Ions are pulsed into the TOF analyser at right angles to their path out of the quadrupoles (see Figure 1.2). Collisional cooling within the quadrupoles has transformed the pulsed packet of ions produced on the sample surface into a narrow ion beam within which the ions have similar momentum[186]. The high frequency pulsing (around 10kHz for small molecule ions[186]) samples from this ion beam preserving the natural distribution of position and momentum orthogonal to the beam path. As the ion beam current is small and the pulsing frequency high the number of ions per pulse is low. The ions drift through the flight tube as described in Section 5.3.2 until they impact the MCP. An incoming ion from a pulse excites an
electron shower within the charged plates of the MCP which then fall on the TDC causing a voltage spike. Providing the voltage spike is higher than a threshold (100mV on the QStarElite [186]) a discrete ‘count’ is recorded and stored in a hardware bin. The detector then has some small delay during which it is unable to record another pulse (around 5 ns). Many pulses are performed and the final spectrum read out. Providing that the pulsed ion current results in less than one ion per dead time the normal distribution is preserved through detection and the final peaks will have a Gaussian profile (more advanced models can account for some detector saturation[214]):
f (t) = 1 κ√2πe
−(t−µ)2
2σ2 (5.7)
where the variance term, κ, determines the peak width and the distribution mean, µ, is the ion’s mass (the peak is symmetric around the peak centroid m/z ). The function is parametrised by t as it is applied to the time domain (the square root of the m/z ).
It is known from literature and simulation that the peak width (controlled by the variance σ in Equation 5.7) increases and maximum peak intensity decreases as a function of m/z but that peak area remains constant[43]. The normalised Gaussian has unit area so it takes into account the relationship between peak width and maximum peak intensity but a single convolution does not allow for the peak width to vary.
In the case where multiple analysers are present (e.g. an ion guide preceding a time-of-flight analyser) their effects can be merged into a single experimentally determined convolution.
Fitting a Gaussian to spectral peaks
Figure 5.4: Profiling the statistical properties of peak shapes across a MALDI MSI dataset a. 35 peaks selected from the mean spectrum for profile evaluation which cover the mass range considered. b. plotting the ratio of peak height to width shows there is no m/z dependent relationship. c. Average peak standard deviation for all peaks shown in the time domain (pm/z) shows a linear relationship between standard deviation and time.
5.3. MODELLING THE QSTAR ELITE INSTRUMENT 125 was made from a mean spectrum, see Figure 5.4, that cover the mass range considered. For an instrument of this resolution (mass resolving power 6000 @ m/z 700) it cannot be guaranteed that isobaric ions are not present within a single peak. However, we assume this to be the case for most ions and trust that averages over a number of intense peaks remove any error due to the merging of peaks. From these measurements a good estimate for function parameters can be made.
The image was traversed pixel-by-pixel and for each of these peaks a Gaussian distribution was fitted by iteratively minimising the Root Mean Square (RMS) error between the Gaussian estimate and the data. The peak width is largely independent of the apex height, see Figure 5.4, but as expected[43] there is a broadening of the peaks as m/z increases which is linear in the time domain (i.e. pm/z). The fit generated parameters of standard deviation, peak height and peak area. The standard deviation controls the peak width, the full width half maximum (FWHM) is FWHM u 2√2 ln 2κ u 2.354κ The relationship between κ and p(m/z ) was obtained by fitting a straight line to Figure 5.4c giving κ = 0.01m/z −0.001.
Implementation with Convolution
Convolution applies a shaped filter, f (such as the Gaussian peak shape) to a vector x. x0= x ∗ f , effectively moving the filter along the vector and calculating the joint area at every point. As the filter is a function of the bin index f = f (i) it is slow to apply this directly to a spectrum as the filter must be recalculated at every point. If a constant filter is used then the convolution can be performed very quickly via the fourier transform[23]. As a compromise to enable MSI datasets with tens of thousands of spectra to be produced the m/z axis is split into portions and a fixed width filter applied to each. As the filter width changes quite slowly over the range considered 5 portions were heuristically determined to be sufficient.
Coombes et al [43] asserted that the reciprocal of the peak height varied linearly with m/z . There was found to be a quadratic fit using the integrated convolution approach, see Figure 5.5, but detailed comparison to Figure 6 from[43] also showed similar periodicity in the residuals from their linear fit. In both our case and the published work the trend is dominated by the first order linear relationship even if in general a quadratic fit may be more appropriate.