# 2013 Laboratory E: Inverse Problems

**GOAL: Solve Spatially-Resolved Reflectance inverse problems.**

**I. Impact of Noise and Initial Guess on Spatially-Resolved Reflectance Measurements**

- Select the
**Inverse Solver Panel**. - For
**Fwd Solver Engine:**select "Scaled Monte Carlo - NURBS (g=0.8, n=1.4)", for**Inv Solver Engine:**select "Standard Diffusion (Analytic - Isotropic Point Source)". - In
**Solution Domain:**select "Steady State(*R*(ρ))". - Set begin and end locations to 0.5 and 9.5 mm, respectively with 10 points (every 1 mm).
- Set
**Optimization Parameters**to: μ_{a}and μ'_{s}. - Simulate measured data: set "Forward Simulation Optical Properties:" to: μ
_{a}= 0.01 mm^{-1}, μ'_{s}= 1 mm^{-1}, g = 0.8 and n = 1.4 and 2% noise. - Confirm the
**Hold On**checkbox is checked. - Click the
**Plot Measured Data**button. - Set
**Initial Guess Optical Properties:**to: μ_{a}= 0.05 mm^{-1}, μ'_{s}= 1.5 mm^{-1}, g = 0.8 and n = 1.4. - Click the
**Plot Initial Guess**button. - Click the
**Run Inverse Solver**button.

**Questions:**

- To what optical property values did the inverse solver converge? (Scroll to the bottom of the page to see the output).
- Why are the converged values not exactly the forward simulation optical properties?
- Perform the same analysis with 0% noise added to the simulated measured data. How accurate are the converged properties now?
- Perform the same analysis with initial guess μ
_{a}= 0.001 mm^{-1}, μ'_{s}= 0.5 mm^{-1}, g = 0.8 and n=1.4. How accurate are the converged properties now?

**II. Impact of Inverse Solver Model on Optical Property Recovery**

- Perform the same analysis changing the
**Inv Solver Engine**to "Scaled Monte Carlo - NURBS(g=0.8, n=1.4)". - Which Model Engine provided the more accurate converged values?

**III. Impact of Number of Measurements on Optical Property Recovery**

- Repeat II using only 2 detectors. Can you strategically place the two detectors to obtain the same accuracy in the converged values as you obtained with 10 detectors?
- Use the plots generated in Lab C, Section VII that showed the sensitivity of spatially-resolved diffuse reflectance to optical properties to help guide their placement.

**GOAL: Solve Spatial Frequency Domain Reflectance inverse problems.**

**IV. Sensitivity of Spatial Frequency Domain Reflectance to Optical Properties**

##### First let us examine the sensitivity of Spatial Frequency Domain Reflectance to optical absorption

- Go to the
**Forward Solver/Analysis Panel** - For
**Fwd Solver Engine:**select "Scaled Monte Carlo - NURBS (g=0.8, n=1.4)" - In
**Solution Domain:**select "Steady State(*R*(*f**x*))". - Set start and stop locations to 0 and 0.5 /mm, respectively with 51 points (every 0.01/mm).
- In
**Optical Properties:**enter μ_{a}= 0.01mm^{-1}, μ'_{s}=1mm^{-1}, n=1.4. - Click the
**Plot Reflectance**button. - Confirm the
**Hold On**checkbox is checked. - Now plot the spatial frequency domain reflectance when the tissue absorption is 10% higher i.e., for
**Optical Properties:**of μ_{a}= 0.011mm^{-1}, μ'_{s}=1mm^{-1} - Click the
**Plot Reflectance**button. - Now, in the
**Plot View**window, click the**Curve**Radio Button in the**Normalization**Controls. This operation divides all spatial frequency domain reflectance results in the plot view window by the first*R*(*f**x*) result. Thus each result becomes*R*_{i}(*f**x*) /*R*_{1}(*f**x*) where*R*_{1}(*f**x*) is the first reflectance result and*R*_{i}(*f**x*) is any successive reflectance result. This results in the first*R*_{1}(*f**x*) result getting transformed to a series of '1' values. - Note what spatial frequency regime shows the most sensitivity to μ
_{a}changes?

##### Now let us examine the sensitivity of Spatial Frequency Domain Reflectance to optical scattering

- Click
**Clear All**and set**Normalization**to**None**. - In
**Optical Properties:**enter μ_{a}= 0.01mm^{-1}, μ'_{s}=1mm^{-1}, n=1.4. - Click the
**Plot Reflectance**button. - Confirm the
**Hold On**checkbox is checked. - Now plot the spatial frequency domain reflectance when the tissue scattering is 10% higher i.e., for
**Optical Properties:**of μ_{a}= 0.01mm^{-1}, μ'_{s}=1.1mm^{-1} - Now, in the
**Plot View**window, click the**Curve**Radio Button in the**Normalization**Controls. This operation divides all spatial frequency domain reflectance results in the plot view window by the first*R*(*f**x*) result. Thus each result becomes*R*_{i}(*f**x*) /*R*_{1}(*f**x*) where*R*_{1}(*f**x*) is the first reflectance result and*R*_{i}(*f**x*) is any successive reflectance result. This results in the first*R*_{1}(*f**x*) result getting transformed to a series of '1' values. - Note what spatial frequency regime shows the most sensitivity to μ'
_{s}.

**V. Impact of Noise and Initial Guess on Spatial Frequency Domain Reflectance**

- Click
**Clear All**and set**Normalization**to**None**. - Select the
**Inverse Solver Panel**. - For
**Fwd Solver Engine:**select "Scaled Monte Carlo - NURBS (g=0.8, n=1.4)", for**Inv Solver Engine:**select "Standard Diffusion (Analytic - Isotropic Point Source)". - In
**Solution Domain:**select "Steady State(*R*(*f**x*))". - Set begin and end locations to 0 and 0.5 /mm, respectively with 11 points (every 0.05/mm).
- Set
**Optimization Parameters**to: μ_{a}and μ'_{s}. - Simulate measured data: set "Forward Simulation Optical Properties:" to: μ
_{a}= 0.01 mm^{-1}, μ'_{s}= 1 mm^{-1}, g = 0.8 and n = 1.4 and 2% noise. - Confirm the
**Hold On**checkbox is checked. - Click the
**Plot Measured Data**button. - Set "Initial Guess Optical Properties:" to: μ
_{a}= 0.05 mm^{-1}, μ'_{s}= 1.5 mm^{-1}, g = 0.8 and n = 1.4. - Click the
**Plot Initial Guess**button. - Click the
**Run Inverse Solver**button.

**Questions:**

- To what optical property values did the inverse solver converge? (Scroll to the bottom of the page to see the output).
- Perform the same analysis with 0% noise added to the simulated measured data. How accurate are the converged properties now?
- Perform the same analysis with initial guess μ
_{a}= 0.001 mm^{-1}, μ'_{s}= 0.5 mm^{-1}, g = 0.8 and n=1.4. How accurate are the converged properties now?

**VI. Effect of Measurement Range on Sensitivity to Optical Absorption and Scattering in Spatial Frequency Domain Reflectance**

- Go to the
**Inverse Solver Panel**. - Follow the instructions provided in Section V
**except**modify the**Spatial Frequency**begin and end values to those obtained in Section IV. "Impact of Optical Properties on Spatial Frequency Domain Reflectance". - Rerun the inverse solver.

**Questions**

- Were you able to improve the μ
_{a}and μ'_{s}converged properties? - In what spatial frequency domain is reflectance most sensitive to μ
_{a}? Why is this? - In what spatial frequency domain is reflectance most sensitive to μ'
_{s}? Why is this?