Understanding allometry

Steve, Thanks for your comments and questions on theory based allometry. I have cross-posted it to nmusers because this topic has been of interest there over many years. I am aware that you wrote an editorial on this topic with Denis Fisher which you titled “Allometry, Shallometry!”. Having read your editorial and your comments I don’t think the implication that allometry is shallow is appropriate. On the contrary, I get the impression (see below) that you and Denis do not really understand the biological principles underlying allometry and seem to be unaware of the substantial literature supporting theory based allometry and its application in humans. Would your journal be willing to receive a rejoinder to your editorial with a deeper explanation of the science and the literature? Best wishes, Nick Nick Holford, MBChB, FRACP Professor of Clinical Pharmacology, University of Auckland Adjunct Professor of Bioengineering and Therapeutic Sciences, UCSF

Nick Your condescending tone is not appropriate — Steve did not insult you nor should you insult him. An apology is due. You, Steve, and I had the same mentor — Lewis Sheiner. His most important teaching was LET THE DATA SPEAK. When theory and evidence clash, Lewis would not blindly stick to theory. In this instance, Steve asked you to provide EVIDENCE to support your claim that allometric scaling would yield markedly better fits than weight-normalization. You offered the McCune article to support your argument (without mentioning your vested interest as an author). In that article, you wrote: all clearance (CL,Q) and volume (V1,V2) parameters were scaled for body size and composition using allometric theory and predicted fat free mass (FFM).(19–21) It does not appear that you evaluated a weight-normalized model. If you don’t look, you never see! I also note that the table reports the following: CL Clearance L/h/62kg NFM CL 11.4 (1.1) This brings up the issue of SAFETY. I was a clinician for several decades and Steve continues to be an active clinician. I don’t know if you see patients. However, many participants in this mailing list have never selected a dose of a drug, then administered it to a patient. I venture to say that most clinicians when faced with a dosing regimen that requires raising weight to the 3/4 power would run in the opposite direction (or, make a dosing error). The entry in the table is even more problematic. A clinician first needs to calculate NFM, then they need to realize that the dose is not proportional to 11.4 by a factor of weight/62. If the goal of PK is publishing journal articles about pure science, your approach might be OK. But, as far as I know, the goal is to improve patient safety. If there were a strong (or even moderate) preference for allometrically-scaled models, I would support their use. But, Steve asked you to provide EVIDENCE for this and you failed to do you. You also cited Eleveld’s article. Although his manuscript did include one weight-normalized model, the allometric model required multiple additional terms to fit the data. In particular, that article (as yours) required a “maturation” term to describe younger patients. In other words, neither of these articles demonstrates that allometric models are sufficient to describe the range of sizes in humans. Had Eleveld added those extra terms to the weight-normalized model, it might have performed as well as the model he published. Over the past two decades, I have analyzed data from > 200 studies including many in infants and children. In many of these, I have compared weight-normalized and allometric (and, in adults, unscaled) approaches. In virtually all cases, the difference in fit between the weight-normalized and allometric approaches was trivial and often favored the weight-normalized approach. Can you cite cases in which the allometric approach fares much better? I also have had the opportunity to be an editor for a journal and to review articles for many journals (and I have reviewed submissions by many people who participate in this mailing list). In many instances, authors refuse to evaluate a weight-normalized model, citing you. In many of these instances, I have insisted that the authors conduct that analysis and (as far as I can recall) there has never been strong evidence to support the allometric model. I repeat — if you don’t look, you don’t see. I suspect that Steve will have more to add. Dennis Dennis Fisher MD P < (The "P Less Than" Company) Phone / Fax: 1-866-PLessThan (1-866-753-7784) www.PLessThan.com

Dennis, I do not accept that my commentary and constructive criticism of the public remarks made by Steve Shafer should be considered an insult. Public discussion and criticism is essential to detect mistaken ideas and fraud ( http://onlinelibrary.wiley.com/doi/10.1111/bcp.12992/full). I am proud to be a co-author of the McCune et al (2014) article. I don't understand what you are implying with your "vested interest" remark. My name was clearly written as a co-author in the reference list. Are you suggesting I should not cite the science in this area that I have been involved in for over 20 years? Thank you for looking at the McCune article. However, you are mistaken in suggesting that the "weight-normalized model" was not evaluated. This model (based on total body weight with an allometric exponent of 1) is a nested sub-model of size model described in the McCune article. When the allometric exponents are estimated the values provide no support for the "weight-normalized model" (see supplementary table 4). The objective function improves by 532.12 when a non-linear allometric model with 4 additional parameters is compared to the linear model. I find it strange that you offer anecdotally "Over the past two decades, I have analyzed data from > 200 studies" to support your "weight-normalized" view yet insist I provide EVIDENCE. If you wish to make a credible statement to support "weight-normalization" then I ask you to publish the DATA with analysis details showing that you account for factors such as maturation that are correlated with mass but are not part of theory based allometry, and apply allometric scaling to all clearance and volume parameters. With regard to SAFETY-- my work with allometric scaling applied to prediction of clearance is not an ivory tower activity. You will find details in the response I will send to Steve on PharmPK who raises a similar point in his comments. Best wishes, Nick McCune JS, Bemer MJ, Barrett JS, Scott Baker K, Gamis AS, Holford NHG. Busulfan in Infant to Adult Hematopoietic Cell Transplant Recipients: A Population Pharmacokinetic Model for Initial and Bayesian Dose Personalization. Clin Cancer Res. 2014;20(3):754-63.

Dear Steve, Thanks for your responses of 22 May and 24 May PDT. Due to some unexplained technical problem I have not been receiving posts from PharmPK since 23 May and only just became aware of your 24 May post. I have tried twice to post a response to PharmPK but it has not appeared on the list as far as I can tell. I am therefore transferring this discussion to nmusers. On 22 May you wrote “I am not aware of any theory that supports scaling by weight to the 3/4 power WITHIN A SPECIES”. Having read the 22 May post from Douglas Eleveld and my own reply do you now accept that the theory of West et al. supports 3/4 power weight scaling within a species including humans? My comments below are in response to your PharmPK post of 24 May. Best wishes, Nick SS: Dear Nick: You stated in your comments to me, both on NMUSERS and on PHARMPK, that a paper of yours with Dr. McCune offered a validation of the use of allometric scaling. Dennis Fisher has already commented on this paper. Dennis said I might have additional comments. As usual, Dennis is right. The paper is McCune et al, Busulfan in infant to adult hematopoietic cell transplant recipients: a population pharmacokinetic model for initial and Bayesian dose personalization.. Clin Cancer Res. 2014;20:754-63. Based on my reading of your paper, and the supplementary material, I would like to offer several observations. 1. Quoting from page 755: “To characterize busulfan pharmacokinetics over the entire age continuum, all clearance (CL,Q) and volume (V1, V2) parameters were scaled for body size and composition using allometric theory and predicted fat-free mass.” In other words, you assumed allometric theory at the outset of your analysis. NH: It is correct that theory based allometry was used at the outset to guide the analysis. This was based not only the biological plausibility of this theory but also the consistent use of non-linear weight scaling in all previous reports of IV busulfan PK (see McCune Supplementary Table 1). The results of non-linear regression using NONMEM are well known to be better when starting from a priori more reasonable models and parameters. A deeper understanding of the problem is always helpful. SS: 2. I think your testing of this assumption is described on page 759: “we estimated the allometric exponents for each of the 4 main pharmacokinetic parameters (Supplementary Table S4). Initial estimates of 2/3 and 1.25 were used for the clearance and volume exponents.” However, you tested this with bootstrap, not with log-likelihood profiles. Why? NH: A recognized limitation of log-likelihood profiles is the focus on just one parameter at a time plus interpretation of the confidence interval is dependent on the assumption that the difference in -2LL is Chi-squared distributed. As I am sure you know that is a questionable assumption when using the NONMEM approximation to the likelihood. While the bootstrap cannot be considered perfect it has useful properties compared with log-likelihood profiles because it allows the uncertainty for all exponents to be estimated simultaneously. The log-likelihood profile method provides a rather superficial view because it only allows the uncertainty for one parameter at time while fixing other parameters at some other values. SS: 3. If allometry makes little difference, then it is an expected result that your final estimates would be close to the starting parameters. This might especially be the case where there are 10 parameters in the calculation of clearance (see item 10 below), compromising the “navigability” of the model away from the starting estimate of PWR when the other 9 parameters start at the value determined by assuming that PWR=0.75. NH: The initial estimates for the bootstrap were deliberately set to values that were quite distant from the theory based values (page 759, para 1). This would be expected to allow the exponent estimates to “navigate” more freely. As you point out it is possible that there would be some kind of “memory” in the other parameters of the model determined using theory based exponents.This is a good idea. I have tried to test it by fitting the data with initial estimates assuming that total body weight is the allometric mass with an allometric exponent for CL and V parameters equal to 1 (linear weight model). The results of such a fit should have no “memory” of theory based values. I then refit the data starting from the final estimates of this linear weight model and estimated the allometric exponents for CL, V, Q and V2 (empirical allometric model). The bootstrap confidence intervals for these exponents estimated from the empirical allometric model with total body weight are shown here: Parameter Description Bootstrap Average 2.5% ile 97.5% ile Bootstrap RSE PWR_CL Allometric exponent for CL 0.764 0.733 0.798 2.3% PWR_V Allometric exponent for V1 1.011 0.871 1.115 5.6% PWR_Q Allometric exponent for Q 0.838 0.734 0.957 6.7% PWR_V2 Allometric exponent for V2 0.930 0.885 0.988 2.6% It is clear that the exponent estimates for CL and V consistent with theory based allometry and inconsistent with a linear weight model. Not accounting for body composition probably explains why the confidence interval for V2 just misses the theory based prediction. This is not due to a “memory” problem because very similar results were obtained starting from theory based initial estimates and total body weight as shown in McCune supplementary table 4. The objective function value (OFV) difference between the linear model and the empirical allometric model is 532.12 with 4 estimated parameters. This difference in OFV would usually be interpreted to show that the non-linear allometric model is superior to the linear function of weight. This is letting the data speak and it is shouting out that the linear model is inconsistent with the data. SS: 4. I stated in my comments that allometric theory did not account for the upper extreme of obesity. You agree, since you found it necessary to corrected allometric scaling with an additional parameter to account for the effects of obesity. NH: I am happy to see that we agree that “obesity” is not part of theory based allometry. However, please recognise that the mass associated with “obesity” would be expected under theory based allometry to have an influence. A deeper, biological question is what is this mass? The West et al.theory does not specify how to account for mass, such as excess fat, that probably has metabolic properties different from mass with “normal” body composition. The NFM method estimates the combination of fat free mass and fat mass that best agrees with allometric theory. The assumption of allometric theory then allows insights into the contributions of these components to “normal” allometric size. SS: 5. I stated in my comments that allometric theory did not account for maturation. You agree, since you found it necessary to add an additional parameter for maturation. NH: I am very happy to see that we agree that allometric theory does not account for maturation. That is why a separate model component (with 2 parameters TM50CL and HIllCL) is included to account for maturation of clearance. SS: 6. You had actual body weight for only 133 subjects in this study, of which only 24 subjects were less than 18 years of age (supplementary table 2). Although your model has 1610 individuals, you only estimated the allometric portion of your model from 24 children. This allometric scaling parameter was assumed to be true for all 1407 subjects (calculated from supplementary table 2). Since your allometric parameter, Ffat, was derived from just 24 children, and applied to all 1407 children, your testing (supplementary table S4) may be a tautology. NH: It is indeed a limitation of this data analysis that most of the hospitals who contributed data did not supply actual body weight but a “dosing weight” (see below). Ffat contributes to the allometric part of the model but recall that allometric theory does not include maturation. The Ffat parameters were estimated from all 133 subjects , irrespective of age, with actual body weight recorded. Therefore I do not agree with your assertion that I “only estimated the allometric portion of your model from 24 children”. SS: 7. You state on page 755 that you used the dosing weight in reference 18. Reference 18 is Gibbs, et al, The Impact of Obesity and Disease on Busulfan Oral Clearance in Adults, Blood 1999;93:4436-40. Reference 18 discusses actual body weight, body surface area, adjusted ideal body weight, and ideal body weight, all calculated from standard formulae. There is no reference to Dosing Weight in this publication. NH: You are correct. There is no reference describing DWT for patients outside Seattle. This is because the method used to calculate DWT was calculated using each institution’s own practice (see page 755, col 2, end of 2nd paragraph describing the study population). The exact method was not supplied with the data provided for each patient by the institution. The dosing weight in Seattle was not used for model development because actual body weight was recorded. SS: 8. Dennis pointed out the potential safety concerns of allometric scaling. I suggest that interested readers look at page 756 of your paper. If that does not scare clinicians, then the exact math for dose calculation appears in supplementary table 7. Would you be comfortable if the oncologist treating your child had to calculate dose based on the complex, interlocking equations required to estimate body size? What theoretical advantage in dose calculation justifies the potential for computational error inherent in supplementary table 7? The risk vs. benefit of allometric scaling cannot be determined from the data in the paper. NH: You bring up an interesting issue that was never the subject of this paper.Unfortunately, the literature abounds with evidence that doctors e.g. anesthetists (Nanji, Patel et al. 2016), often make dosing errors. My work with allometric scaling applied to prediction of clearance and dosing is not an ivory tower activity.Safe dosing of dangerous medicines requires good science and validated methods to ensure the correct dose is administered. The calculation of the dose for busulfan is complex. This is the only drug I know of where the FDA has recommended detailed dose individualization including calculation of an AUC in order to use the drug (FDA 2015). At the request of my clinical colleagues in Auckland, I was involved in the implementation and testing of a tool to guide busulfan dosing in children and adults. The tool was initially based on an FDA study (Booth, Rahman et al. 2007) describing an allometrically scaled model for clearance and was subsequently updated using the results in the McCune paper when an audit showed the predictions were better. The web based dosing tool (www.nextdose.org) has been in use for over 4 years to guide dosing of all patients in New Zealand receiving high dose busulfan for bone marrow ablation. The tool is available for use by anybody who has access to the internet. I understand that the model will be used to guide dosing of all patients in the USA who have samples submitted to the national laboratory in Seattle for measurement of busulfan concentrations. It is my personal hope that dosing decisions will be taken out of the hands of doctors who rarely recognize the principles of rational dosing and continue to use ad hoc empirical methods. To quote from your “Allometry, Shallometry!” editorial, with an example based on what you claim is a simple approach using the linear weight model: “It is OK if you skipped the math. As a clinician, all you need to know is the punchline”. You have been a pioneer in this area with target controlled infusions so I don’t think I have to convince you that this is the way of the future. Doctors should be responsible for providing the data to decide on an appropriate dose and after that a science based computation tool should work out the dose. SS: 9. You have no data showing how well your model predicts individual patients. The closest you come are the visual predictive checks (figure 1) and the prediction corrected visual predictive check (supplement 2). This tells me that the cloud of points is about right. That’s fine, but the average patient does not die. It is patient at the extremes of prediction accuracy who are at increased risk. The data, as presented, does not provide this information. NH: This paper is based on pharmacokinetic data. There is no effectiveness or safety data to judge risk. However, we have described the expected fraction of patients that would be expected to be within an acceptable range of concentrations using a predicted initial dose. Our model performs better than other methods in nearly all the scenarios we tested and is never worse to a clinically important degree. As noted above a web based dosing system using this approach has been used by clinicians in Auckland for over 4 years. SS: 10. Clearance (page 757) is calculated 10 parameters: a population estimate, which is adjusted for F(size), F(maturation), and F(sex). F(size) is based on dosing weight (not explained, see 7 above), height, WHS(50), WHS(max), F(fat), FFEM(DW), and PWR (your allometric parameter, fixed at ¾). F(maturation) is based on PMA, TM(50), and the Hill coefficient. F(sex) is a further adjustment for sex. When clearance is a function of 10 parameters, I do not see how this tests allometric scaling. Indeed, if allometric scaling were hurting your fit (unlikely – more likely it makes no difference, see below), other parameters might compensate to fit the data. NH: Please look at Table 2 to count the parameters in the fixed effect model. There are 12 estimated parameters. The number of parameters is not a “test of allometric scaling”. It is a measure of the complexity of variability of busulfan PK. These parameters identify predictable sources of variability that can be used to aid initial dosing. SS: 11. You compare this model to models by Trame, Paci, and Bartelink, noting that your model performs much better than these models. You are comparing your model with 12 structural parameters to models with 2 (Trame), 4 (Paci), and 5 (Bartelink) structural parameters. Your 12 parameter model better described your data than these simpler structural models fit to your data. Did you expect anything else? NH: I certainly expected to find our model would do better because it has a stronger mechanistic and biological basis. It is more complex than others because it goes more deeply into biological understanding and does better over a wide range of human size and age. SS: 12. You state on page 762: “The model is based on principles that have already been shown to be robust for predictions with other small molecule agents from neonates to adults.” I don’t see that. If “robust” means that it allometric helps describe PK at the extremes of weight, then the allometric model was not robust. It required adjustments for both maturation and for obesity. Between these extremes, say 30-100 kg, any optimal coefficient times weight to the ¾ power will differ by less than 10% from an optimal coefficient times weight alone. This will be invisible given the order of magnitude variability in clearance (your figure 2). NH: Robust refers to principles which recognize the major role of size and maturation (the key components of our model) in explaining variability in PK for many drugs (see (Holford, Heo et al. 2013)).The influence of body composition as a predictor of allometric size has fewer examples but it is only by digging below the surface that we can discover new things and evaluate their importance. It is no surprised that over a narrow range (30-100 kg) a linear model is a reasonable approximation to theory based allometry. But this is not true over the range of TBW (3 to 140 kg; see Figure 1) in the patients in this study (see below). SS: I see little to no evidence that your paper with Dr. McCune demonstrated superiority of allometry. Rather, your paper demonstrated that even a model with 12 parameters could not reduce the variability of busulfan estimated clearance beyond an order of magnitude. NH: If the model can predict how to reduce variability by an order of magnitude for a very toxic drug such as busulfan then I think this is a major advance. It makes no difference how many parameters are needed. The important thing is to be able to predict differences in PK which can then be applied to achieve safe and effective dosing. If this was my child faced with a bone marrow transplant I would want to use every means possible to improve the chances of a successful graft and reduce the substantial risk of serious toxicity and death. The simulations you provide in your “Allometry, Shallometry!” editorial replicate what I demonstrated 20 years ago (Holford 1996). I pointed out at that time the underestimation of doses predicted from adults if a linear weight model was assumed. You appear to propose using the same mg/kg dose in a children as in adults for computational convenience. But clinically recommended dosing regimens for busulfan use a higher mg/kg dose in younger and lighter children with lower mg/kg doses for older and heavier children. The allometric and maturation model we have developed predicts and explains this pattern of mg/kg dosing recommendations for busulfan and all other drugs used in humans (Holford, Heo et al. 2013). SS: You also demonstrated that allometric models require specific adjustments for maturation and dosing. You will recall this was one of the points that I made in my comments, which are also discussed in the Allometry Shallomatry! editorial. NH: I think we are in agreement that theory based allometry can only explain variability due to differences in body mass. Other factors also explain variability such as maturation, organ function, drug interactions, genotypes, etc. These factors have no influence on the allometric component of the model. I do not agree with you when you say that allometric models requires “specific adjustments” using factors such as maturation. If you tried to understand more deeply the allometric model you would realize it is not based on these other factors. SS: Perhaps there are other analyses of these data that would demonstrate a significant benefit of allometric scaling of data. If you are willing to share with me your data on the 133 subjects for whom you have actual body weights, I would be happy to address the question directly. NH: I understand that Jeannine McCune has contacted you and offered to work with you to obtain permission to use the data. SS: Respectfully, Steve -- Steven L. Shafer, MD Professor of Anesthesiology, Perioperative and Pain Medicine, Stanford University Adjunct Associate Professor of Bioengineering and Therapeutic Sciences, UCSF NH: References Booth, B. P., A. Rahman, R. Dagher, D. Griebel, S. Lennon, D. Fuller, C. Sahajwalla, M. Mehta and J. V. Gobburu (2007). "Population pharmacokinetic-based dosing of intravenous busulfan in pediatric patients." _J Clin Pharmacol_ *47*(1): 101-111. FDA (2015). "Busulfex Product Label http://www.accessdata.fda.gov/drugsatfda_docs/label/2015/020954s014lbl.pdf." ; Holford, N., Y. A. Heo and B. Anderson (2013). "A pharmacokinetic standard for babies and adults." _J Pharm Sci_ *102*(9): 2941-2952. Holford, N. H. (1996). "A size standard for pharmacokinetics." _Clin Pharmacokinet_ *30*(5): 329-332. Nanji, K. C., A. Patel, S. Shaikh, D. L. Seger and D. W. Bates (2016). "Evaluation of Perioperative Medication Errors and Adverse Drug Events." _Anesthesiology_ *124*(1): 25-34. -- Nick Holford, Professor Clinical Pharmacology Dept Pharmacology & Clinical Pharmacology, Bldg 503 Room 302A University of Auckland,85 Park Rd,Private Bag 92019,Auckland,New Zealand office:+64(9)923-6730 mobile:NZ+64(21)46 23 53 FR+33(6)62 32 46 72 email:[email protected] http://holford.fmhs.auckland.ac.nz/ "Declarative languages are a form of dementia -- they have no memory of events" Holford SD, Allegaert K, Anderson BJ, Kukanich B, Sousa AB, Steinman A, Pypendop, B., Mehvar, R., Giorgi, M., Holford,N.H.G. Parent-metabolite pharmacokinetic models - tests of assumptions and predictions. Journal of Pharmacology & Clinical Toxicology. 2014;2(2):1023-34. Holford N. Clinical pharmacology = disease progression + drug action. Br J Clin Pharmacol. 2015;79(1):18-27.