RefModel

 The metabolic rate decreases in proportion to the local drug concentration as given by equation (1), where k is a first order rate constant [12]

                dC/dt=-kC(t)                                      (1)

Intrinsic clearance is a measure of enzyme activity, and is independent of physiological factors such as the liver blood flow or drug binding in the blood. In vitro, the intrinsic clearance of a drug is commonly expressed by equation 2, where C_s is the concentration of the unbound drug at the enzyme site [12].

Rate of metabolism:        V = CL_int C_s                        ( 2 )

Rate of metabolism is generally defined by the Michaelis-Mention enzyme kinetics relationship. When the drug concentration is much smaller than K_m (the Michaelis-Mention constant), CL_int  becomes:

                               CL_int = V/C_s                                    ( 3 )

What we need for our mathematical reference model, is an equation that describes the concentration of the unchanged drug as a function of time. Researchers use the following equation to express the concentration of unchanged drug for in vitro experiments, when keeping the initial drug concentration much smaller than K_m:

                         dC/dt=-CL*D*C(t)                   ( 4 )

where C(t) is the concentration, CL the in vitro clearance of the drug and D the cell density.

By solving differential equation (4) the amount of drug remaining after incubation time T is expressed as follows:

             C(T)=C(0)*exp(-CL*D*T)                ( 5 )

The basic equations explained above cannot be applied to all drugs, but they are applicable to the drugs and conditions selected for this study, and so we use equation 5 to express in vitro clearance and to be the reference model for IS-HIC.

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DatModel

The DatModel represents the real biological system. It contains the data obtained from in vitro experiments, and is used to validate the in silico model (ArtModel).  The validation process involves measuring the output of the in silico model and comparing it with the data provided by DatModel. In this work the data was obtained from Fig.  2 of [11], which depicts the time course for nine unchanged compounds in cell culture media containing freshly isolated rat hepatocytes different compounds. The data points were carefully obtained from the graphs using computer design tools. The DatModel interpolates the data points using a linear interpolation method to estimate the drug concentration at each time step of the simulation. 

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The Similarity Measure

After each in silico experiment a similarity measure algorithm compares the output of that experiment with the data provided by DatModel and assigns a score to the output based on the degree of their similarity. Several similarity measure algorithms are surveyed in [13]. The similarity measure used in this work was the measure calculated by the “global standard deviation” method [13],  with a wider envelope:  
upper_i = m_i (1 + sd) + 10, lower_i = m_i (1 – sd) – 10, where m_i and sd are the nominal mean and standard deviation of the time series (for additional detail see [13]).  This score is calculated by counting the number  of observations of the candidate time series that fall within the envelope and dividing that by the total observations in the series [13], as a result similatiry_score is in [0,1].

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 MODEL OPTIMIZATION

By model optimization we mean finding a set of model parameter values for ArtModel which produce the maximum possible similarity score. Several simulation optimization methods are surveyed in [17], including the Nelder and Mean simplex search method.

The Nelder and Mead algorithm, introduced in [18] for the first time, has been used widely to solve parameter estimation problems for almost 40 years. Despite its age it is still the method of choice for many practitioners in the fields of statistics, engineering and the physical and medical sciences because it is straightforward to code and easy to use. Particularly, it’s been used widely by researchers for simulation optimization [18-22]. It belongs to a class of methods which do not require derivatives and which are often claimed to be robust for problems with discontinuities or where function values are noisy. This property makes it a good candidate for optimizing our stochastic in silico simulation.

There are several different versions and extensions of this optimization algorithm. We used the one described in [16] to optimize our parameters.

Figure 3 shows the simplex algorithm employed in this paper. q­_best,q_worst and q_­next-worst are the best, worst and next worst vertex of the simplex. There are four basic operations used in this algorithm: reflect, contract,  expand and shrink each of which is depicted for a 2D simplex in Figure 4. The general heuristic in this search method is to move away from the worst point toward the best.

Fig 3

Fig 4

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REFERENCES

[1] Hunt, C.A., G.E.P. Ropella, M.S. Roberts, and L. Yan, 2004, “Biomimetic In Silico Devices. Computational Methods in Systems Biology,” Second International Workshop, CMSB 2004 (Paris, France, May 26-28, 2004) Proceedings.  Lecture Notes in Bioinformatics, Springer (in press); available at http://biosystems.ucsf.edu/Researc/RecentPapers/HuntCMSB04b.pdf.

[2] Leahy, D.E., 2004, “Drug Discovery Information Integration: Virtual Humans for Pharmacokinetics,” DDT: Biosilico. 2, no. 2: 78-84.

[3] Lipscomb, J.C., M. Meek, K. Krishnan, G.L. Kedderis, H. Clewell, and L. Haber, 2004, “Incorporation of Pharmacokinetic and Pharmacodynamic Data Into Risk Assessments,” Toxicology Mechanisms and Methods, 14, no. 3: 145-158.

[4] Gunaratna, C., 2001, “Drug Metabolism and Pharmacokinetics in Drug Discovery: A Primer for Bioanalytical Chemists, Part II,” Current Separations, 19, no. 3 (www.currentseparations.com/issues/19-3/19-3e.pdf.

[5] Venkatakrishnan, K., L.L. von Moltke, and D.J. Greenblatt, 2001, “Human Drug Metabolism and the Cytochromes P450: Application and Relevance of In Vitro Models,” Journal of Clinical Pharmacology, 41, no. 11: 1149-1179.

[6] Takahashi, K., K. Kaizu, B. Hu, and M. Tomita, 2004, “A Multi-algorithm, Multi-timescale Method for Cell Simulation,” Bioinformatics, 20, no. 4: 538-46.

[7] Meng,T.C., S. Somani, and P. Dhar, 2004, “Modeling and Simulation of Biological Systems with Stochasticity,” In Silico Biology, 4, 0024.

[8] S. Eklins, 2003, “In Silico Approaches to Predicting Drug Metabolism, Toxicology and Beyond,” Biochemical Society Transactions, 31,(Pt 3): 611-4.

[9] Ropella, G.E.P., and C.A. Hunt, 2003, “Prerequisites for Effective Experimentation in Computational Biology,” 25th Annual Conference of the IEEE Engineering in Medicine and Biology Society (Cancun, September 17-21, 2003); available at http://128.218.188.153:8080/~gepr/furm/docs/EMBC03Paper1.pdf.

[10] Daniels, M., 1999, “Integrating Simulation Technologies with Swarm,” Agent Simulation: Applications, Models and Tools Conference (University of Chicago, October 1999); available at http://www.santafe.edu/~mgd/anl/anlchicago.html .[1]

[11] Naritomi,Y., S. Terashita, A. Kagayama, and Y. Sugiyama, 2003, “Utility of Hepatocytes in Predicting Drug Metabolism: Comparison of Hepatic Intrinsic Clearance in Rats and Humans In Vivo and In Vitro,” Drug Metabolism and Disposition, 31, no. 5: 580-588.

[12] Shibata, Y., H. Takahashi, and Y. Ishii, 2000, “A Convenient In Vitro Screening Method for Predicting In Vivo Drug Metabolic Clearance Using Isolated Hepatocytes Suspended in Serum,” Drug Metabolism and Disposition, 28, no. 12: 1518-1523.

[13] Ropella, G.E.P., D.A. Nag, and C.A. Hunt, 2000, “Similarity Measures for Automated Comparison of In Silico and In Vitro Experimental Results,” ibid; available at http://128.218.188.153:8080/~gepr/furm/docs/EMBC03Paper2.pdf .

[14] Treijtel, N., A. Barendregt, A.P. Freidig, B.J. Blaauboer, and J.C.H. van Eijkeren, 2004, “Modeling the In Vitro Intrinsic Clearance of the Slowly Metabolized Compound Tolbutamide Determined in Sandwich-Cultured Rat Hepatocytes,” Drug Metabolism and Disposition, 32, no. 8: 884-891.

[15] Haenen, B., C. Rompelberg, K. Van Twillert, M. Hamzink, J. Dormans, and J. Van Eijkeren, 2002, “Utility of Rat Liver Slices to Estimate Hepatic Clearance for Application in Physiologically Based Pharmacokinetic Modeling: A Study With Tolbutamide, a Compound with Low Extraction Efficiency,” Drug Metabolism and Disposition, 30, no. 3: 307-313.

[16] H.G. Neddermeijer & G.J. van Oortmarssen & N. Piersma & R. Dekker, 2000, "Adaptive extensions of the Nelder and Mead Simplex Method for optimization of stochastic simulation models," Econometric Institute Report 199, Erasmus University Rotterdam, Econometric Institute.

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