Abstract

A quantitative method for determining the depth of burn eschar would aid surgeons in determining whether to excise and subsequently graft a burn wound. We hypothesize that tissue viability could be assessed by an analysis of the spatial modulation of near-field laser speckle by flowing blood. A feasibility study of the technique was performed with two-layer tissue phantoms used to simulate a burn wound. A sheet of polytetrafluoroethylene (PTFE) was used to simulate nonperfused burn eschar, and tissue perfusion within deeper layers was represented by Brownian motion from a scattering solution. A low-power He–Ne laser was focused onto the target, and the resulting speckle image was captured with a CCD camera and stored on a computer for further processing. The diameter of the speckle pattern was found to be directly proportional to the thickness of the overlying layer. These data suggest that the thickness of PTFE can be determined to ±100-μm accuracy with 95% confidence and may be suitable for burn depth detection in vivo.

© 1996 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  33. H. J. van Staveren, C. J. M. Moes, J. van Marle, S. A. Prahl, M. J. C. van Gemert, “Light scattering in Intralipid-10% in the wavelength range of 400–100 nm,” Appl. Opt. 30, 4507–4514 (1991).
    [CrossRef] [PubMed]

1993 (2)

B. Ruth, J. Schmand, D. Abendroth, “Noncontact determination of skin blood flow using the laser speckle method: application to patients with peripheral arterial occlusive disease (PAOD) and to type-I diabetics,” Lasers Surg. Med. 13, 179–188 (1993).
[CrossRef] [PubMed]

S. A. Prahl, M. J. C. van Gemert, A. J. Welch, “Determining the optical properties of turbid media by using the adding-doubling method,” Appl. Opt. 32, 559–568 (1993).
[CrossRef] [PubMed]

1992 (4)

Y. Aizu, K. Ogino, T. Sugita, T. Yamamoto, N. Takai, T. Asakura, “Evaluation of blood flow at ocular fundus by using laser speckle,” Appl. Opt. 31, 3020–3029 (1992).
[CrossRef] [PubMed]

B. Ruth, “Superposition of two dynamic speckle patterns,” J. Mod. Opt. 39, 2421–2436 (1992).
[CrossRef]

D. Heimbach, L. Engrav, B. Grube, J. Marvin, “Burn depth: a review,” World J. Surg. 16, 10–15 (1992).
[CrossRef] [PubMed]

H. A. Green, D. Bua, R. R. Anderson, N. S. Nishioka, “Burn depth estimation using indocyanine green fluorescence,” Arch. Dermatol. 128, 43–49 (1992).
[CrossRef] [PubMed]

1991 (3)

H. J. van Staveren, C. J. M. Moes, J. van Marle, S. A. Prahl, M. J. C. van Gemert, “Light scattering in Intralipid-10% in the wavelength range of 400–100 nm,” Appl. Opt. 30, 4507–4514 (1991).
[CrossRef] [PubMed]

S. Feng, P. A. Lee, “Mesoscopic conductors and correlations in laser speckle patterns,” Science 251, 633–639 (1991).
[CrossRef] [PubMed]

Y. Aizu, T. Asakura, “Bio-speckle phenomena and their application to the evaluation of blood flow,” Opt. Laser Technol. 23, 205–219 (1991).
[CrossRef]

1990 (3)

B. Ruth, “Blood flow determination by laser speckle method,” Int. J. Microcirc. Clin. Exp. 9, 21–45 (1990).
[PubMed]

W. F. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

R. P. Cole, S. G. Jones, P. G. Shakespeare, “Thermographic assessment of hand burns,” Burns 16, 60–63 (1990).
[CrossRef] [PubMed]

1989 (4)

1988 (1)

M. A. Afromowitz, J. B. Callis, D. M. Heimbach, L. A. DeSoto, M. K. Norton, “Multispectral imaging of burn wounds: a new clinical instrument for evaluating burn depth,” IEEE Trans. Biomed. Eng. 35, 842–850 (1988).
[CrossRef] [PubMed]

1987 (1)

M. A. Afromowitz, G. S. Van Liew, D. Heimbach, “Clinical evaluation of burn injuries using an optical reflectance technique,” IEEE Trans. Biomed. Eng. 34, 114–117 (1987).
[CrossRef] [PubMed]

1986 (1)

T. K. Wachtel, G. R. Leopold, H. A. Frank, D. H. Frank, “B-mode ultrasonic echo determination of depth of thermal injury,” Burns 12, 432–437 (1986).
[CrossRef]

1984 (2)

J. Micheels, B. Alsbjorn, B. Sorensen, “Clinical use of laser Doppler flowmetry in a burns unit,” Scand. J. Plast. Reconstr. Surg. 18, 65–73 (1984).
[CrossRef] [PubMed]

J. H. Cantrell, W. T. Yost, “Can ultrasound assist an experienced surgeon in estimating burn depth?” J. Trauma 24, 64–70 (1984).

1983 (2)

P. Hlava, J. Moserova, R. Konigova, “Validity of clinical assessment of the depth of a thermal injury,” Acta Chir. Plast. 25, 202–208 (1983).
[PubMed]

J. E. Gatti, D. LaRossa, D. G. Silverman, C. E. Hartford, “Evaluation of the burn wound with perfusion fluorometry,” J. Trauma 23, 202–206 (1983).
[CrossRef] [PubMed]

1980 (1)

P. Newman, “A practical technique for the thermographic estimation of burn depth: a preliminary report,” Burns 8, 59–63 (1980).
[CrossRef]

1979 (1)

A. M. Kalus, “Application of ultrasound in assessing burn depth,” Lancet 1, 188–189 (1979).
[CrossRef] [PubMed]

1968 (1)

M. Dobrkovsky, P. Malek, V. Zastava, “Evaluation of tissue destruction using the fluorescence of tetracycline antibiotics,” Ann. N. Y. Acad. Sci. 150, 560–561 (1968).
[CrossRef] [PubMed]

1966 (2)

D. Goulian, H. Conway, “Dye differentiation of injured tissues in burn injury,” Ann. N. Y. Acad. Sci. 150, 554–559 (1966).
[CrossRef]

R. Mladick, N. Georgiade, F. Thorne, “A clinical evaluation of the use of thermography in determining degree of burn injury,” Plast. Reconstr. Surg. 38, 512–518 (1966).
[CrossRef] [PubMed]

1965 (1)

D. Goulian, H. Conway, “Dye differentiation of injured tissues in burn injury,” Surg. Gynecol. Obstet. 121, 3–7 (1965).
[PubMed]

1964 (1)

J. G. Randolph, L. L. Leape, R. E. Gross, “The early surgical treatment of burns, I: Experimental studies utilizing intravenous vital dye for determining degree of injury,” Surgery 56, 193–202 (1964).
[PubMed]

1957 (1)

J. E. Bennett, R. O. Dingman, “Evaluation of burn depth by the use of radioactive isotopes: an experimental study,” Plast. Reconstr. Surg. 20, 261–272 (1957).
[CrossRef]

1943 (1)

J. A. Dingwall, “A clinical test for differentiating second from third degree burns,” Ann. Surg. 118, 427–429 (1943).
[CrossRef] [PubMed]

Abendroth, D.

B. Ruth, J. Schmand, D. Abendroth, “Noncontact determination of skin blood flow using the laser speckle method: application to patients with peripheral arterial occlusive disease (PAOD) and to type-I diabetics,” Lasers Surg. Med. 13, 179–188 (1993).
[CrossRef] [PubMed]

Achauer, B.

K. Waxman, N. Lefcourt, B. Achauer, “Heated laser Doppler flow measurements to determine depth of burn injury,” Am. J. Surg. 157, 541–543 (1989).
[CrossRef] [PubMed]

Afromowitz, M. A.

M. A. Afromowitz, J. B. Callis, D. M. Heimbach, L. A. DeSoto, M. K. Norton, “Multispectral imaging of burn wounds: a new clinical instrument for evaluating burn depth,” IEEE Trans. Biomed. Eng. 35, 842–850 (1988).
[CrossRef] [PubMed]

M. A. Afromowitz, G. S. Van Liew, D. Heimbach, “Clinical evaluation of burn injuries using an optical reflectance technique,” IEEE Trans. Biomed. Eng. 34, 114–117 (1987).
[CrossRef] [PubMed]

Aizu, Y.

Y. Aizu, K. Ogino, T. Sugita, T. Yamamoto, N. Takai, T. Asakura, “Evaluation of blood flow at ocular fundus by using laser speckle,” Appl. Opt. 31, 3020–3029 (1992).
[CrossRef] [PubMed]

Y. Aizu, T. Asakura, “Bio-speckle phenomena and their application to the evaluation of blood flow,” Opt. Laser Technol. 23, 205–219 (1991).
[CrossRef]

Alsbjorn, B.

J. Micheels, B. Alsbjorn, B. Sorensen, “Clinical use of laser Doppler flowmetry in a burns unit,” Scand. J. Plast. Reconstr. Surg. 18, 65–73 (1984).
[CrossRef] [PubMed]

Anderson, R. R.

H. A. Green, D. Bua, R. R. Anderson, N. S. Nishioka, “Burn depth estimation using indocyanine green fluorescence,” Arch. Dermatol. 128, 43–49 (1992).
[CrossRef] [PubMed]

Asakura, T.

Y. Aizu, K. Ogino, T. Sugita, T. Yamamoto, N. Takai, T. Asakura, “Evaluation of blood flow at ocular fundus by using laser speckle,” Appl. Opt. 31, 3020–3029 (1992).
[CrossRef] [PubMed]

Y. Aizu, T. Asakura, “Bio-speckle phenomena and their application to the evaluation of blood flow,” Opt. Laser Technol. 23, 205–219 (1991).
[CrossRef]

Bauer, J. A.

J. A. Bauer, T. Sauer, “Cutaneous 10 MHz ultrasound B scan allows the quantitative assessment of burn depth,” Burns 15, 49–51 (1989).
[CrossRef]

Bennett, J. E.

J. E. Bennett, R. O. Dingman, “Evaluation of burn depth by the use of radioactive isotopes: an experimental study,” Plast. Reconstr. Surg. 20, 261–272 (1957).
[CrossRef]

Bua, D.

H. A. Green, D. Bua, R. R. Anderson, N. S. Nishioka, “Burn depth estimation using indocyanine green fluorescence,” Arch. Dermatol. 128, 43–49 (1992).
[CrossRef] [PubMed]

Callis, J. B.

M. A. Afromowitz, J. B. Callis, D. M. Heimbach, L. A. DeSoto, M. K. Norton, “Multispectral imaging of burn wounds: a new clinical instrument for evaluating burn depth,” IEEE Trans. Biomed. Eng. 35, 842–850 (1988).
[CrossRef] [PubMed]

Cantrell, J. H.

J. H. Cantrell, W. T. Yost, “Can ultrasound assist an experienced surgeon in estimating burn depth?” J. Trauma 24, 64–70 (1984).

Cheong, W. F.

W. F. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Cole, R. P.

R. P. Cole, S. G. Jones, P. G. Shakespeare, “Thermographic assessment of hand burns,” Burns 16, 60–63 (1990).
[CrossRef] [PubMed]

Conway, H.

D. Goulian, H. Conway, “Dye differentiation of injured tissues in burn injury,” Ann. N. Y. Acad. Sci. 150, 554–559 (1966).
[CrossRef]

D. Goulian, H. Conway, “Dye differentiation of injured tissues in burn injury,” Surg. Gynecol. Obstet. 121, 3–7 (1965).
[PubMed]

DeSoto, L. A.

M. A. Afromowitz, J. B. Callis, D. M. Heimbach, L. A. DeSoto, M. K. Norton, “Multispectral imaging of burn wounds: a new clinical instrument for evaluating burn depth,” IEEE Trans. Biomed. Eng. 35, 842–850 (1988).
[CrossRef] [PubMed]

Dingman, R. O.

J. E. Bennett, R. O. Dingman, “Evaluation of burn depth by the use of radioactive isotopes: an experimental study,” Plast. Reconstr. Surg. 20, 261–272 (1957).
[CrossRef]

Dingwall, J. A.

J. A. Dingwall, “A clinical test for differentiating second from third degree burns,” Ann. Surg. 118, 427–429 (1943).
[CrossRef] [PubMed]

Dobrkovsky, M.

M. Dobrkovsky, P. Malek, V. Zastava, “Evaluation of tissue destruction using the fluorescence of tetracycline antibiotics,” Ann. N. Y. Acad. Sci. 150, 560–561 (1968).
[CrossRef] [PubMed]

Engrav, L.

D. Heimbach, L. Engrav, B. Grube, J. Marvin, “Burn depth: a review,” World J. Surg. 16, 10–15 (1992).
[CrossRef] [PubMed]

Feld, M. S.

Feng, S.

S. Feng, P. A. Lee, “Mesoscopic conductors and correlations in laser speckle patterns,” Science 251, 633–639 (1991).
[CrossRef] [PubMed]

Frank, D. H.

T. K. Wachtel, G. R. Leopold, H. A. Frank, D. H. Frank, “B-mode ultrasonic echo determination of depth of thermal injury,” Burns 12, 432–437 (1986).
[CrossRef]

Frank, H. A.

T. K. Wachtel, G. R. Leopold, H. A. Frank, D. H. Frank, “B-mode ultrasonic echo determination of depth of thermal injury,” Burns 12, 432–437 (1986).
[CrossRef]

Gatti, J. E.

J. E. Gatti, D. LaRossa, D. G. Silverman, C. E. Hartford, “Evaluation of the burn wound with perfusion fluorometry,” J. Trauma 23, 202–206 (1983).
[CrossRef] [PubMed]

Georgiade, N.

R. Mladick, N. Georgiade, F. Thorne, “A clinical evaluation of the use of thermography in determining degree of burn injury,” Plast. Reconstr. Surg. 38, 512–518 (1966).
[CrossRef] [PubMed]

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), p. 12.

Goulian, D.

D. Goulian, H. Conway, “Dye differentiation of injured tissues in burn injury,” Ann. N. Y. Acad. Sci. 150, 554–559 (1966).
[CrossRef]

D. Goulian, H. Conway, “Dye differentiation of injured tissues in burn injury,” Surg. Gynecol. Obstet. 121, 3–7 (1965).
[PubMed]

Green, H. A.

H. A. Green, D. Bua, R. R. Anderson, N. S. Nishioka, “Burn depth estimation using indocyanine green fluorescence,” Arch. Dermatol. 128, 43–49 (1992).
[CrossRef] [PubMed]

Gross, R. E.

J. G. Randolph, L. L. Leape, R. E. Gross, “The early surgical treatment of burns, I: Experimental studies utilizing intravenous vital dye for determining degree of injury,” Surgery 56, 193–202 (1964).
[PubMed]

Grube, B.

D. Heimbach, L. Engrav, B. Grube, J. Marvin, “Burn depth: a review,” World J. Surg. 16, 10–15 (1992).
[CrossRef] [PubMed]

Hartford, C. E.

J. E. Gatti, D. LaRossa, D. G. Silverman, C. E. Hartford, “Evaluation of the burn wound with perfusion fluorometry,” J. Trauma 23, 202–206 (1983).
[CrossRef] [PubMed]

Heimbach, D.

D. Heimbach, L. Engrav, B. Grube, J. Marvin, “Burn depth: a review,” World J. Surg. 16, 10–15 (1992).
[CrossRef] [PubMed]

M. A. Afromowitz, G. S. Van Liew, D. Heimbach, “Clinical evaluation of burn injuries using an optical reflectance technique,” IEEE Trans. Biomed. Eng. 34, 114–117 (1987).
[CrossRef] [PubMed]

Heimbach, D. M.

M. A. Afromowitz, J. B. Callis, D. M. Heimbach, L. A. DeSoto, M. K. Norton, “Multispectral imaging of burn wounds: a new clinical instrument for evaluating burn depth,” IEEE Trans. Biomed. Eng. 35, 842–850 (1988).
[CrossRef] [PubMed]

Hlava, P.

P. Hlava, J. Moserova, R. Konigova, “Validity of clinical assessment of the depth of a thermal injury,” Acta Chir. Plast. 25, 202–208 (1983).
[PubMed]

Jacques, S. L.

Jones, S. G.

R. P. Cole, S. G. Jones, P. G. Shakespeare, “Thermographic assessment of hand burns,” Burns 16, 60–63 (1990).
[CrossRef] [PubMed]

Kalus, A. M.

A. M. Kalus, “Application of ultrasound in assessing burn depth,” Lancet 1, 188–189 (1979).
[CrossRef] [PubMed]

Keijzer, M.

Konigova, R.

P. Hlava, J. Moserova, R. Konigova, “Validity of clinical assessment of the depth of a thermal injury,” Acta Chir. Plast. 25, 202–208 (1983).
[PubMed]

LaRossa, D.

J. E. Gatti, D. LaRossa, D. G. Silverman, C. E. Hartford, “Evaluation of the burn wound with perfusion fluorometry,” J. Trauma 23, 202–206 (1983).
[CrossRef] [PubMed]

Leape, L. L.

J. G. Randolph, L. L. Leape, R. E. Gross, “The early surgical treatment of burns, I: Experimental studies utilizing intravenous vital dye for determining degree of injury,” Surgery 56, 193–202 (1964).
[PubMed]

Lee, P. A.

S. Feng, P. A. Lee, “Mesoscopic conductors and correlations in laser speckle patterns,” Science 251, 633–639 (1991).
[CrossRef] [PubMed]

Lefcourt, N.

K. Waxman, N. Lefcourt, B. Achauer, “Heated laser Doppler flow measurements to determine depth of burn injury,” Am. J. Surg. 157, 541–543 (1989).
[CrossRef] [PubMed]

Leopold, G. R.

T. K. Wachtel, G. R. Leopold, H. A. Frank, D. H. Frank, “B-mode ultrasonic echo determination of depth of thermal injury,” Burns 12, 432–437 (1986).
[CrossRef]

Malek, P.

M. Dobrkovsky, P. Malek, V. Zastava, “Evaluation of tissue destruction using the fluorescence of tetracycline antibiotics,” Ann. N. Y. Acad. Sci. 150, 560–561 (1968).
[CrossRef] [PubMed]

Marijnissen, J. P. A.

Marvin, J.

D. Heimbach, L. Engrav, B. Grube, J. Marvin, “Burn depth: a review,” World J. Surg. 16, 10–15 (1992).
[CrossRef] [PubMed]

Micheels, J.

J. Micheels, B. Alsbjorn, B. Sorensen, “Clinical use of laser Doppler flowmetry in a burns unit,” Scand. J. Plast. Reconstr. Surg. 18, 65–73 (1984).
[CrossRef] [PubMed]

Mladick, R.

R. Mladick, N. Georgiade, F. Thorne, “A clinical evaluation of the use of thermography in determining degree of burn injury,” Plast. Reconstr. Surg. 38, 512–518 (1966).
[CrossRef] [PubMed]

Moes, C. J. M.

Moserova, J.

P. Hlava, J. Moserova, R. Konigova, “Validity of clinical assessment of the depth of a thermal injury,” Acta Chir. Plast. 25, 202–208 (1983).
[PubMed]

Newman, P.

P. Newman, “A practical technique for the thermographic estimation of burn depth: a preliminary report,” Burns 8, 59–63 (1980).
[CrossRef]

Nishioka, N. S.

H. A. Green, D. Bua, R. R. Anderson, N. S. Nishioka, “Burn depth estimation using indocyanine green fluorescence,” Arch. Dermatol. 128, 43–49 (1992).
[CrossRef] [PubMed]

Norton, M. K.

M. A. Afromowitz, J. B. Callis, D. M. Heimbach, L. A. DeSoto, M. K. Norton, “Multispectral imaging of burn wounds: a new clinical instrument for evaluating burn depth,” IEEE Trans. Biomed. Eng. 35, 842–850 (1988).
[CrossRef] [PubMed]

Ogino, K.

Prahl, S. A.

Randolph, J. G.

J. G. Randolph, L. L. Leape, R. E. Gross, “The early surgical treatment of burns, I: Experimental studies utilizing intravenous vital dye for determining degree of injury,” Surgery 56, 193–202 (1964).
[PubMed]

Richards-Kortum, R. R.

Ruth, B.

B. Ruth, J. Schmand, D. Abendroth, “Noncontact determination of skin blood flow using the laser speckle method: application to patients with peripheral arterial occlusive disease (PAOD) and to type-I diabetics,” Lasers Surg. Med. 13, 179–188 (1993).
[CrossRef] [PubMed]

B. Ruth, “Superposition of two dynamic speckle patterns,” J. Mod. Opt. 39, 2421–2436 (1992).
[CrossRef]

B. Ruth, “Blood flow determination by laser speckle method,” Int. J. Microcirc. Clin. Exp. 9, 21–45 (1990).
[PubMed]

Sauer, T.

J. A. Bauer, T. Sauer, “Cutaneous 10 MHz ultrasound B scan allows the quantitative assessment of burn depth,” Burns 15, 49–51 (1989).
[CrossRef]

Schmand, J.

B. Ruth, J. Schmand, D. Abendroth, “Noncontact determination of skin blood flow using the laser speckle method: application to patients with peripheral arterial occlusive disease (PAOD) and to type-I diabetics,” Lasers Surg. Med. 13, 179–188 (1993).
[CrossRef] [PubMed]

Shakespeare, P. G.

R. P. Cole, S. G. Jones, P. G. Shakespeare, “Thermographic assessment of hand burns,” Burns 16, 60–63 (1990).
[CrossRef] [PubMed]

Silverman, D. G.

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Figures (9)

Fig. 1
Fig. 1

Schematic diagram of light propagation within the burn model. Photons that travel to depth x1 in the static layer produce a speckle pattern upon returning to the surface, whereas photons that travel to depth x2 in the modulated layer produce a time-averaged blurred pattern at the surface. The probability of a photon returning to the surface further from the point of incidence increases with the depth that it travels. Integrating over depth yields the speckle line profiles P(r), which transitions from speckle to a blurred pattern at some value r. The value r defines the speckle pattern radius that is dependent on the thickness of the static layer. The transition point is found from an empirically defined threshold (see text).

Fig. 2
Fig. 2

Experimental setup used to measure speckle images. The polarization of the laser beam is reinforced with a polarizer, d. The beam is expanded by a telescope composed of 130-mm and 500-mm focal length lenses, a and b, respectively. The expanded beam is tightly focused onto the PTFE surface with a 300-mm focal length lens, c. The resultant speckle image is collected through a crossed polarizer, d, by a CCD camera. The image is digitized and stored for further processing.

Fig. 3
Fig. 3

Sixteen frame-averaged speckle images for various static layer thicknesses: A, 0.13-mm-thick PTFE sheet; B, 0.41-mm-thick PTFE sheet; C, 0.87-mm-thick PTFE sheet; D, 1.3-mm-thick PTFE sheet.

Fig. 4
Fig. 4

Line profiles taken through the center of the speckle images in Fig. 3: (a) 0.13-mm-thick PTFE sheet; (b) 0.41-mm-thick PTFE sheet; (c) 0.87-mm-thick PTFE sheet; (d) 1.3-mm-thick PTFE sheet. Regions where camera saturates are not shown.

Fig. 5
Fig. 5

Three-pixel value moving standard deviation line profiles from the speckle images of Fig. 4: (a) 0.13-mm-thick PTFE sheet; (b) 0.41-mm-thick PTFE sheet; (c) 0.87-mm-thick PTFE sheet; (d) 1.3-mm-thick PTFE sheet. Regions where camera saturates lead to standard deviations equal to zero and are not shown.

Fig. 6
Fig. 6

Speckle pattern diameter as a function of static layer thickness for different threshold values. Error bars represent the standard error determined after the 72 line profiles were processed.

Fig. 7
Fig. 7

Speckle pattern diameter for a 1.3-mm-thick PTFE sheet as a function of the number of line profiles (rotations) used.

Fig. 8
Fig. 8

Speckle pattern diameter as a function of PTFE thickness. The solid line represents a linear least-squares fit to the data. The dotted lines represent the 95% confidence prediction intervals.

Fig. 9
Fig. 9

(a) Line profile through the center of a single frame speckle image of the lipid solution at room temperature, showing little indication of a speckle pattern. (b) Line profile through the center of a single frame speckle image of the frozen lipid solution. A distinct speckle pattern is evident when Intralipid is frozen.

Tables (1)

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Table 1 Linear Regression Results of the Threshold Analysis Shown in Fig. 6a

Equations (3)

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t 0 = ( n k 2 D p ) - 1 ,
k = 4 π n 0 λ sin θ 2 ,
D p = k B T / 6 π η r .

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