Calibrating models of cancer invasion: parameter estimation using approximate Bayesian computation and gradient matching

We present two different methods to estimate parameters within a partial differential equation model of cancer invasion. The model describes the spatio-temporal evolution of three variables—tumour cell density, extracellular matrix density and matrix degrading enzyme concentration—in a one-dimensional tissue domain. The first method is a likelihood-free approach associated with approximate Bayesian computation; the second is a two-stage gradient matching method based on smoothing the data with a generalized additive model (GAM) and matching gradients from the GAM to those from the model. Both methods performed well on simulated data. To increase realism, additionally we tested the gradient matching scheme with simulated measurement error and found that the ability to estimate some model parameters deteriorated rapidly as measurement error increased.

3. The authors say in a few places that ABC is not suitable for assessing uncertainty in the parameter estimates. Could the authors expand on what type of uncertainty they wish to capture here? From a Bayesian or maximum likelihood viewpoint, if the likelihood is a spike at a single parameter value then there is no uncertainty! Other sources of uncertainty exist, such as model misspecification, but are quite hard to capture by any method 4. Page 2 says "Under the assumption that the differential equation model is correct and that the observations arise independently from a normal distribution then the least squares parameter estimates are maximum likelihood estimates." I think it is not necessary for the differential equation model to be correct for the least squares estimates and MLE to coincide. 5. Page 6 says "the algorithm will eventually converge on the best fitting values with no variation between parameter sets". As well as the bandwidth issue mentioned above, another issue with this is the use of summary statistics. There's an implicit assumption here that an exact match of summary statistics implies an exact match of the full data. This should be mentioned, with a brief discussion of how reasonable this is for the application under investigation. 6. Page 9. Mention which optimization algorithm was used when using the optim function in R. 7. Page 11 says "Constraining certain parameters when the gradients are inaccurate will make the estimates even worse, since this makes the optimization scheme loses its ability to adjust itself." Can you give some more explanation or reword here? It's not obvious to me that this is true or exactly what "the ability to adjust itself" means. 8. Page 12 says "If we look at the PDE model itself, the parameters in the tumour cells equation are all associated with numerically complex terms, and thus the information hidden behind the PDE model that can help to improve the accuracy of their estimates is therefore limited." Reword this? Currently it seems to say that numerical complexity implies lack of informativeness, which seems incorrect. 9. In Figure 4, on page 13, it's hard to make out different lines. This is better in the larger sized versions of these plots at the end of the document. Maybe mention that these are available in the caption to Figure 4? Also, the ^ character seems to be misplaced in several of the plot headings.
10. Page 13 says "However, this difference here should be accounted for the errors in the tumour cell-related estimates when no perturbation was added." I didn't understand what this sentence meant. Is it possible to explain further, or is there a typo in the sentence? 11. Page 15 says "Overall, we believe the parameter estimates for PDE models using statistical approaches is a strong alternative to searching high-dimensional parameter space by handtuning". What is meant by "hand-tuning" here? I couldn't find this phrase anywhere earlier in the paper.
Possible typos ============== 1. Page 6: "suitable for estimating parameter values but **not not** for assessing uncertainty on the estimates" 2. Page 8: "The discrepancies between the two sides of the equations **was** then calculated" 3. Page 10: "but in general, they were quite insensitive to the increase in measurement errors, except for ∂m/∂t, the deviations at the left tail seem to increase with the CV." Is there a word missing somewhere here e.g. should it be "...**where** the deviations..."? 4. Page 11: "The complex spatial gradients (e.g. second order spatial derivatives, haptotaxis term) also **shown** obvious deviations" Decision letter (RSOS-202237.R0) We hope you are keeping well at this difficult and unusual time. We continue to value your support of the journal in these challenging circumstances. If Royal Society Open Science can assist you at all, please don't hesitate to let us know at the email address below.

Dear Mr Xiao
The Editors assigned to your paper RSOS-202237 "Calibrating models of cancer invasion: parameter inference using Approximate Bayesian Computation and gradient matching" have now received comments from reviewers and would like you to revise the paper in accordance with the reviewer comments and any comments from the Editors. Please note this decision does not guarantee eventual acceptance.
We invite you to respond to the comments supplied below and revise your manuscript. Below the referees' and Editors' comments (where applicable) we provide additional requirements. Final acceptance of your manuscript is dependent on these requirements being met. We provide guidance below to help you prepare your revision.
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Please submit your revised manuscript and required files (see below) no later than 21 days from today's (ie 02-Mar-2021) date. Note: the ScholarOne system will 'lock' if submission of the revision is attempted 21 or more days after the deadline. If you do not think you will be able to meet this deadline please contact the editorial office immediately.
Please note article processing charges apply to papers accepted for publication in Royal Society Open Science (https://royalsocietypublishing.org/rsos/charges). Charges will also apply to papers transferred to the journal from other Royal Society Publishing journals, as well as papers submitted as part of our collaboration with the Royal Society of Chemistry (https://royalsocietypublishing.org/rsos/chemistry). Fee waivers are available but must be requested when you submit your revision (https://royalsocietypublishing.org/rsos/waivers). Comments to the Author(s) This paper investigates statistical inference for a PDE cancer model using approximate Bayesian computation (ABC) and gradient matching. It provides an interesting comparison of these methods on a challenging application.
I have one major concern about this paper: the correctness of the ABC algorithm used. Therefore I recommend major corrections. Otherwise the paper performs a good simulation study and is well written, so I have only a few other minor comments. Details of major and minor comments are below.
Also if the authors have time in future, it would be interesting to extend this study, especially to investigate how ABC performs in the case of observation error. Furthermore I wonder if ABC is the best algorithm to use for optimization when the simulator is deterministic (i.e. when there is no observation error). For instance one could use stochastic approximation optimization methods (of the Kiefer-Wolfowitz type). However exploration of these issues is beyond the scope of what's needed for this paper to be accepted.

Major comment =============
The ABC algorithm used has some non-standard features compared to standard ABC-SMC algorithm (for example, see the review article of Marin et al 2012 https://doi.org/10.1007/s11222-011-9288-2). One of these features seems like it could invalidate the correctness of the algorithm for this application (see first point below). The others seem problematic for general use of ABC, but not this particular application (see second point below).
ABC bandwidth -------------The paper's ABC algorithm uses weight = distance^(-1/2). ABC-SMC algorithms would typically include a bandwidth variable h here e.g. weight = distance^(-1/2h) which is reduced as the iterations of the algorithm progress (early papers used a pre-specified schedule of values, but reducing h adaptively is now more common). This is necessary for the algorithm to converge on the true parameter values. I suggest using a reducing bandwidth here and rerunning the simulations. Alternatively the authors could explain why they wish to use a fixed bandwidth, and adjust this discussion accordingly i.e. note that the algorithm does not converge on the true parameters, but instead gives approximate results, and justify why this is good enough for the current purposes. However in the latter case, I think the issues in the next point become more important.
Other issues with algorithm ---------------------------Some other unusual features of the ABC algorithm are: * The weights do not include a term for the density of the proposed parameters. * Weights are not rescaled by the standard formula (dividing by the sum of the weights).
This means that the usual sequential Monte Carlo / population Monte Carlo analysis of the algorithm does not hold and so this algorithm does not output samples from the usual ABC approximation to the posterior density. It is somewhat unclear therefore what distribution the algorithm samples from. In general it would be preferable to have some reassurance that this algorithm does not have some undesirable properties compared to standard ABC e.g. large extra approximation error, or no guarantee of convergence.
However for their particular application, the authors simply wish to use ABC as an optimisation algorithm. Thus, so long as it finds parameter values which approximately minimise the distance, then this is sufficient for the desired behaviour. (However to do so, the bandwidth issue above probably needs to be addressed.) Minor comments ============== 1. It would be preferable to show numerical results using mathematical notation e.g. $6.9 \times 10^6$ rather than 6.9e6. This is particularly useful for numbers which can simply be represented as decimals e.g. 0.012 is much easier to interpret in a table than 1.2e-2.
2. It might be worth briefly discussing the recent paper "A comparison of approximate versus exact techniques for Bayesian parameter inference in nonlinear ordinary differential equation models" by Alahmadi et al (https://doi.org/10.1098/rsos.191315). They discuss how ABC methods can fail for differential equation models if badly implemented. Although this problem applies mainly to the case of observation error, it might be interesting to point this out to readers of your paper.
3. The authors say in a few places that ABC is not suitable for assessing uncertainty in the parameter estimates. Could the authors expand on what type of uncertainty they wish to capture here? From a Bayesian or maximum likelihood viewpoint, if the likelihood is a spike at a single parameter value then there is no uncertainty! Other sources of uncertainty exist, such as model misspecification, but are quite hard to capture by any method 4. Page 2 says "Under the assumption that the differential equation model is correct and that the observations arise independently from a normal distribution then the least squares parameter estimates are maximum likelihood estimates." I think it is not necessary for the differential equation model to be correct for the least squares estimates and MLE to coincide. 5. Page 6 says "the algorithm will eventually converge on the best fitting values with no variation between parameter sets". As well as the bandwidth issue mentioned above, another issue with this is the use of summary statistics. There's an implicit assumption here that an exact match of summary statistics implies an exact match of the full data. This should be mentioned, with a brief discussion of how reasonable this is for the application under investigation. 6. Page 9. Mention which optimization algorithm was used when using the optim function in R. 7. Page 11 says "Constraining certain parameters when the gradients are inaccurate will make the estimates even worse, since this makes the optimization scheme loses its ability to adjust itself." Can you give some more explanation or reword here? It's not obvious to me that this is true or exactly what "the ability to adjust itself" means. 8. Page 12 says "If we look at the PDE model itself, the parameters in the tumour cells equation are all associated with numerically complex terms, and thus the information hidden behind the PDE model that can help to improve the accuracy of their estimates is therefore limited." Reword this? Currently it seems to say that numerical complexity implies lack of informativeness, which seems incorrect. 9. In Figure 4, on page 13, it's hard to make out different lines. This is better in the larger sized versions of these plots at the end of the document. Maybe mention that these are available in the caption to Figure 4? Also, the ^ character seems to be misplaced in several of the plot headings.
10. Page 13 says "However, this difference here should be accounted for the errors in the tumour cell-related estimates when no perturbation was added." I didn't understand what this sentence meant. Is it possible to explain further, or is there a typo in the sentence? 11. Page 15 says "Overall, we believe the parameter estimates for PDE models using statistical approaches is a strong alternative to searching high-dimensional parameter space by handtuning". What is meant by "hand-tuning" here? I couldn't find this phrase anywhere earlier in the paper.
Possible typos ============== 1. Page 6: "suitable for estimating parameter values but **not not** for assessing uncertainty on the estimates" 2. Page 8: "The discrepancies between the two sides of the equations **was** then calculated" 3. Page 10: "but in general, they were quite insensitive to the increase in measurement errors, except for ∂m/∂t, the deviations at the left tail seem to increase with the CV." Is there a word missing somewhere here e.g. should it be "...**where** the deviations..."? 4. Page 11: "The complex spatial gradients (e.g. second order spatial derivatives, haptotaxis term) also **shown** obvious deviations" ===PREPARING YOUR MANUSCRIPT=== Your revised paper should include the changes requested by the referees and Editors of your manuscript. You should provide two versions of this manuscript and both versions must be provided in an editable format: one version identifying all the changes that have been made (for instance, in coloured highlight, in bold text, or tracked changes); a 'clean' version of the new manuscript that incorporates the changes made, but does not highlight them. This version will be used for typesetting if your manuscript is accepted. Please ensure that any equations included in the paper are editable text and not embedded images.
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Recommendation?
Accept as is

Comments to the Author(s)
The authors have substantially addressed every concern that I raised in my initial report. Thus, I am of the view that this manuscript is now suitable for publication in Royal Society Open Science.

Are the interpretations and conclusions justified by the results? Yes
Is the language acceptable? Yes

Do you have any ethical concerns with this paper? No
Have you any concerns about statistical analyses in this paper? No

Recommendation?
Accept with minor revision (please list in comments)

Comments to the Author(s)
The authors have addressed all the points from my review and I am happy to recommend that the paper is accepted for publication.
I have two minor comments which the authors could perhaps incorporate in their final draft. These are listed below using the numbering scheme from the authors' response.
R2.2 It might be worth mentioning that there are some differences between your algorithm and standard ABC-SMC, so that readers are aware that some modifications are needed if they are interested in inferring the full posterior.

Decision letter (RSOS-202237.R1)
We hope you are keeping well at this difficult and unusual time. We continue to value your support of the journal in these challenging circumstances. If Royal Society Open Science can assist you at all, please don't hesitate to let us know at the email address below.

Dear Mr Xiao
On behalf of the Editors, we are pleased to inform you that your Manuscript RSOS-202237.R1 "Calibrating models of cancer invasion: parameter estimation using Approximate Bayesian Computation and gradient matching" has been accepted for publication in Royal Society Open Science subject to minor revision in accordance with the referees' reports. Please find the referees' comments along with any feedback from the Editors below my signature.
We invite you to respond to the comments and revise your manuscript. Below the referees' and Editors' comments (where applicable) we provide additional requirements. Final acceptance of your manuscript is dependent on these requirements being met. We provide guidance below to help you prepare your revision.
Please submit your revised manuscript and required files (see below) no later than 7 days from today's (ie 18-May-2021) date. Note: the ScholarOne system will 'lock' if submission of the revision is attempted 7 or more days after the deadline. If you do not think you will be able to meet this deadline please contact the editorial office immediately.
Please note article processing charges apply to papers accepted for publication in Royal Society Open Science (https://royalsocietypublishing.org/rsos/charges). Charges will also apply to papers transferred to the journal from other Royal Society Publishing journals, as well as papers submitted as part of our collaboration with the Royal Society of Chemistry (https://royalsocietypublishing.org/rsos/chemistry). Fee waivers are available but must be requested when you submit your revision (https://royalsocietypublishing.org/rsos/waivers). Comments to the Author(s) The authors have addressed all the points from my review and I am happy to recommend that the paper is accepted for publication.
I have two minor comments which the authors could perhaps incorporate in their final draft. These are listed below using the numbering scheme from the authors' response.
R2.2 It might be worth mentioning that there are some differences between your algorithm and standard ABC-SMC, so that readers are aware that some modifications are needed if they are interested in inferring the full posterior.

Reviewer: 1
Comments to the Author(s) The authors have substantially addressed every concern that I raised in my initial report. Thus, I am of the view that this manuscript is now suitable for publication in Royal Society Open Science.

===PREPARING YOUR MANUSCRIPT===
Your revised paper should include the changes requested by the referees and Editors of your manuscript. You should provide two versions of this manuscript and both versions must be provided in an editable format: one version identifying all the changes that have been made (for instance, in coloured highlight, in bold text, or tracked changes); a 'clean' version of the new manuscript that incorporates the changes made, but does not highlight them. This version will be used for typesetting. Please ensure that any equations included in the paper are editable text and not embedded images.
Please ensure that you include an acknowledgements' section before your reference list/bibliography. This should acknowledge anyone who assisted with your work, but does not qualify as an author per the guidelines at https://royalsociety.org/journals/ethicspolicies/openness/.
While not essential, it will speed up the preparation of your manuscript proof if you format your references/bibliography in Vancouver style (please see https://royalsociety.org/journals/authors/author-guidelines/#formatting). You should include DOIs for as many of the references as possible.
If you have been asked to revise the written English in your submission as a condition of publication, you must do so, and you are expected to provide evidence that you have received language editing support. The journal would prefer that you use a professional language editing service and provide a certificate of editing, but a signed letter from a colleague who is a native speaker of English is acceptable. Note the journal has arranged a number of discounts for authors using professional language editing services (https://royalsociety.org/journals/authors/benefits/language-editing/).

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Decision letter (RSOS-202237.R2)
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Dear Mr Xiao, I am pleased to inform you that your manuscript entitled "Calibrating models of cancer invasion: parameter estimation using Approximate Bayesian Computation and gradient matching" is now accepted for publication in Royal Society Open Science.
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A review of "Calibrating models of cancer invasion: parameter inference using Approximate Bayesian Computation and gradient matching" by Y Xio, L Thomas and MAJ Chaplain.
This is an interesting and exciting piece of Mathematical Biology that has been a pleasure to read. The authors review the construction of a Partial Differential Equation (PDE) model of cancer invasion, and they then proceed to fit parameters to this model. Parameter fitting is carried out with both an Approximate Bayesian Computation (ABC) and a gradient matching technique. The thrust of this manuscript is in demonstrating a successful parameter inference pipeline. This approach is likely to be emulated by other authors.
There are clear strengths to this expository work, and these are set out, section-by-section, in the following paragraphs. As this work introduces well-described ABC inference techniques to the Mathematical Biology literature, a number of directions for future work are suggested for the author's consideration. Whilst this reviewer has considered the manuscript in its entirety, this feedback focuses on the ABC inference component of the work.
Introduction . The authors start by explaining that "a common issue" that arises when using systems of differential equations to model various phenomena is that "some or all model parameters are not known". They then say that model fitting techniques can provide the missing parameters. On a first reading, it appears that only point parameter estimates are required for the Mathematical Biology case study that is to follow. For many modelling applications, such as the study eloquently set out in the next section, this is perfectly sensible. Then, the authors raise the ambition of operating in a fully Bayesian setting --setting priors on the parameters, observing experimental data, and then arriving at a posterior distribution to fully characterise the parameters. The authors might wish to explore the possibility of using the entire posterior distribution to mark the uncertainty in the parameters, for example, by considering the posterior predictive distribution. The authors might wish to highlight the circumstances where a fully Bayesian approach would be advantageous.
The authors might wish to make a small number of technical changes to this section. For example, the authors ought to formalise their references to "the best parameter values", perhaps by explaining that they make use of Bayes estimators. Further details, such as the appropriate loss function the authors have considered, could be specified to ensure a complete description of their work.
Methods. The PDE system is set out with stunning clarity and a standard non-dimensionalisation is carried out. The description of the ABC method could, perhaps, be edited to make full use of the opportunities offered by Bayesian inference. Specifically: The authors might wish to acknowledge that there are specific circumstances where it is necessary to first impose priors in the dimensional PDE setting, and then to translate these priors into the non-dimensional PDE formulation. This is a consequence of not all priors being invariant under a change of coordinates.
(b) The observant reader might recognise the similarities between the author's approach, and the ABC Sequential Monte Carlo (ABC-SMC) method contained in reference [15]. The authors might wish to include a few sentences discussing the similarities and differences.
(c) The authors do not seek the entire posterior distribution, but rather, they zoom in on a mean value. This has the advantage of providing a point estimate parameter for the model. For certain models this would mean the opportunity to fully characterise the uncertainty in the parameter-set is passed up, and a valid posterior density is not provided. This arises as the parameters are sampled according to a resampling probability (see part (iii)(a)(iv)) rather than the prior distribution, and presumably not re-weighted accordingly. The authors are up-front about this: for example, in Figure 2, an "initial density" and a "final density" are shown.
(d) The authors have computed resampling weights by first determining ⍴ -0.5 , and then linearly re-scaling. It is clear that the authors arrived at this choice after exhaustive investigations. To ensure that full credit is given to their findings, the authors might wish to provide a brief motivation for this choice (especially of the exponent shown above). It might also be worth mentioning that K can be j-dependent.
(e) The authors are right to acknowledge that a "stopping rule" has not been specified. A very simple option would be to plot the mean parameter as the algorithm unfolds, and to use this as a basis for a simple stopping rule. It is not necessary to address this in full in this manuscript, and a brief description of how a stopping rule could be implemented would probably satisfy most readers.
The reviewer has no specific comments concerning the gradient matching scheme set out in this section.
Results. This section carefully describes the outputs of the authors' investigations. In short, the authors picked specific parameters, generated in silico data, and then sought to recover the parameters with the inference algorithm. To ensure that readers fully appreciate the benefits of ABC inference, the authors might wish to consider: (a) The difference between the picked and recovered parameters is referred to as a "percentage error". This is a snap-shot of the algorithm's performance at a specified parameter-set, and, in this case, in the absence of any measurement error. The reviewer would like to raise the possibility of considering the mean squared error of the outputs, so that the error can be naturally decomposed into bias and variance components.
(b) Only the marginal "final densities" are shown. An interested reader might wish to see pairwise heatmaps of "final densities" too.
(c) It is unfortunate that, due to computational constraints, it has not been possible to demonstrate the performance of the ABC method on a noisy dataset. Building the computer code to carry out this research must have been a formidable task, and the reviewer is satisfied that every effort would have been made to ensure its efficiency.
Discussion . This final section is refreshingly realistic in its description of the manuscript's findings. The authors are right to raise "the model selection problem" as the next frontier in mechanistic modelling. Whilst it would be unfair to ask the authors to detail their future research plans, it might be helpful if, by way of example, they are proposing to make use of Bayes factors or similar tools to conduct this analysis.

Dear Professor Rogers,
Thank you for your initial decision on our manuscript, and for the very helpful comments and suggestions provided by the two reviewers. We attach a revised version that addresses all the points made. Below we reproduce each comment (in regular font) and how we dealt with it (in red italic font, with underlining used to indicate text changes in the manuscript). We have numbered the comments (R1.1, R1.2, … for reviewer 1 and R2.1, R2.2, … for reviewer 2) to make them easier to track. We look forward to hearing your thoughts on our revised submission.
With best wishes,

Reviewer 1
A review of "Calibrating models of cancer invasion: parameter inference using Approximate Bayesian Computation and gradient matching" by Y Xiao, L Thomas and MAJ Chaplain.

R1.1
This is an interesting and exciting piece of Mathematical Biology that has been a pleasure to read.
The authors review the construction of a Partial Differential Equation (PDE) model of cancer invasion, and they then proceed to fit parameters to this model. Parameter fitting is carried out with both an Approximate Bayesian Computation (ABC) and a gradient matching technique. The thrust of this manuscript is in demonstrating a successful parameter inference pipeline. This approach is likely to be emulated by other authors.
There are clear strengths to this expository work, and these are set out, section-by-section, in the following paragraphs. As this work introduces well-described ABC inference techniques to the Mathematical Biology literature, a number of directions for future work are suggested for the author's consideration. Whilst this reviewer has considered the manuscript in its entirety, this feedback focuses on the ABC inference component of the work.

Thank-you for your helpful and constructive comments.
R1.2 Introduction. The authors start by explaining that "a common issue" that arises when using systems of differential equations to model various phenomena is that "some or all model parameters are not known". They then say that model fitting techniques can provide the missing parameters. On a first reading, it appears that only point parameter estimates are required for the Mathematical Biology case study that is to follow. For many modelling applications, such as the study eloquently set out in the next section, this is perfectly sensible. Then, the authors raise the ambition of operating in a fully Bayesian setting -setting priors on the parameters, observing experimental data, and then arriving at a posterior distribution to fully characterise the parameters. The authors might wish to explore the possibility of using the entire posterior distribution to mark the uncertainty in the parameters, for example, by considering the posterior predictive distribution. The authors might wish to highlight the circumstances where a fully Bayesian approach would be advantageous. Page 17: … We also side-stepped the important issue of uncertainty quantification. A realworld application will at least wish to consider quantifying uncertainty arising from measurement errors (see, e.g., [45]) and may also attempt to quantify other sources of uncertainty such as model mis-specification error.

R1.3
The authors might wish to make a small number of technical changes to this section. For example, the authors ought to formalise their references to "the best parameter values", perhaps by explaining that they make use of Bayes estimators. Further details, such as the appropriate loss function the authors have considered, could be specified to ensure a complete description of their work. R1.6 (c) The authors do not seek the entire posterior distribution, but rather, they zoom in on a mean value. This has the advantage of providing a point estimate parameter for the model. For certain models this would mean the opportunity to fully characterise the uncertainty in the parameter-set is passed up, and a valid posterior density is not provided. This arises as the parameters are sampled according to a resampling probability (see part (iii)(a)(iv)) rather than the prior distribution, and presumably not re-weighted accordingly. The authors are upfront about this: for example, in Figure 2, an "initial density" and a "final density" are shown.
Yes, our current ABC scheme is not designed for assessing uncertainties on the parameter estimates, it was built to retrieve the parameters which give the best fit to the reference data. We agreed the calculation of resampling probabilities needs to be improved, the weights are now rescaled in a standard manner (weights/sum of weights) to obtain resampling probabilities, which is often seen in traditional ABC methods. We hope the changes made in subsection 2.c)iii) "ABC-BCD (Approximate Bayesian Computation -Bhattacharyya distance) optimization scheme" give a better clarification on the issue of resampling probability.
Actions: Text revised in subsection 2.c)iii). , where ε is a positive number less than 1. |ρ y j -ρ y 1 | < ϵ * ρ y 1 Page 8: … Here we used K = 10000. We checked the Monte Carlo error by undertaking two additional runs on the same reference dataset and computing the standard error of the parameter estimates across the three runs. Overall accuracy is governed by the stopping criterion and the increasing bandwidth t in weight calculations. Different |ρ y j -ρ y 1 | < ϵ * ρ y 1 ε's were used in the evaluations of the three different density profiles. It was set to be 0.8 when evaluating the ECM density profile alone, to prevent particle depletion in early rounds when only one parameter was being estimated. We then raised it to 0.9 in the evaluations of ECM and MDE profiles. Finally, it was set to be 0.98 when all density profiles are being evaluated. In our opinion, particle depletion is a concerning issue in early rounds, but should not cause any troubles in the last few rounds since our goal is to obtain accurate parameter point estimates. The bandwidth of weights t was chosen to start from 0.5 and increased by 50% in every subsequent round, it was reset to 0.5 when we proceeded to evaluate the next equation. By setting such stopping criterion and adaptive bandwidths for weights, the resampling surface became steeper in later rounds, parameter sets had minor Bhattacharyya distances to the reference values could then be resampled with heavier weights and convergence to the true values could be guaranteed.
The reviewer has no specific comments concerning the gradient matching scheme set out in this section.
We sincerely thank the reviewer for the suggestions on future directions. We have now added a few sentences to explain some possible ideas that can be extended from the current work.
Actions: Text revised in the Discussion.
Edits: Page 15-16: … Lastly, our current ABC scheme was not designed to measure the uncertainties in the estimates --no analytic methods exist and bootstrapping is computationally infeasible. In future work, observation errors may be introduced to the reference dataset so uncertainties in parameter estimates can be assessed using a full posterior inference under the Bayesian framework. However, we realize the incorporation of observation errors can be a challenging problem. Alahmadi et al. [45] have argued the posterior distributions derived from those ABC methods which fail to adequately model the measurement errors may not accurately reflect the epistemic uncertainties in parameter values. Hence it is necessary to model and incorporate the errors in the correct way when using ABC methods.

Reviewer 2
Comments to the Author(s) This paper investigates statistical inference for a PDE cancer model using approximate Bayesian computation (ABC) and gradient matching. It provides an interesting comparison of these methods on a challenging application.
I have one major concern about this paper: the correctness of the ABC algorithm used. Therefore I recommend major corrections. Otherwise the paper performs a good simulation study and is well written, so I have only a few other minor comments. Details of major and minor comments are below.
Also if the authors have time in future, it would be interesting to extend this study, especially to investigate how ABC performs in the case of observation error. Furthermore I wonder if ABC is the best algorithm to use for optimization when the simulator is deterministic (i.e. when there is no observation error). For instance one could use stochastic approximation optimization methods (of the Kiefer-Wolfowitz type). However exploration of these issues is beyond the scope of what's needed for this paper to be accepted.
Thank-you for these helpful comments. Yes, we are planning to investigate both the performance of ABC in the presence of observation error (for parameter estimation and for uncertainty quantification, as mentioned above) as well as other optimization methodsthanks for the suggestion of Kiefer-Wolfowitz-like methods.
Major comment ============= R2. 1 The ABC algorithm used has some non-standard features compared to standard ABC-SMC algorithm (for example, see the review article of Marin et al 2012 https://doi.org/10.1007/s11222-011-9288-2). One of these features seems like it could invalidate the correctness of the algorithm for this application (see first point below). The others seem problematic for general use of ABC, but not this particular application (see second point below).
In the fully Bayesian setting, the ABC methods aim at not only obtaining the parameter estimates but also drawing inference on the posterior distributions and assessing uncertainties. But in our application, due to the absence of observation errors, our algorithm was designed to retrieve the parameter estimates in the most efficient way rather than drawing a full inference on posterior distributions, hence there were some unusual features in our algorithm compared to the traditional ABC methods. We have made certain adjustments to the algorithm to ensure the correctness of the ABC application and focus on retrieving parameter point estimates at the same time. We hope the responses to the reviewer's later comments can clarify these adjustments.