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Monte Carlo Simulation of Low-Count Signals in Time-of-Flight Mass Spectrometry and its Application to Single-Particle Detection Alexander W. Gundlach-Graham, Lyndsey Hendriks, Kamyar Mehrabi, and Detlef Günther Anal. Chem., Just Accepted Manuscript • DOI: 10.1021/acs.analchem.8b01551 • Publication Date (Web): 21 Sep 2018 Downloaded from http://pubs.acs.org on September 22, 2018

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Analytical Chemistry

Monte Carlo Simulation of Low-Count Signals in Time-ofFlight Mass Spectrometry and its Application to SingleParticle Detection Alexander Gundlach-Graham*, Lyndsey Hendriks, Kamyar Mehrabi, and Detlef Günther Department of Chemistry and Applied Biosciences, ETH Zurich, 8093 Zurich, Switzerland *Please address correspondence to [email protected]

keywords: Time-of-Flight mass spectrometry, ion detection, electron multiplier, micro-channel plate, analog-to-digital conversion, single-particle ICP-MS, Monte Carlo simulation, compound Poisson distribution, critical value, detection limit

Abstract

Many modern time-of-flight mass spectrometry (TOFMS) instruments use fast analogto-digital conversion (ADC) with high-speed digitizers to record mass spectra with extended dynamic range (compared to time-to-digital conversion). The extended dynamic range offered by ADC detection is critical for accurate measurement of transient events. However, the use of ADC also increases the variance of the measurements by sampling the gain statistics of electron multipliers (EMs) used for detection. The influence of gain statistics on the shape of TOF signal distributions is especially pronounced at low count rates and is a major contributor to measurement variance. Here, we use Monte Carlo methods to simulate low-ion-count TOFMS signals as a function of Poisson statistics and the measured pulse-height distribution (PHD) of the EM-detection system. We find that a compound Poisson distribution calculated via Monte Carlo simulation effectively describes the shape of measured TOFMS signals. Additionally, we apply Monte Carlo simulation results to single-particle inductively coupled plasma TOFMS (sp-ICP-TOFMS) analysis. We demonstrate that subtraction of modeled TOFMS signals can be used to quantitatively uncover particle-signal distributions buried beneath dissolved-signal backgrounds. Based on simulated signal distributions, we also calculate new critical values (LC) that are used as decision thresholds for the detection of discrete particles. This new detection criterion better accounts for the shape of dissolved signal distributions and therefore provides more robust identification of single particles with ICP-TOFMS.

Introduction

Time-of-flight mass spectrometry (TOFMS) is a leading design for high-speed, fullspectrum, moderate-to-high resolution mass spectrometry.1-4 TOFMS is particularly well suited for analytical applications in which full mass spectra need to be measured from short transients, for example from single particles,5-7 high-speed separations,8-9 or fast-transient sample analysis, such as from laser-assisted methods.10-11 In fact, interest in high-speed MS analysis continues to drive the development of new TOFMS systems. For high-speed TOFMS applications in which low numbers of ions (i.e. low-count signals) are recorded, basic understandings of the origins of measured signal variation and subsequent characterization of signal distributions are required to accurately develop decision rules, like critical values (LC) and detection limits (LD).

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Here, we use Monte Carlo methods to model TOFMS signal as a compound Poisson distribution,12-13 in which TOFMS signals are the sum of Poisson-distributed numbers of ions that each independently sample the gain distribution of microchannel-plate (MCP) detection. We find that this model adequately describes the major sources of variance for low-count TOFMS signals, which is in agreement with previous studies of noise sources in analog-todigital conversion (ADC)-based MS detection systems.14-17 Further, we apply the compound Poisson model to the measurement of analyte-containing microdroplets by inductively coupled plasma (ICP)-TOFMS and demonstrate that TOFMS signal can be modeled well enough to subtract the contribution of dissolved-analyte signals from single microdroplets. This work builds off previous research into deconvolution of single-particle ICP-MS (sp-ICP-MS) signals from dissolved background,18 but extends the approach with characterization of signal distributions from TOFMS and through validation of distribution-based particle counting with dual-element microdroplet detection. While interest in sp-ICP-MS motivated our work, the fundamentals of MS signal distributions explored herein are not restricted to atomic mass spectrometry or single-particle detection. Rather, basic understandings of the influence of MCP signal structure on recorded MS signals is broadly applicable to electron-multiplier (EM)based MS detection systems that digitize analog detector currents.

Ion Detection Modes in TOFMS

In mass spectrometry, EM detectors are typically used for the detection of low-tomoderate count rates (1 to 109 ions sec-1).19 For TOFMS, EMs are always used for ion detection because they provide sufficient speed to register ions at mass-to-charge (m/z)dependent flight times. A commonly used EM detector in TOFMS is a chevron assembly of two MCPs. In this device, m/z-separated ions strike the MCP surface and produce secondary electrons that are amplified through the MCP stack to produce a pulse of 106-107 electrons for each particle-striking event. However, not all ions that strike the MCP detector produce exactly the same number of secondary electrons at the output. Instead, these detectors produce output signals with a distribution, called the pulse-height distribution (PHD) or pulse-amplitude distribution.20-21 Because PHDs of MCPs and other EM detector outputs are not easily modeled with a single function,22-23 they are usually described empirically by their pulse-height resolution (PHR), which is the full-width at half maximum (FWHM) signal divided by the mean signal. Chevron-configuration MCP stacks used for TOFMS detection, often have PHRs from 100-150%. TOFMS signal from an EM detector at a given time bin (i.e. flight time) can either be counted using time-to-digital conversion (TDC) or recorded with a fast digitizer, i.e. with analog-to-digital conversion (ADC). In the TDC approach, an ion-detection threshold is set above the electronic noise of amplifier electronics and slightly lower than the signal range of the PHD. With TDC, multi-ion strikes the TOF detector within a single time bin are recorded as a single count. In this way, TDC limits the dynamic range of TOFMS measurements and can be problematic for transient analysis;24-25 though dead-time correction approaches26-28 or multi-anode detection29 can extend the dynamic range of TDC systems. On the other hand, TDC filters electronic noise and MCP noise effectively: if TOFMS instruments are operated with low enough ion fluxes, TDC (or pulse-counting) detection enables MS precision limited only by Poisson counting statistics.30 Despite advantages of TDC for reducing detection noise, modern TOFMS instruments often employ fast ADC detection systems because these systems enable greater dynamic range. In Figure 1, we provide a schematic of a typical ion detection approach with ADC for TOFMS. In recent years, fast digitizer technology has improved dramatically, and continues to do so. In our TOFMS instrument (icpTOF 2R, TOFWERK AG, Switzerland), the digitizer has 14-bit ADC with 625-ps sampling intervals (ADQ1600, Teledyne spDevices, Sweden). 2 ACS Paragon Plus Environment

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Fast digitization of the electron currents from an MCP or discrete-dynode EM allows for quantization of high ion currents (up to ~1000 ions per time bin with our digitizer). The dynamic range of fast ADC detection is especially required for the accurate determination of MS signals from temporally short, intense signals, such as from fast chromatography or single particles. However, direct digitization of ion current from an electron multiplier also means that the inherent amplification distribution due to gain statistics of the EM will be measured and contribute to the ion-signal variation. Contribution of the PHD on measured ion signal elevates the relative standard deviation (RSD) compared to an ion-counting detection system.14,31

Single-Particle ICP-MS

We were motivated to model low-count signals from our ICP-TOFMS instrument because we routinely use this instrument in a single-particle mode to count and quantify nanoparticles (NPs).7,32-33 Single-particle ICP-MS (sp-ICP-MS) is an emerging method for the quantification of inorganic nanoparticles at environmentally relevant concentrations, and has been recently described in a number of reviews.34-38 In sp-ICP-MS, a dilute suspension of NPs is introduced into the ICP. When an NP passes through the ICP, it is efficiently vaporized, atomized, and ionized, and produces a temporally short burst of ions that is transferred into the mass analyzer. The typical duration of ion signal from an NP event is 200-500 µs, and can be detected as an intensity spike on the ICP-MS signal time trace. The frequency of discrete NP-induced signals is proportional to particle number concentration (PNC), and the magnitude of these signal spikes correlates to the mass of the analyte element in each particle.39-40 A lingering question in sp-ICP-MS is how best to separate the signals from single NPs and dissolved content, especially when the NP signal intensity is close to that of the dissolved analyte. When individual entities, such as NPs, are measured by sp-ICP-MS, the sample needs to be dilute enough so that NP-signal events are absent most of the time. Sufficient temporal separation between signal spikes and high signal-to-background ratios enable relatively simple identification of signal spikes from “big” nanoparticles. However, a challenge arises for “small” NPs near the detection limits. In this case, NP signal distributions may overlap with that of the background and signals from these “small” NPs can easily be mistaken to be from dissolved content, or vice versa. Currently, most researchers define a signalintensity threshold above which all signals are considered to be from NPs.40-43 This threshold is typically based on the standard deviation of the dissolved-background signal (3σdissolved or 5σdissolved). Other researchers have proposed deconvolution strategies to separate distributions of NP- and dissolved-element-derived signals.18 Most previous work on sp-ICPMS uses pulse-counting based MS detection, which, as will be discussed, should not be directly applied to sp-ICP-TOFMS with fast ADC detection.

Experimental Dual Sample Introduction Experiment Design

To study approaches to quantitatively measure NPs by ICP-TOFMS in variable concentrations of dissolved analyte matrix, we used a dual-sample introduction setup as depicted in Figure 2.44 In this setup, variable concentrations of dissolved analyte (i.e. cerium) are introduced via a conventional pneumatic nebulizer while microdroplets with known concentrations of the analyte, as well as a tracer element (indium), are produced by a microdroplet generator (50-µm diameter Autodrop Pipette, AD-KH-501-L6, Microdrop Technologies, GmbH, Germany) and introduced to the ICP via a falling tube. The falling tube is filled with helium to accelerate solvent evaporation, so that dry salt-residue particles from each droplet are transferred to the ICP.45-46 Dissolved analyte from the nebulizer and dried droplet residues are introduced simultaneously into the ICP. In this experiment, analyte in the 3 ACS Paragon Plus Environment


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droplet residues can be thought of as a NP proxy, and the nebulized dissolved matrix solution represents the dissolved analyte that might be present in an environmental sample. Through altering the concentration of cerium in the droplets and in the dissolved matrix, we can change the relative signal intensities of particle versus the dissolved background. For each set of droplet and matrix conditions, we collected ICP-TOFMS signals, fit the dissolved background signal distribution with our Monte Carlo method, and then determined the number of analyte particles measured. Because a tracer element (i.e. indium) is present in each droplet, we know the true number of particles introduced into the plasma for each experiment.

TOFMS Detector Calibration

Operating parameters of the dual-sample introduction system and the ICP-TOFMS instrument are provided in Table S1. As already mentioned, signal on the ICP-TOFMS instrument is acquired with a high-speed signal digitizer. This digitizer converts the analog voltages from the pre-amplifier to a digital number per time bin (mV ns) and stores the data in memory to average (sum) multiple TOF spectra. The digitizer measures the voltage per sampling time; however, in order to relate this voltage to a physical number of ions, a calibration factor (mV ns ion-1) must be used: this factor is called single-ion signal (SIS) in the TOFWERK data acquisition software. To measure the SIS, we used a program that estimates the PHD of the MCP detection system (SingleIon2.3, TOFWERK AG, Switzerland); further details on how we measured the SIS are in the supporting information. Hereafter, we refer to the SIS histogram as the “measured PHD” (see Fig. S1) and we use this histogram to represent the EM gain distribution for all Monte Carlo simulations.

Monte Carlo Simulation

To evaluate how EM gain statistics (i.e. the PHD) impact measured ion signals, a Monte Carlo simulation was performed with a custom written program in LabVIEW (ver. 17.0f2, National Instruments, USA). A detailed description of our Monte Carlo approach is provided in the supporting information. Briefly, in this simulation, we model TOFMS signal as a compound Poisson process in which the signal in each TOFMS acquisition is the sum of a Poisson-distributed number of ions that each sample the measured PHD of the detection system. Mathematically, this relationship is described by equation 1, where Sacq is the signal measured in one TOF acquisition, k is the number of ions in that acquisition, and PHD() is the measured pulse-height distribution. The Poisson distribution is well known and is defined in equation 2, in which λ is the average ion-count rate (ions acq-1) and P(k) is the probability of k events occurring. 𝑆𝑆𝑎𝑎𝑎𝑎𝑎𝑎 = ∑𝑘𝑘𝑖𝑖=1 𝑃𝑃𝑃𝑃𝑃𝑃(𝑖𝑖)

(1)

𝜆𝜆𝑘𝑘

𝑃𝑃(𝑘𝑘) = 𝑒𝑒 −𝜆𝜆 𝑘𝑘!

(2)

We used Monte Carlo simulation to generate outcomes of equation 1 because the PHD is an empirical (measured) distribution. By running the Monte Carlo simulation a defined number of times, which we refer to as the number of acquisitions (Nacq), we can simulate the output signal histogram of the TOFMS instrument based on just two parameters: average count rates (λ), and numbers of acquisitions (Nacq). Additionally, to find optimum λ and Nacq values that describe the TOFMS-signal distributions, we developed a sum of the squared error fitting procedure—details of this fitting operation are provided in the supporting information. Importantly, the Monte Carlo method we use here only considers signal variation as a result of two noise sources: counting statistics and the measured PHD from electron multiplier detection. As shown by others,16 additional noise sources, such as electronic noise, time discretization error, digitization error, and instrument drift may all contribute to MS signal distributions; we did not explicitly model these additional noise sources. Nonetheless, our 4 ACS Paragon Plus Environment

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relatively simple compound Poisson model for TOFMS signals provides a satisfactory match with experimental data.

Results and Discussion Accuracy of TOFMS Signal Modelling

In Figure 3, we provide the time trace of 140Ce+ signal from continuous sample introduction of a 100 ng L-1 solution of Ce. In Fig. 3b, we provide the histogram of these signals with an overlay of the Monte Carlo simulated signal distribution; as seen, the match between the measured and modeled signal distributions is good, with an R2 value of 0.9997. This suggests that Poisson noise and PHD-based signal variation are dominant sources of signal fluctuation in our measurement. Moreover, our compound Poisson model is applicable for a broad range of average ion currents. In Figure S5, we provide results for the match between Monte Carlo simulations and measured histograms for samples with Ce concentrations of 10 and 1000 ng L-1, which have average count rates of 0.2 and 19.1 counts per acquisition, respectively. In all cases, the model effectively describes ion-signal distributions. As seen in Figure 3b, for ADC detection of low-count signals, an unusually shaped distribution is obtained that has a large proportion of acquisitions in the zero counts bin and then a right-skewed distribution with a maximum around 1.2 counts and a long tail out to ~10 counts. Because the PHD of the MCPs is a continuous distribution, we observe estimated ion counts with non-integer numbers of ions in each acquisition. Clearly, the shape of this TOFMS signal distribution cannot be described solely by conventional statistical distributions, such as Poisson or Gaussian. Instead, the distribution of measured TOFMS signals partly depends on the shape of the PHD of the TOF detector. The influence of variation due to PHD is most pronounced at low ions count rates because at higher count rates (e.g. > 5 ions per acquisition) the measured distribution can be approximated as a Gaussian distribution.

Measuring Single-Particle Signal Distributions

We used a dual-sample introduction system (see Fig. 2) to investigate sp-ICP-TOFMS analysis and test capabilities of separating nanoparticle signals from a dissolved background. In this setup, analyte-doped microdroplets serve as a proxy for NPs and are introduced concurrently with dissolved analyte. In addition to Ce, each droplet also contained 100 ng L-1 indium (15 fg In per droplet), and 115In+ signal served as a tracer signal to unambiguously record the number of droplets introduced into the plasma (see Fig. 4a). In Figure 4, we provide representative results for measurement of microdroplets with 10 µg L-1 Ce introduced to the ICP on top of a background from 100 ng L-1 dissolved Ce. As shown, the majority of TOFMS signals come from dissolved Ce; results from Monte Carlo simulation of the background signal distribution match that of the dissolved fraction. With these conditions, there is no overlap between distributions of the dissolved and microdroplet signal distributions, so the 99.95% recovery of Ce-microdroplet signal is expected. In Figure 5, we provide histograms of two “challenging” combinations of analyte concentrations in the microdroplet and nebulized solutions that cause the distributions of signals from the dissolved and microdroplet fractions to overlap. In the first case (Fig. 5a), Ce signal from each microdroplet (5 µg L-1 Ce, 750 ag Ce per droplet) has an average signal of less than 10 counts per droplet and is only apparent as a slight shoulder on the histogram from the dissolved-analyte signals (100 ng L-1 Ce). To fit the dissolved-signal distribution, we truncated the dataset to include only signals up to 3 counts acq-1 and binned at the data at 0.2-count intervals (see SI for details). In Fig. 5a, the histograms of TOFMS data and Monte Carlo simulation data are binned at 1-count intervals to smooth over the distributions and to improve ability to directly subtract them; however, this relatively large bin size also obscures 5 ACS Paragon Plus Environment


the double-peaked distribution (as in Fig. 3b), which would otherwise be visible. As shown, by subtracting the Monte Carlo-simulated signal distribution from the total measured signal, the distribution of signals attributed to single droplets is uncovered. This difference distribution predicts 3869 Ce-droplet signals, which is a 98.9% recovery of droplet number. In Fig. 5b, we show results from a case in which high dissolved Ce signal (1000 ng L-1) causes overlap with the signal distribution from Ce-containing microdroplets (10 µg L-1, 1.56 fg Ce per droplet). The measured signal distribution from 0 to 30 counts acq-1 was used to fit the dissolved-signal distribution. As seen, Monte Carlo simulation of the dissolved Ce signal again enables subtraction of the modeled distribution to uncover the low-count tail of the Ce-microdroplet distribution. After background subtraction, we obtain a Ce-droplet count of 4038 (99.4% recovery). In addition to quantitative recovery of microdroplet number, subtraction of the modeled background distribution, followed by subtraction of the mean background signal (λbkgd), allows for estimation of signal distributions from microdroplets only. In the supporting information, we show schematically how measured distributions are treated to remove the contribution of dissolved background signals (see Fig. S6) and demonstrate recovery of droplet-signal intensity distributions for a range of dissolved background concentrations (see Fig. S7). In Table 1, we provide the recovery of droplet numbers and mean microdroplet signals for different droplet and dissolved Ce concentrations. For all matrix concentrations, the background-subtracted mean droplet signal is within 10% of the “true” droplet-signal mean, which is determined by measuring Ce signals concomitant with droplet-tracer signal (115In+) and subtracting the remaining average dissolved signal. Monte Carlo simulation followed by subtraction of the dissolved-signal distribution allows for accurate droplet counting and reproduction because the shape of the dissolvedsignal distribution is predictable: microdroplet signals alter the shape of the dissolved-signal distribution and so can be recovered. However, this distribution subtraction approach cannot be used to identify and classify discrete microdroplet signals because, for overlapping regions of the dissolved and microdroplet signals, we cannot state the origin of particular signals. For these regions we can only describe the fraction of signals that are attributable to dissolved and microdroplet species. Nonetheless, it is often necessary in sp-ICP-MS to be able to detect and classify individual nanoparticles. For example, classification of NP type based on multielemental composition or stoichiometry determination of diverse NPs requires accurate detection of individual nanoparticles.32

Critical Values, Detection Limits, and Detection of Individual Particles To classify discrete signals as originating from a NP or not, one must make a decision rule: in sp-ICP-MS, this is usually a threshold intensity level above which all signals are considered to come from NPs. Commonly, the threshold level in sp-ICP-MS is defined as an integer multiple of the standard deviation of the dissolved-background signal (i.e. 3σdissolved or 5σdissolved).41,43,47 As defined by Currie,48-49 and adopted by the IUPAC,50 there are two accepted detection criteria: the critical value (LC), which is the minimum detectable signal, and the detection limit (LD), which is the minimum signal level that results in reliably detected signals. These detection criteria are defined by false positive errors (α) and the false negative errors (β). The critical value is a one-sided detection criteria and depends only on the background distribution: it is the value above which only some fraction (α) of the noise signals are present. With Gaussian-distributed noise, LC is defined as equation 3, where σbkgd is the standard deviation of the background (i.e. noise) and z1–α is the 1–α quantile of the standard normal distribution. For routine measurements, a commonly used false positive rate is 5% (α=0.05).49 For Poisson distributed noise—like that typical of ion-counting mass 6 ACS Paragon Plus Environment

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spectrometers—LC can be defined based on the mean count rate of the background signal (λbkgd),48 as given in eqn. 4. Unlike LC, the detection limit (LD) considers both false positives and false negatives, and is defined as the value at which only a certain fraction (β) of analyte signals are below the critical value. With Gaussian-distributed signal and noise, LD is given in eqn. 5, where z1–β is the 1–β quantile of the standard normal distribution. A typical value for β is 0.05 and, for Poisson-distributed data, LD can be approximated as shown in eqn. 6.48 𝐿𝐿𝐶𝐶 = 𝑧𝑧1−𝛼𝛼 𝜎𝜎𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏

(3)

𝐿𝐿𝐷𝐷 = 𝐿𝐿𝐶𝐶 + 𝑧𝑧1−𝛽𝛽 𝜎𝜎𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏

(5)

𝐿𝐿𝐶𝐶(𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃) (𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐) = 𝑧𝑧1−𝛼𝛼 �𝜆𝜆𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 → 1.64�𝜆𝜆𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 [α=0.05]

(4)

𝐿𝐿𝐷𝐷(𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃) (𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐) = 3.29�𝜆𝜆𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 + 2.71 [𝛼𝛼=0.05, β=0.05]

(6)

𝐿𝐿𝐶𝐶(𝐴𝐴𝐴𝐴𝐴𝐴) (𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐) = 1.80�𝜆𝜆𝑏𝑏𝑏𝑏𝑔𝑔𝑑𝑑 + 0.32 [α=0.05]

(7)

While equations 4 and 6 are suitable for pulse-counting-based analyses,51-52 these approximations are not adequate for ADC-based detection from electron multipliers of lowcount signals because signal variation from both Poisson noise and the PHD of the detection system should be considered. To determine the combined influence of Poisson distribution and the measured PHD, we used a Monte Carlo approach to simulate background and signal distributions for a range of λ values, then calculated LC(ADC) and LD(ADC) based on distribution quantiles, and approximated the relationship of (λbkgd)1/2 vs. LC(ADC) and (λbkgd)1/2 vs. LD(ADC) as linear fits. Details of this calculation are provided in the supporting information, and our results are given in equations 7 and 8. As seen, both LC(ADC) and LD(ADC) are increased compared to those from Poisson-only conditions.

𝐿𝐿𝐷𝐷(𝐴𝐴𝐴𝐴𝐴𝐴) (𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐) = 3.64�𝜆𝜆𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 + 3.10 [α=0.05, β=0.05]

(8)

Figure 6 shows results from our empirical approximation of LC. In Fig. 6a, we compare LC values for Poisson-only distributed background and signals and for the ADC case of Poisson noise plus signal variation due the measured PHD. Our Poisson-only results nearly match equation 4, which indicates our approximation method performs as expected. In Figure 6b, we apply calculated LC(ADC) values for a range of α values to threshold signal distributions from measurement of solutions with varying Ce concentration. In almost all cases, our LC(ADC) values accurately predict the quantile of the measured data above the critical value, which demonstrates again that our compound Poisson model accurately accounts for the shape and spread of our ICP-TOFMS data. Equation 7 is adequate for establishing a critical value for continuous sample introduction. However, for sp-ICP-MS, the reasonable false positive rate must be reduced because of a high ratio of background events to NP events. In sp-ICP-MS, it is common to have 10% or fewer data points occupied by NP signals, so, for instance, a 5% false positive rate from the background would result in close to 50% more detected “nanoparticle” signals. Thus, for sp-ICP-MS, the critical value should be calculated with a conservative false positive rate; here, we suggest α=0.001 as a reasonable rate.43 (A false positive rate of 0.1% actually matches quite well with the commonly used 3σdissolved threshold for NP detection, for which α=0.0015 if dissolved signal follows a Gaussian distribution). To use LC(ADC) as a decision rule for detection of NPs in sp-ICP-TOFMS, we consider now the gross signal (signal + background) instead of the net signal (signal – background). Gross signal detection criteria are useful because they can be directly applied to measured data. We define SC(ADC) to be the gross signal critical value, which is net critical value (LC(ADC)) 7 ACS Paragon Plus Environment


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plus the mean of the background, λbkgd; likewise, SD(ADC) is LD(ADC)+λbkgd. In eqn. 9, we report SC(ADC) for a false-positive rate of 0.1% (α=0.001); this critical value was calculated via Monte Carlo approach as described above and in the supporting information. Additionally, in eqn. 10, we calculate SD(ADC) by setting α=0.001 and β=0.05. For this calculation, we assume that NP signal distribution has the same shape as the background distribution. SD(ADC) is the value for which 95% of measured signals from NPs would be registered as “detected.” 𝑆𝑆𝐶𝐶(𝐴𝐴𝐴𝐴𝐴𝐴) = 𝜆𝜆𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 + (3.41�𝜆𝜆𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 + 1.6) [𝛼𝛼=0.001]

𝑆𝑆𝐷𝐷(𝐴𝐴𝐴𝐴𝐴𝐴) = 𝜆𝜆𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 + (5.24�𝜆𝜆𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 + 5.54) [𝛼𝛼=0.001, β=0.05]

(9) (10)

It is important to note that SC(ADC) and SD(ADC) depend on the shape of the measured PHD of the detection system and can change from instrument-to-instrument or with time as an EM detector ages. For this reason, calibration of MS detection setups and routine measurement of a system’s PHD is required. However, for a given set of MS instrument conditions, SC(ADC) and SD(ADC) are quasi-stable. To investigate the accuracy of SC(ADC) for NP detection, we used SC(ADC) to threshold Ce signals from our experiment with 5 µg L-1 Ce in microdroplets and a 100 ng L-1 Ce solution introduced as the dissolved background. With SC(ADC) as the NP-detection threshold, we found 2667 microdroplets, which is a 68% recovery. For comparison, we also thresholded the data at 3σdissolved and 5σdissolved, which are common approaches in sp-ICP-MS.40,42-43,47 The σdissolved–based NP-detection thresholds result in more detected microdroplet events; however, they also produce more false positives. In Figure 7, we plot the histograms of 140Ce+ above the 3σdissolved, 5σdissolved, and SC(ADC) values, and we also plot the 115In+ signal for each of these detected events. Because each microdroplet contains 100 µg L-1 indium in addition to cerium, presence of 115In+ signal concomitant with “detected” Ce-microdroplet signal indicates true detection of a microdroplet. On the other hand, Ce-microdroplet signals with associated 115In+ signal < 20 counts are droplet misidentifications or false positives. As seen in Table 2, for the 3σdissolved and 5σdissolved NP-detection thresholds, 23% and 5.8% of found microdroplet signals are actually noise spikes, i.e. false positives. On the other hand, the SC(ADC) method produces 1.3% false positives, which is in line with α=0.001 and a 10:1 number ratio of background-todroplet events. The σdissolved–based NP detection thresholds produce more false positives for our ICP-TOFMS signals because standard deviation is not a good descriptor of the shape of the background distribution of our data. SC(ADC) accounts for distribution shape due Poisson noise and the measured PHD, and so is a more representative detection criterion for sp-ICPTOFMS, especially for low-count signals. 140

+

Conclusions

Here, we investigated fundamentals of signal distributions for low-count signals in mass spectrometry with detection by ADC. We found that the major sources of signal variation are adequately described by a compound Poisson distribution of the measured PHD and Poisson-distributed ion arrival at the TOF detector. Through Monte Carlo simulations, we demonstrated effective modelling of MS signal distributions; this approach should be extendible to other MS systems that use EM detection followed by ADC. Our approach depends on measurement of the PHD of the MCP-based detection system. In the future, mathematical description of this PHD could be incorporated to directly solve for the contribution of gain statistics on low-count-rate signals, which would enable for direct calculation of LC(ADC) and LD(ADC). It is also likely that the PHD changes with the m/z value of ions striking the EM detector—while incorporation of m/z-specific PHDs was not included here, it should be considered, especially for MS methods with a large m/z range, such as is encountered in molecular MS. Finally, the Monte Carlo approach we developed nicely 8 ACS Paragon Plus Environment

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complements recent studies for moderate-count-rate ADC-based ion detection.16-17 Continued research into the fundamentals of MS signal distributions supports the field and is essential for MS approaches that depend on high sensitivity, large dynamic range, and robust MS signal identification. Apart from defining the major sources of variation for low-count signals, we applied our model to the specific case of sp-ICP-TOFMS analysis. We demonstrated that Monte Carlo simulation of the dissolved signal background is an effective way to separate distributions of dissolved and particle/droplet signals. We also calculated critical values and detection limits for sp-ICP-TOFMS that may be used for detection and identification of NPs on a particle-byparticle basis. Importantly, our method of dissolved background fitting is developed from first principles and only considers detector response shape and average signal count rate— therefore, our method can be applied regardless of particle-mass distribution. As long as Poisson counting statistics and measured PHD control signal variation (i.e. signal drift does not contribute appreciably to measurement noise), the approach outlined here is applicable. sp-ICP-TOFMS is a promising method for simultaneous quantification of diverse inorganic nanoparticles at environmentally relevant concentrations,53 our studies support its continued advancement and application.

Acknowledgements

We would like to thank four anonymous reviewers for excellent and thoughtful reviews that helped us improve the science and presentation of this work. We also thank Dr. Bodo Hattendorf for manuscript comments and fruitful discussions regarding critical values and detection limits. A. Gundlach-Graham and K. Mehrabi acknowledge financial support through an Ambizione grant of the Swiss National Science Foundation (SNSF), Project number PZ00P2_174061. This work was also supported by supported by SNSF Project no. 200021 162870/1. Thank you to Roland Mäder from the ETH mechanical workshop for manufacturing custom pieces necessary for the microdroplet introduction system and the dual inlet setup.

Supporting Information Available: • • • • •

ICP-TOFMS Operating Conditions Details of the Single Ion Signal (SIS) Measurement Detailed description of Monte Carlo simulation Comparison of background-subtracted signal-intensity distributions with true (no background) distributions Description of approach used to calculate critical value and detection limit

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Figures

Figure 1. Schematic diagram of the detection scheme in the ICP-TOFMS instrument. a) Ions strike the MCP and an electron current is produced as a result of the secondary electron cascade. The output current for many events has a distribution of amplitudes due to gain statistics of the MCP detector. b) The electron current is amplified and converted to a voltage by high-speed pre-amplifier; a typical amplification factor is 5-10. c). The amplified ioncurrent signal is digitized at a given sample interval with a high-speed ADC. Error in digitized signal is ±½ bit. In our system, 14-bit ADC is used to minimize digitization error and the sampling interval is 625 ps. d) Digitized signals across TOFs are stored in digitizer memory to average spectra from consecutive TOF experiments. e) After the acquisition of digitized TOFMS spectra, recorded voltages are converted to estimated ion counts by dividing the digitized signals by the single-ion signal (SIS) value. This SIS value must be measured for given MCP detector settings.

Figure 2. Dual sample-introduction setup for simultaneous introduction of microdroplets (nanoparticle proxy) and dissolved analyte into the ICP.



Figure 3. a) Time trace of 140Ce+ signal from continuous aspiration of 100 ng L-1 Ce solution; no microdroplets were introduced during this measurement. b) Histogram of 140Ce+ signal shows unusual shape of signal distribution at low ion currents, which is the result of Poisson counting statistics and ion-signal response variation due to gain statistics of the MCP detector. At low ion currents (

Monte Carlo Simulation of Low-Count Signals in ... - ACS Publications

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