5.2 Investigation of parametric deviations to the reference case
5.2.5 Variation of artery stiffness - Comment on aortic waveforms
Fig. 5.17 shows augmentation of pressure and diminishing of flow at the network’s root as the age related to the cases in Table 5.3 advances, meaning that the arterial impedance, when viewed from the heart, becomes greater in term of amplitude as the arterial system grows older. This is in perfect agreement with a simplified impedance analysis, sinceRc ∝β1/2, in whichβis the equivalent elasticity for the vessel’s wall as defined in Chapter 2, and with experimental results taken from invasive and non invasive methods in human subjects, (McVeigh et al., 1999). The hypertensive states achieved are
case pmin pmax qmax qmean
+20 years 90% 109% 87% 92%
+40 years 82 % 117 % 76 % 86%
Table 5.4: Alteration of pressure intervals, maximum and mean flow, caused by the changes in Table 5.3, relative to Table 5.1
.
6 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7 0.1
0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
norm. t
norm. W1
6 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7
−0.4
−0.35
−0.3
−0.25
−0.2
−0.15
−0.1
−0.05
norm. t
norm. W2
Figure 5.18: Separated variables at the aorta (vessel 1) plotted for the reference case (solid black) and for deviate casesageref erence+ 20years(dashed black) andageref erence+ 40years(solid red), described in Table 5.3. Plotted values are normalised by the the reference case’sc0root.
6 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
6 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
6 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Figure 5.19: Comparison between pressure waveforms obtained at the network’s root (smaller peak) and at the extremity of artery 54 (larger peak). From left to right: Cases ofageref erence,ageref erence+20years andageref erence+ 40years. Curves normalised by peak at the root for each case.
possibly exaggerated by the elastic model of the walls, and by the fact that we assume that the left ventricle elastance function’s amplitude doesn’t decay with age. The aortic pressure pulse becomes significantly wider, as is evidenced by the numbers in Table 5.4. The pressure peak in the aorta is significantly delayed with the increasing age, as the interactions between ejected and reflected waves intensify earlier, when the ventricle is still in a more contracted state, closer to the elastance peak.
Flow peaking patterns at the proximal sites remain very similar however, with ejection time periods fairly unchanged when reference and deviate cases are compared.
5.2.6 Variation of artery stiffness - Comment on global waveforms
As observable in Table 5.4, contrarily to the previous case, the diastolic and systolic pressure values now diverge from one another, forming wider intervals as the age advances. This is caused by the rise in the c0speeds that accompanies the elasticity augmentation, and causes for a shorter pulse presence in a certain site. Values for the undeformed vessels wave speeds rise to110%and120%, when compared
to the reference case, allowing for a faster propagation of the pulse, and for an earlier arrival of the reflected waves at the aortic root, that mitigates the ejection, causing for a reduction of the gross input flows into the network. If we move on down to the periphery, we can observe in the arm and leg vessels an even larger amplitude magnification. Wave patterns remain more constant from case to case than in the case of varying heart rate, possibly because once again we chose not to change the aortic valve’s closing speed and velocity threshold.
In fig. 5.19, we can also see that, the rising peripheralc0causes for the wave interactions to intensify, as they are reflected more often during the cardiac cycle. This confers a smoother character to the solution, indicating that the strong tapers that are present at these sites will contribute to the proper dissipation of the pulse before it reaches the capillaries.
On a global scale, even though decays might seem more smooth and physiological for the aged variants of the 55 artery network, we can conclude from the larger pressure intervals registered that this system has become less viable and efficient at damping the strong oscillations imposed by the ventricular func- tion upon the arterial network, and at delivering high volume rates of blood to the periphery with the loss of its compliance.
Chapter 6
Conclusions
The past pages exposed the work that was done in terms of building, validating and verifying a compu- tational 1D pulse replicator, and evidenced the benefits its usage can bring in fast hemodynamic time domain solutions (when compared to full 3D treatment), that can encompass a large space scale, for many seconds. A quantitative evaluation was obtained, and it was concluded that the Maccormack scheme adopted for solving the 1D domains lacked in order of accuracy, producing significant phase errors, only eliminated when computed using very fine descriptions of the domains, which proved to be inefficient, as computing times for a cycle of a mesh with minimal phase error, when compared to a higher order spectral hp solution were registered at about1100s. Amplitude errors however were more permissive in terms of the roughness of the methods. We revert this discussion to Section 6.2.
Despite the minimal phase errors, waveforms remain similar to the ones observed in past works, fromin vivoandin silicostudies, and two different descriptions of the physiological arterial system were assembled, each producing waveforms consistent with in vivo systems, featuring typical arterial flow characteristics, such as increasing peaking pressures as we move to the periphery, the presence of the dicrotic notch, and its convection through the aorta, and the smooth decay in pressure when flow ceases during diastole. Each of the models adopted, namely the 1D domains, the terminal RCR model of the periphery and the LV heart were concluded to be essential in obtaining a still simplistic, but already phys- iologically relevant description of pulsatile flow in human arteries. While most 1D works already adopt a RCR WK terminal BC, fewer are the studies in which a LV model is coupled to the system. We found that the inclusion of this model is important, as different network properties will produce different reflected waves transit times and amplitudes, that in reality are determinant in the timing of the beginning of ejec- tion, so, approaches in which the root solution is pre determined, byin vivomeasurement, or application of a simple function, are suited only to interact in a physiologically coherent way with vessel systems that match the imposed velocity function in terms of reflections, otherwise, a non natural ejection pattern is obtained.
The explicit method adopted was also responsible for the slow computation of systems’ solutions,
however, its simplicity allows for easy editing, as it facilitates the inclusion of BC blocks that rely on non linear equation solving such as the present ones. Through the methodology adopted, many other mod- els can be assembled and coupled to 1D domains, one only needs to relate variables to external ones at predetermined nodes. Although convenient for academic uses, the iterative behaviour of the BCs slows down the already slow system, this is especially true at bifurcations in which a 6 by 6 system must be solved at all times.
Still regarding the computational method, it is concluded that it was not the optimal solution, but it re- vealed to be a satisfactory one. Numeric uncertainties are almost irrelevant for some of the educational uses for this kind of tool, and even rough, fast solutions can be of value to students of the medical sci- ences, and for example, for the designing of biomedical devices, such as ventricular assistance pumps etc, in which quantifications of the arterial system’s input impedance are required. The truth is that, despite the phase errors, the waveforms are minimally different when refined and rough solutions of the same network are compared, presenting similar peaking patterns and signal amplitudes.
Based in the findings of Chapter 4 and 5, we can conclude that the cardiovascular system, even for such a reduced description of its components, is a rich system, with wave reflection patterns that are not trivial. The propagation of pulses through the arteries is conditioned by each vessel’s properties, and at the same time, by the rest of the network. This causes parameters to heavily influence wave- forms. A correct set up of a model representative of a real circulatory system is possible for a network or single vessel configuration, and both are capable of delivering results consistent with mammalianin vivopulse forms. However, the properties must be carefully selected, and a clear idea of what we want to see (i.e. pressure intervals, systolic decay) must exist before the parameters can be compiled from the sources. Note that these parameters also vary widely from subject to subject, some in seemingly unrelated manners. All of this adds difficulty to assembling a set of parameters that are representative of the cardiovascular system.
Still based on the results of Chapter 5, it is important to discuss the appearance of some oscillations that are not featured in physiological flows, or in more detailed descriptions of the cardiovascular system.
These oscillations where attributed to excessively elastic terminal vessels, caused by the absence of ta- per in the present formulation of the 1D equations of blood flow. This indicates that more physiological waveforms can still be achieved, by considering the dissipative effects of reductions of radius towards the periphery. Another natural source of dissipation, which was also ignored in the present work is the viscoelasticity of the vessel’s walls. However, along larger radius vessels as the branches of the aorta, the signals decay smoothly, and are almost perfectly matched with alternative solutions obtained from methods where the aforementioned dissipative effects of the taper of the arteries and the viscoelastic behaviour of their walls were included.
6.1 Achievements
In the work that has been described in this document, a computational 1D pulse duplicator, habilitated to treat networks of vessels connected by bifurcations, and with physiologically relevant BCs, represen- tative of theLV, at the proximal root node, and of the peripheral smaller radius vasculature, has been developed (Chapter 3) , verified and validated (Chapter 4), using the MATLAB environment. The dis- cretisation of the equations was provided by an explicit Maccormack second order method.
6.2 Future Work
The task of assembling a pulse-duplicating computational system that overcomes the shortcomings of the present method, in our present knowledge, can be split into several topics.
• For faster computing times, the method should be translated to a compiled language such as FOR- TRAN or C. Certain algorithms adopted, such as solving the 6 by 6 non linear equation system that constitutes the bifurcations, with a Netwon-Raphson iterative scheme, should be revised as they are clearly very inefficient, when employed in a repetitive way, as was the case. Also of interest are the parallelisation techniques that could be adopted to further improve the method’s performance.
• The inclusion of tapering vessel geometries, and of viscoelastic wall behaviour will further approx- imate the reduced model to the biological system. The inclusion of such source terms encourages the changing of method’s paradigm. A high order implicit method, like the ones adopted to obtain the solutions that served as comparative reference in Chapter 4 is advisable to minimise phase error. This will also contribute to lowering the performance handicap, as it will permit the usage of fewer spatial nodes and time steps per cycle.
• An experimental validation, by means of experiencesin vitro, would also be of great importance, as well as a systematic, complete and quantitative study of errors in the model.
• Parametric uncertainty is the rule in rich biological systems such as the arterial vasculature. An adaptation of the method to accommodate stochastic variations of the properties would be inter- esting, especially after the inclusion of tapering and viscoelasticity.
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Appendix A
Main blood1D loop
Figure A.1: The main solver loop, in a summary way. BC’s are only applied at the end of the predictor- corrector sequence, and before the the friction correction takes place.
Appendix B
Control Network Parameters
B.1 1D arterial network properties
V. ID. Rproximalt Rdistalt V. ID. Rproximalt Rdistalt
1 0.00000 0.02925 29 -0.78860 0.08807
2 -0.19898 0.10700 30 -0.38526 0.19878
3 -0.83027 0.02658 31 -0.70280 0.00000
4 -0.34484 0.20362 32 -0.63381 0.00000
5 -0.68173 0.22028 33 -0.56498 0.00000
6 -0.90997 0.00000 34 -0.66394 0.00000
7 -0.29365 0.09594 35 -0.30265 -0.05246
8 -0.54035 0.00000 36 -0.85610 0.00000
9 -0.55559 0.27802 37 -0.09145 -0.06889
10 -0.75939 0.00000 38 -0.85253 0.00000
11 -0.51863 0.00000 39 -0.07858 0.06933
12 -0.61686 0.00000 40 -0.94355 0.00000
13 -0.60342 0.00000 41 -0.12577 0.06550
14 -0.17973 0.01699 42 -0.53275 0.20204
15 -0.92728 0.22028 43 -0.53275 0.20204
16 -0.61686 0.00000 44 -0.36233 -0.09450
17 -0.60342 0.00000 45 -0.83972 0.00000
18 -0.18062 0.22912 46 -0.45057 0.10809
19 -0.83637 0.15099 47 -0.45493 0.00000
20 -0.91241 0.00000 48 -0.26644 0.00000
21 -0.23858 0.09287 49 -0.84165 0.00000
22 -0.53879 0.00000 50 -0.36233 -0.0945
23 -0.55408 0.27802 51 -0.83972 0.00000
24 -0.75939 0.00000 52 -0.45057 0.10809
25 -0.51863 0.00000 53 -0.45493 0.00000
26 -0.96706 0.00000 54 -0.26644 0.00000
27 -0.26207 0.01919 55 -0.84165 0.00000
28 -0.23059 -0.03341 – – –
Table B.1: Reflection coefficients derived from bifurcation data, for the control case.
V. ID. Rc(GP as/m3) Rp(GP as/m3) Cp(1e−8m3/P a) Rt
6 1.0327 4.5075 0.023216 0.62719
8 0.3138 3.9600 0.026426 0.85314
10 1.7241 63.2250 0.001655 0.94691
11 0.8618 3.9600 0.026426 0.64254
12 0.5064 3.0000 0.010035 0.71114
13 0.4892 3.0000 0.010035 0.71956
16 0.5064 3.0000 0.010035 0.71114
17 0.4892 3.0000 0.010035 0.71956
20 1.1513 4.5075 0.023216 0.59309
22 0.3138 3.9600 0.026426 0.85314
24 1.7241 63.2250 0.001655 0.94691
25 0.8618 3.9600 0.026426 0.64254
26 0.6528 1.0425 0.100380 0.22988
31 0.5144 2.7225 0.038438 0.68217
32 0.8141 4.0575 0.025792 0.66578
33 0.6853 1.7400 0.060143 0.43488
34 0.1110 0.6975 0.150040 0.72535
36 0.3523 0.8475 0.123480 0.41269
38 0.3523 0.8475 0.123480 0.41269
40 1.0683 5.1600 0.020281 0.65696
45 1.1796 5.9475 0.017582 0.66898
47 0.4926 3.5775 0.029252 0.75796
48 0.7382 3.5775 0.029252 0.65792
49 3.4195 4.1925 0.024961 0.10156
51 1.1796 5.9475 0.017582 0.66898
53 0.4926 3.5775 0.029252 0.75796
54 0.7382 3.5775 0.029252 0.65792
55 3.4195 4.1925 0.024961 0.10156
Table B.2: Windkessel model parameters for the control network and equivalent linear reflection coeffi- cientRt.
V. ID Lenght(cm) r(cm) h(cm) E(MPa) c0(m/s)
1 4.00 1.470 0.163 0.4 5.28
2 2.00 1.263 0.126 0.4 5.01
3 3.40 0.620 0.080 0.4 5.70
4 3.40 0.500 0.067 0.4 5.81
5 17.70 0.370 0.063 0.4 6.54
6 14.80 0.188 0.045 0.8 10.97
7 42.20 0.404 0.067 0.4 6.46
8 23.50 0.300 0.043 0.8 8.49
9 6.70 0.300 0.046 0.8 8.78
10 7.90 0.160 0.028 1.6 13.27
11 17.10 0.203 0.046 0.8 10.68
12 17.60 0.250 0.045 0.8 9.52
13 17.70 0.250 0.042 0.8 9.19
14 3.90 1.100 0.115 0.4 5.13
15 20.80 0.370 0.063 0.4 6.54
16 17.60 0.250 0.045 0.8 9.52
17 17.70 0.250 0.042 0.8 9.19
18 5.20 1.000 0.110 0.4 5.26
19 3.40 0.474 0.066 0.4 5.92
20 14.80 0.180 0.045 0.8 11.22
21 42.20 0.403 0.067 0.4 6.47
22 23.50 0.300 0.043 0.8 8.49
23 6.70 0.300 0.046 0.8 8.78
24 7.90 0.160 0.028 1.6 13.27
25 17.10 0.203 0.046 0.8 10.68
26 8.00 0.200 0.049 0.4 7.85
27 10.40 0.800 0.100 0.4 5.61
28 5.30 0.700 0.090 0.4 5.69
29 2.00 0.390 0.064 0.4 6.43
30 1.10 0.300 0.054 0.4 6.73
31 6.60 0.220 0.049 0.4 7.49
32 7.10 0.180 0.045 0.4 7.93
33 6.30 0.200 0.054 0.4 8.24
34 5.90 0.435 0.069 0.4 6.32
35 1.00 0.600 0.080 0.4 5.79
36 3.17 0.260 0.053 0.4 7.16
37 1.00 0.590 0.080 0.4 5.84
38 3.49 0.260 0.053 0.4 7.16
39 10.60 0.580 0.075 0.4 5.71
40 5.00 0.160 0.043 0.4 8.22
41 1.00 0.520 0.065 0.4 5.61
42 5.80 0.368 0.060 0.4 6.40
43 5.80 0.368 0.060 0.4 6.40
44 14.40 0.320 0.053 0.8 9.13
45 5.00 0.200 0.040 1.6 14.19
46 44.30 0.259 0.050 0.8 9.86
47 12.60 0.255 0.047 0.8 9.63
48 32.10 0.247 0.045 1.6 13.54
49 34.33 0.130 0.039 1.6 17.37
50 14.40 0.320 0.053 0.8 9.13
51 5.00 0.200 0.040 1.6 14.19
52 44.30 0.259 0.050 0.8 9.86
53 12.60 0.255 0.047 0.8 9.63
54 32.10 0.247 0.045 1.6 13.54
55 34.30 0.130 0.039 1.6 17.37
Table B.3: Artery properties for the reference case (Section 5.1). νP oisson= 12 for all of the tubes. The names of the arteries can be consulted in Table 1 of Wang and Parker (2004).