\documentclass{midl} % Include author names
% \documentclass[anon]{midl} % Anonymized submission

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\usepackage{mwe} % to get dummy images
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\jmlrvolume{-- 103}
\jmlryear{2024}
\jmlrworkshop{Full Paper -- MIDL 2024}
\editors{Accepted for publication at MIDL 2024}

\title[deep blind aif]{Deep blind arterial input function: signal correction in perfusion cardiac magnetic resonance}


% More complicate cases, e.g. with dual affiliations and joint authorship
\midlauthor{\Name{Habib Rebbah\nametag{$^{1}$}} \Email{habib.rebbah@olea-medical.com}\\
  \Name{Magalie Viallon\nametag{$^{2, 3}$}} \Email{magalie.viallon@creatis.insa-lyon.fr}\\
  \Name{Pierre Croisille\nametag{$^{2, 3}$}} \Email{Pierre.Croisille@creatis.insa-lyon.fr}\\
  \Name{Timothé Boutelier\nametag{$^{1}$}} \Email{timothe.boutelier@olea-medical.com}\\
  \addr $^{1}$ Research and Innovation department, Olea Medical, La Ciotat, France \\
    \addr $^{2}$ Univ Lyon, UJM-Saint-Etienne, INSA, CNRS UMR 5520, INSERM U1206, CREATIS, F-42023, Saint-Etienne, France\\
    \addr $^{3}$ Department of Radiology, University Hospital Saint-Etienne, Saint-Etienne, France
}

\begin{document}

\maketitle

\begin{abstract}
  \textit{Objectives}: The non-linear relationship between gadolinium concentration and the signal in perfusion 
  cardiac magnetic resonance (CMR) poses a significant challenge for accurate quantification of pharmacokinetic 
  parameters. This phenomenon primarily impacts the arterial input function (AIF), causing it to appear saturated 
  in comparison to the temporal concentration profile. This study aims to leverage a blind deconvolution strategy 
  through a deep-learning approach to address the saturation in the AIF.\\
  \textit{Methods}: We propose the utilization of a convolutional neural network (CNN) architecture with the 
  saturated AIF and a set of myocardial tissue signals as inputs, generating the corrected AIF as the output. 
  To train the network, a dataset comprising over $3 \times 10^6$ simulated AIFs with associated signals from 
  five simulated tissues response for each instance was employed. To assess the effectiveness of the approach, 
  the trained network was evaluated using a dual-saturation sequence to compare the corrected AIF with the 
  unsaturated version. The clinical dataset encompassed scans from 43 patients.\\
  \textit{Results}: The mean square error (MSE) for the testing subset of the simulated database was 0.69\% of the 
  peak. In the in vivo dataset, the coefficient of determination R2 was 0.26 and 0.86 for the saturated and 
  corrected AIF, respectively, in comparison to the unsaturated AIF.\\
  \textit{Conclusion}: The proposed network successfully corrects the acquisition-induced effects on the AIF. 
  Moreover, the extensive simulated database, featuring diverse acquisition parameters, facilitates the robust 
  generalization of the network's application.
\end{abstract}

\begin{keywords}
  Perfusion, MRI, AIF, CNN.
\end{keywords}

\section{Introduction}

The intravenous administration of gadolinium-based contrast agent (CA) in cardiovascular magnetic resonance (CMR) 
during its initial vascular transit has become a clinical standard for assessing myocardial perfusion states 
\citep{jerosch-heroldMRIMyocardialPerfusion2006}. Specifically, dynamic contrast enhanced (DCE) imaging in the cardiac domain has demonstrated 
its efficacy in evaluating various diseases \cite{montalescot2013ESCGuidelines2013,nagelMagneticResonancePerfusion2019}. Perfusion quantification typically involves 
employing a deconvolution approach based on a parametric tracer-kinetic model and an arterial input function (AIF) 
\cite{sourbronTracerKineticModelling2011,davillerDirectComparisonBayesian2021a}. The AIF is commonly derived from time-signal profiles within a region of 
interest (ROI) placed in the left ventricular (LV) blood pool. 

However, the signal obtained is not linearly dependent on the concentration of the CA. 
Such a linearity can only be assumed for the lowest CA concentrations, which, in practice, 
can be expected for myocardial signals. The cardiac image acquisition sequence introduces effects 
often described as saturation. Typically, common sequences involve a saturation preparation followed 
by multiple acquisition pulses of spoiled gradient-recalled echo (SPGR) \cite{barkhausenImagingMyocardialPerfusion2004b,slavinFirstPassMyocardialPerfusion2001a}. 
The long recovery time ($\sim$90-110ms) used in CMR perfusion sequence to improve the contrast between remote to 
ischemic region in ﬁrst-pass imaging lead non-linear attenuation of the AIF. Relying on signals from the 
LV blood pool could consequently lead to the misestimation of pharmacokinetic (PK) parameters. 
The signals acquired in such sequences can be theoretically described by acquisition and tissue parameters, 
including T1, T2, repetition time (TR), and flip angle \cite{hanickeSignalStrengthSubsecond1990b,hsuNonlinearMyocardialSignal2008b}.

One can utilize a derived equation for the inverse problem to estimate the concentration profile from the 
LV blood pool signal \cite{hsuNonlinearMyocardialSignal2008b}. However, this approach relies on a detailed description of the sequence 
used for acquisition, which may not always be available. Additionally, 
certain model limitations, such as the water exchange effect, can complicate the methodology \cite{landisDeterminationMRIContrast2000b}.

Various solutions have been proposed to address these limitations. One popular approach involves using a 
population-based Arterial Input Function (AIF). This entails deriving a constant AIF by averaging AIFs from a 
population of volunteers or patients who share characteristics similar to those of the current patient under analysis 
\cite{parkerExperimentallyderivedFunctionalForm2006a}. However, this method has the obvious drawback of not capturing individual variations, 
which can be substantial.

Some authors have suggested modifying the acquisition scheme or sequence. The dual-bolus method involves two injections: 
one, dedicated to the AIF acquisition, with a lower dose of contrast agent (CA) to limit saturation effects and another 
with a higher dose to maximize tissue response \cite{christianAbsoluteMyocardialPerfusion2004a}. However, the dual-bolus protocol remains complicated 
to implement for clinical routine with extensive care to be taken to ensure the reproducibility of the boluses, 
especially during stress.

The dual-saturation acquisition proposes obtaining two types of images during the same breath-hold \cite{gatehouseAccurateAssessmentArterial2004a, kimMultisliceDualimagingSequence2006}. 
The first image, with a short recovery time ($\sim$10ms), aims to limit the acquisition's impact on the signal, 
promoting linearity in the signal-concentration relationship but with low resolution. The second image follows 
common parameters of a perfusion sequence in CMR. The dual-imaging approach is still proposed as prototype sequence 
by most vendors. Hence, this solution is limited to research areas, and its clinical routine application requires 
specific training. Both dual-bolus and dual-saturation methods are constrained to their own acquisitions and cannot 
be generalized to the extensive dataset of perfusion acquisitions over time. Most of the perfusion acquired data in 
routine clinical practices are still obtained without satisfying solution and considering the potential pitfalls of 
using various elaborated perfusion pulse sequences in research studies, an attenuation resilient AIF estimation method 
based on conventional perfusion imaging protocols with high-dosage gadolinium boluses is of high interest.

Other research has focused on estimating the correct AIF directly from acquired images using a blind deconvolution approach 
\cite{schabelModelConstrainedMonteCarlo2010b}. This assumes a shared AIF among various tissues with different tissular responses, 
with the redundancy of information allowing simultaneous estimation of tissular pharmacokinetic parameters and 
the corresponding AIF. However, the practical application of such a method requires a sufficiently diverse range of 
tissue responses to ensure stability. One frequently suggested approach is to constrain the AIF to a specific parametric 
time-course model, such as Parker’s AIF. Bayesian approaches have shown also promising results \cite{schmidAttenuationResilientAIF2007a}. 
Despite the stability limits, these methods can be computationally intensive due to the conventional iterative 
resolution of the deconvolution problem for tissue response and AIF.

To overcome these limitations, recent studies propose using a deep learning approach to correctly estimate the AIF from 
the acquisition. One approach involves using unsaturated AIFs from dynamic susceptibility contrast (DSC) acquisitions
and their corresponding DCE profiles to train a conditional generative adversarial network for correcting 
DCE saturated AIFs \cite{choiImprovingReliabilityPharmacokinetic2020}. Another approach uses a similar method based on unsaturated AIFs 
obtained using the dual-saturation sequence to train a 1D U-Net for correcting input DCE AIFs \cite{scannellAIAIFArtificialIntelligencebased2023}. 
Although both methods present promising results, their generalization is limited by the variability in their databases, 
primarily due to differences in magnetic field strength (1.5 vs. 3T) and MRI machine characteristics.

To the best of our knowledge, the latest proposition introduces a form of blind deconvolution into a physics-informed 
neural network (PINN) to constrain AIF estimation \cite{vanhertenPhysicsinformedNeuralNetworks2022a}. Despite the extensive dataset and the absence of 
the need for labeled data, the main limitation is the considerable computation time required for this approach 
(1 hour per slice for a standard personal computer).

In this study, we propose a deep learning solution to estimate the AIF while maximizing generalization, 
based on simulations and blind deconvolution principle, taking saturated AIF and tissue signals as inputs to return 
the estimation of the unsaturated AIF. The objective is to develop a user-friendly and efficient solution that circumvents 
the need for complex modeling of acquisition sequences and can be extended to diverse acquisition parameter settings.

\section{Methods}
\subsection{Simulated database}

A large simulated database of AIFs and their corresponding tissue responses of over $3\times10^6$ cases is used to train the network.
We adopted Parker’s simplified form to generate the unsaturated AIFs, as described in \cite{parkerExperimentallyderivedFunctionalForm2006a}.

\begin{equation}
  \text{AIF}_{unsat}=A_0 \frac{\exp(-kt)}{1+\exp(-s(t-\tau))} + \sum_{n=1}^{2} A_n \exp\left({-\frac{t-T_n}{2\theta_n}}\right)
  \label{eq:ParkerAif}
\end{equation}

Five different tissue responses were generated for each AIF using the Toft-Kety model \cite{toftsMeasurementBloodbrainBarrier1991, toftsEstimatingKineticParameters1999a}.
The concentration observed for each case can be computed as follows:


\begin{align}
  &C_{tissue}(t)=\text{AIF}(t)\star R(t) \notag\\
  &R(t)=K^{trans} \times \exp\left(-k_{ep}t\right)
  \label{eq:ConvTissue}
\end{align}

A uniform, independent, and identically distributed random sampling of the parameters in \equationref{eq:ParkerAif} and \equationref{eq:ConvTissue} 
enabled the generation of 3,906,250 unique cases, with the upper and lower bounds described in Annex \ref{Annex:simulation}. 
Eventually, the simulated AIF is converted into a saturated signal using a simplified set of Bloch equations 
tailored for cardiac perfusion acquisition, involving perfect saturation followed by SPGR pulses 
(equation 1.2.7 in \cite{rebbahQuantificationOptimisationImagerie2019}). Annex \ref{Annex:conversion} provides details on the conversion process. 
Tissue concentrations were converted by assuming a linear approximation with their corresponding signals, 
employing a proportionality coefficient set arbitrarily to 1.

Due to the simulation aspect, time step units were used rather than absolute time length, 
employing a signal length of 40 steps for both AIF and tissue responses. Additionally, 
to avoid any scaling issues, we normalized all inputs and outputs—accurate AIF, saturated AIF, 
and tissue signals—to the $[0, 1]$ range.


\subsection{In vivo database}
The evaluation of the network was conducted using in vivo data obtained from a set of 43 patients/volunteers 
with dual-saturation acquisitions extracted from the in vivo database described in \cite{davillerDirectComparisonBayesian2021a}. 
The perfusion sequences were acquired on a 3T MAGNETOM Prisma (Siemens Healthineers, Erlangen, Germany) 
at 3 to 5 short-axis locations for every heartbeat with a bolus injection (6 ml/s) of gadoterate meglumine 
(0.2 mmol/kg) (Dotarem, Guerbet, Paris, France). 

The unsaturated AIF was computed by avering blood time curves derived from ROIs positionned in the cavity on 
the short recovery time acquisition series of the dual perfusion sequence acquisition. 
For the saturated acquisitions, an automatic registration was provided by the manufacturer 
(MOCO series, Siemens Healthcare). LV blood pool time curves, and myocardial time curves were obtained 
from ROIs manually delineated at the optimal contrast perfusion image to obtain the saturated AIF for 
correction and a set of myocardial signals. A linear interpolation is used if necessary to match inputs 
length of the proposed model. A K-means algorithm was employed to select 5 main signals, 
computed as the average signal of each cluster. To match the simulated signal length, all time profiles were 
linearly interpolated to obtain signals with a length of 40 time steps.


\subsection{Network architechture and training}
The proposed network is illustrated in \figureref{fig:CnnModel}. It consists of a two-branch Convolutional Neural Network (CNN):
a 1D branch for the saturated Arterial Input Function (AIF) and a 2D branch for the 5 tissue signals. 
The outputs of the two branches are flattened, combined, and processed through dense layers. 
The final output is a correction of the input saturated AIF. Annex \ref{Annex:ModelConstruction} provides
a comparison with a fully blind deconvolution model (without saturated AIF input).

\begin{figure}[htbp]
  \floatconts
  {fig:CnnModel}
  {\caption{Deep blind AIF: CNN model. the model consists in two branches of CNN: 
  1D for the saturated AIF and 2D for the 5 main myocardial signals.
   The output is the estimated unsaturated AIF. Inputs and outputs are normalized in $[0, 1]$.}}
  {\includegraphics[width=0.9\textwidth]{figures/fig1.png}}  
  % \label{fig:CnnModel}
\end{figure}

The randomly allocated subdivision of the simulated dataset between training and validation follows an 80/20\% ratio, 
corresponding to 3,198,093/781,250 cases. Training was conducted using the ADAM optimizer with 
a learning rate of 0.001 and Mean Squared Error (MSE) loss between the proposed corrected AIF and 
the unsaturated simulated one. This training process spanned 30 epochs, with each epoch consisting 
of 20,000 cases per batch. The study was implemented using Python and the TensorFlow 2 framework.


The model is completed with a scale correction process. The output of blind deconvolution typically yields
an estimate of the AIF that is scale\-free, as all multiples of the proposed AIF satisfy the deconvolution problem,
allowing the flow or $K^{trans}$ to absorb the variation. To estimate the correct scale factor, 
we followed recommendations in the literature and chose to match the area under the curve (AUC) 
of the tail of the correct AIF with the AUC of the tail of the saturated one \cite{schabelModelConstrainedMonteCarlo2010b}. 
Specifically, we selected the last 10 time points to represent the tail of the AIF.  


\section{Results}
\subsection{Training results}

\begin{figure}[htbp]
  \floatconts
  {fig:TrainingResults}
  {\caption{Training results. Few randomly selected examples from the validation dataset. 
  The green line representing the unsaturated AIF when it is not apparent, is under the blue line, representing 
  an accurate prediction. The orange arrows highlight the zeros points.}}
  {\includegraphics[width=1\textwidth]{figures/fig2.png}}  
\end{figure}

\noindent After 30 epochs, the model converged with a MSE of $0.69\times10^{-2}$ for both the training and validation subsets. 
\figureref{fig:TrainingResults} showcases a few randomly selected results from the validation dataset. The unsaturated AIF (green line) 
is barely discernible, hidden by the prediction of the proposed algorithm (blue line). 

However, we observed a peculiar phenomenon where, in each training iteration, certain temporal positions of the 
corrected AIF consistently exhibited values of zero. These specific temporal positions varied with each new 
training process but remained constant within the same iteration, independent of the input conditions. 
The cause of this behavior is unclear, but a straightforward solution was implemented. Following a successful 
training, the temporal positions where the corrected AIF exhibited zeros were identified and a post-processing 
step to interpolate the values at those time points was added. In the algorithm presented here, there were two 
instances of zero values. Refer to Annex \ref{Annex:zeros} for further details.

\subsection{In vivo results}
The comprehensive evaluation of the entire algorithm (comprising the network and the post-processing correction 
for scale and zero values) using dual-saturation sequences was conducted by comparing discrepancies in the 
unsaturated AIF (AIF$_{unsat}$) with the saturated one (AIF$_{sat}$) and the discrepancies of the AIF$_{unsat}$ and the 
corrected AIF estimated by our algorithm (AIF$_{db}$ for deep blind). 
The discrepancies were evaluated as MSE and coefficient of determination R2.
Moreover, myocardial blood flow (MBF) was estimated using a deconvolution algorithm with a free-form residue function 
(Olea Medical), for each AIF along with the extracted main myocardial signals.

\begin{figure}[htbp]
  \floatconts
  {fig:InvivoQuantResults}
  {\caption{In vivo results. To compare the accuracy of the saturated AIF$_{sat}$ and the correction proposed in this 
  study AIF$_{db}$. The accuracy is tested against the AIF$_{unsat}$ in case of dual-saturation sequences, in term of MSE and coefficient of determination R2.
  The upper right plot presents the relative error (in comparison to AIF$_{sat}$) of the estimated MBF using the AIF$_{db}$ against the AIF$_{unsat}$.}}
  {\includegraphics[width=1\textwidth]{figures/fig3.png}}  
\end{figure}

The results presented in \figureref{fig:InvivoQuantResults} and \figureref{fig:InvivoSelectedResults} showcase some selected cases. The median MSE for 
the AIF$_{sat}$ was 0.15 +/- 0.05 compared to 0.07 +/- 0.03 for the AIF$_{db}$. The median R2 was 0.27 +/- 0.12 for 
the AIF$_{sat}$ compared to 0.86 +/- 0.45 (+/- 0.10 with outlier exclusion for two cases) of the AIF$_{db}$. The median relative error of MBF estimations
was 3.78 +/- 1.99 for the AIF$_{sat}$ compared to the AIF$_{db}$ was 0.34 +/- 0.57. More in vivo results are gathered in 
Annex \ref{Annex:quantitative}.


\begin{figure}[htbp]
  \floatconts
  {fig:InvivoSelectedResults}
  {\caption{Selected in vivo results. Each case is represented by a cropped image around the heart 
  with its associated AIFs at the right. For each case the MSE and R2 of the AIF$_{db}$ against 
  the AIF$_{unsat}$ is provided. The AIFs are scaled such as the peak value of  AIF$_{sat}$ is 1. }}
  {\includegraphics[width=.88\textwidth]{figures/fig4.png}}  
\end{figure}
 

\section{Discussion}
The proposed approach demonstrated its capability to estimate the unsaturated version of the AIF accurately. 
Training the model with simulated data allows to easily explore perfusion parameters, enhancing its robustness 
and generalizability compared to strategies using limited number of in vivo data to train the network (Annex \ref{Annex:ModelConstruction}). Figure 2 
highlights that some simulated AIFs differ from real acquired AIFs. However, this variability promotes learning 
the underlying convolution operation between the AIF and tissue responses to derive tissue signals.

While the principle of deconvolution in perfusion imaging traditionally involves only tissue signals, our 
approach deviates from this by incorporating the saturated AIF, thereby constraining the deconvolution. 
In conventional literature \cite{schabelModelConstrainedMonteCarlo2010b}, this is achieved by choosing a parametric time-course model; 
however, our approach constrains the solution through the effect of the acquisition. The challenge then becomes 
finding an AIF that can explain both the tissue signals and the transformation induced by the acquisition leading 
to the observed saturated AIF. Previous research suggested that ignoring tissue signals and using only the 
saturated AIF could accurately predict its unsaturated version \cite{choiImprovingReliabilityPharmacokinetic2020,scannellAIAIFArtificialIntelligencebased2023}. 
However, we argue that such results primarily reflect the low variability of the datasets used for training 
in those studies, as they utilized perfusion acquisitions from very few centers with limited variations in 
acquisition parameters. Incorporating both the saturated AIF and tissue signals in our approach allows for 
better constraint of the solution. Moreover, choosing simulation with high parameter variations supports the 
validity of the results.

The solution presented here is preliminary work intended to demonstrate the network's ability to correct 
acquisition effects. However, the model proposed generates fixed zeros whose origin cannot be currently explained. 
Preliminary tests with the addition of multiple dropouts seem to address 
this issue effectively (Annex \ref{Annex:zeros}), warranting further investigation. Nevertheless, the proposed 
post-processing successfully passed the evaluation test with real acquisition data. Notably, we did not introduce 
noise to the training data. This choice is derived from the practical pipeline induced by our method. 
Indeed, all inputs are averaged from a group of signals, using ROIs of the LV blood pool for the saturated 
AIF and myocardial signal classification for the five main tissue signals. This process significantly enhances 
the signal-to-noise ratio. Future work will focus on refining this aspect to enable the use of direct signals. 
However, the successful evaluation on real acquired data supports our initial intuition.

An intriguing open question concerns the performance dependency on input tissue signals. Specifically, 
the decision to solely utilize myocardial signals for deriving the five principal signals warrants consideration. 
Notably, a homogeneous myocardium may likely decrease the algorithm's performances, as evidenced by the outlier 
results observed in vivo. Furthermore, the adoption of K-means clustering to extract the main signals as 
cluster centroids is noteworthy. If the chosen number of clusters is restricted relative to the variability of 
the acquired signals, erroneous main signals may be generated through the averaging of disparate signals within 
the same cluster. Subsequent efforts will predominantly concentrate on precisely quantifying thus constraints 
and exercising greater discretion in selecting inputs.

\section{Conclusion}
A CNN model was proposed to correct the impact of acquisition on the signal to concentration relationship in 
perfusion CMR. The suggested model simulates blind deconvolution while imposing constraints on the desired AIF 
based on the concentration-to-signal conversion derived from simplified Bloch equations. 
The proposed strategy only needs a saturated AIF and myocardial signals to estimate an unsaturated AIF. 
Training the algorithm with simulated data allows testing a wide range of acquisition parameters. 
The evaluation against AIF derived from short recovery time acquisitions of dual-saturation acquisition 
demonstrates the validity of the approach.


\newpage
\nocite{*}
\bibliography{midl24_103}

\newpage
\appendix

\section{Simulations description}
\label{Annex:simulation}

\newcommand{\BlochEq}{
  \begin{align*}
    M_0&=1\\
    T1 &\in \{ x \in \mathbb{R} \mid 1400 \leq x \leq 2000, \Delta x=120 \}\\
    T2 &\in \{ x \in \mathbb{R} \mid 200 \leq x \leq 300, \Delta x=1 \}\\
    r1 &\in \{ x \in \mathbb{R} \mid 3 \leq x \leq 6, \Delta x=3/5 \} \times 10^{-3}\\
    r2 &\in \{ x \in \mathbb{R} \mid 4 \leq x \leq 7, \Delta x=3/5 \} \times 10^{-3}\\
    TI &\in \{ x \in \mathbb{R} \mid 80 \leq x \leq 200, \Delta x=24 \}\\
    TR &\in \{ x \in \mathbb{R} \mid 1 \leq x \leq 3, \Delta x=2/5 \}\\
    TE &\in \{ x \in \mathbb{R} \mid 1 \leq x \leq 3, \Delta x=2/5 \}\\
    \alpha &\in \{ x \in \mathbb{R} \mid 10 \leq x \leq 35, \Delta x=5 \}\\
    N &\in \{64, 65, 66, 128, 129, 130\}
  \end{align*}
}

\newcommand{\ToftEq}{
  \begin{align*}
    k_{ep} &\in \{ x \in \mathbb{R} \mid 1 \leq x \leq 5, \Delta x=1 \} \times 10^{-2}\\
    K^{trans} &\in \{ x \in \mathbb{R} \mid 1 \leq x \leq 3, \Delta x=0.4 \} \times 10^{-2}
  \end{align*}
}

\newcommand{\DelayEq}{
  \begin{align*}
    &\text{disperssion} =0.5\\
    &\text{delay} \in \{ x \in \mathbb{R} \mid 0 \leq x \leq 10, \Delta x=2 \}
  \end{align*}
}

\newcommand{\AifEq}{
  \begin{align*}
    p_0 &= 1\\
    p_1 &\in \{ x \in \mathbb{R} \mid 0.1 \leq x \leq 0.75, \Delta x=0.13 \}\\
    p_2 &\in \{ x \in \mathbb{R} \mid 0.1 \leq x \leq 0.9, \Delta x=0.16 \}\\
    p_3 &\in \{ x \in \mathbb{R} \mid 0.05 \leq x \leq 0.3, \Delta x=0.05 \}\\
    p_4 &\in \{ x \in \mathbb{R} \mid 0.05 \leq x \leq 1.5, \Delta x=0.15 \}\\
    p_5 &\in \{ x \in \mathbb{R} \mid 0.75 \leq x \leq 1, \Delta x=0.05 \}\\
    p_6 &\in \{ x \in \mathbb{R} \mid 2 \leq x \leq 7, \Delta x=1 \}\times 10^3\\
    p_7 &\in \{ x \in \mathbb{R} \mid 1 \leq x \leq 2, \Delta x=0.2 \}\\
    p_8 &\in \{ x \in \mathbb{R} \mid 1000 \leq x \leq 10^6, \Delta x=10^6/5 \}\\
    p_9 &\in \{ x \in \mathbb{R} \mid 100 \leq x \leq 10^4, \Delta x=10^4/5 \}\\
  \end{align*}
}

\subsection{AIF simulation}
\noindent The AIFs were generated using a fixed time length (94s) with 40 evenly spaced samples. The equation \ref{eq:ParkerAif} was rewrite in the following form:
\begin{multline*}
  \text{AIF}_{unsat}(t) = p_0 \exp\left( -\frac{t-t_{max}(p_3 + .1)}{2p_6}\right) + p_1 (1-p_2) \exp\left(-\frac{t-(p_3 + t_{max}(1+p_4)/10)}{2p_6p_7}\right) + \\ p_1p_2\frac{\exp(-t/p_8)}{1+\exp\left(-(t-t_{max}(p_4 + (1 + p_4 p_5)/10))/p_9\right)}
\end{multline*}

\noindent with the following values:
\AifEq
\noindent Generating AIFs from all the combinations leads to a dataset of 3,906,250 unique cases.

\subsection{Delay simulation}
As described in \cite{schabelModelConstrainedMonteCarlo2010b}, we simulated the delay using a convolution with a gaussian by randomly sampling the following parameters:
\DelayEq

\subsection{Tissue signal simulation}
We simulated the tissues signals using the Toft-Ketty model, by randomly sampling the following parameters:
\ToftEq

\subsection{Conversion simulation}
Using the equation \ref{eq:Conversion} in Annex \ref{Annex:conversion}, we simulated the conversion of AIF$_{unsat}$ to AIF$_{sat}$ by randomly sampling the following parameters:
\BlochEq

\section{Conversion concentration to signal}
\label{Annex:conversion}
\noindent To convert concentration to signal for generating saturated AIF, the following equation was used:
\begin{align}
  \label{eq:Conversion}
  &S([C])=P_{T2}\times[(M_{z,1} - P_{eq} )\times P_{T1} + N\times P_{eq} ] \\
  &P_{T2}=\sin(\alpha) \exp(-TE/T2) \notag\\
  &M_{z,1}=M_0\times [1 - \exp(-TI/T1)] \notag\\
  &E_1= \exp(-TR/T1) \notag\\
  &P_{eq}=M_0 \frac{1 - E_1}{1 - E_1\cos\alpha} \notag\\ 
  &P_{T1}=\frac{1-(E_1  \cos\alpha)^N}{1-E_1  \cos\alpha} \notag
\end{align}

\noindent The link with concentration used the usual relaxivities equations:

\begin{align}
  &R1=1/T1=1/(T1_0 )+r_1\times[C] \notag\\
  &R2=1/T2=1/(T2_0 )+r_2\times[C] \notag
\end{align}

% \noindent with:

\begin{table}[!htb]
  \centering
  \begin{tabular}{l | l}
    \bfseries Parameter & \bfseries Description \\
    \hline
    $M_0$ & Steady state magnetization \\ 
    $T(1/2)_0$ & Initial $T(1/2)$ before bolus arrival\\
    $r_{(1/2)}$ & relaxivities of $T(1/2)$\\ 
    $TI$ & Recovery time between saturation pulse and acquisition\\
    $TR$ & Repetition time\\
    $TE$ & Echo time\\
    $\alpha$ & Flip angle\\
    $N$ & Number of acquired line in k-space
    \end{tabular}
\end{table}


\noindent To obtain this equation, we assumed a perfect saturation, followed by a series of SPGR pulses after a recovery 
time of TI. The process can be reconstructed following the methodology of \cite{hanickeSignalStrengthSubsecond1990b}. More detailed, 
step by step, approach is available in chapters 1.2.a. and 1.2.b. of \cite{rebbahQuantificationOptimisationImagerie2019}.

\newpage
\section{Quantitative perfusion analysis}
\label{Annex:quantitative}
All the analysis use the dual saturation acquisition described in the in vivo database section. 
The quantitative perfusion analysis was performed using a deconvolution algorithm with a free-form residue function (Olea Medical).

\subsection{Main myocardial signals} 
The analyze were performed using the different AIFs and the 5 main myocardial signals. The results were evaluated through:

\begin{itemize}
  \item Comparison with the extracted main signals: MSE and R2 of the estimated reconstructed myocardial signals
  \item Comparison with the residue function derived from AIF$_{unsat}$: MSE of the estimated residue function with AIF$_{sat}$ and AIF$_{db}$
  \item Comparison with the residue function derived from AIF$_{unsat}$: relative error of the estimated MBF using AIF$_{sat}$ and AIF$_{db}$
\end{itemize}

\noindent All results are depicted in \figureref{fig:InvivoMainSignalsResults}. It is observed that the estimation achieved with AIF$_{db}$ 
demonstrates greater concordance with AIF$_{unsat}$ compared to AIF$_{sat}$. Specifically, there is a factor 10 improvement favoring AIF$_{db}$ 
in terms of relative error of the estimated MBF.

\begin{figure}[htbp]
  \floatconts
  {fig:InvivoMainSignalsResults}
  {\caption{Quantification of the extracted in vivo main myocardial signals obtained with AIF$_{unsat}$, AIF$_{sat}$ and AIF$_{db}$.
  The results are presented in term of MSE, R2 and relative error of the estimated MBF. 
  The table presents the average and the standard deviation of the results.}}
  {\includegraphics[width=1.\textwidth]{figures/fig5.png}}  
\end{figure}

\subsection{Mapping results}
We selected one volunteer from the in vivo database to present the mapping results that are gathered in \figureref{fig:InvivoMappingResults}.

\begin{figure}[htbp]
  \floatconts
  {fig:InvivoMappingResults}
  {\caption{Mapping results: quantification of MBF using AIF$_{unsat}$, AIF$_{sat}$ and AIF$_{db}$.
  AIF$_{sat}$ and AIF$_{db}$ are compared to AIF$_{unsat}$ in term of relative error of MBF (last row), and the distributions
  of MBF in the selected bounding box are plotted for each AIF. R2 was computed between the acquired and estimated signals.}}
  {\includegraphics[width=1.\textwidth]{figures/fig6.png}}
\end{figure}

The mapping results reinforce the earlier observations, indicating stronger agreement between AIF$_{unsat}$ and AIF$_{db}$ 
than between AIF$_{unsat}$ and AIF$_{sat}$. Notably, the error associated with AIF$_{db}$ is lower compared to the findings 
observed for the main myocardial signals dataset. This variance might stem from the extraction procedure of the main 
signals, potentially introducing erroneous signals through the averaging of divergent signals within the same cluster. 
This aspect warrants further investigation.

\newpage
\section{Building model and comparison of training strategies}
\label{Annex:ModelConstruction}

\subsection{Full blind deconvolution model}
Theoretically, the blind deconvolution problem  could be performed solely using the main signals. We propose here to
compare the results of the full blind deconvolution model with the proposed model. The full blind deconvolution model is constructed
using the same architecture as the proposed model, except that the layers associated with AIF$_{sat}$ are omitted, utilizing only the 
main signals as input. The model is trained using the same simulated database as the proposed model. 
Dropout layers with a rate of 0.2 are incorporated to prevent zero values in the output. Both models are trained with a maximum of 150 
epochs, with a stopping criterion based on the loss of the validation dataset.

\begin{figure}[htbp]
  \floatconts
  {fig:FullBlindModel}
  {\caption{Comparison between the proposed model and the full blind deconvolution model. The MSE is computed using the validation dataset.}}
  {\includegraphics[width=1.\textwidth]{figures/fig7.png}}
\end{figure}

\noindent The results are presented in \figureref{fig:FullBlindModel}. Notably, we observe a higher MSE for the full blind deconvolution model 
compared to our proposed model. Such behavior is anticipated, as the input signals may exhibit limitations such as low variability or significant 
delay. Introducing an initial AIF$_{sat}$ constraint in our proposed model helps mitigate these limitations.

In \cite{choiImprovingReliabilityPharmacokinetic2020} and \cite{scannellAIAIFArtificialIntelligencebased2023},  the authors advocated for using only 
the AIF$_{unsat}$ as input to train the network, demonstrating the feasibility of such an approach. However, these results are likely influenced 
by the limited dataset variability, as these studies utilized perfusion acquisitions from a few centers with restricted protocol parameter 
variations. Additional inputs such as the main signals or the AIF$_{sat}$ can provide constraints to enhance the solution. One could also experiment 
with fewer primary signals, theoretically achieving accurate AIF$_{unsat}$ estimation. Nevertheless, such approaches will expose the process 
to a higher risk of error.

\subsection{Comparison of training strategies}
In this section, we compare the same proposed model (with dropout rate of 0.2) trained with:
\begin{itemize}
  \item The simulated database as described in the main text
  \item The dual saturation acquisitions from the in vivo database
  \item The dual saturation acquisitions from the in vivo database to fine tune the model trained with the simulated database
\end{itemize}

The performances of the models were evaluated using 10 cases from in vivo database and 10,000 cases of the simulation training dataset.
To train with in vivo data, we split them in two sets: 26 for training and 7 for testing and managing the training process. The performances
were evaluated in terms of MSE and R2. The results are presented in \figureref{fig:ModelComparison}.

\begin{figure}[htbp]
  \floatconts
  {fig:ModelComparison}
  {\caption{Comparison of the proposed model trained with different datasets. The fine-tuned approach is initially trained with the simulated 
  database and then fine-tuned with the in vivo database. 
  MSE and R2 are computed using subsets of the in vivo (last row) and testing simulation (upper row) datasets.}}
  {\includegraphics[width=1.\textwidth]{figures/fig8.png}}
\end{figure}

The results highlight the advantage of the models trained with the simulated database. The fine-tuned model exhibits a slight improvement
in terms of MSE and R2 compared to the model trained solely with the in vivo database. However, the model trained with the simulated database
outperforms the fine-tuned model for in vivo and simulation dataset. These results lend support to the proposed strategy and its ability to generalize.

\newpage
\section{Analyze of zeros outputs}
\label{Annex:zeros}
The proposed model exhibits a peculiar phenomenon where certain temporal positions of the corrected AIF consistently 
display zero values in each training iteration. These specific temporal positions vary with each new training seed, 
which is a variation of the randomly selected cases for training. The cause of this behavior remains unclear; however,
we propose to investigate the impact of the dropout layers on a such behavior. 

The model was trained with different dropout rate (0 and 0.2) with two different seeds. The results are presented in \figureref{fig:DropoutImpact}.

\begin{figure}[htbp]
  \floatconts
  {fig:DropoutImpact}
  {\caption{Impact of the dropout layers on the zero outputs. The sub-figures represent the average AIFs across the 
  testing dataset for two models trained with different seeds. The orange arrows highlight some zeros points.
  1: The positions with zero values are excluded from the MSE computation}}
  {\includegraphics[width=1.\textwidth]{figures/fig9.png}}
\end{figure}

The positions with zero values vary with the selected seeds, and the addition of dropout layers effectively addresses 
the issue. However, the MSE computed without considering the zero positions is higher for the case with dropout layers.
Such results suggest that if a robust post-processing step is implemented to handle the zero points, the dropout 
layers could potentially be removed. In the algorithm presented in the main text, we proposed detecting these zero 
positions and interpolating the values at those time points.


\end{document}
