Geometric Characterization of Local Changes in Tungsten Microneedle Tips after In-Vivo Insertion into Peripheral Nerves
Round 1
Reviewer 1 Report
In the manuscript entitled “Geometrical characterization of local changes in tungsten microneedles tips after insertion into in vivo peripheral nerves”, authors present a mathematical model for both the indentation force seen by a tungsten microneedle upon insertion into a peripheral nerve, and the resulting tip geometry after insertion. In this work the authors compared the modeled data of deformation to the experimental data and found two trends of different geometries that resulted in either bending or bulging of the microneedle tip. Although the work presented is very interested, it is a bit limited in scope for wider applicability. Therefore, I recommend major revisions to the manuscript before acceptance based on the following comments:
1. Some of the introduction of tungsten is irrelevant to the microneedles that are discussed (i.e. high melting temp use in construction etc) and should be removed. In its place, the addition of more introduction about insertion forces (potentially even the addition of a diagram explaining the variables on the insertion) would be beneficial for general understanding of the manuscript.
2. Although the manuscript does a convincing job validating experimental observations, there is no discussion about how these findings impact the application / use of tungsten microneedles to peripheral nerves. Therefore, the manuscript would be significantly improved by taking the framework developed forward to investigate alternative microneedle tip geometries to potentially ameliorate the issues of bending and bulging seen upon implantation.
3. Given the concern raised in comment number two, it is unclear how this tip optimization would change existing implantation methods of tungsten microneedles on a practical level. Is it just for the reuse of the microneedle? Expanding upon this point in the discussion would be beneficial for the overall scope of the manuscript.
4. Consider adding more descriptive identifiers for the mathematical variables as they are mentioned in the text (especially in the results section). It would minimize the back and forth looking at equations to determine which variable was being modified.
5. There is some switching back and forth between using “tungsten” and “W”, just read through again to be consistent.
6. Specific comments:
Page 2 Line 56: is “narrowed” the correct word choice here?
Page 2 Line 56: “entered” could be changed to “penetrated” to be consistent with typically used words for penetrating electrodes
Page 2 Line 58: is “narrowed” the correct word choice here?
Page 2 Line 64: missing a space between “force” and “was”
Page 3 Line 101: consider replacing “goodness” for a more scientific word
Figure 3: is it possible to put units for the variables being plotted here? It would make the figure read more clearly. Also, the yellow in (a) is difficult to read, consider changing to another color
Figure 7: it is unclear what is meant by in vitro here. Is that referring to the mathematical approximations? If so, it would be more correct to state that as in silico.
Page 14 sentence starting at 264: Please revise for clarity
Page 18 Line 329: “underwent to” should be changed to “experience” or “undergo”
Page 18 Line 329: is “narrowed” the correct word choice here?
Page 19 (no line number): “Indeed, if from a side,” should be edited to improve grammar
Page 19 (no line number): “In other words, phenomenological changes in the geometry of the tip shape were described due to damages near the microneedle tip” should be combined with the previous sentence to improve readability
Page 19 (no line number): “The genesis of these permanent inelastic damages” – “damages” should be changed to “damage”
Page 19 (no line number): “These issue could be” – either change to “these issues” or “this issue”
Author Response
The authors thank this reviewer for the time spent to improve the manuscript and for the useful comments and suggestions. All the comments have been answered and the answers are listed below.
Comments and Suggestions for Authors
Legend: Cx =comment x) Ax) Answer to the X comment
C0. In the manuscript entitled “Geometrical characterization of local changes in tungsten microneedles tips after insertion into in vivo peripheral nerves”, authors present a mathematical model for both the indentation force seen by a tungsten microneedle upon insertion into a peripheral nerve, and the resulting tip geometry after insertion. In this work the authors compared the modeled data of deformation to the experimental data and found two trends of different geometries that resulted in either bending or bulging of the microneedle tip. Although the work presented is very interested, it is a bit limited in scope for wider applicability. Therefore, I recommend major revisions to the manuscript before acceptance based on the following comments:
A0) The authors thank this reviewer for the words “ the work presented is very interesting”. In the following all the comments of this reviewer have been fully answered.
C1. Some of the introduction of tungsten is irrelevant to the microneedles that are discussed (i.e. high melting temp use in construction etc) and should be removed. In its place, the addition of more introduction about insertion forces (potentially even the addition of a diagram explaining the variables on the insertion) would be beneficial for general understanding of the manuscript.
A1) Yes. The authors agree with this reviewer. Therefore, a novel part has been added within the “Introduction” section to provide to the readers several references to works related to the puncture of soft tissues, as requested by this reviewer. In addition, a scheme (which is now the Figure1 in the revised version of the manuscript) has been added to explain all the used variables, as requested by this reviewer. However, since the journal also ranks in the category “Materials Science, Multidisciplinary” the general part on the tungsten is still within the main text, since it could be of interest for the readers coming from the material science side.
C2. Although the manuscript does a convincing job validating experimental observations, there is no discussion about how these findings impact the application / use of tungsten microneedles to peripheral nerves. Therefore, the manuscript would be significantly improved by taking the framework developed forward to investigate alternative microneedle tip geometries to potentially ameliorate the issues of bending and bulging seen upon implantation.
A2) The authors thank this reviewer for the words “ the manuscript does a convincing job validating experimental observations” and for this interesting comment. As reported in the references provided into the “Introduction” section, the deformation of the tip is also present in needles for aesthetic filler injections and cannulas for local anesthesia in human patients. So the “tip damage” is likely common for all geometries and materials when the tip has a sharp profile. As a consequence, the only way to eliminate this kind of damage is to use a microneedle with a flat geometry of the tip. Therefore, the use of microneedles with a flat tip has been investigated. In particular, the “Materials and Methods” section has been enlarged to analyse the case of flat tip (with the help of recent literature data). Similarly, the “Results” section has been improved by adding a further plot assessing the predicted diameter of W microneedles with a flat tip. Finally, the in the “Discussion” section, the flat geometry of the tip has been discussed as a possible alternative geometry, as requested by this reviewer.
C3. Given the concern raised in comment number two, it is unclear how this tip optimization would change existing implantation methods of tungsten microneedles on a practical level. Is it just for the reuse of the microneedle? Expanding upon this point in the discussion would be beneficial for the overall scope of the manuscript.
A3) The authors thank this reviewer for this interesting comment. Yes, we agree with this reviewer, some way to improve the insertion procedure have been discussed within the “Discussion” section of the revised version of the manuscript, as requested by this reviewer. The tip damage has been reported in literature as related to patient discomfort and anaesthesia complication due to a lack a precision. Similarly, the use of a “deformed” microneedle during implantation could be likely related to a lack of precision in insertion of soft neural interfaces. In other words, the contact sites of the neural interfaces could be misplaced and both stimulation and registration from nervous fibers could be compromised.
C4. Consider adding more descriptive identifiers for the mathematical variables as they are mentioned in the text (especially in the results section). It would minimize the back and forth looking at equations to determine which variable was being modified.
A4) The authors agree with this reviewer. Some more descriptive identifiers (i.e., the name of the authors of know Equations,..) have been added to reduce the effort of the reader.
C5. There is some switching back and forth between using “tungsten” and “W”, just read through again to be consistent.
A5) Yes, the authors agree with this reviewer. The main text has been corrected accordingly.
C6. Specific comments:
A6) Several parts of the main text of the manuscript have been changed and improved so some pages and comments related to the old version of the text may no match with the novel version of the manuscript. Nevertheless, the authors corrected all the issues listed below.
C7) Page 2 Line 56: is “narrowed” the correct word choice here?
A7) The text has been changed.
C8) Page 2 Line 56: “entered” could be changed to “penetrated” to be consistent with typically used words for penetrating electrodes
A8) Yes, the authors agree with this reviewer.
C9) Page 2 Line 58: is “narrowed” the correct word choice here?
A9) The text has been changed
C10) Page 2 Line 64: missing a space between “force” and “was”
A10) Corrected
C11) Page 3 Line 101: consider replacing “goodness” for a more scientific word
A11) Yes, this word has been replaced with “accuracy”
C12)Figure 3: is it possible to put units for the variables being plotted here? It would make the figure read more clearly. Also, the yellow in (a) is difficult to read, consider changing to another color
A12) The figure has been changed and more informative labels have been inserted
C13) Figure 7: it is unclear what is meant by in vitro here. Is that referring to the mathematical approximations? If so, it would be more correct to state that as in silico.
A13) Yes, within the caption of the Figure 8, the difference between in vivo and in vitro specimens of nerves has been clarified
C14) Page 14 sentence starting at 264: Please revise for clarity
A14) The text has been changed and clarified.
C15) Page 18 Line 329: “underwent to” should be changed to “experience” or “undergo”
A15) The text has been changed and clarified.
C16) Page 18 Line 329: is “narrowed” the correct word choice here?
A16) The text has been changed and clarified.
C17)Page 19 (no line number): “Indeed, if from a side,” should be edited to improve grammar
A17) Yes, this issue has been corrected along the main text.
C18)Page 19 (no line number): “In other words, phenomenological changes in the geometry of the tip shape were described due to damages near the microneedle tip” should be combined with the previous sentence to improve readability
A18) The text has been changed and clarified.
C19) Page 19 (no line number): “The genesis of these permanent inelastic damages” – “damages” should be changed to “damage”
A19) Corrected
C20)Page 19 (no line number): “These issue could be” – either change to “these issues” or “this issue”
A20) Corrected
Reviewer 2 Report
The authors describe the morphometric analysis of titanium microneedle electrodes that were implanted in animal peripheral nerves in a previous study. In particular, the authors fit equations for the measured indentation force, the two-dimensional shape of unimplanted microneedles, and the two-dimensional shape of microneedles deformed by implantation.
Overall, the manuscript was well-written and there were no technical issues in the study. My recommendation is to accept the manuscript with minor revisions.
Nevertheless, the manuscript appears overly complex for the simple things that were done. At multiple points, simple ideas are made overly complex for no apparent reason. In the following list, I make recommendations where the technical content can be simplified to help readers under the work.
A. Equation 1: This is a simple power function and should be described as such with a_m being the scaling factor and m being the exponent. Further, it should be stated what F is in the equation, including the direction of the force (normal to the nerve surface?) It is acceptable to provide references to others that have fit data to the same function.
B. Line 98-101 and Equation 2: No reason is provided why m must be an irrational number is provided and why it needs to be parameterized with an integer value. It is good enough to say that m is a real number. Removing parameterization of m with k will impact text in multiple places and figures.
C. Line 101-109: The entire search for the best performing values of a_m and m is overly complex. Why not perform a non-linear optimization to find the values of these two parameters that maximize R^2? Further, one could simply state that the optimization was done by some MATLAB or Python library function and direct readers to the documentation of that function to understand the method.
D. Line 105 and Equation 3: It is unclear why it is necessary to provide an implicit equation or to parameterize R^2 with y.
E. Section 2.2: It is never stated how many samples were used for this part of the study. I recommend providing that number so the reader does not have to assume the number based on the number of traces in various figures.
F. Section 2.3: This whole section appears overly complex to say that you tried fitting to 2nd order polynomials, 8th order polynomials, and piecewise polynomials. The text in the methods section makes it appear that generality was maintained but the results suggest you tried only a couple of options.
G. Equation 8: This equation appears never to be used in the results section. I recommend using it or removing it.
H. Section 2.4: This whole section is overly complex to say that the authors fit the change in two-dimensional shape from implantation to a 15th order polynomial and the residual. There is no reason that the fitting function must be a continuous function that is 15 times differentiable.
I. Figure 10 and Section 3.4: The value of this figure and this section is unclear. Nothing is provided to suggest why the R_L, R_H, and D metrics are related to any physical item. I recommend the removal of both or providing a better explanation of the physical relationship of the equations in Section 2.5.
There were few grammatical or syntactic issues with the manuscript but, broadly, I recommend that a native English-speaking person review and revise the text to bring it closer to common usage. (Perhaps the senior author could perform this role.) In many instances, the word choice was poor and sentence structure incorrect. In the following list, I mention presentation issues that stood out; however, this list is not comprehensive.
1) Line 2: I recommend changing neural interfaces to peripheral nerve interfaces as this study is only related to work with peripheral nerves.
2) Line 7: the phrase also after ... insertions is confusing. Is the intent ‘even after successful insertions?
3) Line 7: In English, one does not pluralize damage. I recommend that damages be replaced with damage throughout the manuscript.
4) Line 53: The mechanical performance of W is compared to peripheral nerves. I recommend rephrasing as one typically does not refer to the mechanical performance of peripheral nerves.
5) Line 64: Put a space between force and was.
6) Line 74: The verb form arising during to pierce is an incorrect usage.
7) Line 75-77: This description of the sharpening process is not adequately described.
8) Line 88: It is not clear what item is being filtered. I recommend changing the text to make it clear what signal is filtered.
9) Line 95-96: The sentence starting with In particular makes no sense. One does not test for references. I recommend changing the sentence to explain what was done.
10) Line 114: I believe the word trough should be through.
11) Figure 1 and 2: The x-axis label is too small and missing in some panels. What is beig plotted in panels a, d, … is never provided. Are these the fit or are these the original data. The red lines in panel a, d, … is never explained. It is unclear what is the x-axis label in panel b, e, … The value of plotting panels c, f, … is never made clear. What does this mean?
12) Figure 3: The upper and lower panels are basically the same information with different parameterizations. Why show both? Why show box and whisker plots – nothing is made of these plots. Why show a non-parametric plot but then use a parametric parameter (mean) to further analyze the data? It would be better to use panel b with error bars on each data point. (However, as I stated earlier, I recommend not using this optimization approach so this figure would be removed.)
13) Line 227: Just because a statistically significant difference is not found between two data sets does not mean the two are identical (or similar). I recommend revising the text to indicate that no reason to consider them different was found.
14) Figure 4: It is never made clear in the figure that the presented data are for the optimum value of m found. I recommend providing such (but I earlier recommended that a different optimization approach be taken that would remove this figure).
15) Line 284 and Figure 6’s caption: It is stated that the 8th order polynomial is unstable. However, the selection of parameters for this polynomial is completely stable in that it reduced the mean-squared error between the data provided and the fit. However, the results do not meet the authors’ desires. I recommend revising the text to use a better explanation of why the results were not adequate.
Author Response
The authors thank this reviewer for the time spent to improve the manuscript and for the useful comments and suggestions. All the comments have been answered and the answers are listed below.
Legend Cx) Comment X Ax) Answer to the X comments
C0) Comments and Suggestions for Authors
The authors describe the morphometric analysis of titanium microneedle electrodes that were implanted in animal peripheral nerves in a previous study. In particular, the authors fit equations for the measured indentation force, the two-dimensional shape of unimplanted microneedles, and the two-dimensional shape of microneedles deformed by implantation.
Overall, the manuscript was well-written and there were no technical issues in the study. My recommendation is to accept the manuscript with minor revisions.
Nevertheless, the manuscript appears overly complex for the simple things that were done. At multiple points, simple ideas are made overly complex for no apparent reason. In the following list, I make recommendations where the technical content can be simplified to help readers under the work.
A0) The authors thank this reviewer for this comment and in particular for the words “Overall, the manuscript was well-written and there were no technical issues in the study. My recommendation is to accept the manuscript with minor revisions”. In the following all the comments of this reviewer have been answered by authors:
C1) Equation 1: This is a simple power function and should be described as such with a_m being the scaling factor and m being the exponent. Further, it should be stated what F is in the equation, including the direction of the force (normal to the nerve surface?) It is acceptable to provide references to others that have fit data to the same function.
A1) The authors agree with this reviewer. This is simple power function where a_m are the scaling factors and m the exponent. This is the well know Oliver-Pharr approach to indentation. To better explaining the meaning of all factor Figure 1 has been introduced, where the reaction force F is plotted. In addition, some lines have been added to better describe the nature and the orientation of the F force, as requested by this reviewer. Relevant references have been added, as requested by this reviewer.
C2) Line 98-101 and Equation 2: No reason is provided why m must be an irrational number is provided and why it needs to be parameterized with an integer value. It is good enough to say that m is a real number. Removing parameterization of m with k will impact text in multiple places and figures.
A2) The authors agree with this reviewer. In the main text of the revised version of the manuscript is said that m is a real number, as requested by this reviewer. However, this is still parametrized through some integer values of k. Indeed, even if there are many way to perform a parametrization, this is a quite simple one, which is able to provide also theoretical values (i.e.,m=2 for k=9, etc). This work aimed at understanding whether the reaction to the action of the W microneedles during in vivo penetration was accounted for through Eq(1), where m was a function of some integer value k. However, it has been found that the optimal value was real and not an integer number (the Omega value). As a consequence, in the revised version the reader could refer to the values of k, or directly to the real value of m, as requested by this reviewer.
C3) Line 101-109: The entire search for the best performing values of a_m and m is overly complex. Why not perform a non-linear optimization to find the values of these two parameters that maximize R^2? Further, one could simply state that the optimization was done by some MATLAB or Python library function and direct readers to the documentation of that function to understand the method.
A3) The authors thank this reviewer for this question. The optimization of the value of m, which was needed to provide the best match of the experimental values, has been performed in a simple way. Indeed, Eq(1) was used for some values of m (depending of the value of k). The value of the standard statistics R^2 has been computed for each value of k (or m). Then a simple correlation function between the value of R^2 and the value of m has been provided. Finally, the global maximum of this function has been found to optimize the value of m. Yes, the authors agree with this reviewer, this could be done also with functions in Python or Matlab libraries. Nevertheless, the provided approach is completely general and every reader could reproduce it computationally by pen (without using Matlab or Phyton), or with standard programs.
c4) Line 105 and Equation 3: It is unclear why it is necessary to provide an implicit equation or to parameterize R^2 with y.
A4) The author thank this reviewer for this comment. The value of y has been inserted into the main text to avoid confusion between the value (R^2), which has been derived by matching experimental data with Eq (1) for some values of m, and the Rp^2 statistic, which has been used to assess the global effectiveness of Eq (3). Indeed, Eq (3) has been phenomenologically found by fitting the values of R^2 (y in the main text) as a function of the m values. This function could be provided both in implicit and in explicit form, the authors simply chosen the implicit form. The following lines have been inserted to clarify the main text in revised version: “A suitable functional form for
234 Eq. (3) was found as:
F(y, m) = y - am2 - bm - c = 0 (13)
235 Eq. (13) was able to effectively match data (R2p = 0.99) for the values of a, b, c listed in Table 1
C5) Section 2.2: It is never stated how many samples were used for this part of the study. I recommend providing that number so the reader does not have to assume the number based on the number of traces in various figures.
A5) Yes, the authors agree with this reviewer. The number of insertions of W microneedles (into the in vivo peripheral nerves) was 11. This information is clearly stated in the “Materials and Methods” section within the revised version of the manuscript, as requested by this reviewer.
C6) Section 2.3: This whole section appears overly complex to say that you tried fitting to 2nd order polynomials, 8th order polynomials, and piecewise polynomials. The text in the methods section makes it appear that generality was maintained but the results suggest you tried only a couple of options.
A6) The authors thank this reviewer for this comment. Yes, this framework has been used to maintain the generality. Indeed, also simple ideas should be expressed in a more formal way to maintain a general point of view. Then, the lower (2-th order) and higher (without numeric instability) order (8-th) of suitable polynomials have been used to match the experimental shape of the intact W microneedles.
C7) Equation 8: This equation appears never to be used in the results section. I recommend using it or removing it.
A7) The authors thank this reviewer for this comment. As explained within the “Materials and Methods” section Eq (8) has been coupled to Eq (7) to obtain Eq (9), which has been used to provide plots in the “Results” section. Eq (8) is needed to clarify the overall process, since here the force Fi explicitly appears. Then a simple substitution through Eq (7) has been done to provide (in Eq(9)) an assessments of the needed diameter to avoid elastic buckling of the main shaft of the W microneedles. In this way the authors hope that the readers could easily understand that also W microneedles avoiding global failure due to elastic buckling show tip damage, as requested by this reviewer.
C8) Section 2.4: This whole section is overly complex to say that the authors fit the change in two-dimensional shape from implantation to a 15th order polynomial and the residual. There is no reason that the fitting function must be a continuous function that is 15 times differentiable.
A8) The authors thank this reviewer for this comment. In the revised version of the work Figure2 has been inserted to better explain the origin of the track of the geometrical changes of the tip shape (G(x) in the revised version ). This track has been divided in low frequency and high frequency contributions. The low frequency contribution has been fitted through continuous and smooth polynomial functions with the lowest (possible) degree. The lowest possible degree was 15, and a polynomial of 15-th order is, by definition, a function differentiable 15 times, continuous and smooth.
C9) Figure 10 and Section 3.4: The value of this figure and this section is unclear. Nothing is provided to suggest why the R_L, R_H, and D metrics are related to any physical item. I recommend the removal of both or providing a better explanation of the physical relationship of the equations in Section 2.5.
A9) Yes, the authors agree with this reviewer. The relevant parts of the main text has been revised accordingly. In particular, Figure 9 and Figure 10 have been changed in the revised version of the main text and the mathematical and the physical meaning of R_L and R_D has been both clarified within the main text (Methods, Results, Discussion), as requested by this reviewer.
C10) There were few grammatical or syntactic issues with the manuscript but, broadly, I recommend that a native English-speaking person review and revise the text to bring it closer to common usage. (Perhaps the senior author could perform this role.) In many instances, the word choice was poor and sentence structure incorrect. In the following list, I mention presentation issues that stood out; however, this list is not comprehensive.
A10) Yes, the authors agree with this reviewer. Several parts of this text have been rewritten and the whole main text has been improved and corrected, as requested by this reviewer.
C11)Line 2: I recommend changing neural interfaces to peripheral nerve interfaces as this study is only related to work with peripheral nerves.
A11) Done in abstract. However, neural interfaces are used to interface both for the central and to peripheral nervous system.
C12) Line 7: the phrase also after ... insertions is confusing. Is the intent ‘even after successful insertions?
A12) Done, thanks.
C13) Line 7: In English, one does not pluralize damage. I recommend that damages be replaced with damage throughout the manuscript.
A13) Done, thanks.
C14) Line 53: The mechanical performance of W is compared to peripheral nerves. I recommend rephrasing as one typically does not refer to the mechanical performance of peripheral nerves.
A14) Done, thanks.
C15)Line 64: Put a space between force and was.
A15) Done, thanks.
C16) Line 74: The verb form arising during to pierce is an incorrect usage.
A16) Yes, there was a typo. Corrected thanks.
C17) Line 75-77: This description of the sharpening process is not adequately described.
C18) Line 88: It is not clear what item is being filtered. I recommend changing the text to make it clear what signal is filtered.
A17+18) The description has been improved, as requested by this reviewer.
C19) Line 95-96: The sentence starting with In particular makes no sense. One does not test for references. I recommend changing the sentence to explain what was done.
A19) The main text has been changed and the errors corrected.
C20) Line 114: I believe the word trough should be through.
A20) Yes, corrected, thanks.
C21) Figure 1 and 2: The x-axis label is too small and missing in some panels. What is beig plotted in panels a, d, … is never provided. Are these the fit or are these the original data. The red lines in panel a, d, … is never explained. It is unclear what is the x-axis label in panel b, e, … The value of plotting panels c, f, … is never made clear. What does this mean?
A21) The figures 1,2 has been changed. The Figure 3 has been now corrected in the revised version of the work, thanks.
C22) Figure 3: The upper and lower panels are basically the same information with different parameterizations. Why show both? Why show box and whisker plots – nothing is made of these plots. Why show a non-parametric plot but then use a parametric parameter (mean) to further analyze the data? It would be better to use panel b with error bars on each data point. (However, as I stated earlier, I recommend not using this optimization approach so this figure would be removed.)
A22) Yes, the authors agree with this reviewer. Only the lower panel has been kept in the revised version with the mean and errors. (Note that the number of the figure is now Figure 4 ).
C23) Line 227: Just because a statistically significant difference is not found between two data sets does not mean the two are identical (or similar). I recommend revising the text to indicate that no reason to consider them different was found.
A23) Yes, the authors agree with this reviewer. In the revised text both the lack of significant difference and the similarity of the other conditions have been indicated to infer that no reason to consider them different has been found, as requested by this reviewer.
C24) Figure 4: It is never made clear in the figure that the presented data are for the optimum value of m found. I recommend providing such (but I earlier recommended that a different optimization approach be taken that would remove this figure).
A24) Yes, the authors agree with this reviewer. In the caption of the Figure 5 and in the main text has been explicitly said that all the plot are referred to the optimum value of m, as requested by this reviewer.
C25) Line 284 and Figure 6’s caption: It is stated that the 8th order polynomial is unstable. However, the selection of parameters for this polynomial is completely stable in that it reduced the mean-squared error between the data provided and the fit. However, the results do not meet the authors’ desires. I recommend revising the text to use a better explanation of why the results were not adequate.
A25) The authors thank this reviewer for this question. In the revised version of the work has been clearly indicated that the 8-th order polynomial was able to lower the value of the R^2 statistic only up to 20 microns from the origin of the abscissas (0 value, theoretical position of the symmetry axis of the intact tip shape), while for greater values instability phenomena arise, as requested by this reviewer.
Reviewer 3 Report
Dear authors,
Although the manuscript was informative and lead to useful references, it is not clear what is the main objective of the work, why is it important, what gap are you closing with the analysis provided. Also, re-consider whether the methodology used was necessary or if there is a more simple way to reach to the conclusions you expose in the manuscript. Please read further comments in the attached file.
Kind regards
Comments for author File: 
Comments.docx
Author Response
See the attached file
Author Response File: 
Author Response.pdf
Round 2
Reviewer 1 Report
Authors have adequately addressed the comments / revisions that were suggested during the previous review, therefore I suggest accepting the manuscript in its present form.
