The study group consisted of 24 premenopausal women who were planed to undergo laparotomy because of symptomatic myomas. Sixteen of them were characterised with large myomas and 8 with small ones. The diagnosis of myomas was made by use of bimanual gynecologic examination as well as with both transabdominal and transvaginal gray-scale sonography. The transabdominal examination was necessary to adequately measure uterine size in women with large uterus. Transvaginal examination was performed with a woman in the lithotomy position. Myoma volume was expressed in cm3 and was calculated according to the formula length (cm) × depth (cm) × width (cm) × 0.5. A myoma was considered large if at least one of its diameters was > 5 cm; otherwise it was characterized a small one. If more than one myoma was found in the pelvis, the largest myoma was examined. The uterine arteries and myoma vascularization were visualized by the color Doppler technique. Blood flow velocity waveforms from both uterine arteries were obtained by placing the Doppler gate over the colour areas and activating the pulsed Doppler function. The main stem of the uterine arteries was examined lateral to cervix at the level of the internal os. The mean value from the PI obtained from the right and left uterine artery of each patient was recorded and correlated with the myoma volume and with the corresponding biomagnetic measurements. All women underwent hysterectomy or excision of the myoma and histologic diagnosis of a benign uterine myoma was made for all of them.
Biomagnetic recordings were obtained by a single channel second order gradiometer DC-SQUID (MODEL 601; Biomagnetic Technologies Inc., San Diego, USA) [5–7]. During the recording procedure the patient was relaxed lying on a wooden bed free of any metallic object so as to decrease the environmental noise and get better signal to noise ratio. The recordings were performed after positioning the SQUID sensor 3 mm above the exact position of myomas in order to allow the maximum magnetic flux to pass through the coil with little deviation from the vertical direction. Five points were selected for examination according to the myoma topography made by use of gynecologic and ultrasound examination. Point 5 was located at the very center of the myoma, whereas points 1–4 were located at the periphery of the examined area. The measured magnetic field was at the order of 10-12 Tesla. By convention the maximum of the 5 values was used when assessing each myoma. For each point 32 recordings of 1-second duration each were taken and digitized by a 12 bit precision analogue – to – digital converter with a sampling frequency of 256 Hz. The duration of the above recordings is justified because the chosen time interval is enough to cancel out, on the average, all random events and to allow only persistent ones to remain. The biomagnetic signals were band-pass filtered, with cut-off-frequencies of 0.1–100 Hz. The associated Nyquist frequency limit, with the above-mentioned sampling frequency, is therefore 128 Hz, which is well above the constituent frequency components of interest in biomagnetic recordings and avoids aliasing artifacts. Informed consent for the study was obtained from all the patients prior to the procedure.
Theory and algorithm
Nonlinear analysis was used to estimate the complexity index of the strange attractor characterizing the biomagnetic time series obtained from the patient. According to Grassberger and Procaccia [8], the dynamics of the system can be experimentally reconstructed from the observed biomagnetic time series Bi = B(ti) (i = 1, 2, ..., 8192) and the vector construction of Vi is given by the following equation:
Vi = {Bi,Bi+τ,...,Bi+(m-1)τ} (1)
This equation gives a smooth embedding of the dynamics in an m-dimensional phase space. The evolution of the system in the phase space – once transients die out – settles on a submanifold, which is a fractal set called the strange attractor. The strange attractor can be described by a geometrical parameter: the complexity index D. This parameter is related to the number of variables required to define the attractor within the phase space and it can be estimated from an experimental time series by means of the correlation integrals C(r,m) defined as follows:
where Θ (u) is the Heaviside function defined as Θ(u) = 1 for u>0 and Θ(u) = 0 for u≤ 0); m is the embedding dimension; n is the number of vectors constructed from a time series with N samples, and is given by the formula n = N-(m-1)τ (where τ is a delay parameter which is equal to the first zero crossing of the autocorrelation time of the biomagnetic signal); and B is a correction factor for spurious influences of autocorrelation [9]. The correlation integral C(r,m) measures the spatial correlation of the points on the attractor and it is calculated for different values of r in the range from 0 to rmax, where rmax is the maximum possible distance of two random selected points of the attractor of the selected time series. The rmax is equal to (m)1/2 (xmax -xmin), (assuming that xmax and xmin are the maximum and minimum recorded values in the time series). The complexity index D of the attracting submanifold in the reconstruction phase space is given by:
In the case of a chaotic signal exhibiting a strange attractor, there is a saturation value (plateau) in the graph of the function D vs. ln(r(2-r)). This value remains constant, although the signal is embedded in successively higher-dimensioned phase spaces. Using the above method the complexity index D of the selected time series was estimated for the biomagnetic activity of the large and small myomas.
