Patent Application
20230081060
Venous grafts or fistulas are tubular members used to move blood from one part of the body to another; for instance, an AV graft or AF fistula used as a shunt or bridge, moving blood from the arterial system to the venous system, such as used in dialysis. A fistula employs natural materials, such as a harvested vein, for the tubular member. For a graft, synthetic non-elastomeric but flexible materials such as plastics (Dacron, polyesters, PVC, polyurethanes, PTFE or Teflon, or eTeflon) are used to form the tubes. Flow rates in these shunts and fistulas are important, as variable flow rates can cause problems in the procedure using the graft. It is desired to have a graft that has improved flow properties, such as constant conductance flow along the length of the graft, for greater flow rates than those present in uniform diameter grafts which exist in the prior art.
The invention includes grafts whose radius grows monotonically along the length of the graft. It is preferred that the radius r grows with length L, so that r4/L remains constant. The invention includes grafts where the radius grows so that rn/L that remains constant, where n≥4.
The invention includes grafts that grow piecewise linear; for instance, a graft can have a setoff radii ri along the length at positions Li, where each radius and associated length is such that rin/Li is a constant, where n≥4 and the radius between sequential ri grows linearly with length. The grafts can include fitted radial plastic rigid rings at various positions along the length to maintain the shape of the graft. This piecewise growth can also be an approximation of rn growth with length.
Blood flow is normally modeled with Poiseuille's law and hydrodynamic relationships, including the expression between flow, pressure, and 1/resistance (or conductance), are as follows:
Poiseuille equation, Fluid flow, Q=ΔP/R where 1/R is the conductance and where R=8 μL/(π4), and where μ is the fluid viscosity
Rearranging, and combining the two equations:
Q=ΔP*(π/8)*(r4/L)*(1/μ)
where L represents the length of the cylinder (graft) (measured from the start of the graft), the last three terms represent the numeric, geometric, and viscosity factors, respectively. Q, or volumetric fluid velocity (m3/l) or flow, in the venous system, is generally measured in ml/sec or liters/min. These relationships can also be used for modeling volumetric flow. For instance, we constructed a constant conductance flow conduit or “unitary conduit” with an initial diameter of 11.2 mm expanding to an end diameter of 17.7 mm at a length of 6 cm. We compared volumetric fluid flow through this unitary conduit to a second uniform diameter conduit (non-unitary flow). A total of 4.5 liters of fluid with the same viscosity as blood were allowed to run through each conduit with ahead pressure of 25 mm Hg. The time taken to empty the 4.5 liters is shown in Table 1 below.
Thus, the flow rate for the unitary flow conduit was 103 mL/sec and was 68 mL/sec averaged) for the uniform radius conduit (constant radius). As can be seen, errors in the conduit or graft radius can have significant consequences on blood flow. The flow equations can be simplified further by inserting known values for π and μ (the viscosity of blood). For instance, for a conduit of 1 cm in length with a diameter of 11.3 mm at 1 cm, the formula further reduces to: Q=ΔP*1=ΔP. Such a “unitary conduit” will have a conductance of exactly 1, allowing flow to be directly proportional to the pressure head. This shows the power of expanding the radius of the conduit (the graft). Such a graft, of constant conductance flow (e.g., r4/l=constant=K). will be referred to as a unitary graft or a constant conductance flow graft.
A constant conductance flow graft is particularly useful in grafts that are short in length, typically 20 cm or less. Initial diameters of 6 mm to 14 mm are typically useful in some applications. An example is shown in
Some uses for grafts follow:
Too much flow or too little flow is a problem with grafts in the standard configuration of a uniform diameter graft. Too much flow may result in heart failure, putting too much load on the heart. Too little flow may not clear creatinine, urea, and other substances during dialysis, requiring longer dialysis times or more frequent dialysis. Too little flow also may result in graft thrombosis, a major problem with dialysis grafts at the present time. This requires surgery or interventions to reopen the graft and may ultimately lead to need for a new graft. Some patients run out of places where you can put such grafts in (you need a good artery and vein)—sometimes an indirect cause of fatality.
A low flow situation is suspected to cause a proliferation of fibrous tissue at the site of graft-vein anastomosis in some patients. This proliferation is similar to ISR and is suspected to be the cause of dialysis graft thrombosis in over 80% of the occurrences.
These surgically created artificial fistula are used to increase flow to keep a venous section, bypass, or graft open. Typically, they are used when inflow into a bypass (e.g., Palma femoro-femoral venous bypass) or graft (e.g., iliac vein graft with poor inflow) is considered poor. The A-V fistula may also be used after clearing a clot from a vein (thrombectomy) or scar tissue from a post-thrombotic vein (endophlebectomy). The fistula is meant to be temporary and is closed after a period of 6 weeks or so. Like dialysis grafts, the A-V flow can cause heart failure, but a more common problem is venous hypertension—increased venous pressure in the leg, the correction of which is often the original goal of the primary operation. Using an increased flow graft with a higher rate of flow in the above situations will help prevent thrombosis which may occur from low flow. The variability of flow is worsened where the length of the graft is set by the surgeon, possibly shorten the graft by cutting as desired. A change in graft length creates a change in flow delivered by the graft.
Venous or Arterial Grafts with Poor Inflow or High Rate of Thrombosis
While arterial grafts enjoy high pressure inflow, venous grafts do not because venous pressure is naturally lower. A constant conductance flow graft makes sense in order to preserve the pressure energy from degrading by use of the standard configuration grafts. Examples of such grafts include porta-caval, mesenteric-caval, axillary subclavian, femoro-femoral, femoro-iliac and iliac-caval venous bypasses.
The constant conductance flow grafts may also be useful in short length arterial bypasses where no venous substitute is available. A constant conductance flow graft will also be useful in arterial-arterial grafts where the “run off” or the downstream bed that it flows into is poor.
Grafts with Support
Grafts with rigid skeleton support, such as formed by placing rigid or semi-rigid plastic rings spaced along the graft length and glued to the graft exterior or interior, are useful as endoprosthesis. After deployment, the resident vessel assumes the standard configuration of these devices. They are usually deployed in distressed situations where flow is problematic, and the chance of thrombosis is high. A constant conductance flow configuration makes better sense in these locations.
One such example is a TIPS (trans jugular intrahepatic porta-systemic) shunt performed for portal hypertensions. The device used has a high thrombosis rate from low operating pressures. Using a constant conductance flow graft, employing external rings whose diameter also varies with position, would improve patency.
The concept is to keep the conductance or flow constant in the graft, which can be achieved by maintaining the geometric factor (r4/L) a constant value K in the graft. All examples used herein will have the graft grow after the first 1 cm of length, where the first cm has a constant radius, thus avoiding the ambiguity of examining r4/L as L→0. In this instance, the constant K the radius at 1 cm, R1, to the chosen geometric factor, divided by 1 cm, for instance, (R1)4/1. In reality, a graft starts in an existing conduit. That implies that the initial conduit/graft combination can be viewed as a single conduit. To determine the “effective length” of that portion of the conduit before the onset of the graft, we have L=ΔPr4*π)/(Q*8*μ). Consequently, the combined “conduit” at the beginning of the graft, has a length, and the length of the graft will never be zero. Use of a 1 cm starting length is arbitrary, but not unreasonable, as L should be small. Indeed, L can be estimated using r as the graft radius, the measured Q (such as from doppler sonar), and ΔP, within biological systems other than arterial, should be small, so L will not be very large. Hence, using a 1 cm constant diameter starting graft is arbitrary, but not unreasonable, as it unlikely changes L significantly.
As an example, consider a graft having a diameter of 18 mm (9 mm radius) with a starting length of 1 cm with a constant radius graft, then at the start of the constant conductance flow (at length 1), r4/L=(1.8/2)4/1=0.6561=K. This number K will be used as the constant K or constant value used in the reminder of the graft. that is, r4/L=(1.8/2)4/1=0.6561 in the remainder of the graft. Consequently, for a 2 cm long graft, the terminating radius would be (0.6561*2)−0 25=1.070 cm or a diameter of 21.4 mm. A 3 cm long graft would have a terminating radius of (0.6561*3)−0 25=1.18, cm, or diameter of 23.6 mm; a 4 cm long graft would have an ending radius of 12.7 cm, or a diameter of 254 mm; and for a graft length of 5 cm, the terminating radius would be 13.45 cm, or a diameter of 257 mm (an overall increase in cross-sectional area of about 123.3% (1.34/0.9)*2.
As used herein, the “downstream” end of the graft is larger. The downstream direction of flow in the venous system is closer to the heart. Alternatively, downstream is the lower pressure end of the graft. Downstream in the venous system is closer to the heart; in the arterial system, downstream is further from the heart. The “downstream” direction of the graft is the direction that increases in radius. This increase in diameter with length will assist to maintain improved flow in the graft, to prevent graft malfunctions like in-graft restenosis caused by ingrowth of clot/tissue which accumulates and lines the wall of the graft. In addition, as the graft delivers constant conductance flow, the initial upstream end can be a smaller diameter than would be needed in a uniform diameter graft, as a smaller initial diameter constant conductance flow graft can produce the same flow at the outfall or downstream end as that of a uniform diameter graft. We have calculated the length necessary for various diameters in grafts up to 5 cm length, in Table 2. The first cm in length of these grafts is of constant diameter.
As described, for a lengthwise cross section though a graft, the outer envelope preferably creases as a 4th order polynomial with the length. However, in some applications, such as in long grafts, fourth order graft growth (R4 growth or r4/l=constant) may present an ending diameter that is too large for the landing site. In this case, a graft that preferably expands monotonically with length, but less than R4 growth, will still provide a benefit, as the flow loss through such a graft will be less than that from R4 growth, but greater than a constant diameter graft. For instance, growth of “near constant conductance flow” is such that rn/n is constant, where n>4, provides such slower growth and flow than R4 growth.
For instance, grafts can be achieved with r5/l, r6/l or r7/l being constant. Faster growth, where 3≥n≥1, in parts of or all of the graft, can also provide benefits, as flow will be greater than that in constant conductance flow, and can be useful in areas of a graft where deposits are a concern.
Additionally, growth in the radius with length can occur piecewise. For instance, the graft's outer envelope may linearly increase between set graft radii, ri in piecewise steps. For instance, there could be a series of lengths Li where the radius ri is such that ri4/Li=constant. Between such 4th order radii, the graft could grow linearly or by other growth rates. Faster growth, and faster flow, such as linear growth (R1) can be used. Such a step wise growth approximates fourth order growth and may be more efficient to manufacture. However, this is not preferred as the flow will inconsistently change in the graft. While a 4th order polynomial increase (R4) is preferred for the outer envelope of the graft, any preferably monotonically increasing graft diameter with length will provide a benefit, in that radius growth with length helps offset reduced flow with length in the existing constant diameter grafts. An example is depicted in
In one embodiment, for a graft of selected length L1, the upstream and downstream terminating diameters are chosen, and the length and the growth factor Rn are determined to best fit the selected diameters. Alternatively, the outflow and ending diameter are chosen, and the length and growth factor selected to fit the selected parameters. Variations are possible, for a particular ending diameter, a variety of starting diameters, lengths and growths can produce the desired outflow.
Expansion with the chosen growth factor is depicted in
In one embodiment, for a graft of selected length L1, the upstream diameter areas well as the growth factor is chosen to provide the desired flows and ending diameter. For a particular ending diameter, a variety of starting diameters, lengths and growths factors can produce the desired outflow. Additionally, the invention includes grafts that have a portion that does not grow, for instance, the starting end of the graft, or the terminating end of the graft can be constant diameter, or both. Preferably the graft diameter is monotonically increasing over the length of the graft. Typical starting diameters can be 4 mm, 6 mm or 8 mm such as, for instance, dialysis. The smaller diameters are useful when tapping small veins.
Graft Growth with Length
Flow, as used herein, is fluid velocity. For instance, the growth of each graft segment in
Radius growth with length is within the scope of the invention. All such will provide a benefit, as the increasing graft diameter helps offset reduced flow with length. For instance, increasing radii with rn/L constant, where n>4 is constant overall or at specified intervals, and connecting those intervals linearly as shown in the graft in
For instance, fabrication of a graft with R4 growth will yield a gradually expanding tube that will double its radius at 16 cm length. In many applications, this growth is too fast, resulting in an ending radius that is too large for the application. A more practical formulation is to keep r5/l or even r6/l or r7/l constant over the length of the conduit after the first cm. This will yield longer tube lengths before the radius doubles (Table 3); the conductive performance (flow) will be less than the R4 growth, but still with improved flow, greater than that of existing uniform cylindrical grafts. The fractional constants necessary for the various integer radius exponents vs length are shown in Table 4 for a number of selected initial calibers. Note, n does not have to be an integer.
To demonstrate performance of slower growth grafts, the following experiment was done:
To test less aggressive growth graft designs, r° growth conduits, n>4. were designed using engineering software (Autodesk, Inc.; San Rafael, Calif.) and fabricated in a commercial 3D printer (Stratasys; Eden Prairie, Minn.).
The basic flow model consisted is of a header tank A with outflow controlled by a calibrated ball valve A1 (
The flow rates of expanding caliber conduits (r4-6) compared to traditional constant radius cylindrical conduits are shown in Table 5 and
In closed flows where the fluid completely fills the conduit and the flow is driven by a pressure gradient, the incidence of transition to non-laminar flow should be reduced, as in an expanding pipe, where rn/l is constant and where n≥4, the fluid velocity declines with length, reducing the potential for turbulence.
Accretive manufacturing (3-D printing) makes it much easier to fabricate expanding caliber grafts for biological use. There is a practical limit to the length of the graft depending upon location and use. Examination of Table 3 (length of conduit when initial radius doubles) and Table 5 (measured flow rated for the conduits tested without an air trap) suggests that up to ˜16 cm is practical for graft designs keeping r4/L constant. As shown in Table 6, for common iliac vein grafts of 14 mm diameter, the ending radius is calculated for various length conduits for different rN N=4, 5, or 6. with all conduits having an initial starting length (1 cm) of constant radius. As shown, combinations up to 64 cm length appear practical for grafts keeping r5/L constant. Longer lengths may be required for particular applications and are possible keeping r6/L or r7/L or higher r values constant. Fabrication techniques described above or known to those of ordinary skill in the art may be used to construct the Rn expanding stent/conduit.
aFlow separation did not occur in the constant radius conduits; Penrose air-traps did not affect these flows.
The Rn expanding caliber grafts may have an advantage over the traditional cylindrical prosthetics in the following areas of vascular surgery:
Inadequate flow for effective dialysis is a common problem with legacy dialysis grafts. A 3 mm radius 31 cm length conduit (which approximates commonly used dialysis grafts) yields 294 and 401 cc/min (10 &25 mm Hg input pressure) in the test bed. In comparison, the expanding configurations yield a 28-38% more flow (nearly 50% more flow with the Penrose air-trap). These flows are for 10- and 25-mm Hg input pressures, respectively. Quantitative flows in patients may be different for higher input pressures. While quantitative duplication of clinical flows in the test bed is not to be expected, the relative flow advantage is a useful indicator. Graft thrombosis and intimal hyperplasia likely will be reduced. Grafts are used to correct intimal hyperplasia at the venous end of dialysis grafts and fistulas. An expanding caliber graft may function better in these locations. Grafts are also used to correct intimal hyperplasia at the venous end of dialysis grafts and fistulas.
Venous grafts were used in veno-venous bypasses 50 years ago but have largely fallen out of use since the introduction of venous stents. Even then, it was known that grafts to be used in the venous system had to be larger (lesser resistance) than the ones used in the arterial system, as the pressure gradient is lower in the venous system. Prosthetic grafts are still occasionally required in complex reconstructions involving the large central veins. The expanded configuration may function better in these situations. Grafts are also used to correct intimal hyperplasia at the venous end of dialysis grafts and fistulas. An expanding caliber graft may function better in these locations.
Prosthetic use in arterial applications has declined as well, substituted by stents or autogenous material. Arterial grafts are still used to a greater extent in arterial bypasses than in the venous system. The expanded configuration is likely more efficient and may have improved long-term patency than the legacy design in these applications. Prosthetic grafts are avoided in peripheral arterial bypasses in general and particularly when the inflow or outflow is marginal reducing the pressure gradient to maintain flow. The expanded configuration may find a place for use in these challenging situations. Use of prosthetics is faster (less operating time) and less laborious than harvesting or constructing autogenous grafts from autologous material (which may be unavailable).
These composite stent/grafts are used in specific anatomic locations where the prosthetic is subject to external compression/stress. Flow characteristics of the expanded configuration may function better where both the stent and graft expand equally. Indeed, the stent expansion may be greater, but the graft will control/limit the expansion of the stent. Both should be expanding in diameter with length.
As will be understood by one of skill in the art, the length in in rn/l in constant conductance flow grafts (r=4√l), or the near constant conductance flow conduits, is measured from the start of the conduit, not the start of the expanding section. This provides for smooth transition of flow through into the expanding section. Conduits (r=n√l where n>4) can have endless applications where the flow rate through the application is an important factor to the functioning of the system, including the arterial system. In biological systems, use of expanding radii grafts should greatly reduce restenosis in these grafts.
In one embodiment, for a graft of selected length L1, the upstream and downstream terminating diameters are chosen (for instance, the upstream diameter>4 for slower growth and slower flow rates than a constant flow design. If the resulting outflow is too fast, a short ending segment might have a constant diameter, or a reducing diameter section, to slow the flow. Alternatively, the starting and ending diameters can be chosen, the selected length, and then solve for the best filling value of n in rn/l, over the selected length of non-constant section. Note that n does not have to be an integer value. A similar design can be utilized to construct long grafts in steps or segments, by choosing the segment ending diameters and selecting the geometric expansion to be used between the segments.
Grafts can have one or more portions that are constant conductance flow or near constant conductance flow. Consider a graft that has an initial radius r1 of 15 mm (or 1.5 cm) and remains constant for two cm. For the next 5 cm, the graft is unitary, with the starting radius of 1.5 cm. In other words, for the next 5 cm, r4/L remains constant, where L is the distance from the starting point of the graft, (here L starts at 2 cm), that is, to the 7th cm of the graft, L=7 cm. At the end or the unitary graft (or near unitary) portion, the radius in the final portion may further increase, remain constant or even decrease (not preferred). For instance, at the end of the unitary portion described above, the graft may continue for another 2 cm but over that 2 cm, the radius may smoothly decline, such as linearly (e.g., a first order polynomial), to end at the normal radius of the resident vein, and thus allow for a smooth flow transition from the end of the unitary portion to the end of the graft.
The considerations above for long grafts, that is, having a portion of the graft with near constant conductance flow, are also applicable to long venous stents, or stent stacks (consecutive stents placed end to end possibly with overlap).
A stenosis is often treated with a stent. Stents are generally cylindrically shaped devices which function to expand when deployed. Stents may be balloon expandable or self-expanding. The balloon expandable stent is a stent that is usually made of a coil, mesh, or zigzag design. The stent is pre-mounted on a balloon and the inflation of the balloon plastically expands the stent with respect to the balloon diameter. Self-expanding stents are tubular devices stored in an elongated configuration in what is called a delivery system or applicator. The applicator is introduced percutaneously into the body into a vessel and guided through the vessel lumen to the location where the stent is to be released. Upon release, the stent material auto expands to a predetermined size. Auto expansion is rather weak in many self-expanding stents. This may require pre-dilatation of the stenotic lesion with a balloon of appropriate size before the stent is deployed to enable it to expand to its intended size. In some stents, auto expansion must be assisted with ‘post dilatation’ for full expansion of the stent to occur.
Commonly used self-expanding stents are braided stents, or laser cut stents. A braided stent is a metal stent that is produced by what is called a plain weaving technique. It is composed of a hollow body, which can stretch in the longitudinal direction and whose jacket is a braid made up of a multiplicity of filament-like elements which, in the expanded state of the braided stent, intersects a plane perpendicular to the longitudinal direction at a braid angle. Laser cut stents are constructed from a tube of material (most frequently, nitinol, a nickel titanium alloy), and stainless steel, cobalt, etc. that is laser-cut during production to create a meshed device. The tube is comprised of sequential aligned annular rings that are interconnected in a helical fashion. The tube is compressed and loaded into the delivery device and expands to original size when released. Nitinol, which has thermal memory, may help stents made of this material expand into position when exposed to body temperature after delivery. Compared with self-expanding braided stents, laser cut stents provide more accurate stent deployment with less foreshortening. Laser cut stents are much less subject to foreshortening but are probably less rigid than braided stents.
The stent, after expansion, is intended to restore the occluded vessel to normal or near normal flow conditions in the stented area. In the arterial and venous system, the stented area should have smooth laminar blood flow of uniform velocity. To help avoid restenosis, or the depositing of material in the stented vein, and the resultant re-occurrence of an occlusion, maintaining adequate flow through the stent is desirable. A stent with a growth portion that is placed over the stenotic area should alleviate these issues.
The iliac veins are the most common location for stent placement. The common femoral vein is ≈12 mm in diameter, the external iliac vein is ≈14 mm in diameter; and the common iliac vein is slightly larger at ≈16 mm diameter (venous caliber naturally scales up as tributaries coalesce). A gradual increasing growth configuration will likely provide greater flow than current constant radius cylindrical designs. In-stent restenosis is a substantial problem in iliac vein stents and correlates with low inflow. It is believed that the expanding stent will ameliorate these problems of legacy designs. The most frequent cause of stent thrombosis is poor inflow; outflow problems are less frequent causes. In either case, the pressure gradient (ΔP) is reduced, causing flow decrease. A greater volumetric flow rates may be possible with a reduced pressure gradient if the expanded configuration stent is used.
In the iliac system, a stent conduit extending from the common femoral vein (12 mm diameter) to the external iliac (14 mm diameter), to the common iliac (16 mm diameter) can be as much as 18 or 19 cm long. As seen in Table 3, at R5 growth, the stent or stent stack through the iliac veins, starting at the common femoral vein, would double to 24 mm at 32 cm. Hence R5, R6 or R7 stent growth could easily be used for stenting the entire iliac vein system. A first stent in the stack would have a starting diameter of 12 cm, and overlapped with the second stent, would have an ending diameter of about 14 cm. Given the length to be covered by the first stent, solve for the best fitting value of N. Repeat for the second and third stents in the stack, the solved for N for each will likely not be the same. Note that N in RN growth does not have to be an integer. Such long stents or stent stacks may jail the hypogastric vein, which is well tolerated.
As with grafts, stents can be constructed which are piecewise approximation of stents of R5 and lesser growth, where certain radius are sized to fit rn/L constant, n>4, and these radii connected by linear radius growth between adjacent radii. This linear approximation can be readily constructed with existing Z stents, as is shown in
Indeed, longer stents can be constructed with sections of different growth, such as constant radius (no growth) sections to an R5 growth or higher section and ending with a no growth or R4 growth section. The preferred growth pattern is monotonic growth (non-decreasing growth) over the length of the stent or graft. This piecewise approximation in RN growth may make manufacturing easier, for instance, building a long stent using sections of Z stents.
Stents (r=n√l where n>4) can have endless applications where the flow rate through the application is an important factor to the functioning of the system, including the arterial system. Stents are also used to correct intimal hyperplasia at the venous end of dialysis grafts and fistulas. An expanding caliber stent may function better in these locations. In biological systems, use of expanding radii stents should greatly reduce restenosis in these stents. Note for stents—stent balloons 500 are needed that, when expanded, match the form of the expanded stent. See
As described, it is preferred that in a growth section where RN/L is constant, that L is measured from the beginning of the conduit system (graft or stent). If you measure the length from the start of the growth conduit, then the growth in this case is not identical to that when length is measured from the start of the system. This occurs because r=n√(K1) in the growth section. The radius is smaller in a growth section when L is measured from the start of the conduit system. Note also that the growth constant K is a different value in the two systems, as K=(rs)n/Ls, where rs is the radius at the start of the growth section, and Ls is the conduit length at the start of the growth section.
As an example, consider a two-conduit system, each 10 cm length, with a 1 cm overlap, where the first conduit is constant, radius of 2 cm, the second conduit grows at R4 after the 1 cm overlap.
4{square root over (KL)} = 4{square root over (1.6 * 19)} = 2.34
4{square root over (KL)} = 4{square root over (16 * 10)} = 3.55
Clearly, the two measurements of L result in a different growth profile.
Measuring L in a growth section from the start of the conduit in a growth section is more manufacturer friendly. Otherwise, the manufacturer will have to custom build each conduit, with an understanding of the length of the conduit system prior to the conduit in question. Measuring L from the start of the conduit system more closely emulates a single conduit system, particularly in performance.
You can build a conduit system using growth conduit sections where the growth is referenced from the start of the growth conduit. Such a conduit system will have different growth profile and different performance characteristics than one where length L is measured from the system start. Care should be taken understanding length measurement which system was used.
As can be seen, Constant conductance and near constant conductance flow grafts and stents allow the surgeon to design a graft and/or stent with suitable starting and ending diameters, with an expanding portion in the graft or stent therebetween, providing for improved flow grafts or stents. The variations are almost limitless, particularly as n in rn does not have to be an integer value. Indeed, if n is an integer, rn can also represent a nth order polynomial. A graft or stent can be constructed with two or more portions, each portion expanding, but with possibly different growth factors; or, if the final flow is too fast, a final section may have a greater decreasing diameter section to produce the desire outfall or outflow, but with possibly greater growth factors, such as r2 or r3. It will be apparent to those skill in the art that here are other biological applications for the improved flow graft and long stents.
This application is a continuation of PCT/US/2021/071126, filed Aug. 5, 2021, which application claimed the priority benefit of provisional application No. 63/062,764, filed on Aug. 7, 2020, both of which are incorporated by reference.
| Number | Date | Country | |
|---|---|---|---|
| 63062764 | Aug 2020 | US |
| Number | Date | Country | |
|---|---|---|---|
| Parent | PCT/US2021/071126 | Aug 2021 | US |
| Child | 17817567 | US |
