispheric ratio (HR).

For this purpose, we generated synthetic data by performing axisymmetric one-
dimensional CFD simulations of pulsatile blood flow for the same CoW arterial
tree with and without vessel stenosis. We used the outputs of the CFD
simulations (velocity, area, and pressure in the entire CoW vasculature) for
pre-spasm baseline as well as six levels of vessel spasm from $10\%$ (mild
vasospasm) to $60\%$ (severe vasospasm) diameter reduction [32, 8], which are
equivalent to $19\%$ to $84\%$ cross-sectional area reduction, respectively.
For all the cases, we used the same inlet and outlet boundary conditions as
explained in section 3.1, which although not realistic, still exhibits the
capability of our framework to detect the vasospasm by computing the local
vessel cross-sectional areas employing only the updated velocity data.
Moreover, since the PINN model is agnostic with respect to the boundary
conditions, any physiologically change in the boundary conditions due to the
vessel stenosis would reflect in the velocity TCD measurements and therefore
the PINN model learns how to update areas accordingly. We imposed the vessel
narrowing in the M1 segment of the left MCA of the CoW network since it is the
most commonly occurring region for CVS [16, 8]. Furthermore, as previously
reported, we assume that narrowing is uniform across the lumen in the affected
branch, with no changes in vessel shape [8]. We then solve an inverse problem
by retraining our PINN models to estimate and update the vessel cross-
sectional area maps in the entire CoW after the vessel stenosis and compared
its predictions against the CFD ground-truth (GT) simulations in terms of
vessel cross-sectional area and hemodynamic variables. Our main assumption
here is that the blood pressure does not substantially alter after vessel
stenosis and thus we can utilize the pressure distributions in the vasculature
before CVS to recompute the vessel cross-sectional areas after CVS (see
appendix S 0.1 Fig. S3). We took the CFD velocity time series at the same nine
points as shown in Fig. 4a to mimic the clinical TCD measurements, which we
used in order to train the spastic network (Fig. 6a). We observed that the
blood flow velocities in the MCA artery are significantly altered compared to
other vessels (red box), given its proximity to the location of stenosis. On
the other hand, the blood velocities in the vessels that are sufficiently away
from the stenosed MCA (blue and purple boxes) are less affected.

To further demonstrate the performance of the PINN model, cross-sectional
areas and blood velocities at systolic phase after vasospasm, recomputed by
PINN, are projected on the vessel centerlines (Fig. 6b-c). We found good
agreement between the predictions of PINN and the ground-truth data provided
by the CFD simulations, indicating the capability of the presented approach in
updating the local vascular cross-sectional areas and hemodynamics in the
entire vasculature. In addition, the area and velocity space-time maps for the
affected vessel (M1 segment of the left MCA) at $30\%$ reduction in diameter
are compared between the PINN predictions and ground truth (Fig. 6d-e). To
better visualize the two maps, the cross-sectional areas are non-
dimensionalized by subtracting and normalizing by the equilibrium cross-
sectional areas $A_{0}$ at each spatial position $x$, i.e.,
$\hat{A}\left(x,t\right)=\left(A\left(x,t\right)-A_{0}\left(x\right)\right)/A_{0}\left(x\right)$.
We further compared the non-dimensionalized cross-sectional area $\hat{A}$ in
the affected MCA vessel at three positions $(x/L\in[0,0.5,1])$ corresponding
to beginning, middle, and end points along the vessels longitudinal axis and
at peak systolic phase (Fig. 6f). The data points are tightly distributed
along the 45-degree line (shown by red dashed line), indicating good agreement
between the PINN and the GT. We presume that the pressure distributions remain
unchanged from baseline condition. Nonetheless, they deviate from baseline
conditions for severe vasospasm, i.e., $60\%$ lumen diameter reduction after
vessel stenosis (see appendix S 0.1 Fig. S3). Consequently, the predictions of
PINN are prone to inaccuracies in the case of severe vasospasm. To further
demonstrate the clinical feasibility of our approach, we compare the cross-
sectional area predictions to the hemispheric ratio (HR) also known as
“Lindegaard ratio” [33] for baseline condition and all stages of vasospasm
(Fig. 6g). The HR is the ratio between mean flow velocities through the MCA
and ipsilateral ICA [33]. This ratio was recommended to improve the accuracy
of TCD measurements for vasospasm detection at the bedside and ratios larger
than three indicate anterior vasospasm [33]. This ratio has also been well-
captured by the PINN model as Fig. 6g demonstrates the HR values are larger
than 3 (corresponding to $10\%$ reduction in vessel lumen diameter) for all
vasospasm cases. As expected, the vessel’s local diameter shrinkage due to
vasospasm leads to an increase in blood flow velocities and higher HR values.

##  4 Discussion

Developing a technology that is able to generate rapid and reliable estimates
for brain hemodynamics as well as local vessel diameters with high
spatiotemporal resolution can significantly improve diagnosis and treatment of
various cerebrovascular diseases. Currently, TCD ultrasound is the most common
technique that is utilized in clinical practice to approximate hemodynamics
albeit at only a few cross-sections accessible through the cranial acoustic
windows, which might not be sufficient to gather comprehensive insight for
clinical diagnosis. In addition, CTA and ToF scans, although highly reliable
in terms of extracting vascular geometrical features, are limited for repeated
measurements. In this work, we put forth a deep learning framework that
augments the sparse clinical data, including baseline admission angiograms and
repeated daily TCD measurements, with simplified physical laws of fluid flow
in compliant arteries to recover brain hemodynamics and changes in local
vessel cross-sectional areas in the entire brain vasculature. We developed a
physics-based neural network model called “ASPINN” in order for use in
cerebrovascular hemodynamics with commonly collected clinical data. We
validated our approach by comparing the predicted blood velocity time series
at several locations in the CoW network against in-vivo velocity data acquired
from 4D flow MRI. Our framework can lead to a significant paradigm shift in
quantitative approaches towards clinical sciences, wherein by augmenting the
highly accurate (yet sparse) clinical measurements with the more easily
available computational models in order to increase the spatiotemporal
resolution while maintaining the same level of accuracy of the clinical
measurements.

We further showed the clinical significance of our developed deep learning
model by utilizing it to diagnose CVS, which is one of the most devastating
cerebrovascular diseases by solving an inverse problem. We demonstrated this
clinical value by synthesizing blood flow through the CoW for the baseline
condition (no vasospasm) and after vasospasm via CFD simulations. We found
that the PINNs can accurately estimate the vessel cross-sectional areas and
brain hemodynamics maps in the entire vascular domain using only the baseline
angiograms (representing ToF MRI or CTA), updated velocities (representing TCD
measurements) at the same nine locations as shown in Fig. 4, and pressure
distributions given by the PINNs from the baseline condition. Although we
generated synthetic data to show the clinical value of our developed tool, the
same tool can be applied in the clinical case where we do have access to the
admission CTA and daily TCD data. Eventually, the estimated vessel local
diameters resulting from the PINN predictions can be compared against a
secondary CTA, which is usually acquired after vessel spasm in the current
clinical workflow.

We found that although the CFD results qualitatively agree with in-vivo
clinical measurements, they tend to have substantially larger deviations in
the distribution of the hemodynamic variables compared to PINN results.
Conversely, the PINN approach trained on the clinical data, despite the
physics-deficiencies of the same underlying axisymmetric model, recovers the
brain hemodynamics with higher accuracy. Furthermore, the deep learning model
does not suffer from the major limitations imposed by the pure computational
models such as precise prescription of boundary conditions, hefty mesh
generation, or even constitutive laws [19, 20, 22]. In hemodynamics
simulations, in particular, inaccurate prescription of inflow/outflow boundary
conditions can severely degrade the simulation predictions [35, 36]. Although
blood velocity measurements via TCD are available, the spatially sparse
measurement points usually lie in the interior of the domain of interest and
not at the boundaries. Incorporating these measurements into the CFD code is
challenging because: (i) one has to solve an inverse problem to infer the
boundary conditions from these measurements, which can be cast as an
optimization problem and it is in general costly to solve, and often becomes
ill-conditioned or under-determined in the absence of sufficient measurements,
(ii) the TCD measurements are noisy, precluding their utility in numerical
techniques that demand smooth profiles for differentiation. On the other hand,
our PINN approach does not require the exact prescription of the boundary
conditions and they compute derivatives and solve inverse problems using noisy
data. The results from our PINN model will further improve the axisymmetric
CFD simulations of pulsatile blood flow in deformable walls. Indeed, it
appears that the purely physics-based CFD models can predict the general trend
of hemodynamic variables rapidly. However, they fail to capture the velocity
peak and mean values and induce a time delay in a cardiac cycle. This is due
to the combined effects of physics deficiencies and uncertainties in outlet
boundary condition subscription. The former is the result of model
simplifications and the latter is due to the lack of knowledge in internal
blood pressure distribution in efferent vessels. Conversely, our PINN model,
which is trained to both fit the clinical data and satisfy the physical laws
of unsteady blood flow, can successfully capture blood velocity peak and mean
values as well as its spatiotemporal distribution. Therefore, our findings in
this paper may be used as a benchmark for refining the CFD models to better
capture brain hemodynamics.

The current study has a number of limitations that should be considered when
interpreting the results. In this work, for the vessel wall compliance, we
resorted to the simplest constitutive law that correlates pressure and area,
i.e., the elastic tube law, neglecting the effects of the vessel wall
viscoelasticity on pressure distribution [37]. In addition, the axisymmetric
model that we have adopted neglects 3D effects such as vessel tortuosity and
curvature which are expected to play a major role in predicting hemodynamic
factors for cerebral arteries. Although incorporating more sophisticated
models into the PINN framework could lead to more accurate results [22, 38,
39], we found that, despite using the simple axisymmetric model, our approach
is capable of generating hemodynamic results on par with in-vivo data in terms
of accuracy, while ensuring much less computationally-intensive and faster
results. Furthermore, in order to train the PINNs properly, the spatial
position of the velocity training data in the arterial tree should be
specified. Since velocity data is obtained via TCD, that information is
elusive. In this work, in order to determine the location of TCD velocity
measurements, we used the 4D flow MRI measurements, which is usually
unavailable in clinical measurements. Having further studied the effect of
precise knowledge of the location, we found that for the vessels with almost
constant cross-sectional area in space, the model predictions are fairly
robust, but for the vessels with varying cross-sectional areas, furture
studies should be performed to test the robustness of the PINNs. We also
assumed that the vessel areas resulting from segmentation of the ToF scan is
at the equilibrium position due to the lack of information on the exact time
of image acquisition in a cardiac cycle. However, we found that this
assumption does not significantly affect the predictions of PINNs as wall
displacements of intracranial arteries in a cardiac cycle are small (less than
$3\%$) due to the high vessel stiffness [26]. Additionally, as the TCD angle
of insonation is unknown, the TCD velocity values are not corrected by the
cosine of this angle, which is common clinical practice. We presume that this
angle for major arteries in the CoW e.g. MCA is small [40], so the measured
values correspond well to true velocity. Finally, to update the vessel local
diameters and detect the vasospasm, we assumed that the pressure distribution
in the affected vessel does not change significantly as compared to the
baseline condition. However, this is not the case for all the CVS cases,
especially for the severe vasospasm, i.e., $60\%$ lumen diameter reduction.
Similarly, our algorithm relies on an assumption that the vessel shape is not
greatly affected by the vasospasm and the vessel diameter is reduced uniformly
across the lumen. Although previous researchers made the same assumption in
their simulations [8] or observed the same pattern in their experiments [41],
uniform diameter reduction may not always occur in the SAH patients.
Consequently, the pressure distribution will alter after vessel spasm as
compared to the baseline conditions and cannot be used to retrain the PINNs
and update the area maps. Hence, the PINN framework with its current format
may generate inaccurate vessel cross-sectional areas and brain hemodynamic
parameters due to the lack of sufficient training data for solving an inverse
problem.

In summary, this study opens an avenue in potentially addressing many clinical
problems regarding cerebrovascular diseases and introduces a new technology
that can be readily transferred to the clinical workflow and help clinicians
to diagnose and make better decisions for the subsequent treatment. Moreover,
this study sheds light on the physics deficiencies of conventional reduced-
order computational models of pulsatile blood flow and highlights the
importance of accurate boundary condition subscription in predicting brain
hemodynamics. We demonstrated that by combining the most simple reduced-order
computational model with the in-vivo clinical data through deep learning
algorithms, we can recover the hemodynamics and wall displacements of brain
arteries with an accuracy comparable to clinical data, albeit with higher
spatiotemporal resolution. The pressure distribution predicted by the PINNs
can be employed to discover the physiologically correct outlet boundary
conditions and hence eliminates one of the sources of uncertainties in
conventional CFD models.

###  S 0.1 MRI data analysis

We acquired the arterial geometry of the CoW from a healthy male volunteer
(age = 30 years, weight = 94 kg, height = 185 cm) by measuring 200 slices of
$0.25mm$ thickness with a 3D-TOF MRI sequence. The in-plane resolution was
$0.4mm\times 0.4mm$, the repetition time (TR) and the echo time (TE) were
$18ms$ and $3.57ms$, respectively, and the flip angle was 15°. In the same
scanning session, a prospectively ECG-gated 4D flow MRI scan was performed
using a 3.0 Tesla MRI scanner (Skyra, Siemens Healthcare, Erlangen, Germany)
and a 32-channel head coil. Scanning parameters were as follows: flip angle =
7°, repetition time/echo time=44.56/2.78 ms, bandwidth=445 Hz/pixel, velocity
encoding (VENC)=90 cm/s, voxel size =1.5x1.26x2 $mm^{3}$, temporal resolution
= 41.36 ms over 14 cardiac phases.

From the 3D-TOF slices, the 3D geometry was segmented with a model
construction pipeline of SimVascular. The threshold for the basic segmentation
was set to one-fourth of the maximum signal intensity value obtained in the
measurements, which showed the best results for segmenting arterial
structures. Everything above this value was considered to be an artery.
Additional segmentation was performed manually adding or removing voxels from
the geometry based on anatomical knowledge.

Before extracting velocity from 4D flow MRI phase difference images, we
perform image pre-processing including noise-masking, eddy-currents correction
as well as phase unwrapping following the method explained in. Noise masking
was performed by thresh-holding of the signal intensity in the magnitude data
to exclude regions with low signal intensity. Eddy current correction
consisted of three steps: 1. separation of static regions from blood flow, 2.
fit a plane with least-squares method to the static regions from the last time
frame (late diastole), 3. Subtraction of the fitted plane to the MRI data in
every time frame. Finally, we correct the velocity aliasing effect, i.e, if
adjacent pixels velocities in temporal or slice direction differ by more than
VENC. If aliasing occurs, we unwrap the phase difference images based on the
following equation:

$V\left(x,y,z,t\right)=V\left(x,y,z,t\right)-2\operatorname{sign}\left(V\left(x,y,z,t\right)\right)\operatorname{VENC}.$
(11)

Once, all the pre-processing steps are done and the image, we remap the phase
difference images in the range of $-\pi\leq\Delta\phi\leq\pi$ and compute the
blood velocity via:

$V\left(x,y,z,t\right)=\frac{\Delta\phi\left(x,y,z,t\right)}{\pi}\operatorname{VENC}.$
(12)

All the medical image processing steps are performed using an in-house MATLAB
code.

Figure S1: Validation of the developed area surrogate physics-informed neural
networks (ASPINN) that approximate the subject-specific velocity-area
correlation on flow through the aorta/common carotid bifurcation of a healthy
human volunteer. a) 4D flow MRI magnitude image along with the positions of
velocity and area measurements in the aorta/carotid bifurcation of a healthy
subject. Area time-series measurement at Aorta 3 (yellow cross-marker) is used
to validate the ASPINN predictions. b) Vessel centerlines and the velocity and
area time evolution in two cardiac cycle resulting from one-dimensional
computational modeling at four spatial positions in the arterial tree (marked
by blue cross-markers) which include inlet of the aorta (Aorta 1), aorta and
left common carotid bifurcation, outlet of the left common carotid artery, and
outlet of aorta (Aorta 4). c) Comparison of velocity-area correlations at
three spatial positions along the vessel that connects Aorta 2 and Aorta 3
between the ASPINN predictions and one-dimensional computational fluid
dynamics (CFD) modeling of the pulsatile blood flow. d) Velocity measurement
from 4D flow MRI at Aorta 3 (yellow cross-marker in panel a)) over one
pulsatile cycle. e) Comparison of area time evolution resulting from the
ASPINN prediction and the 4D flow MRI measurements at Aorta 3. f) Pressure
time-series at Aorta 3 resulting from the ASPINN prediction. Clearly, the
optimized ASPINN that is trained to recover subject-specific velocity-area
correlation, can estimate the wall displacement with high accuracy. Further,
while we cannot validate the pressure distribution due to the lack of clinical
data, the range of predicted pressure by the ASPINN is physiological. Figure
S2: a) Area surrogate physics-informed neural networks (ASPINN) predictions of
velocity-area correlations in different segments and at three spatial
positions in each segment of the circle of Willis arterial network. The
predicted correlations using the ASPINN model are also compared against one-
dimensional CFD modeling. The correlations are shown at the beginning, middle
and endpoints of the left and right vertebral arteries (LVA, RVA), left and
right sides of internal carotid arteries (LICA, RICA), basilar artery (BA),
left and right sides of M1-segment of middle cerebral arteries (LMCA-M1,
RMCA-M1), and A2-segment of the left anterior cerebral artery (LACA-A2). b)
ASPINN model predictions of area time-series in vessels of the CoW network
(marked by orange cross-markers) that we have access to the Transcranial
Doppler (TCD) ultrasound velocity measurements. These generated data will be
subsequently used as training data in the PINN model to predict the brain
hemodynamics in the entire vasculature. Note that, in total, we have the
velocity measurements in nine segments of the vasculature. However, here we
show the area predictions of the ASPINN model at six vessels for the sake of
clarity.  Figure S3: Pressure distributions in the MI-segment of the left MCA
of the CoW arterial network at baseline condition (no vasospasm) and six
stages of vasospasm i.e, $10\%$ (mild vasospasm), $20\%$, $30\%$, $40\%$,
$50\%$, and $60\%$ (severe vasospasm) of lumen diameter reduction. In all of
the vasospasm cases, narrowing is uniform across the lumen, with no alter in
vessel shape. a) Comparison of blood pressure colormaps in the $x-t$ plane
between baseline condition and the vasospasm cases. b) Comparison of pressure
time-series at the vessel endpoint i.e, the bifurcation point of M1-segment
and M2-segments (daughter branches) of the left MCA between baseline condition
and vasospasm cases. c) Comparison of blood pressure distribution at diastolic
phase along the vessel between baseline condition and vasospasm cases. d)
Probability density functions (PDFs) of blood pressure during one cardiac
cycle at the vessel endpoint for the baseline and six stages of vasospasm
cases along with the mean and median values. e) PDFs of the blood pressure
along the vessel length at the diastolic phase for the baseline and vasospasm
cases. The mean and median values are indicated. Figure S4: Flow through the
CoW network of a healthy human subject: Values of the loss functions versus
number of stochastic gradient descent epochs during the training of the PINN
model. The solid green line corresponds to the interface loss
$\mathcal{L}_{I}$. The solid blue line denotes the measurement loss of
velocity measurements, and the area generated by the area surrogate physics-
informed neural networks (ASPINN) model $\mathcal{L}_{m}$. The solid red line
represents the residual loss $\mathcal{L}_{r}$.

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