1. Lecture 10

2. The time-dependent transport equation Spatial photon gradient Photons scattered to direction ŝ' Absorbed photons Photons scattered into direction ŝ from ŝ' Light source q

3. Time-Dependent Transport Equation Typically the transport equation is expressed in terms of the radiance (I(r,ŝ,t) =N(r,ŝ,t)h c), and after dropping the integrals

4. Time-Independent Transport Equation For the steady-state situation, we assume that radiance is independent of time, and the transport equation becomes

5. Approximations The transport equation is difficult to solve analytically.In order to find an analytical solution we need to simplify the problem. Discretization methods –Discrete ordinates method –Kubelka-Monk theory –Adding-doubling method Expansion methods –Diffusion theory Probabilistic methods –Monte Carlo simulations

6. Diffusion approximation Expand the photon distribution in an isotropic and a gradient part Where  (r,t) is the photon density And J(r,t) is the photon current density (photon flux)

7. Fick’s 1 st law of diffusion Movement or flux in response to a concentration gradient in a medium with diffusivity  Photon flux (J cm -2 s -1 ) in response to a photon density gradient, characterized by the diffusion coefficient D, defined as

8. Diffusion approximation Transport equation: Photon distribution expansion: Photon source expansion: Diffuse intensity is greater in the direction of net flux flow

9. Diffusion approximation Plug in, integrate over , and assume only isotropic sources (refer to supplementary material for full derivation) Assume a constant D and use the relation for the fluence rate To get

10. Types of diffuse reflectance measurements Continuous wave (CW)Time domain (TD) I0I0 ItIt I0I0 ItIt t=0~ns intensity frequency domain (TD) phase shift tissue t (ns) intensity dc ac

11. Point source solution: time- domain The solution to the diffusion equation for an infinite homogeneous slab with a short pulse isotropic point source S(r,t)=  (0,0) is This is known as the Green’s function solution and can be used to solve more complicated problems

12. Point source solution: frequency domain Harmonic time dependence is given by factor exp(-i  t), so that ∂/∂t -i  Diffusion equation takes the form

13. Point source solution: frequency domain Green’s function for homogeneous, infinite medium containing a harmonically modulated point source of power P(  ) at r=0 is frequency domain (TD) phase shift tissue t (ns) intensity dc ac ln(r 2 *I dc ) r intercept (  s ’) slope (  a,  s ’) phase r intercept = 0 slope (  a,  s ’)

14. Frequency domain measurements The slope of r*I DC as a function of r and the slope of the phase as a function of r depend on  a and  s '. Find the slopes and extract the optical properties

15. Medical applications of reflectance spectroscopy Pulse Oximetry Frequency domain NIR spectroscopy and imaging Steady-state diffuse reflectance spectroscopy

16. The Pulse oximeter Function: Measure arterial blood saturation Advantages: –Non-invasive –Highly portable –Continuous monitoring –Cheap –Reliable

17. The pulse oximeter How: –Illuminate tissue at 2 wavelengths straddling isosbestic point (eg. 650 and 805 nm) Isosbestic point: wavelength where Hb and HbO 2 spectra cross. –Detect signal transmitted through finger Isolate varying signal due to pulsatile flow (arterial blood) Assume detected signal is proportional to absorption coefficient (Two measurements, two unknowns) Calibrate instrument by correlating detected signal to arterial saturation measurements from blood samples

18. The pulse oximeter Limitations: –Reliable when O 2 saturation above 70% –Not very reliable when flow slows down –Can be affected by motion artifacts and room light variations –Doesn’t provide tissue oxygenation levels

19. Department of Biomedical Engineering Tufts University, Medford, MA Near-infrared spectroscopy and imaging of tissue Sergio Fantini

20. outline Near-infrared spectroscopy and imaging of tissues applications to skeletal muscles hemoglobin oxygenation (absolute) hemoglobin concentration (absolute) blood flow and oxygen consumption applications to the human breast detection of breast cancer spectral characterization of tumors applications to the human brain optical monitoring of cortical activation intrinsic optical signals from the brain volume probed by near-infrared photons source detector source detector

21. Why near-infrared spectroscopy and imaging of tissues? Non-invasive Non-ionizing Real-time monitoring Portable systems Cost effective

22. Dominant tissue chromophores in the near infrared ultravioletnear infrared 410 nm600 770 nm 1300 wavelength (nm) Hb, HbO 2 from: Cheong et al., IEEE J. Quantum Electron. 26, 2166 (1990) H 2 O from: Hale and Querry, Appl. Opt. 12, 555 (1973) absorption coefficient (cm -1 )

23. Diffusion of near-infrared light inside tissues low power laser biological tissue optical detector optical fiber

24. high scattering problem is there a car in front of me? is there a cookie in the milk?

25. Frequency-domain spectroscopy (FD) phase shift tissue t (ns) intensity (a.u.) dcac

26. Diffusion equation: frequency domain Harmonic time dependence is given by factor exp(-i  t), so that ∂/∂t -i  Diffusion equation takes the form

27. Point source solution: frequency domain Green’s function for homogeneous, infinite medium containing a harmonically modulated point source of power P(  ) at r=0 is frequency domain (TD) phase shift tissue t (ns) intensity dc ac ln(r*I dc ) r intercept (  s ’) slope (  a,  s ’) phase r intercept = 0 slope (  a,  s ’)

28. TISSUE OXIMETRY

29. Time-domain oximetry Miwa et al., Proc. SPIE 2389, 142 (1995)

30. Configuration for tissue oximetry H e m o g l o b i n S a t u r a t i o n ( % ) T i m e ( m i n ) measuring probe main box 2.0 cm laser diodes source optical fibers detector optical fiber laser driver detector RF electronics multiplexing circuit

31. HbO 2 and Hb (  M) Frequency-domain oximetry 750nm time (min)  a (1/cm) 830nm 750nm 830nm ischemia time (min) ischemia  s ’ (1/cm) time (min) saturation (%) time (min) ischemia