Clinical Assays for Biological Macromolecules

Abstract

Assessments of pharmacokinetics (PK), pharmacodynamics (PD) and immunogenicity are indispensable parts of the development process for therapeutic biologics. It is, therefore, essential to develop suitable assays for these assessments. However, development of assays for biological macromolecules poses unique challenges. In this chapter, we review the scientific background of clinical assay development and validation, as well as the common statistical methods used during the life-cycle of assay development.

A.1 Appendix

1.1 A.1.1 Generalized Least Squares Estimation for Nonlinear Regression Model with No-Constant Variance

The GLS algorithm is an iterative procedure. At the beginning, let k = 0 and set the least square estimator ${\hat{\upbeta}}_{\mathrm{LS}}$ as ${\hat{\upbeta}}^{(0)}$. In general, for the k-th iteration, perform as follows:

• 1. Given ${\hat{\upbeta}}^{\left(\mathrm{k}\right)},$ estimate θ and σ² by the method of pseudo-likelihood (PL). The pseudo-likelihood function is:

  $$ \mathrm{PL}\!\left({\hat{\upbeta}}^{\left(\mathrm{k}\right)},\uptheta, {\upsigma}^2\right)\!\,{=}\,-\mathrm{Nlog}\left(\upsigma \right)-\!{\displaystyle \sum } \log \mathrm{g}\left({\mathrm{x}}_{\mathrm{i}},{\hat{\upbeta}}^{\left(\mathrm{k}\right)},\uptheta \right)\!-\frac{1}{2{\upsigma}^2}{\displaystyle \sum_{\mathrm{i}}}\frac{{\left({\mathrm{y}}_{\mathrm{i}\mathrm{j}}-\mathrm{f}\left({\mathrm{x}}_{\mathrm{i}},{\hat{\upbeta}}^{\left(\mathrm{k}\right)}\right)\right)}^2}{{\mathrm{g}}^2\left({\mathrm{x}}_{\mathrm{i}},{\hat{\upbeta}}^{\left(\mathrm{k}\right)},\uptheta \right)} $$

  The variance parameters are estimated by minimizing the pseudo-likelihood using numerical method. The weights are thus: ${\hat{\mathrm{w}}}_{\mathrm{i}}={\mathrm{g}}^{-2}\left({\mathrm{x}}_{\mathrm{i}},{\hat{\upbeta}}^{\left(\mathrm{k}\right)},\hat{\uptheta}\right).$

• 2. Use the estimated weights from step 1 to obtain an updated estimator of β by minimizing

  $$ {\displaystyle \sum_{\mathrm{i}}}{\hat{\mathrm{w}}}_{\mathrm{i}}\ {\left[{\mathrm{y}}_{\mathrm{i}\mathrm{j}}-\mathrm{f}\left({\mathrm{x}}_{\mathrm{i}},{\hat{\upbeta}}^{\left(\mathrm{k}\right)}\right)\right]}^2 $$

Denote the resultant estimator as ${\hat{\upbeta}}^{\left(\mathrm{k}+1\right)}$ _.

The procedure stops after certain steps when the parameter estimates converge.

When there are multiple runs, estimate β same as in a single run. Thepseudo-likelihood, however, becomes:

$$ \mathrm{PL}\left(\uptheta, {\upsigma}^2\right)={\displaystyle \sum }\ \mathrm{PL}\left({\hat{\upbeta}}_{\mathrm{i}}^{\left(\mathrm{k}\right)},\uptheta, {\upsigma}^2\right). $$

The other part of the algorithm remains the same as in the single run.