Spatial intensity distribution model of fluorescence emission considering the spatial attenuation effect of excitation light

Yuchao Fu; Meizhen Huang; Wanxiang Li

doi:10.1364/OE.416452

1. Introduction

Quantitative analysis of solution by fluorescence is widely used in biochemical analysis, clinical medicine testing, environmental monitoring, food safety and other fields. [1–5] It has the advantages of high sensitivity, good selectivity and simplicity. [4,6,7] In the conventional fluorescence quantitative analysis method, the solution of the fluorescent substance to be measured is usually placed in the cuvette, and the excitation light of a certain wavelength goes into the cuvette, and the fluorescence is collected at a certain angle direction (such as the direction perpendicular to the incident light as shown in Fig. 1(a)). [8] The fluorescence intensity of fluorescent solution F is equal to the product of the light absorbed by the fluorescent substance ${\textrm{I}_\textrm{a}}$ and its fluorescence quantum yield is ${\varphi }$. The light absorbed by the measured solution ${\textrm{I}_\textrm{a}}$ is equal to the difference of the incident light intensity ${\textrm{I}_0}$ and the transmission light intensity ${\textrm{I}_\textrm{t}}$.

Fig. 1. Schematic diagram of element analysis method. (a) The fluorescent solution in the actual cuvette is irradiated by excitation light to produce fluorescence. (b) The horizontal cross section of fluorescence emission in the cuvette.

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According to Lambert-Beer's law: [8]

(1)$$\frac{{{\textrm{I}_\textrm{t}}}}{{{\textrm{I}_0}}} = {\textrm{e}^{ - 2.303{\varepsilon }cx}},$$

Therefore, the relationship between fluorescence intensity and concentration of fluorescent solution in the conventional fluorescence quantitative analysis method can be written as follows: [8]

(2)$$F = \varphi ({{I_0} - {I_t}} )= \varphi {I_0}\left( {1 - \frac{{{I_t}}}{{{I_0}}}} \right) = \varphi {I_0}({1 - {\textrm{e}^{ - 2.303\varepsilon cx}}} ),$$

where, $\varepsilon $ is the molar extinction coefficient, c is the concentration of the fluorescent substance, and x is the optical path of the excitation light in the solution.

In Eq. (2), ${\textrm{e}^{ - 2.303{\varepsilon }cx}}$ can be expanded into Taylor series. When ${\varepsilon }cx \ll 0.05$, the higher-order term of concentration c in Taylor series expansion can be omitted in Eq. (3). [8]

(3)$${\textrm{e}^{ - 2.303{\varepsilon }cx}} = 1 - 2.303{\varepsilon }cx + \frac{{{{({ - 2.303{\varepsilon }cx} )}^2}}}{{2!}} + \frac{{{{({ - 2.303{\varepsilon }cx} )}^3}}}{{3!}} + \ldots ,$$

Then the linear relationship between the fluorescence intensity of the solution and the concentration of the fluorescent substance can be obtained:[8]

(4)$$F = 2.303\varphi {I_0}\varepsilon cx,$$

This is the equation on which the conventional fluorescence quantitative analysis is based. Considering the fluorescent solution in the cuvette, x is the width of the cuvette, which is usually 10 mm, 5 mm or 1 mm. According to the derivation process of Eq. (4), there are two conditions are presupposed: first, the concentration of the solution is not too high, thus the condition of ${\varepsilon }cx \ll 0.05$ is satisfied; second, assuming that when the excitation light passes through the solution, the excitation light is uniformly absorbed by fluorescent substance, and the fluorescence is uniform in all direction. [9–12] Therefore, the fluorescence quantitative analysis method is only applicable to fluorescent solution within a certain narrow concentration range. [10] These two premises seriously limit the application scope of fluorescence quantitative analysis. On the one hand, when the solution of fluorescent substance exceeds a certain concentration range, the higher-order term with respect to c in Taylor series expansion of ${\textrm{e}^{ - 2.303{\varepsilon }cx}}$ cannot be omitted. On the other hand, due to the absorption of fluorescent substance, the excitation light attenuates along the propagation direction, namely the spatial attenuation effect of excitation light, which makes the fluorescence emission non-uniformly distributed in space. Consequently, errors will inevitably occur when the conventional fluorescence quantitative analysis Eq. (4) is applied. [8,13]

The spatial attenuation effect of excitation light, also known as inner filter effect in some researches, was first proposed and considered by Stokes. [14,15] When concentrated quinine was irradiated with ultraviolet light, he found that the blue fluorescence only came from the surface of the solution. [15] Usually, the spatial attenuation effect of excitation light is considered as a source of error in fluorescence quantitative analysis, which greatly limits the application of this effect to on-line measurement and high-throughput analysis. Researchers attempt to compensate the spatial attenuation effect of excitation light to maintain the linear relationship between the fluorescence intensity and the concentration of fluorescent solution. [16–19] In 1977, based on the study of the spatial attenuation effect of excitation light and fluorescence intensity by Parker and Barnes, Holland et al. deduced a formula to modify the fluorescence intensity. And correction equation is verified and analyzed in detail by Yapper and Ingle. The corrected fluorescence intensity can be expressed as follows: [14]

(5)$$\frac{{{F_{ideal}}}}{{{F_{obs}}}} = \frac{{2.3d \cdot Ab{s_{Ex}}}}{{1 - {{10}^{ - d \cdot Ab{s_{Ex}}}}}} \cdot {10^{\textrm{g} \cdot Ab{s_{Em}}}} \cdot \frac{{2.3s \cdot Ab{s_{Em}}}}{{1 - {{10}^{ - \textrm{s} \cdot Ab{s_{Em}}}}}},$$

where ${F_{obs}}$ represents the fluorescence intensity received by the detector. ${F_{ideal}}$ represents the modified fluorescence intensity. $Ab{s_{Ex}}$ represents the absorbance of the excitation light passing through the sample cell. $Ab{s_{Em}}$ represents the absorbance of the emitted fluorescence passing through the sample cell, which is a secondary factor and will not be taken into consideration in the following analysis. s is the diameter of the exciting beam with a uniform cylindrical cross section. $\textrm{g}$ is the distance from the edge of the excitation beam to the edge of the sample cell. $\textrm{d}$ is the width of the sample cell. In 1994, Albinsson et al. also proposed a modifying formula with similar correction effect and being extensively used. [14] The modified fluorescence intensity can be approximately expressed as:

(6)$${F_{ideal}} = {F_{obs}} \times {10^{\frac{{Ab{s_{Ex}} + Ab{s_{Em}}}}{2}}},$$

Due to the spatial attenuation effect of excitation light and fluorescence, there is a degree of deviation between the observed value ${F_{obs}}$ and the ideal value ${F_{ideal}}$. Within the linear range of Lambert-Beer's Law (i.e., under the condition that the absorbance of the detected fluorescent substance can be accurately determined), the above modification method can correct the received fluorescence intensity. [14] However, the relationship between absorbance and concentration in high concentration solution deviates from the linear due to the limitation of Lambert-Beer's law's linear range. [20–22] Hence, the modification of the received fluorescence intensity by absorbance is failed, and the modified fluorescence quantitative analysis method is still not applicable to some on-line quantitative analysis of high-concentration samples. For example, in the textile industry, high concentration of thick fluorescent dyes need to be measured online, for which consumed chemical sensors are used now. [15] And for another example, the total concentration of free amino acids in normal adult plasma is $350\sim 650\; \textrm{mg}/\textrm{L}$ with a periodic difference of about 30% between days and such a large number of plasma samples need to complete a tests at one time. [23,24] When the concentration of free amino acids is detected by conventional fluorescence quantitative analysis, the plasma needs to be diluted to a low concentration, the process of which will change the state of some substances in the plasma. [23,25]

Absorption of excitation light for fluorescence measurements at high concentrations investigated before is focused on how to obtain the ideal fluorescence value by the absorbance parameter, which are either based on the absorbance parameter to compensate the fluorescence value [14], or are committed to exploring the relationship between absorbance and fluorescence to obtain the ideal fluorescence value [15]. But the fatal flaw of these studies is that the absorbance needs to be accurately determined. In the measurement of high concentration solution, due to the accuracy limitation of the sensor itself, it is difficult to accurately determine the absorbance beyond the linear range of Lambert Beer's law. More importantly, a theoretical analysis and numerical simulation on the distribution of excitation intensity and fluorescence intensity is discussed in this paper, which is a new way of thinking. A fluorescence emission model is established based on the interaction and propagation law between the excitation light and the fluorescent substances. Considering the spatial attenuation effect of excitation light, the spatial intensity distribution of fluorescence emission with a wide concentration range are derived and the relationship between the received fluorescence intensity and the concentration of fluorescent substance are also obtained. On this basis, the simulation and experimental verification are carried out. The quantitative analysis method designed in this paper did not directly use the absorbance parameter, but only took it as a threshold. The parameter involved in the quantitative analysis is the received fluorescence intensity. This will greatly expand the scope of quantitative analysis.

2. Theory and model

Considering an actual certain volume of fluorescent solution, for example, the solution in the cuvette, the microelement analysis method is carried out to divide the solution into a series of micro-units, and then the microelement is selected as the concrete object. Considering the interaction between the excitation light and the micro-unit fluorescent solution, the fluorescence emission and its distributing disciplinarian is described in detail. The advantage of dividing the solution into a series of micro-units for analysis is that the light path of the excitation light passing through can be very short, even if the concentration of the fluorescent solution is high, ${\varepsilon }cx \ll 0.05$ is always satisfied, so that the higher-order term with respect to c in Taylor series expansion of ${\textrm{e}^{ - 2.303{\varepsilon }cx}}$ can still be omitted. At the same time, microelement analysis can also accurately analyze the change of spatial attenuation caused by the absorption of excitation light. So the intensity distribution of fluorescence in the whole solution can be strictly calculated. Then, the received fluorescence intensity within a certain scope (the detected fluorescence intensity) is calculated from the intensity distribution of fluorescence, and the relationship between the received fluorescence intensity and the concentration of fluorescent substance is analyzed.

Figure 1 shows the schematic diagram of analysis. The fluorescent solution in the cuvette is irradiated by excitation light to produce fluorescence [Fig. 1(a)]. Assuming the propagation direction of excitation light as $x$-axis, the initial incident excitation light is a parallel Gaussian light source, the intensity of the fluorescence emission at any point $({x,y} )$ on the cross section is $F({x,y} )$, and the total fluorescence on the column $x = k$ is $F(k )$. Figure 1(b) shows the horizontal cross section of fluorescence in the cuvette.

2.1 Theoretical derivation

As shown in Fig. 1(a) and Fig. 1(b), the sample cell is filled with a certain concentration of fluorescent solution. A parallel Gaussian excitation beam passes through the sample cell along the x direction. To simplify the analysis, a fluorescence emission plane $({x,y} )$ is intercepted for analysis and calculation. Considering the excitation wavelength ${\lambda _{Ex}}$ and fluorescence emission wavelength ${\lambda _{Em}}$. It is assumed that the distribution of excitation light intensity in the plane is $E({x,y} )$, and the distribution of fluorescence intensity in the plane is $F({x,y} )$. Given that the intensity of the parallel Gaussian excitation beam before incident into the fluorescence emission plane (at x=0) is $n(y )$ and the fluorescence quantum yield is $\varphi $, the transmission function of the excitation light defined by any microelement $\Delta\textrm{x}\Delta\textrm{y}$ in the fluorescence emission plane is:

(7)$$m({\Delta}{x})= 10^{ - \varepsilon c{\Delta}{x}} = {\textrm{e}^{\zeta c{\Delta}x}},$$

where $\varepsilon $ is the molar extinction coefficient, c is the concentration of fluorescent substance, $\zeta ={-} 2.303\varepsilon $. The unit of light intensity is an arbitrary unit, and the micro-unit of sample cell length is a normalized unit. For each microelement $\Delta\textrm{x}\Delta\textrm{y}$ on the fluorescence emission plane in Fig. 1(b), the fluorescence emission intensity and excitation light transmission intensity can be expressed in combination with the above assumption, and then the fluorescence intensity distribution on the fluorescence emission plane can be analyzed.

Based on the above analysis and assumptions, the following difference equations can be set up:

(8)$$E({x + \varDelta x,y} )= E({x,y} )\cdot m({\varDelta x} ),$$

(9)$$F({x,y} )= \varphi \cdot E({x,y} )\cdot({1 - m({\varDelta x} )} )\; ,$$

(10)$$boundary\; condition:\; \; E({0,y} )= n(y )\; ,$$

The intensity of the excitation light on the boundary is exactly the intensity distribution of the initial parallel Gaussian excitation beam. Taking the limit $\lim \Delta\textrm{x} \to 0$ on the difference equation above, the partial differential equation of the intensity distribution of excitation light can be written as:

(11)$$\frac{{\partial E({x,y} )}}{{\partial x}} = E({x,y} )\cdot m^{\prime}(0 ),$$

$m^{\prime}(0 )$ means the derivative of $m({\mathrm{\Delta }x} )$ at $\Delta {x} = 0.$ Combined with the boundary conditions, the partial differential equation can be solved:

(12)$$E({x,y} )= n(y )\cdot {\textrm{e}^{\zeta cx}},$$

(13)$$F({x,y} )= \varphi \cdot n(y )\cdot {\textrm{e}^{\zeta cx}} \cdot (1 - {\textrm{e}^{\eta c}}),$$

where $\eta = \zeta \mathrm{\Delta }x$, which means the absorption capacity of each unit of fluorescent substance. If the fluorescence receiving area is within the range of $x = a$ to $x = b$, the received fluorescence intensity detected by the detector is as follows:

(14)$${F_D} = \mathop \int \nolimits_a^b \mathop \int \nolimits_{ - \infty }^{ + \infty } F({x,y} )\textrm{d}x\textrm{d}y = \varphi \cdot \mathop \int \nolimits_{ - \infty }^{ + \infty } n(y )\textrm{d}y \cdot \mathop \int \nolimits_a^b {\textrm{e}^{\zeta cx}} \cdot (1 - {\textrm{e}^{\eta c}})\textrm{d}x = \varphi \cdot N \cdot \frac{{({1 - {k^c}} )}}{{\zeta c}} \cdot [{{\textrm{e}^{\zeta cb}} - {\textrm{e}^{\zeta ca}}} ],$$

where k is a constant that characterizes the absorbance capacity of fluorescent substance and N is the excitation light power constant. Thus, the expression of the received fluorescence intensity varying with the concentration of the fluorescent substance can be written under the condition of single wavelength excitation:

(15)$${F_R} = \mathop \int \limits_{{\lambda _{Em}}} \varphi ({{\lambda_{Em}}} )\cdot N \cdot \frac{{({1 - {k^c}} )}}{{\zeta c}} \cdot [{{\textrm{e}^{\zeta cb}} - {\textrm{e}^{\zeta ca}}} ]\textrm{d}{\mathrm{\lambda }_{\textrm{Em}}} = \varPhi \cdot N \cdot \frac{{({1 - {k^c}} )}}{{\zeta c}} \cdot [{{\textrm{e}^{\zeta cb}} - {\textrm{e}^{\zeta ca}}} ],$$

where $\varPhi $ is a constant that characterizes the fluorescence conversion efficiency of fluorescent substance.

Therefore, the fluorescence distribution of a certain volume of fluorescent solution is deduced. And the relationship between the fluorescence intensity observed in a certain range and the concentration of fluorescent substance is expressed as Eq. (15).

Equation (15) can be utilized to establish the model of received fluorescence intensity for specific fluorescent substance. Firstly, the standard gradient solutions of fluorescent substance is used to obtain the data for the received fluorescence intensity and concentration as the training set. Then, the Levenberg-Marquardt algorithm is applied to optimizing the fitting under the estimation criteria of the least squares. [26] Thus, the model of received fluorescence intensity versus concentration can be established. Any fluorescence quantitative analysis in ultra-wide concentration range can apply this analytical expression and determine the constant parameters in the analytical expression by Levenberg-Marquardt optimization fitting algorithm to establish the relationship curve of the received fluorescence intensity versus concentration.

2.2 Algorithm model

When solving the spatial intensity distribution of fluorescence emission, the corresponding difference equation is listed. According to the difference equation, a simulation algorithm model suitable for computer calculation can be designed. The simulation algorithm model can quickly calculate the fluorescence intensity distribution in different concentrations of fluorescent solution, which can be used to compare with fluorescence intensity distribution in experiments. Moreover, the simulation algorithm model also reveals the influence of different fluorescence receiving range on the curve of the received fluorescence intensity versus fluorescence concentration. There are great differences between them. In this paper, the simulation results of the algorithm model will be verified by experiments, which also checks the correctness of the theory from another perspective.

According to the flow chart of algorithm model [Fig. 2(a)], the received fluorescence intensity between $x = a$ to $x = b$ can be calculated, and the relationship curves of the received fluorescence intensity versus the concentration of the fluorescent substance are simulated for the same fluorescent substance under the same excitation conditions. The initial condition of iteration is the excitation light on the first row of microelements, which is Gaussian distributed beam. The amount of excitation light absorption, fluorescence emission and the residual excitation light intensity of each microelement are calculated by iteration. After the iteration, the numerical simulation results of fluorescence intensity distribution can be obtained. The aggregate of received fluorescence intensity detected by the detector is the sum of fluorescence emission in the range from $x = a$ to $x = b$. The algorithm model simulates and calculates the relationship between the received fluorescence intensity versus concentration of different fluorescence receiving range. For instance, for the same fluorescent substance under the same excitation conditions, changing the parameters of the fluorescence receiving range $x = a$ and $x = b$, the relationship curve of the received fluorescence intensity versus the concentration of the fluorescent substance are shown in Fig. 2(b). It can be seen that the relationship between the received fluorescence intensity and the concentration of fluorescent substance is not a simple linear relationship in an ultra-wide range of concentration, not even a positive correlation anymore.

Fig. 2. Algorithm model and the simulation of the fluorescence intensity – concentration relationship. (a) Flow chart of algorithm model for calculating fluorescence intensity. (b)The simulated relationship curve of the received fluorescence intensity between $x = a$ to $x = b$ versus the concentration of the fluorescent substance.

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2.3 Simulation result

The algorithm model based on the theoretical derivation process has been given in Section 2.2. The six figures in Figs. 3(a)–3(f) show the simulation results of the fluorescence intensity distribution of tryptophan concentrations $c = 0.1mg/L$,$c = 3mg/L$,$c = 10mg/L$,$c = 30mg/L$,$c = 60mg/L$,$c = 120mg/L$ respectively under the same excitation light intensity.

Fig. 3. Spatial intensity distribution of fluorescence emission by simulation. (a) ∼ (f) The simulation algorithm results of the fluorescence intensity distribution of tryptophan concentrations $\textrm{c} = 0.1\textrm{mg}/\textrm{L}$, $\textrm{c} = 3\textrm{mg}/\textrm{L}$, $\textrm{c} = 10\textrm{mg}/\textrm{L}$, $\textrm{c} = 30\textrm{mg}/\textrm{L}$, $\textrm{c} = 60\textrm{mg}/\textrm{L}$, $\textrm{c} = 120\textrm{mg}/\textrm{L}$ respectively under the same excitation light intensity.

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3. Experiment and results

3.1 Experimental setup and method

Tryptophan is selected as the fluorescent substance to carry out the verification experiment. The optical configuration of fluorescence measurement system is shown in Fig. 4(a). The excitation light source is a deep ultraviolet LED with 280 nm center wavelength and $2\; mW$ power. Previous studies have drawn out that fluorescence emission wavelength range of tryptophan is about $340 \pm 40$ nm [27]. The LED light source is collimated by a convex lens with a $\phi = 5$ mm diameter circular aperture and a parallel Gaussian beam is generated as the excitation beam.

Fig. 4. Experimental hardware and data processing. (a) The optical configuration of fluorescence measurement system. (b) Spectral processing method. (c) The measured spectra of different concentrations of fluorescent solution.

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A quartz cuvette with a cross section side length $\textrm{L} = 10$ mm is placed in the measuring chamber to hold the solution of tryptophan. The concave mirror A is placed on the back side of the cuvette to reflect and focus the transmission light to the collection fiber. And the concave mirror B is placed on the left side of the cuvette, which reflects and focuses the fluorescence on the other side of the collection fiber then to the detector, so as to improve the efficiency of fluorescence reception. The absorbance of tryptophan solution is calculated by this transmission light. Two convex lenses and a band-pass filter are placed on the fluorescence receiving side to obtain the focused $340 \pm 40\,\textrm{nm}$ tryptophan fluorescence and filter the stray light. The fluorescence and residual excitation light received by the optical fiber probe are transmitted into the UV-Visible spectrometer (QE65000, Ocean Optics Company). The spectral resolution of the spectrometer is 0.8 nm, the spectral range is 200-900 nm, and the integration time for measurement is 100 ms.

Forty-three groups of tryptophan standard solutions with concentration of $0.02 - 250\,\textrm{mg}/\textrm{L}$ are measured, 60 samples in each group. The variation of the received fluorescence intensity measured from these concentration gradient solutions can be fitted by the Eq. (15) derived from Section 2.1. For the same excitation light source, the same fluorescence receiving range and the same CCD integration time, the analytical expression given in the Eq. (15) has a good characterization ability for the experimental data. In the concentration range of $0.02 - 250\,\textrm{mg}/\textrm{L}$, the Levenberg Marquardt algorithm is utilized to optimize the fitting under the 95% confidence limit. The maximum number of iterations is 1000 rounds.

The original spectrum obtained from the test needs to be processed by the flow chart shown in Fig. 4(b). And the increasing concentration of tryptophan is shown in the processed spectra from the front to the back [Fig. 4(c)]. The figure does not show all the spectra, but selects several spectra with different tryptophan concentration as samples to display. As can be seen from the figure, with the increase of tryptophan concentration, the residual excitation intensity at 280 nm wavelength gradually decreases until almost completely disappears, while the received fluorescence intensity increases at first and then decreases. This change can be fitted by the analytical expression derived from the theoretical part.

3.2 Results and discussions

Figures 5(a)–5(f) show the experimental results of the fluorescence intensity distribution of tryptophan concentrations $c = 0.1\; mg/L$, $c = 3\; mg/L$, $c = 10\; mg/L$, $c = 30\; mg/L$, $c = 60\; mg/L$, $c = 120\; mg/L$ respectively under the same excitation light intensity. The photo of cuvette is the original camera photo without any image processing. The results are consistent with the simulation results in Figs. 3(a)–3(f).

Fig. 5. Spatial intensity distribution of fluorescence emission by experiment. (a) ∼ (f) The experimental results of the fluorescence intensity distribution of tryptophan concentrations $\textrm{c} = 0.1\,\textrm{mg}/\textrm{L}$,$\textrm{c} = 3\,\textrm{mg}/\textrm{L}$, $\textrm{c} = 10\,\textrm{mg}/\textrm{L}$, $\textrm{c} = 30\,\textrm{mg}/\textrm{L}$, $\textrm{c} = 60\,\textrm{mg}/\textrm{L}$, $\textrm{c} = 120\,\textrm{mg}/\textrm{L}$ respectively under the same excitation light intensity.

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When tryptophan concentration is $\textrm{c} = 0.1\,\textrm{mg}/\textrm{L}$, the fluorescence excited by the solution is relatively weak. When the tryptophan concentration is $\textrm{c} = 3\,\textrm{mg}/\textrm{L}$ and $\textrm{c} = 10\,\textrm{mg}/\textrm{L}$, the concentration is still in the linear range of absorbance - concentration described by Lambert-Beer's law. The spatial attenuation effect of excitation light is not significant, and the fluorescence intensity overall distribution is a bright passband. The higher the tryptophan concentration is, the wider and brighter the bright passband is. When tryptophan concentration is $\textrm{c} = 30\,\textrm{mg}/\textrm{L}$, the absorbance-concentration relationship gradually go beyond the linear range described by Lambert Beer's law, and the spatial attenuation effect of excitation light begins to show gradually. It can be observed that the fluorescence intensity at the front end of the sample cell is obviously stronger than that at the back end of the sample cell. When the tryptophan concentration is $\textrm{c} = 60\,\textrm{mg}/\textrm{L}$, the sample absorbance increase is no longer obvious with the increase of tryptophan concentration, and the relationship between absorbance and concentration is completely beyond the linear range described by Lambert Beer's law. Due to the existence of spatial attenuation effect of excitation light, the distribution of fluorescence intensity has obvious layered structure with the propagation direction of excitation light, also due to the Gaussian distribution of excitation light intensity, the fluorescence distribution of the same intensity appears a triangle shape. When the tryptophan concentration is $\textrm{c} = 120\,\textrm{mg}/\textrm{L}$, with the further increase of tryptophan concentration, the absorbance of the sample almost does not change, and the spatial attenuation effect of excitation light is very significant. The layered structure of fluorescence intensity distribution becomes more and more obvious with the propagation direction of excitation light, and the interval between layers becomes shorter, and the fluorescence is mainly concentrated in the front end of the sample cell. The fluorescence distribution of each concentration is consistent with the fluorescence intensity distribution calculated and analyzed above.

This can be explained intuitively why the analytical expression fitted by the standard solution exists phenomenon that, with the increase of tryptophan concentration, the received fluorescence intensity first increases and then decreases:

i. Limited by the linear range of Lambert-Beer's law, when the concentration is higher than a certain range, the absorbance of the solution is no longer proportional to the concentration of the solution;
ii. The spatial attenuation effect of excitation light in high concentration solution becomes significant, and the fluorescence intensity distribution presents a layered structure phenomenon;
iii. The fluorescence intensity distribution has the property of spatial distribution, but the receiving range of fluorescence does not completely cover the entire distribution range of fluorescence intensity.

Figure 6 shows the experimental results that the received fluorescence intensity varies with the change of the concentration of fluorescent substance in solution when changing the fluorescence reception range. The reception range of fluorescence can be changed by precisely adjusting the position of fiber probe. Changing the fluorescence reception range will change the concentration-fluorescence peak position, and also change the overall received fluorescence intensity. Furthermore, the linear range of the received fluorescence intensity with respect to the concentration of fluorescent substance will also change. The smaller the reception range of the x is, the stronger the fluorescence intensity is, and the more backward the position of the concentration-fluorescence peak is. The results are consistent with the simulation results in Fig. 2.

Fig. 6. The results of the relationship between received fluorescence intensity and fluorescent substance concentration by changing the fluorescence receiving range.

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Using the model between received fluorescence intensity and concentration with determined parameters, the solution with unknown concentration is analyzed as test set. However, there are two prediction results of tryptophan concentration under a certain received fluorescence intensity. To solve this problem, the following segmented fluorescence quantitative analysis is proposed. As shown in the Fig. 7(a), the concentration corresponding to the maximum received fluorescence intensity is taken as the concentration threshold, and the corresponding absorbance is the absorbance threshold. If the absorbance of the measured solution is less than the absorbance threshold, the concentration of tryptophan can be predicted by the fluorescence intensity segment which is less than the concentration threshold; if the measured absorbance is greater than the absorbance threshold, the concentration of tryptophan can be predicted by the fluorescence intensity segment which is greater than the concentration threshold. This concentration prediction method allows the interference substance in the solution to have a certain influence on the absorbance, because the absorbance value is only used as the threshold for judgment, and its value is not directly related to the predicted results.

Fig. 7. The illustration and result of segmented fluorescence quantitative analysis method. (a) The segmented fluorescence quantitative analysis method. (b) ∼ (d) The result of segmented fluorescence quantitative analysis method compared with conventional method.

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Figure 7(b) shows the received fluorescence intensity - tryptophan concentration data points in the test set as compared to the curve of the received fluorescence intensity - tryptophan concentration with determined parameters. Combined with the above method for predicting tryptophan concentration, the prediction of tryptophan concentration can be obtained as shown in Fig. 7(c). Among them, the black line represents the results of the concentration prediction method proposed in this paper. The red line represents the prediction within the linear range of Lambert-Beer's law by ultraviolet spectrophotometry, and the blue line represents the prediction obtained by conventional fluorescence quantitative analysis method which satisfies the fluorescence - concentration linear law. Figure 7(d) shows the relative error of the prediction obtained by quantitative analysis method based on the analysis of fluorescence intensity distribution, ultraviolet spectrophotometry and conventional fluorescence quantitative analysis under each tryptophan concentration. The goodness of fit of ${\textrm{R}^2}$ and root mean square relative error of $\textrm{RRMSE}$ of various prediction methods are shown in the Table 1 below.

Table 1. Comparison of the absorbance quantitative analysis method, conventional fluorescence quantitative analysis method, and segmented fluorescence quantitative analysis method.

View Table

4. Conclusion

To summarize, we have constructed the fluorescence emission model based on the microelement analysis method. Considering the spatial attenuation effect of excitation light, the spatial intensity distribution of fluorescence emission of an actual certain volume of fluorescent substance in a wide range concentration are revealed and the relationship between received fluorescence intensity and fluorescent substance concentration is theoretical derived and calculated. As an example, based on this model, an ultra-wide concentration range ($0.02 - 250\; \textrm{mg}/\textrm{L}$) detection of tryptophan with high accuracy (${\textrm{R}^2} = 0.9994$, $\textrm{RRMSE} = 0.0356$) is realized. It greatly broadens the scope of conventional fluorescence quantitative analysis and its applications.

Funding

Key Research and Development Program of Guangdong Province, China (No. 2020B1111020001); National Natural Science Foundation of China (No. 61775133).

Acknowledgments

This work was supported by grants from the Key research and development program of Guangdong Province (No. 2020B1111020001) and the National Natural Science Foundation of China (No. 61775133).

Disclosures

The authors declare no conflicts of interests.

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Quantitative analysis method	Analysis range (mg/L)	R²	RRMSE
Absorbance analysis method	5∼90	0.9909	0.3556
Conventional quantitative analysis method	0.02∼17.5	0.9968	0.1691
Segmented fluorescence quantitative method	0.02∼250	0.9994	0.0356

Spatial intensity distribution model of fluorescence emission considering the spatial attenuation effect of excitation light

Abstract

1. Introduction

2. Theory and model

2.1 Theoretical derivation

2.2 Algorithm model

2.3 Simulation result

3. Experiment and results

3.1 Experimental setup and method

3.2 Results and discussions

4. Conclusion

Funding

Acknowledgments

Disclosures

References

Cited By

Figures (7)

Tables (1)

Equations (15)

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