BACKGROUND OF THE INVENTION Vascular diseases are often manifested by reduced blood flow due to atherosclerotic occlusion of vessels. For example, occlusion of the coronary arteries supplying blood to the heart muscle is a major cause of heart disease. Invasive procedures for relieving arterial blockage such as bypass surgery and balloon dilatation with a catheter are currently performed relying on estimates of the occlusion characteristics and the blood flow through the occluded artery. These estimates are based on measurements of occlusion size and/or blood flow. Unfortunately, current methods of occlusion size and blood flow measurement have low resolution, are inaccurate, are time consuming, require expertise in the interpretation of the results and are expensive. Thus, decisions on whether or not to use any of the blockage relieving methods and which of the methods should be used, are often based on partial information. The evaluation of therapeutic success is also problematic, where both occlusion opening and stent position have to be evaluated.

Pressure, flow and geometry are three variables often measured in the cardiovascular system. Recent progress in probe miniaturization, improvements of the frequency response of probe sensors and computerized processing have opened a whole new range of pressure, flow and geometrical measurements that have been previously impossible to perform.

Typically, the physician first selects the appropriate treatment method from among medication therapy, transcatheter cardiovascular therapeutics (TCT), coronary artery bypass grafting (CABG), or non-treatment. Atherosclerotic lesions may have different characteristics. Some lesions exhibit a variable degree of calcification while others have a fatty or thrombotic nature. Lesion characteristics together with vessel condition proximal and distal to the lesion are the major factors for determining the therapeutic procedure needed. Recently, increasing numbers of patients have been directed toward TCT. TCT starts with an interventional diagnosis procedure (angiography), followed by the treatment of the patient with medication therapy, CABG or continuation of the TCT procedure with adequate interventional treatment.

Numerous methods are currently available for treating various lesion types. Some of these methods which are sequenced from"softer"to"heavier", relating to their ability to open calcified lesions are as follows; percutaneous transluminal angioplasty (PTCA), "Cutting balloon"angioplasty, directional coronary atherectomy (DCA), rotational coronary atherectomy (RCA), Ultrasonic breaking catheter angioplasty, transluminal extraction catheter (TEC) atherectomy, Rotablator atherectomy, and excimer laser angioplasty (ELCA). Often, stents are placed within the lesion so as to prevent re-closure of the vessel (also known as recoil). If the stent is malpositioned, it interrupts the flow and may initiate restenosis.

Lesion characteristics, together with vessel condition proximal and distal to the lesion, are used to determine the medically and economically optimal treatment method or combination of methods of choice. Angiography has been the main diagnostic tool in the catheter laboratory. The physician uses the angiographical images in order to identify and locate the lesions, evaluate the occlusion level (percentage of normal diameter) and qualitatively estimate the perfusion according to"thrombolysis in myocardial infarction" (TIMI) grades, determined according to the contrast material appearance. TIMI grades 0,1,2,3 represent no perfusion, minimal perfusion, partial perfusion and complete perfusion, respectively.

Among the more sophisticated diagnostic tools are qualitative coronary angiography (QCA), intravascular ultrasound (IVUS) intravascular Doppler velocity sensor (IDVS) and intravascular pressure sensor (IPS). QCA calculates geometrical properties from angiographic images, in image zones that are chosen by the physician.

IVUS provides accurate geometrical data regarding cross section area and accurate information regarding the vessel wall structure and composition. IDVS provides velocity measurements, enabling discriminating various degrees of occlusion according to coronary flow reserve (CFR) criteria. IDVS suffers from inaccuracy problems resulting from positioning errors within the vessel. IPS provides pressure measurements enabling discriminating various degrees of occlusion according to the FFR (fractional flow reserve) criteria. Angiography and the sophisticated techniques discussed above may be employed prior to and after the therapeutic procedure (the last for the evaluation of the results and decision about correcting actions). Among the noninvasive diagnostic devices for blood flow rate and vessel walls properties there exists an apparatus named "Vascular Echo Doppler"developed at the"Institute of fundamental Technological Research-Polish Academy of Sciences"Swietokrzyska 21,00-049 Warsaw, Poland.

The device is used for Noninvasive Examination of Input Vascular Impedance, elasticity of Arterial Walls and volumetric blood flow. (Impedance is defined here as the ratio between blood pressure and blood flow in the linear approximation of blood flow). The apparatus is claimed to be a computerized Ultrasonic system consisting of a double ultrasonic probe locates at the same position along the artery. Both probes operate simultaneously, the first probe in the Echo-mode measures instantaneous vessel diameter and the second probe is in a CW-Doppler mode and measures blood flow (i. e., blood velocity). Being a single position probe, elasticity of the artery cannot be calculated by vessel diameter alone. It is claimed that it is possible to infer systolic and diastolic pressure in other parts of the body (e. g. common carotid) from pressure measurements in the brachial artery. (Pol et al., 1994). Assuming this correlation always valid, pressure and vessel diameter are known at the same location, thereby giving the value of the compliance or distensibility at the same location.

The need for extraneous pressure measurement is a disadvantage since the use of the brachial pressure curve might not be useful for distant or small arteries (Nichols and O'Rourke, 1998 chapter 8).

SUMMARY OF THE INVENTION The invention discloses a method and devices for determining volume flow, blood velocity profile, artery wall properties of a tubular conduit system and stenosis identification and localization by introducing an artificial pressure or flow excitation signal (a single signal or multiple signals) into the blood vessel (or in any other tubular flowing fluid conduits), and or using natural heart beat signals measurement and analysis of the vessel wall displacements or vessel diameter changes.

This invention provides an apparatus, and system for determining volume flow, blood velocity profile, and/or artery wall properties of a tubular conduit system and stenosis identification and localization, said apparatus comprising: a means for generating an artificial pressure signal or an artificial pressure wave velocity; a means for measuring artery diameter or displacement resulting from said artificial pressure signal or an artificial pressure wave velocity or using natural heart beat signals; a processor unit operatively connected to said measuring means; a program for controlling the processor unit, wherein said processor unit is operative with said program to receive said signal which is represented of the measure of said artery diameter or displacement resulting from said artificial pressure signal or an artificial pressure wave velocity or using natural heart beat signals and determine volume flow, blood velocity profile, and/or artery wall properties from analysis of the vessel wall displacements or vessel diameter changes.

In one embodiment the processor unit is operative to select a method from a plurality of methods to generate artificial pressure signal or an artificial pressure wave velocity or using natural heart beat signals. In another embodiment, the tubular conduit system is a blood vessel system and said processor unit is operative to detect changes in arterial characteristics. In another embodiment, the processor unit is operative to determine

volume flow. In another embodiment, the processor unit is operative to detect stenosis. In another embodiment, the processor unit is operative to determine blood velocity profile.

In one embodiment, the measuring means is by ultrasound or magnetic motion sensor.

In another embodiment, the measuring means is by ultrasound. In another embodiment, the measuring means is by magnetic motion sensor. In another embodiment, the means for generating a pressure signal is in by a non-invasive means. In another embodiment, the means for generating a pressure signal is by an exciter. In another embodiment, the non-invasive means for generating a comprises a mechanical and/or electrical means. In another embodiment, the mechanical non-invasive means for generating a comprises applying pressure to the carotid.

In one embodiment, the signal is a plurality of discrete signals said processor unit is operative to sample said discrete signals and receive pressure wave velocity data. In another embodiment, the processor unit is operative to perform a single pressure function using said discrete signals and said pressure wave velocity data.

This invention provides a method for determining volume flow, blood velocity profile, and/or artery wall properties of a tubular conduit system and stenosis localization, comprising: generating an artificial pressure signal or an artificial pressure wave velocity or using natural heart beat signals; and measuring artery diameter or displacement resulting from said artificial pressure signal or an artificial pressure wave velocity; thereby determining volume flow, blood velocity profile, and/or artery wall properties of a tubular conduit system.

In one embodiment the processor unit is operative to select a method from a plurality of methods to generate artificial pressure signal or an artificial pressure wave velocity. In another embodiment, the tubular conduit system is a blood vessel system and said processor unit is operative to detect changes in arterial characteristics. In another embodiment, the processor unit is operative to determine volume flow. In another

embodiment, the processor unit is operative to detect stenosis. In another embodiment, the processor unit is operative to determine blood velocity profile.

In one embodiment, the measuring means is by ultrasound or magnetic motion sensor.

In another embodiment, the measuring means is by ultrasound. In another embodiment, the measuring means is by magnetic motion sensor. In another embodiment, the means for generating a pressure signal is in by a non-invasive means. In another embodiment, the means for generating a pressure signal is by an exciter. In another embodiment, the non-invasive means for generating a comprises a mechanical and/or electrical means. In another embodiment, the mechanical non-invasive means for generating a comprises applying pressure to-the carotid.

In one embodiment, the signal is a plurality of discrete signals said processor unit is operative to sample said discrete signals and receive pressure wave velocity data. In another embodiment, the processor unit is operative to perform a single pressure function using said discrete signals and said pressure wave velocity data.

This invention provides a method of determining the geometrical shape of the stenosis.

Such determination as provided for herein is determined by comparing the pressure signal proximal to the stenosis to the pressure signal distal to the stenosis so as determine the geometrical shape of the stenosis. In another embodiment the reflection method as disclosed herein is determined and then based on the reflection the geometrical shape of the stenosis is determined.

This invention provides a method and devices may also serve for evaluating the success of medical treatment. For example tracking sufficient opening for determining volume flow, blood velocity profile, and/or artery wall properties of a tubular conduit system occlusion or malpositioning of a stent. It may also serve for the characterization of vascular bed, downstream the vessel.

The present invention provides a method for further analysis of the response to the excitation signal yielding a quantitative determination of elastic properties of blood vessel walls for characterizing, inter alia, the distensibility and the compliance of lesioned and non-lesioned parts of blood vessels. The derived elastic properties may be further used to determine the degree of calcification of lesioned and non-lesioned parts of blood vessels.

BRIEF DESCRIPTION OF THE DRAWINGS The present invention will be understood and appreciated more fully from the following detailed description taken in conjunction with the appended drawings in which like components are designated by like reference numerals: Figure 1 illustrates an ultrasound measuring system for measuring vessel wall displacements or vessel diameter changes caused by heartbeats pressure signals.

Figure 2 illustrates an ultrasound measuring system for measuring vessel wall displacements or vessel diameter changes caused by artificial pressure signals Figure 3 illustrates an exciter use for generating artificial pressure signal inside a blood vessel.

Figure 4 illustrates an in vitro test apparatus used to verify the analytical results Figure 5 is a zoom on part of the in vitro test apparatus illustrated in Fig. 4.

Figure 6 is a schematic description of the in vitro test apparatus.

Figure 7 illustrates the Magnetic Motion Sensor incorporated within the in vitro test apparatus and used to measure tube wall displacements caused by pressure signal.

Figure 8 illustrate the MMS together with the in vitro test apparatus.

Figure 9 illustrates the test set up used in the experiment for flow determination.

Figure 10 illustrates an in vitro test performed in order to determined the distensibility of the latex tube.

Figure 11 illustrates the results of the in vitro test for determining the distensibility of the latex tube.

Figure 12 illustrates test results of pressure measurements versus MMS output voltage.

Figure 13 illustrates the pulses measured by the MMS in experiment for determining pressure wave velocity.

Figure 14 is a graph illustrating comparison of the volume flow rate as a function of time as computed using model 2 and the flow as obtain from flow meter measurements.

Figure 15 is a graph illustrating the tube wall shear stress as a function of time as calculated from data obtained from two experiments with different flow rate.

Figure 16 illustrates flow axial velocity profile as calculated from the data obtained from an in vitro test.

Figure 17 is a plot of computed axial centerline velocity using model 1.

Figure 18 is a graph illustrating comparison of the volume flow rate as a function of time as computed using model 1 and the flow as obtain from flow meter measurements.

Figure 19 illustrates comparison between the flow calculated from model 1 and 2for two different flow rates.

Figure 20 illustrates a test setup for identifying and localizing stenosis.

Figure 21 is a plot of the proximal and distal signal measured by the MMS in an in vitro test for identifying and localizing of a stenosis.

Figure 22 is a zoom on the peaks of the pulses illustrated in Fig. 21.

DETAILED DESCRIPTION OF THE INVENTION The present invention gives a method for deducing the volume flow rate, velocity profile, properties of artery walls (compliance and distensibility) and stenosis detection and localization. The method uses measurement of blood vessel cross section area (or diameter if the artery is quasi-circular) as a function of time. Then pressure and pressure wave velocity are derived from vessel area (or diameter) measurements enabling the determination of flow, wall properties and stenosis location. The method is designed to serve as a diagnostic tool for the physician when emphasis is put on optimizing measurement duration, computer resources and data processing time.

Arteries distend during systole and store volume energy during this phase of the cardiac cycle thereby limiting the increase in blood pressure. The local area of the artery, at a point x is a function of time A (x, t) directly dependent on the Pressure pulse. It has been established that distension is practically normal to the axis of the artery since longitudinal motion is strongly hindered (strong tethering), and that pressure is constant in a cross-section so that pressure is a function of x and t alone. Also the local instantaneous area is a function of the instantaneous local pressure which in mathematical terms can be written as A (x, t) =f (P (x, t)); the function f determining the elastic properties of the artery (f is monotonically increasing). Several forms have been proposed for f that will be described herein below. Thus, knowing the area A (x, t), or diameter, D (x, t), is equivalent to knowing pressure.

In an elastic tube, there are two agencies for driving the flow. One is the pressure gradient (when pressure is transversally constant) and the other is the pulsatile motion of the tube, which acts on the fluid through the no-slip condition enforcing zero fluid velocity (axial, radial and circumferential) at the wall. Hence, by measuring areas and pressures at Nx points xi i=l... N, ; long a short segment of an artery of length L, at time intervals t+jTs j=O... NT for at least one cardiac cycle, thereby producing a matrix A (xj, tj), it is possible to reconstruct with reasonable accuracy the relevant flow and wall elastic properties..

Flow, stenosis location and vessel wall properties are computed from geometrical measurements of blood vessel (wall displacement or artery diameter). Vessel diameter is measured using an ultrasound device or any other noninvasive device as for example described below. If an ultrasound apparatus is used, it must have a probe with at least two ultrasound lines at two different neighboring locations with known distance between them. The probe operates the two lines simultaneously in Echo-mode and measures vessel diameter. Pressure is derived from diameter measurements. The data is processed to yield two diameters which are functions of time.

Reference is now made to Fig. 1 illustrating the measuring system. An ultrasound-measuring device (1) is attached to a processor unit (3) and measures changes in blood vessel diameter (2) as a function of time. The changes in diameter are due to pressure pulses caused by heartbeats. Measurements are performed at two neighboring points. A stenosis (4) might be present, distally, in which case it will be detected. From these geometrical measured data, flow, stenosis location (if present) and artery wall properties are computed.

In another embodiment the displacements of artery walls may be determined using more than two measuring points along the artery. In such a case P (x, t) is derived from the multi number of measurements.

In another embodiment the displacements of artery walls may be determined using ultrasound measurement of a single artery section. In such case, a second measurement of another section, located at a known distant from first section is performed later. Then time synchronization of the measured results has to be performed.

In another embodiment the displacements of artery walls may be determined using Magnetic Motion Sensor (MMS) which measures the vessel wall displacements. Such a device capable of measuring wall displacements of latex tube is incorporated within an in vitro test system. The laboratory MMS device is illustrated in Figs 7 and 8. A device, based on the same principles, may be adjusted for clinical purposes and used for measuring vessel wall displacements. This is possible in cases of arteries located close enough to the body external skin, such as the carotid.

In another embodiment an exciter (a short duration pulses generator) may be used.

Reference is now made to Fig. 2 illustrating this embodiment. The exciter (5) generates an artificial pressure pulse that propagates along the vessel. The pressure pulse is introduced into the vessel by a catheter (6). The changes in vessel diameter (2) caused by the excited pressure signal going forward and the reflected signal coming backward from a stenosis (4) are measured by an ultrasound device (1). The changes in diameter are better detected and measured than those caused by heartbeats. This improves the accuracy of measurements and results.

Reference is now made to Fig. 3 demonstrating such an exciter. The exciter is a device for the excitation of the pressure waves inside the tube (catheter). It consists of a hammer (7) and conical chamber (8). Low voltage is applied to the solenoid (not shown) that pushes the weight (hammer). This weight strikes the membrane (9) of the conical chamber thus initiating a short pressure pulse. The opening of the cone is connected to a catheter (6). In the initial state the chamber and the catheter are filled with the fluid. The movement of the membrane is allowed only in one direction, so only positive pressure pulse is produced. The membrane returns to its initial position by the effect of a spring.

Displacement of the membrane and pressure wave generation could be achieved as well by using an actuator based on piezoelement or other pressure generating device.

In one embodiment, the methods provided herein are based on introducing an artificial pressure signal into the blood vessel. In one embodiment the pressure signal originates from a pressure signal generator (PSG). For example, a PSG of the type suitable for this purpose is a"blood pressure systems calibrator"model 601 A, commercially available from Bio-Tek Instruments Inc., Highland Park, Box 998, Winooski, VT-05404-0998, U. S. A. Catheterization laboratory injection systems are known to those skilled in the art. For example, a system of the type suitable for this purpose is a"Mark V Plus Injection System"from Medrad, inc. 271 Kappa Drive, Pittsburgh, PA 15238-2870 U. S. A. Other examples include but are not limited to the following: a pressure signal generated within the catheter or in its distal tip (e. g. piezoelectrically or by another form of energy burst introduction e. g. AcolysisSystem, ultrasound thrombolysis selective lysis of fibrin, by Angiosonics Inc., NC, U. S. A.); is by the movement of an hydrodynamic surface, activated either manually or by a special mechanism (e. g catheter used for removing malpositioned or embolized stents, for example Amplatz Goose Neck Snare GN 500 and Microsnare SK200 from Microvena corporation, Minnesota USA and catheters which prevent plaque debris from moving downstream); a pressure signal caused by an external controlled pressure applied on an organ, transmitted into a pressure signal within the vessel; a pressure signal caused by a non-invasive energy transmission into the vessel (e. g. ultrasound) in which the artificial pressure/flow signal may be either controlled or measured (within the catheter or the vessel).

In one embodiment, the methods provided herein are based on using natural heart beat signals measurement and analysis of the vessel wall displacements or vessel diameter changes.

In another embodiment the signal generator is a pressure signal generator; said signal sensor is a pressure signal sensor; and said processor unit is operative to receive a heart

beat signal; and synchronize receipt of said probe signal with said heart beat signal. As contemplated herein, the signal sensor includes at least two sensing transducers disposed in spaced apart relation. The signal sensor is movable between at least two positions relative to said tubular conduit system and said processor unit is operative to calculate a pressure wave velocity from signals received from said two positions. In another embodiment the signal sensor includes a signal conditioner. The signal may be derived from either one of the internally generated waveforms or excited electronically from a separate signal generator either a stand alone unit. A stand alone unit is of the type suitable is a multifunction synthesizer model HP8904A from HP Test and measurement Organization, a Hewlett Packard company USA) or integrated within a system computer, otherwise used for data acquisition and analysis. For example, an electric impulse generator is shown in Figure 29.

In another embodiment the pressure signal may originate from an impact mechanism system. Such impact mechanisms are known to those skilled in the art. For example, the impact mechanisms may be of the spring loaded or electronically activated mechanical impact system types, applying pressure on either the catheter or on a container attached to it. A signal generation apparatus (in vitro and in vivo) of the impact mechanism type makes use of a pistol hammer mechanism, where the pistol hammer hits directly on the catheter, lying on a rigid surface. The pistol used was a P230 semiautomatic pistol from Sig Sauer, Switzwerland. Alternatively, the same pistol hammer hit the head of a standard 5 ml syringe, where the syringe was connected to the catheter through a standard manifold.

The artificial pressure flow signal may also be synchronized with heart beats, either by gating to ECG or to system measurements (pressure or flow) in which the ECG device measures heart heat signals upon reaching a desired time in the heart beat triggering the artificial pressure flow signal. The pressure signal advances through a catheter lumen into the blood vessel. The catheter may be a guiding catheter. A guiding catheter of the type suitable for this purpose is a 8F Archer coronary guiding catheter from Medtronic Interventional Vascular, Minneapolis, U. S. A.. A diagnostic catheter of the type suitable

for this purpose is a Siteseer diagnostic catheter, from Bard Cardiology. U. S. A. A balloon catheter of the type suitable for this purpose is a Supreme fasr exchange PTCA catheter, by Biotronik GMBH & Co, U. S. A. It should be noted that almost any type of hollow catheter may be used. The presence of occlusion or aneurysm downstream creates reflection of pressure and flow waves. By extracting data of the reflected pressure waves, originated in the occluded site, the location and degree of occlusion can be determined using signal processing methods.

The performance of pressure wave velocity and reflection site parameters (position and reflection coefficient) estimation is based on two-point pressure measurement carried out inside of an artery. The pressure is measured either simultaneously by two pressure sensors or in different time by single sensor with some additional synchronization mechanism. In the case of the simultaneously pressure measurement two pressure sensors are placed throw known distance d. In the single sensor case the pressure is firstly measured in point a (upstream) and after that-in point b (downstream).

Computation of Pressure Wave Velocity (PWV: There are several methods for computing the PWV. One of the most usual and noninvasive methods is to compute the time, At, between arrivals of the area or diameter pulse between two positions at distance L apart. The PWV is then equal to PWV = L/At.

In order to isolate the part of the incoming pulse, without contributions of the reflected pulse the foot of the pulse at systole is considered.

Compliance and Distensibility: Compliance and distensibility are related to Pressure Wave Velocity as follows: Consider an element of volume dV= A. dl of length dl and cross-section area A in which the instantaneous pressure equals P. Then compliance, C, is defined as the derivative of the volume with respect to pressure per unit of length:

since arteries are strongly axially tethered. Distensibility, Dst, is the relative change of the area with respect to pressure: <BR> <BR> <BR> <BR> <BR> <BR> 1 dA<BR> <BR> <BR> Dst =<BR> <BR> <BR> <BR> A dP (2) and the compliance is: C=A Dst (2-1) The pressure velocity c is related to the measured Pressure wave velocity (PWV) through: c=PWV-u (3) Where u is the blood velocity in the artery. Since u is of the order 30 cm/s or less, and c is 7 m/s, we may ignore the distinction between c and PWV (but should be kept in mind if higher accuracy is needed) so that c=PWV (3-1) In the present invention the measured parameter is the inner diameter D. Assuming a circular vessel diameter measurement is equivalent to area measurement since A = n. D2 /4 and <BR> <BR> <BR> <BR> <BR> 2 dD<BR> <BR> <BR> Dst =<BR> <BR> <BR> <BR> D dP (4) The obvious way to calculate the elastic properties of the artery is to measure the diameter as function of pressure. In the absence of any pressure measurements, the distensibility is calculated through the measurement of the Pressure Wave Velocity using the relation (p is the blood density):

<BR> <BR> <BR> <BR> 1<BR> C² =<BR> pDst(5) If one assumes a constant distensibility, then integrating Eq. (2) yields: Ao x) (6> where Ao is the area corresponding to an internal reference pressure Po. Human arteries are stiffer at higher pressure. Several empirical models have been proposed and may be used for more accurate results.

For example, Streeter et al. (1963) consider the artery as a thin-walled vessel made of incompressible elastic material (Poisson's ratio = 0.5) and derived the theoretical expression where ho and Do are the wall thickness and arterial diameter at reference pressure Po, respectively and E is the effective elastic modulus. A popular model is the one of Langewouters et al., (1984) using an arctangent function and three optimal fit parameters: The difficulty with these formulae is in the determination of the parameters.

It appears that Eq. (6) is in error for systolic pressures. Since these large pressure last for but a small fraction of the cardiac cycle, the error in the dynamics is not significant. If one desires a more accurate result, the calibration of the parameters should be carried out concurrently to the measurement. It is easy to incorporate artery stiffening in the fluid dynamical algorithm presented below.

In arteries, pressure is an almost periodic function. During the short measurement time pressure is quasi periodic. Therefore the minimal value of the pressure exhibits very small variations. Therefore it is legitimate to take as the reference pressure Po the minimum of the pressure in an cardiac cycle, i. e., the diastolic pressure. The corresponding area at position x is thus Ao (x). When the PWV can be measured at several points along a cardiac cycle thereby obtaining a good approximation (by numerical interpolation) to the function c (A) where A varies between diastolic to systolic Area. Then formula (5) can be generalized to <BR> <BR> <BR> c2(A}=-----=A(p)-dp-<BR> <BR> <BR> <BR> <BR> <BR> <BR> <BR> <BR> <BR> <BR> p-Dst Dst (p) p dA (5-1) and upon inverting the latter formula Eq. (6) now reads In case the artery can be considered circular the latter formula is replaced by Blood Rheology: In large arteries, blood can be modeled as a Newtonian fluid (Y. C Fung, chapter 3)

where v is the velocity vector, T the stress tensor and V the shear rate tensor. The symbol p is the pressure field and t = 0.0036-0.0040 the coefficient of viscosity which depends on the hematocrit. However, for sufficiently small shear rate blood is no longer Newtonian. A reasonable model incorporating the high and low shear rate is Casson's model. The effect of the non-Newtonian behavior is mainly felt in regions of low shear which are mainly in the center of the artery. The total effect of the non-Newtonian model is to make the velocity profile blunter than otherwise. Near the wall, the shear rate is large so that the fluid is Newtonian there. In this invention the blood is modeled as a fully Newtonian fluid, although its non-Newtonian behavior can be incorporated, as well.

Model 1: Transversally averaged Fluid Dynamics Equations: This model is a quasi-linear model in which the fluid dynamical equations are integrated over the cross section. It is especially useful for arteries in which the cross-section is manifestly non-circular. In effect it transforms the full three dimensional flow equation to one-dimensional ones in the couple Q and P where Q (x, t) is the volume flow rate or alternatively A and Q since A and P are related through compliance. The price of such a drastic simplification is the introduction of empirical constants. Fortunately, these constants have a small effect so that they need not be known with great accuracy.

The basic equations are the continuity and momentum equations (Navier-Stokes). It is convenient to use cylindrical coordinate where x is the axial coordinate along the tube, r is the radial coordinate and 9 is the circumferential angle.

BasicEquations: Continuity Equation: Axial component of the momentum equation (A is the Laplacian, p is the pressure,, u is the blood viscosity and p is the blood density) : The momentum equation is simplified by making the long wavelength approximation where a is the angular velocity, R is the radius (or an average transverse dimension) and c is the wave velocity. It follows that the axial viscous transport term is negligible and pressure is constant throughout a cross-section. The momentum equation reads then: Our interest is in the flow (i. e. the volume flux), Q (x, t) which is the volume of blood crossing a point x at time t. Clearly, 0 (x, Q= \\Vx, r, 0) rdrd6 crosss-section It should be emphasize that we are not interested in the velocity field but in the integrated field over a cross-section. First, the continuity equation is integrated over the cross-section at point x: f a ( vr) +ve +vX a d e = o t tFS t0X JJ,. v Q GX) \) cross-section

The first integral over r is trivial and one obtains: 2 A-rV-'), b, dO fV (xJ0), O) R (O) dO= Jr (") (15) cross-section 0 where R (9) is the radial distance of the boundary to the center of the tube at angle 9 and A (x, t) is the cross-section area at point x at time t.

The second integral vanishes identically (upon performing the angular integral, first) and the third integral yields: (/lJ Vx Y Gt Y Gl D = x rdr dO=- (16) ffx JJ. r x (16) cross-sec tion All in all, the-continuity equation becomes: 9AffQ<BR> <BR> ~+=0 Next, the continuity equation (Eq. 10) is multiplied by Vx and then add the momentum equation (Eq. 12) obtaining: and upon integrating the latter equation, one obtains: rdrd0=- \r {6) d6 fft 0x Jr (S) d XdO (19) cross-sec tion perimeter Without loss of generality, one can define a parameter/ ? (x, t) such that: rdrdet}-1- A x t (20) cross-sec tion The parameter varies between 1 for a blunt velocity profile and 4/3 for a parabolic velocity profile. In arteries it is observed that ft is very close to one. In addition, the quadratic term is very small, except possibly at severe constrictions (stenoses).

The integral on the r. h. s. represents the force exerted by the wall on the fluid. The wall shear stress (WSS) conventionally denoted by r having dimensions of a pressure is definedas: v (X, t) = fr (9) x dO (21) tA (x, t) primeter The Momentum equation finally reads: Eqs (17) and (22) are the one-dimensional equations for A and Q in the long-wave-length approximation. Clearly, by ignoring the quadratic term in Q and the stress one recovers the well-known linear model. It remains to find an appropriate model for the WSS as well as/ As is often the case in fluid dynamics, empirical parameters must be adjusted for the specific application at hand. Nevertheless, choosing ß = 1 introduces a negligible error.

The pressure drop in a pulsatile flow is mainly due to both viscous effects and to a pressure force for flow acceleration. The Poiseuille WSS for a steady flow is <BR> <BR> <BR> <BR> proportional tu Q whereas the inertial force is proportional to the time derivative<BR> <BR> <BR> <BR> <BR> <BR> pA<BR> <BR> <BR> <BR> <BR> <BR> of Q.

According to Young and Tsai (1974), the following expression is the preferred one: The coefficients cu and c, are slowly varying coefficients depending on the Womersley number:

The first term in the stress expression is the usual Poiseuille stress and the second one is an unsteady contribution. In the aorta c,, 1. 1 and c" 1.6. For each type of artery (e. g. carotid) a computational and clinical study should yield an optimal set of coefficients.

Area-Pressure Relation: <BR> <BR> <BR> A(x,t)<BR> = f(p(x,t)) (25)<BR> <BR> <BR> <BR> A 0(x) Compliance and distensibility are explicit functions of x.

The Hyperbolic System: It is more convenient (although not mandatory) to use variables P and Q. Clearly, <BR> <BR> <BR> <BR> DA QPQQ<BR> <BR> <BR> + = 0 (26)<BR> #P #t #x and using (Eq 25) the continuity equation becomes: <BR> <BR> <BR> <BR> 1 df 9P8Q<BR> <BR> <BR> <BR> <BR> <BR> <BR> Ao dP at ax The second equation is obtained using the substitution in Eq. (22): one obtains the second equation

(A in the latter equation stands for f (P) Ao so that A can be fully eliminated) (Stergiopulos et al. 1992).

Initial Conditions: The initial value of A as well as Q must be specified (i. e. A (x,0) and Q (x, 0) for all x in the segment [0, L]). The difficulty resides in the absence of datum concerning the initial flow. However, since the system is periodic, the initial conditions rapidly die off. The system of equations is run with arbitrary initial conditions (e. g., zero flow) with repeating the datum for a few periods (up to ten periods, typically) until the system settles in the periodic regime.

Model 2: Axisymmetric Flow Equations: When flow is axisymmetric, and in the absence of swirl, Vo = 0, the axial momentum 02v<BR> equation (the term is neglected in the long wave-length approximation) reads and the continuity equation <BR> <BR> 1 #rr Vr) #Vx<BR> + = 0<BR> r #r #x (31) together with the elasticity equation R (t, x-/ (32)<BR> <BR> ----={P-PJ(32)<BR> <BR> ho Note that D = 2 (Po} and C = (pD)-l/2. The boundary conditions are assuming no slip at the wall and axisymmetric flow: <BR> <BR> # Vx<BR> =0, ver=0 at r=0<BR> a r

Vx-0 vr=Qj- atr=R (t, x) Where t denotes time, P the pressure, Po mean pressure. Now, we define the transverse coordinate = r/R (t, x) with the property that # = 1 on the boundary. Next, by non-dimensionalizing the equations and using the non-dimensional variables: <BR> <BR> <BR> X = cot, z = cox/C, w = Vx/C, u = Vx/coRo, p = (P - P0)/pC² and the Womersley<BR> <BR> <BR> <BR> <BR> <BR> parameter a = Ro (p/ppl/2 and co is the angular frequency. Finally, after tedious but straightforward algebra Eq. (33) is obtained: and a = a (p). The boundary conditions assume the much simpler form: <BR> <BR> <BR> <BR> <BR> # w<BR> <BR> =0, u=0, at=0 a w= 0 u = @dat at 4 = 1 The system simplifies considerably by noting that upon integrating the second equation over # from 0 to # (after multiplying by ): the radial component of the velocity is an explicit function of the axial component.

Because of the term-the equations require the knowledge of the axial velocity at both aZ ends of the vessel. Since the axial velocity profile does not vary significantly in the axial direction, a natural approach is to attempt a solution in powers of the axial variable z: w (z,#,#) = w0(#,#) + zw1(#,#) + z²w2(#,#) +... (35a)

u (z, 4, r) = u0 (#,#) + zu1(#,#) + z²u2(#,#) +... (35b) a (z,#,#) = a0(#,#) + za1(#,#)+z²a2(#,#) +... (35c) ap (z,#) = g0(#) + zg1(#) + z²g2(#,#) +... (35d) Since the equations are non-linear, the order n terms depends on the term of order n+1.

Normally one cuts the hierarchy at some low order (the closure problem) and oftentimes the scheme yields a good approximation (a known counter-example is turbulence). It is sufficient to assume a linear tapering on the measured segment. The continuity equation decomposes into two equations: The axial equation becomes to zeroth order in z (Eq37a): and to first order in z (Eq37b) The extra complication comes from the tapering term al.

No Tapering: Model 2 relates to a generalized case where the artery is with or without tapering. The case of no tapering is a simplified case of model 2.

It is simpler to return to the dimensional variables (Johnson et al. 1992). To linear order in x: <BR> <BR> <BR> <BR> Vx (r, x,t) = f0(r,t)+xf1(r,t) (38a)<BR> <BR> <BR> <BR> <BR> <BR> #P(z,t) = G0(t) + xG1(t) (38b)<BR> <BR> <BR> <BR> <BR> <BR> <BR> ox The continuity equation reads: <BR> <BR> 1 (a- v)<BR> r Introducing a stream function solves the continuity equation <BR> <BR> <BR> V (, t)=ltl<BR> <BR> <BR> <BR> <BR> and<BR> <BR> <BR> <BR> <BR> <BR> <BR> <BR> <BR> 1 ##<BR> f1 = -<BR> jazz (41) The two axial equations read: where

The boundary conditions are along the centerline and along the wall r= R (t) (no tapering): <BR> <BR> y (R (t), t) = R (t) U (R (t),t)(46b)<BR> <BR> <BR> <BR> <BR> <BR> aWQ)..nn<BR> a -. fo (R (t), t) = 0 (46c) the flow is calculated using the following equation: Preparing the Pressure Signals for Numerical Computation: 1. The minimal areas at the Nx positions are recorded by selecting the minimal area in time at each of those positions.

2. Pressure signals are computed according to the compliance model. For example: from the formula in the constant distensibility mode one obtains the Nx signals: <BR> <BR> <BR> P(x,,/,)=+/21og A(xi,tj) (48)<BR> <BR> <BR> <BR> <BR> <BR> <BR> A0(xi) 3. From the Nx values of (x ;, tj), NT interpolating polynomial of order NX-1 Pcj (x) are constructed through the Nx positions points on the measured segment. This is necessary for solving the numerical equations by finite differences or finite elements.

For example, in the case of two measuring points xl and xl+L

p (x) =P(x,,t,)+----------'-(x-x,) and the pressure gradient is interpolated as: #P = P(xl,tj) -P (x, + L, tj) (50) 9x ~L5 and for three measuring points xl and x2 and X3 the pressure gradient is: Detection and Localization of a Downstream Stenosis: The non invasive sensors measure diameters and upon using the Pressure-Area (or diameter) relation the pressure signal are obtained. Assuming that the linear wave theory is valid and that the superposition principle is applicable, the pressure wave can be linearly decomposed into a forward moving and backward moving wave, Pf (x, t) and Pb (x, t), respectively. Hence, the pressure wave is a sum of a forward and backward moving wave: <BR> <BR> <BR> <BR> P (x, t) =Pf{x,t) + Pb(x,t) (53)<BR> <BR> <BR> <BR> <BR> <BR> For a forward wave (propagating in the direction of the flow: <BR> <BR> <BR> <BR> <BR> Pf(x,t)=Pf(0,t-x/c) (54) Fourier transforming the function P, {0, t):

from Eq. (54) In order to incorporate attenuation, a function CC (co) (complex c) is introduced: <BR> <BR> <BR> 1 1 + y. Att ().<BR> <BR> <P> (s)<BR> CC () c () the real part and the imaginary part of 1/CC being the inverse of the PWV and the attenuation of the wave. Therefore The PWV is practically constant for frequencies except for the first few harmonics. Also attenuation vanishes at zero frequency and can be estimated by keeping the linear term in a series expansion of the attenuation function, that is: Att (where ce is the attenuation coefficient.

The backward wave originates from reflection of the forward wave at the stenosis and in the linear approximation the forward wave of angular frequency o yields a reflected wave of the same frequency multiplied by a (complex) reflection coefficient R (co) so that : Pb(#) = R(#) . Pf(#20) (59)

The dependence of the reflection coefficient on the frequency is very weak in the range of physiological pulses and can be ignored so that R (co) ~ R.

Let the proximal and distal sensors have coordinate"0"and"Z", respectively. The backward wave at a given sensor is equal to R times the path from the sensor to the stenosis and back; the path equals 2L for the proximal sensor and (2L-Z) for the distal sensor. Hence: Let F (co) and G (co) denote the Fourier transforms of P (O, t) and P (Z, t), respectively. They are equal to the expressions multiplying exp (jcot) under the integral sign. Upon dividing F (OD) by G (co) it follows that: Thus the ratio of the Fourier transforms of the proximal and distal pressure pulses is a known function of OD, R, L and Z and CC (co). Clearly, Z, the distance between probes is known. In addition, the PWV can be computed using the method described here in above. There remain, three parameters R, L (the distance to the stenosis) and the attenuation coefficient, a. To be computed. In practice, a least-square fit procedure is

the easiest method, as shown next (the left and right hand side of Eq. (62) are denoted, rat (co) and t (cl)), respectively). Then R, L and a are given by: and the integral is minimum at the physical values of R, L and a (the integral is taken over a frequency interval such that the proximal and distal power spectra are, say, at least 20 times larger than the power spectrum of the noise). For clinically relevant stenoses, exceeding 70%, (percent stenosis is defined as 1-A/As where A is the"healthy artery cross section area"and As is stenosis area) the reflection coefficient may be considered as a real number independent of frequency [Stergiopulos et al. 1996).

The main clinical application of the method is in detecting downstream stenosis which would otherwise be hidden from the other modalities. For example, in certain individuals, the internal carotid is situated behind the jaw and thus opaque to B-mode ultrasound. One may thus envision a screening process by which the common carotid is scanned using two probes as described which measure the parameter L (distance of the stenosis from the proximal probe) thereby detecting both the presence and the location of the stenosis in the internal carotid.

Detecting and localizing a stenosis by tracking the motion of the anterior wall of the artery: The wall of an artery closest to the skin is called the anterior wall, while the side farthest from the skin is the posterior wall. It has been observed that the anterior wall of a carotid undergoes a greater displacement than the posterior wall during a cardiac cycle (Reneman, Hoeks and Westerhof 1996). The same phenomenon might be observed in other peripheral arteries as well. Let ant (t, x) and pos (t, x) be the radii of the anterior and posterior walls at time t at position x along the artery, measured from the centerline of the artery (the exact position of the centerline being arbitrary). Then the corresponding diameter, D (t, x) is given by D (t, x) = ant (t, x) +pos (t, x). Upon arrival of a small pulse, the anterior and posterior undergo a small additional displacement Aant (t, x) and Apos (t, x)

which, in a linear approximation (in view of the smallness of the pressure pulse amplitude), are proportional to the pressure pulse AP (t, x). That is Aant (t, x) = aAP (t, x) and Apos (t, x) = bAP (t, x) with a and b proportionality factors.

Assume one measures the anterior wall movement alone (similar considerations hold for the posterior wall, mutatis mutandis). The PWV can be measured by measuring the time between arrival of the peaks of Aant (t, xi) and Aant (t, X2) measured at times t, and t2, respectively. The PWV, c, is then given by: c = x1 - x2 ti-t2 The method is demonstrated, below, using the MMS. Moreover, one can use Eqs. (55-61) with the substitution -/\ant (t, x) P (t, x) a The results do not depend on the value of a, only on the fourier transform of Aant (t, 0) and Aant (t, Z) which are measured at"0"and"Z".

In Vitro Tests and Results: In vitro experiments were performed in order to verify this method. The in vitro test apparatus described in Fig. 4 was used to perform the experiments. The pulse generator (exciter) used in the experiments is described in Fig. 3.

Reference is now made to Figs. 4,5and 6 4 illustrating an in-vitro experimental apparatus constructed and operative for determining flow characteristics in simulated non lesioned and lesioned blood vessels,.

Reference is now made to Fig. 4. The fluidics system 51 of Fig. 4 is a recirculating system for providing pulsatile flow. The system 51 includes a pulsatile pump 42 model 1421 A pulsatile blood pump, commercially available from Harvard Apparatus, Inc., Ma,

U. S. A., however other suitable pulsatile pumps can be used. The pump 42 allows control over rate, stroke volume and systole/diastole ratio. The pump 42 re-circulates glycerin solution from a reservoir 15 to a reservoir 14.

The system 51 further includes a flexible tube 43 immersed in a water bath 44, to compensate for gravitational effects. The flexible tube 43 is made from Latex and has a length of 120 cm. The flexible tube 43 simulates an artery. The flexible tube 43 is connected to the pulsatile pump 42 and to other system components by Teflon tubes. All the tubes in system 51 have 4 mm internal diameter. A bypass tube 45 allows flow control in the system and simulates flow partition between blood vessels. A Windkessel compliance chamber 46 is located proximal to the flexible tube 43 to control the pressure signal characteristics. A Windkessel compliance chamber 47 and a flow control valve 48 are located distal to flexible tube 43 to simulate the impedance of the vascular bed. The system also includes a flowmeter 11 connected distal to the flexible tube 43 and a flowmeter 12 connected to the bypass tube 45. The flown meters 11 and 12 are suitably connected to A/D converter (not shown). The flow meters 11 and 12 are model 111 turbine flow meters, commercially available from McMillan Company, TX, U. S. A..

In certain cases, an ultrasonic flow meter model T206, commercially available from Transonic Systems Inc., NY, U. S. A is used. An exciter (5) is connected to the flexible tube (43) through a catheter (6) enable introducing pressure pulses inside the tube.

Reference is now made to Fig. 5, which is a schematic cross sectional view illustrating a part of the fluidics system 51 in detail. An artificial stenosis made of a tube section 55, inserted within the flexible tube 43 is described. The tube section 55 is made from a piece of Teflon tubing. The internal diameter of the artificial stenosis 55 may be varied by using artificial stenosis sections fabricated separately and having various internal diameter.

Reference is now made to Fig. 6. The system 41 includes the system 51. The system 41 also includes a signal conditioner 23 model TCB-500 control unit commercially available from Millar Instruments.

Data acquisition was performed using a PC-Pentium 586 (20 with an E series Instruments multifunction I/O board model PC-MIO-16E-4, commercially available from National Inc., TX, U. S. A. The I/O board was controlled by a Labview graphical programming software, commercially available from National Instruments Inc., TX, U. S. A. 10 sec interval of pressure and flow data were sampled at 5000Hz, displayed during the experiments on the monitor and stored on hard disk. Analysis was performed offline using Matlab version 5 software, commercially available from The MathWorks, Inc., MA, U. S. A.

Reference is now made to Figs. 7 and 8 illustrating the magnetic motion sensor (MMS) device attached to the in vitro test system 51. The device is used for measuring wall displacement of the tube simulating a blood vessel.

The sensor array measures small changes in diameter along a latex tube while the tube inner diameter is under water pressure variation caused by a pump simulating a human heart. Tests show that the sensors are able to sense movements as small as a few microns in real time and that these movements correspond well to pressure measurements obtained simultaneously using a Radi pressure sensor inside the tubing.

In order to minimize the effects of the sensor array on the latex tube Hall Effect sensors were arranged to measure movements of small magnets cemented to the surface of the latex tubing.

A total of six Hall Effect sensors are spaced at 2 CM intervals along the latex tubing and several MM above the tubing. Small rare earth magnets are placed above and below the sensors with there like magnetic poles facing each other. This arrangement creates a magnetic field that is almost linear near the center point between the magnets. The top magnet is fixed relative to the Hall effect sensor, while the lower magnet is free to move with the changing diameter of the tubing. The opposite side of the tubing diameter is fixed to 2 MM diameter posts opposite each sensing position.

Each Hall Effect sensor generates an output voltage proportional to the change in magnetic field strength that is in turn proportional to the change in tubing diameter. The sensor voltage is amplified by a low noise instrumentation amplifier followed by a differential output line driver which is drives a set of cables to the nearby data acquisition system.

Experiment for Flow Determination: The experiment was performed in order to verify the method described herein above for determination of PWV, Volume Flow Rate, axial and radial velocity and other relevant fluid dynamical parameters.

Test Setup : The experiment was performed using the in vitro test apparatus illustrated in Figs. 4,5,6 using a flexible tube under the action of an external pulsating pump simulating heart beat. The test configuration is illustrated in Fig. 9.

The pump is a pulsating HarvardTM pump producing pressure pulses simulating heartbeats. The flexible tube is made of latex simulates blood vessel wall characteristics of 70 years old man. The latex tube is filled by fluid consists of a 60 % water and 40 % glycerine mixture with a density of 1103 Kg/m3 and a measured viscosity close to the one measured in blood.

The data was acquired at a sampling rate equal to 5000 Hz and digitized using an A/D board. The data acquisition was monitored with a LABVIEWTM program and then processed through a MatlabTM program (FID). (the sampling rate is much higher than needed for computations) The MMS system output is voltage signal that follows the tube diameter changes caused by pressure variation in the vessel produced by the pump. The voltage signal is translated into pressure units by means of sensor calibration. For the purpose of calibration we use a single pressure sensor Millar in a static configuration. The sensor characteristic obtained was practically linear.

Distensibility of the latex tube: Reference is now made to Fig 10 illustrating a test carried out In order to obtain the Pressure-Area relationship. A pressure sensor 21 (Millar) is positioned in the experimental setup without'pump. The pressure sensor is calibrated to zero pressure at atmospheric pressure. Both ends of the tube are closed whereas at one end a syringe is attached to the tube. The length of the tube is 1.31 m and the diameter of the tube filled with fluid at atmospheric pressure is 3.6 mm. The volume of the tube, Vo, (neglecting the Y-connectors) is 13.34 ml. The Syringe capacity is 2.4 ml. Small quantities of fluids are injected into the tube and the pressure read by the Millar sensor is plotted as a function of the logarithm of the volume in the tube, Vo + AV. If the latex tube exhibits constant distensibility the following holds <BR> <BR> <BR> <BR> <BR> <BR> V0 + #V<BR> P.P..pc2 logv----<BR> <BR> <BR> <BR> <BR> <BR> <BR> <BR> vo<BR> <BR> <BR> <BR> <BR> <BR> <BR> <BR> which is linear with slope equal to pc2 Reference is now made to Fig. 11 illustrating this test results. The plot is indeed linear. Where the plot is linear as should. From the slope the value of c = 14.12 was deduced. Since the latex tube was stretched, the length of the tube was practically constant. Therefore the increase in volume, AV = LxAA where AA is the increase in cross-sectional area and L the length of the tube. Therefore the latter relation between pressure and volume can be replaced by the pressure-Area (or pressure-diameter) relation (upon simplifying by L the right hand side of the equation): <BR> <BR> <BR> <BR> <BR> Ao + DA<BR> <BR> <BR> P = PO + pc1log-----<BR> <BR> <BR> <BR> <BR> Ao Pressure-Voltage Characteristics: In addition to pressure and injected volume, MMS output voltage is recorded simultaneously. A plot of pressure as measured by the Millar versus MMS voltage is presented in Fig 12 In the physiological range, it appears that the Pressure-Voltage

relation is linear. On the other hand the Pressure-Area is logarithmic and so is the Area-Voltage relation. However, expanding the logarithm in a Taylor series, the relations are practically linear, the higher order terms being negligible. Thus, voltage yields area and pressure. It was found that accuracy is increased upon calibrating the apparatus by means of the volume flow rate, as follows. Through the latex tube, steady flow at a precise value of volume flow rate is applied. The experiment is repeated for several distinct values Q ; with APi the corresponding pressure drop is given by Poiseuille'law <BR> <BR> <BR> -"<BR> <BR> <BR> <BR> <BR> <BR> <BR> <BR> <BR> r D Assuming a linear relation between pressure and voltage: <BR> <BR> Pi1 = K¹Vi1 + P01; Pi2 = K²Vi2 + P02<BR> <BR> <BR> <BR> The following system is solved for Kl, K2, APo = Pol-Po2 by a least-square method The value of Pol is the pressure at zero flow.

Pressure Wave Velocity: Since the pressure-voltage relation is linear, the PWV can be computed directly from the MMS Voltage as a function of time. Several methods may be used. Basically the method computes the shift in time between arrivals of the pulses at the magnetic sensors. Due to imperfections in the systems such as tube instabilities, pump variations and other factors, there is a spread in the PWV values. A train of 44 pulses was selected and outliers eliminated (less than 8 m/s and more than 20 m/s).

The computed PWV is c= 13.79 1.65 m/s.

The pulses measured by the MMS are illustrated in Fig. 13

Flow Computation: the results of two measurements are presented. The measured mean volume flow rates using a graduated cylinder was 110 ml/min and 204 ml/min. In addition a Flow-meter, Mac-Millan model Fui 11 was inserted distally for validation. The Flow-meter consists in a turbine, through which the liquid flows. Due to inertia of the turbine, the full amplitude of the flow is effectively truncated. Numerical computations using Model 1 and model 2 were applied to the two configurations.

Results using Model 2 The computed volume flow rate using model 2, for the two cases, is plotted in Fig 14.

For comparison, the output of the Flow-Meter (FM) appears on the same plot. The volume flow rate as measured by the flow-Meter has a smaller amplitude due probably to inertia. Plots of the Wall Shear Stress as function of time for volume flow rate 120 ml/min (top) and 200 ml ! sec (below). are illustrated in Fig. 15for the two experiments.

Reference is now made to Fig. 16 illustrating a plot of the axial velocity profile for a single period with volume flow rate 120 ml/min. It is convenient to measure time by an angular variable, a whole period being 360 degrees. Velocity profiles as function of time for volume flow rate 120 ml/min The maximum (or peak) velocity is seen to be situated along the centerline of the tube.

In addition to the flow-Meter peak Doppler velocity was measured using a 0.0014 inch Doppler guide wire (Flowire, Cardiometrics, Inc., Mountain View CA). A proprietary real-time fast Fourier analyzer (FlowmapTM Cardiometrics, Inc.) with automated tracking of the maximal velocity of the Doppler spectrum was employed to measure peak Doppler velocity. In an angle of view oriented downstream and covering the whole tube, the Doppler shift reflected by red cells is recorded (we use corn-flour starch, instead).

Reference is now made to Fig. 17 illustrating a plot of the calculated centerline axial velocity using Model 1 (continuous curve) and the peak velocity as measured by the Flow ire. The volume flow rate is 120 ml/min.

Results using Model 1: The computed volume flow rate using model 1 is plotted in Fig. 18 together with the output of the Flow-Meter. The parameters chosen were: p =1, C"=1, C,, =1. The results are quite insensitive to small changes in these parameters. Model 1 is more limited than Model 2, since its only output is volume flow rate. In the present situation were the tube is circular and there is no swirl, it is not advantageous to use Model 1.

Comparing Model 1 and Model 2: Reference is now made to Fig. 19. The volume flow rates given by Model 1 and Model 2 are compared and are almost identical.

Expriment for detecting and localizing stenosis : In order to verify the method for stenosis detection an localization, in vitro experiments were performed using the in vitro test apparatus illustrated in Figs 4.5 and 6. Tube diameter was measured at two points using the MMS device illustrated in Figs. 7 and 8.

The MMS, due to lack of stability was found unsuitable for detecting stenosis using physiological pressure signal generated by a pump simulating physiological pressure signal caused by heart beats. Instead, an excited short duration pressure pulse generated by the exciter illustrated in Fig. 3 was used. The test set up is illustrated in Fig. 20 Reference is now made to Fig. 21 which is a plot of the proximal and distal signal measured by the MMS. In this experiment the stenosis has a diameter of 1 mm and a length of 1 cm and is located at a distance of 16 cm from the proximal magnet.

Reference is now made to Fig 22 which is a zoom about the peaks of the pulses illustrated in Fig. 21."A"and"B are the pulses measured by proximal and distal magnet, respectively. The time between peaks"A"and"B"is 7.4 msec. Since the distance between magnet is exactly 10.4 cm, it follows that the PWV, c, is equal to c=10.4cm/7.4msec= 14.0541 m/sec.

Expriment Results: Experiments were performed on blunt stenoses with inner diameter (d) of 1 mm and 1,5 mm for distances (L) of 13,16,20 and 25 cm from the proximal magnet (3,6,10,15 cm from the distal magnet). The results are summarized in the table 1 (a is the attenuation coefficient).

Table 1: 1 mm diameter Stenosis

distance L 13 cm 16 cm 20 cm 25 cm in test L (cm) 12.8 16.4 19.6 44 COMPUTED d (mm) 0.9 1 0.60 1 COMPUTED a (X 10-3) 12 20 15 19 COMPUTED Table 2: 1.5 mm diameter Stenosis

distance L 13 cm 16 cm 20 cm 25 cm intest L (cm) 14 15.40 19.2 26.3 COMPUTED R (mm) 1 1 0.5 0.5 COMPUTED a (X 10-3) 12 28 38 12 COMPUTED The presence of stenosis is always detected and its distance well estimated. The test data, (Teflon tube diameter) was measured using the MMS which used as a demonstration device. By using Medical Ultrasound systems or other well-developed systems the result will be more accurate.

References G. A. Jolmson, H. S Borovetz and J. L. Anderson"A Model of Pulsatile Flow in a Uniform Deformable Vessel", J. Biomechanics 1992,25,91-100.

Tadeusz Powalowski, Zbigniew Trawinski"Noninvasive Evaluation of the elasticity of common Carotid artery Wall"Archives of Acoustics (1994), 19,451-465.

Fung Y. C. Biomechanics : Mechanical Properties of Living tissues", Springer-Verlag, New York, 1981.

R. S. Reneman, A. P. G. Hoeks and N. Westerhof"Non-invasive Assesment of Artery wall properties in humans-methods and interpretation"Journal of Vascular Investigation (1996) 2: 2: 53-64.

Arnold P. G. Hoeks and Robert Reneman"Biophysical Principles of Vascular diagnosis"J. Clin. Ultrasound (1995) 23: 71-79.

Streeter, V. L, W. F. Keitzer and D. F. Bohr"Pulsatile pressure and flow through distensible vessels" (1963) Cir, c. Res. 13: 3-20.

Langewouters G. J., Wesseling K. H., goedhard K. H."The static elastic properties of 45 human thoracic and 20 abdominal aortas in vitro and the parameters of a new model"J.

Biomechanics1984,17,425-435.

Stergiopulos N., Young D. F. and Rogge T. R..,"Computer simulations of arterial flow with applications to arterial and aortic stenoses"J. Biomechanics 1992,25,1477-1488.

Nichols W. W. and O'Rourke M. F., (1998)"McDonald's Blood flow in arteries: Theoretical, experimental and clinical Principles", fourth Edition