A graphical method for practical and informative identifiability analyses of physiological models: A case study of insulin kinetics and sensitivity
 Paul D Docherty^{1}Email author,
 J Geoffrey Chase^{1},
 Thomas F Lotz^{1} and
 Thomas Desaive^{2}
https://doi.org/10.1186/1475925X1039
© Docherty et al; licensee BioMed Central Ltd. 2011
Received: 29 March 2011
Accepted: 26 May 2011
Published: 26 May 2011
Abstract
Background
Derivative based apriori structural identifiability analyses of mathematical models can offer valuable insight into the identifiability of model parameters. However, these analyses are only capable of a binary confirmation of the mathematical distinction of parameters and a positive outcome can begin to lose relevance when measurement error is introduced. This article presents an integral based method that allows the observation of the identifiability of models with twoparameters in the presence of assay error.
Methods
The method measures the distinction of the integral formulations of the parameter coefficients at the proposed sampling times. It can thus predict the susceptibility of the parameters to the effects of measurement error. The method is tested insilico with Monte Carlo analyses of a number of insulin sensitivity test applications.
Results
The method successfully captured the analogous nature of identifiability observed in Monte Carlo analyses of a number of cases including protocol alterations, parameter changes and differences in participant behaviour. However, due to the numerical nature of the analyses, prediction was not perfect in all cases.
Conclusions
Thus although the current method has valuable and significant capabilities in terms of study or test protocol design, additional developments would further strengthen the predictive capability of the method. Finally, the method captures the experimental reality that sampling error and timing can negate assumed parameter identifiability and that identifiability is a continuous rather than discrete phenomenon.
1. Background
A number of physiological phenomenon have been modelled by formulating mathematical representations of the relevant interactions. These models frequently incorporate variable parameters that can be identified to match the model representation to the observed behaviour. The value of these parameters is then used to characterise or quantify the response. However, with complex or large models, variable parameters can be selected that seem mathematically distinct, but in reality define the same observable effect and identification failure is certain. Thus, model identifiability analyses are used to test the selection of model parameters and ensure that they are mathematically distinct.
Approaches for the analysis of model identifiability typically assume continuous perfect input data [1–3]. However, these derivativebased identifiability methods can produce false assurances of identifiability. Recent dissatisfaction with the classical algorithms has resulted in the development of new methods that recognise assay error and discrete measurements as critical to identifiability [4, 5]. The limitation of discrete data that is often subject of assay error often causes parameter tradeoff [6–9] and thus limitation of the identified metrics clinical value. Thus, not only should a model be checked for identifiability in the classical apriori sense, but the susceptibility of parameters to mutual interference should also be tested.
For example, the Minimal Model of insulin sensitivity [10] has been shown to be identifiable using such methods [11–13]. However, with discrete data that is subject to assay error, parameter identification has sometimes failed [6, 7, 14]. Numerous Bayesian techniques have had success in limiting this failure [7, 15–17], but they tend to force the parameters to diverge away from their true least square values, limiting the relevance of the model and exaggerating the influence of population trends on an individual test's identified parameter values. Thus, widespread clinical application of these models has been limited by the ambiguity of results.
This article presents a novel graphical method for identifiability analysis that allows an identifiability analysis with consideration of noise and assay error. Furthermore, the method highlights areas for potential improvements to protocols and sampling times that would improve practical identifiability. At this stage of development, the method is limited to firstorder, twoparameter models that allow a separation of parameters, but are typical of those found in pharmacokinetic (PK) and pharmacodynamic (PD) modelling.
2. Method and Study Design
The proposed method will be evaluated insilico using clinically validated models of insulin kinetics and the dynamic between insulin concentration and glucose decay. The method's ability to predict the variation behaviour of identified parameters in a Monte Carlo analysis will be tested.
2.1 Proposed method process
 1.
 2.
 3.
 4.
 5.
 6.
where: μ is a proportionality factor that incorporates factors such as the relative contribution of the parameter to the derivative of the relevant species concentration in the governing differential equation, and the absolute magnitude of the noise at the sampled times in relation to the relative magnitude of the parameter coefficient.
Thus, the method cannot accurately predict the coefficient of variation that a Monte Carlo analysis may find. However, it can predict the change of variation that might be observed when changes are made to the test sampling or stimulus protocols.
2.2 The dynamic insulin sensitivity and secretion test (DISST) model
2.2.1 Insulin pharmacokinetics
Symbol  definition  units 

I  Plasma insulin concentration  mU/L 
Q  Interstitial insulin concentration  mU/L 
U _{ N }  Endogenous insulin production rate profile  mU/L/min 
U _{ X }  Exogenous insulin bolus  mmol 
n _{ K }  Renal clearance of plasma insulin  1/min 
n _{ L }  Hepatic clearance of plasma insulin  1/min 
n _{ C }  Insulin clearance to cells  1/min 
n _{ I }  Transition of insulin between plasma and interstitium  L/min 
α _{ I }  Saturation of hepatic insulin clearance  L/mU 
x _{ L }  First pass clearance of insulin  1 
V _{ P }  Distribution volume of plasma insulin  L 
V _{ Q }  Distribution volume of interstitial insulin  L 
The model is used in the dynamic insulin sensitivity and secretion test (DISST) to define the PKs of insulin due to test stimulus [18, 20]. The model assumes that plasma insulin (I) is sampled. A traditional derivative based identifiability analysis of the model presented in Equations 2 and 3 using Ritt's pseudodivision algorithm [1, 21] is presented in Appendix 1.
2.2.2 Pharmacodynamics of glucose and insulin
Nomenclature from Equation 4
Symbol  definition  units 

G  Glucose concentration  mmol/L 
Q  Interstitial insulin concentration  mU/L 
G _{ b }  Basal glucose concentration  mmol/L 
Q _{ b }  Basal interstitial insulin concentration  mU/L 
p _{ G }  Glucose dependant glucose clearance  1/min 
S _{ I }  Insulin sensitivity  L/mU/min 
P _{ X }  Exogenous glucose bolus  mmol 
V _{ G }  Glucose distribution volume  L 
Although the structural identifiability of Equation 4 is trivial it is also presented in Appendix 1.
2.3 Participants
Anatomical and identified parameter values
Glucose tolerance  Sex  Age  BMI  U _{ N } ^{†}  I clearance  V _{ G }  SI  

U _{ b }  U _{ 1 }  U _{ 2 }  n _{ L }  x _{ L }  
NGT  M  22  21.5  26.6  487.7  9.7  0.218  0.797  9.75  20.95 
IGT*  F  57  33.9  115.5  233.6  150.7  0.064  0.822  13.35  2.236 
2.4 Simulated test protocol
The simulated protocol is similar to the DISST test; a 10 g glucose bolus is administered at t = 7.5 and a 1 U insulin bolus at t = 17.5. The test duration is 60 minutes with a 5 minute sampling frequency. The U _{ N } profile is defined as a step function with three stages including basal, first and second phase production rates. The first phase of insulin production has a fiveminute duration and begins with the glucose bolus. Simulations of plasma and interstitial insulin are completed using Equations 2 and 3, the parameter estimation equations from [22], n _{ L } and x _{ L } values from Table 3 and an a _{ I } value of 0.001 L/mU. Glucose is simulated using Equation 4 and the interstitial insulin profile obtained in the evaluation of Equations 2 and 3.
2.5 Iterative integral method
 1.
The method converts the model governing equations to their integral formulation.
 2.
The parameter coefficients and the remainder terms are evaluated between t = 0 and the sample times.
 3.
The coefficients form the LHS of a matrix equation in terms of the parameters, while the remainder terms form the RHS. This matrix is evaluated to get parameter values.
 4.
These parameter values are used to update the coefficient evaluations in step 2, which enables a more accurate matrix equation in step 3.
 5.
Step 4 is iterated until convergence is achieved.
2.6 Analysis
A series of parameter and sampling scenarios were analysed using the models presented in a Monte Carlo analysis. Clinically measured physiological parameters of the NGT participant presented in Table 3 were used to define simulated responses to the test protocol described in Section 2.4. Samples were obtained from the simulated profiles at the defined times. Each iteration of the Monte Carlo analysis adds normally distributed assay error (the error magnitude is defined within each section). 100 iterations are used for each analysis with parameter identification by the iterative integral method. Only simulated profiles from the NGT participant are used until Section 3.2.2. Each scenario will present the mean value of those identified in the Monte Carlo simulation normalised by the simulation value from Table 3, and the coefficient of variation (CV) of each identified parameter (i.e. A mean of 1 and standard deviation of 0 implies perfect identification). The CV values are the paramount indicator of parameter tradeoff during identification. The CV values are thus was compared to the ε/Δ_{2} value defined using the methods of Section 2.1 to obtained values for μ that linearise Equation 1.
2.7 Cases tested
A series of indicative cases will be investigated to show how the selection of model parameters, sample placement, participant behaviour and protocol dosing can have an effect on model identifiability.

The insulin pharmacodynamic model will include five variable parameters to be identified to confirm the robustness and convexity of the iterative integral method, and to confirm the findings of the traditional derivativebased identifiability analysis.

Parameter interference of n _{ L }and n _{ K }will be analysed. The nature of identifiability will be explored by alteration of the α _{ I }term that moderates the denominator of n _{ L }in Equation 2.

The effect of sample selection on parameter identification of n _{ T }and V _{ P }will be measured. (n _{ T }is the addition of n _{ L }and n _{ K }).

The effects of sample omissions on SI and V _{ G }identification are measured.

The disparity of parameter identifiability in insulin resistant and sensitive individuals will be assessed using p _{ G }and SI as variable model parameters.

The protocol proposed in Section 2.4 will be altered to see if the identifiability of insulin resistant participants can be improved.

Measured samples that the proposed method claims are not valuable to stable identification are ignored and the identification process is repeated to confirm the prediction.
All cases will be tested with 1% and 3.5% normally distributed noise added to the virtually obtained data (to a maximum of 3 standard deviations).
3. Results
3.1 Analysis of the insulin pharmacokinetic model
3.1.1 Confirmation of global model identifiability
3.1.2 Hepatic and renal clearance rate identification
Variation in identified n _{ K } and n _{ L } values
Noise  0%  1%  3.5%  

Saturation  α _{ I } = 0.001  α _{ I } = 0.05  α _{ I } = 0.001  α _{ I } = 0.05  α _{ I } = 0.001  α _{ I } = 0.05 
n _{ K }  1(0)  1(0)  0.952(0.297)  0.999(0.017)  0.946(0.836)  1.004(0.059) 
n _{ L }  1(0)  1(0)  1.035(0.214)  1.001(0.008)  1.041(0.595)  1.000(0.028) 
where: i= 5, 10, 15, ..., 60 minutes, matching sampling times.
Thus, the parameters identified with the exaggerated α _{ I } term should have approximately 15 times smaller variability than those identified with the accepted α _{ I } value. Table 4 shows the effect of the α _{ I } distinction on the identified parameter values.
Table 4 shows the distinction between the effects of noise on identified parameters when the α _{ I } term is changed. Although the 0% noise case indicates that the parameters are uniquely identifiable, at 1% noise the variation in the identified values limits their clinical viability. At 3.5% noise, which may be expected in a real clinical setting, the parameters are no longer uniquely identifiable. This is illustrated by the very large CVs of the parameters. However, when the α _{ I } term is significantly increased, unique identifiability is once again possible, even with 3.5% noise. The mean ratio of variation caused by the disparate a _{ I } values was approximately 1:20. This is larger than the ratio predicted by the method (1:15), but still represents a positive outcome in terms of predicting the relative magnitude of the change.
The difference between the curves at the sample times indicates the identifiability of the model parameters in this twoparameter case. Thus, when the saturation term is increased, the coefficients of the parameters are more distinct and identifiability is increased. Despite the positive findings of the typical identifiability analysis, a saturation value of 0.001 L/mU causes n _{ K } and n _{ L } to become uniquely unidentifiable in a real clinical setting. This outcome may be considered as an elementary finding that should be inferred with a quick observation of Equation 2. However, it points to a failing of typical apriori identifiability tests that this approach can negate with a quick graphical analysis.
3.1.3 Plasma insulin distribution volume and insulin clearance identifiability
Δ_{2} values from three sampling protocols defined in Section 3.1.3
Δ _{2}  

Protocol 1  0.731 
Protocol 2  0.990 
Protocol 3  0.045 
The Δ_{2} values indicate that Protocol 3 will be comparatively unable to reproduce the simulation values. Contrary to the expected result that parameter identification is best with frequently sampled Protocol 1, the method predicts that the sparsely sampled Protocol 2 will be slightly more accurate.
Protocol dependence of identified n _{ T } and V _{ P } values
Noise  1%  3.5%  

Protocol  1  2  3  1  2  3 
n _{ T }  1(0.004)  1(0.003)  0.973(0.012)  1.003(0.012)  1.001(0.014)  0.958(0.045) 
V _{ P }  1.001(0.014)  0.995(0.012)  1.231(0.175)  0.999(0.047)  0.995(0.051)  1.190(0.463) 
It is evident that although Protocols 2 and 3 contain the same number of samples, the resolution of the identified parameters is considerably reduced in Protocol 3. In effect, was unidentifiable with Protocol 3. This result occurs because of the lack of distinction in the coefficients of the parameters at the sample times as indicated in Table 5 and Figure 3.
Thus, the method predicted the poor performance of the third protocol while it predicted much lower variability for both Protocols 1 and 2. However, it also suggests that Protocol 2 would improve slightly upon Protocol 1, which was not the case as both protocols performed equally in terms of parameter identifiability. It is expected that it's equality of variance is an artefact of the normalisation as a function of mean coefficient at the sample value, artificially lowering the magnitude of the Δ_{2} terms in Protocol 1.
Overall, these findings highlight the inefficiency, extreme clinical burden and intensity of frequent sampling in contrast to wellpositioned and infrequent sample timing. More specifically, Protocol 1 used 9 more samples than Protocol 2 with significant added clinical intensity and assay cost (^{~}200% more!) for absolutely no information gain. This outcome was successfully predicted by the identifiability analysis method presented.
3.2 Analysis of the glucose PD model
Equation 4 will be used in the analysis of identifiability of terms frequently used to model the PDs of insulin and glucose. All analyses in this section will simulate insulin concentration profiles for the plasma and interstitium only once for each Monte Carlo analysis. Thus for clarity and simplicity, it is assumed that insulin is not subject to assay error here. Furthermore, while glucose assay error from a blood gas analyser is approximately 2%, errors of 1% and 3.5% will be used for consistency with Section 3.1. As such, the resultant coefficients of variation should not be considered fully applicable clinically, but merely as an indication of parameter tradeoff during identification. The Monte Carlo analysis method with the NGT participant described in Section 2.3 is repeated for the glucose PD model. The IGT participant will be used in tandem with the NGT participant from Section 3.2.2.
3.2.1 Insulin sensitivity and distribution volume
Sample resolution dependence of identified SI and V _{ G } values
Noise  1%  3.5%  

Protocol  1  2  3  1  2  3 
SI  1.001(0.007)  1.002(0.008)  1(0.016)  1.002(0.028)  1.002(0.033)  1.010(0.054) 
V _{ G }  1(0.012)  1(0.014)  1.001(0.029)  0.998(0.046)  1.001(0.049)  0.990(0.087) 
Table 7 shows that parameter stability is generally very high even in a sparsely sampled data set with a relatively high level of noise. This result was expected due to the relatively large difference in the coefficient integrals shown in Figure 4 for each of the sampling protocols. Thus, like the case of V _{ P }and n _{ T } , intelligent sample timing can significantly reduce clinical burden and study cost with negligible loss of information. Furthermore, the proposed method successfully predicted this outcome.
3.2.2 Insulin sensitivity and glucose dependent decay
Modelbased studies of insulin sensitivity frequently identify a parameter synonymous with p _{ G } in addition to SI and V _{ G } as parameters when using the Minimal Model [10] or similar. However, although these parameters are mathematically distinct, they are known to tradeoff during identification and practical identifiability is not generally assured. It has been reported that these issues can be exacerbated for insulin resistant (IR) individuals [7, 8]. Thus, the second, IGT participant defined in Table 3 will also be analysed.
As mentioned, this limitation of the Minimal Model has been reported, but not explained in the literature until now.
Participant dependence of identified SI and p _{ G } values
Noise  1%  3.5%  

Participant  NGT  IR  NGT  IR 
SI  1(0.008)  1.001(0.161)  0.998(0.032)  0.876(0.449) 
p _{ G }  1.009(0.320)  0.959(0.756)  1.109(0.802)  1.573(1.244) 
It is apparent that the insulin resistant individual's parameter identifiability is much lower than the NGT participant despite the identical PD model, test protocol and identification process. Table 8 highlights this result, as well as the increasing loss of identifiability as assay error increases. This is in accordance to published findings and the proposed method's prediction.
3.2.3 A hypothetical protocol to enable p_{G} identification in dynamic tests
To forcibly remove this ambiguity introduced by the comparable coefficients of the IR individual, the protocol of the DISST could be altered. After an initial observation of the effect of the insulin bolus on glucose concentration, more insulin could be introduced to ensure that the participant's glucose concentration is maintained approximately 0.5 mmol/L below the basal concentration. Such a protocol may include an extension of the protocol described in Section 2.4 wherein a period of slight hypoglycaemia is achieved for each participant with a series of participantspecific insulin boluses administered with feedback control.
To allow a fair comparison between the variability of the parameters of the proposed protocol and the protocol used in Section 3.2.2 the sampling regimen and test duration will be maintained. Thus, the additional insulin is administered as a bolus t = 32.5 minutes. In a clinical setting, the magnitude of the bolus would be participantspecific and dependent on the glucose concentration response alone (as glucose can be assayed in realtime). Note that this task would be very difficult and potentially dangerous in a regular clinical setting. In particular, the amount of insulin required would vary between participants, and must be estimated in ignorance of endogenous insulin production. This may cause a high incidence of potentially harmful hypoglycaemia. This protocol is only mooted to illustrate the ability of the method, and in reality the introduction of additional insulin could be applied slowly and safety would be assured.
For the case of the IR participant presented here, reducing glucose sufficiently would require a 3U bolus at t = 32.5 minutes.
Effect of the added insulin bolus on the identifiability of SI and p _{ G } from the IR participant
Noise  1%  3.5% 

SI  1.002(0.029)  0.996(0.070) 
Δ SI  +0.001(0.132)  +0.120(0.379) 
p _{ G }  0.989(0.369)  0.992(0.969) 
Δ p _{ G }  +0.030(0.387)  0.581(0.275) 
Although the inhibitive CV values for the parameters indicate that the proposed protocol could not be used clinically, the variability decreased in the order predicted by the identifiability method presented. The hypothetical protocol presented has confirmed the reasons discussed for the poor parameter identification observed in many clinical studies which utilise these two competing parameters [7, 8]. Furthermore, it demonstrates a limitation of traditional identifiability methods, which provide an evaluation of identifiability in ignorance of probable participant behaviour or test protocol design.
The protocol presented would be virtually impossible to apply clinically in the 60 minute duration as it is shown here. However, this analysis was limited by the need for a comparable duration and sampling regimen to Section 3.2.2. The method could thus be used to define similar protocols that could yield data that enables unique identification of these parameters. If such protocols are pursued, they would most likely require two sections in a much longer test. The first section would involve the protocol defined in Section 2.4, and would be followed immediately by an infusion of insulin designed to safely bring the participant's glucose concentration to 0.5 mmol/L below the basal level. Robust results would be most assured if the participant's glucose concentration was maintained below this level for approximately 30 minutes, and thus, the protocol would most likely require about 2 hours. However, a stable result both in terms of SI and p _{ G } would be generally assured.
3.2.4 Removal of redundant points
The effect of targeted sample omission
Noise  1%  3.5%  

Participant  NGT  IR  NGT  IR 
SI  0.999(0.010)  0.960(0.182)  1(0.029)  0.852(0.496) 
Δ SI  0.001(+0.002)  0.041(+0.021)  +0.002(0.003)  0.024(+0.047) 
p _{ G }  1.019(0.360)  1.152(0.693)  1.400(0.762)  1.231(1.156) 
Δ p _{ G }  +0.010(+0.040)  +0.193(0.063)  +0.311(0.040)  +0.342(0.088) 
The findings of this analysis imply that the omission of the samples that were assumed to be obsolete had little effect on the outcome of the identification process. Most changes were very small and only those for the particularly unstable parameters showed any significant changes.
3.3 The value of μ
μ values from the various analyses
Analysis  Parameter  μ 

Section 3.1.2  n _{ L }  0.560 
n _{ K }  0.801  
Section 3.1.3  n _{ T }  0.081 
V _{ P }  0.683  
Section 3.2.1  SI  1.48 
V _{ G }  2.41  
Sections 3.2.2 and 3.2.4  SI  6.45 
p _{ G }  26.0 
Table 11 presents the μ values identified by the gradients of the regression lines
It can be observed that no single value for μ can be applied across all models and that different parameters are considerably more susceptible to the distinction of the parameter coefficients. However, the general adherence to the linear relationships observed in most examples implies that the form of Equation 1 is accurate for this purpose with the possible exception of SI in Sections 3.2.2 and 3.2.4.
4. Conclusions
The method presented was able to predict and explain the parameter identifiability behaviour exhibited in models of insulin kinetics and insulin/glucose dynamics. This capability is in direct contrast to the traditional derivativebased identifiability analysis [1, 3, 11–13] that can only provide a confirmation of an infinitesimal distinction of the parameters in the governing equations. In addition to the ability to predict parameter identifiability, the method has been shown to be able to aid sample selection and explain the nonidentifiability of the p _{ G } term in common insulin sensitivity tests for IR participants of dynamic tests. It was also used to derive and justify a novel protocol to make this parameter more identifiable in this subgroup. Furthermore, the method also accurately predicted sample redundancy.
Like the proposed method, the method proposed by Raue et al. [4] recognises the ability of apparently identifiable parameters to become practically nonidentifiable in the presence of assay error and discrete sampling. The Raue et al. method requires numerous simulations and parameter identification processes to characterise the model sensitivity to variations in each model parameter. In contrast, the proposed method allows a quick graphical analysis, which can produce immediately apparent and intuitive results. Furthermore, the proposed method can quickly appraise protocol variants, and provide indications of the reason for practical nonidentifiability. The method of Raue et al. is a more general method as it can appraise most model configurations, whereas the proposed method is currently limited to models with two separable parameters.
In reality, many models utilise more than two parameters to describe physiological kinetics and seek to identify all at once. It is expected that development of the proposed method will enable identifiability analyses of such models. However, more care must be taken to construct the coefficient integral formulations as combinations of parameters that may come into conflict. This task would require the contrast between the most deleterious combinations of integral coefficients to be measured. However, this point was not explored in this study, as the goal was to introduce the overall approach.
Only cases with separable parameters in terms of measured species were analysed. In reality, some model parameters are intrinsically linked and this method will not work. An example of linked parameters would be insulin sensitivity and insulin effect saturation that requires a MichaelisMenten formulation. In addition, some parameters effect remote, unmeasured concentrations or masses and are thus not able to be identified with this method. For example, the n _{ C } term in Equation 3 could not be evaluated for identifiability using this method without the inclusion of measurements of interstitial insulin [26]. As such, this method should not replace the traditional identifiability analysis, but be used in tandem with it (or methods such as Raue et al. [4] or similar) to produce both theoretical and practical investigations of identifiability.
Furthermore, the model assumes that an expected range of parameter values is known prior to the commencement of the clinical study. This knowledge is important, as the method requires that species simulations are available to define the coefficients of the parameters. However, in most cases, the researcher will be able to obtain an indication of the likely range of parameter values in a cohort from a brief literature search, and likely test outcomes could be cases that can be safely evaluated prior to clinical testing.
There is also an assumption that the model captures all of the dynamics of the system perfectly. In reality no model can provide such accuracy. In particular, Figures 4, 5, 6, 7 show a sample taken at t = 10 for glucose. In reality, this sample will be affected heavily by error caused by incomplete mixing, and although the method presented indicates that this is a valuable sample, if it is used in the glucose pharmacodynamic model of Equation 5, the resultant parameters will be overly influenced by an unmodelled mixing effect.
Although the method has limitations and potential for improvement, it can provide valuable information at a study design stage, as well as valuable identifiability information not available from typical methods. It can differentiate between the applicability of different dynamic tests based on cohorts, and also help to define optimal sample timing and frequency. In particular, it explained the observation of poor parameter convergence in the Minimal Model for insulin resistant participants that has been widely reported but without clearly defining the cause. Thus, despite the method's limitations, it should still be used in the design stage of a study to ensure that the resultant clinical data can provide usable results, and time and money is not wasted.
Finally, the method has highlighted the limitation of discrete binary identifiability analyses as providing potentially misleading assurances of parameter identifiability in real clinical applications, and shown that identifiability is instead a continuous artefact of sample timing and the distinction between parameter coefficients.
Appendix 1: Structural identifiability
A1.1 Structural identifiability of the insulin model
if: Y = I
Using arbitrary values for the coefficients confirms global identifiability of the insulin kinetic model of Equations 2 and 3.
A1.2 Structural identifiability of the glucose model
and observation confirms that p _{ G } , SI, G _{ b } and V _{ G } are globally identifiable using derivativebased identifiability analysis.
Declarations
Acknowledgements
The authors would like to thank Professor Jim Mann and Dr Kirsten McAuley of the Edgar National Centre for Diabetes Research who organised the clinical investigation that allowed this research. This project was funded by the Health Research Council of New Zealand (HRC) and the Foundation for Research, Science and Technology (FRST).
Authors’ Affiliations
References
 Bellu G, Saccomani MP, Audoly S, D'Angio L: DAISY: A new software tool to test global identifiability of biological and physiological systems. Comput Methods Programs Biomed 2007, 88: 52–61. 10.1016/j.cmpb.2007.07.002View ArticleGoogle Scholar
 Bellman R, Åström KJ: On structural identifiability. Mathematical Biosciences 1970, 7: 329–339. 10.1016/00255564(70)90132XView ArticleGoogle Scholar
 Pohjanpalo H: System identifiability based on the power series expansion of the solution. Mathematical Biosciences 1978, 41: 21–33. 10.1016/00255564(78)900639MathSciNetView ArticleGoogle Scholar
 Raue A, Kreutz C, Maiwald T, Bachmann J, Schilling M, Klingmüller U, Timmer J: Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics 2009, 25: 1923–1929. 10.1093/bioinformatics/btp358View ArticleGoogle Scholar
 Schaber J, Klipp E: Modelbased inference of biochemical parameters and dynamic properties of microbial signal transduction networks. Current Opinion in Biotechnology 2011, 22: 109–116. 10.1016/j.copbio.2010.09.014View ArticleGoogle Scholar
 Erichsen L, Agbaje OF, Luzio SD, Owens DR, Hovorka R: Population and individual minimal modeling of the frequently sampled insulinmodified intravenous glucose tolerance test. Metabolism 2004, 53: 1349–1354. 10.1016/j.metabol.2004.04.011View ArticleGoogle Scholar
 Pillonetto G, Sparacino G, Magni P, Bellazzi R, Cobelli C: Minimal model S(I) = 0 problem in NIDDM subjects: nonzero Bayesian estimates with credible confidence intervals. Am J Physiol Endocrinol Metab 2002, 282: E564–573.View ArticleGoogle Scholar
 Quon MJ, Cochran C, Taylor SI, Eastman RC: Noninsulinmediated glucose disappearance in subjects with IDDM. Discordance between experimental results and minimal model analysis. Diabetes 1994, 43: 890–896. 10.2337/diabetes.43.7.890View ArticleGoogle Scholar
 Caumo A, Vicini P, Zachwieja JJ, Avogaro A, Yarasheski K, Bier DM, Cobelli C: Undermodeling affects minimal model indexes: insights from a twocompartment model. Am J Physiol 1999, 276: E1171–1193.Google Scholar
 Bergman RN, Ider YZ, Bowden CR, Cobelli C: Quantitative estimation of insulin sensitivity. Am J Physiol 1979, 236: E667–677.Google Scholar
 Audoly S, Bellu G, D'Angio L, Saccomani MP, Cobelli C: Global identifiability of nonlinear models of biological systems. IEEE Trans Biomed Eng 2001, 48: 55–65. 10.1109/10.900248View ArticleGoogle Scholar
 Audoly S, D'Angio L, Saccomani MP, Cobelli C: Global identifiability of linear compartmental models: a computer algebra algorythm. IEEE Trans Biomed Eng 1998, 45: 36–47. 10.1109/10.650350View ArticleGoogle Scholar
 Chin SV, Chappell MJ: Structural identifiability and indistinguishability analyses of the Minimal Model and a Euglycemic Hyperinsulinemic Clamp model for glucoseinsulin dynamics. Computer Methods and Programs in Biomedicine 2010, in press. Corrected Proof Google Scholar
 McDonald C, Dunaif A, Finegood DT: Minimalmodel estimates of insulin sensitivity are insensitive to errors in glucose effectiveness. J Clin Endocrinol Metab 2000, 85: 2504–2508. 10.1210/jc.85.7.2504Google Scholar
 Pillonetto G, Sparacino G, Cobelli C: Numerical nonidentifiability regions of the minimal model of glucose kinetics: superiority of Bayesian estimation. Math Biosci 2003, 184: 53–67. 10.1016/S00255564(03)000440MathSciNetView ArticleGoogle Scholar
 Cobelli C, Caumo A, Omenetto M: Minimal model SG overestimation and SI underestimation: improved accuracy by a Bayesian twocompartment model. Am J Physiol 1999, 277: E481–488.Google Scholar
 Denti P, Bertoldo A, Vicini P, Cobelli C: Nonlinear mixed effects to improve glucose minimal model parameter estimation: A simulation study in intensive and sparse sampling. IEEE Trans Biomed Eng 2009, 56: 2156–2166.View ArticleGoogle Scholar
 Lotz TF, Chase JG, McAuley KA, Shaw GM, Docherty PD, Berkeley JE, Williams SM, Hann CE, Mann JI: Design and clinical pilot testing of the model based Dynamic insulin sensitivity and secretion test (DISST). J diabetes Sci Technol 2010, 4: 1195–1201.View ArticleGoogle Scholar
 Lotz T, Chase JG, Lin J, Wong XW, Hann CE, McAuley KA, Andreassen S: IntegralBased Identification of a Physiological Insulin and Glucose Model on Euglycaemic Clamp Trials. 14th IFAC Symposium on System Identification (SYSID 2006); March 29–31; Newcastle, Australia. IFAC 2006, 0000: 6pages.Google Scholar
 Docherty PD, Chase JG, Lotz T, Hann CE, Shaw GM, Berkeley JE, Mann JI, McAuley KA: DISTq: An iterative analysis of glucose data for lowcost realtime and accurate estimation of insulin sensitivity. Open Med Inform J 2009, 3: 65–76. 10.2174/1874431100903010065View ArticleGoogle Scholar
 Ritt JF: Differential Algebra. Providence, RI: Amer. Math. Soc; 1950.Google Scholar
 Van Cauter E, Mestrez F, Sturis J, Polonsky KS: Estimation of insulin secretion rates from C peptide levels. Comparison of individual and standard kinetic parameters for Cpeptide clearance. Diabetes 1992, 41: 368–377. 10.2337/diabetes.41.3.368View ArticleGoogle Scholar
 Hann CE, Chase JG, Lin J, Lotz T, Doran CV, Shaw GM: Integralbased parameter identification for longterm dynamic verification of a glucoseinsulin system model. Comput Methods Programs Biomed 2005, 77: 259–270. 10.1016/j.cmpb.2004.10.006View ArticleGoogle Scholar
 Thorsteinsson B: Kinetic models for insulin disappearance from plasma in man. Dan Med Bull 1990, 37: 143–153.Google Scholar
 Lotz T, Chase JG, McAuley KA, Shaw GM, Wong J, Lin J, Le Compte AJ, Hann CE, Mann JI: Monte Carlo analysis of a new modelbased method for insulin sensitivity testing. Computer Methods and Programs in Biomedicine 2008, 89: 215–225. 10.1016/j.cmpb.2007.03.007View ArticleGoogle Scholar
 Sjostrand M, Holmang A, Lonnroth P: Measurement of interstitial insulin in human muscle. Am J Physiol 1999, 276: E151–154.Google Scholar
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