Manifold-based denoising for Ferumoxytol-enhanced 3D cardiac cine MRI

Anna Andrews; Pezad Doctor; Lasya Gaur; F. Gerald Greil; Tarique Hussain; Qing Zou; Anna Andrews; Pezad Doctor; Lasya Gaur; F. Gerald Greil; Tarique Hussain; Qing Zou

doi:10.3934/mbe.2024163

Mathematical Biosciences and Engineering

2024, Volume 21, Issue 3: 3695-3712. doi: 10.3934/mbe.2024163

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Research article Special Issues

Manifold-based denoising for Ferumoxytol-enhanced 3D cardiac cine MRI

1.
Department of Biomedical Engineering, Mercer University, Macon, USA
2.
Division of Pediatric Cardiology, Department of Pediatrics, The University of Texas Southwestern Medical Center, Dallas, TX, USA
3.
Department of Radiology, The University of Texas Southwestern Medical Center, Dallas, TX, USA
4.
Advanced Imaging Research Center, The University of Texas Southwestern Medical Center, Dallas, TX, USA

Received: 08 November 2023 Revised: 21 December 2023 Accepted: 11 January 2024 Published: 18 February 2024

The two-dimensional (2D) cine cardiovascular magnetic resonance (CMR) technique is the reference standard for assessing cardiac function. However, one challenge with 2D cine is that the acquisition time for the whole cine stack is long and requires multiple breath holds, which may not be feasible for pediatric or ill patients. Though single breath-hold multi-slice cine may address the issue, it can only acquire low-resolution images, and hence, affect the accuracy of cardiac function assessment. To address these challenges, a Ferumoxytol-enhanced, free breathing, isotropic high-resolution 3D cine technique was developed. The method produces high-contrast cine images with short acquisition times by using compressed sensing together with a manifold-based method for image denoising. This study included fifteen patients (9.1 $\pm$ 5.6 yrs.) who were referred for clinical cardiovascular magnetic resonance imaging (MRI) with Ferumoxytol contrast and were prescribed the 3D cine sequence. The data was acquired on a 1.5T scanner. Statistical analysis shows that the manifold-based denoised 3D cine can accurately measure ventricular function with no significant differences when compared to the conventional 2D breath-hold (BH) cine. The multiplanar reconstructed images of the proposed 3D cine method are visually comparable to the golden standard 2D BH cine method in terms of clarity, contrast, and anatomical precision. The proposed method eliminated the need for breath holds, reduced scan times, enabled multiplanar reconstruction within an isotropic data set, and has the potential to be used as an effective tool to access cardiovascular conditions.

Keywords:

Citation: Anna Andrews, Pezad Doctor, Lasya Gaur, F. Gerald Greil, Tarique Hussain, Qing Zou. Manifold-based denoising for Ferumoxytol-enhanced 3D cardiac cine MRI[J]. Mathematical Biosciences and Engineering, 2024, 21(3): 3695-3712. doi: 10.3934/mbe.2024163

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Abstract

1. Introduction

Very recently the concept of fractal differentiation and fractional differentiation have been combined to produce new differentiation operators ^[1,2,3,4]. The new operators were constructed using three different kernels namely, power law, exponential decay and the generalised Mittag-Leffler function. The new operators have two parameters, the first is considered as fractional order and the second as fractal dimension.

The operators were tested to model some real world problems ^[9,10,11] surprisingly, the new differential and integral operators were found to be powerful mathematical tools able to capture even hidden complexities of nature ^[5,6,16]. Very detailed attractors could be captured when modelling with such differential and integral operators ^[8]. One of the great advantage of these new operators is that they can at the same time capture processes following power law, exponential decay law, fading memory, crossover and self-similar processes. This unique and outstanding capacity makes these new operators suitable tools to model many complex real world problems. Additionally when the fractal dimension is one, we recover all the existing fractional differential and integral operators, if the fractional order is one, we recover the so-call fractal differential and integral operators. Finally if the fractal and fractional orders tend to 1 we recover classical differential and integral operators. Therefore, it is believed that these new operators are the future for modeling.

On the other hand, modeling real-life problems have been always of interest of researchers. Mathematicians especially have introduced and developed a variety of well-known and sophisticated tools to model problems arising from other sciences such as biology, finance, epidemiology, and geology. The list is not exhaustive. Moreover, they used these tools to investigate solutions to many important and crucial problems that humans are challenging in their daily life. Among these tools are differential equations and fractal differential equations for instance. Nevertheless, the continuous improvement leads constantly their researches in order to obtain better solutions that aim at reaching the reality. Fractal-fractional differential operators can be seen as an enhanced instrument for modeling important practical problems. One of the main motivations for this article was to introduce methods for solving fractal-fractional problems that have not been considered before. By using the proposed methods, approximate solutions to these problems can be determined more easily and with higher accuracy. In addition, the methods can be applied to real-world problems. In this work, we present applications of such numerical schemes in solving chaotic models that involve the new class of differential operators. We consider some well-known chaotic models with fractal-fractional differential operators, so that we can compare our results with those in the literature. The article is organized as follows: In section 2, we give a brief overview of some basic definitions of fractal-fractional differential calculus. Two efficient and effective numerical methods for determining approximate solutions to fractal-fractional problems are presented in section 3. The first is for the Caputo fractal derivative second is for Caputo-Fabrizio-Caputo fractal-fractional derivative and the thierd is for the Atangana-Baleanu-Caputo fractal-fractional derivative. The kernels used in these two definitions are singular and non-singular, respectively. Several numerical simulations for chaotic systems are described in section 4 we find the numerical analysis for Duffing attractor. Numerical analysis for EL-Ni $\hat{n}$ o Southern Oscillation is find in section 5. In section 6–9, we analysis numerical analysis for Ikeda system, numerical analysis for Dadras attractor and numerical analysis for Aizawa attractor, numerical analysis for Thomas attractor and numerical analysis for 4 Wings attractor. In section 9 we introduce a new Atangana Sonal attractor and find the solution of this. The results obtained are accurate, interesting, and meaningful. Finally, we present our overall conclusions.

2. Preliminaries on fractal-fractional calculus

Definition 2.1. ^[3] The fractal fractional derivative of f(t) with order $\varpi-\kappa$ in the Riemann-Liouville sense is defined as follows:

$\begin{equation} ^{FF-RL}\mathfrak{D}_{0, t}^{\kappa, \varpi}\left\lbrace f(t)\right\rbrace = \frac{1}{\Gamma(m-\kappa)}\frac{d}{dt^\varpi}\int_{0}^{t}(t-s)^{m-\kappa-1}f(s)ds, \end{equation}$

(2.1)

where $m-1 < \kappa, \varpi\leq m\in\mathbb{N}$ and $\frac{df(s)}{ds^\varpi} = \lim\limits_{t\rightarrow s} \frac{f(t)-f(s)}{t^\varpi-s^\varpi}.$

Definition 2.2. ^[3] The Caputo-Fabrizio fractal-fractional derivative of $f(t)$ with order $\varpi-\kappa$ in the Riemann-Liouville sense is defined as follows:

$\begin{equation} ^{FF-CFR}\mathfrak{D}_{0, t}^{\kappa, \varpi}\left\lbrace f(t)\right\rbrace = \frac{M(\kappa)}{(1-\kappa)}\frac{d}{dt^\varpi}\int_{0}^{t}\exp\left( -\frac{\kappa}{1-\kappa} (t-s)\right)f(s)ds, \end{equation}$

(2.2)

where $\kappa > 0, \; \varpi\geq m\in\mathbb{N}$ and $M(0) = M(1) = 1.$

Definition 2.3. ^[2] The Atangan-Baleanu fractal-fractional derivative of $f(t)$ with order $\varpi-\kappa$ in the Riemann-Liouville sense is defined as follows:

$\begin{equation} ^{FF-ABR}\mathfrak{D}_{0, t}^{\kappa, \varpi}\left\lbrace f(t)\right\rbrace = \frac{AB(\kappa)}{(1-\kappa)}\frac{d}{dt^\varpi}\int_{0}^{t}E_\kappa\left( -\frac{\kappa}{1-\kappa} (t-s)^\kappa\right)f(s)ds, \end{equation}$

(2.3)

where $0 < \kappa, \; \varpi\leq 1$ and $AB(\kappa) = 1-\kappa+\frac{\kappa}{\Gamma(\kappa)}.$

Definition 2.4. ^[3] The fractal-fractional derivative of with order $\varpi-\kappa$ in the Liouville-Caputo sense is defined as follows:

$\begin{equation} ^{FF-C}\mathfrak{D}_{0, t}^{\kappa, \varpi}\left\lbrace f(t)\right\rbrace = \frac{1}{\Gamma{(m-\kappa)}}\frac{d}{dt^\varpi}\int_{0}^{t}(t-s)^{m-\kappa-1}\left(\frac{d}{ds^\tau}f(s)\right)ds, \end{equation}$

(2.4)

where $m-1 < \kappa, \; \varpi\leq m\in\mathbb{N}$ and $\frac{df(s)}{ds^\varpi} = \lim\limits_{t\rightarrow s} \frac{f(t)-f(s)}{t^\varpi-s^\varpi}.$

Definition 2.5. ^[2] The Caputo-Fabrizio fractal-fractional derivative of $f(t)$ with order $\varpi-\kappa$ in the Liouville-Caputo sense is defined as follows:

$\begin{equation} ^{FF-CFC}\mathfrak{D}_{0, t}^{\kappa, \varpi}\left\lbrace f(t)\right\rbrace = \frac{M(\kappa)}{(1-\kappa)}\int_{0}^{t}\exp\left( -\frac{\kappa}{1-\kappa} (t-s)\right) \left( \frac{d}{ds^\varpi}f(s)\right)ds, \end{equation}$

(2.5)

where $\kappa > 0, \; \varpi\geq m\in\mathbb{N}$ and $M(0) = M(1) = 1.$

Definition 2.6. ^[4] The Atangana-Baleanu fractal-fractional derivative of $f(t)$ with order $\varpi-\kappa$ in the Liouville-Caputo sense is defined as follows:

$\begin{equation} ^{FF-ABC}\mathfrak{D}_{0, t}^{\kappa, \varpi}\left\lbrace f(t)\right\rbrace = \frac{AB(\kappa)}{(1-\kappa)}\int_{0}^{t}E_\kappa\left( -\frac{\kappa}{1-\kappa} (t-s)^\kappa\right)\left( \frac{d}{ds^\varpi}f(s)\right) ds, \end{equation}$

(2.6)

where $0 < \kappa, \; \varpi\leq 1$ and $AB(\kappa) = 1-\kappa+\frac{\kappa}{\Gamma(\kappa)}.$

Definition 2.7. ^[2] The Liouville-Caputo fractal-fractional integral of $f(t)$ with order $\kappa$ is defined as follows:

$\begin{equation} ^{FF-C}\mathfrak{I}_{0, t}^{\kappa}\left\lbrace f(t)\right\rbrace = \frac{\varpi}{\Gamma(\kappa)}\int_{0}^{t}(t-s)^{\kappa-1}s^{\varpi-1}f(s)ds. \end{equation}$

(2.7)

Definition 2.8. ^[3] The Caputo-Fabrizio fractal-fractional integral of $f(t)$ with order $\kappa$ is defined as follows:

$\begin{equation} ^{FF-CF}\mathfrak{I}_{0, t}^{\kappa, \varpi}\left\lbrace f(t)\right\rbrace = \frac{\kappa\varpi}{M(\kappa)}\int_{0}^{t}s^{\kappa-1}f(s)ds+\frac{\varpi(1-\kappa)t^{\varpi-1}}{M(\kappa)}f(t). \end{equation}$

(2.8)

Definition 2.9. ^[3] The Atangana-Baleanu fractal-fractional integral of $f(t)$ with order $\kappa$ is defined as follows:

$\begin{equation} ^{FF-AB}\mathfrak{I}_{0, t}^{\kappa, \varpi}\left\lbrace f(t)\right\rbrace = \frac{\kappa\varpi}{AB(\kappa)}\int_{0}^{t}s^{\kappa-1}(t-s)^{\kappa-1}f(s)ds+\frac{\varpi(1-\kappa)t^{\varpi-1}}{AB(\kappa)}f(t). \end{equation}$

(2.9)

3. Numerical approximation of fractal-fractional calculus

In this section, we have given three numerical schemes for Caputo-fractal-fractional, Caputo-Fabrizio-fractal fractional and the Atangana-Baleanu fractal-fractional derivative operators ^[7].

3.1. Caputo-fractal-fractional derivative

Consider the following differential equations in the fractal-fractional Liouville-Caputo sense

$\begin{equation} ^{FF-C}\mathfrak{D}_{0, t}^{\kappa, \varpi}\left\lbrace u(t)\right\rbrace = f(t, u(t), v(t)). \end{equation}$

(3.1)

Equation (3.1) can be converted to the Volterra case and the numerical scheme of this system using a Caputo-fractal-fractional approach at $t_{n+1}$ is given by

$\begin{equation} u(t_{n+1}) = u^0+\frac{\varpi}{\Gamma(\kappa)}\int_0^{t_{n+1}}s^{\varpi-1}(t_{n+1-s})^{\kappa-1}f(s, u, v)ds. \end{equation}$

(3.2)

We can approximate the above integral to

$\begin{equation} u(t_{n+1}) = u^0+\frac{\varpi}{\Gamma(\kappa)}\sum\limits_{j = 0}^{n}\int_{t_j}^{t_{j+1}}s^{\varpi-1}(t_{n+1-s})^{\kappa-1}f(s, u, v)ds. \end{equation}$

(3.3)

With in the finite interval $[t_j, t_{j+1}]$ , we approximate the function $s^{\varpi-1}, f(t, u, v)$ using the Lagrangian piecewise interpolation such that

$\begin{equation} \mathbb{E}(s) = \frac{s-t_{j-1}}{t_j-t_{j-1}}t_j^{\varpi-1}f(t_j, u^j, v^j)- \frac{s-t_{j}}{t_j-t_{j-1}}t_{j-1}^{\varpi-1}f(t_{j-1}, u^{j-1}, v^{j-1}). \end{equation}$

(3.4)

So we obtain

$\begin{equation} u(t_{n+1}) = u^0+\frac{\varpi}{\Gamma(\kappa)}\sum\limits_{j = 0}^{n}\int_{t_j}^{t_{j+1}}s^{\varpi-1}(t_{n+1-s})^{\kappa-1}\mathbb{E}(s)ds. \end{equation}$

(3.5)

Solving the integral of the right hand side, we obtain the following numerical scheme

$\begin{equation} \begin{split} u(t_{n+1}) & = u^0+\frac{\varpi(\Delta t)^\kappa}{\Gamma(\kappa+2)}\sum\limits_{j = 0}^{n}\left[t_{j}^{\varpi-1}f(t_{j}, u_{j}, v^{j})\left( (n+1-j)^\kappa(n-j+2+\kappa)\right.\right.\\ &\left.\left.-(n-j)^\kappa(n-j+2+2\kappa)\right)-t_{j-1}^{\varpi-1}f(t_{j-1}, u_{j-1}, v^{j-1}) \left( (n+1-j)^{\kappa+1}\right.\right.\\ &\left.\left.-(n-j)^\kappa(n-j+1+\kappa)\right)\right]. \end{split} \end{equation}$

(3.6)

3.2. Caputo-Fabrizio-Caputo fractal-fractional derivative

Consider the following differential equations in the Caputo-Fabrizio-fractal-fractional derivative

$\begin{equation} ^{FF-CFC}\mathfrak{D}_{0, t}^{\kappa, \varpi}\left\lbrace u(t)\right\rbrace = f(t, u(t), v(t)). \end{equation}$

(3.7)

Applying the Caputo-Fabrizio integral, we obtain

$\begin{equation} u(t) = u^0+\frac{\varpi t^{\varpi-1}(1-\kappa) }{M(\kappa)}f(t, u, v)+\frac{\kappa \varpi}{M(\kappa)}\int_0^{t}s^{\varpi-1}f(s, u, v)ds. \end{equation}$

(3.8)

Here we present the detailed derivation of the numerical scheme. Thus, at $t_{n+1}$ we have

$\begin{equation} u(t_{n+1}) = u^0+\frac{\varpi t_n^{\varpi-1}(1-\kappa) }{M(\kappa)}f(t_n, u^n, v^n)+\frac{\kappa \varpi}{M(\kappa)}\int_0^{t_{n+1}}s^{\varpi-1}f(s, u, v)ds. \end{equation}$

(3.9)

Taking the difference between the consecutive terms, we obtain

$\begin{equation} \begin{split} u(t_{n+1}) & = u^n+\frac{\varpi t_n^{\varpi-1}(1-\kappa) }{M(\kappa)}f(t_n, u^n, v^n)-\frac{\varpi t_{n-1}^{\varpi-1}(1-\kappa) }{M(\kappa)}f(t_{n-1}, u^{n-1}, v^{n-1})\\ &+\frac{\kappa \varpi}{M(\kappa)}\int_{t_{n}}^{t_{n+1}}s^{\varpi-1}f(s, u, v)ds. \end{split} \end{equation}$

(3.10)

Now using the Lagrange polynomial piece-wise interpolation and integrating, we obtain

$\begin{equation} \begin{split} u(t_{n+1}) & = u^n+\frac{\varpi t_n^{\varpi-1}(1-\kappa) }{M(\kappa)}f(t_n, u^n, v^n)-\frac{\varpi t_{n-1}^{\varpi-1}(1-\kappa) }{M(\kappa)}f(t_{n-1}, u^{n-1}, v^{n-1})\\ &+\frac{\kappa \varpi}{M(\kappa)}\times\left[\frac{3}{2}(\Delta t)t_n^{\varpi-1}f(t_n, u^n, v^n) -\frac{\Delta t}{2}t_{n-1}^{\varpi-1}f(t_{n-1}, u^{n-1}, v^{n-1})\right]. \end{split} \end{equation}$

(3.11)

3.3. Atangana-Baleanu-Caputo fractal-fractional derivative

Consider the following differential equations in the Atangana-Baleanu fractal-fractional derivative in the Liouville-Caputo sense

$\begin{equation} ^{FF-ABC}\mathfrak{D}_{0, t}^{\kappa, \varpi}\left\lbrace u(t)\right\rbrace = f(t, u(t), v(t)). \end{equation}$

(3.12)

Applying the Atangana-Baleanu integral, we have

$\begin{equation} u(t) = u(0)+\frac{\varpi t_n^{\varpi-1}(1-\kappa) }{AB(\kappa)}f(t, u, v)+\frac{\kappa \varpi}{AB (\kappa)\Gamma(\kappa)}\int_{0}^{t}s^{\varpi-1}(t-s)^{\kappa-1}f(s, u, v)ds. \end{equation}$

(3.13)

At $t_{n+1}$ , we have the following

$\begin{equation} \begin{split} u(t_{n+1}) & = u(0)+\frac{\kappa \varpi}{AB (\kappa)\Gamma(\kappa)}\int_{0}^{t_{n+1}}s^{\varpi-1}(t_{n+1}-s)^{\kappa-1}f(s, u, v)ds\\ & +\frac{\varpi t_n^{\varpi-1}(1-\kappa) }{AB(\kappa)}f(t_n, u^n, v^n), \end{split} \end{equation}$

(3.14)

the above system can be expressed as

$\begin{equation} \begin{split} u(t_{n+1}) & = u(0)+\frac{\varpi t_n^{\varpi-1}(1-\kappa) }{AB(\kappa)}f(t_n, u^n, v^n)\\ &+\frac{\kappa \varpi}{AB (\kappa)\Gamma(\kappa)}\sum\limits_{j = 0}^{n}\int_{t_{j}}^{t_{j+1}}s^{\varpi-1}(t_{n+1}-s)^{\kappa-1}f(s, u, v)ds. \end{split} \end{equation}$

(3.15)

Approximating $s^{\varpi-1f(s, u, v)}$ in $[t_j, t_{j+1}]$ , we have the following numerical scheme

$\begin{equation} \begin{split} u(t_{n+1}) & = u(0)+\frac{\varpi t_n^{\varpi-1}(1-\kappa) }{AB(\kappa)}f(t_n, u^n, v^n)+\frac{\varpi\Delta t^\kappa}{AB(\kappa)\Gamma(\kappa+2)}\\ &\times\sum\limits_{j = 0}^{n}\left[t_j^{\varpi-1} f(t_j, u^j, v^j)[(n+1-j)^\kappa(n-j+2+\kappa)-(n-j)^\kappa(n-j+2+2\kappa)]\right. \\ &\left.-t_{j-1}^{\varpi-1}f(t_{j-1}, u^{j-1}, v^{j-1})\left( (n+1-j)^{\kappa+1}-(n-j)^\kappa(n-j+1+\kappa)\right) \right]. \end{split} \end{equation}$

(3.16)

4. Numerical analysis for Duffing attractor

The Duffing Pendulum is a kind of a forced oscillator with damping, the mathematical model represented in state variable as when we apply Caputo-fractal-fractional derivative is given by

$\begin{equation} ^{FF-C}\mathfrak{D}_{0, t}^{\kappa, \varpi}\left\lbrace u_1(t)\right\rbrace = Au_2(t), \end{equation}$

(4.1)

$\begin{equation} ^{FF-C}\mathfrak{D}_{0, t}^{\kappa, \varpi}\left\lbrace u_2(t)\right\rbrace = u_1(t)-u_1^3(t)-au_2(t)+b\cos(\omega t), \end{equation}$

(4.2)

where initial conditions are $u_1(0) = 0.01$ , $u_2(0) = 0.5$ and the constant are $\kappa = 0.25,$ $b = 0.3$ , $\omega = 1$ , $h = 0.01$ , and $t = 100.$ The numerical scheme is given by

$u_1(t_{n+1}) = u_1^0 +\frac{A\varpi(\Delta t)^\kappa}{\Gamma(\kappa+2)}\sum\limits_{j = 0}^{n}\left[t_{j}^{\varpi-1}u_2(t)^j\left( (n+1-j)^\kappa(n-j+2+\kappa)\\ -(n-j)^\kappa(n-j+2+2\kappa)\right)\right.\\ \left.-t_{j-1}^{\varpi-1}Au_2(t)^{j-1} \left( (n+1-j)^{\kappa+1}-(n-j)^\kappa(n-j+1+\kappa)\right)\right], \\ u_2(t_{n+1}) = u_1^0+\frac{\varpi(\Delta t)^\kappa}{\Gamma(\kappa+2)}\sum\limits_{j = 0}^{n}\left[t_{j}^{\varpi-1}\left( u_1(t)^j-u_1^3(t)^j-au_2(t^j-\tau)+b\cos\omega t^j\right)\right. \\ \left. \times \left( (n+1-j)^\kappa(n-j+2+\kappa)-(n-j)^\kappa(n-j+2+2\kappa)\right)\right.\\ \left.-\left( t_{j-1}^{\varpi-1}u_1(t)^{j-1}-u_1^3(t)^{j-1}-au_2(t^{j-1}-\tau)+b\cos\omega t^{j-1} \right)\right. \\ \left. \left( (n+1-j)^{\kappa+1}-(n-j)^\kappa(n-j+1+\kappa)\right)\right].$

(4.3)

Numerical simulation of the Caputo-power law case are depicted in – for different values of fractional order $\kappa$ and fractal dimension $\varpi$ .

Figure 1. Numerical simulation for

$\kappa = \varpi = 1$ and A = 1 with classical differentiation for Duffing attractor.

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Mathematical Biosciences and Engineering

Manifold-based denoising for Ferumoxytol-enhanced 3D cardiac cine MRI

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Abstract

1. Introduction

2. Preliminaries on fractal-fractional calculus

3. Numerical approximation of fractal-fractional calculus

3.1. Caputo-fractal-fractional derivative

3.2. Caputo-Fabrizio-Caputo fractal-fractional derivative

3.3. Atangana-Baleanu-Caputo fractal-fractional derivative

4. Numerical analysis for Duffing attractor

5. Numerical analysis for EL-Niˆn \hat{n} o Southern Oscillation

6. Numerical analysis for Ikeda system

7. Numerical analysis for Dadras attractor

8. Numerical analysis for Aizawa attractor

9. Numerical analysis for Thomas attractor

10. Numerical analysis for 4 Wings attractor

11. AS attractor

12. Conclusions

Acknowledgment

Conflict of interest

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5. Numerical analysis for EL-Ni $\hat{n}$ o Southern Oscillation