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On-chip, time-correlated, fluorescence lifetime extraction algorithms and error analysis Day-Uei Li,1,* Eleanor Bonnist,2 David Renshaw,1 and Robert Henderson1 1
Institute for Micro and Nano Systems, School of Engineering and Electronics, University of Edinburgh, Faraday Building, King’s Buildings, Edinburgh EH9 3JL, Scotland, UK 2 School of Chemistry, University of Edinburgh, Joseph Black Building, King’s Buildings, Edinburgh EH9 3JJ, Scotland, UK *Corresponding author:
[email protected] Received September 21, 2007; revised February 27, 2008; accepted March 2, 2008; posted March 4, 2008 (Doc. ID 87744); published April 30, 2008 A new, simple, and hardware-only fluorescence-lifetime-imaging microscopy (FLIM) is proposed to implement on-chip lifetime extractions, and their signal-to-noise-ratio based on statistics theory is also deduced. The results are compared with Monte Carlo simulations, giving good agreement. Compared with the commonly used iterative least-squares method or the maximum-likelihood-estimation- (MLE-) based, general purpose FLIM analysis software, our algorithm offers direct calculation of fluorescence lifetime based on the collected photon counts stored in on-chip counters and therefore delivers faster analysis for real-time applications, such as clinical diagnosis. Error analysis considering timing jitter based on statistics theory is carried out for the proposed algorithms and is also compared with MLE to obtain optimized channel width or measurement window and bit resolution of the time-to-digital converters for a given accuracy. A multi-exponential, pipelined fluorescence lifetime method based on the proposed algorithms is also introduced. The performance of the proposed methods has been tested on mono-exponential and four-exponential decay experimental data. © 2008 Optical Society of America OCIS codes: 000.5490, 030.4280, 030.5260, 040.1345, 170.2520, 170.3650, 170.6920.
1. INTRODUCTION Fluorescence lifetime measurements have been widely used to study various scientific and practical applications in optics, chemistry, biology, medicine, including medical diagnosis. Up to now, a large number of different techniques including time-domain and frequency-domain methods have been well developed for measuring fluorescence lifetimes. In time-domain methods, the fluorescence intensity decay is measured through a time-correlated, single photon-counting (TCSPC) card after excitation with a short pulse of laser light [1–3], whereas in frequency-domain methods, the fluorescent sample is illuminated with a periodic light source to obtain a measured phase difference between the light source and the fluorescent emission. Irrespective of the methods used [4–10], the lifetime extraction is done using computer software. For general purpose time-domain analysis tools, for scientific research demanding high accuracy down to the picosecond timescale, or for practical medical/clinical diagnostic applications demanding fast results, it is necessary to cover a wide range of faint multi-exponential fluorophores, and the lifetime resolution is expected to be lower than 50 ps. Because of the inability of the least-squares method (LSM) or maximum likelihood estimation (MLE) to resolve a small lifetime with a coarse channel width, the bit resolution of the time-to-digital converters (TDCs) in the photon counting card is therefore expected to be larger than 11-bit [3]. Moreover, to use LSM or MLE properly, the measurement window is usually taken to be as large as possible; otherwise the software would interpret the measured data as having a DC offset. As a conse1084-7529/08/051190-9/$15.00
quence the laser pulse repetition rate must be kept slow, which will further lower the photon collection speed. Further, because fluorescence lifetimes in imaging are determined on a pixel-by-pixel basis, iterative methods can be quite time-consuming and make real-time image processing impossible. Although the requirement for short laser pulses can be avoided by using frequency-domain methods [5,6], lifetime extraction still relies on software analysis, which also makes real-time image processing difficult to accomplish. For photon detection non-solid-state photomultiplier tubes (PMTs) have been the detectors of choice because of their high dynamic range and high-speed performance [11]. However, cost and size considerations, as well as the need for cumbersome optical scanning, have prevented wide adoption of PMT-based bioimagers. To achieve parallelism, larger arrays of miniaturized single photon detectors are necessary. As complementary metaloxide semiconductor (CMOS) process technology advances, the compact design of single-photon-avalanchediode (SPAD) detector arrays provides better and better performance [12,13], and because of its low cost and high potential for integration, it has become a good candidate for fluorescence-lifetime-imaging microscopy (FLIM) system-on-chip (SOC) design. In this paper, we start by considering a singleexponential decay and propose a novel and simple direct integration for lifetime extraction method (IEM) and derive its error equation to determine the optimal channel width and the bit-resolution of the TDCs. The singleexponential assumption allows a proper comparison of the © 2008 Optical Society of America
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various fitting techniques. Also we derive the error equation for the M-gate rapid-lifetime-determination algorithm (RLD-M) and calculate its optimal measurement window. We compare our proposed methods with the theoretically most accurate estimator, MLE [7,8], to check the performance and efficiency of an on-chip implementation using the IEM and the 2-gate RLD (2-RLD). Based on single-exponential extraction methods, we also propose a new pipelined IEM algorithm (PL-IEM) for on-chip FLIM implementation. For initial publication of the results, the PL-IEM lifetime extraction method is demonstrated on four-exponential decay experimental data. To the best of our knowledge, IEM is the first proposed algorithm that can be integrated within a SPAD or a pixel cell.
2. THEORY The recorded fluorescence histogram f共t兲 is related to the true decay function I共t兲 through the integral
f共t兲 =
冕
t
I共t − 兲 · IRF共兲d ,
共1兲
0
where IRF共t兲 is the instrumental response function (IRF), or the convolution of the transition spread of the detector and the pulse function of the laser source. The IRF can be obtained by making observations in the absence of any fluorophore sample. The IRF is first stored in memory and, after a second measurement, a fluorescence response is stored in a second memory. The true response I共t兲 can be obtained through an on-chip digital deconvolution calculation. However, we need to evaluate whether the enhanced precision can justify the cost of extra chip area for digital deconvolution. Here we assume I共t兲 = A exp共−t / 兲, and the ratio of the FWHM of IRF共t兲 over the lifetime is denoted by r. For simplicity, we tried four different forms for the IRF, each with the same FWHM: a rectangular, a symmetric triangular, an exaggerated asymmetric shape, and a Maxwell distribution with a longer tail (to model the realistic case). All four IRFs have their center of mass located at t = 0. The recorded response f共t兲 is obtained from Eq. (1) and is shown in Fig. 1. It shows that as r becomes larger than 1, it is difficult to obtain a clear response, because the pure exponential behavior for the Maxwell IRF, starting from the circular mark, is too small and contaminated with noise, making it inefficient to accumulate enough photon counts for a given SNR. The smaller the ratio r, the more efficiently and accurately the lifetime can be extracted. Considering the jitter of the light source to be about 10 ps, the transition spread of the detector structure to be about 80 ps, and the jitter of gate transitions to be about 30 ps, the overall FWHM is approximately 100 ps. Thus, without an on-chip deconvolution function, the smallest lifetime that can be obtained is of the order of 200 ps. For initial implementation, we simplify to illustrate the principle and carry out tests on samples with longer lifetime. For these the assumption that f共t兲 is a single exponential is reasonable.
Fig. 1. (Color online) Convolution of a single decay and artificial IRFs with different ratios of FWHM over lifetime.
A. Integration for Extraction Method The concept of using numerical integration to extract the fluorescence lifetime dates back to 1970 [9] for multiexponential decay curve fitting. However, owing to the limited accuracy of high-order numerical integration, the precision is dubious and is not applicable to current applications that require high lifetime resolution. Further, this method needs to deal with many complicated arithmetic operations and is therefore not suited to real-time hardware implementation. Moreover, so far as we are aware, error analysis of this method has not appeared in any available literature. We propose an algorithm suitable for on-chip lifetime extraction by introducing highaccuracy numerical integration and call it the integration for extraction method (IEM). As for multi-exponential lifetime extraction, we proposed a pipelined IEM algorithm based on the single-exponential IEM. Figure 2 shows a single-exponential fluorescence decay histogram f共t兲 = A exp共−t / 兲 generated by using M time bins. For each laser pulse, we assume there is a probability of P (in a real case, P Ⰶ 0.1.) for detecting a photon, and that P = DCR/LPR = 1/NP ,
Fig. 2.
共2兲
(Color online) Concept of the single-exponential IEM.
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where DCR is the detector count rate, and LPR is the laser repetition rate. This is equivalent to saying that one photon is detected on average for Np laser pulses. If P is larger than 0.1, then the pile-up effect should be considered. This is not so simple as Eq. (2), and it should be corrected as in [1, p. 333]. For a photon to be detected by a SPAD, it should first enter the silicon without being reflected at the SPAD surface. Then, the photon has to be absorbed to give a primary electron–hole pair. Finally, the photogenerated carriers should induce a Geiger pulse. Therefore, the probability P is a complicated function of the wavelength , excess voltage Ve applied to the diode, the quantum efficiency and doping profile Nd of the diode, and also the laser intensity IL [12,13]. Therefore we can write Np = Np (, Ve, Nd, IL). Assume that we have optimized the performance of the diode by choosing a proper wavelength, an optimized excess voltage and doping profile considering the dark count rate, and the after-pulsing effect; then Np = Np 共IL兲. Assume there are M time bins with bin width of h within the measurement window. For a total count Nc, we need Nt = NcNp laser pulses to collect Nj counts in the jth time bin. At the kth laser pulse, the probability that a detected photon is located in the jth time bin 共tj ⬍ t ⬍ tj+1兲 is 1 − e−h/ Pj,k = P
1−e
−Mh/
冉
e−jh/ 1 +
hj,k h
冊
共3兲
,
where j = 0 , 1 , . . ., M − 1; k = 1 , 2 , . . ., Nt; hj,k is the timing jitter for the jth time bin at the kth laser pulse, and we assume that hj,k (j = 0 , 1 , 2 , . . ., M − 1; k = 1 , 2 , . . ., Nt) are independent and asymptotically normally distributed. The jitter source comes from the jitter of the kth laser pulse and also those of TDC transitions. The recorded variables Nj are independently Poisson distributed with 共j+1兲h respective mean values ENj = 兰jh f共t兲dt and standard de1/2 viations Nj = 共ENj兲 . We also have Nt
Nj =
兺P
1 j,k
=
k=1
Nt
兺
1 − e−h/
Np k=1 1 − e−Nh/
冉
= Np−1P0e−jh/ Nt +
冉
⬵ENj 1 +
Nj ENj
+
hj,0 h
e
冑Nt
hj,0 h冑NcNp
冊
−jh/
冊
冉
1+
hj,k h
冊
冕
tM−1
f共t兲dt =
t0
⬵
h 3
冕
IEM ⬵
=
h 共EN0 + 4EN1 + 2EN2 + ¯ + 4ENM−2 + ENM−1兲 3
EN0 − ENM−1
共M−3兲/2 −2jh/ e h 共1 + 4e−h/ + e−2h/兲兺j=0
1 − e−共M−1兲h/
3
h 1 + 4e−h/ + e−2h/ =
1 − e−2h/
3
=
5 3 17 6 − 6␣ + 4␣2 − 2␣3 + ␣4 − ␣5 + ␣6 + ¯ 6 10 180 4 4 8 6 − 6␣ + 4␣2 − 2␣3 + ␣4 − ␣5 + ␣6 + ¯ 5 15 105
冋
= 1+
1 180
␣4 −
1 1512
册 冉 冊
␣6 + O共␣8兲 = 1 +
,
共6兲
M−1
IEM ⬵
j=0
共7兲
,
N0 − NM−1
where Cj = 关1 / 3 , 4 / 3 , 2 / 3 , . . . , 4 / 3 , 1 / 3兴. If Romberg’s rule is used for the numerical integration, Cj = 关1 / 2 , 1 , 1 , . . . , 1 , 1 / 2兴. Comparing this with the RLD-M algorithm [10], which gives M−1
共4兲
h 兺 共CjNj兲
M 兺 tj2 − ,
⌬
where ␣ = h / , ENj = NcP0 exp共−jh / 兲, and Taylor’s series expansion is used in the derivation. This gives the extracted lifetime expressed in terms of recorded counts as
,
=−
j=0 M−1
冉 冊 M−1
2
兺 tj
j=0
M−1 M−1
,
共8兲
M 兺 关tj ln共Nj兲兴 − 兺 tj 兺 ln共Nj兲
considering Poisson shot noise and where P0 = 共1 − e−h/兲 / 共1 − e−Mh/兲 is the probability of a detected photon being located in the first time bin. This indicates that collecting more photon counts can minimize the effect of timing jitter. With the assumption of a single-exponential decay, the lifetime constant is related to the area under the histogram function as
共f0 − fM−1兲 =
(M is odd) is used to calculate the numerical integration. Then multiply Eq. (5) on both sides by the factor 共1 − e−h/兲 to obtain
tM−1
A exp共− t/兲dt
t0
共f0 + 4f1 + 2f2 + ¯ + 4fM−2 + fM−1兲, 共5兲
where A = NcP0 / 关共1 − e−h/兲兴, fj = f共tj兲, and Simpson’s rule
j=0
j=0
j=0
the number of arithmetic operations for Eq. (7) is much smaller and easier to compute, and it contains no natural logarithmic function, which makes on-chip implementation of lifetime extraction possible. It might be argued that using RLD-M could provide better resolution, but we will show later that the resolution limit will reach a lower bound worse than Eq. (7) no matter how much we increase the number of channels. With IEM, we are trying to balance the complexity of hardware and the precision of lifetime extraction by choosing optimal parameters for measurement. Considering the impact of Poisson noise and timing jitter, Eq. (4) is applied to Eq. (6) to give
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Vol. 25, No. 5 / May 2008 / J. Opt. Soc. Am. A
h
冉
IEM ⬵
EN0 1 +
冋
M−1 j=0
N0
+
EN0
Combining with Eq. (6), we rewrite Eq. (9) as
IEM ⬵
h共U + u兲 V +
⬵
冉
兺 CjENj 1 +
hU V
h0 h冑Nt
冉
冊
ENj
hj
+
h冑Nt
冉
− ENM−1 1 +
u
1+
Nj
−
U
V
冊册
NM−1
+
ENM−1
冊 冉
⬵ 1+
⌬
+
hM−1 h冑Nt
u U
−
V
冊 冊
1193
.
共9兲
,
共10兲
where M−1
U=
兺 C EN , j
共11兲
V = EN0 − ENM−1 ,
j
j=0
M−1
u =
兺C j=0
j
冉
冉
= N0 + u U
−
V
冊 冊冉
ENjhj
Nj +
h冑Nt
EN0h0 h冑Nt
,
− NM−1 +
ENM−1hM−1 h冑Nt
b =
冑冉 冊 C0 U
1
−
V
N02 +
j=1
冑冋冉 冊 1
C0
1 h冑Nt
冊 冉 兺冉
M−2
2
−
U
V
Cj U
2
Nj
+
M−2
2
h02EN02
+
兺 j=1
冉
Substituting ENj = NcP0 exp共−jh / 兲 = Nc exp共−jh / 兲共1 − e−h/兲 / 共1 − e−Mh/兲 and Nj = 共ENj兲1/2 into Eq. (13) and assuming Nj, hj are independent, we have
冋 冉冊
冉
= 1+
1
h
180
⌬
共12兲
,
= 冑a2 + b2 ,
a =
IEM = 1 +
冊
+
4
冊
1 −
冉冊 h
1512
6
+ 冑a2 + b2
册
CM−1 U Cj U
1 +
V
冊
2 2 NM−1 ,
冊 冉 2
hjENj
+
CM−1
a = 2
Q共x兲 =
U
冑
b =
where
V
冊
册
2
2 2 hM−1 ENM−1 .
共13兲
Q共x兲
共15兲
,
Nc
共1 − xM兲共4x + 5x2 + 5xM+1 + 4xM+2兲 共1 − x兲共1 − xM−1兲2共1 + 4x + x2兲2
共14兲
,
1 +
2h0 h
冑
G共x兲 N cN p
and x = exp共−h / 兲. The term ⌬ / in Eq. (14) is the error of the Simpson’s numerical integration. Then we define Precision ⬅
IEM
IEM
= 冑a2 + b2 =
2
冑N c
冑
Q共x兲 +
G共x兲 h02 Np h2
,
Accuracy ⬅
⌬IEM
IEM
共18兲
,
共1 + x2兲共1 − xM−1兲2共1 + 4x + x2兲2
1 =
共16兲
共17兲
,
8x2 + 4x3 + 2x4 + 4x5 + 4x2M−1 + 2x2M + 4x2M+1 + 8x2M+2 G共x兲 =
,
冉冊 h
180
4
1 −
冉冊 h
1512
6
.
共19兲
Before the light intensity reaches a high critical value—where the SPAD presents a saturation behavior
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due to the dead time—the detection probability is propor˜ / I , and K ˜ is a tional to the light intensity [13]; then Np = K L constant. Precision of Eq. (19) can be rewritten as Precision =
冑N c
冑
M = 15 2
10
Q共x兲 +
ILG共x兲 h02
共20兲
h2
˜ K
Q(x), Q(x)/G(x)
2
10
for a given number of photon counts, or as 2 Precision =
冑N t
冑
˜ Q共x兲 K +
IL
h2
−1
M−1
M−1
+M
兺
冉兺 冊 M−1
tj2 −
j=0
M−1
兺 兺
关tj ln共Nj兲兴 −
tj
j=0
ln共Nj兲
j=0
冎
2
tj
共22兲
= 0,
j=0
where tj = jh + hj, Nj is expressed in Eq. (4), and again x = exp共−h / 兲 or h / = −ln共x兲. Also
g共x兲 = xg⬘共x兲, x x
=h
2
共24兲
=
Ch
冑
S共x兲 N cP 0
+
2Ch02 N cN p 2
1 共x
−1
− 1兲3
1.5
1.8
IEM
冑
2 IEM +
2 ⌬IEM
共27兲
.
h
90
,
共25兲
关共M − 1兲2共x−M−2 − 1兲 + 共6 − 2M2兲共x−M−1
− x−1兲 + 共M + 1兲2共x−M − x−2兲兴,
1.2
The optimal point marked as point A is located at Mh = 10, or h = 2 / 3 = 0.67 for the 15-gate IEM with maxi-
2
where S共x兲 =
0.9
h/τ
100
冉 冊 1+
0.6
Precision ⬅
共26兲
2
70
/∆τ
IEM
τ
IEM,Cal
B
A
/∆τ
IEM
IEM
60
τ
40 30 20 0
/στ
IEM
50
and C = M共M − 1兲 / 6. C. Optimized Conditions To determine optimized conditions first we check the impact of the timing jitter on precision. Figure 3 shows the functions Q共x兲 and Q共x兲 / G共x兲 versus h / for gate number M = 15 and 255. It shows Q共x兲 becomes smaller as the bin
τ
80
τ/στ, τ/∆τ (dB)
RLD−M
0.3
width h increases. The value of Q共x兲 / G共x兲 is greater than 2.9 over the whole range of h / . Using the photon count data taken from [13] as an example, we consider Nc ⬇ 共1.21⫻ 1011兲ILTM (Sc ⬇ 共1.21⫻ 1011兲ILPAT in [13]) with TM = 20 ms and IL ⬍ 10−5W / cm2 (in linear region), which gives Np Ⰷ 100 共P Ⰶ 0.01兲 with LPR = 50 MHz. Moreover, in most cases, h02 / h2 Ⰶ 1. As a result the first term in Eq. (19) is at least four-hundred-fold larger than the second term. This means the impact of timing jitter is averaged out and is therefore much smaller than that of Poisson noise. Figure 4 shows the precision and accuracy curves versus measurement window 共Mh兲 in for the 15-gate IEM 共M = 15兲 and the 2-gate RLD 共M = 2兲 with Nc = 217. The theoretical results marked as solid curves are compared with Monte Carlo simulations marked with open circles, giving good agreement and proving the correctness of Eq. (19). For IEM, the accuracy gets worse and a DC offset exists, while the precision gets better as the measurement window increases. There exists an optimal point. We define a new precision value as
共23兲
.
0
Fig. 3. (Color online) Q共x兲 and Q共x兲 / G共x兲 versus h / for gate number M = 15 and 255.
Combining Eqs. (4) and (22)–(24), we have
RLD−M
Q(x)
共21兲 10
M−1
j=0
Q(x)/G(x) 0
B. Deriving RLD-M To derive the standard deviation for the RLD-M, we rewrite Eq. (8) as
再兺
1
10
10
h02G共x兲
˜ −1 for fixed measurement time, since Nc = Nt / Np = NtILK −1 ˜ = K TMLPRIL, with TM being the measurement time. For better accuracy, the channel width should be smaller than the lifetime; however, for higher precision, the channel width should be made as large as possible. There exists an optimal trade-off between the accuracy and precision, and it is also clear that at higher light intensity, the timing jitter term of Eq. (20) or Eq. (21) becomes more significant. If the jitter term of Eq. (19) is negligible (as shown later), then the precision is inversely proportional to the square root of laser intensity, / ⬀ I−1/2 L .
g共x兲 = M
M = 255
2
IEM
1/2
2
τIEM/(στIEM+∆τIEM)
τRLD/στRLD 5
10
15
20
Measurement window M×h (τ)
25
Fig. 4. (Color online) Precision and accuracy curves for the 15gate IEM and 2-gate RLD with Nc = 217.
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mum SNR of 46.9 dB. The optimal point for the 2-gate RLD 共M = 2兲 is at h = 2.4 with maximum SNR of 47.6 dB, which has been verified in earlier literature [14]. We use the F value introduced in [15] to quantify the performance of a lifetime imaging technique. The F value is defined as F = N1/2 c / , where is the standard deviation of repeated measurements of the lifetime value . The F value for the IEM is given by 1.6, regardless of the number of time bins, and that for the 2-gate RLD is given by 1.5. For IEM, since the accuracy term in Eq. (14) is a DC offset and can be predicted and so calibrated, a closer value to the real lifetime IEM,Cal is obtained by calculating
冋 冉 冊
IEM,Cal ⬵ IEM 1 −
1
h
180 IEM
4
1 +
冉 冊册 h
1512 IEM
6
. 共28兲
The Monte Carlo and theoretical accuracy curves with the above calibration technique are marked in Fig. 4 with squares and solid lines, respectively. We can choose another optimal measurement window of 25 (marked as point B with h = 25/ 15 = 5 / 3 = 1.67) to obtain a precision of 49.6 dB or F value of 1.2. This means that in a fixed measurement window, the minimum lifetime that can be resolved with calibration is 40% of that without calibration. A measurement window much larger than 25 gives unstable results, because Eq. (14) is divergent. Compared with RLD-2, IEM with calibration provides better precision and much greater insensitivity to the choice of measurement window. Figure 5 shows the precision plot in dB in terms of the measurement window and log2共M兲 for Nc = 217 without calibration. The optimal point occurs at h = 2 / 3 regardless of the value of M. It is clearly shown that the region of SNR larger than 40 dB (or F = 3.6) is getting larger as M increases. In some applications, we would like to know the range of lifetimes that a predictor can resolve when the laser repetition rate or the measurement window is fixed. Fig-
Fig. 5. (Color online) SNR plot in terms of measurement window and log2共M兲 with Nc = 217.
Fig. 6. (Color online) SNR comparison plot for the IEM, RLD-M, and the MLE with Nc = 217.
ure 6 shows such comparison results between the IEM, RLD, and MLE in terms of normalized by measurement window 共Mh兲. The standard deviation of the lifetime using the MLE is provided in [7,16] as
MLE
MLE
= h
冑冦 Nc
2 M
x 共1 − x兲
M x 2
−
M 2
共1 − x 兲
冧
,
x=e
−h/
.
共29兲 This provides the best result, and it has been used in most iteration-based commercial software. Can multigate RLD bring better resolvability? The answer is, No. No matter how large the number of gates is, the result converges to a saturated value shown in the Fig. 6, rapidly beyond M ⬎ 8, and it cannot resolve those with / 共Mh兲 ⬍ 0.1. That is, for a measurement window of 12.5 ns, the 77-gate RLD cannot resolve lifetimes less than 1.25 ns, although it provides the best and closest result to the MLE for a longlifetime region. The optimal performance occurs at = 0.3Mh (F = 1.8 or SNR= 46.0 dB). The results for IEM are different. Denser time bins do provide better resolvability for short lifetimes, although this sacrifices some precision for the long-lifetime region. Figure 7 shows SNR contours in terms of / Mh and M. This clearly shows that a higher resolution TDC provides better resolvability for short lifetimes, although it sacrifices a little precision for the long-lifetime region. The reason for this is the relative shot noise getting larger for each denser time bin, and the subtraction operation in the denominator of Eq. (9). However, this problem can be alleviated by decreasing the size of the measurement window. D. Algorithm for Multi-Exponential Decay The above analysis is based on the assumption that the fluorescence emission is a single-decay function. When trying to extract a multi-decay fluorescence, we need a simple algorithm. Figure 8 shows an algorithm, called the pipelined IEM (PL-IEM), for extracting a bi-exponential
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Table 1. Comparison of Lifetimes (ns) Extracted by the IEM, RLD-M, LSM, and Edinburgh Instruments F900 Software 共ns兲 11-IEM 11-RLD 11-LSM F900
Fig. 7. (Color online) SNR contour plot in terms of / 共Mh兲 and log2共M兲.
Fig. 8.
= 370 nm
= 390 nm
10.35 10.33 10.14 10.3
10.36 10.28 10.17 10.3
was passed through a pulse picker to reduce the repetition rate to 4.75 MHz and then frequency tripled to give an output at 320 nm. The emission from the sample was collected orthogonal to the excitation direction through a polarizer. The fluorescence was passed through a monochromator and detected by a Hamamatsu PMT (R3809U-50). The instrument response was 50 ps FWHM. Fluorescence decay curves were recorded at emission wavelengths of 370/ 390 nm on a timescale of 50 ns, then resolved into 4096 channels to a total 104 counts in the peak channel. The decay curve was analyzed using the 11-gate IEM, 11-gate RLD, 11-gate LSM, and the iteration-based Edinburgh Instruments F900 software. The extracted lifetimes are listed in Table 1. The table shows the extracted lifetimes differ within 1%. Figure 9 shows the logarithmic plot for the measured photon counts starting from the channel with the peak intensity of 104 counts and the fitted data using the proposed IEM. The normalized re-
(Color online) Concept of the pipelined IEM.
fluorescence histogram similar in concept to pipelined analog-to-digital converters. The lifetime-extraction procedure uses the IEM to extract the longer lifetime and intensity with the first memory, and subtract the extracted extrapolation function from the photon counts stored in the second memory to obtain the second lifetime and intensity. Pipelined algorithms for a higher number 共⬎2兲 of decays can follow this procedure until the shortest lifetime has been obtained.
3. EXPERIMENTAL RESULTS A. Single-Exponential Decay The first example is used to test the proposed singleexponential IEM algorithm. It is the decay of 2-aminopurine riboside dissolved in an enzyme buffer solution [50 mM KOAc, 20 mM Tris/HOAc, 10 mM Mg(OAc)2, and 1 mM DTT] measured in an Edinburgh Instruments spectrometer equipped with TCC900 photoncounting electronics. The excitation source was a Ti-Sapphire femtosecond laser system producing pulses of ⬃200 fs at 76 MHz repetition rate. The output of the laser
Fig. 9. (Color online) Measured single-exponential data and fitted and residual data using IEM.
Li et al.
Vol. 25, No. 5 / May 2008 / J. Opt. Soc. Am. A
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ments F900 software, assuming a multi-exponential decay function in the equation 4
I共t兲 =
i
i=1
Fig. 10. (Color online) Measured 4-exponential data and fitted and residual data using PL-IEM.
Table 2. Comparison of Lifetimes (ns) and Fractional Amplitudes (%) Extracted by PL-IEM and Edinburgh Instruments F900 Software i共ns兲 / Ai共%兲 1 / A1 2 / A2 3 / A3 4 / A4
PL-IEM
F900 Software
0.135/ 26.3 0.478/ 56.6 2.225/ 11.4 8.235/ 5.7
0.14/ 47 0.47/ 39 2.19/ 9 8.15/ 5
sidual is defined as R = 共Nr − Nf兲 / N1/2 r , where Nr and Nf are the measured and fitted count number, respectively. The residual plot reveals that the proposed method fits well with the experimental data.
B. Multi-Exponential Decay The second example uses multi-exponential data to test the proposed pipelined IEM algorithm. This example examines the different conformations of the free DNA duplex with the intention of studying the phenomenon of base flipping induced by DNA methyltransferase enzymes [3]. A 14 base-pair, double-stranded DNA sample containing a single adenine analogue, 2-aminopurine (2AP), buffered in 100 mM sodium chloride, 50 mM Tris, and 5 mM EDTA was measured using the same measurement setup as in the first example. The decay curve was analyzed using the proposed PL-IEM and the standard iterationbased deconvolution method from the Edinburgh Instru-
冉 冊
兺 A exp
t
−
i
共30兲
,
where Ai is the fractional amplitude and i is the fluorescence lifetime of the ith decay component. PL-IEM is an example of an automatic curve-fitting algorithm. It is impossible for different algorithms to generate the same residuals while calculating very close lifetimes. Since the residual distribution is sensitive to the extracted lifetimes, the flatness of the normalized residual can indicate if the program has extracted the correct lifetimes. Figure 10 shows the logarithmic plot for the measured photon counts starting from the channel of peak intensity (104 counts) and the fitted data using the proposed PL-IEM. The normalized residual plot reveals that the proposed method fits well with the experimental data. The extracted lifetimes and fractional amplitudes using the PL-IEM and the F900 software are listed in Table 2. The table shows the extracted lifetimes differ by less than 4% whereas the amplitudes differ significantly. The likely reason for such differing amplitudes is that tail curve fitting is performed without doing any deconvolution with the IRF. However, the accuracy of lifetime is acceptable. This is consistent with the claim in recent literature [5,17] suggesting that fluorescence lifetime measurements offer better precision than amplitude measurements of the presence of fluorophores. This table suggests there is potential in applying PL-IEM in on-chip multiexponential lifetime extraction if adaptive gating can also be introduced. The error analysis of the pipelined IEM is not provided in this paper but will be published later.
4. CONCLUSIONS We have proposed a very simple fluorescence-lifetimeimaging method called IEM as Eq. (7) and derived the theoretical lifetime deviation equation for easily obtaining the best recording parameters, such as measurement window, width of a time bin, and bit resolution of the TDC. The method has the potential of using the available photons efficiently, provided that the recording parameters are correctly optimized. For single-exponential lifeTable 3. Comparison Summary of the IEM, RLD, and MLE Method Standard RLD-2 Standard RLD-M 共M Ⰷ 2兲 MLE IEM wÕo Calibration IEM with Calibration
F value @h共兲
/ MW⬍ 0.1 Resolvability
On-Chip Feasibility
1.5 @2.4 ⬃1.8 @共3.3/ M兲
No
YesÕLookUp Table YesÕLookUp Table
⬃1.0 1.6 @0.67 1.2 @1.67
Yes Yes
No Yes
Yes
Yes
No
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Li et al.
time imaging, the method provides almost the same precision level with the available 2-gate RLD with F of 1.6 without any calibration, and offers better precision level with F of 1.2 with calibration. An interesting result of our study is that optimum performance can be obtained at
h=
冦
2 3 5 3
, w/o calibration
冧
3.
4.
,
共31兲
, with calibration
regardless of the bit number of the TDC. This means that for a lifetime of 3 ns, an h of approximately 2 ns (or 5 ns with calibration) is optimal. Table 3 lists the comparison summary for the IEM, the RLD, and the MLE. It clearly shows the merits of the IEM in terms of F value, lifetime resolvability, and on-chip feasibility. For multi-exponential decay, we have also proposed a pipelined IEM method and presented its performance on experimental data from a four-exponential time-constant decay. It provides better precision for the lifetimes than for the fractional amplitudes, as is suggested in the latest literature. We have also demonstrated that the FWHM of the excitation waveform can be quite important in determining the range of lifetimes that are to be extracted.
ACKNOWLEDGMENTS This work has been supported by the European Community within the Sixth Framework Programme of the Information Science Technologies, Future and Emerging Technologies Open MEGAFRAME project (contract 029217-2, www.megaframe.eu). The measurements have been performed using the COSMIC laboratory facilities at the University of Edinburgh. Disclaimer. This publication reflects only the authors’ views. The European Community is not liable for any use that may be made of the information contained herein.
5. 6. 7. 8.
9. 10.
11.
12. 13. 14. 15.
16.
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