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Similarly to the explicit method, implicit BC models require the assignment of patient- specific parameters to the peripheral vascular network. In 0D models, they include the vascular resistance and compliance and for 1D models, it is the branching topology, length, diameter, and material properties of vessel segments.

In [114], Olufsen proposed a 1D structured tree outflow BC representing the peripheral impedance for blood flow simulations in large systemic arteries. This model is capable of predicting blood flow at any level of the arterial vascular network, which generally consists of up to 20 generations before the level of the precapillary arterioles. An example of an asymmetric binary structured tree, which defines the peripheral network of smaller systemic arteries is given in Figure 2.15.a. At each bifurcation, the daughter vessel radii are scaled by factors α and β (where 0 < α, β < 1) and the branching terminates at a specified minimum radius. Other factors determining the vascular tree geometry are length, wall thickness, and Young’s modulus. The morphometric parameters of 1D structured trees can be summarised as follows:

- The scaling law for daughter and parent vessel radii is defined through the scaling parameters and asμ rp = rd1 + rd2 , = (rd1/rd2)2 , where rp , rd1, rd2 are the parent and two daughter radii, respectively, is the radius exponent, and is the ratio of the daughter vessel cross-sectional areas.

- Correspondingly, the daughter vessel radii rd1, rd2 can be expressed through rp : rd1= α·rp and rd2= β·rp , where the ratios are given by α = (1+ /2)-1/ and β = α·√ .

- The radius exponent can vary in the range of 2.33 – 3 depending on the flow assumption (i.e., 2.33 for turbulent flow or 3 for laminar flow in accordance with Murray’s law) with 2.76 suggested as the optimal value for the branching structure peripheral vasculature. - The vessel length is defined though the constant ratio of its radius.

a. Asymmetric structured tree model for systemic arteries

b. Pressure in a single vessel with 1D BC as a function of time and vessel length x

Figure 2.15 1D structured tree for outflow BC models [114]

One of the major advantages of 1D over 0D models is the ability to model pulsatile flow wave propagation and reflection phenomena. The comparison of 0D RCR WK and 1D BC models demonstrated that although WK models produce similar results, they cannot implement the high-frequency oscillations associated with physiologically realistic flow waveforms. In both the 0D and 1D domains, vasodilation and vasoconstriction are modelled through changing the radius and elasticity of the vessels. In their further research, Olufsen et al. described a method for estimation of the root impedance of the structured tree, which represents the peripheral vasculature and its application as outflow BC for large arteries [115]. The method was successfully validated with in-vivo MRI-measured aortic and peripheral flow rates. Next, Olufsen and Nadim investigated the use of the 1D axisymmetric N-S equations defining the blood flow model for derivation of RCR parameters for the equivalent 0D models [116]. This method is based on the Laplace transform and is capable of producing a 0D model of the downstream vasculature impedance for a given outlet radius. It was also reported that for an outlet vessel radius < 20 mm, pressure can be considered to be proportional to flow and the effects of inertia can be neglected. Further development of the concept led to an algorithm for computation of the root impedance of the structured trees for small arteries [77]. It is based on the computation of the zero impedance of the N-generation vascular network by the recursive solution of Z(0,w) for each vessel. It requires the specification of the root radius rroot, the bifurcation condition through the scaling parameters α and β, the terminal boundary condition as the terminal resistance Zt , and the radius rmin, at which the vessels are truncated. A comparison between the 0D R, and 0D RCR WK, and 1D BC models showed that while the pure resistance BC affects the overall shape of the curve, the results of the RCR WK and the

structured tree models are similar, except for the lag between the flow rate and pressure in the 1D results.

In [117], Sommer et al. investigated the impact of different outlet BC on the computed blood flow and the corresponding arterial input function (AIF) in a patient-specific LAD with multiple outlets. Knowledge of the AIF is required for estimation of the contrast agent bolus dispersion (i.e., contrast agent transport) in MRI-based quantification of myocardial blood flow (MBF). The types of outlet BC investigated included zero/constant pressure, 0D downstream model, 1D structured tree model, and radius-dependent flow distribution. The fractal-based 1D tree was constructed with the minimum downstream radius rmin = 100 m in accordance with [77]. The RCR parameters for the 0D WK models were estimated as the zero impedance of the 1D structural tree [115]. In the flow distribution BC cases, the outflow ratio was derived from the outlet radius in accordance with Murray’s law qi ~ ri , where =2.76. The results showed that the choice outlet BC significantly affects the computed flow fields and may result in different reactions on artificially introduced stenoses and simulated hyperaemia. The general conclusion is that constant pressure can be reasonable only for single outlet models as it produces erroneous flow distribution between the outlets (by more than 100%) otherwise. Under the resting condition, the resulting flow distributions were relatively similar for resistance and flow distribution BC (20% difference) and RCR WK and 1D BC (maximum difference of 7.9%). However, in the case of the stenosed branch, the radius-dependent flow distribution BC results in the same flow distribution, which is inconsistent with clinical observations. At the same time, the computed blood flow distributions under the 0D WK and 1D BC models were very similar. Additionally, compensation of the stenosis by arteriolar vasodilation was modelled in accordance with the long-term autoregulation mechanisms. With respect to the non-Newtonian viscosity assumption, it was reported that although the velocity pattern did not change there was a decrease in the velocity magnitude during the systole. Still, this can be considered as having negligible effects on the estimation of mean velocities.

Stergiopulos et al. proposed a pulse pressure method for evaluation of the arterial tree compliance for the 2-element WK model, based on the fitting of the RC predicted pressure pulse to the measured flow and pressure waveforms [118]. The model was successfully tested in the estimation of the mean compliance of the major systemic tree arteries (ascending aorta, thoracic aorta, common carotid, and iliac arteries). In the follow-up study, Stergiopulos et al. further evaluated the proposed method in estimating the total arterial compliance from in vivo

measured aortic pressure and flow data in 7 anesthetised dog studies [119]. It was reported that the proximal aorta constitutes for 60% of the total arterial compliance.

In the case of the coronary blood flow simulations, tuning of the R and C patient- specific parameters is required to replicate the long-term autoregulation processes. In [120], Spilker and Taylor proposed a method for iterative tuning of 0D and 1D downstream impedance models based on the feedback of the 3D computed haemodynamics to control pressure and flow rate waveforms in a physiologically realistic range. This method was formalised as the solution of an inverse problem in the form of a nonlinear system of equations of RCR WK parameters and the corresponding 3D computed blood flow features. The tuning function is defined through the flow feature requirements, i.e., min/max values and shape of the pressure waveform, and mean, amplitude, and mean diastolic values of the flow rate waveform. The feasibility of employing coarse meshes and reduced order models for temporal 3D domain representation was also demonstrated. This reduces the computational cost of the tuning procedure, which requires multiple simulations. The application of this method was successfully shown in the idealised carotid bifurcation, the iliac arterial bifurcation, and patient-specific abdominal aorta models, as well as in their later works [44].