Motion of Magnetic Microcapsules Through Capillaries in the Presence of a Magnetic Field: From a Mathematical Model to an In Vivo Experiment

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

1. In the second and third paragraphs of Introduction, a number of samples about in vitro study were listed, but it is unclear why to give such samples. Some comments are necessary.
2. The research on animals can be divided as small animals and large animals.
3. Anyway, too many samples are described in Introduction, which seems not closely related to the theme of this paper.
4. Why to use an inhomogeneous magnetic field but not homogeneous magnetic field in this study?
5. The preparation of Magnetic polyelectrolite capsules is described too simply to be repeated by others. The detailed steps and the amount of each component should be supplied.
6. What is the actual component of CaCO₃@PSS-DOX/(PAH/NpFe₃O₄)₃/ (PAH/DexS)2? How to load NpFe₃O₄?
7. In short, the relationship between in vitro and in vivo experiment result is unclear.

Author Response

We would like to express our gratitude to the reviewer for the careful analysis of our work. We have taken in account all comments and improved our article in accordance with them. We have also improved the language throughout the manuscript. Below are the responses to the comments and changes that we have made.

Comment 1.

In the second and third paragraphs of Introduction, a number of samples about in vitro study were listed, but it is unclear why to give such samples. Some comments are necessary.

Response to comment 1. 

These references about in vitro studies are introduced to highlight the current state of research and to demonstrate the existing problems in the field. We have added a clarifying sentence at lines 120-125 of the revised manuscript:

“Despite the large number of theoretical studies and experiments conducted both in vitro and in vivo, a gap remains between the theory and both types of experiments. This gap is a significant issue, primarily because real in vivo conditions are nearly impossible to replicate in laboratory setups, and secondly, because quantitative analysis of experimental results is most valuable when it can be compared with theoretical predictions”.

Comment 2.  

The research on animals can be divided as small animals and large animals.

Response to comment 2. 

In the revised manuscript, we grouped the references related to studies on small and large animals in the third and fourth paragraphs of Introduction, respectively.

Comment 3.

Anyway, too many samples are described in Introduction, which seems not closely related to the theme of this paper.

Response to comment 3. 

We have removed references to papers in which the assessment of magnetic targeting of nanoparticles is not the main goal of the work. This allowed us to focus the reader's attention on the studies closest to the topic of the article. In particular, references to papers 14-16 of the first version of the manuscript have been removed.

 

Comment 4. 

Why to use an inhomogeneous magnetic field but not homogeneous magnetic field in this study?

Response to comment 4. 

We have considered an inhomogeneous magnetic field in our study because the magnetic force acting on the capsule is proportional to the gradient of magnetic field. It follows from equation (9), which was used previously by many authors. Therefore, the magnetic force vanishes in the homogenous magnetic field.

 

Comment 5. 

The preparation of Magnetic polyelectrolite capsules is described too simply to be repeated by others. The detailed steps and the amount of each component should be supplied.

Response to comment 5.

We have described the method for obtaining magnetic polyelectrolyte capsules in more detail. These are lines 155-198 in the Materials and Methods section of the revised version of the manuscript.

 

Comment 6. 

What is the actual component of CaCO3@PSS-DOX/(PAH/NpFe3O4)3/ (PAH/DexS)2? How to load NpFe3O4?

Response to comment 6.

The method of Fe3O4 nanoparticle loading is described in line 187-190. To clarify the composition of the substance, we have added the following fragment in lines 194-198:

“Thus, we obtained capsules containing a core consisting of calcium carbonate and polystyrene sulfonate with adsorbed doxorubicin. The shell was represented with several layers of oppositely charged polymers (PAH and DexS) and iron nanoparticles incorporated between them. The capsules can be described by the following formula: CaCO3@PSS-DOX/(PAH/NpFe3O4)3/(PAH/DexS)2”.

 

Comment 7.

In short, the relationship between in vitro and in vivo experiment result is unclear.

Response to comment 7.

To clarify this issue, we added the following fragment in lines 570-576:

“It is worth noting that the in vivo and in vitro experimental results are consistent: in both cases, the number of deposited capsules was lower than the theoretically calculated number of capsules reaching the capillary wall under the influence of the magnetic field. In both cases, the introduction of a deposition probability function containing only two fit-ting parameters allowed for a satisfactory approximation of the experimental curves for all measured points”.

Reviewer 2 Report

Comments and Suggestions for Authors

This research articles describes "Motion of magnetic microcapsules through capillaries in the  presence of a magnetic field: from a mathematical model to an in vivo experiment"

 

The subject of the study is unique and novel. It will be helpful if the authors should explain the following comments………

 

• In the abstract section lack any data about the obtained results.
• The authors did not report how iron oxide nanoparticles are obtained.
• The size of administered microcapsule should be specified.
• The conclusion is too long.
• Is the micro sized capsule is suitable for intravenous administration?
•  What about the in vivo distribution of administered system in the targeted organs? A graph presentation should be presented.

Author Response

We are very thankful to the reviewer for the detailed examination of our manuscript and constructive comments. We appreciate feedback and have taken all comments into consideration. We have made the necessary changes to the article based on suggestions. We have also improved the language throughout the manuscript. Below, we have provided responses to each comment and the corresponding edits we have made.

Comment 1

In the abstract section lack any data about the obtained results.

Response to comment 1. 

We have added the results to the abstract. The updated version of the manuscript contains the corrected abstract. 

 

Comment 2

The authors did not report how iron oxide nanoparticles are obtained.

Response to comment 2. 

We have included a detailed description of the method for synthesis iron oxide nanoparticles in the text of the article. The change can be found in lines 145-154 

 

Comment 3

The size of administered microcapsule should be specified.

Response to comment 3. 

We measured the sizes of the synthesized capsules using transmission electron microscopy. The average size was 820 ± 80 nm for submicron capsules and 2.5 ± 0.5 μm for micron ones. These data are contained in subsection 3.2. Magnetite containing capsules of the Results section and in Table 1. We have found a misprint in line 360 and corrected it.

Comment 4

The conclusion is too long.

Response to comment 4. 

We have shortened the conclusion, leaving only a summary of the results of the studies. The updated version of the manuscript contains the corrected conclusion.

 

Comment 5

Is the micro sized capsule is suitable for intravenous administration?

Response to comment 5. 

The microcapsules used in our work can be administered intravenously. This is evidenced by the results of previously conducted and published studies of the safety and acute toxicity of similar polyelectrolyte microcapsules. When administered intravenously, toxic effects develop when the dose of capsules is exceeded by 2.5×10⁹ capsules per kg. 

[Nikita A. Navolokin et al. Systemic Administration of Polyelectrolyte Microcapsules: Where Do They Accumulate and When? In Vivo and Ex Vivo Study. Nanomaterials 2018, 8, 812; doi:10.3390/nano8100812]

In the in vivo experiment conducted in the present work, we administered microcapsules intravenously at a dose of 0.45×10⁹ capsules per kg, which corresponds to a magnetite dose of 0.65 mg/kg. 

 

Comment 6

What about the in vivo distribution of administered system in the targeted organs? A graph presentation should be presented.

Response to comment 6. 

In this work, the lungs was the target organ in which the capsules tissue distribution was studied. A detailed description of the distribution of capsules in the lungs is given in the «3.4 In vivo experiment» subsection and is shown in Figure 10. In this figure black points represent the results of experimental measurements of capsule concentration in the lung tissue.

Reviewer 3 Report

Comments and Suggestions for Authors

In this study, an approach was developed to theoretically predict the feasibility of magnetic targeting in a living organism based on the parameters of the carriers and the characteristics of the magnetic field generated by a cylindrical magnet. A theoretical model describing the motion of magnetic microcapsules in fluid flow under a non-uniform magnetic field was formulated. Experiments conducted in vitro and in vivo demonstrated a correlation between the theoretically predicted and the experimentally observed deposition of capsules, with greater discrepancies observed in weaker fields. The maximum effective distances for particle deposition using cylindrical neodymium magnets were calculated, showing that beyond a certain magnet radius, increased distance is not achieved due to a reduction in the field gradient.

However, the conclusions drawn above are not well supported by the calculations presented in the manuscript. As a result, the applicability of the obtained results remains questionable.

The major comments are as follows:

The magnetic field and its gradient were calculated only along the axis of a cylindrical permanent magnet (Eqs. 32–34), considering solely the axial components of the magnetic field and its gradient, dB_z/dz, in the subsequent analysis. However, it is well known that for a permanent magnet, the magnetic field gradient reaches its maximum near the magnet’s edges (see [1]). Therefore, the results presented in Fig. 11 do not correspond to realistic magnet application scenarios and are of limited practical relevance.

It was concluded that stronger field sources, such as superconducting or actively cooled electromagnets, may enable deeper targeting, although tissues closer to the surface would also be subjected to stronger magnetic forces. This statement is inaccurate. Superconducting magnets and electromagnets use coils that produce a spatial magnetic field distribution similar to that generated by a cylindrical permanent magnet. In practice, it is technically challenging to achieve magnetic field strengths and gradients greater than those produced by high-performance permanent magnets [2].

The concept of magnetic gravity—defined as the ratio of magnetic force to the mass of magnetic material (Eq. 12)—was introduced as a key parameter for evaluating field effectiveness. However, the term “magnetic gravity” (or acceleration) is misleading. According to Eq. 13, the quantity g_m​ depends on spatial coordinates due to the spatial dependence of the magnetic field B=B(x,y,z). Consequently, as a magnetic particle moves, g_m becomes a function of time. Therefore, Eq. 16 is incorrect, which compromises the validity of the subsequent calculations.

[1] Zablotskii V., et al. How a High-Gradient Magnetic Field Could Affect Cell Life. Sci Rep 6, 37407 (2016). https://doi.org/10.1038/srep37407

[2] Blümler, P. Magnetic Guiding with Permanent Magnets: Concept, Realization and Applications to Nanoparticles and Cells. Cells 2021, 10, 2708. https://doi.org/10.3390/cells10102708

Comments on the Quality of English Language

Should be improved. 

Author Response

We are grateful to the reviewer for their valuable comments and helpful suggestions. All of them have been taken into account, and appropriate revisions have been made to the manuscript. Since the responses to the comments include formulas and figures, they are provided in the attached PDF file. Please see the attachment.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The Introduction is too tedious at present. It needs to be simplified before the final acceptance for publication.

Author Response

Dear reviewer! Many thanks for the detailed re-analysis of our manuscript. In response to your comments, we report the following.

Comment 1. The Introduction is too tedious at present. It needs to be simplified before the final acceptance for  

Response to Comment 1. We have shortened the Introduction section and left the material most relevant to the topic of our article. The new version of the article contains a revised Introduction section.

Reviewer 3 Report

Comments and Suggestions for Authors

The authors have provided responses to my previous comments, but I am not fully satisfied with their replies. Below, I outline my specific concerns regarding their answers:

Comment 1:
The authors’ statement that the magnetic gradient is reached at the magnet’s edges, along with the addition of Ref. [37], is not sufficient. This fact imposes a limitation on the obtained results and their applicability to real-experiment drug/cell delivery using permanent magnets. This limitation should be clearly stated in the Conclusions section under a dedicated statement, such as "Limitation of our work."

Comment 2:
Reference [41] deserves an independent citation, for example in the Introduction, as it presents an advanced magnetic system for guiding magnetic particles. I strongly recommend that the authors review recent literature on the medical applications of permanent magnets and include relevant references in the Introduction.
Please note: the newly added sentences in lines 725–729 state obvious facts that do not require citation.

Comment 3:
I still find the concept of "magnetic gravity" misleading. The theoretical description should be done in terms of the magnetic gradient force, and the appropriate equations of motion should be used. According to Eq. (13), the so-called "magnetic gravity" depends on spatial coordinates (specifically x and z), i.e., g = f(x, z). As the particle moves such that x = x(t) and z = z(t), this implies that g = f(x(t), z(t)), i.e., a time-dependent function. Consequently, Eq. (16), which uses a constant g, is not valid.

To clarify this point, consider an analogy: imagine someone wants to calculate the motion of a falling object from a location far from Earth’s surface, where the acceleration due to gravity varies with distance. The question is: can one still use the equation v = v₀ + gt in such a scenario?

Furthermore, the introduced "effective relaxation time" in Eq. (15) lacks physical meaning. At the very least, the relaxation time should account for the transverse capillary size. I strongly recommend that the authors abandon the term "magnetic gravity" altogether and revise the relevant equations accordingly.

Additional Comment (new):
Regarding the newly added sentence in lines 399–400: "The magnetic field was calculated using an approach described in [37]." I question whether the authors truly recalculated the magnetic field distribution using the method described in Ref. [37]. Why? Because Figure 2 in the revised manuscript appears identical to the one in the original version, in which the authors were not aware of Ref. [37]. This inconsistency is puzzling and warrants clarification.

Author Response

We thank the reviewer for carefully re-examining our manuscript. In response to the comments, we report the following.

Comment 1

The authors’ statement that the magnetic gradient is reached at the magnet’s edges, along with the addition of Ref. [37], is not sufficient. This fact imposes a limitation on the obtained results and their applicability to real-experiment drug/cell delivery using permanent magnets. This limitation should be clearly stated in the Conclusions section under a dedicated statement, such as "Limitation of our work."

Response to comment 1.

We have added to the revised version of our manuscript subsection 4.3 "Limitations of our work" with the following content (lines 725-734):

The theoretical approaches used in the work have certain limitations. Theoretical calculations for the in vivo experiment were performed using a model of a straight cylindrical capillary. The liquid velocity profile was assumed to follow Poiseuille’s theory. The curvature of capillaries and disturbing of fluid velocities by the blood cells was not taking into account. Therefore, the values of the parameter  obtained within this model should be considered as approximate to a certain extent. The analytical evaluation of the distances at which  is achieved were carried out only for the magnetic field at the axis of a cylindrical longitudinally magnetized magnet. Therefore, new calculations are needed when using magnets of a different shape or with a different spatial orientation.

 

Comment 2.

Reference [41] deserves an independent citation, for example in the Introduction, as it presents an advanced magnetic system for guiding magnetic particles. I strongly recommend that the authors review recent literature on the medical applications of permanent magnets and include relevant references in the Introduction.

Please note: the newly added sentences in lines 725–729 state obvious facts that do not require citation.

Response to comment 2.

According to this comment and the comment of Reviewer 1 we have revised the introduction of our paper. We have added a new paragraph in the introduction discussing the biomedical application of permanent magnets and provided relevant links. Reference [41] now has number [18] and is citated at line 63. We also deleted citation [41] in line 727 and references [13] and [23] of the previous version of the manuscript and added the following bibliographic sources:

15. Blümler, P.; Friedrich, R. P.; Pereira, J.; Baun, O.; Alexiou, C.; Mailänder, V. Contactless Nanoparticle-Based Guiding of Cells by Controllable Magnetic Fields. Nanotechnol. Sci. Appl. 2021, 14, 91–100, doi:10.2147/NSA.S298003.
16. Baun, O.; Blümler P. Permanent magnet system to guide superparamagnetic particles. J. Magn. Magn. Mater. 2017, 439, 294–304, doi:10.1016/j.jmmm.2017.05.001.
17. Sharma, S.; Katiyar, V.K.; Singh U. Mathematical modelling for trajectories of magnetic nanoparticles in a blood vessel under magnetic field. J. Magn. Magn. Mater. 2015, 379, 102–107, doi:10.1016/j.jmmm.2014.12.012.
18. David, A.E.; Cole, A.J.; Chertok, B.; Park, Y.S.; Yang, V.C. A combined theoretical and in vitro modeling approach for predicting the magnetic capture and retention of magnetic nanoparticles in vivo. J Control Release. 2011, 152, 67–75, doi:10.1016/j.jconrel.2011.01.033.
19. Furlani, E. P. Permanent Magnet and Electromechanical Devices: Materials, Analysis, and Applications. Academic Press, Cambridge, MA 2021.

 

Comment 3

I still find the concept of "magnetic gravity" misleading. The theoretical description should be done in terms of the magnetic gradient force, and the appropriate equations of motion should be used. According to Eq. (13), the so-called "magnetic gravity" depends on spatial coordinates (specifically x and z), i.e., g = f(x, z). As the particle moves such that x = x(t) and z = z(t), this implies that g = f(x(t), z(t)), i.e., a time-dependent function. Consequently, Eq. (16), which uses a constant g, is not valid.

To clarify this point, consider an analogy: imagine someone wants to calculate the motion of a falling object from a location far from Earth’s surface, where the acceleration due to gravity varies with distance. The question is: can one still use the equation v = v₀ + gt in such a scenario?

Furthermore, the introduced "effective relaxation time" in Eq. (15) lacks physical meaning. At the very least, the relaxation time should account for the transverse capillary size. I strongly recommend that the authors abandon the term "magnetic gravity" altogether and revise the relevant equations accordingly.

Response to comment 3.

We agree that the term "effective relaxation time" was not physically meaningful and we decided to remove it along with formula (15). Following the reviewer's recommendation, we abandoned the term “magnetic gravity” and revised the equations using the term “specific magnetic force”.
We also note that the equation , in which the velocity is linearly dependent on time, has no direct relation to our equations. It is only superficially similar in form to equation (16) of the previous version of the manuscript (probably due to the awkward notation ). In our case, the motion is not free, and the acceleration is not a constant.  in equation (16) was not the initial velocity of the capsule.
In the revised version of our manuscript, we represent the particle velocity by equation (15) without magnetic gravity and relaxation time. We note that equation (15) is not our original result. It has been used previously by various authors, in particular, (8) in [27], (10) in [31] and (1) in [34].

Additional comment

Regarding the newly added sentence in lines 399–400: "The magnetic field was calculated using an approach described in [37]." I question whether the authors truly recalculated the magnetic field distribution using the method described in Ref. [37]. Why? Because Figure 2 in the revised manuscript appears identical to the one in the original version, in which the authors were not aware of Ref. [37]. This inconsistency is puzzling and warrants clarification.

Response to additional comment.

The magnetic field was not recalculated. It was initially calculated numerically using formulas equivalent to formulas (16) and (17) from [37]. We derived these formulas ourselves based on the concept of “magnetic charge” described, for example, in the book Fundamentals of the theory of electricity by I.E. Tamm, Moscow, Mir, 1979 (Eq. 5.118) or in ref. [41] of the revised version of our manuscript (Eq. 3.107).  In the current version of the manuscript, we have added the reference [41] and clarified the phrase that cites [42] ([37] in old version) on lines 281-284.

Round 3

Reviewer 3 Report

Comments and Suggestions for Authors

I am satisfied with the revisions made. The work is acceptable for publication.