RESEARCH, DESIGN, AND TECHNOLOGY

BIOPHYSICAL SUBSTANTIATION OF PHOTOPLETHYSMOGRAPHY

IN REFLECTED LIGHT

M. I. Gaiduk, V. V. Grigor'yants, V. P. Zaitsev, V. D. Menenkov, and I. V. Chernousova

UDC 615.471.03:616.1-073.173

An increasingly greater number of methods based on the use of optoelectronic instruments is presently being introduced into medical diagnostics.

Among such methods is the photoplethysmographic method (PPM) making it possible to measure the change in the volume and flow of blood both in large veins and arteries and in peripheral vessels and capillaries.

In comparison with other methods of diagnosing a biological object (BO) on the basis of its optical indices, for example, in comparison with the photoacoustic method, the PPM is distinguished by the simplicity of the devices for its realization, as well as by the fact that by using fiberoptic elements and sources with different wavelengths of the probing radiation in the photoplethysmographic devices, it is possible to rather simply solve prob- lems of photodynamic investigations, remote measurements of the content of various elements in the BO being studied, etc.

At the given stage of introducing photoplethysmography into medical practice, the PPM has still not found wide use for a number of reasons. One of them is the absence of a bio- physical substantiation of obtaining a photoplethysmographic signal.

There are two varieties of the PPM - photoplethysmography in reflected light and photo- plethysmography in transmitted light, Investigations are most often carried out in transmit- ted light, by virtue of the fact that in the given case the change in the blood volume in the investigated part of the BO is estimated directly. But it is often rather difficult to conduct such investigations, for example, for optically poorly transparent BOs or for difficultly accessible parts of the objects. Then the method of photoplethysmography in reflected light is used, which not only makes it possible to estimate the total blood flow in the investigated part but also gives an integral estimation of the properties of the sur- face of investigation.

As for photoplethysmography in transmitted light, the problem of the biophysical nature of the signal is to some extent elucidated in the literature [2-5, I0]. For PPM in transmit- ted light the value of the radiant flux (RF) transmitted through the investigated BO varies according to the Lambert-Beer law and depends on the thickness of the layer of the BO and scattering in it of the flux having a certain spectral range [12]. The change in the optical density of tissues of the investigated BO due to cardiac activity is the main varying physical parameter in this case.

In the case of using photoplethysmography in reflected light, i.e., when the photo- plethysmographic transducer (PT) receives the RF reflected from the tissues of the BO, the nature of the change in the values of the RF is unclear in all the literature sources ex- amined.

It was shown experimentally in our previous work [i] that the PPM makes it possible to record the value of the change in the blood supply of tissues of the BO on the basis of the pulsation of the surface of the BO closest to the PT, i.e., the value of the change in the luminous flux reflected from the investigated tissue of the BO as a function of the ampli- tude of pulsation of the tissue.

The purpose of the present work was to examine the theoretical relation between the

Institute of Radio Engineering and Electronics, Academy of Sciences of the USSR, Moscow. Central Scientific-Research Institute of Stomatology, USSR Ministry of Health, Moscow. Trans- lated from Meditslnskaya Tekhnika, No. 2, pp. 4-7, March-April, 1990. Original article sub- mitted February i, 1989.

0006-3398/90/2402-004552.50 © 1990 Plenum Publishing Corporation 45

/

Fig. i. Graphic explanation of the symbols of the mathematical apparatus.

changes in the two quantities indicated above. In connection with this, let us examine a system consisting of tissue of the BO and PT operating in reflected light. In the given case, the PT represents a radiator and photodetector.

It is known from various literature sources examining the optical characteristics of tissues of a BO [6, 8, ii, 13] that the coefficients of reflection, absorption, scattering, and transmission are different for different tissues and their values depend both on the properties of the investigated tissues of the BO and on the wavelength of the probing radia- tion.

There are also a number of theoretical works examining the propagation of radiation in tissues of a BO [7, 9, 14] in which the radiation incident on the BO is divided into several components, each of which characterizes the value of the reflected, absorbed, and scattered radiation and radiation transmitted by the tissue of the investigated BO. Thus, it was shown in [3] that a part of the radiant flux which was multiply reflected from the internal layers of the investigated tissue and emerged on the investigated surface can be introduced into the flux reflected from the surface of the BO.

The difference introduced due to multiple reflection in the reflection coefficients from the tissue surface and for total reflection is examined in the aforementioned literature in the case of using collimated and high-power laser radiation for certain tissues of the BO.

In the given case the PT uses the radiation (uncollimated with a power up to 5 mW) either of light-emitting diodes or of low-power lasers (up to 15 mW) supplied to the object of in- vestigation by means of fiberoptic light pipes.

It is also known that low-power radiation penetrates into a BO to a small depth (for example, power 1.5 mW, wavelength ~ = 0.89 ~m, depth of penetration into the skin 200 ~m).

Consequently, it can be assumed that for probing radiation with such characteristics, multiply reflected radiation will not make a perceptible contribution to the RF reflected from the surface of investigation.

Therefore, in our case we will limit ourselves to an examination of RF reflected only from the surface of the investigated BO.

Let us assume that the investigated tissue of the BO has diffuse reflection [9], and we will take each element of area of the illuminated surface of the BO as a point source. Let us also assume that during the time of measuring the }IF reflected from the investigated tissue, its reflection coefficient remains constant.

Then the total reflected RF, ~0, is determined by the formula

~0=2~I, I s i n ~ e o s = d = , (i) e = O

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A, tel. units 7,0!

c~s \ .... .~ = Zt ol [ "~ O.g

,,. -A- " ~.s . o.z

0 " 0 a2 04 OG OB 1.o/.z 1.4 /.6 /.8 2.0 z.

A, tel. units Y-O z=O

I\ \ \ V \ \

\ N

0 0,I O,F.I,I l,E ZI ZG ~I ~6 4,1 d.G $,,6" 7,1R,~

Fig. 2 Fig. 3

7 mm

RI~ d = 2.6 mm

R~d = 2 Jm~

Fig. 2 . Change in the coefficient of effectiveness of collecting useful information A as a function of a change in the distance L between the BO and plane of the photodetector (for Rpd/R = const).

Fig. 3. Change in the coefficient of effectiveness of collecting useful information A as a function of the change in the size of the luminous spot in the plane of the photodetector R (for Rpd = const).

where I 0 is the intensity of the RF; ~ is the angle between the normal to the luminous sur- face and direction of propagation of the RF.

Since a real PT makes it possible to collect only a part of the reflected radiation, the efficiency of collecting the reflected RF depends on the design of the PT, consequently, it is necessary to examine the relation between the optical and geometric factors affecting the value of the reflected RF in the form:

O p d = A O ° ' (2)

where A is the coefficient of effectiveness of collecting the energy of the RF reflected from the tissue of the BO.

The illuminance E being created by the diffusely reflecting area is determined by the formula:

I cos il cos i2 . ¢ r-r-----.o, ( 3 ) E = B

J S,

where ii, i 2 are the angles between the direction and normals to the investigated tissues of the BO and PT; B is the brightness of the radiation source.

We will denote the plane in which lies the surface of the investigated tissue of the BO by S(x, y) and the plane in which lies the photodetector and radiator by S'(q, $) and we will assume that these planes are parallel. From the triangle AA'B' we find the distance between point A(x, y) in the plane of the BO and point B'(n, 6) in the plane of the PT:

r-..~3/L2 +(x- -~)~ q-(y--Ti) 2 , (4)

where L is the distance between the plane of the PT and plane of the investigated tissue of the BO (Fig. i).

Since planes S' and S are parallel, it is obvious that

co~ i, = ~ o ~ i ~ = L = L . ( 5 ) r ~ / L ~ + ( x _ ~ ) ~ + ( y _ n ) ' ~

Then, substituting (5) and (4) into (3), we obtain:

E = B L 2 I I dxdy s, [L~ + ( x - - ~)~ +(Y-- ~)~'Y

( 6 )

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To calculate this integral, we change to polar coordinates:

{ x=hcosq~ {~=pcosO y=h sin tp rl=p sin 0 " (7)

Then after appropriate transformations and use of the tabular integral, we obtain:

/= R

2ahdh

,-( L2+h~+p'2!

(8)

Further transformations give the integrand in the form

R

1 = T [r 2 + 402x + 4Q2(L 2 + p2) ]:~ ' ( 9 )

where ~ = L 2 + h 2 + 0 2 , h is the current coordinate in plane S'.

Performing further integration, using the tabular integral, and substituting the limits of integration, we finally obtain:

x L2+h2--R 2 ] I : ~z[ I - - A/h4..+.2h2(L.2 R2)_~(L2_~I~2)2j. (i0)

Here R is the radius of the luminous area of the photodetector, determined by the expression:

R=p-F i tg-~ . (ii)

where p is the current coordinate of the luminous spot on the BO; ~ is the plane angle of expansion of the }iF, increasing with increase of L.

Substituting (i0) into (6), we obtain the value of illuminance in the form:

L~+h2--R= ] " E= l-- ZL-t .x/h, W 2h'2(L2_t~2)+(L'e W R~)'2 (12)

The RF incident on the photodetector is determined by the expression:

~d

a~d=2a I E(h)ah, (13) 0

where Rpd is the radius of the photodetector.

Substituting expression (12) into (13) and calculating the integral, we obtain:

~t'B [R,.+I~d+L,_.x/L,+ (RI~d_R~) z_l_2LZ (R~q_RI~.d)]. (14) * p d : T

The t o t a l RF f rom t h e i n v e s t i g a t e d s u r f a c e ( r e f l e c t i n g d i f f u s e l y ) can be w r i t t e n so :

* o = ~ 2 B R 2. (15)

Then, expressing A from (2) and substituting expressions (14) and (15), we obtain:

~[R +~m -- A : % g * o : ' ' 2 i. (16)

It is obvious from (14) that the value of the RF reflected from the investigated tissue depends on the distance L between the object of investigation and the plane of the PT. Thus the relation obtained confirms the correctness of the experimental conclusion concerning the

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2, tel_ units O,~

0,75 k_

0,74 ~ ~'~ k,,~

0,78 ~ :

o,72 J I

S I0 N ZO 2~ Jo 35 40-Y-~, degrees

Fig. 4. Change in the coefficient of effectiveness of collecting useful in- formation A as a function of the change in the angle of expansion of the beam of ratiation ~ (for L = const).

relation of the change in the RF reflected from tissue during pulsation of the investigated surface (i.e., as a result of a change in the distance between the plane of the PT and sur- face of the tissue of the BO) occurring as a result of cardiac activity.

The factors Rpd, R, h depend directly on the design of the PT, whereas a change in the factor L depends only on the functional state of the object of investigation, which in the given case depends on the blood flow in the investigated tissues and on the reaction of vessels to this flow.

The design of the PT is important when it is used for diagnosing various diseases in various parts of the BO. Thus, for difficultly accessible parts of the BO it is required to minimize the PT, and to obtain an integral picture of large lesions of the BO it is desir- able to have a transducer covering the entire pathologically altered area. Therefore, to optimize the dimensions of the PT for particular problems, we will analyze the effect of the aforementioned factors on the value of the output signal. We will use expression (16) for this purpose.

Calculation of the coefficient A as a function of the change in the parameters figur- ing in formulas (16) was carried out on a computer with a step for L 0 < L < 2 mm of 200 ~m.

Figures 2-4 show graphs of the dependence of A on the distance between tissues of the BO and photodetector, on the distance between the axis of the radiator and axis of the photo- detector, on the radius of the spot createdby the RF reflected from the tissues of the BO on the sensing area of the photodetector, and on the angle of expansion of the beam of radiation due to the presence of the quantity L.

It is seen in Fig. 2 that for any ratio R/Rpd the maximum effectiveness of collecting use- ful information will occur for L = 0. When the ratio R/Rpd > i the coefficient of effective- ness A rapidly decreases (for example, when R/Rpd = 0.5 for L = 0.6 =gn, A amounts to about 0.09 of the maximum value of A = 1 for L = 0).

For the ratio R/Rpd < i, A decreases considerably more slowly (for L = 0.6 mm, A = 0.94). As for R/Rpd = i, here the smaller the components of the ratio, the more rapidly A decreases. Thus, for L = 0.6 mm when R = Rpd = 0.6 mm, A = 0.38, when R = Rpd = 1.6 mm, A = 0.69, and when R = Rpd = 6.6 mm, A = 0.93~

Figure 3 shows the function A = f(R) for various values of Rpd and constant value L = 0. It is obvious that the greater Rpd , the greater the coefficient o~ effectiveness of collect- ing information A.

As for the dependence of A on the angle of expansion of the beam reflected from tissues of the BO and incident on the plane of the photodetector for certain L, it is seen in Fig. 4 that the angular component ~/2 does not make a substantial contribution to A. Thus, for dis- tance L = 0.i mm when R/Rpd = 0.1/0.2, the decrease of A amounts to 4.5% for angle~J2 = 40 ° relative to A for ~ = 0.

CONCLUSIONS

The coefficient of effectiveness of collecting the useful signal A and, consequently, the change in the value of KF reflected from tissues of the BO substantially depend on the

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arrangement of the reflecting surface of these tissues relative to the PT used. An important factor in collecting the useful signal is the design of the PT itself, i.e., the dimensions of its components and their arrangement relative to one another. However, since the same PT, as a rule, is used in experimental investigations, the value of the RF reflected from the investigated tissues will depend practically only on the change in the position of the re- flecting surface relative to the working surface of the PT, which the cardiovascular system introduces into the PPM (with consideration of the assumptions made). Depending on whether this system is normal or various pathologies are present in it, the value of the reflected RF will change.

As for the optimal selection of the real design of a PT, the results obtained in cal- culations by formula (16) make it possible to find the value of the coefficient of effective- ness of collecting useful information practically for all types of radiators and photodetec- tors, including fiber light pipes, being used in instruments (of the photoplethysmograph type).

LITERATURE CITED

i. M. T. Aleksandrov, V. K. Osipov, and I. V. Chernousova, Stomatologiya, No. I, 19-21 (1986).

2. V. P. Zaitsev, N. K. Loginova, and K. N. Zhizhina, Fundamental Diagnostics in Stomatolog- ical Practice [in Russian], Moscow (1980), pp. 19-27.

3. V. B. Moshkevich, Photoplethysmograph: Apparatus and Methods of Investigation [in Rus- sian], Moscow (1979), pp. 41-54.

4. V. V. Orlov, Methods of Investigating Blood Circulation [in Russian], Moscow (1976), pp. 78-81.

5. Scheme of a Light Shutter for Measuring the Blood Supply of Tissue, West German Patent, No. 3040831A 61135/02 [Russian translation].

6. R. Anderson and J. Parrish, J. Invest. Dermatol., ~, No. I, 93-99 (1981). 7. S. L. Jocques and S. A. Prahl, Laser Surg. Med., 6, No. 6, 494-503 (1987). 8. P. Kubelka, J. Opt. Soc. Am., 38, No. 5, pp. 448-547 (1948). 9. L. Reynolds, C. Johnson, and A. Ishimaru, Appl. Opt., 15, No. 9, 5191-5199 (1976).

10. Y. Shah et al., J. Biomed. Eng., ~, No. 4, 326-328 (1985). ii. S. Wan, R. Anderson, and J. Parrish, Photochem. Photobiol., 34, No. i, 93-99 (1984). 12. A. J. Welch, J. Quant. Electr., No. 12, 8732-8740 (1984). 13. G. Yoon, W. F. Gheon, and A. J. Welch, Laser Surg. Med., [, No. i, 43-57 (1987). 14. G. Yoon, A. J. Welch, et al., J. Quant. Electr., 23, No. I0, 7057-7063 (1987).

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