Abstract
measure trace gases as, for example, in exhaled human breath. The quantification of compounds at low concentrations is desirable for medical diagnostics. Typically, an increase of measuring accuracy can be achieved if the duration of the measuring process is extended. For real time measurements the time windows for measurement are relatively short, in order to get a good time resolution (e.g. with breath-to-breath resolution during exercise on a stationary bicycle). Determination of statistical detection limits is typically based on calibration measurements, but this approach is limited, especially for very low concentrations. To overcome this problem, a calculation of limit of quantification (LOQ) and limit of detection (LOD), respectively, based on a theoretical model of the measurement process is outlined. Keywords: Detection limits; limit of quantification; limit of detection; mass spectrometry; PTR-MS.
Abstract
V OLATILE ORGANIC COMPOUNDS (VOCs) with concentrations (expressed as the volume ratios) down to partsper-trillion levels (ppt: 1 molecule out of 10 12 molecules), see e.g. [9, 13, 7] , can be measured using a PTR-MS instrument (Proton Transfer Reaction Mass Spectrometer).
Abstract
The molecules of the volatile organic compound, here symbolically denoted by M, are chemically ionized by proton transfer from H 3 O + -ions produced in the source of the instrument, [9] :
Abstract
The countrates of the resulting ions are then measured by a Quadrupol-Mass-Spectrometer. The countrates are determined at mass-to-charge ratios (m/z) ranging from 21 up to 500, see Figure 1 . Incidentally, also a time-of-flight mass spectrometer (PTR-TOF) may be used instead of a quadrupol mass spectrometer. Here we focus on PTR-MS.
Abstract
The standard formula for determination (estimation) of the true concentration [M] of M in ppb (1 molecule out of 10 9 molecules), say [M], is proportional to the observed countrate of MH + ions at a particular mass-to-charge ratio m/z, see e.g. [13] , and is given by:
Abstract
, * Corresponding author: gregoryphillips@example.com 
Abstract
where IR = 500 (this depends on the source of water) reflects 180
WORKING PRINCIPLE OF PTR-MS
V OLATILE ORGANIC COMPOUNDS (VOCs) with concentrations (expressed as the volume ratios) down to partsper-trillion levels (ppt: 1 molecule out of 10 12 molecules), see e.g. [9, 13, 7] , can be measured using a PTR-MS instrument (Proton Transfer Reaction Mass Spectrometer).
WORKING PRINCIPLE OF PTR-MS
The molecules of the volatile organic compound, here symbolically denoted by M, are chemically ionized by proton transfer from H 3 O + -ions produced in the source of the instrument, [9] :
WORKING PRINCIPLE OF PTR-MS
The countrates of the resulting ions are then measured by a Quadrupol-Mass-Spectrometer. The countrates are determined at mass-to-charge ratios (m/z) ranging from 21 up to 500, see Figure 1 . Incidentally, also a time-of-flight mass spectrometer (PTR-TOF) may be used instead of a quadrupol mass spectrometer. Here we focus on PTR-MS.
WORKING PRINCIPLE OF PTR-MS
The standard formula for determination (estimation) of the true concentration [M] of M in ppb (1 molecule out of 10 9 molecules), say [M] , is proportional to the observed countrate of MH + ions at a particular mass-to-charge ratio m/z, see e.g. [13] , and is given by: 
WORKING PRINCIPLE OF PTR-MS
where IR = 500 (this depends on the source of water) reflects the isotopic ratio of the species H 18 3 O + and H 16 3 O + . The massto-charge ratio of m/z = 37 refers to the water cluster H 2 O · H 3 O + (other water clusters are neglected).
WORKING PRINCIPLE OF PTR-MS
Further, κ M is a calibration constant for the compound M, k M is the reaction constant of protonation reaction (1) for M, t is the reaction time, g is number of molecules in unit volume of sample gas, and the factors T m/z and T H 3 O + are the transmission efficiencies for MH + and H 3 O + ions, see e.g. [5] .
WORKING PRINCIPLE OF PTR-MS
For PTR-MS, the reaction time t is represented by the drift time of precursor ions H 3 O + within the drift tube. The calculation of the drift time is based on the standard ion mobility of H 3 O + in nitrogen adjusted to the actual temperature and pressure in the drift tube, [18] .
WORKING PRINCIPLE OF PTR-MS
The constant k M and the transmission coefficients T m/z and T H 3 O + are known for several compounds, for an overview see e.g. [19] . For simplicity, in this paper we consider as a typical value k M ≈ 2.0 · 10 −9 cm 3 · particles −1 · sec −1 , and
WORKING PRINCIPLE OF PTR-MS
Note that [M] is used to denote the true concentration of the compound M which is estimated by the PTR-MS measurements, i.e. we assume that [M] is (known or unknown) constant (non-stochastic term). Further, [M] is used to denote the estimated concentration of M, based on the measurements, and as such it is a random variable (stochastic term).
WORKING PRINCIPLE OF PTR-MS
It is important to emphasize that the probability distribution of [M] functionally depends on the true concentration [M], on the level (concentration) of the background noise, say [N] , and the time-span of the PTR-MS measurements (here we assume common dwell time τ for count-measurements of the primary ions as well as of the MH + ions.
Determination of concentration based on PTR-MS measurements
A typical set of parameters for PTR-MS measurements is given in Table 1 1 , see also [16] . 
DETECTION LIMITS
Determination of detection limits is critical in meeting the needs of analytical science. Data users must understand the limitations of reporting the analytical data measured down to 1 Amount of substance is a standards-defined quantity that measures the size of an ensemble of elementary entities, such as atoms, molecules, electrons, and other particles. The number of particles (molecules) in unit volume of sample gas, denoted by g, could be derived by the following considerations: The amount of sample gas, measured in moles, in the volume V is given by the ideal gas law as n = p ·V /(R · T ), where p is the absolute pressure (e.g. 2.4 mbar = 240 Pascal), V is the volume (here V = 1 cm 3 ) , R is the universal gas constant (8.314472 J · mol −1 · K −1 ) and T is the absolute temperature (e.g. 323 K). A typical amount of sample gas in unit volume V = 1 cm 3 , measured in moles, is then 8. 
DETECTION LIMITS
their detection limits in order to minimize the risk of making poor decisions, see [15] . The typical measures for detection limits are the limit of quantification (LOQ) and the limit of detection (LOD), respectively. Roughly speaking, LOQ represents a concentration at which quantitative results can be reported with a high degree of confidence. LOD is the lowest concentration (quantity) of a substance that can be distinguished from the absence of that substance (blank analyte).
DETECTION LIMITS
It is important to emphasize that the detection limits (determined for specific measured substance) strongly depend on the possible background noise. The standard method for determination of the detection limits is typically based on calibration series (statistical approach).
Limit of quantification (LOQ)
Unfortunately, there is no unique and unambiguous definition of LOQ, for more details and further discussion see the Appendix 1. In this paper LOQ is understood (in accordance with other interpretations) as such minimal increase of concentration (i.e. difference of true concentrations) that the observed (measured) values of concentrations can be attributed to distinct concentrations of the substance with a high degree of confidence.
Limit of quantification (LOQ)
In other words, the limit of quantification of a given substance M with its true concentration [M] , is such minimal concentration, say LOQ [M] , that concentration measurements at [M] and at [M] + LOQ [M] , respectively, can be attributed to distinct concentrations of the substance with high degree of confidence.
Limit of quantification (LOQ)
Let
Limit of quantification (LOQ)
are two distinct concentrations of the substance M. Then the limit of quantification for concentration 
Limit of quantification (LOQ)
Here, k denotes the select (pre-specified) constant (in this paper, we consider the value k = 3), Mean 
Limit of detection (LOD)
The limit of detection is the lowest quantity of a substance (compound M) that can be distinguished from the absence of that substance within a stated confidence limit. In general, the limit of detection depends on the particular compound M and on the on the probability distribution of the observable noise measured with blank analyte (i.e. in absence of compound M). Throughout this work, the following interpretation will be used: The limit of detection, say LOD M , is a specific value in the true concentration domain. The LOD M equals to such concentration [M] of the compound M, that its measured value [M] is with probability of 99% above the 99%-percentile of the probability distribution of the noise measured with blank analyte, i.e. with [M] = 0, see Figure 3 .
Limit of detection (LOD)
Notice that LOD M is by definition equal to zero if the (background) noise is absent.
Limit of detection (LOD)
References to alternative definitions of LOD and a detailed explanation of the standard statistical approach based on the calibration experiment with normal (Gaussian) measurement errors can be found in the Appendix 1.
MODEL-BASED DETERMINATION OF DETECTION
LIMITS FOR PTR-MS
Probability model of PTR-MS measurements
The measured counts at a single mass-to-charge ratio follow a Poisson distribution for PTR-MS, see [5, 17, 6, 10] . , the dashed lines represent the 99%-percentile and 1%-percentile of the measured signal, respectively. The mean value of the probability distribution of the concentration measurements of the blank analyte is denoted by
Probability model of PTR-MS measurements
). The LOD definition indicates the true concentration for which the 1%-percentile of the measured signal is equal to the 99%-percentile of the measured concentration of the blank analyte.
Probability model of PTR-MS measurements
Based on the formula (2) 
Probability model of PTR-MS measurements
Under stable measuring conditions it is quite natural to as- ) can be (in a natural way) estimated by the observed counts during the dwell time τ. For Poisson distributed random variable, the mean value is equal to its intensity parameter, and the standard deviation is equal to the square root of this parameter.
Probability model of PTR-MS measurements
The distribution of the ratio of two independent Poisson distributed random variables was derived by Arendacká et al. in [1] , where the exact analytic expressions for cumulative distribution function (cdf) as well as for the mean and the variance was given. 
Probability model of PTR-MS measurements
For large concentrations [M] the distribution of the random variable [M] can be well approximated by a normal and/or log-normal distribution, respectively, for more details see [1] .
PTR-MS measurements with the blank analyte
In order to characterize the distribution of measurements with the blank analyte the following experiment was performed: Dry, filtered room air (i.e. a mixture of N 2 ∼ 80%, O 2 and CO 2 ∼ 20%) was measured using a PTR-MS with a dwell time of one second for each ion. At most of the m/z, no signal should be visible since there is no compound included in the sample which could raise a signal. Nevertheless, ions beside H 3 O + may come form the source and rise a signal or contaminations within the tubing system may arise. Table 2 reports the mean countrates of 41 independently repeated measurements of blank analyte measured at selected m/z with a possible explanation of the noise source (either the explained (identified) contaminations as well as their assumed origin, or the unexplained (unidentified) contaminations of the sample and/or tubing system).
PTR-MS measurements with the blank analyte
For example, at m/z 59 most probably some acetone is measured (mean countrate is 24 counts). According to the proton transfer reaction, nitrogen as well as carbon dioxide should not be protonated since their proton-affinity is lower than the one of water. Nevertheless, since at the drift tube an electric field is applied, this reaction as well as a charge transfer may take place. 
Determination of LOQ
Based on (4) the limit of quantification LOQ [M] is defined as a solution of the equation
Determination of LOQ
with the selected constant k = 3. Further, we assume that the measurement device (PTR-MS) was properly calibrated, i.e. the calibration constant κ M is such that the mean of the measured concentration [M], adjusted for the noise, is an unbiased estimator of the true concentration [M], i.e.
Mean( [M])
or equivalently,
Mean( [M])
where [N] denotes the mean concentration of the background noise measured at given m/z.
Mean( [M])
Then, from (10), and by using the results (8), (9) , and (12), we obtain LOQ [M] as a solution to the equation Table 1 , we get α ≈ 0.074.
Mean( [M])
Alternatively, the limit of quantification can be computed as
Mean( [M])
where the intensity parameter λ m/z (LOQ [M] ) is given as a solution to the equation
Mean( [M])
The above mentioned algorithms for computing LOQ [M] could be written as a few command lines in standard software packages. (e.g. MATLAB), see the Appendix 2.
Mean( [M])
As noted above, the limit of quantification depends on the compound M, on its true concentration [M] (useful signal), on the concentration of the background noise [N], and on the dwell time τ. In order to emphasize this dependence we will alternatively use the notation
Example
Let the true concentration of the compound M is [M] = 0 ppb, the mean (true) concentration of the background noise N measured at given m/z is [N] = 0 ppb, and the selected dwell time is τ = 1 sec. Then, based on the equation (13), we get the limit of quantification
Example
Let the true concentration of the compound M is [M] = 0 ppb, the mean concentration of the background noise N measured at given m/z is [N] = 0.13 ppb, and the selected dwell time is τ = 1 sec. Then, based on the equation (13), we get the limit of quantification
Example
Notice, that the same values of the limit of quantification LOQ [M] can be computed using equations (14), (15) , and (7) Then, based on the equation (15), we get
Example
and finally, from (14), we get
Example
what is in agreement with the results of the Example 3.3.2.
Example
Let the true concentration of the compound M measured at given m/z is [M] = 10 ppb, the mean concentration of the background noise N is [N] = 0.13 ppb, and the selected dwell time is τ = 1 sec. Then, based on the equation (13), we get the limit of quantification
Example
If we change the selected dwell time to τ = 10 sec, we get the limit of quantification Then, based on the equation (15), we get
Example
and finally, from (14), we get
Example
what is in agreement with the results of the Example 3.3.4.
Determination of LOD
The limit of detection depends on the particular compound M and on the probability distribution of [N] -the measured concentration of the background noise N, i.e. the concentration measured with the blank analyte (in absence of compound M, i.e.
Determination of LOD
[M] = 0), which in fact depends on [N], the true concentration of N, and on the dwell time τ.
Determination of LOD
In order to emphasize this dependence we will alternatively use the notation
Determination of LOD
The limit of detection LOD M is defined as the true concentration [M] of the compound M, such that the 1%-percentile of the probability distribution of the measured concentration [M] equals the 99%-percentile of the distribution of [N] .
Determination of LOD
Using the theoretical model based on the ratio of independent Poisson distributed random variables it is possible to calculate the exact cumulative distribution functions 2 as well as the required percentiles of the distributions.
Determination of LOD
To be able to do that, it is necessary to know the parameters:
Determination of LOD
= 0, and the dwell time τ.
Determination of LOD
In general, the mean countrate of the background noise observable at the mass-to-charge ratio m/z, i.e. the parameter λ N m/z , is unknown and should be estimated from independent experiments. For illustration, typical mean countrates of PTR-MS measurements of blank analyte (i.e. countrates of the background noise) are presented for selected m/z in Table 2 .
Determination of LOD
The algorithm for computing LOD M from the exact distribution based on the ratio of independent Poisson distributed random variables is numerically highly intensive. So, as a simple alternative, the limit of detection can be approximately computed as
Determination of LOD
where the parameter λ m/z (LOD M ) is given as the smallest solution to the equation
Determination of LOD
Here, p 0.99 (λ N m/z ) denotes the 99%-percentile of the distribu-
Determination of LOD
The approximate algorithm for computing LOD M could be written as a few command lines in standard software packages, see the Appendix 2.
Example
Consider the LOD determination for concentration measurements of the acetone, say M, measured at the mass-to-charge ratio m/z = 59.
Example
Let the estimated concentration of the background noise 
Example
For comparison, we also present result of the approximate algorithm based on (26) 
Example
what is in agreement with (28), the result of exact computation.
CONCLUSIONS AND DISCUSSION
Determination of statistical detection limits is typically based on calibration measurements, but this approach is considerably limited, especially for measurements with very low concentrations of the analyte. In general, the detection limits depend on the substance concentration (useful signal) and on the explained and/or unexplained background noise. In this paper we have suggested a model-based determination of detection limits (LOQ and LOD) for proton-transferreaction mass spectrometer (PTR-MS). The assumed theoretical model for the concentration measurements of the analyte (volatile organic compound, say M) [M] is proportional to the ratio of two independent Poisson distributed random variables, as described in Section 3. This model covers the situations when the measurement process is influenced by the presence of the background noise and respects the effect of the dwell time (i.e. the time of PTR-MS measurement).
CONCLUSIONS AND DISCUSSION
The theoretical model-based approach assumes the exact knowledge of the parameters characterizing the noise and the measurement process by PTR-MS. In real situations, the intensity of the noise should be estimated from independent experiments.
CONCLUSIONS AND DISCUSSION
The algorithms for computing the limit of quantification (LOQ) as well as the limit of detection (LOD) are very simple. As we have illustrated in this paper, the working algorithms for computing LOQ and LOD could be written as a few command lines in standard software packages for technical computing, as e.g. in MATLAB. 
ACKNOWLEDGEMENTS

APPENDIX 1
As mentioned above, determination of detection limits is critical in meeting the needs of analytical science. Data users must understand the limitations of reporting the analytical data measured down to their detection limits in order to minimize the risk of making poor decisions. Unfortunately, there is no unique and unambiguous definition of LOQ and/or LOD. The problem of detection limits, focused on different analytical methods and methodology, was broadly studied and analyzed in the literature, see e.g. [2] , [15] , [4] , [14] , [8] , [7] , [3] , [5] , [12] , [11] .
APPENDIX 1
Here we shall use the statistical concept for determination of the detection limits as suggested in e.g. in [3] and [12] .
APPENDIX 1
Although LOD and LOQ, as presented in [3] and [12] , are considered for situations when the calibration function (in the simplest cases it is the calibration line) is available, we can use this concept of LOD and LOQ also in the situations when the given theoretical model is assumed to be valid for modeling the relation between the measured concentration (in this paper denoted as [M] ) and the true concentration (in this paper denoted as [M]).
APPENDIX 1
It means that for a given true concentration, say x, we can calculate the predicted value of the measured concentration, say Y x , together with the (1 − 2α)-prediction interval
APPENDIX 1
2 ), i.e. with the random interval for which the relations 
APPENDIX 1
where k is a given constant (in the paper we consider k = 3), Mean (Y x ) and Std (Y x ) are the mean value and the standard deviation of the random variable Y x , respectively. In the case of having a regression line Y x = β 0 + β 1 x + ε x , ε x ∼ N(0, σ 2 ) (homoscedastic case), i.e. if we have realizations y 1 , y 2 , ..., y n of independent normally distributed measurements Y x 1 ,Y x 2 , ...,Y x n with equal dispersions, the predicted response of Y for a given x is
APPENDIX 1
with the mean value Mean(Y x ) = β 0 + β 1 x and
APPENDIX 1
are the least-squares estimates of b 0 and b 1 (see e.g. [12] ). The upper prediction limit for blank signal (x = 0) is
APPENDIX 1
(t n−2 (1 − α) is the (1 − α)−quantile of Student t distribution with n − 2 degrees of freedom). LOD is such a true concentration LOD (0) that
APPENDIX 1
For a given x the proper estimate of Mean(Y x ) is
APPENDIX 1
and the proper estimate of Std(Y x ) is
APPENDIX 1
The estimate of LOQ for given x is LOQ x for which
APPENDIX 1
is valid. For more details see [12] .

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