The Control Matrix

As emphasized in Section 8.5, use of a control ( blank ) matrix in the preparation of matrix matched calibrator solutions provides several significant advantages over use 

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Accuracy (systematic error or bias) expresses the closeness of the measured value to the true or actual value. Accuracy is usually expressed as the percentage recovery of added analyte. Acceptable average analyte recovery for determinative procedures is 80-110% for a tolerance of > 100 p-g kg and 60-110% is acceptable for a tolerance of < 100 p-g kg Correction factors are not allowed. Methods utilizing internal standards may have lower analyte absolute recovery values. Internal standard suitability needs to be verified by showing that the extraction efficiencies and response factors of the internal standard are similar to those of the analyte over the entire concentration range. The analyst should be aware that in residue analysis the recovery of the fortified marker residue from the control matrix might not be similar to the recovery from an incurred marker residue. 

The FDA requests that the method exhibit sufficient sensitivity to measure accurately the residue of interest after fortification of the control matrix at half the tolerance concentration. Minimally, the detector response at the tolerance should be at least 10 times the average background response. 

To compute the controllability matrix, we can use the MATLAB function ctrb () ... 

We can find the canonical forms ourselves. To evaluate the observable canonical form Aob, we define a new transformation matrix based on the controllability matrix ... 

The remaining task lies in the determination of the control matrix X and observer matrix Z such that the sufficient condition for robust performance, Eq. (22.28), holds. A Lyapunov-based approach is employed to obtain these two matrices. After some lengthy and complicated manipulations of Eq. (22.29) and the control structure shown in Fig. 22.3, the following two Riccati equations are derived, whose positive-definite solutions correspond to the control and observer matrices, X and Z. 

The rank of E is computed by standard methods, such as counting the number of independent rows after performing elementary row operations. We prefer here to employ MATLAB. Inputting E and computing its rank gives an answer of three. Since n = 3, rank(E) = n. The controllability matrix E has full row rank and the system is therefore controllable. 

The controllability matrix E may again be computed using A and B giving the following matrix ... 

We wish to compute the controllability matrix E for this system, which will be used to determine the existence of a critical DSR trajectory. What does the matrix N look like for this system ... 

Matrix N in Equation 6.13 therefore contains a single vector N = Hi in this instance. From this result, the controllability matrix E for the DSR may be formed from Equation 6.13 ... 

These expressions are then substituted into E along with N to give the controllability matrix for the system resulting in a 4x4 matrix. 

The expression A(C) is found by taking the determinant of the controllability matrix E... 

Hence, using concepts from geometric control theory, we can state that the condition for a critical DSR is when Det(E) = 0, where E is the controllability matrix for the DSR system. 

For a critical CSTR, the controllability condition is related to the critical DSR, resulting in the final requirement that A(C) = 0, where A(C) is the determinant of the controllability matrix specific for the CSTR. 

A(C) is found from the controllability matrix E for the CSTR. To construct E, the set of vectors orthogonal to the stoichiometric subspace must be known. This is done by finding a basis for the null space of the vectors spanned by the reaction system given by the stoichiometric coefficient matrix A. 

Linear combinations of n hence form the set of vectors that are orthogonal to the stoichiometric subspace. From Chapter 6, it is known that the condition for a CSTR to lie on the AR boundary occurs when the controllability matrix E does not contain full rank. An expression may be determined for this by computing the determinant of E and setting it equal to zero. 

Figure 7.4 The determinant of the controllability matrix E as a function of CSTR residence time from the feed point. Two roots exist one at r 36.7 s and the other at the CSTR equilibrium point.

To determine whether a CSTR effluent concentration is a critical CSTR point, A(C) = 0, which depends on the controllability matrix E. To find E, vectors forming the subspace orthogonal to the stoichiometric subspace... 

From Chapter 6, critical a policies depend on the controllability matrix E of the system, which is a function of the concentration vector C. Hence, a is expressed in terms of C and not residence time t. Nevertheless, an explicit function for a in terms of t may be found once the optimal DSR concentration profile is determined, which is accomplished by substituting DSR concentrations, for a certain value of t, into Equation 7.3 and determining the value of a corresponding to the C - T pair. 

As to the controllability of a switched LTI system, tests similar to the ones for observability considered in the previous section, are known. According to Kalman s criterion, a continuous time LTI system is numerically completely state controllable if and only if the rank of the controllability matrix S equals the number n of states. 

This test has the same disadvantage as the observability criterion. The rank of the controllability matrix depends on the actual numerical values of the system parameters. If the test fails, it does not give any indication for which eigenvalue of A the associated transient behaviour is not controllable in case rank(S) < n. That is, the criterion does not identify the uncontrollable modes. The latter ones can be identified by means of the Hautus test which is also based on numerical values of the system parameters and not on structural properties. According to the Hautus test, an nth order LTI system (A, B) is completely controllable if and only if for all eigenvalues X, (/ = 1, 2,..., n) of the state space matrix A... 

Related to the analytical sample is the control matrix that will be used for method development and for preparing QCs and matrix matched calibrators (Section 9.4.7). In addition, the amount of available control matrix will also impact the design of method development and validation studies, especially in instances where a suitable control is not available or in the case where a similar control matrix is very rare and/or extremely costly. 

In the biomedical context another check can involve comparison of LC-MS and/or LC-MS/MS chromatograms obtained for control matrix spiked with analytical standard with those obtained for incurred samples, i.e. post-dose samples from the same individual (human or lab animal) as that from which the control matrix was obtained. In this way the presence of metabolites and/or of matrix components that are peculiar to the test subject (individual) can sometimes be detected (see below). When several different control matrices from different sources are available, it can be useful to compare chromatograms from the individual control matrices with that obtained for a pool (mixture) of aU of them, since this comparison can highlight regions in the chromatogram that are potentially problematic as a consequence of variations in matrix composition. If undetected, such variations in matrix composition can give rise to false positive identifications or to quantitative results that are significantly in error. 

At the onset of method development the appropriate concentrations of the sub-stock and spiking solutions may not be known, but estimates can be based upon the initial method development scheme and by considering what types of spiking solutions will be required for early sensitivity and recovery assessments. Also, to determine the required concentrations for these solutions, consideration must be given to the type of solvent and the volume of solution that will be used for any subsequent dilutions or for spiking the analyte into the control matrix (Section 9.5.6c). 

As previously discussed (Section 9.4.7b), extracted control matrix is used to assess the selectivity for the method under development. The selectivity of the method is a function of the sample preparation (extraction and clean-up), chromatography and mass spectrometry conditions that are used for the method. Assuming that the control matrix is representative of the sample matrix to be analyzed, and that method blanks have been used to demonstrate that the method is free of exogenous interferences due to solvents, or to containers or other apparatus (a source of interferences that is often overlooked in the method development process), an extracted blank is used to demonstrate that the method has sufficient selectivity for the intended analytical purpose. When interferences at the expected retention time of the analyte being quantified are detected, modifications to the sample preparation and chromatography (and sometimes the ions monitored hy the method) can be made to improve the selectivity of the method. Recall (Section 9.4.7b) that only re-analysis of incurred samples can reveal interferences resulting from metahoUtes or degradates of the analyte with either or both of the analyte and SIS. 

A related problem in biomedical analyses (possibly with analogies in other applications) arises from the presence of metabolites of the analyte in incurred analytical samples, but not in the control matrix spiked with analytical and internal standards used in method development and validation. Problems arising from this distinction are most likely to be important when rate of sample throughput is emphasized at the expense of chromatographic efficiency (Section 9.3.4) and can be addressed by re-analysis of incurred samples as described in Section 9.4.7b. The discussion in the present section is concerned with matrix effects during method development, but these must also be addressed in the context of method validation (Section 10.4.1d). 

Another aspect that is common to matrix interferences (direct contrihutions of matrix components to the signal measured for analyte and/or SIS) and matrix effects (suppression or enhancement of ionization) is that of the consequences of the presence of metabolites or other types of degradates when analyzing incurred analytical samples. Such interferences are in principle absent from the control matrix used for matrix matched calibrators, QC samples etc. Thus use of re-analysis of incurred samples to evaluate and consequent matrix effects was discussed in Section 9.4.7b, and applies equally to matrix interferences arising from presence of metabolites. Variations of metabolite levels among samples (e.g. from different time points in a pharmacokinetic study), which can lead to parallel variations in the extent of both matrix effects and matrix interferences, are an example of how some problems can arise unexpectedly despite prior precantions. 

Depending on the analytical range desired, one possible approach to limited matrix availability would be to employ a dilution method whereby the matrix or a homogenate of the control matrix is diluted in a manner consistent with the dilution used for sample preparation. Calibration curve and QC spiking solutions are then added to the diluted matrix. In this fashion, the rare matrix is conserved and the matrix effects will be equivalent. This type of methodology has the added benefit of allowing repeat sample analysis where typically the entire sample would be consumed upon extraction. 

Depending on the apphcation, one or more source of control matrix may be available for selectivity screening. For instance, when testing for herbicides in a soil dissipation study the control matrix may come from the actual field where the study is to take place prior to application of the herbicide or from an adjacent control plot For bioanalytical analysis several lots of control matrix may be screened independently and as a pool (Section 10.4.1c), and additional selectivity data may be obtained from actual subjects prior to dosing (pre-dose samples). When a representative control matrix is not available or if additional confidence is required for analyte identification additional measures such as the inclusion of multiple... 

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