Pressure in arteries is difficult to measure non-invasively. Although computational fluid dynamics (CFD) provides high-precision numerical solutions according to the basic physical equations of fluid mechanics, it relies on precise boundary conditions and complex preprocessing, which limits its real-time application. Machine learning algorithms have wide applications in hemodynamic research due to their powerful learning ability and fast calculation speed. Therefore, we proposed a novel method for pressure estimation based on physics-informed neural network (PINN). An ideal aortic arch model was established according to the geometric parameters from human aorta, and we performed CFD simulation with two-way fluid-solid coupling. The simulation results, including the space-time coordinates, the velocity and pressure field, were obtained as the dataset for the training and validation of PINN. Nondimensional Navier-Stokes equations and continuity equation were employed for the loss function of PINN, to calculate the velocity and relative pressure field. Post-processing was proposed to fit the absolute pressure of the aorta according to the linear relationship between relative pressure, elastic modulus and displacement of the vessel wall. Additionally, we explored the sensitivity of the PINN to the vascular elasticity, blood viscosity and blood velocity. The velocity and pressure field predicted by PINN yielded good consistency with the simulated values. In the interested region of the aorta, the relative errors of maximum and average absolute pressure were 7.33% and 5.71%, respectively. The relative pressure field was found most sensitive to blood velocity, followed by blood viscosity and vascular elasticity. This study has proposed a method for intra-vascular pressure estimation, which has potential significance in the diagnosis of cardiovascular diseases.
Citation: Meiyuan Du, Chi Zhang, Sheng Xie, Fang Pu, Da Zhang, Deyu Li. Investigation on aortic hemodynamics based on physics-informed neural network[J]. Mathematical Biosciences and Engineering, 2023, 20(7): 11545-11567. doi: 10.3934/mbe.2023512
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Pressure in arteries is difficult to measure non-invasively. Although computational fluid dynamics (CFD) provides high-precision numerical solutions according to the basic physical equations of fluid mechanics, it relies on precise boundary conditions and complex preprocessing, which limits its real-time application. Machine learning algorithms have wide applications in hemodynamic research due to their powerful learning ability and fast calculation speed. Therefore, we proposed a novel method for pressure estimation based on physics-informed neural network (PINN). An ideal aortic arch model was established according to the geometric parameters from human aorta, and we performed CFD simulation with two-way fluid-solid coupling. The simulation results, including the space-time coordinates, the velocity and pressure field, were obtained as the dataset for the training and validation of PINN. Nondimensional Navier-Stokes equations and continuity equation were employed for the loss function of PINN, to calculate the velocity and relative pressure field. Post-processing was proposed to fit the absolute pressure of the aorta according to the linear relationship between relative pressure, elastic modulus and displacement of the vessel wall. Additionally, we explored the sensitivity of the PINN to the vascular elasticity, blood viscosity and blood velocity. The velocity and pressure field predicted by PINN yielded good consistency with the simulated values. In the interested region of the aorta, the relative errors of maximum and average absolute pressure were 7.33% and 5.71%, respectively. The relative pressure field was found most sensitive to blood velocity, followed by blood viscosity and vascular elasticity. This study has proposed a method for intra-vascular pressure estimation, which has potential significance in the diagnosis of cardiovascular diseases.
The definition of impulsive semi-dynamical system and its properties including the limit sets of orbits have been investigated [1,9]. The generalized planar impulsive dynamical semi-dynamical system can be described as follows
{dxdt=P(x,y),dydt=Q(x,y),(x,y)∉M,△x=a(x,y),△y=b(x,y),(x,y)∈M, | (1) |
where
I(z)=z+=(x+,y+)∈R2, x+=x+a(x,y), y+=y+b(x,y) |
and
Let
C+(z)={Π(z,t)|t∈R+} |
is called the positive orbit of
M+(z)=C+(z)∩M−{z}. |
Based on above notations, the definition of impulsive semi-dynamical system is defined as follows [1,9,23].
Definition 1.1. An planar impulsive semi-dynamic system
F(z,(0,ϵz))∩M=∅ and Π(z,(0,ϵz))∩M=∅. |
Definition 1.2. Let
1.
2. for each
It is clear that
Definition 1.3. Let
Denote the points of discontinuity of
Theorem 1.4. Let
In 2004 [2], the author pointed out some errors on Theorem 1.4, that is, it need not be continuous under the assumptions. And the main aspect concerned in the paper [2] is the continuality of
In the following we will provide an example to show this Theorem is not true for some special cases. Considering the following model with state-dependent feedback control
{dx(t)dt=ax(t)[1−x(t)K]−βx(t)y(t)1+ωx(t),dy(t)dt=ηβx(t)y(t)1+ωx(t)−δy(t),}x<ET,x(t+)=(1−θ)x(t),y(t+)=y(t)+τ,}x=ET. | (2) |
where
Define four curves as follows
L0:x=δηβ−δω; L1:y=rβ[1−xK](1+ωx); |
L2:x=ET; and L3:x=(1−θ)ET. |
The intersection points of two lines
yET=rβ[1−ETK](1+ωET), yθET=rβ[1−(1−θ)ETK](1+ω(1−θ)ET). |
Define the open set in
Ω={(x,y)|x>0,y>0,x<ET}⊂R2+={(x,y)|x≥0,y≥0}. | (3) |
In the following we assume that model (2) without impulsive effects exists an unstable focus
E∗=(xe,ye)=(δηβ−δω,rη(Kηβ−Kδω−δ)K(ηβ−δω)2), |
which means that model (2) without impulsive effects has a unique stable limit cycle (denoted by
In the following we show that model (2) defines an impulsive semi-dynamical system. From a biological point of view, we focus on the space
Further, we define the section
y+k+1=P(y+k)+τ=y(t1,t0,(1−θ)ET,y+k)+τ≐PM(y+k), and Φ(y+k)=t1. | (4) |
Now define the impulsive set
M={(x,y)| x=ET,0≤y≤YM}, | (5) |
which is a closed subset of
N=I(M)={(x+,y+)∈Ω| x+=(1−θ)ET,τ≤y+≤P(yθET)+τ}. | (6) |
Therefore,
According to the Definition 1.3 and topological structure of orbits of model (2) without impulsive effects, it is easy to see that
However, this is not true for case (C) shown in Fig. 2(C). In fact, for case (C) there exists a trajectory (denoted by
If we fixed all the parameter values as those shown in Fig. 3, then we can see that the continuities of the Poincaré map and the function
Theorem 2.1. Let
Note that the transversality condition in Theorem 2.1 may exclude the case (B) in Fig. 2(B). In fact, based on our example we can conclude that the function
Recently, impulsive semi-dynamical systems or state dependent feedback control systems arise from many important applications in life sciences including biological resource management programmes and chemostat cultures [5,6,10,12,17,18,19,20,21,22,24], diabetes mellitus and tumor control [8,13], vaccination strategies and epidemiological control [14,15], and neuroscience [3,4,7]. In those fields, the threshold policies such as
The above state-dependent feedback control strategies can be defined in broad terms in real biological problems, which are usually modeled by the impulsive semi-dynamical systems. The continuity of the function
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