**Identifiability Analysis and Parameter List Reduction of a Nonlinear Cardiovascular PKPD Model**

Sau Yan Amy Cheung (1), James W. T. Yates (2), Oneeb Majid (3), Leon Aarons (4)

(1) The Centre for Applied Pharmacokinetic Research (CAPKR), School of Pharmacy and Pharmaceutical Sciences, University of Manchester, Manchester; (2) Astrazeneca R&D, Alderley Park, Macclesfield; (3) Medeval Ltd, Skelton House, Manchester Science Park, Manchester

**Objective:** Pharmacokinetic and pharmacodynamic modelling is becoming evermore sophisticated. Specifically there has been a drive to create more mechanistically relevant models. A particular feature of these models is that parameter values have to be inferred from observations of a small subset of the compartments in the model. In this talk, the problems caused by this and solutions, are examined.

**Introduction:** The distribution in the body and the effect on the cardiac system of an alpha 1A/1L partial agonist can be described by adapting a previously published nonlinear cardiovascular PKPD model [1]. This model comprises a linear two-compartment PK model with first order absorption and elimination and a PD model which characterises the regulation of the effects of the increased peripheral resistance induced by the constriction of blood vessels. A Hill-type Emax function is used to link the PK model to the reduction in total peripheral resistance in the PD of the model. The PD model consisted of four hemodynamic variables, total peripheral resistance (TPR), heart rate (HR), mean arterial pressure (MAP) and auxiliary control which described the arterial baroreceptor reflexes in the heart. Within the auxiliary control component a mechanism was incorporated for a first order delay. All these variables are joined together to form a closed loop control system to maintain the arterial pressure.

Experimentally the following measurements may be made: plasma concentration (Cp), mean arterial pressure and heart rate. The pharmacodynamic component of the model has 12 unknown parameters. These parameters can be divided into 4 types: three time constant parameters, three physiological parameters for setting the variables to their equilibrium levels, four control parameters related to heart rate and total peripheral resistance and two parameters for the Emax function. Some of these were found to be not well determined from parameter estimation using NONMEM version 5; these parameters were also found to have little influence during global sensitivity analysis. The sensitivity analysis of the PK-PD model was done using Simlab 1.1 and Matlab. Such a phenomenon suggested a potential unidentifiability status of these parameters within the model. Application of structural identifiability analysis [2] to the model was sought in order to verify the cause and factors of this phenomenon: whether it is due to the estimation technique or lack of structural identifiability of the model.

The goal of structural identifiability analysis is to evaluate the internal structure of a mathematical model based purely on the input-output responses, with the assumption of perfect noise free data. If all the unknown parameters in the model may be uniquely determined, then the model is globally uniquely identifiable. This means that the model is unique and the estimated parameters will be unique. If one or more of the parameters may take more than one of a finite number of values without affecting the goodness of fit then the model is locally identifiable. Finally, if there is one parameter that may take on an infinite number of possible values then the model is unidentifiable; this means there are infinite sets of parameters values that will fit to the model equally well. In this situation, re-design, reparameterisation or model reduction of the original model is necessary.

**Methods:** Generally, different methods are available for structural identifiability analysis of a nonlinear model such as the Taylor series expansion approach [3], the similarity transformation approach [4] and the differential algebra approach [5]. The nonlinear similarity transformation approach was considered in this study due to its robustness in handling complex models. For the analysis to apply, it is necessary for the model to be both controllable and observable. The nonlinear similarity transformation approach is not straightforward as a non-linear mapping is involved in the analysis; therefore a modified nonlinear similarity transformation approach was used. The new modified version makes use of the theorem [6] stating that if the differential equations of the model are polynomial in the state variables and the observation function is linear in terms of state variables, then it is sufficient to consider a linear map in the analysis. Therefore, to simplify the analysis, the PD model was rewritten in polynomial form and an extra state was added to ensure linear observation. The PK and PD parts of the model were analysed simultaneously. All the unknown parameters in the PK model were assumed to be known to decrease the complexity of the analysis and because the parameters were originally estimated in a sequential manner. Using this approach a linear mapping was deduced that altered the parameters in the model, but left the predicted time course for the observed variables unchanged.

**Results:** The analysis was conducted using MATHEMATICA (version 6). The parameters related to the control mechanism were found to be unidentifiable while all the equilibrium, time constant and core PD parameters were found to be globally identifiable. This result leads to an unidentifiable model which confirmed the findings from the sensitivity analysis and parameter estimation.

The reparameterisation [7], also known as parameter list reduction, method was then used as it may be applied using the similarity transformation that had already been deduced. In this method, the Taylor series of the similarity transformation criteria is calculated. It was found that the model is rank deficient by one and this means that the new parameterisation will have one less parameter than the old parameterisation. The model with one less parameter was used for a second structural identifiability analysis. It was confirmed that the new model parameterisation was now globally identifiable.

The improved identifiability of the model was then confirmed by a simulation study. A patient population was simulated in NONMEM and then refitted to confirm improved convergence of the estimation algorithm and that the estimated parameter values were comparable to those used for the simulation.

**Conclusion:** Structural identifiability analysis and parameter list reduction can be a helpful tools for the analysis of mechanistic PK-PD models. The example considered demonstrates how a model may be reduced in complexity while maintaining mechanistic relevance. The parameter list reduction technique allowed the similarity transformation to be used, allowing the two stages of analysis to be more computationally economical. Rewriting the model further reduced the complexity of the analysis required. This approach may be applied to a large class of models and so potentially allows the application of structural identifiability analysis to more complex PKPD models.

**References:**

[1] Francheteau, P., Steimer, J.-L., Merdjan, H., Guerret, M. and Dubray, C., A Mathematical Model for Dynamics of Cardiovascular Drug Action: Application to Intravenous Dihydropyridines in Healthy Volunteers, Journal of Pharmacokinetics and Biopharmaceutics 21(5): 489-510, 1993.

[2] Bellman, R. and Åström K. J., On Strutural Identifiability. Mathematical Biosciences 7:329-339, 1970.

[3] Pohjampalo, H., System identifiability based on the power series expansion of solution. Mathematical Biosciences 41: 21-33, 1978.

[4] Vajda, S., Godfrey, K. R., and Rabitz, H., Similarity Transformation Approach to Identifiability Analysis of Nonlinear Compartmental Models, Mathematical Biosciences 93(2): 217-248, 1989.

[5] Margaria, G., Riccomagno, E., Chappell, M. J., Wynn, H. P., Differential algebra methods for the study of the structural identifiability of rational function state-space models in the biosciencs, Mathematical Biosciences 17:1-26, 2001.

[6] Chappell, M. J., Godfrey, K. R., and Vajda, S., Global Identifiability of Parameters of Nonlinear Systems with Specified Inputs. Mathematical Biosciences. 102(1) : 41-73, 1990.

[7] Chappell, M. J., Gunn, R. N., A procedure for generating locally identifiable reparameterisations of unidentifiable non-linear systems by the similarity transformation approach. Mathematical Biosciences. 148(1): 21-41, 1998.