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IntroductionDiamond based quantum sensing methods have gained attention over the last few years. Through optically detected magnetic resonance (ODMR)[1] the nitrogen‐vacancy defect in diamond possesses fluorescence properties which depend on the surrounding magnetic field.[2] This property is only observed in its negatively charged state (NV−) and not in the neutral state (NV0). Working at room temperature[2,3] and offering nanoscale resolution[4–7] the NV− center (hereafter just noted NV center) offers many tremendous experimental assets[8] as well as noteworthy sensitivities[9–11] for sensing variables like temperature,[12,13] pH,[14] strain,[15] electric[16] or magnetic fields,[4,17] microwave signals[18–20] and electrical current.[21]Among the NV based sensing methods, T1 relaxometry enables sensing magnetic noise without any microwave excitation.[22] As described in Figure 1a, a pulse of light initializes the NV center's electron spin into its the bright (ms=0)$({m_{\mathrm{s}}} = \;0)$ state. Another pulse, applied after different delays (or dark times, τ), probes the proportion of NV centers which have returned to a darker equilibrium between ms=0${m_{\mathrm{s}}} = \;0$, +1 or −1$ - 1\;$states. As initially observed in pioneering works[22–28] and further quantified[29] the time this process takes, called relaxation time or T1, is shortened by magnetic noise. This phenomenon was originally used to detect spin labels such as gadolinium ions[22,24,26] and to monitor chemical reactions.[30–32] In our group, T1 relaxometry has turned out to be particularly useful for measuring free‐radicals (chemicals with unpaired electrons) that are generated by the metabolism of living cells.[33–35]1FigurePrinciple of T1 measurements: a) An observation consists of applying a train of 51 laser pulses (532 nm, ≤50 kW c−1,) of 5μs$5{\mathrm{\;\mu s}}$ each, used both to initialize the and readout the nitrogen‐vacancy (NV) center spin state. The 50 darktimes τ between them are exponentially spanned from 0.2 up to 104μs${10^4}{\mathrm{\;\mu s}}$. Each received photon was timestamped with respect to the laser pulse. b) Each measurement sequence is made of Nacq=104${N_{{\mathrm{acq}}}} = {10^4}\;$successive observations. c) The pulses starting from the second onward were used to determine the T1 by summing the total number of photons received at each timepoint either through the whole NAcq or through the bootstrap or rolling window methods. The black dots indicate the pulse succeeding a dark time of 0.4μs$.4\;{\mathrm{\mu s}}$andthereddotsadarktimeof6.8μs${\mathrm{and\;the\;red\;dots\;a\;dark\;time\;of\;}}6.8{\mathrm{\;\mu s}}$. The intensities of the pulses are integrated over the read window or through pulse fitting. d) The T1 relaxation curve was obtained either by integrating the first 0.6μs$0.6{\mathrm{\;\mu s}}$ of each pulse, or through fitting Equation (1) on it. The photoluminescence was normalized to 1 by dividing by I∞${I_\infty }$ obtained from the fit of the Equations (2–4). c,d) All observations are summed together for averaging.The relaxometry curve for a single NV center is well understood and can be fitted by a single exponential model.[27–29,36] Using a larger amount of NV centers significantly increases the photoluminescent signal and leads to greater reproducibility between particles. However, the summation of the signals from different NV centers in varying magnetic environments renders the situation more complex. While few fitting models such as the single exponential,[23,31] biexponential,[25,32–35] and the stretched exponential[37,38] decays are often use to analyze such data, there is no clear consensus on which should be used under different circumstances.In this work, we use the acquisition procedure developed[32] with predefined laser intensity and pulse duration. Applying it on a set of eight nanodiamonds submitted to different gadolinium concentrations, we acquired a calibration dataset that we used to systematically compare different ways to extract data to optimize the data processing flow. In particular, we investigate how to best extract the data from the raw photoluminescence pulse train. We then quantitatively compare the three fitting models. To that end, we use a bootstrap approach to obtain an SNR that can be directly compared between the different methods. We also investigate the effect of the bootstrap itself on the result precision as well as the use of a rolling window to obtain temporal resolution compared to just obtaining a single T1 value.Experimental SectionExperimental DetailsSampleGadolinium ions (in the form of GdCl3 in solution) which is a common contrast agent in magnetic resonance imaging (MRI) for magnetic noise to lower T1 in a controlled manner were used as samples. This work used oxygen terminated nanodiamonds with hydrodynamic diameter of 70 nm (Adamas Nanotechnology), identical to the ones used in our intracellular experiments.[33–35] These nanodiamonds were produced by the manufacturer by high pressure high temperature synthesis. Then particles were irradiated with 3 MeV electrons at a fluence of 5 × 1019 e cm−2 and annealed at the temperature exceeding 700 °C. According to the manufacturer, they contain around 500 NV centers on average. To perform a measurement, the nanodiamonds were first allowed to attach to the bottom of a glass bottom petri dish before filling the dish with water. Gadolinium (Gd) solution was then added to achieve the desired concentration from 1 × 10−9 m up to 100 × 10−3 m.T1 ProtocolOne diamond nanoparticle was first identified with the homemade confocal microscope described.[39] The laser power was chosen such that the nanodiamonds were submitted to a power density that was similar to the works in biological condition.[33–35] To this end, a 532 nm laser at 50μW$50\;{\mathrm{\mu W}}$, was routed to the sample through a microscope objective with a numerical aperture of 1.4. As a result, the laser power density is 50 kW cm−2, far below the measured saturation intensity of 120 kW cm−2.[40] This work also includes data[32] for (two particles out of eight) where 100μW$100\;{\mathrm{\mu W}}$ was routed to a 0.9 numerical aperture objective, resulting in nearly identical power density.A T1 measurement protocol[32] (see Figure 1a) consisted of a pulse train of 51 pulses of 5μs$5\;{\mathrm{\mu s}}$ each intermitted by 50 dark times exponentially varied from 200 ns to 10 ms. During that sequence, the times at which each photon was detected was stored. One execution of a whole sequence of Npulses=51${N_{pulses}} = \;51$pulses was called an observation. The acquisition of each observation lasted (mostly limited by the dark times. 50 ms).AveragingIn order to extract sufficient statistics, a set of T1 measurements consisted of NAcq=104${N_{Acq}} = {10^4}\;$(10 min) observations. T1 measurement sets were acquired independently for eight nanoparticles for each Gd3+ concentration. Those observations were then aggregated either altogether (Figure 1b), through a bootstrap (Figure 2a) or a rolling window aggregation (Figure 2b) (see details in Section 2.4). A photoluminescence pulse train was obtained by counting how many photons were detected at each instant of the pulse sequence (Figure 1a) over all the aggregated observations. Each of the photoluminescence pulses of the train were extracted and isolated such as presented in (Figure 1c).[25] As described in Section 2.2, a T1 relaxation curve (Figure 1d) was extracted from the first microsecond of each pulse as a function of the preceding dark time. The relevant T1 was obtained by fitting a decaying curve (see Section 2.3).2FigureData aggregation principle a) Bootstrap: A subset of the NSub observations is randomly chosen to generate a T1 relaxation curve from which a T1 value can be fitted. This process is repeated NBoot times generating the same number of T1 values, from which a T1 probability density can be obtained (The most likely value corresponds to the maximum of the curve. Red line). b) Rolling window: We first generate a T1 from the first NWidth observations. We then move the window by Nstep observations to generate the next T1. This process is repeated until the entire dataset is processed to infer a temporal evolution.Pulse ReadingGiven the laser power density applied on the sample, the photoluminescence pulses typically differed only within the few first microsecond.The photoluminescence signal of a T1 curve could be obtained by integrating the photoluminescence pulses over an optimized window. The first 0.6μs$0.6\;{\mathrm{\mu s}}$ was used here as previously determined[32] where this method was used to measure ions in solution and determined the effect from a protein corona.Alternatively, the photoluminescence can be fitted with Equation (1), adapted from elsewhere,[41] which models an ensemble of NV centers.1It=A11−e−k1t+A21−e−k2t$$\begin{equation}I\;\left( t \right) = {A_1}\;\left( {1 - {e^{ - {k_1}t}}} \right) + {A_2}{\mathrm{\;}}\left( {1 - {e^{ - {k_2}t}}} \right)\end{equation}$$Here t is the time after the beginning of the pulse. A1 and A2 are related to the different populations in thems=0${\mathrm{\;}}{m_s} = \;0$ andms=±1${\mathrm{\;}}{m_s} = \; \pm 1\;$states and k1 and k2 are parameters that include but are not limited to the laser power and relaxation rates for the different spin states. To obtain a directly comparable T1 curve, the obtained fitted function is integrating the first 0.6μs$0.6\;{\mathrm{\mu s}}$.Extraction of T1 DynamicsThe spin relaxation was obtained from the relaxation curve in the form of a T1 time resulting from a fit. The different commonly used fit models including single‐, bi‐, and the stretched exponential models were compared. In any case, τ is the dark time between pulses.The simplest investigated model is a single exponential decay (Equation 2). Applied on an ensemble, it computes a global relaxation time of the average of all NV centers.2It=I∞1+C1e−τ/T1$$\begin{equation}I\;\left( t \right) = {I_\infty }\;\left( {1 + {\mathrm{\;}}{C_1}{{\mathrm{e}}^{ - \tau /{{\mathrm{T}}_1}}}} \right)\end{equation}$$I∞${I_\infty }$ is the photoluminescence intensity of the final thermal equilibrium at long dark times τ and C1 is the contrast of the relaxation curves.As introduced earlier, the T1 relaxation curves are often well fitted by a biexponential model (Equation 3) comprising a short TS and a long TL component. TL${T_L}$ was determined to be a good predictor for determining the concentration of paramagnetic species (see Section 3.2).[32]3Iτ=I∞1+CSe−τ/Ts+CLe−τ/TLwithT1=TL$$\begin{equation}I\;\left( \tau \right) = {I_\infty }\;\left( {1 + {\mathrm{\;}}{C_S}{e^{ - \tau /{T_s}}} + {\mathrm{\;}}{C_L}{\mathrm{\;}}{e^{ - \tau /{T_L}}}} \right)\hbox{with }{T_1} = {T_L}\;\end{equation}$$The final model is the stretched exponential in which an additional exponent (β) ranges from 0 to 1[42] (Equation 4). This way, the obtained T1 constitutes the global relaxation time of the system.4Iτ=I∞1+C2e−τ/T1β$$\begin{equation}I\;\left( \tau \right) = {I_\infty }\;\left( {1 + {\mathrm{\;}}{C_2}{e^{ - {{\left( {\tau /{T_1}} \right)}^\beta }}}} \right)\end{equation}$$Potential laser intensity fluctuations between pulses are averaged out as described in the next section such that theT1 relaxation curves are fitted without initial normalization. For easier visual comparisons, the curves displayed in Figures 1d, 3a, and 4a are normalized by I∞${I_\infty }$.3Figurea) The T1 curves from particle 1 in water and 10 × 10−6 m Gd3+ resulting from window integration (gray) and fitting with Equation (1) (all observations summed together). b) Signal to noise ratio S/N as a function of the Gd3+concentration${\mathrm{G}}{{\mathrm{d}}^{3 + }}{\mathrm{\;concentration}}$. The signal and noise are derived from the bootstrap and averaged over eight particles, as defined in Section 2.5.2.4Figurea) Fitted single and bi exponential model for two different concentrations of Gd3+ 1 × 10−9 m1 nM (top) and 100 × 10−9 m bottom. The black dots represent the results of the measurement (all observations summed together). The figure shows fits for the biexponential model (solid black), single exponential model (dashed red), and stretched exponential (dashed blue) and the long (green dashed) and short (purple dashed) components of the biexponential fits. b) T1 values as a function of the Gd3 + concentration. The signal S and error bars N, are derived from the bootstrap, averaged over the eight different particles, according to Section 2.5.2. c) Signal to noise ratio S/N as a function of the Gd3 + concentration.Data Aggregation: Generation of the Raw Photoluminescence Pulse TrainAll observations defined above can be summed together. For each T1 extraction method (Section 2.3), a single T1 value was extracted. Other aggregation methods were also used here.BootstrapAs commonly applied in both medical sciences and signal processing, the data was aggregated using a bootstrap.[43] With this method, this work could obtain the probability density for the fitted parameter,[44] derived its maximum likelihood, and a standard deviation (see below).During a bootstrap a subset of NSub${N_{Sub}}$ randomly chosen observations (each observation could be chosen more than once or not at all) was combined (see Figure 2a and Table 1). In this case,Nsub=104${\mathrm{\;}}{N_{sub}} = {10^4}\;$(same as Nacq${N_{acq}}$). For a given T1 extraction method (Sections 2.2 and 2.3) a T1 value is obtained.1TableBootstrap process1Select Nsub observations randomly among the Nacq allowing to select the same measurement multiple times*2Reconstruct the T1 measurement based on the selected repetitions3Compute the T1 value from the resulting curve4Repeat steps 1–4 Nboot times to obtain the bootstrap samples5Apply kernel density approximation6Compute the applicable statistical properties maximum of likelihood (Signal) and standard deviation (Noise) used in for Figures 3 and 4***The observations are selected randomly, without excluding the ones already selected.**The signal and noise displayed in Figure 5 are derived from the statistic over the particles (See Section 2.5)To obtain the probability density of T1, this procedure was repeated Nboot${N_{boot}}$ times. In our case,Nboot=104${\mathrm{\;}}{N_{boot}} = {10^4}\;$resulting in 104 different values for T1 constituting the probability density using the Kernel Density Estimate (KDE) or Parzen–Rosenblatt window.[45] Here, the same kernel was used as Botev et al.[45] which is based on diffusion equations. The KDE transfers the continuous values into a smooth distribution with a total integral of 1 from which the most likely T1 or contrast values and their confidence interval were extracted.The parameters Nboot,Nsub,andNAcq${N_{boot}},\;{N_{sub}},\;{\mathrm{and}}\;{N_{Acq}}$ (10 000 each in this case) were chosen to ensure sufficient sampling for building the density probability while limiting redundancies.Rolling WindowTo observe the temporal evolution of T1 over the total duration of acquisition, a rolling window (or rolling average) could be applied. This process, often used in econometric studies,[46] is schematically presented in Figure 3. While we previously used the entire Nacq${N_{acq}}$ repetitions (which are collected within about 10 min) to compute one T1 value, we attempted here to further divide the measurement to gain time resolution. To perform a rolling window the first T1 was computed for a defined number of repetitions (here 1 to 7000). Afterwards the window was moved by 10 repetitions (this is called shift) and the second T1 was computed for observation 11 to 7010. The window was moved again and this was repeated until the window for the final T1 was computed up to the final repetition. The steps that are needed to perform a rolling window are shown in Table 2.2TableRolling window process1Select a window size and a shift size2Reconstruct the T1 measurement in the window3Compute the T1 for this window4Move the window by the shift size5Repeat steps 2–4 until the window ends at the last repetitionSignal and Noise Errorbar CalculationStatistics Over ParticlesIn case the bootstrap cannot be applied (Section 3.3, Figure 5), the signal S is obtained for each concentration by taking the absolute value of the subtraction of the T1 obtained at that concentration to the one in water, averaged over eight particles. The noise N is taken from the standard deviation over the eight particles. However, this value includes both the T1 estimation error made by the fit methods and the initial T1 dispersion. Since the second is much larger than the first,[32] the ability to discriminate different concentrations is significantly masked by the dispersion of the initial T1 value.5Figurea) T1 values obtained by fitting either a biexponential or a stretched exponential model over all the observations at once (direct fit) or taking the most likely value from the bootstrap. In all cases we averaged over eight particles. The error bars correspond to the standard deviation of the values obtained above calculated from the data for eight particles (details in Section 2.5.1a). b) The signal to noise ratio as a function of the Gd3 + concentration.Statistics Obtained with BootstrapThe role of the bootstrap is to place the randomization on the selection of the observation when using a particle rather than on selecting that particle. It therefore enables to derive statistics on the effect of the measurement method itself. In case it can be applied (Sections 3.1 and 3.2, Figures 3 and 4), a signal S is taken from the T1 of maximum of likelihood as obtained by the bootstrap at each Gd3 + concentration (subtracted by the one obtained at water condition). The error bars N are also taken from the standard deviation obtained from the bootstrap at the considered concentration. This whole process is repeated and averaged over the eight different particles.Signal to Noise RatioWith above definition of S and N the signal to noise ratio is defined as5SNR=S/N$$\begin{equation}{\mathrm{SNR\;}} = {\mathrm{\;}}S/N\end{equation}$$Results and DiscussionsPulse FittingAs shown in Figure 1c), the pulses start with a rapid build‐up toward a steady value. As shown,[47,48] at high excitation intensity, an overshoot can be visible at the beginning of the pulses. This corresponds to the initial decay of the dark metastable state population to the optically active ground state. In our case however, the laser intensity is low and the absorption–emission cycle remains slower that this decay. As a result this overshoot is not visible. The pulses succeeding longer dark times (red) reach the steady value slower than those after shorter ones (black). As described in Section 2.2, the pulses were either integrated over the read window (gray) or fitted with Equation (1) to better take all the timepoints of the pulse into account and reduce the noise.The difference between pulses is caused by the probability of the decay from the excited to the ground state via the metastable state which does not emit photons in the detected range.[47,49] When the decay goes through the metastable state, the electrons are also shelved there for a few hundred nanoseconds before they can cycle again.[47,48] Furthermore, the decay via the metastable state is more likely to occur for electrons in themS=±1${\mathrm{\;}}{m_{\mathrm{S}}} = \; \pm 1$ state.[47,49] These factors create the build‐up in the pulses and are the basis of the spin readout of the NV center. A major variable in the polarization model is the excitation rate. Related to the laser intensity, it impacts k1 and k2 in Equation (1) directly. While too low pumping does not allow to polarize the NV center fast enough compared to the relaxation, too high energies may ionize the NV− center to NV0.[50,51] In our case, as observed in our previous works,[32–35] the polarization effectively builds up during the first microsecond of the pulse such that the NV centers are efficiently polarized during the pulses. The laser intensity was chosen to be safe for biological samples and kept constant between experiments.[32–35]The fit well captures the build‐up to a saturated steady state (ranging up to ≈200counts$ \approx 200\;{\mathrm{counts}}$). Taking all datapoints into account, the pulse fit should reduce the total relative shot noise. The comparison between the resulting T1 curve obtained from particle 1 exposed to either water or 10 × 10−6 m, fitted with the biexponential model is shown in Figure 3a.However, as observed in Figure 3b the SNR is not significantly improved when the fitted pulse is used.Exponential FitsWe compare the performance of the three different models in differentiating known concentrations of Gd3+. Each of these models has their origin in the population dynamics of the NV center. The most important parameters are the difference in intensity between the mS=0${m_S}\; = \;0$ and mS=±1${m_S}\; = \; \pm 1$ and the transition rate between these states (expressed in the T1). The single exponential model depicted in Equation (2) can be naturally derived from the spin relaxation decay dynamics of individual NV centers.[22,36]Extending the model to ensembles starts with considering that each NV center has different T1 times. Empirically[25,32–34,52] we found a biexponential decay (Equation 3), comprising both a short component Ts${T_s}$, in the microseconds range, and a longer TL${T_L}$ about a hundred time longer. The origin of such distinct dynamics remains unknown. For instance, in ref.[25] TS${T_S}$ is attributed to a group of clustered NV centers dominated by cross‐relaxation. These depend on their orientation[33] and distance to the surface of the nanodiamond,[32,34,35] the number and proximity of both paramagnetic species, and dangling bonds on the surface[27] and nitrogen and 13C within the diamond. As a result, each NV center feels a slightly different magnetic noise intensity. The relaxation curve obtained from an ensemble is therefore the sum off all those contributions. However, the number of NV centers vary for each nanodiamond and fitting over many variables reduces the precision of the fit and complicates the calculation.As shown,[42] such a sum over the different T1 can be modeled with a stretched exponential depicted in Equation (4). This increases the number of fitted parameters to four.Depending on various parameters such as the nitrogen concentration in the diamond,[53] the surface chemistry, the NV center depth,[54] and the laser wavelength[55–57] or intensity,[58] charge transfer may occur between the negatively charged NV− and the neutral NV0.[51,59] This may significantly impact the relaxation curves obtain as above.[60]In previous a work,[34] we compared measurements with or without a microwave pulse placed at resonance with the spin transition. As proposed in [23], this approach can be used to exclude spin independent processes. Due to the multiple possible direction of NV centers quantization axis (four in each nanodiamond) and including possible rotation during acquisitions several Rabi oscillations of different frequencies are summed together such that the contrast of the relaxation curves is significantly decreased by the microwaves (see Supporting Information of [34]). Nonetheless, the longer decay time temporal constants (TL${T_L}$) from biexponential fit and its dependency to the external magnetic noise is preserved. Further evidence that the charge transfer is not dominating in our experimental conditions can be found in our other works[61,62] where we obtained similar relaxation times for particles in a different charge environment or surface termination. In the following we assume that a spin relaxation process is dominating the relaxation curves.The fits of these three models on typical T1 relaxation curves from the first observed particle exposed to 1 × 10−9 and 100 × 10−9 mGd3 + are presented in Figure 4a.A first measure of the performance of the fits can be read through their residuals which considers the total average squared error made by the fit with respect to the original data. The residuals obtained from those fits are presented in the inset. At first, it confirms that the bi and stretched exponential decaying models render the situation better than the single exponential one. Although a clear first shoulder corresponding to the short relaxation decay is often observed (see Figure 4b),[32,33,35] the stretch and biexponential's residuals turned out to be quite similar. We also observed that, weight associated to each decay depends on the nanodiamond, or the Gd3+ concentration. At high Gd3+ concentrations, or for certain particles, only one decay persists. In such a case the biexponential fit approaches the single or stretched exponential ones.The degree to which the models explain the difference in T1 as caused by changes in concentration rather than random noise can be measured by computing the SNR (see Section 2.4). Higher SNR implies that a change in concentration induces better visible differences in the signal with respect to the noise.Figure 4b depicts the most likely T1 values and the standard deviations obtained from the bootstrap at each Gd3+ concentration. In all cases, we averaged over the eight different particles. Figure 4c shows the obtained SNR as defined in Section 2.5. The biexponential fit results in longer time constants than the single and stretch exponential ones. In the first case indeed, the faster dynamics of the decay curves is already reproduced by the shorter exponential decay (in TS${T_S}$) leaving only slower ones toT1=TL${\mathrm{\;}}{T_1} = {T_L}\;$.The standard deviation given by the bootstrap is also larger for the biexponential than for the other models. For the lower concentrations (1 × 10−9 and 10 × 10−9 m Gd3+) the relative change in T1 (signal to noise ratio given in Figure 4c) highlights better sensitivities of the biexponential ‐ and stretch exponential models over the single one. Biexponential and stretched exponential models perform similarly to each other here. While the biexponential has also slightly higher standard deviation, the steepness of the concentration dependency outweighs this disadvantage with respect to the other models.For higher concentrations however, the error bars linked to the biexponential fit increases relatively to the signal. With more fitted parameters, and when the two time constants Ts${T_s}$ and TL${T_L}$ get closer, we observe that the fitting procedure may exchange the role of the two causing instability of the fit. Similarly, a single exponential may become sufficient such that TL${T_L}$ has no longer any effect on the shape of the curves making its value meaningless.Nonetheless, our analysis notably confirms that for most of biologically relevant cases, i.e., when the T1 is larger than 100 μs, the widely used biexponential fit remains well suited.BootstrapWhile the bootstrap can be used to compare the different fit models, it can also be applied on the raw data to determine the most likely T1. The idea behind is that some outlier observations may or may not be considered by the bootstrap in a random manner such that the obtained most likely value could be more robust than the direct fit. In that case however, since the bootstrap cannot be applied on itself. The uncertainty is obtained each time from the standard deviation observed over the eight different particles at the considered Gd3+ concentration (see Section 2.5).The T1 values remain quite similar when using the bootstrap or not. The standard deviations over the eight different particles slightly increased, with the bootstrap. This value is more difficult to interpret as it is significantly enlarged by the initial T1 dispersion making quantitative comparisons difficult. Nonetheless, a possible explanation of the observed decrease in SNR could be the following: The bootstrap selects observations independently from one another. This both enables to take each observation more than once but also not to take some observations at all. Altogether, although both the direct fit and bootstrap take the same amount of observations in total (NSub=NAcq=104)$( {\;{N_{Sub}} = {N_{Acq}}\; = {{10}^4}\;} )$, the number of them that are independent from one another is lower in the bootstrap than in the direct fit. Reducing the sampling may have a more important effect than the possibility to exclude obvious outliers (especially if the dataset contains relatively few of them).Rolling WindowWhile the previous methods aim to detect a concentration that is constant within one measurement, we here attempt to improve time resolution. To this end we created a measurement consisting of three Gd3+ concentrations: water (0 × 10−9 m), 10 × 10−9 m, and 1 × 10−6 m (1000 × 10−9 m). This means that the T1 should be high in the beginning of the measurement and low at the end. The rolling window (shown in Figure 6) used 7000 repetitions with a shift of 10. This corresponds to an acquisition time of approximately 7 min and a shift of half a second such that the displayed curve is largely oversampled.6FigureThe rolling window of three measurements for different concentrations. Each window consisted of 7000 repetitions and are shifted by 10 for each sequential point. The solid lines represent the measured T1 values for each point of the figure for the different models; bi‐(black), single (red), and stretched (blue) exponentials. The colored blocks represent the different concentrations which contributed to the calculation (purple for water and 10 × 10−9 m, green for × 10−9 m, and 1 × 10−6 m). The dashed lines represent the T1 value obtained from the average of a rolling window of 7000 and are used to represent the average T1 for one specific concentration.This window was intentionally moved over the different concentrations. This simulates a case where we do not know when a concentration change occurs. The points in the purple and green areas of Figure 6 are composed of data from two different concentrations. The T1 value in each point can be predicted by using the average T1 value of the rolling window of each measurement with a size of 7000. The concentration prediction is then made by calculating the proportion of each measurement in each point and computing the average T1 based on these proportions. The predicted values are shown in Figure 6 with the dotted lines with different colors representing the different models (black: biexponential, red: single exponential, blue: stretched exponential). Figure 6 shows that all exponential models follow the predicted curve very well.The most important consideration for choosing the right parameters for a rolling window is the window's size. If the timescale of the expected changes is known this can be used as a guideline for selecting the window size. However, when lowering the window size, we have to consider that each resulting T1 value is then based on a smaller number of repetitions (for an optimization of this parameter see Supporting Information).The rolling window acts as a longpass filter. A too long rolling window would average out changes on short timescales. Oppositely, a too short window will lead to unreliable results. Overall, the window size has to be optimized per set of experiments. In our case, we observed that below 7000 observations, the fits are becoming less stable, which reduces the accuracy of the fit.ConclusionIn this paper we compared different methods to analyze T1 data: single, biexponential and stretched exponential models. These are the most commonly used models to fit the T1 curve. We presented pulse fitting and bootstrap to remove noise from T1 data. Lastly, we showed the rolling window method to display a temporal evolution of the data. We compared all these methods based on a calibration dataset which uses NV centers in nanodiamond to measure different concentrations of gadolinium. We demonstrated that all models and methods can be applied successfully to this data.By using a bootstrap, we showed that the stretched and biexponential fit models are better in differentiating between concentrations, with a preference for the stretched one at higher Gd3+ concentrations. For lower concentrations, the biexponential has larger variation between measurements, but is also more sensitive to concentration changes. The T1 resulting from the stretched exponential remains very similar to the single exponential but appears to be more predictable. Furthermore, the fit quality, for the exponentials and stretched exponentials, are significantly better than for single exponential. However, when selecting the best model for the data, the experimental design should be a leading factor as well. While we investigated nanodiamonds with NV center ensembles, the single exponential model may still be best for single NV centers.We also presented two alternative methods to compute the T1 from the measurement. The first is based on modeling the pulses from the pulse train. While the output T1 remain unchanged, the measurement quality is not improved in term of SNR. Thus, it might be useful in datasets of worse quality, or when the optimized integration window is not known.We further used a bootstrap to improve the quality of an output T1. Either masked by initial T1 dispersion, or due to plausible limitations that we identified, we could not observe significant improvements.Lastly, the rolling window was used to show temporal information on the T1. We showed that each model reproduces the decrease in T1 with increasing concentrations. While the T1 is “noisy” as the rolling window moves, the changes induced by combining different Gd3+ concentration is larger.AcknowledgementsThis work was supported by a VIDI grant (016.Vidi.189.002). M.C. acknowledges supports from the Swiss National Science Foundation (SNSF) under grant n 185824.Conflict of InterestThe authors declare no conflict of interest.Author ContributionsF.P.M. had performed the experiments performed in this article. T.V. had performed the data analysis with the help of T.H. R.S. and M.C. had supervised the project. R.S. had acquired the funding and leads the research group. T.V. had written the first version of the manuscript and all authors had edited and approved the final version.Data Availability StatementThe data that support the findings of this study are available from the corresponding author upon reasonable requestD. Suter, Magn. Reson. 2020, 1, 115.A. Gruber, A. DräBenstedt, C. Tietz, L. Fleury, J. Wrachtrup, C. 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Advanced Quantum Technologies – Wiley
Published: Jul 5, 2023
Keywords: diamonds; nanodiamond; nitrogen‐vacancy (NV) centers
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