*2.2.2 Photon absorption and fluorescence phenomena*

Photons that propagate inside a medium can be absorbed by the molecules in the sample.

*Absorption:* During absorption spectroscopy, an incident photon can be absorbed by a molecule, which leads to the photon energy being converted into an excitation of that molecule's electron cloud. This interaction is sensitive to the internal structure of the molecule since the laws of quantum mechanics only allow for the existence of a limited number of excited states of the electron cloud of any given chemical species. Each of these excited states has a defined energy, the absorption of the photon has to bridge the energy gap between the ground state (lowest energy state) and an allowed excited state of the electron cloud. As a consequence, molecules can be identified by their absorption spectrum, their wavelength-dependent capacity for absorbing photons depends on the energy spacing of the states of their electron cloud. If the frequency of the radiation matches the vibration frequency of the molecule, then radiation will be absorbed, causing a change in the amplitude of molecular vibration [41]. Molecules, which strongly absorb Vis light, appear colored to the human eye and are, therefore, called chromophores, that is "carriers of color".

*Fluorescence:* In absorption, the signal of molecules in a sample is direct but it can be done with higher sensitivity by using an indirect approach, fluorescence detection. Then the ingoing light will give the absorbing molecules excited states (higher energy) of their electron cloud as described above. From this state, the molecule can shift, "relax", to the electronic ground state by transforming the excess energy into an outgoing emitted light having longer wavelengths than the ingoing light. Different

molecules have different emitted spectrums, which being applied during the fluorescence measurements [42].

#### *2.2.3 Beer-Lambert law*

The Beer–Lambert law states that the absorbance of light intensity is proportional to the concentration of the substance. This means that the amount of, for example, UV-light absorbed when passing through a cuvette (manufactured by UV-transparent material such as quartz) with spent dialysate, **Figure 2**, is linearly dependent on the concentration c [mol/L] of the absorbing solute, the optical pathlength in (l) [m] (depth of the cuvette) and the extinction coefficient ε [m�<sup>1</sup> (mol/L)�<sup>1</sup> ], even called the molar absorptivity at a certain wavelength [43].

If I0 is the intensity of the incident light and I is the intensity transmitted light through the medium, the absorbance (A), dimensionless, is Eq. (1):

$$\mathbf{A} = \log\_{10} \left[ \frac{\mathbf{I}\_0}{\mathbf{I}} \right] = \mathbf{e} \cdot \mathbf{c} \cdot \mathbf{l} \tag{1}$$

If ε is known for a substance, the absorbance (A) can be calculated by multiplying the path length and the concentration of the substance. If ε is known for a substance and A is obtained from a measurement, it is possible to derive the concentration as Eq. (2):

$$\mathbf{c} = \frac{\mathbf{A}}{\mathbf{e} \cdot \mathbf{l}} \tag{2}$$

In our case, when the spent dialysate contains several different absorbing compounds, the overall extinction coefficient is the linear sum of the contributions of each compound. However, all the components are not identified and probably there is interference between different substances, which makes it difficult to separate and determine the concentrations of each solute. Absorbance of a solution, obtained by a double beam spectrophotometer, is given by the Lambert–Beer law as [44, 45] Eq. (3):

$$\mathbf{A} = \log \frac{\mathbf{I}\_0}{\mathbf{I}\_{\mathbf{r}+\mathbf{s}}} - \log \frac{\mathbf{I}\_0}{\mathbf{I}\_{\mathbf{r}}} = \log \frac{\mathbf{I}\_{\mathbf{r}}}{\mathbf{I}\_{\mathbf{r}+\mathbf{s}}} \tag{3}$$

where I0 is the intensity of incident light from the light source, Ir is the intensity of transmitted light through the reference solution (e.g. pure dialysate) and Ir+s is the

**Figure 2.**

*Cuvette with a sample containing an absorber with concentration, c, ingoing light, I0, and outgoing light, I, after absorption in the sample when passing through the cuvette with the optical path length, l.*

summated intensity of transmitted light through the reference solution mixed with the solution (e.g. pure dialysate + waste products from the blood). The common assumptions to utilize absorbance calculated according to the Beer–Lambert law to determine concentration are: (1) the radiation is monochromatic [34, 43, 46, 47], (2) the irradiating beam is parallel (collimated) across the sample [34, 43], (3) the absorption of radiation for a given species is independent of that of other species [34, 43], (4) only the non-scattered and not-absorbed photons are detected from the medium [40], and (5) the incident radiation and the concentration of the chromophores are not extremely high [46, 48]. The Beer–Lambert law for monochromatic light can be derived by solving a differential equation for a solution with a finite depth containing chromophores and is given in detail in many sources [34, 37, 46]. Analysis of a mixture is based fundamentally on the fact that the absorptions, at each wavelength, of separate components in the mixture are additive, provided that chemical or interfering physical reactions between the components do not occur [43] and the solute concentrations are not very high (not usually found in biological media). In this case, in a medium containing n different absorbing compounds with the concentrations of c1 … cn [mol/L] and the extinction coefficients of ε1… ε<sup>n</sup> [m�<sup>1</sup> (mol/L)�<sup>1</sup> ], the overall extinction coefficient is simply the linear sum of the contributions of each compound (Eq. (4)):

$$\mathbf{A} = \log\_{10} \begin{bmatrix} \mathbf{I}\_0 \\ \mathbf{I} \end{bmatrix} = (\mathbf{e}\_1 \mathbf{c}\_1 + \mathbf{e}\_2 \mathbf{c}\_2 + \dots + \mathbf{e}\_n \mathbf{c}\_n) \mathbf{l} \tag{4}$$

#### *2.2.4 Transmittance and absorbance*

The amount of transmitted and absorbed portion of EM radiation in the medium (e.g. in a dialysate sample) can be characterized by the parameter's transmittance T and absorbance A in the spectrophotometer. In order to utilize EM radiation for measurement of constituents in a fluid, the sample is applied in an optical cuvette, **Figure 2**. Through the calibration and measurement procedures, one determines the amount of the ingoing light illuminating the sample symbolized by I0 and the outgoing light I symbolizing the intensity of the light after passing the sample as the remaining ingoing light is partly absorbed by the sample, **Figure 2**. Having knowledge about those parameters, one can determine transmittance T (Eq. (5)), and absorbance A (Eq. (6)) as:

$$\mathbf{T} = \frac{\mathbf{I}}{\mathbf{I}\_0} \Rightarrow \mathfrak{W}\mathbf{T} = \frac{\mathbf{I}}{\mathbf{I}\_0} \cdot \mathbf{100\%}\tag{5}$$

$$\mathbf{A} = -\log \mathbf{T} = -\log \frac{\mathbf{I}}{\mathbf{I}\_0} = \log \frac{\mathbf{I}\_0}{\mathbf{I}} \tag{6}$$
