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Lecture 10

Lecture 10. The time-dependent transport equation. Spatial photon gradient. Absorbed photons. Photons scattered to direction ŝ'. Light source q. Photons scattered into direction ŝ from ŝ' . Time-Dependent Transport Equation.

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Lecture 10

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  1. Lecture 10

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

  3. Time-Dependent Transport Equation • Typically the transport equation is expressed in terms of the radiance ( I(r,ŝ,t) =N(r,ŝ,t)hnc) , 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(r,t) is the photon density • And J(r,t) is the photon current density (photon flux)

  7. Fick’s 1st law of diffusion Movement or flux in response to a concentration gradient in a medium with diffusivity c 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: Diffuse intensity is greater in the direction of net flux flow Photon source expansion:

  9. Diffusion approximation • Plug in, integrate over w, 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 frequency domain (TD) Continuous wave (CW) Time domain (TD) I0 I0 It It intensity intensity t (ns) tissue t=0 ~ns phase shift intensity ac dc t (ns)

  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)=d(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(-iwt), so that ∂/∂t -iw • Diffusion equation takes the form

  13. Point source solution: frequency domain frequency domain (TD) • Green’s function for homogeneous, infinite medium containing a harmonically modulated point source of power P(w) at r=0 is intensity t (ns) tissue phase shift intensity ac intercept (ms’) t (ns) dc phase ln(r2*Idc) slope (ma, ms’) slope (ma, ms’) r r intercept = 0

  14. Frequency domain measurements • The slope of r*IDC as a function of r and the slope of the phase as a function of r depend on ma and ms'. • 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 HbO2 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 O2 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. Near-infrared spectroscopyand imaging of tissue Sergio Fantini Department of Biomedical Engineering Tufts University, Medford, MA

  20. volume probed by near-infrared photons outline source 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 detector source detector source detector

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

  22. Dominant tissue chromophoresin the near infrared ultraviolet near infrared visible light 770 nm 410 nm 600 1300 absorption coefficient (cm-1) wavelength (nm) Hb, HbO2 from: Cheong et al., IEEE J. Quantum Electron. 26, 2166 (1990) H2O from: Hale and Querry, Appl. Opt. 12, 555 (1973)

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

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

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

  26. Diffusion equation: frequency domain • Harmonic time dependence is given by factor exp(-iwt), so that ∂/∂t -iw • 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(w) at r=0 is frequency domain (TD) intensity t (ns) tissue phase shift intensity ac intercept (ms’) t (ns) dc phase slope (ma, ms’) ln(r*Idc) slope (ma, ms’) r r intercept = 0

  28. TISSUE OXIMETRY

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

  30. 0 5 1 0 2 0 2 5 3 0 Configuration for tissue oximetry ) % ( n o i t a r u t a S n i b o l g o m e H e ( m i n ) T i m RF electronics detector optical fiber detector source optical fibers laser driver 2.0 cm laser diodes multiplexing circuit measuring probe main box

  31. Frequency-domain oximetry HbO2and Hb (mM) saturation (%) ischemia ischemia time (min) time (min) 750nm 830nm ms’ (1/cm) ma (1/cm) 750nm 830nm ischemia ischemia time (min) time (min)

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