A Discontinuous Galerkin Method for Blood Flow and Solute Transport in One-Dimensional Vessel Networks

Abstract

This paper formulates an efficient numerical method for solving the convection diffusion solute transport equations coupled to blood flow equations in vessel networks. The reduced coupled model describes the variations of vessel cross-sectional area, radially averaged blood momentum and solute concentration in large vessel networks. For the discretization of the reduced transport equation, we combine an interior penalty discontinuous Galerkin method in space with a novel locally implicit time stepping scheme. The stability and the convergence are proved. Numerical results show the impact of the choice for the steady-state axial velocity profile on the numerical solutions in a fifty-five vessel network with physiological boundary data.

Appendix A

1.1 A.1 Proof of Lemma 2

Let $\phi _h \in {\mathbb {V}}_h^k$. For readability, let $u = C_h^n$ and $v = C_h^{n+1}$. We have

$$\begin{aligned}&\sum _{e=0}^{N} \left( {\mathcal {J}}_e(u,\phi _h) - {\mathcal {J}}_e(v,\phi _h) \right) \nonumber \\&=\sum _{e=0}^N \int _{x_e}^{x_{e+1}} (f(u)-f(v)) \partial _x \phi _h \mathrm{d}x + \sum _{e=0}^{N+1} \left( f^{nf}(u)\vert _{x_e} - f^{nf}(v)\vert _{x_e}\right) [\phi _h]\vert _{x_e}. \end{aligned}$$

(A 1)

Using a Taylor expansion and assumption (61), there exists $\xi$ between u and v such that

$$\begin{aligned} |f(u) - f(v) | = |f'(\xi )| |u-v| \leqslant L_1|u-v|. \end{aligned}$$

Thus, using Cauchy-Schwarz, the first term is bounded by

$$\begin{aligned}&\left| \sum _{e=0}^N \int _{x_e}^{x_{e+1}} (f(u)-f(v)) \partial _x \phi _h \mathrm{d}x \right| \nonumber \\&\leqslant L_1 \Vert u-v \Vert \,\,\left( \sum _e \Vert \partial _x \phi _h \Vert _{L^2(I_e)}^2\right) ^{1/2} \leqslant L_1 a_0^{-1/2} \Vert u-v\Vert \,\, \Vert \phi _h \Vert _{\mathrm{DG}}. \end{aligned}$$

(A 2)

To bound the second term, we write

$$\begin{aligned}&\sum _{e=0}^{N+1} \vert \left( f^{nf}(u)\vert _{x_e} - f^{nf} (v)\vert _{x_e} \right) \vert \, \vert [\phi _h]\vert _{x_e} \vert \\&= \sum _{e=1}^N \vert f^{nf}(u(x_e^-),u(x_e^+)) - f^{nf}(v(x_e^-),v(x_e^+))\vert \, \vert [\phi _h]\vert _{x_e} \vert \\&+\vert f^{nf}(C_{\mathrm{in}}(t_n),u(x_0^+))-f^{nf}(C_{\mathrm{in}}(t_{n+1}),v(x_0^+))\vert \, \vert [\phi _h]\vert _{x_0}\vert \\&+\vert f^{nf}(u(x_{N+1}^-),C_{\mathrm{out}}(t_n))-f^{nf}(v(x_{N+1}^-),C_{\mathrm{out}}(t_{n+1})\vert \, \vert [\phi _h]\vert _{x_{N+1}}\vert . \end{aligned}$$

Using bound (62), we obtain

$$\begin{aligned}&\sum _{e=0}^{N+1} \vert \left( f^{nf}(u)\vert _{x_e} - f^{nf} (v)\vert _{x_e} \right) \vert \, \vert [\phi _h]\vert _{x_e} \vert \nonumber \\&\leqslant \sum _{e=1}^{N+1} L_2 |u(x_e^-)-v(x_e^-)|\vert [\phi _h]\vert _{x_e} \vert +\sum _{e=0}^{N} L_2|u(x_e^+)-v(x_e^+)| \, \vert [\phi _h]\vert _{x_e} \vert \nonumber \\&+ L_2 \vert C_{\mathrm{in}}(t_n) - C_{\mathrm{in}}(t_{n+1})\vert \, \vert [\phi _h]\vert _{x_0}\vert + L_2 \vert C_{\mathrm{out}}(t_n) - C_{\mathrm{out}}(t_{n+1}) \vert \, \vert [\phi _h]\vert _{x_{N+1}}\vert . \end{aligned}$$

(A 3)

Using trace inequality (65), we attain

$$\begin{aligned}&\sum _{e=1}^{N+1} L_2 |u(x_e^-)-v(x_e^-)|\vert [\phi _h]\vert _{x_e} \vert \nonumber \\&\leqslant \sum _{e=0}^{N} L_2\sigma ^{-1/2}(k+1) \Vert u - v \Vert _{L^2(I_e)} |,\sigma ^{1/2} h^{-1/2}[\phi _h]|_{x_e}|, \nonumber \\&\leqslant \left( \sum _{e=0}^{N} L_2^2 \sigma ^{-1}(k+1)^2 \Vert u-v\Vert ^2_{L^{2}(I_e)} \right) ^{1/2} \left( \sum _{e=0}^{N} \sigma h^{-1} [\phi _h]^2 \vert _{x_e} \right) ^{1/2}, \nonumber \\&\leqslant L_2 \sigma ^{-1/2}(k+1) \Vert u-v\Vert \,\, \Vert \phi _h \Vert _{\mathrm{DG}}. \end{aligned}$$

(A 4)

The same bound on the second sum in (A 3) holds. Combining (A 2) and (A 4) yields the result.

1.2 A.2 Proof of Lemma 4

Let $u,v \in {\mathbb {V}}_h^k$, then

$$\begin{aligned} {\mathcal {B}}(u,v)= \sum _{e=0}^{N+1} \left( \{a\, \partial _x u \}\vert _{x_e}[v]\vert _{x_e} -\epsilon \{ a\,\partial _x v \}\vert _{x_e} [ u]\vert _{x_e} +\frac{\sigma }{h} [u]\vert _{x_e}[v]\vert _{x_e} \right) . \end{aligned}$$

First, we consider the interior points and fix $1\leqslant e \leqslant N$. We employ the trace inequality (65) and inverse inequality (66) to obtain a bound on the first term:

$$\begin{aligned} |\{a\, \partial _x u \}\vert _{x_e}[v]\vert _{x_e}| \leqslant a_1 M_k^{1/2}k h^{-3/2} |[v]\vert _{x_e}| \, (\Vert u \Vert _{L^2(I_e)} + \Vert u \Vert _{L^2(I_{e-1})}). \end{aligned}$$

The second term is bounded by the inverse inequality (66):

$$\begin{aligned} |\epsilon \{a\partial _x v \}_{x_e}[u]_{x_e}|&\leqslant |\epsilon | \frac{a_1}{2 \sqrt{a_0}} k(k+1) h^{-1} \left( \Vert u \Vert _{L^2(I_e)} + \Vert u \Vert _{L^2(I_{e-1})}\right) \\&\quad \times \left( \Vert a^{1/2} \partial _x v \Vert _{L^2(I_e)} + \Vert a^{1/2} \partial _x v \Vert _{L^2(I_{e-1})}\right) , \end{aligned}$$

which becomes (using a simple inequality)

$$\begin{aligned} |\epsilon \{a\partial _x v \}_{x_e}[u]_{x_e}|&\leqslant |\epsilon |\frac{a_1}{\sqrt{2 a_0}} k(k+1) h^{-1} \left( \Vert u \Vert _{L^2(I_e)} + \Vert u \Vert _{L^2(I_{e-1})}\right) \\&\quad \times \left( \Vert a^{1/2} \partial _x v \Vert _{L^2(I_e)}^2 + \Vert a^{1/2} \partial _x v \Vert _{L^2(I_{e-1})}^2\right) ^{1/2}. \end{aligned}$$

For the third term, we have

$$\begin{aligned} \bigg \vert \frac{\sigma }{h} [u]\bigg \vert _{x_e}[v]\vert _{x_e}\vert \leqslant \sigma (k+1)h^{-3/2} |[v]_{x_e}| \, \left( \Vert u \Vert _{L^2(I_e)} + \Vert u \Vert _{L^2(I_{e-1})}\right) . \end{aligned}$$

For the boundary terms, similar arguments result in the following bounds:

$$\begin{aligned} |\{a\, \partial _x u \}\vert _{x_0}[v]\vert _{x_0}|&\leqslant 2 a_1 M_k^{1/2} k h^{-3/2} \Vert u\Vert _{L^2(I_0)} \, |[v]\vert _{x_0}|, \\ |\epsilon \{ a\,\partial _x v \}\vert _{x_0} [ u]\vert _{x_0}|&\leqslant |\epsilon | \frac{a_1}{\sqrt{a_0}} k(k+1) h^{-1} \Vert a^{1/2} \partial _x v\Vert _{L^2(I_0)} \, \Vert u \Vert _{L^2(I_0)}, \\ |\frac{\sigma }{h} [u]\vert _{x_0}[v]\vert _{x_0}|&\leqslant \sigma (k+1) h^{-3/2} |[v]\vert _{x_0}|\, \Vert u \Vert _{L^2(I_0)}. \end{aligned}$$

Similar bounds hold for $e=N+1$. Using the Cauchy-Schwarz’s inequality, we obtain

$$\begin{aligned} |{\mathcal {B}}(u,v)| \leqslant \gamma _k h^{-1} \Vert u \Vert \Vert v \Vert _{\mathrm{DG}} \end{aligned}$$

with

$$\begin{aligned} \gamma _k = 2\left( 4 \frac{a_1^2}{\sigma } M_k k^2 + \epsilon ^2 \frac{a_1^2}{a_0} k^2 (k+1)^2 + \sigma (k+1)^2 \right) ^{1/2}. \end{aligned}$$

1.3 A.3 Proof of Bound (94)

To bound the last term, we use the following trace inequality for functions in $H^1(I_e)$ [22]:

$$\begin{aligned} |u(x)| \leqslant Kh^{-1/2} \left( \Vert u \Vert _{L^2(I_e)} + h \Vert \partial _x u \Vert _{L^2(I_e)}\right) , \quad \forall u \in H^1(I_e), \,\, x = x_e, x_{e+1}. \end{aligned}$$

(A 5)

The last term appearing in the convergence proof is

$$\begin{aligned} \left| \sum _{e=0}^N {\mathcal {J}}_e(C_h^n, \chi ^{n+1}) - {\mathcal {J}}_e(C^n, \chi ^{n+1})\right|&\leqslant \left| \sum _{e=0}^N \int _{x_e}^{x_{e+1}} (f(C^n_h)- f(C^n)) \partial _x \chi ^{n+1} \right| \\&\quad + \left| \sum _{e=0}^{N+1} (f^{nf}(C_h^n)\vert _{x_e} - f^{nf}(C^n)\vert _{x_e})[\chi ^{n+1}] \right| . \end{aligned}$$

The first term is bounded by (A 2) and the triangle inequality:

$$\begin{aligned} \left| \sum _{e=0}^N \int _{x_e}^{x_{e+1}} (f(C^n_h)- f(C^n)) \partial _x \chi ^{n+1} \right| \leqslant L_1 a_0^{-1/2} (\Vert \chi ^n \Vert + \Vert \rho ^n \Vert ) \Vert \chi ^{n+1}\Vert _{\mathrm{DG}}. \end{aligned}$$

(A 6)

We bound the second term using the Lipschitz continuity assumption on $f^{nf}$ (62) and the triangle inequality:

$$\begin{aligned} \left| \sum _{e=0}^{N+1} (f^{nf}(C_h^n)\vert _{x_e} - f^{nf}(C^n)\vert _{x_e})[\chi ^{n+1}]\vert _{x_e} \right|&\leqslant \sum _{e=0}^N L_2(|\chi ^n(x_e^+)|+|\rho ^n(x_e^+)|) [\chi ^{n+1}]\vert _{x_e} \\&\quad +\sum _{e=1}^{N+1} L_2(|\chi ^n(x_e^-)| + |\rho ^n(x_e^-)|)| [\chi ^{n+1}]\vert _{x_e}. \end{aligned}$$

Using the trace inequalities (65) and (A 5), we obtain

$$\begin{aligned} \left| \sum _{e=0}^{N+1} (f^{nf}(C_h^n)\vert _{x_e} - f^{nf}(C)\vert _{x_e})[\chi ^{n+1}] \right|&\leqslant 2L_2\sigma ^{-1/2}(k+1) \Vert \chi ^n \Vert \Vert \chi ^{n+1} \Vert _{\mathrm{DG}}\nonumber \\&\!\!\!\!\!\!\!+ K \left( \Vert \rho ^n \Vert ^2 + h^2 \sum _{e=0}^{N} \Vert \partial _x \rho ^n \Vert _{L^2(I_e)}^2 \right) ^{1/2}\nonumber \\&\!\!\!\!\!\!\!\times \Vert \chi ^{n+1} \Vert _{\mathrm{DG}}. \end{aligned}$$

(A 7)

Using the approximation results of the elliptic projection (82) and (83), we obtain

(A 8)

The final bound, (94), is attained by an application of Young’s inequality.

1.4 A.4 Parameters for Fifty-Five Vessel Network

	Vessel name	L, cm	$A_0, \text {cm}^2$	$\beta$, $\text {dyn}/\text {cm}^3$	$R_2$, $\text {dyn.s}/\text {cm}^5$	$C_{\mathrm{cap}}$, $\text {cm}^{5}/\text {dyn}$
1	Ascending Aorta	4.0	5.983	97 000	–	–
2	Aortic Arch I	2.0	5.147	87 000	–	–
3	Brachiocephalic	3.4	1.219	233 000	–	–
4	R. Subclavian I	3.4	0.562	423 000	–	–
5	R. Carotid	17.7	0.432	516 000	–	–
6	R. Vertebral	14.8	0.123	2 590 000	72 417	$3.129 \times 10^{-6}$
7	R. Subclavian II	42.2	0.510	466 000	–	–
8	R. Radial	23.5	0.106	2 866 000	46 155	$4.909 \times 10^{-6}$
9	R. Ulnar I	6.7	0.145	2 246 000	–	–
10	R. Interosseous	7.9	0.031	12 894 000	191 252	$1.185 \times 10^{-6}$
11	R. Ulnar II	17.1	0.133	2 446 000	46 995	$4.821 \times 10^{-6}$
12	R. Internal Carotid	17.6	0.121	2 644 000	23 041	$9.833 \times 10^{-6}$
13	R. External Carotid	17.7	0.121	2 467 000	37 563	$6.032 \times 10^{-6}$
14	Aortic Arch II	3.9	3.142	130 000	–	–
15	L. Carotid	20.8	0.430	519 000	–	–
16	L. Internal Carotid	17.6	0.121	2 644 000	23 118	$9.801 \times 10^{-6}$
17	L. External Carotid	17.7	0.121	2 467 000	37 696	$6.011 \times 10^{-6}$
18	Thoracic Aorta I	5.2	3.142	124 000	–	–
19	L. Subclavian I	3.4	0.562	416 000	–	–
20	Vertebral	14.8	0.123	2 590 000	76 972	$2.944 \times 10^{-6}$
21	L. Subclavian II	42.2	0.510	466 000	–	–
22	L. Radial	23.5	0.106	2 866 000	45 329	$4.998 \times 10^{-6}$
23	L. Ulnar I	6.7	0.145	2 246 000	–	–
24	L. Interosseous	7.9	0.031	12 894 000	191 945	$1.180 \times 10^{-6}$
25	L. Ulnar II	17.1	0.133	2 246 000	47 905	$4.730 \times 10^{-6}$
26	Intercostals	8.0	0.196	885 000	996 508	$2.274 \times 10^{-6}$
27	Thoracic Aorta II	10.4	3.017	117 000	–	–
28	Abdominal I	5.3	1.911	167 000	–	–
29	Celiac I	2.0	0.478	475 000	–	–
30	Celiac II	1.0	0.126	1 805 000	–	–
31	Hepatic	6.6	0.152	1 142 000	13 394	$1.692\times 10^{-5}$
32	Gastric	7.1	0.102	1 567 000	1 373 574	$1.650\times 10^{-7}$
33	Splenic	6.3	0.238	806 000	18 933	$1.197 \times 10^{-5}$
34	Superior Mesenteric	5.9	0.430	569 000	8 728	$2.596 \times 10^{-5}$
35	Abdominal II	1.0	1.247	227 000	–	–
36	L. Renal	3.2	0.332	566 000	9 051	$2.503 \times 10^{-5}$
37	Abdominal III	1.0	1.021	278 000	–	–
38	R. Renal	3.2	0.159	1 181 000	9 082	$2.495 \times 10^{-5}$
39	Abdominal IV	10.6	0.697	381 000	–	–
40	Inferior Mesenteric	5.0	0.08	1 895 000	95 652	$2.369\times 10^{-6}$
41	Abdominal V	1.0	0.578	399 000	–	–
42	R. Common Iliac	5.9	0.328	649 000	–	–
43	L. Common Iliac	5.8	0.328	649 000	–	–
44	L. External Iliac	14.4	0.252	1 493 000	–	–
45	L. Internal Iliac	5.0	0.181	3 134 000	16 632	$1.362 \times 10^{-5}$
46	L. Femoral	44.3	0.139	2 559 000	–	–
47	L. Deep Femoral	12.6	0.126	2 652 000	13 715	$1.652 \times 10^{-5}$
48	L. Posterior Tibial	32.1	0.110	5 808 000	84 662	$2.676 \times 10^{-6}$
49	L. Anterior Tibial	34.3	0.060	9 243 000	98 131	$2.309 \times 10^{-6}$
50	R. External Iliac	14.5	0.252	1 493 000	–	–
51	R. Internal Iliac	5.1	0.181	3 134 000	16 582	$1.366 \times 10^{-5}$
52	R. Femoral	44.4	0.139	2 559 000	–	–

53	R. Deep Femoral	12.7	0.126	2 652 000	13 707	$1.653 \times 10^{-5}$
54	R. Posterior Tibial	32.2	0.110	5 808 000	84 625	$2.677 \times 10^{-6}$
55	R. Anterior Tibial	34.4	0.060	9 243 000	98 100	$2.310 \times 10^{-6}$