We could constitute the feedback loop for an artificial heart
automatic control algorithm by the use of an autonomic nerve discharges
or Mayer wave in peripheral vascular resistances, and or so. It may be
make a complicated structures in time series data as ﾒcomplex systemﾓ
like chaos or fractal.
However, is it really make complex system like circulatory regulatory system in creatures ?
A few investigators have challenged this problem. For example,
Cavalcanti et al, reported a very interesting article in 1996 [20,21].
The profile was tried to be explained here. Model of baroreceptor reflex
series was constituted with simple mathematics model. Simulation with
various delay in the feed back loop were carried out. Behavior of the
hemodynamics repeats bifurcation from the simple limit cycle vibration.
Before long, it changes to the condition which is chaotic. With
alteration only of delay, time series showed period doubling
bifurcation, and reached to the chaos. This kind of experiment wasn't
carried out at the past. As a result, interest is attracted very much.
If it is considered from a viewpoint of artificial heart control,
baroreflex delay means the control delay of an artificial heart. Of
course, hemodynamics is a signal of time series such as blood pressure
or heart rate. We tried to generate deterministic chaos in the feedback
loop model of an artificial heart control.
Section equivalent to cardiovascular system had been simulated with
the three elements Windkessel represented with simple electric circuit.
Three elements Windkessel is expressed as the systemic vascular
resistance (R), arterial compliance (C) and the characteristic impedance
(r). In this model, input is aortic flow, and the output is aortic
pressure. In other words, it is been similar to cardiovascular system
with electric circuit model. Formula formed with this model is (4) and
(5). Each value of these three elements was adopted from the data
mentioned in the literature of the physiology.
dP(t) = ωｔ[ RQ(t) - Ps(t)], ωｔ = 1 / RC (4)
P(t) = Ps + Q(t) (5)
In this simulation, non-linear curve was used for determination.
Cardiac cycle (T) and Stroke volume (SV) is decided from arterial blood
pressure (P), which is an data of a baroreflex. In this study,
non-linear curve as the sigmoid function was used for that purpose as we
shown in (6) and (7). Concerning the each constant, we sought the value
from the literature in physiology. Figure 14 and 15 showed its
relationship.
T(P) = Ts + (Tm - Ts) / ( 1 + γe -αP/Pn) (6)
SV(P) = Svmax / ( 1 + β( P / Pv - 1 )-k) (7)
Delay t was introduced in the feedback loop in this study. After the
delay time r, it was input into the next element. This was included in
the feed-back loop branch. This time delay r was altered as a parameter,
when we carried out the simulation.
The simulation model.
In this study, deterministic chaos was tried to be caused only with
alteration of delay by the use of this very simple model. Block diagram
was made like a fig.16. Simulation was carried out about this model.
Time series data of the simulated hemodynamic parameters, three
dimensional attractors and frequency spectrum was observed here. We had
used MATLAB for simulation and analysis.
Experimental result
Behavior of the hemodynamic parameters were simulated in the simple
electrical circuit model with a changes of the delay time r. Various
interesting phenomenon were shown in the simulated data. Only some
typical data were picked up and shown in the figures.
Figure 17 showed simulated time series data of blood pressure, RR
interval and stroke volume with time delay 0.6 s. Right side showed
reconstructed attractor of the simulated time series with BP, HR, and
SV. In the time series data of these parameters, all signals showed the
damping oscillation. The whorl which converges in a point was shown in
the reconstructed three dimensional attractor.
Figure 18 showed simulated time series data with time delay 1.5 s. In
the time series data of these parameters, all signals showed the
constant periodic oscillation. In the reconstructed three dimensional
attractor, the orbit converged and drew circle. This showed so called
the limit cycle attractor.
A little complex result with time delay 1.8 s was shown in figure 19,
simulated cardiovascular signals showed interesting time series
suggesting one set periodic oscillation with two mountains. When we
reconstructed these data into the three dimensional map, the orbit
converged and becomes track with two rounds.
And complex time series was shown with time delay 2.5 s, as we shown
in figure 20. Simulated cardiovascular signal showed complex oscillation
without any rule. We tried to embed these time series into the three
dimensions. As we shown in right side of figure, the orbit which isn't
accompanied with convergence is drawn.
We paid attention to the behavior with time delay 2.5 s. Time series
data and the power spectrum analysis of simulated heart rate
variabilities were shown in figure 9.
Time series wave form of HR becomes complex oscillation same as other
cardiovascular signal, too. There is frequency spectrum as the biggest
beak approximately 0.12 Hz with many mountains.
We tried to let initial value change at the time of delay 2.5 seconds
inconsiderably. As we shown in fig.10, a figure repeated time series
wave form of each HR, and it was written. A result draws time series
wave form of oscillation different at all mutually.
Discussion
By the use of information of an autonomic nervous system like
sympathetic nerve activity of the Mayer wave in the peripheral vascular
resistances, we can make feed back loop for an artificial heart
automatic control system. In this study, we make simulation model by the
use of the electrical circuit model. Behavior changes in the simulation
model altering delay time could be shown in figures. When delay in
simulation was small comparatively, a signal of everything showed
damping oscillation. The whorl which converges in a point was shown in
the reconstructed three dimensional attractor. And if we set up larger
delay in the simulation circuit, periodic oscillation, and then, period
doubling phenomenon were shown in the reconstructed attractor. After the
repeating of the period doubling according to the increase of the delay
time, we could get strange attractor suggesting the existence of
deterministic chaos, when we set up the delay time of 2.5 seconds in
this simulation model electrical circuit. This complex oscillation
didn't disappear even if delay was set up more than 2.5 seconds.
There is a point showing the characteristics, which showed
deterministic chaos, when tries to pay attention to this condition. At
first, frequency spectrum of the simulated heart rate variability showed
it. As we shown in the figure, power spectrum has a peak on the section
which wasn't harmonics composition. And a lot of another small
mountains were found. These many spectrums was considered to have made a
complex signal. This spectral pattern was, of course, one of the
characteristics of the deterministic chaos. Only harmonics composition
shows a peak in case of limit cycle.
The second was behavior of the simulated data at having let initial
value alter. As one of fundamental characteristics of chaos, there was
the sensitive initial value dependence. Actually it was clear if watched
a figure. We had tried to let alter the initial value a little and did
simulation. Time series wave form showed oscillation different at all.
This phenomenon backed up the existence of the deterministic chaos.
When a spectrum was tried to be paid attention to, a peak having the
biggest power in 0.12 Hz is found. This was, of course, a peak
equivalent to Mayer wave. Fluctuation of period for approximately 10
seconds was the character which was very special in the creatures. By
the use of model with this simple electrical circuit, this interesting
fluctuation was caused. In this simulation, we might be able to consider
that the delay would cause this phenomenon.
Non-linear curve, which was, of course, common in creatures, was
tries to be considered this time. From blood pressure, determination of
cardiac cycle and SV was performed with non-linear curve of sigmoid
style. From these two curves, the curve which decided flow from blood
pressure was drawn. In this simulation, operating range of blood
pressure was from 70 to 100 mmHg, which was similar to human body. The
section of graph had tried to be paid attention to. In this curve, this
range had drew curve of the convexity which just resembled a logistic
function. With this cause, ﾒstretchingﾓ and ﾒholdingﾓ might be happened.
And it is considered to have caused the condition which was chaotic.
However, of course, this electrical simulation model still includes
some problems, yet. Some important problems were shown below.
1. Complex cardiovascular system was simulated with simple electric circuit model.
2. Stroke volume (SV) was decided with arterial blood pressure.
3, Simulation only of the systemic circulation by left heart
It was thought that the first problems may be important. We need to
consider whether this model was able to simulate the original living
body and which degree. Model of baroreflex series is made from real data
of the creatures, and further simulation needs to be retried in future.
And SV depends on pressure of the atrium which is pre-load. In
electrical simulation model of this study, it is decided only by the
arterial blood pressure, which is afterload. And, of course, behavior of
the pulmonary circulation was important. Accordingly this baroreceptor
reflex model is not enough, yet. Baroreceptor reflex system of the
original living body will not be expressed in only this simple
electrical simulation model.
However, it is considered that chaotic dynamics may be caused with
simple three elements circuit with some delay. These results suggest
that we could achieve the chaotic dynamics like human body by the use of
feed back automatic control algorithm of an artificial heart system.
These results may be very important in future, if we consider the Age,
in which Artificial organs become very common, and QOL of Artificial
organs become important. So, studies will want to be continued with more
refined procedure.
人工心臓は今ではただ生命を維持するだけに用いられた時代は過ぎて、現在は患者様のＱＯＬを目標の一つにあげなくてはならない時代に入りかけてい
る。全身の循環を維持できても、患者様が何か行動を起こすたびに流量の不足から心不全症状をきたすようではＱＯＬを問う段階にはないことになる。ＮＹＨＡ
分類を考えるまでもなく、軽度の労作で心不全発作をきたすようでは、患者様は重症心不全段階から軽症心不全段階に移行しただけで、完治には程遠い。
カオス的なダイナミクスは低自由度の非線形システムに発生する数学的な概念であるが、数多い制御系が関与する心臓血管系に置けるダイナミクスに発生するカオスの成因は明らかにされているとは言い難い。
フィードバック回路には人間の統計データを元に相互の相関に非線形性を加えた。
東北大学は心臓移植認定施設に選ばれようとしている段階です。
大学においても研究所においても動物実験を施行することは益々困難になりつつあり、「動物実験を行うくらいなら、人間の臨床試験の方がよほど簡単だ！」などと言う冗談まで囁かれる程である。
この際、重要なポイントになるのは、候補物質の作用を確かめるために作成される疾患の動物実験モデルである。例えば抗生物質であれば、感染症のモデ
ルがin vivo, in
vitroで必要になる。培養で菌の増殖が抑えられればある程度抗生物質としての作用は証明できるが、生体に投与した場合の安全性はまた全く別の問題であ
る。更に生体に投与した場合、感染症を起こしている組織への移行も問われることは自明である。従ってこの二つの問題だけでも動物実験モデルの存在は不可避
になる。
例えばここでシミュレートした血圧反射を、動物実験で再現しようとすれば、心拍数を変えるだけでは純粋に現象を抽出できない。ペースメーカを用いて
心拍数を変えても、動物の心臓は液性因子の影響から独立ではありえないし、そもそもペーシングによる収縮自体が自然なものではない不自然さが残存してい
る。ペーシングリードの位置により、収縮のダイナミクスの形態が異なってしまうのである。 
カオス的な非線形ダイナミクスはフラクタル次元で定量化されるが、フラクタル次元の様々な計算法があり、情報量エントロピーから計算する方法論は情
報量次元と呼ばれるが本質的には、どの方法論で計算しても数学的にはフラクタル次元は一致する。すなわち、非線形ダイナミクスの情報量の定量化にはフラク
タル次元解析も有効な方法論の一つである。
これに対してある種の高血圧患者においては、その心拍変動のフラクタル次元が有意に低下している例もある。
キーワード:
参照:[ナノテク応用型人工臓器の開発と臨床応用に関する研究]