forked from SimVascular/simvascular.github.io-archive
-
Notifications
You must be signed in to change notification settings - Fork 0
Expand file tree
/
Copy pathdocssvFSI-Fluid.html
More file actions
490 lines (392 loc) · 23 KB
/
docssvFSI-Fluid.html
File metadata and controls
490 lines (392 loc) · 23 KB
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8">
<meta http-equiv="X-UA-Compatible" content="IE=edge">
<meta name="viewport" content="width=device-width, initial-scale=1">
<meta name="description" content="">
<meta name="author" content="">
<title>SimVascular Docs</title>
<link href="css/bootstrap.css" rel="stylesheet" type="text/css" />
<link href="css/shop-item.css" rel="stylesheet" type="text/css" />
<link href="css/codestyle.css" rel="stylesheet" type="text/css" />
<link rel="stylesheet" href="font-awesome-4.1.0/css/font-awesome.min.css">
<link rel="stylesheet" href="https://code.ionicframework.com/ionicons/1.5.2/css/ionicons.min.css">
<link rel="shortcut icon" href="img/favicon.ico">
<!-- HTML5 Shim and Respond.js IE8 support of HTML5 elements and media queries -->
<!-- WARNING: Respond.js doesn't work if you view the page via file:// -->
<!--[if lt IE 9]>
<script src="https://oss.maxcdn.com/libs/html5shiv/3.7.0/html5shiv.js"></script>
<script src="https://oss.maxcdn.com/libs/respond.js/1.4.2/respond.min.js"></script>
<![endif]-->
</head>
<body>
<!-- Navigation -->
<nav class="navbar navbar-inverse navbar-fixed-top" role="navigation">
<div class="container">
<!-- Brand and toggle get grouped for better mobile display -->
<div class="navbar-header">
<button type="button" class="navbar-toggle" data-toggle="collapse" data-target=".navbar-main-collapse">
<i class="fa fa-bars" id="barIcon"></i>
</button>
<a class="navbar-brand" id="brandName" href="index.html">
<img src="img/svlogo/svLogoSmallText.png" alt="...">
</a>
</div>
<!-- Collect the nav links, forms, and other content for toggling -->
<div class="collapse navbar-collapse navbar-right navbar-main-collapse">
<ul class="nav navbar-nav">
<!-- USER GUIDES -->
<li>
<a href="#" id="dropdownMenu1" data-toggle="dropdown">
<b><span class="fa fa-user"></span> User Guides</b>
</a>
<ul class="dropdown-menu" role="menu" aria-labelledby="dropdownMenu1">
<li role="presentation" class="divider"></li>
<li role="presentation"><a role="menuitem" tabindex="-1" href="docsQuickGuide.html"><b><span class="icon ion-ios7-bolt"></span> Getting Started</b></a></li>
<li role="presentation" class="divider"></li>
<li role="presentation"><a role="menuitem" tabindex="-1" href="docsModelGuide.html"><b><span class="icon ion-settings"></span> Modeling</b></a></li>
<li role="presentation"><a role="menuitem" tabindex="-1" href="docsMeshing.html"><b><span class="icon ion-ios7-keypad"></span> Meshing</b></a></li>
<li role="presentation"><a role="menuitem" tabindex="-1" href="docsFlowSolver.html"><b><span class="icon ion-play"></span> Simulation</b></a></li>
<li role="presentation"><a role="menuitem" tabindex="-1" href="docssvFSI.html"><b><span class="icon ion-plus-round"></span> svFSI</b></a></li>
<li role="presentation"><a role="menuitem" tabindex="-1" href="docsSimCardio.html"><b><span class="icon ion-plus-round"></span> SimCardio</b></a></li>
<li role="presentation"><a role="menuitem" tabindex="-1" href="docsROMSimulation.html"><b><span class="icon ion-plus-round"></span> ROM Simulation</b></a></li>
<li role="presentation"><a role="menuitem" tabindex="-1" href="docsGenBC.html"><b><span class="icon ion-refresh"></span> GenBC</b></a></li>
<li role="presentation"><a role="menuitem" tabindex="-1" href="docsPythonInterface.html"><b><span class="icon ion-refresh"></span> Python Interface</b></a></li>
<li role="presentation"><a role="menuitem" tabindex="-1" href="docsReferences.html"><b><span class="icon ion-refresh"></span> References </b></a></li>
</ul>
</li>
<!-- CLINCAL CASES -->
<li>
<a href="#" id="dropdownMenu1" data-toggle="dropdown">
<b><i class="fa fa-stethoscope"></i> Clinical Cases</b>
</a>
<ul class="dropdown-menu" role="menu" aria-labelledby="dropdownMenu1">
<li role="presentation"><a role="menuitem" tabindex="-1" href="clinicalCase3.html"><b><span class="fa fa-user-md"></span> Coronary Normal</b></a></li>
<li role="presentation"><a role="menuitem" tabindex="-1" href="clinicalCase1.html"><b><span class="fa fa-user-md"></span> Aortofemoral Normal - 1</b></a></li>
<li role="presentation"><a role="menuitem" tabindex="-1" href="clinicalCase2.html"><b><span class="fa fa-user-md"></span> Aortofemoral Normal - 2</b></a></li>
<li role="presentation"><a role="menuitem" tabindex="-1" href="clinicalCase4.html"><b><span class="fa fa-user-md"></span> Healthy Pulmonary</b></a></li>
<li role="presentation"><a role="menuitem" tabindex="-1" href="https://simtk.org/projects/sv_tests"><b><span class="fa fa-user-md"></span> All demo projects</b></a></li>
</ul>
</li>
<!-- DEVELOPER GUIDES -->
<li>
<a href="#" id="dropdownMenu1" data-toggle="dropdown">
<b><span class="fa fa-caret-square-o-right"></span> Developer Guides</b>
</a>
<ul class="dropdown-menu" role="menu" aria-labelledby="dropdownMenu1">
<li role="presentation"><a role="menuitem" tabindex="-1" href="https://github.com/SimVascular/SimVascular/wiki/wiki_for_developers"><b><span class="fa fa-file-text-o"></span> Compile Source Code</b></a></li>
</ul>
</li>
<!-- svCOMMUNITY -->
<li>
<a href="#" id="dropdownMenu1" data-toggle="dropdown">
<b><i class="fa fa-users"></i> svCommunity</b>
</a>
<ul class="dropdown-menu" role="menu" aria-labelledby="dropdownMenu1">
<li role="presentation"><a role="menuitem" tabindex="-1" href="https://simtk.org/forums/viewforum.php?f=188"><b><span class="fa fa-users"></span> Public Forum</b></a></li>
<li role="presentation" class="divider"></li>
<li role="presentation"><a role="menuitem" tabindex="-1" href="https://github.com/SimVascular/SimVascular/wiki/"><b><span class="fa fa-file-text-o"></span> Wiki</b></a></li>
<li role="presentation" class="divider"></li>
<li role="presentation"><a role="menuitem" tabindex="-1" href="https://simtk.org/mailman/listinfo/simvascular-news"><b><span class="fa fa-sign-in"></span> Join News Mailing List</b></a></li>
<li role="presentation"><a role="menuitem" tabindex="-1" href="https://simtk.org/pipermail/simvascular-news/"><b><span class="fa fa-pencil-square-o"></span> News Mailing List Archive</b></a></li>
<li role="presentation"><a role="menuitem" tabindex="-1" href="https://github.com/SimVascular/SimVascular/issues"><b><span class="fa fa-bug"></span> Report bugs and request features</b></a></li>
</ul>
</li>
<!-- REFERENCES -->
<li>
<a href="docsRefs.html" id="dropdownMenu1" >
<b><span class="icon ion-document-text"></span>References</b>
</a>
</li>
<!-- Archives -->
<li>
<a href="#" id="dropdownMenu1" data-toggle="dropdown">
<b> Archives</b>
</a>
<ul class="dropdown-menu" role="menu" aria-labelledby="dropdownMenu1">
<li role="presentation"><a role="menuitem" tabindex="-1" href="archiveQuickGuideSV2.html"><b>SimVascular 2.0</b></a></li>
</ul>
</li>
<!-- <li><a href="docsQuickGuide.html" id="btnQuickGuide"><b><span class="icon ion-ios7-bolt"></span> Quick Guide</b></a></li>
<li><a href="docsModelGuide.html" id="btnModelGuide"><b><span class="icon ion-settings"></span> Modeling</b></a></li>
<li><a href="docsMeshing.html" id="btnMeshing"><b><span class="icon ion-ios7-keypad"></span> Meshing</b></a></li>
<li><a href="docsPresolver.html" id="btnPresolver"><b><span class="icon ion-log-in"></span> svPre</b></a></li>
<li><a href="docsFlowSolver.html" id="btnFlowSolver"><b><span class="icon ion-play"></span> svSolver</b></a></li>
<li><a href="docsRefs.html" id="btnRefs"><b><span class="icon ion-document-text"></span> References</b></a></li>
<li><a href="clinicalCase1.html" id="btnRefs"><b>Case Studies</b></a></li> -->
</ul>
</div>
<!-- /.navbar-collapse -->
</div>
<!-- /.container -->
</nav>
<!-- Page Content -->
<!--Nav Bar -->
<div class="row">
<div class="col-xs-1 col-sm-1 hidden-md hidden-lg">
</div>
<!-- ONE COLUMN OF SPACE -->
<nav class="hidden-xs hidden-sm col-md-2 col-lg-2 bs-docs-sidebar">
<ul id="sidebar" class="nav nav-stacked fixed mansvFSI-Fluid"> <!--Nav Bar -->
<p><h4>Hemodynamics <br> Modeling <br> using svFSI</h4></p>
<li><a href="#intro">Introduction</a></li>
<li><a href="#nn">Non-Newtonian flow</a>
<ul class="nav nav-stacked">
<li><a href="#nn_theory">Viscosity models</a></li>
<li><a href="#nn_input">Input file</a></li>
</ul>
</li>
<li><a href="#pres">Prescribed wall motion</a>
<ul class="nav nav-stacked">
<li><a href="#pres_wall">Wall motion</a></li>
<li><a href="#pres_input">Input file</a></li>
</ul>
</li>
<li><a href="#ref">Reference</a></li>
</ul>
</nav>
<!--Main Content -->
<div class="col-xs-10 col-sm-10 col-md-9 col-lg-9" id="manualContent">
<!-- ACTUAL CONTENT -->
<div class="mansvFSI-Fluid"><section id="intro" class="group"><h2>Hemodynamics Modeling using <strong>svFSI</strong></h2>
<p>Modeling of cardiovascular flow is the main function of <strong>SimVascular</strong>. Currently, the default flow solver of <strong>SimVasculr</strong> is <a href="https://simvascular.github.io/archiveFlowSolverSV2.html"><strong>svSolver</strong></a>. All of the features in <strong>svSolver</strong> exists in <strong>svFSI</strong> including RCR boundary condition, Coupled Momentum Method, GenBC etc. Also, we have carefully tested the new solver to make sure it produce the same results as <strong>svSolver</strong>. In this section, we won’t dwell on the similarities between these two solvers, and instead will focus two new features in flow simulation using <strong>svFSI</strong>, i.e., non-Newtonian flow model and simulation of prescribed motion.</p>
</section>
<section id="nn" class="group"><h2>Non-Newtonian flow</h2>
<p>Blood is a complex mixture that consists of plasma, blood cells and platelets. The blood viscosity is a complicated subject. It is strongly dependent on several factors such as temperature, hematocrit and, especially, the shear rate. From experimental studies, it is determined that the blood behaves like Newtonian flow at high shear rate ($>100 s^{-1}$). In most large arteries such as aorta, coronary arteries, the shear rate is well above this number and blood can be treated as a Newtonian fluid, that is the viscosity is a constant. On the other hand, when the shear rate is below this threshold, blood presents strong shear thinning behavior, i.e. the viscosity decreases with increasing shear rate. Many viscosity models have been proposed to represent this non-Newtonian behavior <a href="#ref-1">[1]</a>.</p>
</section>
<section id="nn_theory" class="subgroup"><h3>Viscosity models</h3>
<p>Currently, <strong>svFSI</strong> supports three viscosity models: Newtonian, Carreau-Yasuda and Casson <a href="#ref-2">[2]</a>.</p>
<figure>
<img class="svImg svImgSm" src="documentation/svfsi/fluid/imgs/non-newtonian.png" style="width:100%;height:auto;max-width: 30vw;">
<figcaption class="svCaption" >Non-Newtonian viscosity model<a href="#ref-1">[1]</a>.</figcaption>
</figure>
<p>Carreau-Yassuda model is defined as</p>
<p>$$\eta=\eta_\infty + (\eta_0 - \eta_\infty) \left[ 1 + \left( \lambda\dot(\gamma)^a \right) \right]^{\frac{n-1}{a}}$$</p>
<p>Here:
<ul>
<li>$\eta_\infty$: limiting high shear-rateviscosity;</li>
<li>$\eta_0$: limiting low shear-rate viscosity;</li>
<li>$\lambda$: shear-rate tensor multiplier;</li>
<li>$\dot{\gamma}$: shear rate;</li>
<li>$a$: shear-rate tensor exponent;</li>
<li>$n$: power-law index.</li>
</ul> </p>
<p>Casson model is defined as</p>
<p>$$\eta=\frac{1}{\dot{\gamma}}\left[ k_0 ( c ) + k_1 ( c )\sqrt{\dot{\gamma}} \right]^2$$</p>
<p>Here, $k_0 ( c )$ and $k_1 ( c )$ are functions of the hematocrit $c$. </p>
</section>
<section id="nn_input" class="subgroup"><h3>Input file</h3>
<p>The input file mostly follows the master input file <a href="./svFSI_master.inp"><code>svFSI_master.inp</code></a>. Some specific input options are discussed below:</p>
<p>For Newtonian fluid:</p>
<pre class="highlight plaintext"><code> Viscosity: Constant {
Vsalue: 0.04
}
</code></pre>
<p>For Casson fluid</p>
<pre class="highlight plaintext"><code> Viscosity: Cassons {
Asymptotic viscosity parameter: 0.3953
Yield stress parameter: 0.22803
Low shear-rate threshold: 0.5
}
</code></pre>
<p>For Carreau-Yasuda fluid</p>
<pre class="highlight plaintext"><code> Viscosity: Carreau-Yasuda {
Limiting high shear-rate viscosity: 0.022
Limiting low shear-rate viscosity: 0.22
Shear-rate tensor multiplier (lamda): 0.11
Shear-rate tensor exponent (a): 0.644
Power-law index (n): 0.392
}
</code></pre>
</section>
<section id="pres" class="group"><h2>Simulations with prescribed wall motion</h2>
<p>In this section, we discuss how to set up a simulation with prescribed wall motion. For example, one may wish to extract the motion of the walls of the ventricle from CT/MR scans, and reconstruct the flow fields by solving the Navier-Stokes equations. This is accomplished by computing the displacement trajectory of the endocardial surface and prescribed it as the boundary condition. This must be done offline by the user for their specific problem. This is typically done for a small subset of the total times, and displacements between the specified times are prescribed by using a piece-wise linear interpolant. Some researchers refer to this method as the one-way coupled fluid-structure interaction modeling. But since only fluid equation is solved here, we categorize it as flow simulation. </p>
</section>
<section id="pres_wall" class="subgroup"><h3>Wall motion</h3>
<p>There are many established pipelines to obtain the wall motion from CT/MR scans<a href="#ref-3">[3-5]</a>. Here, we would recommend using the <a href="http://simvascular.github.io/docsSimCardio.html#automatic-cardiac-modeling">cardiac geometric modeling tool</a> developed in the <strong>SimCardio</strong> project.</p>
<p>For the wall motion file, the file format is as follows. First, specify the dimension of the problem (three), the number of times at which to specify the displacements, and the number of vertices in the moving wall mesh. Then specify the times at which the displacements occur. Next, for each vertex, specify its index and then the prescribed displacements for each time. Note that in the case of multiple moving faces, these numbers may not start at one for any given face, as indexing is global. If a vertex is on an edge between two faces, it should have the same index and displacement fields, specified redundantly in both files.</p>
<pre class="highlight plaintext"><code>Dimension n_times m_vertices
t_1
t_2
...
t_n
vertex_1_index
displacement_1_vertex_1
displacement_2_vertex_1
...
displacement_n_vertex_1
vertex_2_index
displacement_1_vertex_2
displacement_2_vertex_2
...
displacement_n_vertex_2
...
</code></pre>
<p>For example,</p>
<pre class="highlight plaintext"><code>3 21 14907
0.000000
3.800E-2
7.600E-2
1.140E-1
1.520E-1
1.900E-1
2.280E-1
2.660E-1
3.040E-1
3.419E-1
3.800E-1
4.180E-1
4.560E-1
4.940E-1
5.320E-1
5.699E-1
6.080E-1
6.460E-1
6.839E-1
7.220E-1
7.600E-1
1
0.000000 0.000000 0.000000
2.800E-1 -8.99E-2 -1.00E-2
8.799E-1 -2.09E-1 -8.10E-1
1.339999 -1.59E-1 -1.43000
1.509999 7.000E-2 -1.59000
1.310000 3.100E-1 -1.52000
1.069999 5.100E-1 -1.41000
1.009999 6.000E-1 -1.28000
9.699E-1 3.600E-1 -1.10000
1.049999 -1.09E-1 -8.80E-1
1.169999 -6.69E-1 -7.60E-1
1.169999 -1.17999 -7.90E-1
1.129999 -1.31999 -9.69E-1
1.049999 -1.25999 -1.34000
1.000000 -9.39E-1 -1.64000
9.899E-1 -4.29E-1 -1.96000
1.000000 3.000E-2 -2.15000
9.299E-1 3.600E-1 -1.90000
6.599E-1 4.200E-1 -1.14000
1.999E-1 1.500E-1 -3.50E-1
0.000000 0.000000 0.000000
2
0.000000 0.000000 0.000000
2.700E-1 -8.99E-2 0.000000
8.500E-1 -1.99E-1 -6.80E-1
1.280000 -1.59E-1 -1.16999
1.430000 5.000E-2 -1.28000
1.230000 2.599E-1 -1.19999
1.010000 4.499E-1 -1.09000
9.500E-1 5.099E-1 -9.69E-1
9.100E-1 2.400E-1 -8.19E-1
1.000000 -2.59E-1 -6.70E-1
1.120000 -8.29E-1 -5.69E-1
1.130000 -1.31999 -5.89E-1
1.090000 -1.44999 -7.50E-1
1.000000 -1.36000 -1.06999
9.400E-1 -1.00999 -1.31000
9.300E-1 -4.69E-1 -1.56999
9.500E-1 0.000000 -1.72999
8.900E-1 3.299E-1 -1.53999
6.200E-1 3.999E-1 -9.39E-1
1.900E-1 1.399E-1 -2.90E-1
0.000000 0.000000 0.000000
...
</code></pre>
<p>Note that in this example, we wish the mesh motion to be periodic in time, and thus the final displacement is zero.</p>
</section>
<section id="pres_input" class="subgroup"><h3>Input file</h3>
<p>To set up the input file, set the equation to be FSI to allow the mesh to move under the ALE framework, even though there is not necessarily a structure. For the moving wall, add the motion file when specifying the wall boundary condition, and turn on the option “Impose on state variable integral”.</p>
<pre class="highlight plaintext"><code> Add BC: moving_wall {
Type: Dirichlet
Time dependence: General
Temporal and spatial values file path: wall_motion.dat
Profile: Flat
Zero out perimeter: 1
Impose flux: 0
#---------------- Add this line to the moving boundary face -----------
Impose on state variable integral: 1
}
</code></pre>
<p>It is recommended to include remeshing if the wall motion is such that the domain undergoes large changes. For example, set</p>
<pre class="highlight plaintext"><code> Remesher: Tetgen {
Max edge size: lumen { val: 3.0 }
Min dihedral angle: 10.0
Max radius ratio: 1.1
Remesh frequency: 100
Frequency for copying data: 1
}
</code></pre>
<p>The max edge size should be consistent with the original mesh size.</p>
<p>Under the mesh equation, we similarly add the motion file.</p>
<pre class="highlight plaintext"><code> Add equation: mesh {
Coupled: 1
Min iterations: 1
Max iterations: 8
Tolerance: 1e-3
Residual dB reduction: -20
Poisson ratio: 0
Output: Spatial {
Displacement: t
}
#---------- Add the BC for the moving_wall to the mesh equation as well ------
Add BC: moving_wall {
Type: Dirichlet
Time dependence: General
Temporal and spatial values file path: wall_motion.dat
Profile: Flat
Zero out perimeter: 1
Impose flux: 0
#---------------- Add this line to the moving boundary face -----------
Impose on state variable integral: 1
}
</code></pre>
<p>Follow this <a href="http://simvascular.github.io/docssvFSI.html#app_restart_after_remesh">instruction</a> if you need to restart your simulation after stoppage.</p>
</section>
<section id="ref" class="group"><h2>Reference</h2>
<p><a id="ref-1">
[1] Chandran KB, Rittgers SE, Yoganathan AP. <strong>Biofluid mechanics: the human circulation.</strong> CRC press; 2006 Nov 15. </a></a></p>
<p><a id="ref-2"> <a href="https://doi.org/10.1063/1.2772250">
[2] Boyd, Joshua, James M. Buick, and Simon Green. <strong>Analysis of the Casson and Carreau-Yasuda Non-Newtonian Blood Models in Steady and Oscillatory Flows Using the Lattice Boltzmann Method</strong>. Physics of Fluids 19, no. 9 (September 2007): 093103. </a></a></p>
<p><a id="ref-3">
[3] Vedula V, Lee J, Xu H, Kuo CC, Hsiai TK, Marsden AL.<strong> A method to quantify mechanobiologic forces during zebrafish cardiac development using 4-D light sheet imaging and computational modeling.</strong> PLoS computational biology. 2017 Oct 30;13(10):e1005828.</a></a></p>
<p><a id="ref-4">
[4] Mittal R, Seo JH, Vedula V, Choi YJ, Liu H, Huang HH, Jain S, Younes L, Abraham T, George RT. <strong>Computational modeling of cardiac hemodynamics: current status and future outlook.</strong> Journal of Computational Physics. 2016 Jan 15;305:1065-82. </a></a></p>
<p><a id="ref-5"> <a href="https://doi.org/10.1115/1.4048032">
[5] Kong, F., and Shadden, S. C. (2020). **Automating Model Generation for Image-based Cardiac Flow Simulation**. ASME. J Biomech Eng. </a> </p>
<p><br><br><br><br><br></p>
</section>
</div>
</div>
</div>
<!-- /.container -->
<nav class="navbar navbar-default navbar-fixed-bottom">
<div class="container-fluid text-center">
<ul class="nav navbar-nav">
<li><a>Copyright © SimVascular Development Team - 2017</a></li>
</ul>
</div>
<!-- /.container -->
</nav>
<script src="js/jquery-1.11.0.js" type="text/javascript"></script><script src="js/bootstrap.min.js" type="text/javascript"></script>
<script type="text/x-mathjax-config">
MathJax.Hub.Config({
tex2jax: {inlineMath: [['$','$'], ['\\(','\\)']]}
});
$('body').scrollspy({
target: '.bs-docs-sidebar',
offset: 40
});
</script>
<script>
(function(i,s,o,g,r,a,m){i['GoogleAnalyticsObject']=r;i[r]=i[r]||function(){
(i[r].q=i[r].q||[]).push(arguments)},i[r].l=1*new Date();a=s.createElement(o),
m=s.getElementsByTagName(o)[0];a.async=1;a.src=g;m.parentNode.insertBefore(a,m)
})(window,document,'script','//www.google-analytics.com/analytics.js','ga');
ga('create', 'UA-55333921-1', 'auto');
ga('send', 'pageview');
</script>
<script type="text/javascript"
src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
</script>
</body>
</html>