Table 2.
Representative “deterministic” inversion methods, example environmental and biological applications and software.
| Method/technique | Example application area | Example available software |
|---|---|---|
| Regression methods | ||
| Non-linear least squares (Dennis et al., 1981; Yano et al., 1989; Kaltenbach and Vistelle, 1994; Bender and Heinemann, 1995) | Estimation of PK parameters (Kaltenbach and Vistelle, 1994; Muzic and Christian, 2006) | MOT (Mathworks, 2006c); NIST (NIST, 2006); Boomer (Bourne, 1989); MultiFilt (Yano et al., 1989) |
| Singular value decomposition (SVD) (Paige, 1986; Anderson, 1999) | Spectrum reconstruction from dose (Armbruster et al., 2004); pathway analysis of gene expression (Tomfohr et al., 2005) | Numerical recipes (NR) (Press, 2002); Netlib-TOMS (ACM, 2006) |
| Extreme (LeBlanc et al., 2006)/logistic Projection pursuit (PPR) (Friedman and Stuetzle, 1981; Lingjaerde and Liestol, 1998) | Survival analysis (LeBlanc et al., 2006) Phenol toxicity mechanism (Ren and Kim, 2003) | Matlab toolboxes R (RFSC, 2006); XTAL (Cherkassky and Gehring, 1996) |
| Multivariate adaptive (MARS) (Friedman and Roosen, 1995); Bayesian MARS (Denison et al., 1998; Holmes and Denison, 2003; Nott et al., 2005) | Phenol toxicity mechanisms (Ren and Kim, 2003); GI absorption (Deconinck et al., 2005); subgroup disease–risk relationships (York et al., 2006) | Netlib-Misc (Netlib, 2006) |
| Gradient methods | ||
| Automatic differentiation (Hovland et al., 2005; Ringrose and Forth, 2005) | Uncertainty in PBPKs (Isukapalli et al., 2000) | ADIFOR (Bischof et al., 1996); ADIC (Bischof et al., 1997); MAD (Kharche and Forth, 2006) |
| Steepest descent (Marquardt, 1963; Conway et al., 1970) | Inverse treatment planning (Hristov et al., 2002) | NR (Press, 2002); TOMS (ACM, 2006) |
| Likelihood methods | ||
| Maximum (MLE); restricted MLE (Elzo, 1994; Jamshidian, 2004); approximate likelihood (Albertini et al., 2006); NONMEM (Overgaard et al., 2005; Tornoe et al., 2005) | Short- and long-term exposure reconstruction using PBPK for chloroform (Georgopoulos et al., 1994; Roy et al., 1996) | Matlab (MathWorks Inc., 2006); NR (Press, 2002); TOMS (ACM, 2006); NONMEM (Beal, 2002) |