3.1.

Model Analysis of the Plasma Layer

Experimental delineation of bounding surfaces of the plasma layer is subject to significant uncertainty due both to practical limitations of imaging techniques and to the inherent roughness of the respective luminal and abluminal boundaries. This uncertainty can hamper one’s ability to estimate the functional extent of the plasma layer from optical imaging. Consequently, there is a growing interest in experimental, theoretical, and numerical studies of deterministic and probabilistic descriptions of such surfaces and of solutions of differential equations defined in the resulting domains.

Early attempts to represent surface roughness and to study its effects on the system behavior were based on simplified, easily parameterizable, deterministic surface inhomogeneities, such as symmetrical asperities, to represent indentations and semispheres to represent protrusions. Alternatively, one can use random fields to represent rough surfaces whose detailed topology cannot be ascertained due to the lack of sufficient information and/or measurement errors. Random representations of rough surfaces are conceptually appealing because of their generality and ability to incorporate experimental data. Moreover, such an approach enables one not only to make predictions of the system behavior, but also to quantify the corresponding predictive uncertainties.

The presence of uncertainty in delineation of uneven boundaries necessitates the development of new approaches for the analysis and numerical solution of differential equations defined on random domains. For example, it has been demonstrated that classical variational formulations might not be suitable for such problems, and finite difference approaches remain accurate only for relatively simple rectangular irregularities. The adoption of a probabilistic framework to describe rough surfaces makes even an essentially deterministic problem stochastic, e.g., deterministic equations in random domains give rise to stochastic boundary-value problems. This necessitates the search for new stochastic analyses and algorithms. For example, one of the very few existing numerical studies have employed traditional Monte Carlo simulations, which have turned out to be so computationally expensive as to become impractical.

Both data analysis and modeling of flow and transport in the plasma layer can be carried out within the probabilistic computational framework^{8, 9, 10} that is applicable to a wide class of deterministic and stochastic differential equations defined on domains with random (rough) boundaries. A key component of this framework is the use of robust stochastic mappings to transform an original deterministic or stochastic differential equation defined on a random domain into a stochastic differential equation defined on a deterministic domain. This enables employing the well-developed theoretical and numerical techniques for solving stochastic differential equations in deterministic domains.

4.1.

Blood Vessel Dimensions by Photometric Scanning

4.2.

In-Line Measurement of Lumen by Image Shearing

Current technological advances resulting in the incorporation of microscopic techniques and television scanning system techniques, enable measure of dynamic changes of microvessel lumen. Although, these methods depend on the light intensity, a comparatively simple system, using the principle of image splitting, enables accurate in-line computation of dynamic changes of the microvessel lumen and wall in rapid sequence. The accuracy of measurement relies on direct visual recognition by a trained observer of the lumen and structures of the wall in the microvessel and the unparalleled ability of humans to match the alignment of two parallel lines.¹¹

The method has been successfully applied in numerous in vivo studies of the microcirculation. The process of measurement with both instruments consists of setting the two images edge to edge, which can be done with great precision. The amount of shear is noted, the images then are crossed over (passing through the position of coincidence), and the amount of shear is noted again. The algebraic difference between the two shears gives the diameter of the object. In practice, the video data that are displayed on a monitor can be “sheared” along the raster lines by delaying the start of the raster line scan by a controllable period. The result is that the image encoded in the delayed raster lines appears shifted and therefore "sheared" and laterally displaced by an amount proportional to the adjustable period, which becomes a measure of the image dimension in a manner analogous to the micrometer adjustment setting of an optical image-splitting eyepiece. Shearing can be done manually or by an automatic feedback control.¹² Other automated methods for measuring diameter changes are also available, however they require that vessel walls present parallel images, and vessels that are straight.¹³

4.3.

Dynamic Diameter Measurements and Characterization

Small arteries and arterioles, under normal conditions, and in the absence of anesthesia visualized in vivo, present a spontaneous, rhythmic activity of contraction and relaxation termed vasomotion.⁴⁵ The precise physiological significance of this effect is not well understood, and it is not completely established whether it is a characteristic of a normal tissue or it is a response of the circulation to physiological stress.⁴⁶ It is, however, an activity present in most organic structures endowed with smooth muscle. It appears to originate at bifurcations that act as pacemakers for this activity, however, these pacemakers are not synchronized and their characteristic frequency is inversely proportional to the diameter of parent and daughter vessels, varying from up to $15\text{cycles}∕\min$ in the smallest arterioles to about $2\text{to}3\text{cycles}$ in the larger vessels, making this activity accurately recordable by manual image shearing.

The quantitative characterization of the time-dependant features of the records of diameter variations is in principle readily accomplished by spectral techniques, because of the recurrent, quasiperiodic nature of the contractions and relaxations. The Fourier spectral series analysis would be the standard technique utilized in such a situation; however, it has certain disadvantages that make its use undesirable. First, Fourier analysis should be used with data where the record length is infinite. In most cases, records are finite, therefore, it is necessary to arbitrarily specify the nature of the data outside the recorded interval, using the assumption that either the data record is repetitive or that the data record is zero outside the recorded interval. When data are assumed to be zero outside the period of interest, the result is equivalent to multiplying an infinite record by a rectangle function. As a consequence, the shorter the data record is, the greater becomes the deviation between the actual spectrum and the calculated spectrum. This is termed energy leakage, where energy from higher frequencies “leak” into lower frequency bands. Similarly, if we assume a repetitive record, frequencies related to the extent of the record appear in the power spectrum, and attention must be given on how the successive repetitive records are connected. A second problem inherent to the Fourier analysis is that resolution is prescribed beforehand. However, it is possible that two frequencies may be present in a signal, and that they differ in frequency by a smaller amount than can be resolved by the method.

A method that is better suited for the analysis of vasomotion is the "Prony" spectral analysis technique¹⁴ that models a finite data record as the sum of a finite number of nonharmonically related sinusoids in uncorrelated noise. This is different from the Fourier analysis, which prescribes the frequencies as harmonics. Also, there is no assumption about the data outside the actual record. The Prony method attempts to make a best fit of a finite number of sinusoids over the described region, and resolution is not dependant on the data length, but is a function of the accuracy of the method of calculation. Because the data record length is no longer an important factor, the Prony method is very effective for short data records.

Any method for obtaining the power spectrum of data that contain noise yields results that are to some extent approximate, in which the degree of approximation is in part dependant on how well the data are related to the properties of the model that the spectral technique is designed to characterize. In the case of vasomotion, we assume that the measured time-dependant effects are the consequence of the activity of a finite number of discrete and localized pacemaker-like groups of cells.⁴⁷ This implies a narrow-band process, where the energy of the time-dependant effects is concentrated at discrete frequencies, further suggesting that the vasoactive effects propagate along the vessels, so that at any given location we observe the additive result of the activity of several sources.

The Prony method is mathematically better suited to the analysis of the kind of time-dependant effects that characterize vasomotion, but it has the defect that from the point of view of the amount of computation required, the algorithm is less efficient than the Fourier method, which has prescribed frequencies and requires only the calculation of the amplitudes and phases at those frequencies. The Prony method first searches for the frequencies, and then performs the required operations to determine amplitudes and phases. However, for short data records, even though the computations are longer, the accuracy and resolution in the spectra is notably better.

A procedure to obtain a limited number of sinusoids whose sum best fits the given data consists of calculating the Prony spectrum of frequencies, amplitudes, and phases, and then computing the correlation coefficient between the waveform reconstructed with the data from the Prony spectrum and the original data.¹⁵ A first-order size approximation contains one sinusoid, and the correlation coefficient is computed at each increment of order size. If the correlation coefficient is larger than the previous order, then this solution is kept and the prior solution is discarded. A maximum is reached by increasing the order size, and a maximum correlation will be seen at a specific order. If the correlation is greater than some acceptable level, a solution has been found. If the maximum correlation is less than the acceptable level, the salient features of the signal have not been determined and the Prony method is not appropriate for that particular signal, as the assumption of narrow bandedness has been violated. As a rule, diameter records of $2\text{to}3\min$ , containing 200 to 300 individual data points, show frequency spectra with three to eight separate frequencies, which when utilized to reconstruct the original waveform, fit the data with a correlation coefficient of the order of 0.85 to 0.90, which is significant with $p<0.001$ .

5.1.

Dual Window Correlation

Blood flow in microscopic vessels is also a parameter that is derived from RBC velocity and diameter measurements obtained optically. A widely used technique is based on determining the delay between photometric signals generated from two axially spaced detectors, measured by cross-correlation as a function of time delay according to the approach developed by Wayland and Johnson.¹⁶ This method is also known as the dual-slit method. The precision of the method is primarily dependent on the quality of signals, namely, their similarity, a quality directly related to the previously described correlation length.⁴⁰ The dual-slit method is relatively insensitive to microscope focus, and yields the same result regardless of which part of the image of the flowing blood is in focus. The measured centerline velocity is corrected by a factor with a numerical value of 1.6 for vessels in the $15\text{-}\text{to}80\text{-}\mu \mathrm{m}$ -diam range, and decreases gradually to the value of 1.0 for narrow capillaries, where cells move single file.¹⁷

The dual-sensor method has also been implemented in a video configuration, enabling us to record images of the microvessel in real time, thus providing a synchronized record of diameter and flow velocity.⁴⁸ This technique is primarily applicable to capillaries, vessels where the optoelectronic signatures caused by the passage of RBCs are well differentiated from the noise.

The principal problem of RBC velocimetry is the method for conversion of the optical signature of the passing RBCs into an electronics signal. The use of nonideal optical signal transductors, the complexity of the microvascular image, and the efficiency of the algorithm that computes the intradetector transit time determine the limits of accuracy of the technique. Continuous progress in the development of optical detectors has improved signal quality; however, this advancement is curtailed by the difficulty in obtaining dedicated correlators that compute the delay between signals as a continuous process.

Continuous-correlation dedicated computers were developed in the 1970s by Princeton Applied Research (PAR, Boston, Massachussets) and a little later by Hewlett Packard (HP, Menlo Park, California). A general-purpose correlator with these features has been developed by TSI Incorporated (Shoreview, Minnesota), however, it has not yet been used in blood measurement applications. Vista Electronics (Ramona, California), manufactures a dedicated correlator for blood flow measurements. These instruments differ on how they delay the upstream signal relative to the downstream signal: the PAR and the Vista instruments use a digital delay line, while the HP instrument used a tapped sound transmission line. However, both instruments calculated the cross-correlation function continuously by means of hard-wired multipliers that process the data at each of the delay increments. This scheme ensures the complete utilization of all the available data. By contrast, modern schemes using general-purpose computers compute the correlation function sequentially from discrete record lengths, causing the computed correlation function to contain products from uncorrelated data.

The interdetector optical signature transit time is currently optimally measured by the variable time base tracking cross-correlation custom built by Vista Electronics (Ramona, California). This varies the frequency that controls the delay line for the upstream signal by means of a servo loop that maintains the maximum delay at a fixed, central location in the delay line. An interesting feature of this process is that the resulting frequency to attain this goal is directly proportional to the velocity of the transiting RBCs, which can be readily seen by noting that both functions are in terms of inverse time.¹⁸

5.2.

Other Methods

Several other approaches have been taken to maximize the utilization of the available data. One of these uses spatial correlation, where optical scans are made along the axis of the blood vessels under investigation. When two of these signals are obtained in succession and transformed into electronic format, they can be cross-correlated to determine the displacement that has taken place in the period between scans. A different approach is the direct frequency analysis of the photometric signals. This can be implemented by projecting the image of the moving RBCs through a grating, which generates a frequency that is proportional to the velocity and the spacing in the grating.¹⁹ The grating-photodetector configuration can be replaced by a linear array of photodetectors⁴⁹ or a sequence of four video photometers, which yield the difference between the photometric signals from two contiguous windows, thus forming two signals, each being the difference of time contiguous signals.²⁰ This constitutes a prespatial differentiation of the signals that significantly improves the cross-correlation measurement of transit time relative to the one obtainable from only two windows, has an improved frequency response, and is less sensitive to motion in the preparation.

5.3.

Lased Doppler Velocimetry

A method that provides an absolute measurement of blood flow velocity is based on the measurement of Doppler-shift frequency spectra of laser light scattered from RBCs in flowing blood. This method was first reported by Riva and consists of illuminating the microvessel of interest with visible laser light and analyzing the power spectrum of the backscattered light.^{21, 22} This spectrum presents two distinct regions identified by their different amplitudes, where the large-amplitude region corresponds to the Doppler-shifted backscatter. A special characteristic of this methodology is that the Doppler-shifted backscatter from the particles moving in a tube where velocities have an approximately parabolic distribution has constant amplitude. This enables a trained operator to readily identify the maximum velocity corresponding to the centerline of the flow.

Simultaneous detection of the RBC scattered light from two directions separated by a known, fixed angle makes the methodology independent of the actual angular relationship between the direction of blood flow and that of each of the two backscatter light beams being analyzed.

Velocity measurements based on this methodology do not appear to have been utilized beyond ophtalmographic investigations primarily because the determination of the maximal frequency of the Doppler-shifted spectrum requires visual inspection by experienced observers. Another problem is that accurate laser Doppler measurements of the centerline velocity depend critically on the centering of the incident illumination on the target vessel.⁵⁰

5.4.

Cellular Nonlinear Network (CNN) Technology

CNN technology may provide the next development for data processing associated with the analysis of microvascular phenomena in a network. These networks are made by means of arrays of identical systems called cells whose local interaction can be programmed.^{51, 52} The CNN structure is particularly suitable for the type of spatially distributed input processing characteristic of image analysis. Each cell is, in practice, the processing unit for an element of the distributed input and the connection of cells can be configured to produce an output that is a function of conditions in neighboring pixels, leading to the design of filters and image enhancing features. The dynamics of CNN circuits are controlled by equations that describe the state of each cell, its output, etc. The time constant for each cell is typically of the order of ${10}^{-7}\mathrm{s}$ .

The significance of the CNN approach is that the system can be programmed to perform operations such as image subtraction, thresholding, and particle tracking. The application of such an algorithm to the capillary network highlights capillaries in which there is RBC flow, providing a means for adding maximal contrast to the path of each RBC. Interrogating neighboring cells enables us to form a display showing in which portions of the tissue there is RBC motion, thus delineating the microvessels. Finally, identifying contiguous pixels in which there is image motion delineates the capillaries in a machine interpretable context, which can the be used to count capillaries and determine functional capillary density.^{53, 54}