# Computational Analysis of Parallel Techniques for Nonlinear Biomedical Engineering Problems
## 3 Computational Efficiency and Numerical Results
### Biomedical Engineering Application 2: Blood Rheology Model  
Blood is a non-Newtonian fluid with changing viscosity under various flow circumstances according to the Blood Rheology Model, which characterizes its flow characteristics. The shear-thinning characteristic of red blood cell alignment, which causes blood viscosity to drop at higher shear rates, is taken into account by this model. Understanding thermodynamics, assisting with medical diagnostics, and enhancing computer simulations of blood flow--especially in vascular illnesses where changing flow can affect health outcomes--all depend on blood rheology models [46] as  
\[h(x)=\frac{1}{441}x^{8}-\frac{8}{63}x^{5}-0.05714285714x^{4}+\frac{16}{9}x^{2} -3.624489796x+0.3. \tag{39}\]  
Non-linear Equation (39) has the following exact roots, accurate up to fifteen decimal places:  
\[\zeta_{1,2} = -2.27743963097943\pm 1.98877297881494i, \tag{40}\] \[\zeta_{3,4} = -1.23790259736887\pm 3.40818022390552i,\] \[\zeta_{5,6} = 1.56054635811839\pm 0.946326198047388i,\] \[\zeta_{7} = 3.82315818240685,\] \[\zeta_{8} = 0.0864335580529672.\]  
To determine the global convergence behavior, we assume a very rough initial approximations to exact roots as  
\[\left[x_{1}^{[0]},x_{2}^{[0]},...,x_{8}^{[0]}\right]=\left[1+5i,2+2i,...,4-6i\right] \tag{41}\]  
Table 3 shows the numerical outcomes of these crude initial approximations. Tables 8 and 9 show that in terms of residual error at rough initial approximation, our method relatively better approximates all the solutions in biomedical applications with higher decimal places of accuracy using the defined stopping criteria. Figure 4a-d graphically illustrates the residual error verse number of iterations using (41) to approximate all roots of (39) simultaneously. The overall performance in terms of consistency and efficiency are given in Table 8 and Figure 4a-d. Table 4 and Figures 4a-d and 5a-d, clearly indicate that in terms of iterations, basic arithmetic operations \([\pm,\times,\div]\), memory visualizations, percentage convergence and computer processing time our method BF\({}^{[*]}\) perform better than existing methods PM\({}^{[*]}\), NN\({}^{[*]}\), NF\({}^{[*]}\). The results for the consistency and stability study of the system for resolving biomedical engineering applications are shown in Tables 10-12 and Figure 6a-d, respectively, to verify the global convergence on randomly generated initial vectors. Figure 4: (**a–d**) Error graph of parallel scheme for solving (39). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Method** & \(\mathbf{e_{1}^{[p]}}\) & \(\mathbf{e_{2}^{[p]}}\) & \(\mathbf{e_{3}^{[p]}}\) & \(\mathbf{e_{4}^{[p]}}\) & \(\mathbf{e_{5}^{[p]}}\) & \(\mathbf{e_{6}^{[p]}}\) & \(\mathbf{e_{7}^{[p]}}\) & \(\mathbf{e_{8}^{[p]}}\) \\ \hline \(\text{PM}^{[s]}\) & \(5.1\times 10^{-15}\) & \(6.1\times 10^{-14}\) & \(8.9\times 10^{-33}\) & 0.0 & 0.0 & \(0.7\times 10^{-24}\) & \(0.3\times 10^{-11}\) & 0.0 \\ \(\text{NN}^{[s]}\) & \(4.3\times 10^{-22}\) & \(8.1\times 10^{-22}\) & \(6.1\times 10^{-18}\) & 0.0 & 0.0 & \(0.9\times 10^{-27}\) & 0.0 & 0.0 \\ \(\text{NF}^{[s]}\) & \(7.1\times 10^{-20}\) & \(8.1\times 10^{-11}\) & 1.0 & 0.0 & 0.0 & \(1.0\times 10^{-24}\) & \(5.5\times 10^{-15}\) & 0.0 \\ \(\text{BF}^{[s]}\) & \(2.1\times 10^{-19}\) & \(0.8\times 10^{-19}\) & \(1.1\times 10^{-18}\) & 0.0 & 0.0 & \(9.1\times 10^{-24}\) & 0.0 & 0.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Error of parallel schemes for solving (39). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Scheme** & **v** & **Memory** & \([\pm,\times,\mp]\) & **Per-Con** & **CPU Time** \\ \hline \(\text{PM}^{[s]}\) & 27 & 39.2343\% & 37 & 37\% & 3.3453776 \\ \(\text{NN}^{[s]}\) & 27 & 38.5674\% & 36 & 35\% & 4.4325767 \\ \(\text{NF}^{[s]}\) & 100 & 48.4563\% & 82 & 13\% & 5.6565456 \\ \(\text{BF}^{[s]}\) & 10 & 18.3456\% & 28 & 100\% & 2.3453556 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Consistency analysis using (41) for solving (39). Figure 6: (**a–d**).