Introduction of IAD

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IAD (Inverse Adding-Doubling) is based on the AD (Adding-Doubling) method, supplemented by a Monte Carlo correction. It allows the intrinsic optical parameters of the sample to be deduced from the measured values of reflectance $R$ and transmittance $T$.

1. Introduction to IAD

Overview

IAD (Inverse Adding–Doubling) builds on the Adding–Doubling (AD) forward model and incorporates a Monte-Carlo–based correction to infer a sample’s intrinsic optical parameters from measured reflectance $R$ and transmittance $T$. The intrinsic parameters are the absorption coefficient $\mu_a$, the reduced scattering coefficient $\mu_s’$, and the anisotropy factor $g$.

The software exposes two complementary workflows:

  • IAD (inversion). Iteratively propose a parameter set, generate $R$ and $T$ via AD, compare to the measurements, and iterate until the error falls below a prescribed tolerance.
  • AD (forward). Given known optical parameters, compute the corresponding $R$ and $T$.

Under appropriate geometries and boundary conditions, this methodology is valid over a broad range of single-scattering albedo, optical thickness, and phase functions, and typically recovers parameters with small error.

Fundamental relations

  • True (non-reduced) scattering coefficient
    $$
    μ_s=\frac{μ_s’}{​1−g}
    $$
  • Single-scattering albedo
    $$
    a=\frac{μ_s}{μ_a+μ_s}
    $$
  • Physical thickness
    $$
    d = 10 mm
    $$
  • Optical thickness
    $$
    b=(μ_a+μ_s) × d
    $$

Notes on $a$ and $b$

Some example programs on GitHub compute $a$ and $b$ directly from $\mu_s’$ because those examples fix $g = 0$, making $\mu_s’ \equiv \mu_s$. In realistic biomedical optics, one often assumes $g \approx 0.9$, in which case $\mu_s’$ and $\mu_s$ differ materially. Since the definitions of $a$ and $b$ depend on $\mu_s$ (not $\mu_s’$), continuing to use $\mu_s’$ is equivalent to implicitly setting $g=0$. This systematically underestimates scattering and overestimates absorption and transmittance, producing results that are not physically consistent.

In the first experiment, I followed the example approach and computed $a$ and $b$ from $\mu_a$ and $\mu_s’$. The resulting $R$ and $T$ were strongly inconsistent with parameters recovered by IAD. After revising the computation to use $\mu_s$ per the definitions, discrepancies between AD predictions and IAD-validated values were markedly reduced, which improved IAD’s practical utility (see the comparative section).
Additionally, in early trials, optical thickness $b$ was mistakenly conflated with physical thickness $d$, which caused errors in AD-mode predictions of $R$ and $T$.

2. MOP-MCML Tests

Choice of optical parameters (units: mm)

Optical properties of human tissues have been reported in prior work. For example, Bashkatov et al. [1] measured reflectance and transmittance of ex-vivo subcutaneous fat (carefully de-blooded and kept moist), then used two integrating spheres with IAD to obtain spectra of $\mu_a$ and $\mu_s$ from 400–2000 nm. Simpson et al. [2] used the single integrating-sphere comparison method with inverse Monte Carlo to retrieve optical parameters of epidermis, subcutaneous fat, and muscle in 620–1000 nm.

Bashkatov also proposed a power-law fit for human fat reduced scattering in 600–1500 nm:
$$
\mu_s’ = \frac {1.05 \times 10^3} {\lambda^{0.68}}
$$
with $\lambda$ in nm and $\mu_s’$ in $\text{cm}^{-1}$.

Below we select representative wavelengths and two tissues (fat and muscle) for testing, with fixed thickness $d = 10~\text{mm}$.

Tissue $μ_a(cm⁻¹)$ $μ_s’(cm⁻¹)$ $μ_s(cm⁻¹)$ $a$ $b(mm)$ $n$ $g$
fat 700 1.11 12.20 122 0.990983673 123.11 1.455 0.9
fat 800 1.07 11.15 111.5 0.990494803 112.57 1.455 0.9
fat 900 1.06 10.29 102.9 0.989803771 103.96 1.455 0.9
mus 700 0.48 8.18 81.8 0.994166262 82.28 1.37 0.9
mus 800 0.28 7.04 70.4 0.996038483 70.68 1.37 0.9
mus 900 0.32 6.21 62.1 0.994873438 62.42 1.37 0.9

MOP-MCML input file

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# File created automatically by create_MCML_input_file...

1.0   # file version
6   # Number of runs

### Specify data for run 1 ###
fat700.mco A  # output filename, ASCII/Binary
200000   # No. of photons
0.001 0.002  # dz, dr
1000 1000 1  # No. of dz, dr & da

1   # No. of layers
1.000000   # n for medium above 
#n     mua    mus    g    d # [DOI : 10.1142/S1793545811001319]
1.455  1.11  122  0.90  1 # Gras
1.000000   # n for medium below

1.2 0.0  0.05  0.8 0.0  0.05  # PD : (Rx, Ry), Rl, (Tx, Ty), Tl (in cm)
1  0.8 0.0  0.05     # light source : param 1: 1.point 2.Gaussian 3.Flat; 

### Specify data for run 2 ###
fat800.mco A  # output filename, ASCII/Binary
200000   # No. of photons
0.001 0.002  # dz, dr
1000 1000 1  # No. of dz, dr & da

1   # No. of layers
1.000000   # n for medium above 
#n     mua    mus    g    d # [DOI : 10.1142/S1793545811001319]
1.455  1.07  111.5  0.90  1 # Gras
1.000000   # n for medium below

1.2 0.0  0.05  0.8 0.0  0.05  # PD : (Rx, Ry), Rl, (Tx, Ty), Tl (in cm)
1  0.8 0.0  0.05     # light source : param 1: 1.point 2.Gaussian 3.Flat; 

### Specify data for run 3 ###
fat900.mco A  # output filename, ASCII/Binary
200000   # No. of photons
0.001 0.002  # dz, dr
1000 1000 1  # No. of dz, dr & da

1   # No. of layers
1.000000   # n for medium above 
#n     mua    mus    g    d # [DOI : 10.1142/S1793545811001319]
1.455  1.06  102.9  0.90  1 # Gras
1.000000   # n for medium below

1.2 0.0  0.05  0.8 0.0  0.05  # PD : (Rx, Ry), Rl, (Tx, Ty), Tl (in cm)
1  0.8 0.0  0.05     # light source : param 1: 1.point 2.Gaussian 3.Flat; 

### Specify data for run 4 ###
musc700.mco A  # output filename, ASCII/Binary
200000   # No. of photons
0.001 0.002  # dz, dr
1000 1000 1  # No. of dz, dr & da

1   # No. of layers
1.000000   # n for medium above 
#n     mua    mus    g    d # [DOI : 10.1142/S1793545811001319]
1.37  0.48  81.8  0.90  1 # Gras
1.000000   # n for medium below

1.2 0.0  0.05  0.8 0.0  0.05  # PD : (Rx, Ry), Rl, (Tx, Ty), Tl (in cm)
1  0.8 0.0  0.05     # light source : param 1: 1.point 2.Gaussian 3.Flat; 

### Specify data for run 5 ###
musc800.mco A  # output filename, ASCII/Binary
200000   # No. of photons
0.001 0.002  # dz, dr
1000 1000 1  # No. of dz, dr & da

1   # No. of layers
1.000000   # n for medium above 
#n     mua    mus    g    d # [DOI : 10.1142/S1793545811001319]
1.37  0.28  70.4  0.90  1 # Gras
1.000000   # n for medium below

1.2 0.0  0.05  0.8 0.0  0.05  # PD : (Rx, Ry), Rl, (Tx, Ty), Tl (in cm)
1  0.8 0.0  0.05     # light source : param 1: 1.point 2.Gaussian 3.Flat;

### Specify data for run 6 ###
musc900.mco A  # output filename, ASCII/Binary
200000   # No. of photons
0.001 0.002  # dz, dr
1000 1000 1  # No. of dz, dr & da

1   # No. of layers
1.000000   # n for medium above 
#n     mua    mus    g    d # [DOI : 10.1142/S1793545811001319]
1.37  0.32  62.1  0.90  1 # Gras
1.000000   # n for medium below

1.2 0.0  0.05  0.8 0.0  0.05  # PD : (Rx, Ry), Rl, (Tx, Ty), Tl (in cm)
1  0.8 0.0  0.05     # light source : param 1: 1.point 2.Gaussian 3.Flat;

Fat

700 nm

1
R = 0.246177   T = 0.00108857

800 nm

1
R = 0.240579   T = 0.00166073

900 nm

1
R = 0.230887   T = 0.00215007

Muscle

700 nm

1
R = 0.345293   T = 0.0236438

800 nm

1
R = 0.406412   T = 0.0631558

900 nm

1
R = 0.364507   T = 0.0644091

3. AD-Mode Tests (predict $R, T$ from $\mu_a, \mu_s’$)

Fat

700 nm

Command

1
./ad -a 0.990983673 -b 123.11 -g 0.9

Result

1
R = 0.2830231429296444   T = 0.0010930802183922405

800 nm

Command

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./ad -a 0.990494803 -b 112.57 -g 0.9

Result

1
R = 0.27539070448930375   T = 0.001622275392566738

900 nm

Command

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./ad -a 0.989803771 -b 103.96 -g 0.9

Result

1
R = 0.26537438334599106   T = 0.0021053395468718567

Muscle

700 nm

Command

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./ad -a 0.994166262 -b 82.28 -g 0.9

Result

1
R = 0.34842633123318656   T = 0.022562949774397136

800 nm

Command

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./ad -a 0.996038483 -b 70.68 -g 0.9

Result

1
R = 0.4080569824320218   T = 0.06126586355782593

900 nm

Command

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./ad -a 0.994873438 -b 62.42 -g 0.9

Result

1
R = 0.3678904511387823   T = 0.06152422024329378

4. IAD-Mode Tests (recover $\mu_a, \mu_s’$ from $R,T$)

Fat

700 nm

Commands

1
./iad -r 0.2830231429296444 -t 0.0010930802183922405 -d 10.0 -g 0.9
1
./iad -r 0.246177 -t 0.00108857 -d 10.0 -g 0.9

Results ($mm^{-1}$)

1
mu_a = 0.1696     mu_s' = 0.8615
1
mu_a = 0.1880     mu_s' = 0.7637

800 nm

Commands

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./iad -r 0.27539070448930375 -t 0.001622275392566738 -d 10.0 -g 0.9
1
./iad -r 0.240579 -t 0.00166073 -d 10.0 -g 0.9

Results ($mm^{-1}$)

1
mu_a = 0.1616     mu_s' = 0.7848
1
mu_a = 0.1789     mu_s' = 0.7021

900 nm

Commands

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./iad -r 0.26537438334599106 -t 0.0021053395468718567 -d 10.0 -g 0.9
1
./iad -r 0.230887 -t 0.00215007 -d 10.0 -g 0.9

Results ($mm^{-1}$)

1
mu_a = 0.1609     mu_s' = 0.7354
1
mu_a = 0.1761     mu_s' = 0.6500

Muscle

700 nm

Commands

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./iad -r 0.34842633123318656 -t 0.022562949774397136 -d 10.0 -g 0.9
1
./iad -r 0.345293 -t 0.0236438 -d 10.0 -g 0.9

Results ($mm^{-1}$)

1
mu_a = 0.0783     mu_s' = 0.5808
1
mu_a = 0.0781     mu_s' = 0.5694

800 nm

Commands

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./iad -r 0.4080569824320218 -t 0.06126586355782593 -d 10.0 -g 0.9
1
./iad -r 0.406412 -t 0.0631558 -d 10.0 -g 0.9

Results ($mm^{-1}$)

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mu_a = 0.0478     mu_s' = 0.5014
1
mu_a = 0.0475     mu_s' = 0.4939

900 nm

Commands

1
./iad -r 0.3678904511387823 -t 0.06152422024329378 -d 10.0 -g 0.9
1
./iad -r 0.364507 -t 0.0644091 -d 10.0 -g 0.9

Results ($mm^{-1}$)

1
mu_a = 0.0542     mu_s' = 0.4526
1
mu_a = 0.0538     mu_s' = 0.4414

5. Comparative Assessment of IAD Results

A cross-comparison shows that AD and MCML produce similar $R$ and $T$, with transmittance nearly identical. Inverting both with IAD indicates that AD → IAD tends to recover the original optical parameters slightly more faithfully. In all cases, the inversion error for $\mu_a$ is consistently smaller than for $\mu_s’$.

Fat

700nm

Parameter MCML MCML -> IAD AD -> IAD
$μ_a$ 0.111 0.1880 0.1696
$μ_s’$ 1.22 0.7637 0.8615
R - 0.246177 0.2830231429296444
T - 0.00108857 0.0010930802183922405

800nm

Parameter MCML MCML -> IAD AD -> IAD
$μ_a$ 0.107 0.1789 0.1616
$μ_s’$ 1.115 0.7021 0.7848
R - 0.240579 0.27539070448930375
T - 0.00166073 0.001622275392566738

900nm

Parameter MCML MCML -> IAD AD -> IAD
$μ_a$ 0.106 0.1761 0.1609
$μ_s’$ 1.029 0.6500 0.7354
R - 0.230887 0.26537438334599106
T - 0.00215007 0.0021053395468718567

Muscle

700nm

Parameter MCML MCML -> IAD AD -> IAD
$μ_a$ 0.048 0.0781 0.0783
$μ_s’$ 0.818 0.5694 0.5808
R - 0.345293 0.34842633123318656
T - 0.0236438 0.022562949774397136

800nm

Parameter MCML MCML -> IAD AD -> IAD
$μ_a$ 0.028 0.0475 0.0478
$μ_s’$ 0.704 0.4939 0.5014
R - 0.406412 0.4080569824320218
T - 0.0631558 0.06126586355782593

900nm

Parameter MCML MCML -> IAD AD -> IAD
$μ_a$ 0.032 0.0538 0.0542
$μ_s’$ 0.621 0.4414 0.4526
R - 0.364507 0.3678904511387823
T - 0.0644091 0.06152422024329378

6. Polyurethane Sample Test

Recovering optical parameters from $R$ and $T$

The material is a one-inch port-type polyurethane sample used as a soft-tissue phantom (skin/fat analog). Data come from the IAD GitHub file vio-A, which provides $R$ and $T$ from 650 nm to 850 nm in 1 nm steps. The source and photodiode are placed in opposition (T-mode) so that a broad angular distribution illuminates the sample and the transmitted flux is collected.

We obtain $\mu_a$ and $\mu_s’$ from measured $R$ and $T$ via:

1
./iad -M 0 -q 4 -g 0.9 test/vio-A

Here, M denotes the model used by iad; only model 0 is publicly documented and commonly used. Parameter q specifies the illumination condition under which $R$ and $T$ were measured. To emulate human tissue, we set $g = 0.9$ and the sample thickness to 6.670 mm. A generated output example is available as vio-A.txt (shared link).

Predicting $R$ and $T$ with IAD and MOP-MCML

Given the data volume, batch execution via iadpython is preferable. After preparing input parameters and exporting the computed $a$ and $b$ to CSV, run:

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import pandas as pd
import iadpython as iad

g = 0.9
n = 1.468
n_above  = 1.00
n_below  = 1.00

df = pd.read_csv("albedo.csv")
a_vals = df["a"].astype(float)
b_vals = df["b"].astype(float)

rows = []
for a_i, b_i in zip(a_vals, b_vals):
    s = iad.Sample(a=a_i, b=b_i, g=g, n=n, n_above=n_above, n_below=n_below)
    UR, UT, URU, UTU = s.rt()
    rows.append({
        "a": a_i, "b": b_i, "g": g,
        "UR_total": UR, "UT_total": UT,
        "URU_diffuse": URU, "UTU_diffuse": UTU
    })

pd.DataFrame(rows).to_csv("ad_results.csv", index=False)

IAD baseline. Comparing IAD-predicted $R$ and $T$ with the input measurements over 650–850 nm yields a mean reflectance difference of $7.34\% \pm 1.11\%$ and a mean transmittance difference of $0.83\% \pm 0.63\%$. Across the band, IAD errors remain modest—peaking at $\sim 8.7\%$ for $R$ and $\sim 2.1\%$ for $T$—with minima near 690 nm for $T$ approaching zero.

MOP-MCML comparison. Using the same preprocessing and evaluation pipeline, we forward-simulated $R$ and $T$ with MOP-MCML at the corresponding wavelengths and computed spectral differences relative to the original data. Overall, the MOP-MCML results give a mean reflectance difference of $29.78\% \pm 7.80\%$ and a mean transmittance difference of $9.65\% \pm 5.57\%$. When the two error sets are juxtaposed, the MOP-MCML curves exhibit larger discrepancies, particularly for reflectance, local maxima around 750-760 nm reaching $\sim 43\%$, and for transmittance up to $\sim 19\%$.

Why IAD fits closer to the measurements? Two factors explain the gap:

  1. Source-model mismatch. The light-source types currently available in MOP-MCML do not exactly match those assumed in the reference documentation/measurements (spatial/angular profile and/or spectrum), leading to systematic deviations in the forward model.
  2. Embedded error feedback in IAD. IAD is an inverse procedure that continuously compares predicted and measured R and T while estimating optical parameters, effectively “fitting to the data” at each iteration. In contrast, MCML uses the retrieved parameters to generate a single forward prediction, so any source or boundary mismatch propagates directly into larger residuals.

We note that commercially available phantoms priced at approximately $1,000 exhibit optical-parameter errors on the order of $40\%$. Accordingly, despite the differences discussed above, the errors produced by MOP-MCML remain within our study’s tolerance and are therefore considered acceptable.

8. References

[1] Bashkatov et al. Optical properties of human skin, subcutaneous and mucous tissues in the wavelength range from 400 to 2000 nm. 2005 J. Phys. D: Appl. Phys. 38 2543. 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. 1998 Sep;43(9):2465-78. DOI: 10.1088/0031-9155/43/9/003.