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Simulation of pulse wave propagation using one-dimensional models of hemodynamics

The models of hemodynamics, corresponding to the inviscid, Newtonian, and non-Newtonian models, are compared. The models are constructed by the averaging of the hydrodynamic system on the vessel cross-section. For the inviscid case, the analytical solution of the problem for pulse propagation is obtained. As the result of the comparison, the deviations of the solutions for non-Newtonian models from the Newtonian and inviscid cases are demonstrated.


Introduction
The one-dimensional (1D) models of blood flow, obtained by the averaging of the incompressible Navier-Stokes system, are widely used as the predictive tool in clinical applications [1,2]. As it is demonstrated in [3], the results obtained using 1D models, are consistent with the averaged results, obtained after the application of 3D models.
The blood is considered as a heterogeneous suspension of different cells in plasma. From the physical viewpoint, it is a viscous incompressible fluid with a non-Newtonian behavior [4]. But in most works on 1D models, the non-Newtonian property is ignored, and blood is modeled as a Newtonian fluid [2,5,6]. Moreover, in some papers (e.g. see [7,8,9,10]), the viscosity is ignored and it is considered as an inviscid fluid. The non-Newtonian models are used in a relatively small number of works [11,12]. In the presented paper, new non-Newtonian 1D models of blood flow are considered and compared with the inviscid and Newtonian models on the problem of pulse wave propagation. For the inviscid model, the analytical solution is obtained by the perturbation method. For the viscid models, the numerical solution is obtained by the Lax-Wendroff scheme.

Mathematical models
The hyperbolic system, obtained after the averaging of incompressibility condition and motion equation, is written as [5,6]: where t is a time, z is a cylindrical coordinate, A(t, z) is the cross-sectional area, Q(t, z) is the flow rate, P (t, z) is the pressure, α is a momentum correction coefficient, ρ is a constant 2 density and f is a frictional term. The system is closed by the so-called equation-of-state, which represents the dependence of P on A. For the arteries, the following equation is used [7]: where P ext is the external pressure, A d and P d are the diastolic cross-sectional area and pressure, β = 4 3 √ πEh, where E is the Young's modulus and h is the vessel wall thickness. So the following system is obtained: The particular model of blood is determined by the expression of f . The case of f ≡ 0 corresponds to the inviscid fluid. For the viscid models it is defined as [12]: f (A, Q) = 2π R T rz | r=R , where R is a vessel radius, T rz is the component of the tangential stress tensor, defined by the following dependence on the component D rz of the strain rate tensor D: The value of α is defined by the expression of the dimensionless velocity profile s(y), where y ∈ [0, 1] is a dimensionless radius [5,13]. Inviscid model corresponds to α = 1, Newtonian model -to α = 4/3. For the non-Newtonian models, the following representation of s(y) is used: The flattening of the profile, typical for the blood, can be obtained by the variation of d.
In the dimensionless variables, system (1) in the inviscid case is rewritten as: where A C and U C are the characteristic cross-sectional area and velocity, respectively. In the viscid case, the system is rewritten as: The expressions of f for the models, used in the paper, are presented in Table 1.

Initial-boundary value problem
Let the semi-infinite interval z ∈ [0, +∞) is considered. The initial values of A and Q are written as: where |ψ(z)|,|ϕ(z)| < +∞ at z → +∞. At the left boundary z = 0 the pulse is initiated by the following way:

Perturbation method for the inviscid system
Let the functions, which represent the conditions (4) and (5), are written as: where ε is a small dimensionless parameter, A 0 > 0 is a given constant. Presented formulas can be considered as the perturbation of the rest state (A 0 , 0), corresponding to the zero initial mean velocity (so-called 'dead man' state [14]). According to these representations, the solution of problem (2), (4), (5) is presented in the following form: The equations for A i and Q i are obtained after the substitution of (6) into (2). It is easy to demonstrate, that A 0 = A 0 . For i ≥ 1 the following equations take place: where F 1 ≡ 0 and for i ≥ 2 F i are dependent on A i−1 ,Q i−1 ,. . .,A 1 ,Q 1 . For example, at i = 2 it is presented as:  From system (7) the following equation for Q i is obtained: where c 2 = 3χ √ A 0 /2 and ν i (t, z) = ∂F i ∂t . For the wave equation (8) the following conditions are stated: where κ i (z) = −c 2 dψ i (z) dz +F i (0, z). The expression for A i (t, z) is obtained from the first equation of system (7). The solution of problem (8)-(9) is written as:

Numerical results
In the presented paper, only terms up and including the second order in (6) are considered. Let the following functions for the initial and boundary conditions are used: ϕ i = ψ i = 0, ∀i, m i = 0 at i ≥ 2, and m 1 (t), which simulates the pulse, is presented as: For the numerical calculations, the spatial interval [0, 10] and the same time interval are used. The values of dimensionless parameters are estimated by the values of A C , L C , U C , and T C for the carotid artery, taken from [3]. For the viscid models, parameters of rheological relations are taken from [4,15,16]. The value of ε for the Newtonian model is estimated as 0.0552, for the non-Newtonian model it is equal to 0.0679 for d = 2, to 0.0815 for d = 4 and to 0.1087 for d = 6. For the comparison with the inviscid model, such values are used in the analytical solution.
For the comparison of solutions, the following criterions are used: where U = (A, Q) T is the solution for the inviscid model, U v is the solution for the viscid model, and t * is a selected time moment. Criterion I estimates the deviation of the solution from the inviscid case, and Q rel estimates the magnitude of damping of the solution, corresponding to the presence of the viscosity. For the considered conditions, the maximum of |Q| is realized at t ≈ 1.3642. The value of t * is taken as 8.7484 -at this value, the propagating pulse has its maximum near the right boundary z = 10. In Fig. 1, the plots of solutions for the inviscid and Newtonian models are presented. In Fig. 2, the same plots are presented for the inviscid and non-Newtonian models.
In Fig. 3, the plots of I and Q rel at different values of d are presented. As it can be seen, the values of I for the non-Newtonian models are larger, than for the Newtonian case. The values of this criterion are increased with the increase of d. The values of Q rel for the non-Newtonian models are lower than for the Newtonian case, so the inclusion of non-Newtonian effects in the model leads to the more intense damping of the solution, than in the Newtonian case. With the increase of d the values of Q rel are decreased, so the effect of damping becomes significant with the increasing of the flattening of the velocity profile.

Conclusion
In the paper, the 1D models of hemodynamics for the inviscid and viscid (Newtonian and non-Newtonian) models are compared on the solution of problem for the single pulse propagation. For the inviscid model, the analytical solution is obtained. As the result of the comparison of solutions, obtained by different models, it is demonstrated, that for the non-Newtonian models the strongest damping in comparison with the Newtonian case takes place. Also, it is demonstrated, that the flattening of the velocity profile leads to the larger deviations of solutions in non-Newtonian case from the inviscid and Newtonian case.