6.1. Studying speckle dynamics in the image plane of a cultured cell monolayer

We studied the features of speckle dynamics caused by an activity of cultured cells on L41 cells discussed above in Section 4.3. After the formation of a monolayer, the substrate with the cells was placed into an optical cuvette filled with nutrient solution. A cell-free substrate of similar thickness was placed near. Next, the cuvette was fixated on the optical device shown in Figure 5 that was placed in liquid thermostat ЗЦ-1125М. The typical magnification by the optical system was 0.25, and the typical linear resolution of the lens was 60 μm.

Figure 7 presents a typical speckle pattern recorded in the image plane of the cuvette with substrates.

After equalization of the temperatures of the cuvette and the thermostat the mechanical system was checked for stability. To do so, we used the above software, selected the segment of the frame near the image of the cuvettes. Using two speckle images of the segment recorded in 1 or 2 min, we determined the value of correlation coefficient η of their digital images. If value η equalled 0.99 or 0.98, the system was considered mechanically stable. Further, we recorded the film of the speckle dynamics lasting 20–40 s with 25 Hz frequency. Using the software discussed in Section 5.3, we obtained dependences of the speckle image fragment correlation coefficient η on the time using the frames of the film. In Figure 7, typical selected fragments are shown by white frames.

Figure 7. Typical speckle image of a cuvette with transparent substrates: the substrate with cells is on the left, and the cellfree one is on the right. The selected image fragments are denoted by numbers.

Figure 8 shows typical dependences of ηðτÞ for the cells in the nutrient solution (dependence 1) and for the nutrient solution (dependence 2) obtained after processing of the film. The dotted lines show theoretical dependences obtained using Eq. (27) for normalized Lorentzian function k12ðτÞ. Analysis of the experimental dependences obtained in different segments of the substrate with cells within its image showed that the deviation of the theory from the experiment was in the range of 8–11%.

As seen from the graphs in Figure 8, in about 5 s dependences ηðτÞ level off. The mean square deviation of the last four points from their mean value does not exceed 1%. In compliance with the theory discussed above in Section 3, levelling off dependence ηðτÞ means that random process Δu ¼ ΔuðtÞ is stationary in time. Value Δu is typical (mean) optical wave path difference within the region with cells of 60-μm diameter. As fragments of about 1-mm size correspond to the selected segments of the speckle images in the object plane (Figure 7), random process Δu ¼ ΔuðtÞ can also be regarded as homogeneous in this fragment.

As was pointed out in Section 3, by values of variable η ¼ η� we can determine the corresponding dispersions of the wave pair optical paths σ<sup>2</sup> <sup>1</sup> and σ<sup>2</sup> <sup>2</sup> in the horizontal segment of dependences 1 and 2. Supposing that optical wave path variations in the cells and in the nutrient solution are uncorrelated, we can show that the optical path dispersion in cells can be determined using Eq. (36):

$$
\sigma\_u^2 = \sigma\_1^2 \text{--} \sigma\_2^2.\tag{36}
$$

Mean square deviation σ<sup>u</sup> of values Δu obtained by Eq. (36) came to 14 nm.

Figure 8. Joint dependences of η on the time for the cells in the nutrient solution (1) and for the nutrient solution (2).

The homogeneity and stationarity of process Δu ¼ ΔuðtÞ in a segment of a 1-mm order can be explained by the fact that in a monolayer, the cells are closely packed, so there is no variation of their shape due to translation and division. In these conditions local deviations of the medium refraction index from its mean value are possible at the structural level. As was discussed above in Section 3, chemical reactions and phenomena of mass transfer can be the reasons for refraction index variation.

### 6.2. Defrosted cells and speckle dynamics

frame near the image of the cuvettes. Using two speckle images of the segment recorded in 1 or 2 min, we determined the value of correlation coefficient η of their digital images. If value η equalled 0.99 or 0.98, the system was considered mechanically stable. Further, we recorded the film of the speckle dynamics lasting 20–40 s with 25 Hz frequency. Using the software discussed in Section 5.3, we obtained dependences of the speckle image fragment correlation coefficient η on the time using the frames of the film. In Figure 7, typical selected fragments are

Figure 8 shows typical dependences of ηðτÞ for the cells in the nutrient solution (dependence 1) and for the nutrient solution (dependence 2) obtained after processing of the film. The dotted lines show theoretical dependences obtained using Eq. (27) for normalized Lorentzian function k12ðτÞ. Analysis of the experimental dependences obtained in different segments of the substrate with cells within its image showed that the deviation of the theory from the experiment

Figure 7. Typical speckle image of a cuvette with transparent substrates: the substrate with cells is on the left, and the cell-

As seen from the graphs in Figure 8, in about 5 s dependences ηðτÞ level off. The mean square deviation of the last four points from their mean value does not exceed 1%. In compliance with the theory discussed above in Section 3, levelling off dependence ηðτÞ means that random process Δu ¼ ΔuðtÞ is stationary in time. Value Δu is typical (mean) optical wave path difference within the region with cells of 60-μm diameter. As fragments of about 1-mm size correspond to the selected segments of the speckle images in the object plane (Figure 7), random process Δu ¼ ΔuðtÞ can also be regarded as homogeneous in this

As was pointed out in Section 3, by values of variable η ¼ η� we can determine the

of dependences 1 and 2. Supposing that optical wave path variations in the cells and in the

<sup>1</sup> and σ<sup>2</sup>

<sup>2</sup> in the horizontal segment

corresponding dispersions of the wave pair optical paths σ<sup>2</sup>

free one is on the right. The selected image fragments are denoted by numbers.

shown by white frames.

124 Optical Interferometry

was in the range of 8–11%.

fragment.

We conducted an experiment with L41 cells precipitated onto a transparent substrate immediately after defrosting. The interest in similar experiment was caused by the fact that, as distinct from a monolayer of cultured cells, after defrosting the cells do not attach to the substrate immediately, being in motion. When a cell is moved for a distance comparable to its size, the sounding wave pair path difference can vary by a value comparable to wavelength of radiation λ and exceeding it. Therefore, in compliance with Eq. (26) variation of speckle image fragment correlation coefficient can be caused both by the cosine argument variation and by variation of the values in the exponent. As a cosine can be both positive and negative, appearance of negative values of the variable close to 1 would speak for correctness of Eq. (26) and our theory.

Besides, the objective of the conducted research was to study the possibility to apply the technique at large magnifications to analyze the processes occurring in different parts of one cell. The experiments were conducted on a laboratory device with a horizontal position of the substrate with cells. When the temperature reached the value near 36°С a glass 0.1-mm thick was placed into a cuvette with nutrient solution, and poured frozen cells from Dewar vessel onto it. After the temperature stabilization in 30–60 min, recording of the speckle dynamics film was started. The optical magnification was about ×8, the exposure time equaled 9 s, and the frames were recorded for several hours.

A typical speckle pattern of cells precipitated on a substrate is shown in Figure 9. Viewing of the films showed that the cells contact other cells being in continuous random motion. The typical shift of a cell in one direction was comparable to its dimensions. There were cells making shifts for a larger distance, and there were also cells that could be visually regarded as stationary.

Figure 9. Speckle image of defrosted cells.

Figures 10–13 demonstrate typical dependences η ¼ ηðτÞ obtained for different sizes of speckle image segments. The graph in Figure 10 corresponded to a 4 × 4-pixel segment or the cell fragment size of about 4 × 4 μm. Originally, the segment was in the center of the cell image. The rest of the dependences were obtained by means of data averaging in segments containing from 4 to 200 cells.

As seen from Figure 10, for a randomly moving cell value η randomly varies in the range from -1 to +1 around zero. The obtained result qualitatively confirms correctness of Eq. (26) containing dependence of value η on the optical wave path difference by the cosine law. Variation of η from 1 to -1 means that the positive image has changed to a negative one, or vice versa. This is possible, for example, if in all the 16 pixels radiation intensity varies by the cosine law with the same period (the same Δu) but with the different initial phase. In Figure 10, value η does not reach +1 and -1 again. It points out that either during the cell motion its shape changing in a 4×4-μm segment is inhomogeneous, or value η varies not only by the cosine law.

If the data averaging region covers plenty of cells (see Figures 11–13), values of variable η are positive and reach a horizontal segment in about 0.5 h. The obtained data can be explained by the fact that at a fixed time point values х included in the cosine argument in Eq. (26) can reach large values with different signs in different cells. But in averaging by a large count of cells (the objects of the ensemble) variables 〈x1〉 and 〈x2〉 have values close to zero. Then in Eqs. (26) and (31) the dependence on the cosine disappears, but the dependence on k<sup>22</sup> remains.

Figure 10. Dependence η ¼ ηðtÞ for fragment inside the cell image.

The experiments were conducted on a laboratory device with a horizontal position of the substrate with cells. When the temperature reached the value near 36°С a glass 0.1-mm thick was placed into a cuvette with nutrient solution, and poured frozen cells from Dewar vessel onto it. After the temperature stabilization in 30–60 min, recording of the speckle dynamics film was started. The optical magnification was about ×8, the exposure time equaled 9 s, and

A typical speckle pattern of cells precipitated on a substrate is shown in Figure 9. Viewing of the films showed that the cells contact other cells being in continuous random motion. The typical shift of a cell in one direction was comparable to its dimensions. There were cells making shifts for a larger distance, and there were also cells that could be visually regarded as stationary.

Figures 10–13 demonstrate typical dependences η ¼ ηðτÞ obtained for different sizes of speckle image segments. The graph in Figure 10 corresponded to a 4 × 4-pixel segment or the cell fragment size of about 4 × 4 μm. Originally, the segment was in the center of the cell image. The rest of the dependences were obtained by means of data averaging in segments containing

As seen from Figure 10, for a randomly moving cell value η randomly varies in the range from -1 to +1 around zero. The obtained result qualitatively confirms correctness of Eq. (26) containing dependence of value η on the optical wave path difference by the cosine law. Variation of η from 1 to -1 means that the positive image has changed to a negative one, or vice versa. This is possible, for example, if in all the 16 pixels radiation intensity varies by the cosine law with the same period (the same Δu) but with the different initial phase. In Figure 10, value η does not reach +1 and -1 again. It points out that either during the cell motion its shape changing in a 4×4-μm segment is inhomogeneous, or value η varies not only by the cosine law. If the data averaging region covers plenty of cells (see Figures 11–13), values of variable η are positive and reach a horizontal segment in about 0.5 h. The obtained data can be explained by the fact that at a fixed time point values х included in the cosine argument in Eq. (26) can reach large values with different signs in different cells. But in averaging by a large count of cells (the

the frames were recorded for several hours.

126 Optical Interferometry

from 4 to 200 cells.

Figure 9. Speckle image of defrosted cells.

Figure 11. Dependence η ¼ ηðtÞ corresponding to four cells.

Dependences ηðtÞ shown in Figures 11–13 can be interpreted in two ways. Supposing that the 9-s speckle averaging time exceeds the correlation time of the fastest processes in the cells, one can suppose that by Eq. (31) dependence ηðtÞ corresponds to an unstable process wherein value k<sup>22</sup> first increases continuously and then levels off. On the other hand, we can suppose that graphs of ηðtÞ correspond to stationary process ΔuðtÞ with the correlation time about 30 min.

To clear up this matter, we selected different segments of the view-field containing about 100 cells. For each of these segments, four dependences ηðtÞ lasting about 1 h were built for different time intervals. The form of dependences was well reproducible, the multiple correlation coefficients of the four masses was in the range from 0.86 to 0.96. We came to the conclusion that in fragments containing hundreds of defrosted cells random processes ΔuðtÞ can be regarded as homogeneous in space and stationary in time for several hours.

Figure 12. Dependence η ¼ ηðtÞ corresponding to 60 cells.

Figure 13. Dependence η ¼ ηðtÞ corresponding to 200 cells.

### 6.3. Application of speckle dynamics for studying the reaction of a cell mass and fragments of individual cells to temperature variations

The theoretically detected and experimentally confirmed relation between the correlation coefficient of speckle images η and the dispersion of the optical wave pair path difference σ<sup>2</sup> u was immediately used in our first experiments studying dependence of σ<sup>u</sup> on temperature Т. The details of the experiment can be found in Ref. [33]. A segment of L41 cell monolayer containing hundreds of cells was the averaging region. Value σ<sup>2</sup> <sup>u</sup> corresponding to the cells was obtained as the difference of values σ<sup>2</sup> <sup>u</sup> corresponding to the cells in the nutrient solution and to the nutrient solution. The cuvette with cells was first heated to a temperature around 40°C. Then the heating was stopped, and when the cuvette cooled to room temperature, speckle dynamics films lasting 20–40 s were recorded with 25-Hz frame rate. The optical system presented in Figure 5 was used. Figure 14 presents dependence of σ<sup>u</sup> on temperature T obtained experimentally. As seen from the given graph, an approximately linear relation between σ<sup>u</sup> and T is found.

Figure 14. Dependence of σ<sup>u</sup> on T for L41 cells.

6.3. Application of speckle dynamics for studying the reaction of a cell mass and fragments

The theoretically detected and experimentally confirmed relation between the correlation coefficient of speckle images η and the dispersion of the optical wave pair path difference σ<sup>2</sup>

was immediately used in our first experiments studying dependence of σ<sup>u</sup> on temperature Т. The details of the experiment can be found in Ref. [33]. A segment of L41 cell monolayer

and to the nutrient solution. The cuvette with cells was first heated to a temperature around 40°C. Then the heating was stopped, and when the cuvette cooled to room temperature, speckle dynamics films lasting 20–40 s were recorded with 25-Hz frame rate. The optical system presented in Figure 5 was used. Figure 14 presents dependence of σ<sup>u</sup> on temperature T obtained experimentally. As seen from the given graph, an approximately linear relation

u

<sup>u</sup> corresponding to the cells

<sup>u</sup> corresponding to the cells in the nutrient solution

of individual cells to temperature variations

Figure 13. Dependence η ¼ ηðtÞ corresponding to 200 cells.

Figure 12. Dependence η ¼ ηðtÞ corresponding to 60 cells.

128 Optical Interferometry

was obtained as the difference of values σ<sup>2</sup>

between σ<sup>u</sup> and T is found.

containing hundreds of cells was the averaging region. Value σ<sup>2</sup>

The purpose of the next experiment was studying the reaction of small fragments inside a cell and a small group of cells to temperature variation. As distinct from the previous experiment, the frames were recorded continuously with heating of the thermostat from the room temperature to 43°С in about 2 h. The averaging time (the frame exposure time) equaled 9 s. We used an air bath of ТСЭ-200 type and an optical system with an upright position of the substrate. An L41 cell monolayer was the research objective. For the experiment, we selected a segment that contained at least a small cell-free area (Figure 15) in the view-field. The magnification was ×8, and about 10–30 pixels of the TV camera matrix fell onto an image of an individual cell.

Figure 15. Photographs of cells in white light. A cell-free fragment is visible in the lower part of the frame.

Next in Figure 16, there is distribution of value η obtained at the temperature of 30°С by 2 speckle cell images shown in Figure 15. The time interval between the frames equaled 18 s. Values η were found in segments of 10 × 10-pixel size.

Figure 16. Distribution of value η obtained by two speckle images.

Figures 17, 18 and 19 show typical dependence σ<sup>u</sup> on temperature and joint dependences of σ<sup>u</sup> and temperature Т on time. Value σ<sup>u</sup> was obtained using two dependences ηðtÞ corresponding to the cells in the nutrient solution and to the nutrient solution. Segments containing 60 cells (Figure 18) and small regions inside the cells were averaging regions (Figure 17). As is seen from the pictures, considerable fluctuations of σ<sup>u</sup> are observed with temperature increase. Fluctuations of value σ<sup>u</sup> differ from one cell part to another. If the temperature gets stabilized in 30 min, σ<sup>u</sup> also stabilizes (Figure 19). The correlation coefficient of masses σ<sup>u</sup> and Т shown in Figure 19 equals 0.88.

Figure 17. Dependences of σ<sup>u</sup> on Т for three segments inside one cell. Red colour – cell edge, gray colour – cytoplasm, blue – cell center.

Dynamic Speckle Interferometry of Thin Biological Objects: Theory, Experiments, and Practical Perspectives http://dx.doi.org/10.5772/66712 131

Figure 18. Joint dependences σ<sup>u</sup> on time and those of temperature on time.

Next in Figure 16, there is distribution of value η obtained at the temperature of 30°С by 2 speckle cell images shown in Figure 15. The time interval between the frames equaled 18 s.

Figures 17, 18 and 19 show typical dependence σ<sup>u</sup> on temperature and joint dependences of σ<sup>u</sup> and temperature Т on time. Value σ<sup>u</sup> was obtained using two dependences ηðtÞ corresponding to the cells in the nutrient solution and to the nutrient solution. Segments containing 60 cells (Figure 18) and small regions inside the cells were averaging regions (Figure 17). As is seen from the pictures, considerable fluctuations of σ<sup>u</sup> are observed with temperature increase. Fluctuations of value σ<sup>u</sup> differ from one cell part to another. If the temperature gets stabilized in 30 min, σ<sup>u</sup> also stabilizes (Figure 19). The correlation coefficient of masses σ<sup>u</sup> and Т shown in

Figure 17. Dependences of σ<sup>u</sup> on Т for three segments inside one cell. Red colour – cell edge, gray colour – cytoplasm,

Values η were found in segments of 10 × 10-pixel size.

Figure 16. Distribution of value η obtained by two speckle images.

Figure 19 equals 0.88.

130 Optical Interferometry

blue – cell center.

Figure 19. Joint dependences σ<sup>u</sup> and temperature on time at small heating rates.

So on the basis of the conducted research, we can conclude that with temperature increase from 25° to 43°C at the rate of about 0.5° a minute, there are fluctuations of value σ<sup>u</sup> in space and time. With decreasing variation rate of temperature Т by an order variations of σ<sup>u</sup> stabilize, and the dependence of σ<sup>u</sup> on Т becomes linear. To study the dependence of σ<sup>u</sup> on Т in small segments inside the cells in detail, further research is needed.

#### 6.4. Comparison of theory and experiment: cell activity parameters

Our experiments on cultured and defrosted cells showed qualitative coincidence of theory and experiment. So in random cell motions on the bases of a 1-μm order, the mean difference in the optical paths of two waves can reach and exceed wavelength λ. Then, in compliance with the theory, the value of the cosine and also value η must accept not only positive but also negative values in a random way. Experimental confirmation of this supposition speaks for correctness of the model applied and the calculations made.

We detected good correlation of dependences ηðtÞ corresponding to hundreds of defrosted cells obtained in different time intervals. Absence of dependence of a random process on selection of the counting origin means its stationarity. According to the theory, levelling off dependence <sup>η</sup>ðt<sup>Þ</sup> speaks for stationarity of processes <sup>~</sup>Iðt<sup>Þ</sup> and <sup>Δ</sup>uðtÞ. Homogeneity and stationarity of the intracellular processes in defrosted cells detected using two methods can find practical application. In particular, studying the reaction of hundreds of defrosted cells to the effect of viruses, bacteria, and searching the optimum drugs that prevent their development can be promising. Cultured cells can serve this purpose as well. The advantage of cultured cells is their immobility. That is why above-noted studies can be conducted on a small cell number and on individual cells. The advantage to application of defrosted cells is simplicity of the research target preparation.

At present we suggest that it is value σu, or mean square deviation of wave pair optical paths obtained by way of averaging by some region that can be regarded as a cell activity parameter.

This selection is well substantiated from the viewpoint of physics. Indeed, if some processes do not occur in the cells, there is no optical path variation, so σ<sup>u</sup> ¼ 0. If the processes connected with small energy absorption or emission and with transfer of small amounts of substances arise in the cells, small random deviations of the refraction index and the cell shape from the mean value appear. Therefore, values σ<sup>u</sup> will be small as well. With intensification of physicalchemical processes in the cells, the values of σ<sup>u</sup> will increase.

The selection of parameter σ<sup>u</sup> is justified from the viewpoint of biophysics as well. It is known that at room temperature, the metabolic processes in cultured cells are weakly manifested. The culture techniques have shown that with increase of temperature Т, the metabolic processes become more distinctly manifested and reach their maximum at 34–37°C. The increase of value σ<sup>u</sup> with increase of Т (Figure 14) in a relatively wide range for hundreds of cells and good linear correlation between σ<sup>u</sup> and Т in the range of 0.4°С (Figure 19) for tens of cells speaks good reason behind applying σ<sup>u</sup> as a cell activity parameter.

That said, it is not clear yet what constituents of cell metabolism affect σu. Further, research is needed to clarify the matter.
