Results of the foregoing paragraph showed that chaotic dynamics in circulation may be generated from the peripheral vascular properties mediated by the sympathetic discharges. If we consider the clinical application of these results, autonomic function mediating cardiovascular function may be the suitable target. In the patients with Diabetes Mellitus or Patients with myocardial ischemia showed cardiac denervation. It is an important problem for these patients. In the clinics, we used heart rate variability to analyze the autonomic function, however, there are some problems.
Clinical medicine goes to the direction of low invasion with shorter time. For example, the therapy of the serious coronary artery disease was only a bypass operation a while ago. PTCA by the catheter intervention became the mainstream now. Operation of low invasion by abdominal scope or laparo scope attracts attention in abdominal surgery. It is such direction even in the invasion care as surgery. In diagnostics, more correct information is demanded by more indirect procedure. Accordingly, it is more difficult when took assumption to apply it to a patient, because non-linear dynamics must be pursued from indirect index.
For example, it is heart rate variability. ECG is the biological electrical phenomenon which can measure with the circuit which is simple and easy. Information of the pathophysiology which contained myocardial ischemia from wave form can be detected. This methodology is, of course, clinically used broadly. From a point of view to detect correct biological system information, this methodology doesn't always fill necessary requirements in completeness. The R-R interval used in the research of the heart rate variability is taken as example. It is that R wave of ECG accords with systole of a heart that become assumption here.
Will this be right?
Unfortunately, accumulation of data in clinical medicine as before suggests that there can be situation of the pathophysiology that this prerequisite isn't right. The state of a disease of electromechanical dissociation may be the typical example.
Correct instrumentation becomes the assumption in a research of the non-linear dynamics of the hemodynamics. Because systole of a heart may not be always accorded with ECG in clinical medicine, the method of RR interval was excluded. Firstly, movement of a heart must be detected to examine the alteration of dynamics of the hemodynamics. However, it isn't always simple to observe behavior of a heart directly. There is procedure of image analysis by conductance catheter, nuclear medicine, MRI or computed tomography. But there is some problems in the non-invasiveness or the observation of long time, either. These methodology are not suitable method which can measure long time with stability.
In this research, instrumentation of the stroke volume by the use of the cardiac ultrasonography was attempted. This methodology was used for analysis of non-linear dynamics that hemodynamics had. From M-mode image of left ventricle in ultrasound heart diagnostics, time series curve of systole dynamics of left ventricle volume is provided. Tracing image of M-mode inside diameter was input in a computer with an image reader. Based on the theory of Takens, time series curve provided was embedded to phase space of high dimension, and the nonlinear analysis was attempted. In other words, value in time t on time base, X(t) is taken in x axis. X(t+T) which is value after T is taken in Y axis. Furthermore, value of X(t+2T) after 2 T is taken in Z axis. In this way, plotting were performed.
A change of trajectory with passage of time in phase space is observed. Of course, it is determination method of time lag that become important conclusively here. There are a lot of procedures proposed in the past, procedure sought for from fundamental period, procedure sought for from attractor, and procedure of auto correlation function. An example of auto correlation function of left ventricle systole dynamics sought for from echo cardiogram in figure 1 is shown.
It is a point of 0.2 second that cross zero axis at first. It is the first point in time when doesn't hold correlation in oneself most. Accordingly procedure to do value after 0.2 second with an amount of the other dimension is applied well.
Attractor of systole of a heart sought for with such procedure is shown in fig.25. The upper section is person who is not physically handicapped, and the lower berth is myocardial infarct patient having the past history of left anterior descending artery occlusion.
At the first glance, we understand that attractor becomes forma moderated. Even two dimensional phase plane is the findings which seems to get possible to be described. Alteration to lower dimension of dynamic system controlling attractor is suggested.
Accordingly application to quantitative diagnosis is prospective. Abnormality of innervation in myocardial infarct patient is reported at the past by procedure of nuclear medicine. It is thought that more interesting result is obtained by comparison with the current procedure.
By the further application of these nonlinear mathematical analyzing methodologies, we can anticipate larger development of the diagnostics for the patients with cardiovascular regulatory system disturbances, especially from the view point of whole system.
The mathematical methodology of this paper was discussed in the workshop "Various approaches to the complex systems" held at the International Institute for Advanced Studies in Kyoto, Japan.
The authors thank Mr. Kimio Kikuchi for experimental cooperation, Miss Rie Sakurai, Mrs. Taeko Akabane, and Mrs. Hisako Iijima for their excellent assistance and cooperation.
This work was partly supported by a Grand-in-aid for Encouradgement of Young Scientists (70241578), Research Grand for Cardiovascular Diseases from the Ministry of Health and Welfare, and Prigram for promotion of Fundamental Studies in Health Sciences of the Organization for Drug ADR relief, R&D promotion and product review of Japan.