Through Vial Impedance Spectroscopy (TVIS): A Novel Approach to Process Understanding for Freeze-Drying Cycle Development

Abstract

Through vial impedance spectroscopy (TVIS) provides a new process analytical technology for monitoring a development scale lyophilization process, which exploits the changes in the bulk electrical properties that occur on freezing and subsequent drying of a drug solution. Unlike the majority of uses of impedance spectroscopy, for freeze-drying process development, the electrodes do not contact the product but are attached to the outside of the glass vial which is used to contain the product to provide a non-sample-invasive monitoring technology. Impedance spectra (in frequency range 10 Hz to 1 MHz) are generated throughout the drying cycle by a specially designed impedance spectrometer based on a 1 GΩ trans-impedance amplifier and then displayed in terms of complex capacitance. Typical capacitance spectra have one or two peaks in the imaginary capacitance (i.e., the dielectric loss) and the same number of steps in the real part capacitance (i.e., the dielectric permittivity). This chapter explores the underlying mechanisms that are responsible for these dielectric processes, i.e., the Maxwell-Wagner (space charge) polarization of the glass wall of the vial through the contents of the vial when in the liquid state, and the dielectric relaxation of ice when in the frozen state. In future work, it will be demonstrated how to measure product temperature and drying rates within single vials and multiple (clusters) of vials, from which other critical process parameters, such as heat transfer coefficient and dry layer resistance, may be determined.

Appendices

Appendix 1: Analysis of Peak Frequency and Peak Magnitude

The relative simplicity of the imaginary capacitance spectrum means that it is possible to follow two features of the peak using proprietary LyoView™ data viewing software, namely the peak frequency (F_PEAK) and the peak amplitude (${C}_{\mathrm{PEAK}}^{{\prime\prime} }$).

LyoView™ Peak Finding Procedure

This procedure is intended for the determination of the Maxwell-Wagner peak parameters such as peak amplitude and frequency position for individual dielectric spectra obtained during the freeze-drying cycle.

There are four stages in peak search procedure:

• 1. Preparation of the spectrum: (a) Smoothing of the dielectric loss spectrum to eliminate sharp spikes originating from scattering due to finite sensitivity or from electromagnetic interference. For this purpose the RMS deviation throughout the curve is calculated. For each point deviation from average of previous and next point is calculated and if the deviation is larger than half of RMS the value point is replaced by the average of previous and next point. After that moving average filter with aperture 3 is applied twice and using free cubic spline the number of points N is increased to 10 × N); followed by (b) interpolation by free cubic spline in order to increase the number of points and accuracy of peak parameters determination.

• 2. Differentiation of the spectrum obtained in step 1 and finding all negative peaks (zero first derivative and positive second derivative) and all positive peaks (second derivative negative) peaks which are present in the smoothed spectrum.

• 3. Rejection of any narrow peaks that originate from residual spikes which are not completely suppressed by curve smoothing.

• 4. Selection of the principle peak, if there is more than one peak that meets all previous criteria. Several criteria are applied depending on the stage of the freeze-drying: (a) at freezing and early stages of primary drying the peak with the largest amplitude is selected, (b) at the middle and late stage of primary drying, the peak with frequency position closest to frequency position of principal peak in previous spectrum is selected.

The F_PEAK parameter can be used to predict the product temperature whereas the ${C}_{\mathrm{PEAK}}^{{\prime\prime} }$ parameter can be used to determine the amount of ice remaining during the primary drying phase from which one can then predict both drying rate and end point. Both parameters lend themselves to the determination of phase behavior (ice formation, eutectic formation, and glass transition events).

Appendix 2: Basic Principles of Impedance Spectroscopy as It Applies to a Liquid Filled in TVIS Vial

The impedance of most real objects (i.e., those which have a combination of characteristics, for example, some resistance and some electrical capacitance) takes on a frequency-dependence largely because the impedance of a capacitance (Fig. 22a) is dependent on the frequency of the applied field, whereas an ideal resistor has zero frequency dependence (Fig. 22b). The impedance spectrum of a capacitor (in terms of the impedance magnitude and phase angle) displays a characteristic negative slow of −1 in the plot of log impedance magnitude vs. log frequency. This is because the impedance is an inverse function of the capacitance and the applied frequency.

$$ Z=\frac{1}{i\omega C} $$

(23)

Fig. 22

Bode plots (impedance magnitude (|Z|) and phase angle (ϑ)) for the impedances of a capacitor (a) and a resistor (b)

In the case of a composite object which has capacitance and resistance then the impedance spectrum that results will be dominated by one or the other element, depending on whether the elements are in series or in parallel and how the frequency of the applied field defines the relative magnitude of the impedance of the capacitance and the resistance.

1.1 In a Series Circuit

At low frequency the capacitor dominates the spectrum (Fig. 23a) because the impedance of the capacitance is so high that the capacitor effectively controls the current that flows through the circuit.

Fig. 23

Bode plots (impedance magnitude (|Z|) and phase angle (ϑ)) for the impedances of a series circuit (a) and a parallel circuit (b)

At high frequency the resistor dominates the spectrum (Fig. 23a) because the impedance of the capacitor has fallen below that of the resistor such that the resistor effectively controls the current that flows through the circuit.

1.2 In a Parallel Circuit

At low frequency the resistor dominates the spectrum (Fig. 23b) because the impedance of the capacitance is so high that all the current flows through the resistor.

At high frequency the capacitor dominates the spectrum (Fig. 23b) because the impedance of the capacitance is now lower than the resistor such that all the current now flows through the capacitor.

More complex composite objects can be considered combinations of impedances. Again, the impedance spectrum that results will be dominated by one or the other impedance.

1.3 In a Complex Circuit

At low frequency (<1 kHz) the resistor R (Fig. 24) dominates the impedance of the RC₁ circuit, because this circuit is in series with a capacitor, C₂ (which has a high impedance at low frequency) then C₂ effectively controls the current that flows through the entire circuit.

Fig. 24

Bode plots (impedance magnitude (|Z|) and phase angle (ϑ)) for the impedances of a complex circuit

At intermediate frequency (1–30 kHz) the impedance of C₂ drops below that of the resistor (Fig. 24), such that the resistor begins to dominate the impedance and therefore the phase angle tends to increase from −90 to 0 (Fig. 24).

At high frequency (>30 kHz) the impedance of the capacitor (C₁) in parallel with the resistor, decreases below that of the resistor (Fig. 24) such that the resistor no longer dominates the impedance of the parallel RC₁ circuit and as a result the circuit behaves like two capacitors in series. The high-frequency impedance is therefore dominated by the inverse of the sum of the capacitances

$$ Z=\frac{1}{i\omega \left({C}_1+{C}_2\right)} $$

(24)

The above model can be mapped onto the physical characteristic of the liquid filled vial, where C₂ is the glass wall capacitance (C_G), C₁ is the sample capacitance (C_s), and R is the sample resistance (R_s).

The complex impedance (Z^∗) of the model can be calculated from Eq. 25:

$$ {Z}^{\ast }=\frac{1}{i\omega {C}^{\ast }}=\frac{1}{i\omega {C}_{\mathrm{G}}}+\frac{1}{\frac{1}{R_{\mathrm{S}}}+ i\omega {C}_{\mathrm{S}}}=\frac{1+ i\omega {R}_{\mathrm{S}}\left({C}_{\mathrm{G}}+{C}_{\mathrm{S}}\right)}{i\omega {C}_{\mathrm{G}}-{\omega}^2{R}_{\mathrm{S}}{C}_{\mathrm{G}}{C}_{\mathrm{S}}} $$

(25)

where ω is the frequency of the applied field in radians per second (this angular frequency can be converted into frequency in cycles per second through the relationship ω = 2πf). The impedance of this model can also be expressed in terms of a complex capacitance (C^∗) according to Eq. 26:

$$ {C}^{\ast }={C}^{\prime }+{i C}^{{\prime\prime} }=\frac{1}{i\omega {Z}^{\ast }}=\frac{C_{\mathrm{G}}+ i\omega {R}_{\mathrm{S}}{C}_{\mathrm{G}}{C}_{\mathrm{S}}}{1+ i\omega {R}_{\mathrm{S}}\left({C}_{\mathrm{G}}+{C}_{\mathrm{S}}\right)} $$

(26)

where C′ is the real part capacitance (or simply, the capacitance) and C″ is the imaginary part capacitance (otherwise known as the dielectric loss).

From the complex capacitance formula, the expressions for real and imaginary capacitance can be calculated. This is achieved by multiplying the nominator and denominator by the complex conjugate of the denominator and then grouping to obtain the real (C′) parts (Eq. 29) and imaginary (C″) parts (Eq. 30).

$$ {C}^{\ast }=\frac{1}{i\omega {Z}^{\ast }}=\frac{\left({C}_{\mathrm{G}}+ i\omega {R}_{\mathrm{S}}{C}_2{C}_{\mathrm{G}}\right)\left(1- i\omega {R}_{\mathrm{S}}\Big({C}_{\mathrm{S}}+{C}_{\mathrm{G}}\right)\Big)}{\left(1+ i\omega {R}_{\mathrm{S}}\Big({C}_{\mathrm{S}}+{C}_{\mathrm{G}}\right)\left)\left(1- i\omega {R}_{\mathrm{S}}\Big({C}_{\mathrm{S}}+{C}_{\mathrm{G}}\right)\right)} $$

(27)

$$ =\frac{C_{\mathrm{G}}+{\omega}^2{R}_{\mathrm{S}}^2{C}_{\mathrm{G}}{C}_{\mathrm{S}}\left({C}_{\mathrm{S}}+{C}_{\mathrm{G}}\right)- i\omega {R}_{\mathrm{S}}{C}_{\mathrm{G}}^2}{1+{\left(\omega {R}_{\mathrm{S}}\left(\Big({C}_{\mathrm{S}}+{C}_{\mathrm{G}}\right)\right)}^2} $$

(28)

$$ {C}^{\prime }=\frac{C_{\mathrm{G}}+{\omega}^2{R}_{\mathrm{S}}^2{C}_{\mathrm{G}}{C}_{\mathrm{S}}\left({C}_{\mathrm{S}}+{C}_{\mathrm{G}}\right)}{1+{\left(\omega {R}_{\mathrm{S}}\left(\Big({C}_{\mathrm{S}}+{C}_{\mathrm{G}}\right)\right)}^2} $$

(29)

$$ {C}^{{\prime\prime} }=-\frac{\omega {R}_{\mathrm{S}}{C}_{\mathrm{G}}^2}{1+{\left(\omega {R}_{\mathrm{S}}\left(\Big({C}_{\mathrm{S}}+{C}_{\mathrm{G}}\right)\right)}^2} $$

(30)