In this paper we consider a model for estimation of spatio-temporal distribution of blood in a cardiovascular system. We intro-duce an approach for analyzing the considered model. We consider a possibility of changing the rate of blood transport.
where \(A(z,t)\) is the area of the axial section in the vessel of the considered system; \(Q(z,t)\) is the average blood flow across the pad; \(\alpha\) is the Coriolis coefficient; \(\rho(z,t)\) is the density of blood; \(K_r = 8\pi \nu\) is the coefficient of friction of the blood against the walls of the vessel; \(\nu\) is the blood viscosity. In accordance with Hooke’s law, for the the elastic wall of the vessel, the dependence of blood pressure in the vessel was used \(p(z,t)\)
\[\begin{equation}\label{eq2} p(z,t) = \frac{hE}{R_0^2} \left[ R(z,t) – R_0 \right], \end{equation} \tag{2}\]where \(E\) is the Young modulus; \(h_0\) is the thickness of wall vessel; \(R_0\) is the radius at zero pressure. Using the relation \(A(z,t) = \pi R^2(z,t)\) we obtained
\[\begin{equation} p(z,t) = \sqrt{\pi} \frac{\sqrt{A} – \sqrt{A_0}}{A_0 h_0 E}, \end{equation}\tag{2a}\]where \(A_0 = A(z,0) = A(0,t) = \pi R^2(z,0) = \pi R_0^2\); \(p(z,0) = 0\); \(Q(z,0) = Q_0\); \(Q(0,t) = Q_0\). Next we transform the above differential Eq. (3) to the following integral form
\[\begin{equation}\label{eq3} \left\{ \begin{aligned} A(z,t) &= A(z,t) + \frac{1}{L} \left[ \int_0^z A(u,t)\,du + \int_0^t Q(z,\tau)\,d\tau + A_0 z + Q_0 \right] \\ Q(z,t) &= Q(z,t) + \frac{1}{L} \left[ \int_0^z Q(u,t)\,du + \alpha \int_0^t \frac{Q^2(z,\tau)}{A(z,\tau)}\,d\tau + \int_0^t \int_0^z \frac{A(u,\tau)}{\rho(u,\tau)}\,du\,d\tau \right. \\ &\qquad \left. + \int_0^t \int_0^z K_r(u,\tau) \frac{Q(u,\tau)}{A(u,\tau)}\,du\,d\tau + Q_0 z + \frac{Q_0^2}{A_0} \right] \end{aligned} \right. \end{equation}\tag{3}\]Now let us solve equations of system (3) by method of averaging of function corrections [7,8,9]. In the framework of the method one shall replace the required functions of Eq. (3) on their not yet known average values \(\alpha_1\) in the right sides of Eq. (3). The replacement gives a possibility to obtain relations for the first-order approximations of the required functions \(A(z,t)\) and \(Q(z,t)\) in the following form
\[\begin{equation}\label{eq4} \left\{ \begin{aligned} A_1(z,t) &= \alpha_{1A} + \frac{1}{L} \left( \alpha_{1A} z + \alpha_{1Q} t + A_0 z + Q_0 \right) \\ Q_1(z,t) &= \alpha_{1Q} + \frac{1}{L} \left[ \alpha_{1Q} z + \alpha t \frac{\alpha_{1Q}^2}{\alpha_{1A}} + \int_0^t \int_0^z \frac{1}{\rho(u,\tau)}\,du\,d\tau \right. \\ &\qquad \left. + \frac{\alpha_{1Q}}{\alpha_{1A}} \int_0^t \int_0^z K_r(u,\tau) \,du\,d\tau + Q_0 z + \frac{Q_0^2}{A_0} \right] \end{aligned} \right. \end{equation} \tag{4}\]Not yet known average values \(\alpha_{1A}\) and \(\alpha_{1Q}\) of the required functions \(A(z,t)\) and \(Q(z,t)\) were calculated by the following standard relations [7,8,9],
\[ \begin{equation}\label{eq5} \alpha_{1A} = \frac{1}{L \Theta} \int_0^\Theta \int_0^L A_1(z,t)\,dz\,dt \quad \text{and} \quad \alpha_{1Q} = \frac{1}{L \Theta} \int_0^\Theta \int_0^L Q_1(z,t)\,dz\,dt. \tag{5}\end{equation}\] Substitution of relation (4) into relations (5) gives a possibility to obtain relations to determine the required average values in the following final form \[\begin{equation}\label{eq6} \alpha_{1Q} = \frac{\sqrt{b^2 – 4ac} – b}{2a}, \quad \alpha_{1A} = -\alpha_{1Q} \Theta L – A_0 L^2 – 2Q_0 L, \end{equation} \tag{6}\] where \begin{align*} a =& L \left( \alpha – \alpha_{1Q}^2 L^2 \right) \frac{\Theta^2}{2}, \\ b =& \Theta L^2 \int_0^\Theta (\Theta – t) \int_0^L (L – z) \frac{dz\,dt}{\rho(L,t)} + \Theta^2 \frac{Q_0 L^2}{2A_0}(A_0 L + 2Q_0)\\ &- \frac{A_0 L^4 \Theta}{2} + \int_0^\Theta (\Theta – t) \int_0^L (L – z) K_r(z,t)\,dz\,dt – Q_0 L^3 \Theta,\\ c =& \int_0^\Theta (\Theta – t) \int_0^L (L – z) \frac{dz\,dt}{\rho(L,t)} \left( A_0 L^2 + 2Q_0 L \right)^2 + \Theta (A_0 L + 2Q_0)^2 \frac{Q_0 L^2}{2A_0}. \end{align*}Approximations of the required functions with higher orders could be obtained in the framework of the standard algorithm of method of averaging of function corrections [7,8,9], i.e. by replacement on the following sums \(A(z,t) \rightarrow \alpha_{nA} + A_{n-1}(z,t)\) and \(Q(z,t) \rightarrow \alpha_{nQ} + Q_{n-1}(z,t)\) in right sides of equations (3). The replacement gives a possibility to obtain the following relations to determine the second-order approximations of the considered functions
\[\begin{equation}\label{eq7} \left\{ \begin{aligned} A_2(z,t) &= \alpha_{2A} + A_1(z,t) + \frac{1}{L} \left[ \alpha_{2A} z + \int_0^z A_1(u,t)\,du + \alpha_{2Q} z + \int_0^t Q_1(u,t)\,du + A_0 z + Q_0 \right] \\ Q_2(z,t) &= \alpha_{2Q} + Q_1(z,t) + \frac{1}{L} \left\{ \alpha_{2Q} z + \int_0^z Q_1(u,t)\,du + \alpha \int_0^t \left[ \frac{\alpha_{2Q} + Q_1(z,\tau)}{\alpha_{2A} + A_1(z,\tau)} \right]^2 d\tau \right. \\ &\qquad \left. + \frac{Q_0^2}{A_0} \int_0^t \int_0^z \frac{\alpha_{2A} + A_1(z,\tau)}{\rho(u,\tau)}\,du\,d\tau + \int_0^t \int_0^z K_r(u,\tau) \frac{\alpha_{2Q} + Q_1(z,\tau)}{\alpha_{2A} + A_1(z,\tau)}\,du\,d\tau + Q_0 z \right\} \end{aligned} \right. \end{equation} \tag{7}\] Not yet known average values in our case were determined by the following standard relations [7,8,9], \[ \begin{align}\label{eq8} \alpha_{nA} =& \frac{1}{L \Theta} \int_0^\Theta \int_0^L \left[ A_n(z,t) – A_{n-1}(z,t) \right] dz\,dt, \notag\\ \alpha_{nQ} =& \frac{1}{L \Theta} \int_0^\Theta \int_0^L \left[ Q_n(z,t) – Q_{n-1}(z,t) \right] dz\,dt. \end{align}\tag{8}\] Substitution of relations (7) into relation (8) gives a possibility to obtain relations to determine average values of the second-order approximations of the considered functions in the following form \[\label{eq9} \begin{aligned} \alpha_{2A} ={} & -\alpha_{1A} \frac{\Theta}{L^2} \int_0^\Theta \int_0^L \left(1 – \frac{z}{L} \right) \frac{dz\,dt}{\rho(z,t)} – 2 \frac{\alpha_{1Q} \Theta}{\alpha_{1A} L^2} \int_0^\Theta \int_0^L \left(1 – \frac{z}{L} \right) K_r(z,t)\,dz\,dt \\ & – \frac{\alpha \alpha_{1Q}^2 \Theta^3}{3 \alpha_{1A} L^2} – \frac{\alpha_{1A}}{3} – \frac{\alpha_{1Q} \Theta}{L} – \frac{Q_0^2 \Theta}{A_0 L^2} – \frac{A_0}{3} – \frac{3Q_0}{2L} – \alpha_{2Q} \frac{\Theta}{2L}, \end{aligned} \tag{9}\] \begin{equation*} \alpha_{2Q} = \alpha_{1Q} – \frac{L A_0}{2 \Theta} + \sqrt{ \frac{1}{9 L^2} \left[ \int_0^\Theta \int_0^L \left(1 – \frac{z}{L} \right) \frac{dz\,dt}{\rho(z,t)} \right]^2 – 4 \Theta \alpha_{1Q} \int_0^\Theta \int_0^L \left(1 – \frac{z}{L} \right) K_r(z,t)\,dz\,dt }. \end{equation*}Analysis of spatio-temporal distributions concentrations of area of the axial section in the vessel of the considered system and average blood flow across the pad has been done analytically by consideration of their second-order approximations. Analytical results were checked by comparison with results of direct numerical simulation.
