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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.

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Acknowledgements

Puelz was supported in part by the Research Training Group in Modeling and Simulation funded by NSF via grant RTG/DMS-1646339. Riviere acknowledged the support of NSF via Grant DMS 1913291.

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Correspondence to Beatrice Riviere.

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Appendix A

Appendix A

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.

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

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

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.

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 \(2.310 \times 10^{-6}\)

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Masri, R., Puelz, C. & Riviere, B. A Discontinuous Galerkin Method for Blood Flow and Solute Transport in One-Dimensional Vessel Networks. Commun. Appl. Math. Comput. (2021). https://doi.org/10.1007/s42967-021-00126-5

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Keywords

  • Reduced models
  • Blood flow
  • Solute transport
  • Coriolis coefficient
  • Vessel networks
  • Junction conditions
  • Locally implicit

Mathematics Subject Classification

  • 65M12
  • 65M15
  • 65M60
  • 76R99
  • 76Z99