Bulk \(\mu_a(\lambda)\) from chromophore concentrations
All concentrations are volumetric averages over the full probed tissue volume. The absorption of each chromophore is its concentration multiplied by its extinction coefficient; the total absorption is the sum of these components.
The bulk absorption coefficient is the sum of contributions from each chromophore, each following Beer's law:
$$\mu_{a,\text{HbO}_2}(\lambda) = \varepsilon_{\text{HbO}_2}(\lambda)\,[\text{HbT}]\,\text{StO}_2$$
$$\mu_{a,\text{Hb}}(\lambda) = \varepsilon_{\text{Hb}}(\lambda)\,[\text{HbT}]\,(1 - \text{StO}_2)$$
$$\mu_{a,\text{water}}(\lambda) = \varepsilon_{\text{water}}(\lambda)\,W$$
$$\mu_{a,\text{lipid}}(\lambda) = \varepsilon_{\text{lipid}}(\lambda)\,L$$
where \([\text{HbT}]\) is total hemoglobin concentration (μM), \(\text{StO}_2\) is tissue oxygen saturation, \(W\) is water volume fraction, and \(L\) is lipid volume fraction. \(\varepsilon_{\text{HbO}_2}\) and \(\varepsilon_{\text{Hb}}\) have units mm⁻¹ μM⁻¹; \(\varepsilon_{\text{water}}\) and \(\varepsilon_{\text{lipid}}\) have units mm⁻¹ (i.e., these are the absorptions of pure water or pure lipid, respectively).
Melanin absorption follows the power-law fit from Jacques (2013), Eq. 8:
$$\mu_{a,\text{mel}}(\lambda) = M_f \times 51.9 \left(\frac{\lambda}{500\,\text{nm}}\right)^{-3.5} \quad \text{(mm}^{-1}\text{)}$$
where \(M_f\) is the melanosome volume fraction averaged over the full probed tissue volume (n.b., within the epidermis specifically \(M_f\) can reach a value of 0.43).
The total bulk absorption coefficient is:
$$\mu_a(\lambda) = \mu_{a,\text{HbO}_2}(\lambda) + \mu_{a,\text{Hb}}(\lambda) + \mu_{a,\text{water}}(\lambda) + \mu_{a,\text{lipid}}(\lambda) + \mu_{a,\text{mel}}(\lambda)$$
Tissue absorption model originally made for:
Blaney, G., Frias, J., Tavakoli, F., Sassaroli, A., and Fantini, S. "Dual-ratio approach to pulse oximetry and the effect of skin tone." J. Biomed. Opt. 29(S3), S33311 (2024). doi:10.1117/1.JBO.29.S3.S33311
Primary sources for extinction coefficients: