Phase modulation spectroscopy

A spectroscopic system for quantifying in vivo concentration of an absorptive pigment in biological tissue includes an oscillator for generating a first carrier waveform of a first frequency on the order of 10.sup.8 Hz, a light source for generating light of at least two selected wavelengths modulated by the carrier waveform, and a detector for detecting radiation that has migrated over photon migration paths in the tissue from an input port to a detection port spaced several centimeters apart. At least one of the wavelengths is sensitive to concentration of an absorptive pigment present in the tissue, while the tissue exhibits similar scattering properties at the two wavelengths. A phase detector compares, at each wavelength, the detected radiation with the introduced radiation and determines therefrom the phase shift of the detected radiation at each wavelength. A processor quantifies the concentration of the absorptive pigment by employing the phase shifts measured at the two wavelengths and also employing a scattering property of the tissue.

BACKGROUND OF THE INVENTION
 The present invention relates to quantitative analyses of absorptive
 constituents in biological tissues by employing a phase modulation
 spectroscopy.
 Continuous wave (CW) tissue oximeters have been widely used to determine in
 vivo concentration of an optically absorbing pigment (e.g., hemoglobin,
 oxyhemoglobin) in biological tissue. The CW oximeters measure attenuation
 of continuous light in the tissue and evaluate the concentration based on
 the Beer Lambert equation or a modified Beer Lambert absorbance equation.
 The Beer Lambert equation (1) describes the relationship between the
 concentration of an absorbent constituent (C), the extinction coefficient
 (.epsilon.), the photon migration pathlength &lt;L&gt;, and the attenuated light
 intensity (I/I.sub.0).
 ##EQU1##
 The CW spectrophotometric techniques can not determine .epsilon., C, and
 &lt;L&gt; at the same time. If one could assume that the photon pathlength were
 constant and uniform throughout all subjects, direct quantitation of the
 constituent concentration (C) using CW oximeters would be possible.
 In tissue, the optical migration pathlength varies with the size,
 structure, and physiology of the internal tissue examined by the CW
 oximeters. For example, in the brain, the gray and white matter and the
 structures thereof are different in various individuals. In addition, the
 photon migration pathlength itself is a function of the relative
 concentration of absorbing constituents. As a result, the pathlength
 through an organ with a high blood, hemoglobin concentration, for example,
 will be different from the same with a low blood hemoglobin concentration.
 Furthermore, the pathlength is frequently dependent upon the wavelength of
 the light since the absorption coefficient of many tissue constituents is
 wavelength dependent. Thus, where possible, it is advantageous to measure
 the pathlength directly when quantifying the hemoglobin concentration in
 tissue.
 SUMMARY OF THE INVENTION
 In general, in one aspect, a spectroscopic system for quantifying in vivo
 concentration of an absorptive pigment in biological tissue includes an
 oscillator constructed to generate a first carrier waveform of a first
 frequency on the order of 10.sup.8 Hz (i.e., in the range of 10 MHz to 1
 GHz), a light source constructed to generate light of at least two
 selected wavelengths modulated by the carrier waveform, and a detector
 constructed to detect radiation that has migrated over photon migration
 paths in the tissue from an input port to a detection port spaced several
 centimeters apart. At least one of the wavelengths is sensitive to
 concentration of an absorptive pigment present in the tissue, while the
 tissue exhibits similar scattering properties at the two wavelengths. A
 phase detector is constructed to compare, at each wavelength, the detected
 radiation with the introduced radiation and determine therefrom the phase
 shift of the detected radiation at each wavelength. A processor is
 constructed to quantify the concentration of the absorptive pigment based
 on the phase shifts measured at the two wavelengths and based on a
 scattering property of the tissue.
 In general, in another aspect, a spectroscopic system for quantifying in
 vivo concentration of an absorptive pigment in biological tissue includes
 an oscillator constructed to generate a first carrier waveform of a first
 frequency on the order of 10.sup.8 Hz (i.e., in the range of 10 MHz to 1
 GHz), a light source constructed to generate light of at least two
 selected wavelengths modulated by the carrier waveform, and a detector
 constructed to detect radiation that has migrated over photon migration
 paths in the tissue from an input port to a detection port spaced several
 centimeters apart. At least one of the wavelengths is sensitive to
 concentration of an absorptive pigment present in the tissue, while the
 tissue exhibits similar scattering properties at the two wavelengths. The
 spectroscopic system also includes a phase splitter, two double balanced
 mixers, and a processor. The phase splitter is constructed to receive the
 carrier waveform and produce first and second reference phase signals of
 predefined substantially different phases. The first and second double
 balanced mixers are constructed to receive from the phase splitter the
 first and second reference phase signals, respectively, and also receive
 from the detector the detector signal to produce therefrom a real output
 signal and an imaginary output signal, respectively. The processor is
 constructed to receive a scattering property of the examined tissue and
 the real output signal and the imaginary output signal and quantify
 therefrom the concentration of the absorptive pigment in the examined
 tissue.
 Different embodiments of this type of the spectrophotometer may include one
 or more of the following features. The processor may calculate, at each
 wavelength, a phase shift of the detected radiation as the inverse tangent
 of the ratio of the imaginary output signal and the real output signal.
 The processor may calculate, at each wavelength, a detected amplitude as
 the square root of the sum of the squares of the real output signal and
 the imaginary output signal.
 In different embodiments, the spectrophotometer may be a dual wavelength,
 single frequency system or a dual wavelength, dual frequency system. Each
 system can measure data for a single source-detector separation (i.e.,
 separation of the input port and the detection port) or for several
 source-detector separations.
 Different embodiments of the spectrophotometer may include one or more of
 the following features.
 The spectrophotometer may include a second oscillator constructed to
 generate a second carrier waveform of a second selected frequency on the
 order of 10.sup.8 Hz, while the tissue exhibits similar scattering
 properties at the selected frequencies. The source of the
 spectrophotometer is operatively coupled to the second oscillator and is
 constructed to generate electromagnetic radiation of the two wavelengths
 modulated by the second carrier waveform. The detector is further
 constructed to detect the radiation modulated by the second carrier
 waveform. The phase detector is further constructed to compare, at each
 the wavelength, the detected radiation of the second carrier waveform with
 the introduced radiation and determine therefrom the phase shift of the
 detected radiation of the second frequency.
 The processor may calculate a ratio of absorption coefficients at the two
 wavelengths, and calculate a value of oxygen saturation based on the
 ratio.
 The processor may calculates the ratio of absorption coefficients by taking
 a ratio of the phase shift and a square root of the frequency for each the
 wavelength and each the frequency.
 The processor may calculate the ratio of absorption coefficients by taking
 a ratio of the phase shifts detected at the two wavelengths. The phase
 shift of each the wavelength may be corrected for .theta..sub.0.
 The spectrophotometer may include a mechanism for positioning the input and
 detection ports at several selected relative distances.
 The spectrophotometer may include a look up table comprising values of the
 scattering property for different tissue types. These values may be the
 effective scattering coefficients, (1-g).mu..sub.s.
 The spectrophotometer may further include a magnitude detector constructed
 to measure an amplitude of the detected radiation. The processor may
 calculate the scattering property based on the measured amplitude. The
 processor may calculate the concentration by employing Eq. 5.
 The absorptive pigment may be an endogenous pigment, such as oxy-hemoglobin
 or deoxy-hemoglobin. The absorptive pigment may be an exogenous contrast
 agent.

DESCRIPTION OF THE PREFERRED EMBODIMENTS
 One preferred embodiment of the pathlength corrected oximeter utilizes
 three LEDs for generation of light at three selected wavelengths intensity
 modulated at a frequency of 50.1 MHz and coupled directly to the examined
 tissue. At each wavelength, the introduced light is altered by the tissue
 and is detected by a wide area photodiode placed against the skin. The
 introduced and detected radiations are compared to determine their
 relative phase shift that corresponds to an average pathlength of the
 migrating photons and, furthermore, the light attenuation is determined.
 Referring to FIG. 1, the oximeter includes a master oscillator 10 operating
 at 50.1 MHz connected to a power amplifier 15 of sufficient output power
 to drive LEDs 22a, 22b, and 22c (for example HLP 20RG or HLP 40RG made by
 Hitachi) that emit 760 nm, 840 nm, and 905 nm (or 950 nm) light,
 respectively. A second local oscillator 14 operating at 50.125 MHz and
 mixer 12 are used to generate a reference frequency 13 of 25 kHz. Each LED
 directly positioned on the skin has an appropriate heat sink to eliminate
 uncomfortable temperature increases that could also alter blood perfusion
 of the surrounding tissue. Three PIN diode detectors 24a, 24b, and 24c are
 placed at a distance of approximately 5 cm from the LEDs and have a
 detection area of about 1 cm.sup.2. Photons migrating a few centimeters
 deep into the tissue are detected by the respective PIN diodes. The
 source-detector separation can be increased or decreased to capture deeper
 or shallower migrating photons. The signals from PIN diodes 24a, 24b, and
 24c are amplified by preamplifiers 30a, 30b, and 30c, respectively.
 The amplified signals (32a, 32b, 32c) are sent to magnitude detectors 36a,
 36b, and 36c and to mixers 40a, 40b, and 40c, respectively. The magnitude
 detectors are used to determine intensity values of detected signals at
 each wavelength to be used in Eq. 1. Each mixer, connected to receive a
 50.125 MHz reference signal (41a, 41b, 41c) from local oscillator 14,
 converts the detection signal to a 25 kHz frequency signal (42a, 42b,
 42c). The mixers are high dynamic range frequency mixers, model SRA-1H,
 commercially available from Mini-Circuits (Brooklyn N.Y.). The detection
 signals (42a, 42b, and 42c) are filtered by filters 45a, 45b, 45c,
 respectively.
 Phase detectors 60a, 60b, and 60c are used to determine phase shift between
 the input signal and the detected signal at each wavelength. Each phase
 detector receives the 25 kHz detection signal (54a, 54b, 54c) and the 25
 kHz reference signal (56a, 56b, 56c), both of which are automatically
 leveled by automatic gain controls 50 and 52 to cover the dynamic range of
 signal changes. Phase detectors 60a, 60b, and 60c generate phase shift
 signals (62a, 62b, 62c) corresponding to the migration delay of photons at
 each wavelength. Each phase shift signal is proportional to the migration
 pathlength used in calculation algorithms performed by processor 70.
 FIG. 2 shows a schematic circuit diagram of a precision oscillator used as
 the 50.1 MHz master oscillator 10 and 50.125 MHz local oscillator 14. The
 oscillator crystals are neutralized for operation in the fundamental
 resonance mode; this achieves long-term stability. Both oscillators are
 thermally coupled so that their frequency difference is maintained
 constant at 25 kHz if a frequency drift occurs.
 PIN diodes 24a, 24b, and 24c are directly connected to their respective
 preamplifiers 30a, 30b, and 30c, as shown in FIG. 3. The oximeter uses PIN
 silicon photodiodes S1723-04 with 10 mm.times.10 mm sensitive area and
 spectral response in the range of 320 nm to 1060 nm. The detection signal
 is amplified by stages 29 and 31, each providing about 20 dB
 amplification. The NE5205N operational amplifier is powered at +8V to
 operate in a high gain regime. The 8V signal is supplied by a voltage
 regulator 33. The amplified detection signals (32a, 32b, and 32c) are sent
 to magnitude detectors 36a, 36b, and 36c, shown in FIG. 4. The magnitude
 values (37a, 37b, and 37c) are sent to processor 70 that calculates the
 light attenuation ratio or logarithm thereof as shown Eq. 1.
 Also referring to FIG. 5, the AGC circuit uses MC 1350 integrated circuit
 for amplification that maintains the input signal of phase detector 60 at
 substantially constant levels. The amount of gain is selected to be equal
 for AGCs, 50 and 52. The signal amplitude is controlled by a feedback
 network 53. The AGCs provide a substantially constant amplitude of the
 detected and reference signals to eliminate variations in the detected
 phase shift due to cross talk between amplitude and phase changes in the
 phase detector.
 Referring to FIG. 6, each phase detector includes a Schmitt trigger that
 converts the substantially sinusoidal detection signal (54a, 54b, 54c) and
 reference signal (56a, 56b, 56c) to square waves. The square waves are
 input to a detector that has complementary MOS silicon-gate transistors.
 The phase shift signal is sent to processor 70.
 The oximeter is calibrated by measuring the phase shift for a selected
 distance in a known medium, i.e., using a standard delay unit, and by
 switching the length of a connector wire to change the electrical delay
 between master oscillator 10 and local oscillator 14.
 Referring to FIGS. 8A and 8B source-detector probe 20 includes several LEDs
 (22a, 22b, 22c) of selected wavelengths and PIN photodiodes (24a, 24b,
 24c) mounted in a body-conformable support structure 21. Structure 21 also
 includes a photon escape barrier 27 made of a material with selected
 scattering and absorption properties (for example, styrofoam) designed to
 return escaping photons back to the examined tissue. The support structure
 further includes a second conformable barrier 28, located between the LEDs
 and the diode detectors, designed to absorb photons directly propagating
 from the source to the detector and thus prevent detection of photons that
 migrate subcutaneously. Support structure 21 also includes electronic
 circuitry 29 encapsulated by an electronic shield 21a.
 Each PIN diode is provided with an evaporated single wavelength film filter
 (25a, 25b, 25c). The filters eliminate the cross talk of different
 wavelength signals and allow continuous operation of the three light
 sources, i.e., no time sharing is needed.
 The use of photodiode detectors has substantial advantages when compared
 with the photomultiplier tube used in standard phase modulation systems.
 The photodiodes are placed directly on the skin, i.e., no optical fibers
 are needed. Furthermore, there is no need to use a high voltage power
 supply that is necessary for the photomultiplier tube. The photodiodes are
 much smaller and are easy to place close to the skin. Advantages of the
 photomultiplier tube are a huge multiplication gain and a possibility of
 direct mixing at the photomultiplier; this cannot be achieved directly by
 a photodiode. This invention envisions the use of several different
 photodiodes such as PIN diode, avalanche diode, and other.
 The processor uses algorithms that are based on equations described by E.
 M. Sevick et al. in "Quantitation of Time- and Frequency-Resolved Optical
 Spectra for the Determination of Tissue Oxygenation," published in
 Analytical Biochemistry 195, 330, Apr. 15, 1991, which is incorporated by
 reference as if fully set forth herein. The photon migration in biological
 tissue is a diffusional process in which the photon fluence rate, .phi.
 (r,t), is distributed from the source. The fluence rate is equal to
 N.sub..alpha. c, or the product of the number of the photon at position r
 and time, t, and the speed of photons through the medium. The fluence
 rate, or the effective "concentration" of photons at position r and time
 t, in the tissue or turbid media may be obtained from the solution of the
 general diffusion equation
 ##EQU2##
 where D is the diffusion coefficient and S a source term. For photon
 migration, the diffusion coefficient is equal to
 ##EQU3##
 where .mu..sub.s is the scattering coefficient (cm.sup.-1) and g is the
 mean cosine of scattering angle. The term (1-g).mu..sub.s is referred to
 as the effective scattering coefficient and is equal to the reciprocal of
 the isotropic, mean scattering length, l* (i.e., when the direction of
 scatter is completely random). The absorption coefficient .mu..sub.a is
 based upon the Napierian extinction coefficient.
 The source at .rho.=0 consists of light whose intensity is sinusoidally
 modulated at a frequency f. The light intensity detected at a distance
 .rho. away from the source is both amplitude demodulated and phase shifted
 with respect to the incident source intensity. The measured phase shift,
 .theta., and the modulation, M, of the detected light with respect to that
 of the incident light characterize the tissue wherein the detected photons
 migrated over a distribution of pathlengths. The phase shift describes the
 pathlength distribution in the frequency domain. It can be directly
 related to the mean of the distribution of pathlengths traveled by photons
 before detection. The modulation of the detected intensity also varies
 with changes in the absorbance and pathlength distribution. Pathlengths
 can be used to detect changes in absorption in strongly scattering media.
 Modulation may also be used to detect changes in absorption in the tissue.
 In phase modulation (frequency modulation), the source term represents a
 sinusoidally modulated photon flux at point .rho.=0;S(.rho.=0,
 t)=A+M.multidot.sin(2.pi.f.multidot.t). Expressions of the phase shift and
 modulation of the detected intensity may also be directly found from Eq.
 2.
 The analytical solution for .theta. and M can be obtained from the sine and
 cosine Fourier transforms of Eq. 2:
 ##EQU4##
 At each wavelength, for low modulation frequencies, i.e.,
 2.pi.f&lt;&lt;.mu..sub.a.multidot.c, the phase shift (.theta..sup..lambda.)
 (62a, 62b, 62c) is used to calculate the pathlength as follows:
 ##EQU5##
 wherein f is modulation frequency of the introduced light which is in the
 range of 10 MHz to 100 MHz; t.sup..lambda. is the photon migration delay
 time; c is the speed of photons in the scattering medium; and
 L.sup..lambda. is the migration pathlength. The modulation frequency of 50
 MHz was selected due to the frequency limitation of the LEDs and
 photodiodes. However, 10 for faster LEDs and photodiodes it may be
 desirable to use higher modulation frequencies that increase the phase
 shift resolution.
 At high modulation frequencies, i.e., 2.pi.f&gt;&gt;.mu..sub.a.multidot.c, the
 phase shift is no longer proportional to the mean time of flight &lt;t&gt;.
 ##EQU6##
 wherein .rho. is the source-detector separation; a=(6.pi./c).sup.1/2.
 sin.pi./4; (1-g).mu..sub.s is the effective scattering coefficient,
 .mu..sub.a.sup..lambda. is the absorption coefficient at wavelength
 .lambda., .alpha..sup..lambda. is the background absorbance at wavelength
 .lambda., and .theta..sub.0.sup..lambda. thus represents background
 scattering and absorption. At two wavelengths, the ratio of absorption
 coefficients is determined as follows:
 ##EQU7##
 The wavelengths are in the visible and infra-red range and are selected to
 have absorbance sensitive (or insensitive) to various tissue components
 such as water, cytochrome iron and copper, oxy- and deoxygenated forms of
 hemoglobin, myoglobin, melanin, glucose and other.
 For oxygenated and deoxygenated hemoglobin, the absorption coefficient
 written in terms of Beer Lambert relationship is as follows:
 ##EQU8##
 wherein .epsilon..sub.Hb.sup..lambda.1 and .epsilon..sub.HbO.sup..lambda.1
 are extinction coefficients for hemoglobin and deoxyhemoglobin that can be
 stored in a look up table; [Hb], [HbO.sub.2 ] are the tissue concentration
 of hemoglobin and oxyhemoglobin, respectively; .alpha..sup..lambda.1 is
 background absorbance at wavelength .lambda..sub.1.
 Tissue hemoglobin saturation can also be determined from dual-wavelength,
 dual-frequency measurements of the phase shift. For high modulation
 frequencies, (2.pi.f.sub.1 {character pullout}.mu..sub.a.sup..lambda.1 c
 and 2.pi.f.sub.2 {character pullout}.mu..sub.a.sup..lambda.2 c) the
 differences in the measured phase shift at one wavelength and two
 frequencies can be expressed as
 ##EQU9##
 The ratio of this difference measured at two wavelengths can thus be
 written
 ##EQU10##
 Since the scattering coefficient is wavelength-insensitive over the
 near-infrared range employed, this dual-frequency, dual-wavelength phase
 modulated spectroscopy can be used to obtain the ratio of absorption
 coefficients.
 Furthermore, as predicted from the diffusion approximation, the magnitude
 of the phase shift increases with the source-detector separation, .rho..
 Thus, in homogeneous tissues, the phase shifts measured for several .rho.
 can be used to calculate the absorption and scattering coefficients. These
 coefficients can be used either by employing Eq. 4 or the equations for
 the high and low approximations. Similarly, the magnitude of the detected
 radiation can be measured for different source-detector separations, and
 the absorption and scattering coefficients can be calculated by using Eq.
 5.
 The hemoglobin saturation is conventionally defined as follows:
 ##EQU11##
 For a three wavelength measurement, the hemoglobin saturation can be
 calculated using Eqs. (12) and (15) as follows:
 ##EQU12##
 Thus, processor 70 determines Y from the above equations for each
 wavelength .lambda..sub.1, .lambda..sub.2, .lambda..sub.3.
 In another embodiment, the spectrophotometer's electronics includes a low
 frequency module suitably and a high frequency module switchably coupled
 to the same source-detector probe 20. The low frequency module and the
 arrangement of the source-detector probe are substantially similar to the
 hemoglobinometer described in a co-pending U.S. patent application Ser.
 No. 701,127 filed May 16, 1991 which is incorporated by reference as if
 fully set forth herein. The low frequency module corresponds to a standard
 oximeter with modulation frequencies in the range of a few hertz to
 10.sup.4 hertz and is adapted to provide intensity attenuation data at two
 or three wavelengths. Then, the LEDs are switched to the high frequency
 phase modulation unit, similar to the unit of FIG. 1, which determines the
 average pathlength at each wavelength. The attenuation and pathlength data
 are sent to processor 70 for determination of a physiological property of
 the examined tissue.
 In another embodiment, the pathlength corrected oximeter utilizes the same
 LED sources (22a, 22b, 22c) sinusoidally modulated at a selected frequency
 comparable to the average migration time of photons scattered in the
 examined tissue on paths from the optical input port of the LED's to the
 optical detection part of the photodiode detectors (24a, 24b, 24c), but
 the electronic circuitry is different. Referring to FIG. 9, this
 embodiment utilizes a 200 MHz precision oscillator 61, which drives two
 laser diodes 62 and 64, again at 760 and 816 nm. The outputs of the laser
 diodes are time shared into filter optic coupling 68 and the head 70.
 Detector 72 provides output to an amplifier 74 and to two wide band double
 balance mixers (DBM) 76 and 78 which are coupled through a 90.degree.
 phase splitter 80 so that real (R) and imaginary (I) portions of the
 signal are obtained. The double balance mixers 76 and 78 preferably
 operate at the modulation frequency. The phase (.theta..sup..lambda.) is
 the angle whose tangent is the imaginary over the real part.
 ##EQU13##
 The amplitude is the square root of the sum of the squares of these values,
 providing the phase shift has been taken out as the residual phase shift
 .theta. 0 set to zero.
EQU A.sup..lambda. =(R.sup..lambda. +L ).sup.2 +L +(I.sup..lambda. +L ).sup.2
 +L (19)
 This embodiment uses summing and dividing circuits to calculate the
 modulation index, which is the quotient of the amplitude over the
 amplitude plus the DC component obtained from a narrow band detector 82.
 ##EQU14##
 The phase processor receives the phase shifts for the phase and amplitude
 values for two or three wavelengths and calculates the ratio of the phase
 shifts. For each wavelength, the phase shift and the DC amplitude are used
 to determine a selected tissue property, e.g., hemoglobin oxygenation.
 To study the influence of variation in the scattering coefficient on the
 quantitation of the absorption measurements, several simulations were
 performed. The simulations assumed the phase shift measurements at two
 wavelengths and several frequencies (10 MHz, 50 MHz, 200 MHz and 500 MHz).
 Hemoglobin saturation levels (Y) were varied in the range of
 5%.ltoreq.Y.ltoreq.100%, and the absorption coefficients were varied in
 the range of 0.5.ltoreq..mu..sub.a.ltoreq.1.5 cm.sup.-1, while the
 scattering coefficient .mu..sub.s '=5 cm.sup.-1 was kept constant; these
 values correspond to typical values for human tissue. FIGS. 10A and 10B
 show simulation results obtained by using the high frequency approximation
 (2.pi.{character pullout}.mu..sub.a c) for modulation frequencies f=50,
 200 and 500 MHz, assuming .theta..sub.0.sup..lambda.1
 =.theta..sub.0.sup..lambda.2 =.theta..sub.0, and .mu..sub.a
 c.apprxeq.2.multidot.10.sup.9.multidot..theta..sub.0. As shown in FIG.
 10A, the calculated saturation error decreases with frequency, but still
 introduces a significant error even for the 500 MHz at low saturation
 values. FIG. 10B shows the influence of added 5% noise for f=500 MHz. Low
 saturation values exhibit greater sensitivity to the introduced noise than
 high saturation values.
 The high sensitivity at low saturation values is expected for the high
 frequency approximation (Eq. 11). While the absorption coefficient for an
 isobestic wavelength does not change with saturation, lower saturation
 values yield lower values of the absorption coefficient for a contrabestic
 oxy-hemoglobin wavelength; this yields lower values of
 .theta..sup..lambda.2 -.theta..sub.0 in the denominator of Eq. 11. Thus,
 the .mu..sub.a ratio, at the two wavelengths, is more sensitive to noise
 at low saturation values.
 FIGS. 11A and 11B show simulation results obtained using the low frequency
 approximation (2.pi.{character pullout}.mu..sub.a c) for modulation
 frequencies f=10, 50 and 200 MHz, assuming .theta..sub.0.sup..lambda.1
 =.theta..sub.0.sup..lambda.2 =.theta..sub.0, and .mu..sub.a
 c.apprxeq.2.multidot.10.sup.9.multidot..theta..sub.0. As shown in FIG.
 11A, the low frequency approximation introduces lower error for the
 "intermediate" frequency of 200 MHz than the high frequency approximation
 shown in FIG. 10A. However, the low frequency approximation is much more
 sensitive to noise as shown in FIG. 11B. The relatively high sensitivity
 is again expected because the ratio of the absorption coefficients at the
 two wavelengths is obtained from the square the phase shift ratio, i.e.,
 .mu..sub.a.sup..lambda.2 /.mu..sub.a.sup..lambda.1 =(.theta..sup..lambda.1
 /.theta..sup..lambda.2).sup.2.
 Thus, when using the high and low frequency approximation, the calculated
 data may need to be corrected. The correction can be made by using look-up
 tables or other methods, such as dual frequency phase modulation
 measurement (Eq. 14) or phase modulation measurements with dual
 source-detector separation, to obtain more accurate information about the
 background phase shift.
 FIG. 12 shows simulation results for the oxygen is saturation obtained
 using Eq. 4 to calculate the ratio of absorption coefficients at the two
 wavelengths. This simulation assumed a correct value of the effective
 scattering coefficient (.mu..sub.s '=7 cm.sup.-1) and varied the
 "selected" tissue saturation (and thus the tissue absorption). For each
 "selected" saturation, the simulation calculated the absorption
 coefficient solving Eq. 4, while numerically varying .mu..sub.s ' from 3
 cm.sup.-1 to 13 cm.sup.-1 using the Newton-Raphson method. For each
 .mu..sub.s ', the error in the calculated saturation Y was calculated by
 subtracting the "selected" saturation from the "back-calculated"
 saturation. As shown in FIG. 12, for example, for a error of 3 cm.sup.-1
 in .mu..sub.s, the mean error in Y is about 2.5%, while the standard
 deviation does not exceed 1.59%. Thus, by employing Eq. 4, the phase
 modulation system can use an approximate value of the effective scattering
 coefficient to measure the oxygen saturation. The oxygen saturation is
 quite insensitive to the selection of the effective scattering coefficient
 as the introduced error is reduced by taking the ratio of the absorption
 coefficients.
 The phase modulation system is calibrated initially and may be recalibrated
 after several measurements to obtain a correct phase reading and an
 average drift. Another type of a phase modulation system is PMD-3000
 (available from NIM Incorporated, Philadelphia, Pa.), which is also
 described in U.S. Pat. No. 5,122,974. This phase modulation system uses
 two laser diodes at 754 nm and 780 nm, each having an average signal power
 5 mW. The two wavelengths are time shared using a mechanical shutter
 before the light is introduced in the tissue and then detected by a
 Hamamatsu R928 PMT detector. The system uses two frequencies of 200.000
 MHz and 200.025 MHz, and the detected signal is demodulated by
 heterodyning the second dynode of the PMT detector. The detected amplitude
 is used in a feed-back loop as an automatic gain control.
 The phase detector of the system provides a voltage output that is
 converted then to the phase as specified by the manufacturer. There are
 several techniques to determine the voltage-to-phase conversion curve,
 which ideally should be linear and the precision should be better that
 0.1.degree.. The conversion curve can be verified by changing the
 pathlength of the electrical or optical signal by changing the physical
 length of an electrical line. Here, one has to watch for a line mismatch
 that can potentially create measurement problems. Alternatively, the
 conversion curve can be verified by changing the source detector
 separation on an optical bench and measuring the corresponding voltage
 difference at the output of the phase detector. One has to prevent the
 phase amplitude cross-talk and operate the system at a proper
 signal-to-noise level.
 Alternatively, one can simulate a real experiment by using a tank
 containing an Intralipid.TM. solution of known absorption and scattering
 properties. (See Sevick et al., Analytical Biochemistry Vol. 195, p. 341.)
 The source-detector geometry resembles the actual tissue measurement
 geometry. The measured absorption coefficient can thus be compared to the
 known absorption coefficient. The voltage-to-phase curve is calibrated by
 taking multiple points at different blood concentrations.
 The phase modulation system also has a reference phase (.theta..sub.instr)
 that of course affects .theta..sub.0. The instrumental reference phase can
 be determined empirically or can be measured by butt-coupling the source
 and detector fibers. In this arrangement, the detected optical signal
 should be attenuated with a neutral density or NTR filter so the detector
 works in the same signal power range as for the in vivo tissue
 measurements.
 The instrumental reference phase can also be measured using a dual channel
 phase modulation system that provides both a phase output and an amplitude
 output. In this measurement, the above model should have similar
 scattering and no absorption, or known scattering and absorbing
 properties. The dual channel phase modulation system can resolve both
 .mu..sub.s ' and .mu..sub.a, which in turn are used to calculate the
 instrumental reference phase. Furthermore, the instrumental reference
 phase can also be determined by measuring the phase shift at different
 source-detector separations.
 The phase modulation system can use the amplitude in a feedback arrangement
 to control the laser intensity. (This type of feedback is similar to the
 automatic gain control (AGC) technique described above.) The intensity is
 adjusted in discrete steps so that no change in the laser intensity occurs
 during the measurement. This feedback system can measure tissue at a wide
 range of source-detector separations or background absorptions; there is
 no need to select an optical attenuator or adjust the gain (high voltage)
 of the detector. Furthermore, the detector can be operated in the optimum
 high voltage for all measurements.
 In an experimental study, six newborn piglets, age one to five days, were
 used (average weight--2.0 kg). After anesthesia and surgery, they were
 randomized either to preexisting mixed acidosis with a pH less than 7.00
 and a pCO.sub.2 larger than 8.0 kPa, or a normal pH and pCO.sub.2. The
 acidosis was induced by infusing lactic acid in a vein, and CO.sub.2 was
 added to the inspired air. Once the piglets were stabilized, the fraction
 of oxygen in the inspired air (the FiO.sub.2) was reduced from 21% to 6%
 for 30-40 minutes and then the piglets were resuscitated. Mean arterial
 blood pressure was kept above 40 mmHg at all times using an intravenous
 adrenaline infusion.
 A PMD-3000 system was used to perform the phase modulation measurements.
 Part of the scull skin was removed and the optical probes were fixed
 directly to the scull. Typical separations used were 1.7-2 cm. FIGS. 13A
 and 13B depict the filtered raw data and saturation calculation from a
 typical measurement. The filtering was done digitally by applying a median
 filter (kernel size 5) twice followed by a smoothing filter (kernel size
 11). The saturation was calculated by numerically solving Eq. 4 for the
 two wavelengths in order to compute the .mu..sub.a ratio as discussed
 above. The .mu..sub.s ' value for the pigs was selected to be 12
 cm.sup.-1.
 During the experimental study, the venous and arterial blood was sampled
 regularly and blood saturation was immediately calculated. Cerebro-venous
 saturation values were obtained through an indwelling superior sagittal
 sinus line and arterial values from a catheter in the femoral artery. The
 influence of the arterial blood sampling can been seen on FIG. 13B, where
 the observable sampling points have been marked with arrows, and the local
 variations are due to the local blood volume changes. The characteristic
 values of hemoglobin saturation for venous (Hbv) and arterial (Hba) blood
 are given in FIG. 13B as individual points.
 The calculated saturation is somewhat higher than what was expected for the
 6% FiO.sub.2 interval and lower for the 21% interval. This discrepancy can
 be correlated by measuring or compensating for water absorption, geometry
 and scull influence. Furthermore, the extinction coefficients were
 linearly interpolated for the used wavelengths from charts, and there are
 random errors introduced in the measurement or derivation of the
 .THETA..sub.instr.sup.754 and .THETA..sub.instr.sup.780 which may lead to
 systematic errors in the calculation.