As there is a great desire to employ artificial blood vessels in the treatment of vascular disease, finding a solution for declotting has challenged researchers^{3}. Although the cleaning of artificial vessels via mechanical, chemical, and pharmacomechanical approaches have gained popularity, they have some inherent limitations that cannot be avoided. In catheter-directed thrombolysis (a chemical process), in which blood clots are dissolved by injecting medicine, an increased risk of bleeding restricts the administration of thrombolytic agents for declotting, especially cases of mural thrombi^{30}. The time window for initiating therapy (between 3 and 4.5 h for tPa^{31}) is one of the strictest limitations^{32}.

Moreover, when a mechanical strategy is used, clots are eliminated by rotating devices. Directional thrombectomy has some demerits, such as distal embolization^{14}, its complexity, and the risk of damaging vessels^{17}. The rotation of abrasive tools at high speeds leads to an increase in heat due to friction or damage to the surface of the vessel^{14}. The most important concern in this regard is related to the size of fragments, which affects the risk of occlusion in other small vessels and requires collection systems to cope with this challenge^{33}. In addition, the tool heads are too large to penetrate small capillary vessels^{34}.

Furthermore, the forces required to guide catheters into the artificial vessels against the flow of blood circulation can lead to side effects^{15}. Most current approaches are not capable of cleaning the mural thrombi or biofilms attached to the vessel wall. Therefore, due to the unavoidable inherent limitations of current approaches, focusing on new approaches is likely more worthwhile than further developing previous ones.

### The possibility of NMT

The NMT process described here is based on a new principle and strategy to overcome the inherent limitations of traditional approaches. Employing nano-sized tools makes it possible to clean mural thrombi in all vessels with different diameters. Meanwhile, the principle underlying NMT and its construction is relatively simple, which makes it easier to use than other methods.

The monitoring of the whole process in the transparent tube reveals the details of the process (Fig. 5A). After injecting the nano-magnetic particles into the fluid flow and reaching the clot site, it was observed that the nano-magnetic particles follow the movement of the rotating magnet inside the artificial vein. After that, the nanoparticles ablated the clot in the position of the rotation magnet. During the process, the external magnetic field propeled the nano-magnetic particles—which served as abrading tools—to special sites that eliminated the need for catheters (the external magnet plays the role of catheter). In addition, NMT could greatly improve fluid flow in artificial vessels. The increase in fluid flow approaching that of a normal artificial vessel confirmed that the clot had been abraded by nano-magnetic particles. However, NMT never stoped the fluid flow in the vessel, while in the traditional approach, bypassing the blockage artificial vessel is required until a surgeon fixes the artificial vessels^{34}. Researchers have shown that magnetic nanoparticles can follow a moving magnetic field^{27}. It has been demonstrated that using a method for transporting the magnetic nanoparticles inside a microfluidic channel, where a rotating permanent magnet is employed to induces the rotation of magnetic field on magnetic nanoparticles, leads to the movement of MNP along the surface of a microfluidic channel^{35}. Magnetic nanoparticle clusters carrying drugs can be rotated across physiologic surfaces in response to a rotating magnet to augment the drug delivery in a special site^{36}. While, the researchers used the movement of nanoparticles to improve drug release in dissolving clots, in this study, the motion of nanoparticles under a rotating magnetic field was used to abrasive the clot.Nano-scale abrasive tools make it possible for particles to penetrate very small vessels to abrade clots. One of the main challenges of traditional thrombectomy is the size of the device head, which limits penetration into vessels^{37}. Meanwhile, because the particle size affects the immune system, NPs should be smaller than 100 nm so that they can exit the immune system^{38}.

### Mathematical model of NMT

It is observed from the initial in vitro tests that several parameters govern the NMT process and that significant output factors depend on them. Therefore, the theoretical study of NMT is vital to understand the mechanisms and physics behind the process. In addition, mathematical modeling would be valuable for evaluating and predicting the effects of individual parameters on the outputs of the process. The below relationship state:

Due to the physics of the process, the magnetic and mechanical equations are employed to describe the process.

$$ {text{MR}};{text{and}};{text{h}};sim ;{text{f }}(omega ,;{text{d}},;{text{N}},;{text{t}},;sigma ,;{text{B}}) $$

(2)

In the above relation, MR the clot removal rate, h is the size of fragments, (omega ) the frequency of magnet rotation, d the diameter of magnetic abrasive particle, N the number of magnetic abrasive particles effective in cutting operations, t time of process, (sigma ) the compression strength of clot, B the magnetic field density.

#### Magnetic model

Classical Maxwell’s equations descript the electromagnetism of modeling^{39} as follow:

$$nabla times mathrm{E}=frac{partial mathrm{B}}{partial mathrm{t}}$$

(3)

where E: electric field vector, B: magnetic flux density vector, H: magnetic flux intensity vector, J: electric current, density, D: electric flux density vector, and t: time.

Expression for force on a magnetic dipole moment in a magnetic field describes by Lorentz force equation^{28} as below:

$$overrightarrow{mathrm{F}}= overrightarrow{mathrm{J }}times overrightarrow{mathrm{B}}$$

(4)

From here, assuming that the moment of the magnetic particle is co-linear with the applied field. This is a reasonable assumption given the small size and high susceptibility of the magnetic particles.

Where F is the magnetic force due to magnetic field B and J is the current density.

The magnetic force Fz can be calculated from the equation^{39}:

$${mathrm{F}}_{mathrm{z}}= {upmu }_{0}mathrm{V}left(mathrm{M}.nabla right)mathrm{H}$$

(5)

The magnetic pressure and magnetic tension can be written as follows^{28}. V is the volume of the particle, M is the magnetization of the particles. The magnetic force can be calculated as below:

$$mathrm{F}=frac{{upmu }_{0}{mathrm{H}}^{2}mathrm{A}}{2}=frac{{mathrm{B}_{1}}^{2}mathrm{A}}{2{upmu }_{0}}$$

(6)

$${mathrm{B}}_{1}=frac{{mathrm{B}}_{0}^{2}}{2{upmu }_{0}}left(1-frac{1}{{upmu }_{mathrm{m}}}right)$$

(7)

where: A is the area of each surface, in m^{2}; μ_{0} is the permeability of space, which equals 4π × 10^{−7} T·m A^{−1}; B_{0} is the flux density, in T.

#### Modeling of cutting forces

To cutting the surface of plaque by magnetic abrasive particles, two prerequisites must be given, one is the penetrating force (normal force) (({F}_{V})), the other is relative to the moving velocity between abrasive particles and plaque surface that acts as a shearing force (horizontal force) (({F}_{H})). It is concluded that these two force components are the primary forces governing the process fluid. However, other force components, such as the capillary viscous force due to the presence of blood, the gravitational force are assumed to be negligible compared the other ones. A schematic representation of forces acting in the abrading process is shown in Fig. 1C.

$${mathrm{F}}_{mathrm{total}}=sqrt[2]{{({mathrm{F}}_{mathrm{H}})}^{2}+{({mathrm{F}}_{mathrm{V}})}^{2}}$$

(8)

$${mathrm{F}}_{mathrm{H}}={mathrm{F}}_{mathrm{total}}*mathrm{cos}left(mathrm{alpha }right)$$

(9)

$${mathrm{F}}_{mathrm{V}}={mathrm{F}}_{mathrm{total}}*mathrm{sin}(mathrm{alpha })$$

(10)

The vertical force leads to a compressive stress defined as below:

$$upsigma =frac{{F}_{V}}{mathrm{A}}$$

(11)

$$upsigma *mathrm{A}le {F}_{V}$$

(12)

$$upsigma *{mathrm{A}}_{mathrm{R}1}le frac{{mathrm{B}}^{2}}{2{upmu }_{0}}.left(1-frac{1}{{upmu }_{mathrm{r}}}right).{mathrm{A}}_{mathrm{R}}$$

(13)

where ({mathrm{A}}_{mathrm{R}1}) is the contact area of the magnetic micro/nanoparticles with the target surface, as shown in Fig. 3, the contact area is:

$${mathrm{A}}_{mathrm{R}1}=uppi (2mathrm{Rh}-{mathrm{h}}^{2})$$

(14)

$$upsigma le frac{{F}_{V}}{uppi (2mathrm{Rh}-{mathrm{h}}^{2})}$$

(15)

The height of a nano-particle penetrates on the plaque surface is defined by :

$$mathrm{h}=mathrm{ R}-left(sqrt{{(mathrm{R}}^{2}-frac{mathrm{Fn}}{uppi .upsigma }}right)$$

(16)

$$mathrm{h}=mathrm{ R}left(1-left(sqrt{1-frac{left(frac{{mathrm{B}}^{2}}{2{upmu }_{0}}.left(1-frac{1}{{upmu }_{0}}right)right)}{upsigma }}right)right)$$

(17)

The volume is cut by one nanoparticle is equal to the amount of penetration nano particles on the plaque surface as below:

$${mathrm{V}}_{mathrm{p}}=frac{1}{3}uppi {mathrm{h}}^{2}left(3mathrm{R}-mathrm{h}right)$$

(18)

The length of the path that the particle travels over the circle depends on the size of its movement on the array-diameter of the vessel-and the size of the nanoparticle as below:

$$mathrm{P}=uppi {mathrm{D}}_{mathrm{w}}mathrm{omega t}$$

(19)

The total removal volume is equal to the volume removed by a particle in the total number of particles effective in cutting operations:

$$Delta mathrm{V}= {mathrm{A}}_{mathrm{p}}.mathrm{P}.mathrm{N}$$

(20)

Therefore, the time required to remove the clot is calculated from the following formula:

$${mathrm{V}}_{mathrm{plaque}}={mathrm{A}}_{mathrm{P}}.left(uppi {mathrm{D}}_{mathrm{w}}.upomega right).mathrm{N}.mathrm{t}$$

(21)

In summary, the proposed mathematical modeling describes the relationship between significant process parameters including the magnetic field (B), the size of the nanoparticles (R) and the mechanical properties of clot ((sigma )) which influence the process.The relationship between the applied force on a nanoparticle and the magnetic field strength and particle size is shown in Fig. 10A. It is clear from Fig. 10A that the magnitude of the applied force increases with increasing nanoparticle size and magnetic field strength. Meanwhile, the relationship between the removal volume of the clot by a single nanoparticle and the magnetic field strength is shown in Fig. 10B. It is also clear from Fig. 10B that there is a direct relationship between the removal volume and the intensity of the clot and magnetic field strength and the size of the nanoparticles.

### Fragment size

Distal embolization is one of the main side effects and great challenges of mechanical thrombectomy^{14,17}. Rotating the abrasive tools at higher speeds produces debris, which increases the risk of occlusion in small vessels; thus, collection systems are required to cope with these challenges^{33}. Fragments should be small enough to prevent blockages in the smallest veins (which are 6–8 μm in diameter; the cells and platelets that can move through the smallest veins are 4 microns in size^{38}). It is clear from Fig. 6 that the average fragments are nano-sized. The fragments are small enough (less than 1 micron) to prevent blockage in the smallest veins. Therefore, the fragments produced during the NMT process will pass through the capillary system and will be taken up by the reticuloendothelial system. Furthermore, the mathematical model states that the fragment size depends on the vertical force ((mathrm{Fn})) and compression stress ((upsigma )), or, in other words, on the size of the magnetic nanoparticles ((mathrm{R})) and the applied magnetic field (B) as shown in Fig. 10C. As the fragment size is one of the main criteria in thrombectomy, other NMT parameters should be selected in such a way that large fragments are avoided.

### The effect of magnetic rotational frequency

Figure 7 shows that declotting in artificial vessels increases by raising the rotational frequency. The rate of clot removal depends on the relative motion between the magnetic abrasive particles and the surface layers of the mural clot. The increase in rotational frequency might cause an increase in tangential cutting force; hence, the peaks of clot layers are sheared faster, thus causing clot removal rate to increase. At the same time, the number of cutting edges per unit of time increases when the rotational frequency is increased. When particles travel at a high speed, the tangential force is great, and the chances of the abrasive particles indenting the clot surface and breaking down the micron hills of clot surface increase. But the increased rotational frequency causes the nanoparticles to fail to follow the magnetic field^{35}, resulting in a slipping or lessening process.

### Study of the probability of artificial vessel damage

Damage to vessels is one of the challenges of traditional atherectomy^{17}. Vessel injury or burning has been reported due to the fraction between the head of the device used and the vessel wall during high-speed rotation^{14}. An evaluation of the surface of artificial vessels by FESEM images show that if nano-magnetic particles contact the walls of vessels instead of the clot for 4 h, the rate of penetrating particles on the target surface (on the vessel wall) is at the nano-scale, as shown in Fig. 8 (different scale in Fig. 8A–D). In addition, no evidence of rupturing was observed in the artificial vessels.

### The fate of nanoparticles

The fate of nanoparticles is crucial to what will eventually happen to them or how they will exit the body. Interestingly, studies confirm that nano magnetic particles undergo metabolism equally in hepatocytes and macrophages^{40}. Researchers have shown that Fe_{3}O_{4} nanoparticles are primarily cleared from the blood by the reticuloendothelial system^{41} or lymph nodes^{40}. Intracellular metabolism plays a significant role in the elimination of nano-magnetic particles by the Kupffer cells in the liver, which are the primary site of iron metabolism^{40}. However, some parameters, such as dose injected, percent initially taken up, and the cellular distribution in the liver, are affected by the rate of iron oxide metabolism in the liver^{40}. The half-lives of different iron oxide nanoparticles in the blood for clinical use are between 1 h and 24–36 h^{41}. However, some particle properties, such as size, morphology and surface characteristics affect the clearance process^{42}. Interestingly, core–shell nano-magnetic particles exhibit different clearance mechanisms^{43}. Evidence confirms that nanoparticles smaller than 100 nm can be cleared from the body. Finally, as the size of nano-magnetic particles is between the 10 to 100 nm diameter in NMT, their fate (both are nano-scales) is likely clearance by the liver, kidney, and lungs. Alternatively, it is possible to design a magnetic needle that can remove magnetic nanoparticles. In the future, animal tests will be performed to determine the fate of nanoparticles in details.

The approach presented in this paper demonstrates the possibility of cleaning the inside of an artificial vessel from outside of the body without damaging the vessel and without producing dangerous fragments. The conceptual principles explained in this study could be used in other vascular depositions, such as the accumulation of lipids, white blood cells, fibrosis, calcification, and other materials in the internal layer of arterial walls; mural thrombi in deep vein thrombosis; and atherosclerosis. The proposed technique, when compared with other atherectomy approaches, faces fewer challenges to translate this technique into in vivo. In conventional atherectomy devices, the presence of a catheter, as well as the head of the tool, creates many constraints. NMT makes it possible to eliminate the need for a catheter, thus simplifying the process and making it non-invasive. The present data suggest that NMT may have several practical and conceptual advantages over current commercially available thrombectomy systems.

### Future work

Since our approach is concept-based, it can be applied not only to artificial blood vessels, but also to any kind of atherosclerosis, mural thrombi, calcification in vein, and other situations. Therefore, more research will be done in the future to make our approach suitable for the in vivo testing of different vascular depositions. In the future, the drug-carrying nanoparticles will be examined to enhance the efficiency of declotting by combining different chemical and mechanical mechanisms.