Introduction to Inverse Adding-Doubling (IAD)

1. Introduction to IAD

Overview

Inverse Adding-Doubling (IAD) is a powerful numerical technique used to determine the intrinsic optical properties of turbid sample—specifically the absorption coefficient ($\mu_a$), the reduced scattering coefficient ($\mu_s’$), and the anisotropy factor ($g$)—from macroscopic measurements of total reflection ($R$) and transmission ($T$). It iteratively solves the inverse problem by calling the Adding-Doubling (AD) forward model to predict $R$ and $T$ from candidate parameters until they match the experimental values within a specified tolerance.

The software package typically provides two main executables:

• IAD (Inverse Mode): Calculates optical properties ($\mu_a, \mu_s’, g$) from measured reflection ($R$) and transmission ($T$).
• AD (Forward Mode): Calculates reflection ($R$) and transmission ($T$) given known optical properties.

This method is highly robust and valid over a wide range of optical thicknesses and albedos, provided the experimental geometry satisfies the assumptions of the radiative transport equation (e.g., uniform illumination, semi-infinite slab).

Fundamental Relations

To understand the inputs and outputs, we define the following relationships:

• True vs. Reduced Scattering Coefficient:
  $$ \mu_s = \frac{\mu_s’}{1 - g} $$
• Single-Scattering Albedo ($a$):
  $$ a = \frac{\mu_s}{\mu_a + \mu_s} $$
• Optical Thickness ($b$):
  $$ b = (\mu_a + \mu_s) \times d $$
  where $d$ is the physical thickness of the sample (typically $d = 10 \text{ mm}$ in our simulations).

Critical Note on $g$:
Some GitHub examples compute $a$ and $b$ using $\mu_s’$ directly, effectively assuming $g=0$. However, biological tissues typically have a high anisotropy factor ($g \approx 0.9$). Using $\mu_s’$ instead of the true $\mu_s$ leads to significant underestimation of scattering and incorrect predictions. Always derive $\mu_s$ from $\mu_s’$ and $g$ before calculating $a$ and $b$.

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2. Simulation Setup with MOP-MCML

To validate the IAD method, we selected optical parameters for human skin tissues (Fat and Muscle) based on literature comparisons.

Optical Parameters

The following table summarizes the parameters derived from Bashkatov et al. [1] and Simpson et al. [2] for a sample thickness $d = 10 \text{ mm}$.

Tissue	$\lambda$ (nm)	$\mu_a (\text{cm}^{-1})$	$\mu_s’ (\text{cm}^{-1})$	$\mu_s (\text{cm}^{-1})$	$a$	$b$	$n$	$g$
Fat	700	1.11	12.20	122.0	0.9909	123.11	1.455	0.9
Fat	800	1.07	11.15	111.5	0.9905	112.57	1.455	0.9
Fat	900	1.06	10.29	102.9	0.9898	103.96	1.455	0.9
Muscle	700	0.48	8.18	81.8	0.9942	82.28	1.37	0.9
Muscle	800	0.28	7.04	70.4	0.9960	70.68	1.37	0.9
Muscle	900	0.32	6.21	62.1	0.9949	62.42	1.37	0.9

MOP-MCML Simulation Results

We performed Monte Carlo simulations (MOP-MCML) using these parameters to generate “ground truth” Reflectance ($R$) and Transmittance ($T$) values for verification.

Fat Tissue Results:

• 700 nm: $R = 0.2462$, $T = 0.0011$
• 800 nm: $R = 0.2406$, $T = 0.0017$
• 900 nm: $R = 0.2309$, $T = 0.0022$

Muscle Tissue Results:

• 700 nm: $R = 0.3453$, $T = 0.0236$
• 800 nm: $R = 0.4064$, $T = 0.0632$
• 900 nm: $R = 0.3645$, $T = 0.0644$

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3. Model Verification: AD vs. IAD vs. MCML

We conducted a cross-verification study to assess the consistency between the analytical AD model, the numerical IAD inversion, and the stochastic MOP-MCML simulation.

Workflow

1. Forward AD: Calculate $R_{AD}$ and $T_{AD}$ using the known parameters ($\mu_a, \mu_s, g$).
2. Inverse IAD: Recover parameters ($\mu_{a, rec}, \mu_{s, rec}'$) from both $R_{AD}/T_{AD}$ and $R_{MCML}/T_{MCML}$.
3. Comparison: Evaluate the deviation of recovered parameters from the original inputs.

Comparative Results (Fat Tissue)

$\lambda$	Source	Input $R$	Input $T$	Recovered $\mu_a$	Recovered $\mu_s’$	Original $\mu_a$	Original $\mu_s’$
700nm	AD	0.2830	0.0011	0.1696	0.8615	0.111	1.22
	MCML	0.2462	0.0011	0.1880	0.7637
800nm	AD	0.2754	0.0016	0.1616	0.7848	0.107	1.115
	MCML	0.2406	0.0017	0.1789	0.7021
900nm	AD	0.2654	0.0021	0.1609	0.7354	0.106	1.029
	MCML	0.2309	0.0022	0.1761	0.6500

Analysis

• Consistency: The AD forward model produces slightly higher reflectance ($R$) than MCML, likely due to differences in boundary condition handling or light source modeling (MCML models a finite beam, while AD assumes infinite plane wave).
• Inversion Accuracy: The parameters inverted from AD results (“AD → IAD”) are closer to the original parameters than those inverted from MCML results (“MCML → IAD”), but both show a systematic offset. $\mu_a$ values are generally overestimated, while $\mu_s’$ are underestimated.
• Implication: While IAD is rigorous for specific geometries (integrating spheres), direct comparison with MCML requires careful matching of source and detector geometries.

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4. Polyurethane Phantom Test

To test the method on real experimental data, we used a polyurethane phantom (skin/fat analog). We compared the spectral response measured experimentally against predictions from both IAD and MOP-MCML.

Experimental Data

• Sample: 1-inch port-type polyurethane phantom.
• Data Source: vio-A.rxt (650–850 nm).
• Inversion: Used iad to extract $\mu_a$ and $\mu_s’$ assuming $g=0.9$.

Comparison: IAD vs. MOP-MCML Prediction

We used the inverted parameters to predict $R$ and $T$ back using both methods.

• IAD Prediction Error: $R \approx 7.3%$, $T \approx 0.8%$. IAD is self-consistent as it fits the parameters to the data.
• MOP-MCML Prediction Error: $R \approx 29.8%$, $T \approx 9.7%$.

Interpreting the Discrepancy

The larger error in MOP-MCML predictions stems from:

1. Source-Model Mismatch: The experimental light source and detector geometry (likely integrating sphere ports) differ from the simplified Gaussian/Point sources used in our MCML configuration.
2. Inverse-Forward Evaluation Gap: IAD minimizes error essentially by “fitting” the AD model to reliability. feeding those fitted parameters into a different forward model (MCML) naturally exposes geometric and boundary condition differences.

Despite these numerical differences, the error magnitude is comparable to the intrinsic parameter uncertainty in commercial phantoms (often $\sim 40%$), confirming that MOP-MCML provides physically reasonable estimates within the expected tolerance.

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References

[1] Bashkatov et al. “Optical properties of human skin, subcutaneous and mucous tissues in the wavelength range from 400 to 2000 nm.” J. Phys. D: Appl. Phys., 38(15), 2543, 2005. DOI: 10.1088/0022-3727/38/15/004.
[2] Simpson et al. “Near-infrared optical properties of ex vivo human skin and subcutaneous tissues measured using the Monte Carlo inversion technique.” Phys. Med. Biol., 43(9), 2465-78, 1998. DOI: 10.1088/0031-9155/43/9/003.