# B-spline-based Modulating Function Method for Arterial Blood Flow's Estimations

Mohamed Boukaf, Abderrahim Akhrouf, Messaoud Chakir, Zehor Belkhatir, Taous Meriem Laleg-Kirati

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Mohamed Boukaf, Abderrahim Akhrouf, Messaoud Chakir, Zehor Belkhatir, Taous Meriem Laleg-Kirati. B-spline-based Modulating Function Method for Arterial Blood Flow's Estimations. IFAC BMS 2024 - Biological and Medical Systems - 12th BMS, IFAC, Sep 2024, Villingen, Germany. pp.573-578, 10.1016/j.ifacol.2024.11.100. hal-04945104

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# B-spline-based Modulating Function Method for Arterial Blood Flow's Estimations

Mohamed Boukaf*,† Abderrahim Akhrouf**

Messaoud Chakir*** Zehor Belkhatir****

Taous Meriem Laleg-Kirati†

* Department of Electrical Engineering, Telecom SudParis, Paris Saclay, France.

** Department of Electrical Engineering, Paris Saclay, France.

*** National Polytechnic School of Algiers, Algeria.

**** Department of Electronics and Computer Science, University of Southampton, UK.

† National Institute for Research in Digital Science and Technology (INRIA), Paris Saclay, France.

**Abstract:** This paper considers the estimation problem of arterial blood flow by using a fractional model of arterial hemodynamics. This model has been introduced as an enhanced version of the conventional integer-order Windkessel model. This advanced model incorporates a fractional-order capacitor to characterize the intricate and the frequency-dependent arterial compliance. A recent algorithm that consists of modulating functions method combined with an iterative Newton approach has been proposed to solve this estimation problem. However, the method's performance was highly dependent on the convergence of the Newton method. In addition, it was affected by the choice of the basis functions. In this paper, we present an enhanced version of this two-stage algorithm, which leverages modulating functions, genetic algorithms, and B-spline basis functions for the simultaneous estimation of both the model's input (the blood flow) and the fractional differentiation order within a finite time frame. To validate the approach, in silico human data is employed, and its performance is compared with that of the former method. The outcomes demonstrate significant potential for accurate calibration of the fractional model and estimation of the blood flow.

**Keywords:** Modulating Function, Fractional Windkessel model, Fractional derivative, two-stage joint-estimation, B-spline

## 1. INTRODUCTION

Arterial models exhibit a spectrum of complexity, ranging from simplistic yet less physically interpretable models to highly intricate but accurate ones. These arterial models can be categorized into two primary classes: macro-scale-based models and micro-scale-based models. Macro-scale models encompass low-dimensional modelling approaches, such as zero-dimensional (0-D) or lumped parametric models. These models are straightforward and computationally efficient but lack physical interpretability, as pointed out by Bahloul and Laleg-Kirati (2020). They are primarily employed to represent the global hemodynamic properties of the arterial system. A widely recognized example of a 0D model is the arterial Windkessel, introduced by Frank (1990), this encompasses both single-compartment and multiple-compartment models, as highlighted by Malatos et al. (2016). From a mathematical perspective, macro-scale models are generally developed using ordinary differential equations (ODEs) to represent hemodynamics purely based on time. Conversely, models at the microscale level seek to offer more accurate physical descriptions of the properties of the regional arterial system than those

provided by models at the macro level. These models delve into the intricate hemodynamics of specific cardiovascular regions. Microscale models are characterized by their high-dimensional nature, which encompasses one-dimensional (1D), two-dimensional (2D), and three-dimensional (3D) representations of arterial trees. However, their complexity requests substantial computational resources, making them less practical for online applications. These models are typically based on partial differential equations (PDEs) and describe hemodynamics as a function of both time and spatial coordinates, as elaborated by Ruan et al. (2018). In practical applications, there is a preference for models that offer interpretable parameters and manageable complexity. In recent decades, fractional-order derivatives, which involve non-integer orders, have gained prominence in various fields, particularly in the modelling of biological systems. Fractional-order models extend the concept of differentiability, incorporating non-local and memory effects through fractional-order space and time derivatives. These properties enable the modelling of phenomena across different time and space scales without the need for excessive compartmentalization. In the context of cardiovascular system modelling, fractional-order calculus has demon-
strated its utility in characterizing the viscoelastic properties of elastic aortas, arterial dynamics, collagenous tissue viscoelasticity, arterial blood flow, red blood cell membrane mechanics, and heart valve cusps, as highlighted by Perdikaris and Karniadakis (2014); Bahloul and Laleg-Kirati (2019); Craiern and Magin (2010); Doehring et al. (2005); Liu et al. (2013), respectively. Recently, fractional-order theory has been applied to enhance the arterial Windkessel model by replacing the ideal capacitor with a fractional-order capacitor. Preliminary investigations have confirmed the suitability of fractional-order impedance for accurate aortic input impedance assessment, with a strong correlation observed between the fractional differentiation order (α) and hemodynamic determinants, as reported by Bahloul and Laleg-Kirati (2021, 2020). Fractional order enables a more precise description of the transition between elasticity and viscosity. Calibrating fractional-order models requires estimation-based methods, which can be categorized as asymptotic (e.g., methods based on observers) and non-asymptotic (e.g., algebraic methods), as outlined by Belkhatir et al. (2015). While asymptotic techniques are reliable, they are computationally intensive, initialization-dependent, and susceptible to noise. Non-asymptotic methods offer computational efficiency and enable joint estimation of parameters and unknown inputs, as elaborated by Liu et al. (2014). In previous work by Bahloul and Laleg-Kirati (2020), a finite-time estimation method has been introduced to simultaneously estimate arterial blood flow and parameters at a particular location of the arterial network. It is based on the modulating functions (MFs) combined with a newton approach. However, the performance of the method was highly dependent on the convergence of the Newton algorithm and the choice of basis functions involved in the estimation procedure. In this paper, we propose an enhanced version of the method which uses B-spline functions as a basis and genetic algorithms (GA), allowing us to avoid local-minima problem faced when using Newton's algorithm. The remainder of this paper is structured as follows: Section 2 provides preliminary mathematical formulations for the fractional-order arterial Windkessel representation, and introduces modulating functions and a B-spline basis. Section 3 presents the modulating functions algorithm, which is based on two-stages for the simultaneous estimation of arterial blood flow and fractional orders. Section 4 offers numerical examples illustrating the algorithm's performance, followed by a discussion of the results. Finally, Section 5 concludes the paper and outlines potential future research directions.

## 2. PRELIMINARIES

### 2.1 Fractional order derivative

Definition 1. The Grünwald-Letnikov (GL) fractional derivative is defined as:

$$D_t^\alpha f(t) = \lim_{h \to 0} h^{-\alpha} \sum_{j=0}^{[\frac{t}{h}]} (-1)^j \binom{\alpha}{j} f(t - jh), \quad (1)$$

where $h > 0$ is the time step and $\binom{\alpha}{j} = \frac{\Gamma(\alpha+1)}{\Gamma(j+1)\Gamma(\alpha-j-1)}$, $\Gamma(*)$ being the gamma function.

### 2.2 Windkessel fractional order model

The Windkessel representation concept draws inspiration from the analogy of electrical circuits. In this analogy, arterial blood pressure is related to electrical voltage, whereas blood flow within the arteries acts as an electrical current. The roles of vascular resistance and compliance are played by electrical resistor and capacitor elements, respectively.

The fractional-order arterial Windkessel model is similar to the conventional Windkessel model in structure. However, it diverges by incorporating a fractional-order capacitor (FOC) in place of a traditional capacitor. This FOC represents arterial compliance with enhanced accuracy. Unlike conventional components, the FOC encompasses both capacitive and resistive attributes, using a power-law equation to capture the intricate interplay between elasticity and viscosity. By adjusting the fractional differentiation order, the model more precisely depicts the shift from viscous to elastic behavior. The subsequent content will outline the formulation of the Windkessel fractional two-element model.

Using the mass conservation principle, the arterial blood flow can be expressed as follows:

$$q_{in}(t) = q_{out}(t) + q_{stored}(t), \quad (2)$$

where $q_{stored}(t)$ corresponds to the blood stored in the arterial tree, and $q_{out}(t)$ is the flow out of the arterial system:

$$q_{out}(t) = \frac{1}{R_p} p_a(t), \quad (3)$$

where $R_p$ corresponds to the peripheral vascular resistance and $p_a$ represents the aortic pressure. Concerning the signal $q_{stored}$, which is commonly evaluated using the standard definition, it can be computed by taking the initial derivative of the equation for volume concerning time. However, given the fractional characteristics of both red blood cells and the collagenous tissues comprising the arterial structure, we introduce the possibility of a real differentiation order for changes in blood volume over time. This allows us to apply fractional-order differentiation to this differential equation.

$$\begin{aligned} q_{stored}(t) &= D_t^\alpha V(t) = \frac{d^\alpha V(t)}{dt^\alpha}, \\ &= \underbrace{\frac{d^\alpha V(t)}{d^\alpha p_a(t)}}_{C_\alpha} \frac{d^\alpha p_a(t)}{dt^\alpha}, \end{aligned} \quad (4)$$

where $C_\alpha$ represents a fractional order proportionality constant, which reflects the fractional-order compliance, The latter has a unit of $[l \times sec^{\alpha-1}/mmHg]$.

Substituting (3) and (4) into (2) yield:

$$q_{in}(t) = \frac{1}{R_p} p_{\alpha}(t) + C_{\alpha} \frac{d^{\alpha} p_{\alpha}(t)}{dt^{\alpha}}, \quad (5)$$

which can be written as:

$$y(t) + \tau D_t^\alpha y(t) = u(t), \quad (6)$$

where $y(t) = p_a(t)$, $u(t) = R_p q_{in}(t)$, and $\tau = R_p C_\alpha$.
## 2.3 Modulating Function (MF)

**Definition 2.** A function $\phi(t)$ defined in the interval $[0, T]$, $T \in \mathbb{R}_+$ is a modulating function of order $n$, $n \in \mathbb{N}^*$ if the following properties are satisfied:

(-) $\phi(t) \in C^n([0, T])$, where $C^n([0, T])$ is the space of $n$ times differentiable functions over $[0, T]$,

(-) $\phi^{(i)}(0) = \phi^{(i)}(T) = 0$, for $i = 0, ..., n - 1$,

(-) $D_t^\alpha \phi(t)$ exists $\forall 0 \le \alpha \le n$,

(-) $D_t^\alpha \phi(t)|_{t=0} = 0, \forall 0 \le \alpha \le n$.

## 2.4 B-spline basis functions

**Definition 3.** Given a node vector $(t_0, ..., t_k)$, where $t_j, j = 0, ..., k$ are called knots, we refer to the B-spline functions of order $m^*$ associated with this node vector as the functions $\beta_{m^*}^i(t)$, for $0 \le i \le k - m^* - 1$, defined by the following recursive formula for $m^* \ge 1$:

$$ \beta_{m^*}^i(t) = \frac{t - t_i}{t_{i+m^*} - t_i} \beta_{m^*-1}^i(t) + \frac{t_{m^*+i+1} - t}{t_{m^*+i+1} - t_{i+1}} \beta_{m^*-1}^{i+1}(t), \quad (7) $$

with

$$ \begin{cases} \beta_0^i(t) = 1 & \text{if } t \in [t_i, t_{i+1}], \\ \beta_0^i(t) = 0 & \text{otherwise.} \end{cases} \quad (8) $$

# 3. MATERIALS & METHODS

## 3.1 Two-stage based MFs algorithm

The MF-based estimation method is a non-asymptotic approach that has demonstrated its efficacy in parameter estimation and identification for both fractional order systems, as demonstrated by Belkhatir et al. (2015), and integer order systems, as evidenced by Co and Ydstie (1990); Balestrino et al. (2000). The core concept underlying this method involves the transformation of the estimation problem associated with integer or fractional order differential equations into an algebraic problem, entailing the search for a solution to a set of equations. The MF-based estimation approach is prized for its fastness and ease of implementation, attributed to its intrinsic properties. Moreover, leveraging the principles of integration by parts, and harnessing the modulating function properties outlined in Definition 2, the MF-based method has proven to be resilient in the face of contaminating noise. To concurrently estimate the fractional differentiation order and the input, we propose a two-stage algorithm. This algorithm combines the modulating functions method with genetic algorithms.

Functionally, the two-stage algorithm operates as an iterative procedure, comprising two distinct steps. In the initial step, the algorithm addresses the input estimation problem at each iteration through the MF technique. The primary rationale behind this step is to project the input onto a suitable space (the basis functions space). The coefficients of this projection are treated as unknown parameters, which are simultaneously determined alongside the vector representing fractional differentiation orders.

Subsequently, in the second step, the algorithm tackles a nonlinear system of equations by employing a genetic algorithm to estimate the fractional differentiation orders.

Taking into consideration the linear time-continuous non-commensurate fractional-order system below:

$$ y(t) + \sum_{i=1}^{N} a_i D_t^{\alpha_i} y(t) = u(t), \quad (9) $$

where $t \in [0, T]$ is the time, $y(t)$ represents the output, $y(t) \in \mathbb{R}$, $u(t)$ represents the input, $u(t) \in \mathbb{R}$ and $\alpha_i \in \mathbb{R}^+$, $i = 1, ..., N$, are the unknown fractional differentiation orders to be estimated such that $\alpha_i \in (n_{i-1}, n_i)$. We assume that $0 < \alpha_1 < \dots < \alpha_n$. We denote $\theta = (a_1, ..., a_N)^T$, $\alpha = (\alpha_1, ..., \alpha_N)^T$ and its estimate $\hat{\alpha}$ is denoted as $\hat{\alpha} = (\hat{\alpha}_1, ..., \hat{\alpha}_N)^T$. The associated estimation problem is then formulated as follows:

**Problem Statement:** Find the estimates $\hat{u}(t)$ and $(\hat{\alpha})$ for the unknown input $u(t)$ and the fractional orders vector $(\alpha)$, knowing the output $y(t)$ for the vector parameter $\theta$ and for $t \in [0, T]$.

To use the MF method, algebraic transformations must be applied to the system, with the first step involving the decomposition of the input signal to be estimated, denoted as $u(t)$, into a basis (space) $\{\beta_j(t)\} \in C^1$. Consequently, the input is transformed into:

$$ u(t) = \sum_{j=1}^{\infty} \xi_j \beta_j(t), \quad (10) $$

where $\{\xi_j\}_{j=1}^\infty$, are the unknown projection parameters. Let us perform a V-order truncation of the previous decomposition, where $V \in \mathbb{N}^*$, to proceed with the estimation as follows:

$$ u(t) = \sum_{j=1}^{\infty} \xi_j \beta_j(t) \approx \sum_{j=1}^{V} \xi_j \beta_j(t). \quad (11) $$

We introduce the following notation for the truncation of the signal $u(t)$:

$$ \bar{u}(t) = \sum_{j=1}^{V} \xi_j \beta_j(t). \quad (12) $$

Equation (9) is rewritten as:

$$ y(t) + \sum_{i=1}^{N} a_i D_t^{\alpha_i} y(t) = \sum_{j=1}^{\infty} \xi_j \beta_j(t), \quad (13) $$

with $t \in [0, T]$.

We also introduce the signal estimation:

$$ \hat{u}(t) = \sum_{j=1}^{V} \hat{\xi}_j \beta_j(t). \quad (14) $$

Following the above projection, the estimation problem is reformulated as: Find $(\{\hat{\xi}_j\}_{j=1}^V, \hat{\alpha})$ knowing the output $y(t)$ and the value of $\theta$ for $t \in [0, T]$.

In the subsequent sections, we outline the distinct procedures involved in each phase of the proposed algorithm, which collectively address the resolution of the reformulated problem statement.

**Proposition:** Given the fractional order system (9), the joint estimation of the fractional order vector $\alpha \in \mathbb{R}_+$
and the input $u(t) \in \mathbb{R}$ is processed by the following two-stages algorithm, where the first stage estimates the input $u(t)$ and the second one estimates the fractional order $\alpha$ iteratively.

**Stage 1:** In this stage, we fix the value of the fractional differentiation order $\alpha$ and estimate the input through the estimation of the input's weights. Let us assume that the constant initialization function: $f_0(t) = c_0$ is used for the fractional-order derivative of the output $y(t)$, where $t \in [-t_0, 0]$, with $t_0 < 0$. We proceed by multiplying (13) by $\phi_m(t)$, $m = 1, \dots, M$ (a set of $M$ linearly independent Modulating Functions, or MFs) and integrating over the interval $[0, T]$:

$$ \int_{0}^{T} \phi_{m}(T-t)y(t)dt + \sum_{i=1}^{N} a_{i} \int_{0}^{T} \phi_{m}(T-t)D_{t}^{\alpha_{i}}y(t)dt = \sum_{i=1}^{V} \xi_{i} \int_{0}^{T} \phi_{m}(T-t)\beta_{j}(t)dt, \quad (15) $$

for $m = 1, \dots, M$.

We obtain the following linear equation:

$$ \sum_{j=1}^{V} \xi_j A_{mj} = b_m(\alpha), \quad (16) $$

where

$$ \left\{ \begin{aligned} A_{mj} &= \int_{0}^{T} \phi_{m}(T-t)\beta_{j}(t)dt, \\ b_{m}(\alpha) &= \int_{0}^{T} \phi_{m}(T-t)y(t)dt + \\ & \qquad \sum_{i=1}^{N} a_{i} \int_{0}^{T} D_{t}^{\alpha_{i}} \phi_{m}(t)y(T-t)dt. \end{aligned} \right. \quad (17) $$

Injecting equation (7) in equation (17) gives the following equation:

$$ A_{mj} = \int_{0}^{T} \phi_{m}(T-t) \left[ \frac{t-t_j}{t_{m^*+j}-t_j} \beta_{m^*-1}^{j}(t) + \frac{t_{m^*+j+1}-t}{t_{m^*+j+1}-t_{j+1}} \beta_{m^*-1}^{j+1}(t) \right] dt. \quad (18) $$

If the knots are equally distant, i.e. $t_j = t_0 + j\Delta t$, the previous equation can be written as:

$$ A_{mj} = \frac{1}{m^* \Delta t} \int_{0}^{T} \phi_{m}(T-t) [(t-t_j)\beta_{m^*-1}^{j}(t) + (t_{m^*+j+1}-t)\beta_{m^*-1}^{j+1}(t)]dt, \quad (19) $$

which allows us to highlight the reason B-spline bases are more efficient than other ones for this estimation problem, as they dynamically consider the entirety of available data through a translational process akin to a mobile (moving) window.

Thus, the estimates of the input's projections weights $\hat{\xi} = \{\hat{\xi}_1, \dots, \hat{\xi}_V\}$ for a given fractional order vector $\alpha$, can be determined by solving the linear system below:

$$ A\hat{\xi} = b(\alpha). \quad (20) $$

**Stage 2:** To estimate fractional orders, the genetic algorithm (GA) was chosen for several reasons, mainly to escape the local minima of the objective function. An estimate of $\hat{\alpha}$, the vector of fractional differentiation orders,

is calculated through a standard genetic algorithm. The objective function $J(\hat{\alpha}^k) \in \mathbb{R}^N$ used in the algorithm is defined by:

$$ J(\hat{\alpha}^k) = L(\hat{\alpha}^k) - R, \quad (21) $$

with $L(\hat{\alpha}^k)$, $R \in \mathbb{R}^N$ and for $j = 1, \dots, N$:

$$ \left\{ \begin{aligned} L_j(\hat{\alpha}^k) &= \sum_{i=1}^{V} \hat{\xi}(i) \int_{0}^{T} \phi_{M+j}(T-t)\beta_i(t)dt \\ & \quad - \sum_{i=1}^{N} a_i \int_{0}^{T} D_t^{\hat{\alpha}_i} \phi_{M+j}(t)y(T-t)dt, \\ R_j &= \int_{0}^{T} \phi_{M+j}(T-t)y(t)dt, \end{aligned} \right. \quad (22) $$

Injecting equation (7) in equation (22) gives the following set of equations:

$$ \left\{ \begin{aligned} L_j(\hat{\alpha}^k) &= \sum_{i=1}^{V} \hat{\xi}(i) \int_{0}^{T} \phi_{M+j}(T-t) \left[ \frac{t-t_i}{t_{m^*+i}-t_i} \beta_{m^*-1}^{i}(t) + \frac{t_{m^*+i+1}-t}{t_{m^*+i+1}-t_{i+1}} \beta_{m^*-1}^{i+1}(t) \right] dt \\ & \quad - \sum_{i=1}^{N} a_i \int_{0}^{T} D_t^{\hat{\alpha}_i} \phi_{M+j}(t)y(T-t)dt, \\ R_j &= \int_{0}^{T} \phi_{M+j}(T-t)y(t)dt. \end{aligned} \right. \quad (23) $$

The nonlinear system (23) is solved using the genetic algorithm with enough populations and generations to explore a variety of solutions to find the best one. For this specific application, we used $y = 30$ populations, and $z = 40$ generations. The choice of a meta-heuristic algorithm lies in the fact that it affords the possibility to test several solutions to find the best fitting of the fitness function, with the ability to escape solutions that cause singularity in the resolution of the problem.

## 4. RESULTS

The two-stage algorithm was assessed using a simulated dataset derived from a validated 1D numerical model of the arterial network created by Willemet et al. (2015). This dataset encompasses hemodynamic signals such as pressure, flow, and distension waveforms recorded at various arterial locations, effectively representing the hemodynamics of virtual adult individuals with cardiac and arterial parameters falling within physiological ranges. This simulated dataset accurately replicates the key hemodynamic properties observed in vivo. In this study, we hand-picked three virtual subjects with distinct arterial characteristics, as summarized in Table 1. Subject 1 and subject 2 are assumed to be healthy virtual subjects, when subject 3 is assumed to be a hypertensive virtual subject. The hemodynamic measurements were provided at the ascending aorta. We assume that the initialization function, denoted as $f_0(t)$, equals zero for all $t$ in the interval $(-\infty, 0]$. Grünwald-Letnikov (GL) formula has been used.

To allow comparison with the classical methods, we used polynomial modulating functions as follows:

$$ \phi_m(t) = (T-t)^{q+m}t^{M+q-1-m}, \quad (24) $$
where $q \in \mathbb{R}^+$ represents the degree of freedom, and $M \in \mathbb{N}$ is the number of MFs used.

In this study, we opt for a selection of 3rd order B-spline basis functions as our approach to decomposing the input signal associated with blood flow. To assess the performance of the proposed algorithm, we employ the relative error, denoted as R.E., as our evaluation metric:

$$ \epsilon_{\xi} = \frac{\|u(t) - \hat{u}(t)\|_2}{\|u(t)\|_2}. \qquad (25) $$

Table 1 also displays the parameters and metrics employed in our two-stage algorithm. This algorithm is designed to concurrently estimate both aortic blood flow ($F_{in}(t)$) and the fractional parameter ($\alpha$). This parameter is associated with the differentiation order and input of FOS (9). Additionally, we provide information on the relative error, which assesses the dissimilarity between the actual input and the estimated input, along with the estimation of the fractional differentiation order for three distinct subjects.

Estimations results of the input blood flow for the 3 subjects have been illustrated in Fig. 1, in a noise-free case scenario. A 3rd order B-spline basis has been used for the three Subjects. For these estimations, we used ($M = 26, V = 16$) for the first subject, ($M = 23, V = 15$) for the second subject, and ($M = 33, V = 18$) for subject 3, where $M$ is the number of MFs and $V$ the truncation order. We observe from the reconstruction and absolute error plots for subject 2 is slightly affected at the boundary, but the error is relatively small compared to previous studies.

<table><caption>Table 1. Hemodynamic characteristics of three virtual subjects and the metrics employed in the two-stage algorithm for estimating the blood flow signal and the estimated fractional differentiation, $\hat{\alpha}$.</caption><thead><tr><th></th><th>Subject 1</th><th>Subject 2</th><th>Subject 3</th></tr></thead><tbody><tr><td colspan="4">Hemodynamic determinants</td></tr><tr><td>T<sub>C</sub>[s]</td><td>0.83</td><td>0.83</td><td>0.96</td></tr><tr><td>SV[ml]</td><td>66.40</td><td>83.00</td><td>96.60</td></tr><tr><td>SP[mmHg]</td><td>98.04</td><td>105.07</td><td>103.93</td></tr><tr><td>DP[mmHg]</td><td>67.22</td><td>78.52</td><td>69.25</td></tr><tr><td>τ</td><td>1.26</td><td>1.53</td><td>1.38</td></tr><tr><td colspan="4">Metrics</td></tr><tr><td>Noise</td><td>0%</td><td>10%</td><td>0%</td><td>10%</td><td>0%</td><td>10%</td></tr><tr><td>M</td><td colspan="3">26</td><td colspan="3">23</td><td colspan="3">33</td></tr><tr><td>V</td><td colspan="3">16</td><td colspan="3">15</td><td colspan="3">18</td></tr><tr><td>q</td><td colspan="3">2.8</td><td colspan="3">2.6</td><td colspan="3">1.9</td></tr><tr><td>RE[%]</td><td>2.12</td><td>2.96</td><td>1.56</td><td>1.75</td><td>2.69</td><td>2.81</td><td colspan="3"></td></tr><tr><td colspan="10">Estimates of fractional differentiation orders</td></tr><tr><td>α̂</td><td>0.96</td><td>0.96</td><td>0.98</td><td>0.98</td><td>0.95</td><td>0.95</td><td colspan="3"></td></tr></tbody></table>

Concerning Table 1, $T_C$ is the cardiac period, $\tau$ is the time constant, the stroke volume is represented by $SV$, and $DP$, $SP$ are the diastolic and systolic blood pressure.

Based on the previous results, we can assert that the performance of the joint estimator is highly commendable. This assertion holds concerning the precision, where relative errors remain within a maximum range of 2.96%

for the estimated input $F_{in}(t)$, as well as in terms of robustness. Table 2 summarizes the comparison between the results obtained by Bahloul et al. (2021) and the results obtained in this paper.

<table><caption>Table 2. Comparison of the relative errors of the arterial blood flow estimation: * results by Bahloul et al. (2021), ** results found using genetic algorithms and sinusoidal basis functions, and *** denotes results found using genetic algorithm and B-spline basis functions. All results are expressed in terms of R.E. (%).</caption><thead><tr><th rowspan="3"></th><th colspan="2">Results in *</th><th colspan="2">Proposed Method**</th><th colspan="2">Proposed Method***</th></tr><tr><th colspan="6">Noise level</th></tr><tr><th>0%</th><th>10%</th><th>0%</th><th>10%</th><th>0%</th><th>10%</th></tr></thead><tbody><tr><th>Subject 1</th><td>11.77</td><td>13.51</td><td>5.62</td><td>7.14</td><td>2.12</td><td>2.96</td></tr><tr><th>Subject 2</th><td>11.19</td><td>12.69</td><td>5.35</td><td>5.79</td><td>1.56</td><td>1.75</td></tr><tr><th>Subject 3</th><td>9.94</td><td>11.31</td><td>6.19</td><td>6.46</td><td>2.69</td><td>2.81</td></tr></tbody></table>

We can affirm that the results have significantly improved, reducing the estimation error in terms of R.E. from a minimum of 9.94% (Subject 3, noise-free scenario) to a maximum of 2.96% (Subject 1, noisy measurements). These results are partly due to the choice of the database, the type of modulating functions, and parameter tuning, but primarily due to the use of the genetic algorithm for fractional order estimation. Indeed, the initialization problem associated with fractional order estimation does not affect genetic algorithms due to their global convergence. Since the estimation of the fractional order and the input are interdependent, the improvement in fractional order estimation also enhances the estimation of the system's input.

## 5. CONCLUSION

Assessing the aortic blood flow and the factors that influence it is crucial for diagnosing and treating heart-related conditions. In this paper, we have developed a method that combines B-spline-based modulating functions method with genetic algorithms to estimate the aortic blood flow and the fractional Windkessel model's orders simultaneously from arterial blood pressure waveform. The proposed approach has been validated using an in silico database. The results demonstrate that the proposed method is highly effective in accurately estimating both the blood flow and the models' parameters. The results show also that the proposed algorithm is also robust to measurements noise. We also conducted a comparative study against previous method in Bahloul et al. (2021) which confirmed the superiority of the proposed method. Moreover, through extensive numerical simulations, we discovered that the method's performance depends on the number of MFs used and the weighting of the mathematical basis functions used for input estimation. In addition, the initial value chosen for the fractional differentiation order has little impact on the accuracy and convergence of the estimation process, thanks to the global convergence property of the genetic algorithm we employed. Future research will focus on the validation of the proposed approach using real-world in vivo hemodynamic data from individuals with various physiological characteristics.
<figure><img src="image_2.png" /><figcaption>Fig. 1. Aortic blood flow ($F_{in}(t)$) estimations in noise-free-scenario for the three previous subjects using 3rd order B-spline basis and genetic algorithm for the second stage.</figcaption></figure>

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