WO2001013779A2 - A method and system for stenosis identification, localization and characterization using pressure measurements - Google Patents

Abstract

The present invention is directed towards an apparatus (100) for pressure based lumen mapping. The apparatus includes a flexible, elongated member (30) having a portion insertable into the lumen, a device (40) for measuring pressure inside the lumen operatively connected to the insertable portion of member (30), a device (50) for moving the insertable portion (30) in a predetermined manner, wherein pressure is measured at various points during movement, and a processor (2) for processing pressure measurements, wherein the processor (20) uses pressure measurements to inversely obtain parameters of a fluid dynamics equation for mapping the lumen.

Description

A METHOD AND SYSTEM FOR STENOSIS IDENTIFICATION,

LOCALIZATION AND CHARACTERIZATION

USING PRESSURE MEASUREMENTS

FIELD OF INVENTION

The present invention relates to the field of medical diagnostic and therapeutic devices in general and to a system for intravascular characterization of blood vessel lesions.

DESCRIPTION OF THE RELATED ART

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 rely on estimates of occlusion characteristics and 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.

Stenosis geometry is also important in the therapeutic phase when balloon angioplasty, stenting or drug delivery procedures are subsequently performed. For example, precise stent placement is critical for reducing the risk of re-stenosis. 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.

One of the most reliable functional assessments of the severity of coronary stenoses is obtained using hemodynamical indices based on pressure and flow measurements. Examples of these are myocardial fractional flow reserve (FFR) and coronary flow reserve (CFR). FFR is ideally suited for clinical decision making regarding stenosis revascularization, whereas CFR is an indicator of microcirculatory disease. FFR is the ratio of distal to proximal stenosis on either side of a stenosis during maximum hyperemia, and this index is measured using a pressure sensor located at the tip of a thin wire. CFR is the ratio of flow during vasodilatation to flow at rest and can be measured using a velocity sensor (e.g. Doppler). A method by which both FFR and CFR can be measured simultaneously using a pressure wire alone has been developed recently by the present inventors (U.S. Provisional Application Number 60/123,499, filed on March 9, 1999). This advance simplifies and reduces the cost of intravascular intervention by requiring the insertion of only a single pressure wire. However, these indices are poor indicators of stenosis geometry, ^' έuch as minimal lumen diameter (MLD), lesion length (LL), and percent area stenosis.

Intravascular Ultrasound (IVUS) and angiography are the preferred modalities for deducing geometrical parameters. However, IVUS requires the insertion of an additional catheter subsequent to the removal of the pressure sensor, increasing cost, duration, and risks of the procedure. Moreover, the relatively large diameter of the IVUS catheter does not allow the catheter to cross severe lesions at all. As a consequence, IVUS is often not used, even when available. Angiography maps the internal lumen but leaves out information about the nature of the vessel wall (e.g. plaque, lipid pool, vessel compliance). In addition, despite sophisticated computerized graphical interfaces such as Quantitative Coronary Angiography (QCA), angiography is sometimes misleading because of curvature of the artery. 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. A method for determining geometrical data of a stenosis using pressure measurements is described in Pijls, N.H.J and De Bruyne, B. "Coronary Pressure", Kluwer Academic Publishers, 1997, incorporated herein by reference. The authors describe the method including two relevant figures (Figs. 5.7 and 5.8). In this method, a guide wire having a pressure sensor at its end is slowly advanced and pulled-back across a stenosis. A plot of the instantaneous pressure is recorded on paper at low speed. In addition, the average pressure over each individual cardiac cycle is computed and the interpolated curve plotted on the same paper. The resulting curve is called a "pull-back curve." The authors assert that the exact location of the stenosis can be obtained visually from this curve. However, they do not explain how the location, with the exact start and end positions of the stenosis, can be ascertained.

Even if it is assumed that the start of the stenosis is identified as the point on the "pull-back curve" where there is a sudden drop in the pressure, the end of the stenosis is not clear. An additional limitation of Pijls and De Bruyne's method is the need for inducing h^'yperemia by special drugs in order to obtain a discernible pressure change. Hyperemia isHusually of very short duration which may have a negative impact on the accuracy of the measurement. Finally, neither the geometrical profile nor the diameter at maximal constriction, called Minimal Lumen Diameter (MLD), are measured.

SUMMARY OF THE INVENTION

The present invention provides a method and system for stenosis identification by set of pressure measurements, stenosis localization in the artery (start and end points), determining length, minimal diameter, and severity of stenosis (percent stenosis), hemodynamic reconstruction of stenosis geometry, calculating average flow rate in the occluded artery, determination of average blood velocity, , and determination of pulsatile index from the known flow in the artery. In addition, the present invention provides for a method of determining the diameter of a healthy unstenosed artery or the diameter of an artery proximal or distal to stenosis.

These data are essential for optimal selection of stent (length and diameter), optimal selection of balloon for the required inflated pressure and contact area and for optimal drug delivery. Further, an image or data chart reconstructing the lesion shape can be incorporated with angiography or QCA pictures, either by importing QCA or angiography picture or by exporting calculated lesion geometrical data.

Further, this invention may be used in combination with methods for calculating the flow-based clinical characteristics, coronary flow reserve (CFR), fractional flow reserve FFR and diastolic to systolic velocity ratio (DSVR), using pressure measurements across a stenosis. In addition coronary flow reserve in the same vessel without stenosis (CFR₀) may be estimated as well as aneurysms. FFR, CFR, CFR₀ and DSVR may be simultaneously calculated for a complete characterization of the vessel of interest.

Further, this invention may be used in ^'combination with methods for the determination of a hemodynamic condition of the artery~by determining the vascular bed index (VBIo) which is equal to the ratio of mean shear to mean pressure; methods of determining/detecting microvascular disease due to the abnormal ratio of FFR to CFR based on either or proximal and/or distal pressure; balloon procedures; post PTCA evaluation (prior to stenting); determination or validation of dilatation success by subsequent CFR increase after PTCA, and indication of whether a stent is needed. Lastly, it is a further feature to calculate and report a ratio of the relevant parameter determined in normal unstressed condition compared with a highly stressed condition where such a highly stressed condition maybe vasodilatation induced by injection of vasodilator such as Papaverine. The use of 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 may be used in combination with the present invention.

There is provided in accordance with an embodiment of the present invention, an apparatus for pressure based determination of. at least one parameter of a region of obstruction in a fluid filled tube. The apparatus includes a flexible elongated member with a portion insertable into the fluid filled tube, means for measuring pressure inside the fluid filled tube, operatively connected to the insertable portion; means for moving the insertable portion in a predetermined manner, wherein pressure is measured at various points during movement, and a processor for processing pressure measurements, wherein the processor uses pressure measurements to inversely obtain parameters of a fluid dynamics equation, thereby providing a parameter of the region of obstruction.

There is also provided, in accordance with another embodiment of the present invention, an apparatus for optimizing treatment for an obstructed blood vessel. The apparatus includes a flexible elongated member having a portion insertable into the blood vessel, means for measuring pressure inside the blood vessel, operatively connected to the insertable portion, means for moving the insertable portion in a predetermined manner," wherein pressure is measured at various points during movement, and a processor for processing the pressure measurements to obtain geometrical parameters of the obstructed blood vessel, wherein the processor uses pressure measurements to inversely obtain parameters of a fluid dynamics equation thereby providing optimal treatment for the obstructed blood vessel.

There is provided, in accordance with another embodiment of the invention, a method for determining at least one parameter of a region of obstruction in a fluid filled tube. The method includes inserting a portion of a flexible elongated member with means for measuring pressure into the fluid filled tube, moving the portion from one point along the fluid filled tube to another point along the fluid filled tube while simultaneously measuring pressure along the way, obtaining pressure measurements taken during the movement, and using pressure measurement to inversely obtain parameters of a fluid dynamics equation, thereby providing the parameter. 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 is a schematic isometric view of a system for lesion identification and determination of lesion severity and geometrical shape, maximal flow, maximal velocity, constructed and operative in accordance with an embodiment of the present invention;

Figure 2 is a schematic isometric view of a system for lesion identification and determination of lesion severity and geometrical shape, maximal flow, maximal velocity, constructed and operative in accordance with another embodiment of the present invention; Figure 3 is a schematic functional block diagram illustrating the details of the system of Figure 1 ;

Figure 4 is a schematic functional block diagram illustrating the details of the system of Figure 2;

Figure 5 is a schematic depiction of a stenosis inside a blood vessel; Figure 6 is a schematic depiction of a stenosis;

Figure 7 is a schematic depiction of flow lines due to a stenosis inside a blood vessel and a corresponding pressure curve;

Figure 8 is a schematic illustration of a system used to measure a non-stenosed artery, constructed and operative in accordance with another embodiment of the present invention;

Figure 9 is a schematic illustration of a system used to measure a non-stenosed artery, constructed and operative in accordance with another embodiment of the present invention;

Figure 10 is a graphical depiction showing pressure drop during rest flow and hyperemic flow;

Figure 1 1 is a graphical depiction of a curve derived from Fig. 10;

Figure 12 is a schematic representation of streamlines in a blunt stenosis; Figure13 is a schematic depiction of a blunt stenosis used in an in vitro experiment;

Figure 14 is pressure curve obtained using the system depicted in Fig. 13; Figure 15 is a flowchart summary of the PLM method;

Figure 16 is a picture of a stenosis as seen on an angiogram;

Figure 17 is a plot of pressure as a function of sample number;

Figure 18 is a graphical depiction of average pressures over a cardiac cycle; Figure 19 is a corrected pressure curve resulting from the curve depicted in Fig. 18;

Figure 20 is a plot of velocities in the MLD:

Figure 21 is a graphical comparison of a computed and an angiographic profile; Figure 22 is a picture of a stenosis as seen on an angiogram;

Figure 23 is a graphical depiction of cardiac cycle averaged pressure curves;

Figure 24 is a corrected pressure curve, resulting from the curve depicted in Fig. 23; Figure 25 is a graphical comparison of a computed and an angiographic profile;

Figure 26 is a schematic depiction of a fluidics system, constructed and operative in accordance with an embodiment of the present invention;

Figure 27 is a schematic cross sectional view illustrating one part of the fluidics system shown in Fig. 26 in greater detail;

Figure 28 is a schematic depiction of an in vitro testing apparatus, including the fluidics system of Fig. 26, constructed and operative in accordance with an embodiment of the present invention;

Figure 29 is a depiction of the shape of an artificial stenosis; Figure 30 is a pressure curve related to the artificial stenosis shown in

Fig. 29.

Figure 31 is a graphical depiction of computed and actual diameters of the artificial stenosis of Fig. 29. Figure 32 is a corrected pressure curve obtained from an in vitro test; and

Figure 33 is a graphical depiction of computed and actual diameters of an artificial stenosis.

DETAILED DESCRIPTION OF THE INVENTION

The present invention is directed towards a system and method of Pressure-based Lumen Mapping (PLM). Pressure signals from the heart are measured and are used for determining geometric and flow parameters in an artery with or without stenosis. In addition, volume flow rate in the vessel may be determined. It will be appreciated that the described invention may also be used for non-biological lumens, such as water pipes or other piping systems, or any fluid filled tube with flow. ^~~

Reference is now made to Figs. 1 , 2, 3 and 4. Figs.1 and 2 present schematic isometric views of systems 100 and 200 for lesion identification, localization and determination of lesion minimal diameter, length and geometric profile. The system is constructed and operative in accordance with embodiments as shown in Figs. 1 and 2. Figs. 3 and 4 are schematic functional block diagrams illustrating the details of system 100 of Fig. 1 and system 200 of Fig. 2.

Systems 100 and 200 include a pressure wire 60, which may be either a catheter or guide wire, inserted into a blood vessel directly (not shown) or via a lumen catheter 30 for measuring the pressure inside a blood vessel. Lumen catheter 30 may be a guiding catheter (e.g. 8F Archer coronary guiding catheter from Medtronic Interventional Vascular, Minneapolis, U.S.A.) or a diagnostic catheter (e.g. Siteseer diagnostic catheter, from Bard Cardiology, U.S.A.), a balloon catheter (e.g. Supreme fast exchange PTCA catheter, by Biotronik GMBH & Co, U.S.A.) or any other hollow catheter. A pressure sensor 40 is mounted on pressure wire 60 for measuring the pressure inside a blood vessel. Pressure sensor 40 may be the 3F one pressure sensor model SPC-330A or dual pressure catheter SPC-721 , commercially available from Millar Instruments Inc., TX, U.S.A., or any other pressure catheter suitable for diagnostic or combined diagnostic / treatment purposes such as the 0.014" guidewire mounted pressure sensor product number 12000 from Radi Medical Systems, Upsala, Sweden, or Cardiometrics WaveWire pressure guidewire from Cardiometrics Inc. an Endsonics company of CA, U.S.A.

A Fluid Filled (FF) pressure transducer 31 is connected to the end of lumen catheter 30 outside of the patient's body, and measures the pressure at the opening of lumen catheter 30 inside the vessel. A pull back mechanism 50 is attached to pressure wire 60 in order to retract back pressure wire 60 over a precisely controlled distance. Pull back mechanism 50 may be but is not limited to the mechanism integrated with the Scout 45 MHz catheter of Hewlett Packard company.

Systems 100 and 200 also include a signal conditioner 23, such as a model TCB-500 control unit commercially available from Millar Instruments, or Radi Pressure Wire Interface Type PWI10, Radi Medical Systems, Upsala, or other suitable signal conditioner. Signal conditioner 23 is suitably connected to pressure sensor 40 and FF pressure transducer 31 for pressure signal amplification. System 100 further includes an analog to digital (A/D) converter 28 (i.e. Nl E Series Multifunction I/O model PCI-MIO-16XE-10 commercially available from National Instruments, Austin, TX) connected to signal conditioner 23 receiving the analog signals. Signal conditioner 23 may be integrated in a data acquisition card of a computer 20, or may not be included at all, depending on the specific type of pressure sensor 40 and/or FF pressure transducer 31 used.

System 200 of Fig. 2 further includes a standard cardiac catheterization system 22, such as Nihon Kohden Model RMC-1100, commercially available from Nihon Kohden Corporation, Tokyo, Japan. Signal conditioner 23 and FF pressure transducer 31 are directly connected to catheterization system 22. System 200 further includes an analog to digital (A/D) converter 28 connected to the output of catheterization system 22 through a shielded I/O connector box 27, such as Nl SCB-68 or BNC-2090 commercially available from National Instruments, Austin, TX.

Systems 100 and 200 also include a signal analyzer 25 connected to A/D converter 28 for receiving digitized conditioned pressure signals from A/D converter 28. Signal analyzer 25 includes computer 20 and optionally a display 21 connected to computer 20 for displaying text numbers and graphs representing the results of the calculations performed by computer 20. Signal analyzer 25 further includes a printer 26 suitably connected to computer 20 for providing hard copy of the results for documentation and archiving. A/D converter 28 can be a separate unit or can be integrated in a data acquisition computer card installed in computer 20. Computer 20 using suitable software (such as Matlab version 5 software, commercially available from The MathWorks, Inc., MA, U.S.A.), processes the pressure data, sensed by pressure sensor 40 and FF pressure transducer 31 and acquired by A/D converter 28 or the data acquisition card, and generates textual, numerical and/or graphic data that is displayed on display 21.

In the case of a coronary lesion, lumen catheter 30 is positioned in the coronary ostium. Pressure wire 60, introduced into lumen catheter 30, is advanced to a location where pressure sensor 40 is close to the tip of lumen catheter 30, and both pressure sensor 40 and FF pressure transducer 31 are calibrated to the same pressure reading. Pressure at lumen catheter 30 serves as a reference pressure, thus eliminating physiological variations having a bearing on the measured pressure (such as arrhythmia, changes in vascular bed resistance and breathing). Pressure wire 60 is then advanced distally to the stenosis where the retraction procedure (pull-back, for short) can commence. The axial positions of pressure sensor 40 and FF pressure transducer 31 must be carefully monitored (e.g., visually on an angiogram).

Reference is now made to Fig. 5, which is a schematic depiction of a stenosis 111 inside a blood vessel, such as an artery 109. The clinician inserts pressure wire 60 with pressure sensor 40 inside stenosed artery 109, and positions pressure sensor 40 distal to stenosis 111. Incoming pressure pulses 102 from the heart are measured during a few heartbeats. Pullback mechanism 50 is used to retract pressure wire 60 continuously in the proximal direction, that is, from a point distal to stenosis 111 to a point proximal to stenosis 111. The velocity of retraction is carefully monitored, and is, for example, 0.5 -1 mm per second. Pressure is continuously sampled at a rate of up to 5000 Hz for commercial pressure sensors. The pull-back operation is normally done at constant speed using an external pull-back mechanism 50, basically comprising a wheel with a constant winding angular velocity.

In another embodiment pressure wire 60 is retracted by a precisely controlled distance, for example, Δx, through stenosis 111 where pressure is measured for a duration of a few heartbeats. The procedure is repeated until pressure sensor 40 is located sufficiently proximal to stenosis 111 , where a last pressure measurement is carried out. The resulting data is a set of pressure measurements P(Xj,t) at points Xj . In another embodiment the wire is pulled back manually by a physician and pressure is measured. Pressure sensors 40 mounted on pressure wire 60 measure and record time and velocity so that location of pressure measurements is known.

The pull-back procedure is relatively quick. For example consider a 2 cm long stenosis. Assume a retraction velocity of 0.5 mm/sec, a starting point at 3 cm distally (from the end of the stenosis) and a terminating point 3 cm proximally (from the beginning of the stenosis). The total length to be covered is 8 cm and the time required is thus 80/0.5 = 160 sec. Adding the time for data processing, the whole procedure lasts only about 5 minutes. This time constraint is important in a clinical setting, since lengthy procedures may add risk and discomfort to the patient. PROCEDURE

Calculation using the system is accomplished using the Pressure-based Lumen Measurement (PLM) method. Using this method, pressure, which is a scalar value, can produce an axisymmetric, hemodynamically equivalent profile of a stenosis. In short, the main goal of the PLM method is to produce a diameter value as a function of axial position at various points.

The PLM method is based on the well-known fluid dynamics Navier-Stokes equations in the parabolized (or in the boundary-layer) approximation. Normally, in computational fluid dynamics, a geometry is given as well as appropriate initial and boundary conditions. In an axisymmetric problem, the radius as a function of axial position is given and one of the requirements is to recover the pressure field. In the present invention, the Inverse Problem is solved. That is, the pressure is given and the radius is calculated therefrom. The final output is the geometrical profile of the stenosis, i.e. the radius as a function of the axial position R(z), the mean volume flow rate (Q in ml/min), the velocity profiles, the mean (over a cardiac cycle) axial and radial velocity w(z) and u(z) respectively, and the mean wall shear stress (WSS) τ(z). The geometrical profile is then compared geometric measurement taken from an angiogram. In principle, one could derive a Reference Diameter (RD), as well, which is the diameter of the healthy section of the artery that is outside the stenosis, distal or proximal. However, the RD is difficult to derive since the pressure gradient is generally exceedingly small; for example, in a coronary of 3.5 mm diameter, the pressure gradient is in the order 0.1mm Hg/cm.

Reconstructing the shape of stenosis from pressure measurements can be done only for laminar flow. Once the flow begins to lose its stability the data becomes inadequate. For this reason, the PLM method fails to compute the flow properties in the exit section of the stenosis, since instabilities and vortices appear as soon as the channel diverges. Thus, a different approach is used for the exit section of the stenosis, involving physical insight and the general conservation laws (e.g., momentum and energy). The PLM method consists of the following steps: measurement, data processing, and computational fluid dynamics (CFD). /. Measurement

Pressure wire 60 is inserted into lumen catheter 30 until pressure sensor 40 is positioned distally to the stenosis. This distal position should be downstream to the reattachment point, as defined hereinbelow; in practice it is approximately 3 cm distal to the end of the stenosis for mid-sized arteries. Pressure wire 60 is connected to a pull-back motor whose pull-back velocity is carefully monitored. Pressure wire 60 is then slowly pulled back. Three variants are possible, as follows: 1 ) Pressure wire 60 is pulled back by fixed distances, (on the order of

0.5 -1 mm), after which the motor is stopped for a few cardiac cycles. During the hold time, pressure measurements are taken. The procedure is repeated at each motor stop. 2) Pressure wire 60 is continuously pulled back at constant speed (on the order of 0.5-1 mm/sec) during which time pressure measurements taken.

3) Pressure wire 60 is pulled back manually using a velocity sensor which records the instantaneous velocity of retraction and time, thereby enabling the computation of the instantaneous position of pressure sensor 40. Two pressures are simultaneously measured: a pressure vector as a function of sampling number P_GC(Π) obtained from FF pressure transducer 31 connected to lumen catheter 30, and the pressure of pressure sensor 40 on pressure wire 60. The first variant as described above yields a matrix of pressure sensor 40 in the form of matrix P(i,n) where the first index corresponds to position, whereas the other variations yield a single vector P(n). The index n is the sampling number in all three variations.

Data Processing

Using PGC(Π) as a reference, the physiological pressure fluctuations and noises (in vivo) are weighted by suitable factors and subtracted from P(i,n) or P(n) depending on the particular variant. The output is a vector P(i) which is the average pressure over one (or more, in the case of variation 1) cardiac cycle at position i. The details of the procedure are presented hereinbelow for a human measurement in the Experimental Section. In addition, the Pulsatiliy Index PI is computed as described hereinbelow.

CFD

The inputs are the vectors P(i) and its corresponding position z(i), blood viscosity μ, density p, and the diameter of pressure wire 60. It is possible, also, to measure the RD using a device and method described hereinbelow. If, however, the diameter of the non-stenosed part of the vessel, RD, is not measurable, it is estimated from anatomical tables. It should be noted that the output is not very sensitive to the RD so that the estimated value of the RD (e.g. from an angiogram) is generally sufficient. The output quantities come from two sources, the pressure curve and Computational Fluid Dynamics (CFD). For the pressure curve, outputs are positions of the start and end of the stenosis, estimate of the position of the Minimal Lumen Diameter (MLD), and position of the reattachment point. For CFD, the outputs are a geometrical profile of the equivalent axisymmetric stenosis from the start of the stenosis to the separation point (the point at which the flow separates), the flow rate Q, the mean velocity at the MLD, the axial and radial velocity profiles, the separation point and the diameter at that point.

CFD Analysis

Reference is now made to Fig. 6, which illustrates a simple stenosis 111 as a three-part structure, an entry cone 105, a throat Ϊ06 and an exit cone 107. This schematic structure is convenient for defining the terminology that follows. "Cone" refers to an axisymmetrical constriction or enlargement. Throat 106 is a transitional structure (i.e. a cone with zero angle) and may not exist at all. However, from angiographic data it appears that there generally is a region of a few mm which can be classified as such. A multiple stenosis can be viewed as consecutive simple Cone-Throat-Cone structures, for which the methods presented below should be used in succession. The reference diameter RD, the diameter of the stenosis D_s, and the lengths at entry, exit and along the stenosis (L, L_s, and L_e) are all indicated. PLM pressure curve

Reference is now made to Fig. 7, which illustrates a schematic depiction of a stenosis, the centerline of artery 109, and half of the MLD 110. in an artery. A corresponding pressure curve is presented directly beneath the schematic illustration. In the proximal segment, points A and B are situated outside stenosis 111 and are within the RD. The pressure drop between points A and B is calculated for a "healthy" part of artery 109. For mid-sized arteries, the pressure drop ΔPAB=PA-PB may be too small to be used.

At entry cone 105, corresponding to point B, the pressure drops sharply since the flow is accelerated, thereby increasing the kinetic energy according to the well-known Bernoulli's formula. M usually occurs at the point of MLD, although sometimes the MLD is shifted. The typical pressure drop between points B and M, is on the order of 10-20 mm Hg in a coronary vessel. In hyperemia it can reach 25-50 mm Hg. For a distance BM of 10 mm, for example, the gradient exceeds 1 mmHg/cm, which is minimally a ten-fold increase in the pressure gradient. As a general rule, the point where the pressure gradient exceeds 0.5 mm Hg/cm is usually identified as the start of stenosis 111.

At exit cone 107, point M represents the beginning of an expansion and the point of MLD. From point M and on, there is pressure recovery, that is, the pressure increases. The fact that the pressure does not fully recover is due to losses, mainly from friction and internal vortices. However, there are two distinct regions within the pressure recovery. From M to E the pressure rises relatively quickly with a slope on the order of the initial pressure drop from B to M, while from E to R the pressure increase is very small, only up to a maximum of 1 mm Hg. The breaking point E is identified as the end of stenosis 111 , and is obtained by a method explained hereinbelow.

In the distal region, point S is the separation point where the wall shear stress (WSS) vanishes and the flow separates. At point R the pressure reaches a maximum and starts to drop slowly again. Point R is the reattachment point. At point R, the WSS vanishes again and the streamlines become parallel to the wall of the tube or artery, as they were originally. From point R on, wall friction enters into play, explaining the small pressure drop due to Poiseuille flow.

The region ESR contains a "trapped" vortex where energy is dissipated, whereas the rest of the flow is akin to jet flow with a boundary SR, as shown by a dividing line 108. Computational Fluid Dynamics (CFD) Method

To a very good approximation, blood can be assumed to be a homogeneous, incompressible Newtonian fluid. Although blood is actually non-Newtonian, the Newtonian approximation has been demonstrated to be a valid approximation in mid-sized arteries.

The flow is laminar in entry cone105. A typical healthy coronary artery has a diameter of approximately 3 mm with a mean flow rate of about 50 ml/min corresponding to a velocity of 15 cm/sec. The Reynolds number is defined as Re=WD/v where W is the instantaneous mean velocity. For a typical coronary artery with a diameter of 3 mm, and flow in the range of 50-120 ml/min, Re is approximately 90-210 and the flow is laminar. In a severe stenosis with an MLD of, for example, 0.75 mm, Re is about 350-830. In this range one expects instabilities, but not turbulence. Convergent channels are known to stabilize the flow so that in entry cone 105, the flow is laminar.

Effect of wire

Consider the case of a wire with radius a inserted along the centerline of a circular tube with radius R resulting in an annulus of aspect ratio α = a/R. The steady state flow for such an annulus is

having the shape of a double parabola with zero velocity on the sides of the wire. In a circular tube it is the well-known Poiseuille flow with a parabolic profile.

The Momentum and Energy correction factors, β and γ, where A is the area and W is the mean velocity, are defined as

When the flow is completely blunt β = γ = 1 * Table 1 shows some values of β and γ as a function of aspect ratio α.

Table 1

If a fluid enters a tube or a pipe from a large vessel in which the pressure is maintained constant and in which the velocity is negligible, then at the entrance to the tube the velocity is blunt. Immediately after entering the tube or pipe, the boundary layer starts to grow from the wall inwards towards the center, which is where the flow is fully developed. At some distance Xe called Entry Length, the boundary layer will fill the whole tube. At that point the profile will assume its steady state profile, blunter than without wire. The entry length for a tube with wire is much shorter than without a wire. For Re defined in terms of the mean inlet velocity U, Re=dU/v, where d is the diameter of the tube. The entry length has been calculated by Blasius and many others (see Goldstein "Modern developments in fluid dynamics" Dover Books , New York 1965 ed., S . Mc Comas Journal of Basic Engineering , Dec 1967) . To a good approximation, the following formula is obtained: Xe/Re d=0.0260. For a flow of 50 ml/min, a MLD of 1 mm and a viscosity of 3.6x10^"3, Re=295 and Xe = 7.7 mm. In the presence of a wire of 0.36 mm, and for a typical MLD 0.5 mm - 2 mm, the aspect ratio α is 0.18-0.72. Table 2 is taken from S . Mc Comas cited above.

Table 2

In the range of interest, the entry length with a wire is 4 times shorter than without wire.

The wire has a very large stabilizing effect manifesting itself by shortening the entry length, and making a blunt velocity profile. Experiments show that the wire brings reattachment point R closer to separation point S, as well. In human measurements, reattachment point R lies at a distance of 1.5-2.5 cm from the MLD when a wire is present and is thus observable. Without the wire it would be on the order of 10 cm. The possibility of detecting reattachment point R will be crucial in later calculations as described herein.

Pulsatilitv Index (PI)

The cardiac pressure and flow signals are time dependent and periodic. The advantage of time averaging over a cycle is that it eliminates time derivatives and simplifies the calculations considerably. In the following discussion, time-averaging is denoted by a bracket. In general, <X(t) Y(t)> is not equal to <X(t)> <Y(t)>. In what follows, a good approximation to <Q²> in terms of <Q>² is obtained. It can be shown that in the case of coronaries (and probably mid-sized arteries) one may define a Pulsatility Index parameter PI. It is well known that the instantaneous pressure drop across a stenosis, ΔP(t), is closely related to the instantaneous flow through the formula Δ = Kβ(t f + K₂Q(t) + K

_Equation ,

the K's being functions of the geometry of the stenosis. (See, for example, Young DF and Tsai FY: "Flow characteristic in models of arterial stenosis: 1. Steady flow." Journal of Biomechanics 6: 395-410, 1973 and Young DF and Tsai FY: "Flow characteristic in models of arterial stenosis: 2. Unsteady flow" J of Biomechanics 6: 547-559, 1973).

Averaging this formula over a cardiac cycle and assuming periodicity (or very close to it) over the cardiac cycle, it follows that

(AP) = K, (Q²) + K₂(Q) The Pulsatility Index, PI, is now defined as the ratio of the mean square flow to the square of the mean flow. Clearly,

where

<Δ ) It is difficult to estimate a. However since — —j- = 1 - 1 , it follows that to

a very good approximation

since the pulsatility is a number close to unity in any case. For example, from a published plot of a left coronary artery and blood flow rates it was found that Pl=1.15 (Fig.4 in Back LD, Radbill JR and Crawford DW:

"Analysis of pulsatile viscous blood flow through diseased coronary arteries of man" Journal of Biomechanics 10:339-353). Entry Cone Calculations

In entry cone 105, the flow is accelerated and the velocity profile becomes blunt. The CFD method presented hereinbelow is essentially a parabolized Navier-Stokes, and the simulations are consistent with the profile of steady state flow in an annulus with β ___ 1.2 and γ ≡ 1.5 consistent with the values appearing in Table 1. The wire effect compensates for the thinning of the boundary layer caused by the acceleration. In the case of very blunt stenoses (i.e. close to 90 degrees slope), the velocity profile is blunt and vortices appear in entry cone 105 which invalidate the CFD method. A calculation method is described for these stenoses later on.

The CFD method includes receiving as input the pressure curve, flow Q and pulsatility index PI (plus density, viscosity and wire diameter) and computing the geometrical profile and the velocity profiles in entry cone 105 up to separation point S in exit cone 107. For each value of flow Q, a different geometry is produced which cannot be deduced from entry cone 107. The information which enables the computation of flow Q comes from exit cone 107 as explained further hereinbelow.

When flow Q is independently measured, for example by introducing an additional wire such as a Doppler wire, the geometry is unambiguous. If the geometrical parameters of entry cone 105 of stenosis 111 are known, for example, by angiogram, then flow Q can be obtained solely from a segment of the pressure curve containing the maximal and absolute pressures.

Exit Cone Calculations An expansion of stenosis 111 leads to instabilities of flow Q and to separation beyond the MLD. Only in the case of very slow flow and a small angle do the streamlines remain attached at all times. Once the flow has separated, it reattaches further downstream, the distance of which is on the order of centimeters. Separation occurs because the adverse pressure gradient appearing in the expansion repels the streamlines close to the wall. Mathematically, at separation point S, the wall shear stress WSS initially vanishes, and crosses zero again at reattachment point R. The following equation shows that three additive effects determine the slope of the pressure curve: a change in area A, a change in velocity profile and a frictional force on the boundary (wall and wire), F(z).

Separation point S and reattachment point R are points at which the frictional force vanishes. S is detected by the CFD method as the point where the derivative of the tangential velocity is zero at the wall. Between E and R, there is no area change so that dP _{2 nτ} dβ 1 1 — = -Q p - PI — = F(z) dz dz A² (z) A(z)

At point S, β is larger than its fully developed value since not only is the velocity zero at the wall but so is its tangential derivative. Therefore β should be a decreasing function. The first and second terms are both positive since the velocity close to the wall in the vortex is negative. Thus, it is apparent that pressure increases. Near point R, the first term is very small and the second term tends to zero due to reattachment. Beyond R the wall again exerts a force on the fluid and the pressure gradient becomes positive again. Hence, the reattachment point is the absolute maximum following the absolute minimum of the pressure curve. Next, at E the first term in the equation defining the slope of the pressure curve (dP/dZ) vanishes since there is no area change. Distal to E, the following holds:

The pressure gradient is observed to drop from an order of 10 mm Hg/cm before E to an order of 1 mm Hg/cm after E. There is a small but easily identifiable maximum at E. The preceding analysis leads to the following criterion for detecting the end of a stenosisThe end of the stenosis is located at the first maximum following the absolute minimum in the pressure curve. When this maximum does not exist, the end of the stenosis is defined as the leftmost point on the recovery plateau, but this occurs in a minority of cases.

Detection of this point can be accomplished using the following method. First the absolute minimum of the pressure curve and the absolute maximum following the absolute minimum (point R) must be determined. Starting from point R, step intervals are taken in the negative direction (i.e. to the left) until the point of absolute minimum of the pressure curve is reached. At each interval (for example, every 1 mm along the length of the tube) least square line is drawn from that interval to R (line 1) through the pressure curve and a straight line is drawn to the absolute minimum (line 2). The end of the stenosis is defined as the interval at which the ratio of angular coefficients of line 1 to line 2 is a maximum.

Computation of Flow Q

Using only the pressure curve pertaining to entry cone 105, the mean velocity in the MLD leads to a derived value for flow Q. A relationship, based on exit data is thus derived, thereby determining Q.

In addition to the kinematic conditions of zero velocity at the wall and on the wire and the integral of the axial velocity being equal to Q, the velocity profile must also have zero WSS at the wall. Thus, dw/dr = 0 at S and R. This leads to an assumption that S and R have similar velocity profiles. If one also assumes that dividing line 108 acts as a line of zero WSS, then the vortex trapped between the dividing line and the wall is a region of "dead water" isolated from the "jet" flowing outside. Therefore, it can be assumed that in the jet situated between the dividing lines the following three conditions are satisfied:

1. Conservation of profile between S and R;

2. No energy loss; and

3. Laminar flow.

Let subscripts S, R, and MLD denote quantities at S and R, respectively. For example, A_s and w_s denote the area and axial velocity at S.

The latter assumptions imply that the rate of work done by the pressure at S

+ rate of influx of kinetic energy at S = the rate of work done by the pressure at R + rate of influx of kinetic energy at R. or

Neglecting the kinetic energy contribution at R and rearranging:

Since A_s - W_s = A_R ^■ W _R = Q , and the expression in brackets is γs which is a quantity computed in the CFD method and depends on Q, that is, γs = γs (Q)> one obtains:

The density should be multiplied by PI as shown hereinbelow. Finally, an equation for the mean velocity at S is obtained:

Using conservation of flow,

Note the explicit dependence on Q. As mentioned, for each value of Q, the MLD area and the area at S are computed. In the above equation, WMLD(Q) is the mean MLD velocity as computed from the exit part of the pressure curve which is independent of the data in the entry cone. Since the MLD velocity computed in both manners (entry and exit) must be equal, Q can be determined.

Thus, the steps for computing flow Q are as follows: 1. For n values of Q(i): i=1..n, the following quantities are computed: Diameter at MLD(i), Diameter at S(i), mean axial velocity at MLD(i), mean axial velocity at S(i), gamma at S(i), the position of S(i). Other quantities may be computed as well. This step is done at the entry phase of the curve. 2. The pressure at R is read.

3. W_s(Q(i)) is computed.

4. On a Q-velocity graph, the following points are plotted: Q(i), mean axial velocity at MLD(i) as obtained from entry phase (curve 1), and W_MLD(Q( ) (curve 2). 5. The abscissa at the intersection of curve 1 and curve 2 is flow Q. Measurement of a non-stenosed artery

The following is a method for measuring the diameter of a non-stenosed artery. In particular, the reference diameter (RD) proximal to the stenosis is obtained. A difficulty with pressure wires is that the measured pressure gradient proximal to a stenosis is extremely small: on the order of 0.1 mm Hg/cm. Such a small pressure gradient is impractical for deducing the RD. It is therefore desirable to increase the pressure gradient.

Increasing the pressure gradient can be achieved by increasing the aspect ratio of the annulus. One way to accomplish this is to place two pressure transducers a few centimeters apart on the external side of lumen catheter 30, or to place a differential pressure transducer which can measure the pressure between the two points. The diameter of an artery or vessel which is not stenosed is determined by inserting into the artery or vessel a catheter and/or a wire which contains one or more pressure transducers. Examples of such a catheter or wire are described hereinbelow.

Reference is now made to Figs. 8 and 9, which are schematic illustrations of the system used to measure a non-stenosed artery or vessel based on pressure measurements within the artery or vessel. As shown in Fig. 8, a pressure wire 60 (which may be a catheter and/or a guide wire) comprises two pressure transducers A and B. In another, embodiment as shown in Fig. 9, pressure wire 60' has a single pressure transducer A inside the external wall of pressure wire 60' used as proximal measurement point. A second pressure transducer B is located at the opening at one end of pressure wire 60' and is used as a distal measurement point.

The following computations are performed. The Poiseuille formula ΔP =128μ/π LQ/d⁴ yields the pressure drop Δp between two points a distance L apart along a cylindrical conduit (in this case, an artery) whose diameter is d in which a mean volume flow rate Q of blood with viscosity μ is flowing. Inverting the above formula yields d = (128μ/π LQ/ΔP)^1M. For a typical coronary artery with d = 3 mm, Q = 50 ml/min, μ =3.6-10³ (SI units), L = 2 cm, one obtains Δp = 0.23 mm Hg which is well beyond the present accuracy of a pressure wire.

However, if a tube in the form of a catheter or a pressure wire is inserted co-axially along the centerline of the artery, the pressure drop is multiplied by a factor F depending on the aspect ratio x of catheter diameter to artery diameter. The F factor increases without limit as the latter ratio approaches 1 where

The diameter of the catheter which is inserted co-axially may be 2.00-3.70 mm thick, 2.00-3.00 mm thick, 2.00-3.25 mm thick, 2.00-2.85 mm thick, 2.00-2.40, or 2.4-2.85 mm thick.

Reference is now made to Fig. 10, which is a graphical depiction showing pressure drop over the diameter of the catheter at rest flow and hyperemic flow. Pressure drop is shown for a rest flow of Q = 50 ml/min and hyperemic flow of 120 ml/min. Hyperemia is obtained by administering drugs that reduce the resistance of the vascular bed (e.g. papaverine or adenosine). For a guiding catheter of 7 french (= 2.3 mm) for example, the pressure drop is 20 mm Hg at rest and 50 mm Hg at hyperemia. Applying the "F-modified" Poiseuille formula, the diameter satisfies the following equation:

Eq. 4

Reference is now made to Fig. 11 , which shown the graphical solution of the above equation. Assuming that the true value of Q is 50 ml/min, and knowing Q with an accuracy of 40%, Q may vary between 30 ml/min and 70 ml/min. The lowest value of d is then given at the intersection of the 30 ml/min line and the 70 ml/min line.

The value of d is calculated to be between 2.85 mm and 3.15 mm, which has an error of 0.15/3 = 5%. Thus, measuring the pressure drop is not sufficient for deducing the diameter if the volume flow rate Q is not known. Thus, the RD must first be estimated either from, for example, an angiogram of the artery, or just by using known values. Since the PLM method is not very sensitive to the RD, one can expect ah accuracy for Q of greater than 30%. Further, due to the ^lA th power in the diameter formula, an accuracy for the diameter of at least 8% is obtained. Clearly, the better the accuracy for Q, the better the accuracy in diameter. A 20% accuracy in Q yields a 5% accuracy in RD, which is sufficient in a clinical setting, e.g. for stent implantation.

Blunt Stenosis Reference is now made to Fig. 12, which shows a schematic representation of streamlines in a blunt stenosis. Proximal to the stenosis, at station 1, the velocity is small (on the order of 0.1 m/s) and the kinetic energy, which is proportional to the square of the velocity, is about 2% and can thus be ignored in a first approximation. At some point before the stenosis, the streamlines bend in preparation for entry into the constricted area. At station 2, the profile is essentially blunt and must cross a constriction that is smaller than the entrance area, which is termed the Vena Contracta. At station 3, the flow expands suddenly in a new enlargement. Eventually the streamlines fill the stenosed section at station 4.

Expansions and contractions entail losses which are difficult to estimate. Qualitatively, the situation can be described as follows. Between stations 1 and 2, the flow accelerates which causes a pressure drop. Between stations 2 and 3, the constriction is relatively smooth, but the pressure drops further. Beyond the Vena Contracta, the flow decelerates and the pressure should increase. However, at station 3 the profile is practically blunt and the viscous forces are very large. In the range of the present Reynolds numbers (i.e. 100-300) the size of the Vena Contracta is very small. Beyond station 4 the flow development region begins in which the viscous forces transform the blunt profile into a parabolic profile.

The problem can be described as the determination of the pressure of a fluid entering a tubular annular conduit from a large tank in which the pressure is constant.

When the flow is highly accelerated, it is believed that the velocity profile is completely blunt upon entering the conduit. The problem of computing the pressure drop from entrance to the entry length has received considerable attention. The pressure drop can be expressed in simple analytical terms without the need for CFD calculations. One of the simplest formulae for the pressure drop can be found in the article by Mc Comas: "Hydrodynamic entrance lengths for ducts of arbitrary cross sections", Journal of Basic Engineering, pp. 847-850, 1967, as follows:

p - U Re d

'2 where Pi is the pressure at the entrance to the throat, where the velocity profile is assumed to be blunt, and P_e is the pressure at entry point (i.e., where the flow is fully developed). The constants C and K appear Table 1. The range of aspect ratios of interest is 0.15-0.8. Table 1 shows that for an aspect ratio 0.8, the right hand side of the previous formula is 95.920 times 5.96+0.6935=1.265 and for an aspect ratio of 0.15 the result is 1.400.

In the absence of wire the value is 3 (see the last row in Table 1). At the entrance to the stenosis, the mean velocity is on the order of 0.15 m/sec whereas in the throat of the stenosis it is, typically, close to 1 m/sec. The ratio of kinetic energies is thus (0.15/1)² = 2% so that one may assume zero velocity at the start of the stenosis. There is an additional pressure drop of 1/2-p-U² between the entry to the stenosis and entrance to the throat. Finally, there is a pressure drop due to friction in the entry cone ΔP_f and P ≡ ΔP = V p - U ² PI c ^Xe + κ + \ + ΔP_f

Re d

The factor in bracket varies between 2.26 to 2.40. It is difficult to estimate, a priori, the pressure drop due to friction AP_f. Based on the examination of a large number of stenoses, the friction can be included by choosing 3.1. However, other assumptions are possible as well. It is emphasized that this approach is valid only for very blunt constructions, akin to brutal occlusions with a sufficiently long throat (at least 3 mm). For normal stenoses, the friction increases to 50% for small flow and the present approach is not valid. The above equation can be inverted to yield a simple expression for the Mean Velocity:

The present formula can be used by the physician as a quick estimate of the velocity in any stenosis without the need for CFD. In-Vitro Experimental Analysis

Reference is now made to Fig. 13, which is a schematic depiction of a blunt stenosis used in an in vitro experiment. Variation 2 was performed where the pressure wire is advanced by 2 mm discontinuous steps. The fluid was a 40% glycerine/ 60% water solution with a density of 1100 kg/m.sec and the viscosity was 4.3-10^"3 kg m^"3. At each station, the wire was stopped for a few cycles of the pump, and the average pressure was computed yielding the pressure curve.

Reference is now made to Fig. 14, which shows the pressure curve obtained from the experimental setup depicted in Fig. 13. Here, the minimum of the pressure is also the end of the stenosis (12 mm). The break in the pressure curve (at 20 mm) is identified as the point of entry length, signaling the beginning of fully developed flow. The start of the stenosis is at 22 mm.

Thus the entry length is 2 mm. The pressure drop along the entry length is 37 mm Hg and the velocity in the stenosis for Pl=1.13 is: ΔP 37 - 133.4

W = = 1.60w / sec

/₂ P ' PI - ?>Λ 1 100 - 1.13 - 3.1

In the remaining part of the pressure curve, extending from 20 mm to 12 mm, the pressure drop is pure annulus flow. The pressure drop there is ΔP =72 mm Hg. Equation 4 is used, where the wire is in place of the catheter, and Q has been replaced by π/4-(d²-d_Wιre²)*

Table 2 contains a summary of the experimental results. Table 2

The techniques of this and the preceding section can be applied in industrial applications for measuring the flow in a pipe as well as its diameter. Basic Fluid Dynamics Equations for the CFD Method

The geometry and the flow are assumed to be axisymmetrical. In a cylindrical coordinate system (z,r,θ) the axial velocity (along z) is w(r,z,t), the radial velocity is u(r,z,t) and the azimuthal velocity is v, which is zero in the axisymmetric case. The Navier-Stokes equations read as follows: Continuity: du u dw 1 dv

— + — + + = 0 (6) dr r dz r d θ

Axial Momentum:

where p is the density (in blood p = 1000kg/m ), v= μ/p is the kinematical viscosity and μ is the viscosity coefficient (in blood p = 1000kg/m³, v = 3.6- 10^"6 m²/sec).

An order of magnitude analysis shows that for a "smooth" stenosis (where transverse variations of the stenosis are small compared to the length of the stenosis), axial variations of w are negligible with respect to radial variations of w. This implies that one may drop the last term in the axial momentum equation. In the same approximation, the radial momentum equation reduces to dP

= 0 dr

That is, pressure is assumed constant over the cross-section. Assuming the pressure to be periodical, one obtains upon averaging over a cycle the axial momentum equation

where a bracketed quantity is a time average (e.g., < w ). Note

that the time derivative has been eliminated automatically. The continuity equation reads d(u) (u) d(w) 1 dv _Λ dr r dz r dθ Clearly, (w ² \ ≠ (w )² , in general, so a new equation is needed for (w ² \ .

This is the classical closure problem. If one considers pulsatile flow in the Womersley approximation with a steady state and sinusoidal component, it can be demonstrated that if the Pulsatiliy Index is close to 1 (as in arteries), then there is a number π for which the approximate equality holds. An exception to this is at a location close to the wall or wire, where in any case the velocity is very small.

2

(w² (z, r, t)) = π ' (w(z, r, t))

Since Q is an integral of w over the cross-section, one has π=PI, the Pulsatility Index. Since the continuity equation is linear, it follows that u(z,r,t) is a linear function of w(z,r,t). Therefore, the first term in Equation 7a is quadratic in w as well so that

All in all, the momentum equation becomes (for the sake of readability brackets will be dropped) for axisymmetric flow

All the quantity are time-averages. Thus the sole effect of Pulsatility on the equation is to multiply the density by PI. Equations integrated over a cross-section The continuity equation and the axial momentum equation for general flow (including non-axisymmetric flow) assume pressure to be constant over a cross-section. Multiplying the continuity equation (Equation 6) by w and adding to the momentum equation yields:

Next the latter equation is integrated over the cross-section. The integral over r eliminates the second term identically because of the no-slip condition at r=a and r=R and by angular periodicity the fourth term vanishes in the left hand side. Hence

Clearly, the first term is 5Q/dt. The third term can be transformed using the definition of the momentum correction factor:

R (z)

\w²rdrdθ = β(_Zit)- -

In the integral of the right hand side an integration of r yields

The last term is proportional to the frictional shear force of the wall and the wire acting on the fluid which is denoted by F(z). Finally upon averaging over a cycle, one obtains, using the definition of the Pusatility Index an equation for the pressure gradient

assuming <Q(t)²β(z,t)> ≤ <Q(t)²><β(z,t)>.

When the profile is constant along a segment [z z₂] one obtains upon integrating the latter equation a formula for the pressure drop, ΔP: *p _p ^.pι^.β.ϋ* Λ? dz p-PI' β- U² +AP

2 A{z) f_t

where the frictional pressure drop ΔP_f has been defined. Stretched variable

The axisymmetric case is considered. It is difficult to solve the equations in curved boundaries. The equations are therefore mapped on a stretched grid in which the boundary becomes a straight annulus. A new variable, η, is defined such that at r=a (the radius of the wire) η=0 and at r=R(z) (radius of the stenosis) η=1 :

η = αr + β α = ; β = -α α _{Eq 10}

R - α Since u=u(η,z) and w=w(η,z) then

dw _ dw dη dw _ dw dα dβ dw dw -i dα dw r — + ^ dz dη dz dz dη dz dz dz x dHη >^*-αJ- dz dz using

dw R dw + dw dz (R-a) dη dz and du du dη du 1 du

— = '- = a — = dr dη dr dη R - a dη

The continuity equation bcomes

from which u can be explicitly calculated and the axial momentum equation reads dw 1 \ dP u d\nw

+ dz p-PIwdz R-a dη

R dw 1 d²w 1 dw

R-a dη p-Pl(R-a)² w[dη² η + a/(R-a) dη] Eq. 12

The axial momentum is parabolic in z and can be solved using a forward marching scheme and finite differences or finite elements. In this case, the finite differences method was chosen. The inputs to the algorithm are the pressure vector Pressure (Pos) and the corresponding axial position vector Pos as well as PI, μ, p and Q. Then, normally, at step n of the marching scheme one has schematically:

where w_n and u_n are vectors with indices in the radial direction. Contrary to the problem solved herein, usually the shape is given, i.e., R(n) and R'(n), whereas the pressure is unknown. The pressure can be determined by requiring that at each station z(n) the flow should be conserved. Thus pressure is fixed by the additional constraint

a since Q is known. Here the inverse problem must be solved: the pressure is known but the shape is not. Now the role of the constraint is to determine the shape of the stenosis. In other words:

There is only one pressure field (P) compatible with a given Q and geometry (R) compatible with a given Q and pressure field (P).

In order to solve the equation, the following steps must be taken. First, for k=1..length(pos)-1, the segment [pos(k), pos(k+1)] is divided into m equal sub-segments, [z((k-1) m+j),z((k-1) m+j+1)], j=1..m (e.g., z((k-1) •m+1)=pos(k) and z(k m+1)=pos(k+1)). Next, a vector of P is constructed, of length equal to z, by linearly interpolating pressure at the values of z. The index of z is denoted n. Once step n has been reached, w_n, u_n, P'(n) = (P(n+1)-P(n))/(z(n+1)-z(n)), and R(n) are known. The following steps must then be repeated until the desired accuracy is attained. On the first iteration wold= w_n, uold= u_n, Rold=R(n-1), R'= R'(n-1) (since one does not know R(n)), otherwise R'= (Rold-R(n-I))/ (z(n+1)-z(n)).

Equation 13 must then be solved. One way is by using the fully implicit Scheme, Cranck-Nicholson with θ =1 using the derivative of R of the previous step: R'(n) = R'(n-1) (since one does not know R(n)).

The constraint for step n+1 is solved using wnew , the best estimate of w_n+ι:

(Rnew- af = 0

which is a quadratic equation for Rnew.

Next the continuity equation for unew is solved using Rnew and wnew and R'= (Rnew-R(n-I))/ (z(n+1)-z(n)). New values are compared with the old: If I wnew -

< ε and \Rnew - Rold\ < ε then

W_n+ι = wnew, u_n+ = unew, R(n+1)=Rnew. At this point, the iteration is stopped. The method ends shortly after separation.

The following outputs are obtained: the vectors R and z, the axial and radial velocity field in the stenosis, MLD, the position of the MLD, β and γ at the MLD and the mean velocity at the MLD, the diameter at separation, the position of separation and the mean velocity at separation, and the wall shear stress.

PLM METHOD SUMMARY

Reference is now made to Fig. 15, which is a flowchart summarizing the PLM method. The first step 201 is measurement of the pressure in the stenosis from a distal to a proximal point. After measurement is done, step 202 involves data processing. During the data processing of step 202, physiological variations and noise are eliminated. The next step 203 is analysis of the pressure curve. The start of the stenosis and the exit point of the stenosis are determined visually. The position of the MLD is estimated (the estimation is correct for the majority of stenoses). The reattachment point can be determined as well. Once these parameters are determined, step 204 is the determination of whether the mean flow Q is known or measured independently (e.g. using a Doppler flow wire) or Q is unknown.

If Q is known, branch 205 is followed. In this case, the first step 205A is to measure the RD. The device for measuring non-stenosed arteries can be used since Q is known. Otherwise, the alternative step 205B of estimating RD from anatomical tables is taken.

If Q is unknown, the same steps are taken as above, but the device for measuring non-stenosed arteries cannot be employed. Since Q is unknown, the CFD Method must be repeated for N physiological values of the flow (e.g., for 20 ml/min to 130 ml/min by 10 ml/min intervals), as described in step 207. For each value of Q_n a different output "vector" is obtained: A Diameter curve along the stenosis R_n(z) where z is the axial position vector, axial velocity profile w_n(z) and radial velocity profile u_n(z), and Wall shear Stress WSS_n(z), MLD_n, Diameter at Separation_n. Using the pressure at the Reattachment point obtained in step 203 and the MLD's and Diameters at separation as a function of the Q-vector, the next step 208 is computation of Q. If the device for measuring non-stenosed arteries described hereinabove is available, the RD can be measured since Q is now known (step 208 A). The final step 209 is to apply the CFD method using the known value of Q. The final ouput includes the diameter curve along the stenosis R(z), axial velocity profile w(z) and radial velocity profile u(z), and Wall shear Stress WSS(z), MLD, Diameter at Separation. Experimental Section

In order to validate the foregoing methods of this invention, human trials and in vitro experiments were performed. Human Trials

Two human cases were studied using the invention. The trials relate to stenoses situated in the Left Anterior Descending (LAD) coronary artery. The detailed steps of the PLM method presented in the flowchart will be described for the first patient only since the process is identical in both cases. The human results are then compared with those of an angiogram using an edge detector algorithm (QCA).

Patient 1 Reference is now made to Fig. 16, which is a picture of the stenosis as viewed on an angiogram, together with the geometrical profile of the QCA system. The QCA plot on the right is a plot of diameter as a function of position. The x and y-scales are different and the flow direction is from top to bottom. The QCA system is a Philips Integris™ system and the wire is a Radi™ wire. It may be noted that because of the pre-stenotic dilatation seen on the graph, it is difficult, even with QCA, to determine where the stenosis starts.

The following data were obtained: pressure from Fluid Filled pressure sensor 31 and pressure sensor 40 (Radi pressure sensor) data. The sampling rate was 300 Hz and the pull-back velocity (retraction velocity) was 0.5 mm/sec. No RD was given, and a typical value of 3 mm was chosen for the artery based on a QCA measurement of 3.03 mm). The viscosity of the blood was not measured or otherwise provided. The following typical values for blood were chosen: μ=3.6-10^"3, and p = 1000 SI units. Reference is now made to Fig. 17, which is a plot of the pressure measured by the Radi wire as a function of sample number. The duration of the measurement is determined by dividing the length of the pressure data by the sampling rate. Thus, the duration of measurement is 24000/300 = 80 seconds. It is critical to keep the duration as low as possible and to avoid a second trial, so as to limit the risk to the patient. It should be noted that since the wire is pulled back from a distal position, distal measurements are taken first. There is a segment of corrupted data at around sampling number 22000, which may be ignored. A pressure plot is also obtained for lumen catheter 30 (not shown). Averaging over each period, the cardiac cycle average pressure signals are obtained.

Reference is now made to Fig. 18, which is a graphical depiction of the average pressures over a cardiac cycle. The upper curve and the lower curve are plots of averages over a cardiac cycle of the pressures as measured from lumen catheter 30 and the Radi wire, respectively. The plot has been inverted to conform with the direction of the flow (left to right).

The x axis shows the pulse number which indicates the position of the wire along the artery. The dots are the points used for interpolating the defective pulses using a cubic spline. The lumen catheter 30 records pressure changes caused by physiological changes such as flow variations, baseline and so on. Fluctuations in the pressure data from lumen catheter 30 (upper curve) appear in the pressure data obtained from the wire as well, but with different amplitudes. The fluctuations have the same phase but not the same amplitudes. The pressure measured by the FF enables elimination and correction of these variables.

The pressure curves as shown in Fig. 18 must be corrected for physiological variation, as follows. First, the pressure of the catheter Pc is modified by means of a last square third order polynomial to eliminate the changes on the duration scale of the PLM procedure (63 cycles in this case). This is done in order to rectify the catheter pressure curve, yielding a vector Pc(n). Let m be the average of Pc so that the fluctuation is Pc(n)-m. The fluctuation is amplified (or reduced) to be adapted to the fluctuations of the pressure measured by the wire, P_Wire(n). This is done by centering a window at each point of length equal to the average breathing time of the patient and computing the standard deviations in the window for Pc(n) and P_wir_e(n). Once this is done the two pressure are subtracted from each other. Reference is now made to Fig. 19, which shows a corrected pressure curve resulting from the above manipulation. This curve may then be used to determine the parameters used in the CFD method, such as entrance point, exit point, MLD, reattachment point, and others. The entrance point to the stenosis is usually observable on the graph.

In this case, between 0 and 4 mm the pressure gradient is approximately 2 mm Hg/cm, which is 20 times the typical pressure gradient in a non-stenosed artery. Therefore this part of the curve must be located in the stenosis. Apparently, in this case, the physician did not pull the wire sufficiently proximal to the stenosis. Thus, by default the start of the stenosis is set at position "0". The MLD is estimated as the absolute minimum of the curve, which occurs at 11.4 mm. The end of the stenosis is defined as the first maximum after the MLD. There is some uncertainty in determining the end of the stenosis. It seems to lie somewhere between two small peaks located at 14.5 mm and 15.8 mm. Thus, the main geometrical parameters have been identified.

In addition, two other relevant points can be identified, the reattachment point R and the separation point S. R is clearly identified at 35 mm, as the maximum following the minimum. S seems to be located at the point of inflexion immediately following the MLD. At this point, since the basic geometrical parameters have been determined, and the RD has been measured as described hereinabove, the CFD method may be employed. The CFD method is performed over a physiological range of flow (20 ml/min to 130 ml/min by increments of 10 ml/min). It may be done on a PC, and lasts less than 5 minutes. The results of the CFD method are shown in Table 3.

Table 3

Finally, the method for computing Q is applied, yielding a vector of velocities in the MLD. Reference is now made to Fig. 20, which is a plot of the velocities in the MLD determined by entry cone 105, as shown in curve 122 and by exit cone 107, as shown in curve 121. The intersection of the two curves gives a value for flow Q. In this case, flow Q is determined as 53 ml/min.

The results of the above analysis as compared to measurements obtained from QCA are summarized in Table 4. It should be noted that the start of the stenosis was not clear from the measurements taken. It will be appreciated that the values obtained from the PLM method are very similar to the values obtained from QCA.

Table 4

In addition to the values listed in Table 4, the PLM method also yields the axial and radial velocity fields, the WSS, and the Separation Point. Reference is now made to Fig. 21 , which is a graphical depiction of the profile as computed by the PLM method plotted together with the profile as measured by angiography. The plots show diameter over axial position. The data obtained by the PLM method are plotted on curve 126, and the data obtained by angiography are shown on curve 124. Since this particular angiogram is a projection on a single plane, there may also be angle effects due to curvature of the artery. Patient 2 The second patient was tested for the same artery. The same sampling rate (300 Hz), pullback velocity (0.5 mm/sec), RD, viscosity and density were used.

Reference is now made to Fig. 22, which is a picture of the stenosis as viewed on an angiogram, together with the geometrical profile of the QCA system, as for patient 1.

Reference is now made to Fig. 23, which shows the cardiac cycle averaged pressure curves. Curves 128 and 130 correspond to the Catheter and Radi wire, respectively. The dip in curve 128 demonstrates a slow physiological variation. In both curves 128 and 130, there are short-term variations of a few seconds duration having the same phase but different amplitudes, particularly in the distal portion. These variations were likely caused by breathing.

Reference is now made to Fig. 24, which shows the corrected pressure curve used as the basis for further calculations. In this case, the start of the stenosis is clearly identified at 5 mm. The MLD point is located at 17 mm, i.e., 12 mm from the start of the stenosis, which is in close agreement with the QCA data. The end of the stenosis is determined at a position of 19 mm (point E1) by the PLM method, while it is determined at a position of 21 mm (point E2) by QCA. The final peak, at 24.5 mm, is the reattachment point R. The results are summarized in Table 5.

Table 5

Reference is now made to Fig. 25, which is a graphical depiction of the profile as computed by the PLM method plotted together with the profile as measured by angiography. The plots show diameter over axial position. The data obtained by the PLM method are plotted on curve 132, and the data obtained by angiography are shown on curve 134.

In Vitro Testing

Reference is now made to Figs. 26 and 27, which show schematic depictions of an in vitro testing appraratus, constructed and operative in accordance with an embodiment of the present invention_^ The in vitro testing was done in order to verify the analytical results. Fig. 26 is a schematic diagram representing the fluidics system 51 for determining flow characteristics in simulated non-lesioned and lesioned blood vessels.

Fig. 27 is a schematic cross sectional view illustrating one part of system 51 in greater detail. As shown in Fig. 27, fluidics system 51 is a recirculating system for providing pulsatile flow. Fluidics system 51 includes a pulsatile pump 42 (Model 1421A pulsatile blood pump, commercially available from Harvard Apparatus, Inc., Ma, U.S.A.) Other suitable pulsatile pumps can be used as well. Pump 42 allows control over rate, stroke volume and systole/diastole ratio. Pump 42 recirculates a glycerine/water solution from reservoir 15 to reservoir 14.

Fluidics system 51 further includes a flexible tube 43 immersed in a water bath 44, to compensate for gravitational effects. Flexible tube 43 is made from Latex or Teflon and has a length of 120 cm. Flexible tube 43 is connected to pulsatile pump 42 and to other system components by Teflon tubes. All the tubes in fluidics 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 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.

Fluidics system 51 also includes a flowmeter 11 connected distal to the flexible tube 43 and a flowmeter 12 connected to bypass tube 45. Flowmeters 11 and 12 are suitably connected to an A/D converter 28. Flowmeters 11 and 12 are model 111 turbine flow meters, commercially available from McMillan Company, TX, U.S.A. In certain cases, an ultrasonic flowmeter model T206, commercially available from Transonic Systems Inc., NY, U.S.A is used.

Reference is now made to Fig. 27, which is a schematic cross sectional view illustrating a part of fluidics system 51 in detail. An artificial stenosis 55 made of a tube section, inserted within flexible tube 43 is shown. Tube section representing artificial stenosis 55 is made from a piece of Teflon tubing. The internal diameter 52 of the artificial stenosis 55 may be varied by using artificial stenosis sections fabricated separately and having various internal diameters.

Pressure is measured along flexible tube 43 using a pressure measurement system including pressure wire 60 commercially available from Radi Medical Systems, Upsala. A pressure sensor 40 is connected to a pressure wire interface unit, commercially available from Radi Medical Systems, Upsala. Pressure wire 60 is inserted into the flexible tube 43 via a connector 10, connected at the end of flexible tube 43. A pull back mechanism (not shown) is attached to pressure wire 60 in order to retract back pressure wire 60 by a precisely controlled distance.

Reference is now made to Fig. 28, which is a schematic depiction of fluidics system 51 included within the general system 41, operative and constructed in accordance with an embodiment of the present invention. The system 41 also includes a signal conditioner 23 model TCB-500 control unit commercially available from Millar Instruments. Signal conditioner 23 is suitably connected to pressure sensors for amplifying the pressure signals. Data acquisition was performed using a PC 25 (Pentium 586 Model) , with an E series Instruments multifunction I/O board model PC-MIO-16E-4, commercially available from National Inc., TX, U.S.A. I/O board was controlled by a Labview graphical programming software, commercially available from National Instruments Inc., TX, U.S.A. Pressure and flow data were sampled at 5000Hz at 10 sec intervals, , displayed on a monitor display 21 and stored on hard disk. Analysis was performed offline using Matlab version 5 software, commercially available from The MathWorks, Inc., MA, U.S.A. Experiment 1

Reference is now made to Fig. 29, which shows the shape of artificial stenosis 55 used in the first in-vitro experiment. The length of artificial stenosis 55 was 1 cm, the MLD was 1.5 mm, and it was and located 7.835 mm from the entrance. The measured flow was 89 ml/min and the Pulsatility Index PI was 1.04. The PLM method was performed, and the resulting geometrical parameters were compared with the actual artificial stenosis 55 geometry.

Reference is now made to Fig. 30, which shows ffie resulting pressure curve as measured from system 41 using artificial stenosis 55 as described above. The following values are otained: Start of stenosis occurs at 18.8 mm, end of stenosis occurs at 31.3 mm, and the stenosis length is 12.5 mm. The MLD position is estimated from the absolute minimum of the pressure curve to be at 27.8 mm and is confirmed by the CFD method. The results are summarized in Table 5.

Table 5

Reference is now made to Fig. 31 , which is a graphical depiction of the computed and actual diameter profiles plotted together. Comparison of the plots reveals that the computed diameter measurements were very close to the actual values. Experiment 2

A second in vitro experiment was performed under basically the same conditions as Experiment 1 , but with two changes:

1. The tube was made of latex; and

2. The stenosis was inverted having a short entry cone and a long exit cone.

Reference is now made to Fig. 32, which shows the corrected pressure curve for the second experiment. From the pressure curve the it can be ascertained that the start of the stenosis occurs at 17.7 mm and the end of the stenosis occurs at 27.2 mm. Hence the stenosis length is 9.5 mm.

It will be noted from the graph that the positions of the MLD estimated from the curve (23.5 mm) and computed by the CFD method (20.3 mm) are in disagreement. The computed position is more accurate, as explained hereinbelow. The discrepancy is due to the very large friction in the exit. Since the opening angle is very small, the initial section of the exit cone can be thought of as a throat with parallel walls. In this case a negative pressure gradient of 1.12 mmHg/mm would be obtained. This pressure gradient should compensate for the pressure increase in the initial section of the exit cone. In this particular example, CFD is indispensable for determining the exact location of the MLD. The results are summarized in Table 6.

Table 6

Reference is now made to Fig. 33, which is a graphical depiction of the computed and actual diameter profiles plotted together. The computed profile is shown on curve 138 and the actual profile is shown on curve 136.

It will be appreciated that the embodiments described hereinabove are described by way of example only and that numerous modifications thereto, all of which fall within the scope of the present invention, exist. For example, tubes may include any pipes or tubes in which flow and diameter measurements are important.

It will be appreciated by persons skilled in the art that the present invention is not limited to what has been particularly shown and described hereinabove. Rather, the scope of the present invention is defined only by the claims that follow:

Claims

What is claimed is:

1. Apparatus for pressure based determination of at least one parameter of a region of obstruction in a fluid filled tube, the apparatus comprising: a flexible elongated member having a portion insertable into said fluid filled tube; means for measuring pressure inside said fluid filled tube, operatively connected to said insertable portion; means for moving said insertable portion in a predetermined manner, wherein pressure is measured at various points during said movement; and a processor for processing said pressure measurements, wherein said processor uses said pressure measurements to inversely obtain parameters of a fluid dynamics equation, thereby providing said at least one parameter of said region of obstruction.

2. Apparatus as in claim 1 wherein said at least one parameter includes a geometrical parameter.

3. Apparatus as in claim 2 wherein said geometrical parameter includes a minimal diameter of said tube.

4. Apparatus as in claim 2 wherein said geometrical parameter includes a length of said region of obstruction.

5. Apparatus as in claim 2 whererin said geometrical parameter includes a percent area of obstruction.

6. Apparatus as in claim 1 wherein said at least one parameter includes a location of said obstruction.

7. Apparatus as in claim 1 wherein said at least one parameter includes a location of a minimal diameter of said tube.

8. Apparatus as in claim 1 wherein said at least one parameter includes a velocity.

9. Apparatus as in claim 1 wherein said at least one parameter includes wall shear stress.

10. Apparatus as in claim 1 wherein said insertable portion includes a guide wire.

11. Apparatus as in claim 1 wherein said insertable portion includes a catheter.

12. Apparatus as in claim 1 wherein said means for measuring pressure includes a pressure sensor.

13. Apparatus as in claim 1 wherein said means for measuring pressure includes a fluid filled pressure transducer.

14. Apparatus as in claim 1 wherein said means for moving said insertable portion includes a pullback mechanism.

15. Apparatus as in claim 1 wherein said parameters of said fluid dynamics equation include geometrical parameters.

16. Apparatus for pressure based lumen mapping, the apparatus comprising: a flexible elongated member having a portiorTlnsertable into said lumen; means for measuring pressure inside said lumen, operatively connected to said insertable portion; means for moving said insertable portion in a predetermined manner, wherein pressure is measured at various points during said movement; and a processor for processing said pressure measurements, wherein said processor uses said pressure measurements to inversely obtain parameters of a fluid dynamics equation for mapping said lumen, thereby obtaining a map of said lumen.

17. Apparatus as in claim 16 wherein said lumen map includes a geometrical parameter.

18. Apparatus as in claim 17 wherein said geometrical parameter includes a minimal diameter of said lumen.

19. Apparatus as in claim 17 wherein said geometrical parameter includes a length of a stenosis.

20. Apparatus as in claim 17 wherein said geometrical parameter includes a percent area of stenosis.

21. Apparatus as in claim 16 wherein said lumen map includes a location of a stenosis.

22. Apparatus as in claim 16 wherein said lumen map includes a location of a minimal diameter of said lumen.

23. Apparatus as in claim 16 wherein said lumen map includes a velocity.

24. Apparatus as in claim 16 wherein said lumen map includes wall shear stress.

25. Apparatus as in claim 16 wherein said insertable portion includes a guide wire.

26. Apparatus as in claim 16 wherein said insertable portion includes a catheter.

27. Apparatus as in claim 16 wherein said means for measuring pressure includes a pressure sensor.

28. Apparatus as in claim 16 wherein said means for measuring pressure includes a fluid filled pressure transducer.

29. Apparatus as in claim 16 wherein said means for moving said insertable portion includes a pullback mechanism.

30. Apparatus as in claim 16 wherein said parameters of said fluid dynamics equation include geometrical parameters.

31. Apparatus for determining a geometrical profile of a region of obstruction in a fluid filled tube, the apparatus comprising: a flexible elongated member having a portion insertable into said fluid filled tube; means for measuring pressure inside said fluid filled tube, operatively connected to said insertable portion; means for moving said insertable portion in a predetermined manner, wherein pressure is measured at various points during said movement; and a processor for processing said pressure measurements to obtain geometrical parameters, wherein said processor uses said pressure measurements to inversely obtain parameters of a fluid dynamics equation, thereby enabling creation of a geometrical profile of said fluid filled tube.

32. Apparatus as in claim 31 wherein said geometrical profile includes a minimal diameter of said tube.

33. Apparatus as in claim 31 wherein said geometrical profile includes a length of said region of obstruction.

34. Apparatus as in claim 31 wherein said geometrical profile includes a percent area of obstruction.

35. Apparatus as in claim 31 wherein said insertable portion includes a guide wire.

36. Apparatus as in claim 31 wherein said insertable portion includes a catheter.

37. Apparatus as in claim 31 wherein said means for measuring pressure includes a pressure sensor.

38. Apparatus as in claim 31 wherein said means for measuring pressure includes a fluid filled pressure transducer. ^•

39. Apparatus as in claim 31 wherein said means for moving said insertable portion includes a pullback mechanism.

40. Apparatus as in claim 31 wherein said parameters of said fluid dynamics equation include geometrical parameters.

41. Apparatus for pressure based lumen mapping, the apparatus comprising: a flexible elongated member having a portion insertable into said fluid filled tube; means for measuring pressure inside said fluid filled tube, operatively connected to said insertable portion; means for moving said insertable portion in a predetermined manner, wherein pressure is measured at various points during said movement; means for determining flow changes as a result of insertion of said insertable portion, thereby obtaining flow information; and a processor for processing said pressure measurements and said flow information to obtain parameters for mapping said lumen.

42. Apparatus as in claim 41 wherein said lumen map includes a geometrical parameter.

43. Apparatus as in claim 42 wherein said geometrical parameter includes a minimal diameter of said lumen.

44. Apparatus as in claim 42 wherein said geometrical parameter includes a length of a stenosis.

45. Apparatus as in claim 42 wherein said geometrical parameter includes a percent area of stenosis.

46. Apparatus as in claim 41 wherein said lumen map includes a location of said obstruction.

47. Apparatus as in claim 41 wherein said lumen map includes a location of a minimal diameter of said tube.

48. Apparatus as in claim 41 wherein said lumen map includes a velocity.

49. Apparatus as in claim 41 wherein said lumen map includes wall shear stress.

50. Apparatus as in claim 41 wherein said insertable portion includes a guide wire.

51. Apparatus as in claim 41 wherein said insertable portion includes a catheter.

52. Apparatus as in claim 41 wherein said means for measuring pressure includes a pressure sensor.

53. Apparatus as in claim 41 wherein said means for measuring pressure includes a fluid filled pressure transducer.

54. Apparatus as in claim 41 wherein said means for moving said insertable portion includes a pullback mechanism.

55. Apparatus as in claim 41 wherein said parameters of said fluid dynamics equation include geometrical parameters.

56. Apparatus as in claim 41 wherein said flow changes include blunt flow.

57. Apparatus for optimizing treatment for an obstructed blood vessel, the apparatus comprising: a flexible elongated member having a portion insertable into said blood vessel; means for measuring pressure inside said blood vessel, operatively connected to said insertable portion; means for moving said insertable portion in a predetermined manner, wherein pressure is measured at various points during said movement; and a processor for processing said pressure measurements to obtain geometrical parameters of said obstructed blood vessel, wherein said processor uses said pressure measurements to inversely obtain parameters of a fluid dynamics equation thereby providing optimal treatment for said obstructed blood vessel.

58. Apparatus as in claim 57 wherein said treatment includes stent positioning.

59. Apparatus as in claim 57 wherein said treatment includes drug delivery.

60. Apparatus as in claim 57 wherein said insertable portion includes a guide wire.

61. Apparatus as in claim 57 wherein said insertable portion includes a catheter.

62. Apparatus as in claim 57 wherein said means for measuring pressure includes a pressure sensor.

63. Apparatus as in claim 57 wherein said means for measuring pressure includes a fluid filled pressure transducer.

64. Apparatus as in claim 57 wherein said means for moving includes a pullback mechanism.

65. A system for determining a geometrical profile of a region in a blood vessel, the system comprising: a flexible elongated member having a portion insertable into said blood vessel; means for measuring pressure inside said blood vessel, operatively connected to said insertable portion; means for moving said insertable portion in a predetermined manner, wherein pressure is measured at various points during said movement; a processor for processing said pressure measurements to obtain geometrical parameters, wherein said processor uses said pressure measurements to inversely obtain parameters of a fluid dynamics equation, thereby enabling creation of a geometrical profile of said blood vessel; a display for displaying said geometrical profile.

66. A system as in claim 65 wherein said region includes a stenosed region.

67. Apparatus as in claim 66 wherein said geometrical profile includes a length of said region of obstruction.

68. Apparatus as in claim 66 wherein said geometrical profile includes a percent area of obstruction.

69. A system as in claim 65 wherein said region includes a non-stenosed region.

70. A system as in claim 69 wherein said geometrical profile includes a reference diameter.

71. A system as in claim 65 wherein said insertable portion includes a guide wire.

72. A system as in claim 65 wherein said insertable portion includes a catheter.

73. A system as in claim 65 wherein said means for measuring pressure includes a pressure sensor.

74. A system as in claim 65 wherein said means for measuring pressure includes a fluid filled pressure transducer.

75. A system as in claim 65 wherein said means for moving includes a pullback mechanism.

76. A system as in claim 65 wherein said geometrical profile includes a minimal diameter of said blood vessel.

77. A method for determining at least one parameter of a region of obstruction in a fluid filled tube, the method comprising the steps of: inserting a portion of a flexible elongated member with means for measuring pressure into said fluid filled tube; moving said portion from one point along said fluid filled tube to another point along said fluid filled tube while simultaneously measuring pressure along the way; obtaining pressure measurements taken during said movement; and using pressure measurement to inversely obtain parameters of a fluid dynamics equation, thereby providing said at least one parameter.

78. A method as in claim 77, wherein said step of moving is done using a pullback mechanism.

79. A method as in claim 77 wherein said step of using pressure measurements to inversely obtain parameters of a fluid dynamics equation is done by calculating geometrical parameters from said pressure measurements.

80. A method for pressure based lumen mapping, the method comprising the steps of: inserting a portion of a flexible elongated member with means for measuring pressure into said lumen; moving said portion from one point along said lumen to another point along said lumen, while simultaneously measuring pressure along the way; obtaining pressure measurements taken during said movement; using pressure measurement to inversely obtain parameters of a fluid dynamics equation, thereby providing said at least one parameter of said lumen.