3.1. Homogeneous head model
To separate the errors due to the diffusion approximation, finite geometry, and curvature from the errors due to the layered structure of the head, we first treated the head as a homogeneous medium by setting equal the superficial and brain absorptions, i.e. μa,superf
= [0.05,0.06, . . . ,0.29,0.3]cm−1
. In , each graph presents the retrieved brain absorption for the 7 sets of SDs. The adult model shows better estimate than the young models. This is due to the stronger head curvature for the newborn and infants. In addition, all estimations at 5–20 mm are less accurate because of the diffusion approximation utilized in the FD-MD formulation [41
Fig. 3. Estimated μa,brain when the head is treated as a homogeneous medium, i.e. when μa,superf = μa,brain = [0.05,0.06, . . . ,0.29,0.3]cm−1. For each sub-figure, solid lines represent the 7 groups of SDs used in the estimation (more ...)
In , retrieved absorption coefficients and standard deviation due to experimental noise are shown for three specific cases when μa,superf = μa,brain = [0.05,0.15,0.3]cm−1. While the retrieved values improve with longer SDs, the uncertainty increases due to decreasing signal to noise ratio. In addition, the underestimation of retrieved absorption coefficient for μa = [0.15,0.3]cm−1 is due to the unmatched calibration medium (calibration μa = 0.1cm−1).
Uncertainty and retrieved absorption for three particular cases of the homogeneous head: μa,superf = μa,brain = [0.05,0.15,0.3]cm−1 for the 7 sets of SDs.
presents averages and standard deviations of the retrieved μa,brain over 26 combinations of μa,superf = μa,brain = [0.05,0.06, . . . ,0.29,0.3]cm−1. In newborn and infants, the errors vary between 14–21% and 7–20%, respectively, and decreases to 3–14% for the adult head model. At very short distances (5–20 mm), the error is large for all ages (15–20%) due to the non-validity of the diffusion approximation. At long distances, the error decreases down to less than 5% for the adult head, while the stronger curvature of the head is responsible for additional errors in the case of newborn and infants.
Averages and standard deviations of the relative error on retrieved μa,brain over 26 combinations of head absorption [0.05,0.06, . . . ,0.29,0.3]cm−1, for the 7 sets of SDs.
3.2. Two-tissue head model
depicts the evolution of the recovered brain absorption values as a function of true absorption for the 7 groups of SDs in all head models (row-wise) treated as a two-tissue structure. The superficial absorption coefficient μa,superf was set successively to 0.1 and 0.2 cm−1 (column-wise). In each sub-figure, the dotted line corresponds to the true brain absorption, while the dash-dotted line identifies the superficial absorption. Solid curves represent the 7 groups of SDs.
Fig. 6. Brain absorption coefficients μa,brain estimated across subject age (row-wise) and for two values of superficial absorption μa,superf = [0.1,0.2]cm−1 (column-wise). For each sub-figure, solid lines represent the 7 groups of SDs (more ...)
The effect of the superficial layer is to pull the recovered absorption value of the brain towards the superficial absorption value, especially at short SDs and for the adult case. Because of the specific sets of optical properties we chose, this selection leads to a generally weaker than true value μa,brain, especially for high brain absorption coefficients. The three young head models present similar brain absorption estimations, mostly underestimated, probably due to the head's curvature. The adult head presents a less precise estimation including over- and underestimations, due to the large thickness of the superficial layer which contaminates the signal especially at short SDs.
shows the average error at each age for all SDs groups, over 52 absorption combinations covering the whole range of coefficients for the brain (μa,brain between 0.05 and 0.3 cm−1 in steps of 0.01 cm−1) and the superficial layer (μa,superf =0.1 or 0.2 cm−1).
Fig. 7. Two-tissue head model: averages and standard deviations of the relative error in retrieved μa,brain over 52 combinations of μa,superf = [0.1,0.2]cm−1 and μa,brain = [0.05,0.06, . . . ,0.29,0.3]cm−1 for the 7 sets (more ...)
For the newborn, the optimal SDs group is 10–25 mm and shows a relative error of 12±6%. The 6 and 12 month old infants present a similar trend: the relative error is about 10±5% for all SDs groups, except for the two shortest (5–20 and 10–25 mm). For both infants, the optimal SDs group can be selected from the five longest SDs (from 15–30 to 35–50 mm), since the error is minimized and shows similar values. The adult head shows the highest relative error for almost all SDs groups (with certain combinations of optical properties yielding errors of more than 80%). The optimal SDs group is reached at 35–50 mm with a corresponding error of 19±13%.
The effect of the scattering coefficient was investigated by modifying the value in the superficial layer to μ′s,superf = 5cm−1. The relative error decreases in the newborn (ranging from 6 to 19%) and infant models (2–10%) for all SDs except the shortest group (5–20 mm), and slightly increases in the adult head (from 26 to 45%) for almost all groups (except for 10–25 and 15–30 mm). Again, this is probably due to the thicker superficial layer in the adult model that causes a stronger error in μ′s translating through cross-talk in higher error in μa.
The same analysis was performed with the adult model in which a large vessel was introduced. Relative errors and standard deviations [%] for the 7 SDs (from 5–20 to 35–50 mm) are the following: 51.6 ± 58.2; 48.3 ± 53.9; 35.4 ± 33.4; 26.3 ± 18.1; 23.7 ± 14.7; 22.0 ± 14.5; 19.2 ± 13.2. The error is higher than the case without inhomogeneity in the first 3 groups and then similar for the next 4 groups. The contamination of the shortest estimations is related to the length of the vessel (3 cm) which was located around the beginning of the probe where the light source is positioned. This type of inhomogeneity can be detected experimentally, and data automatically discarded, because of the corresponding low correlation factor r2
of the linear fitting process from Eq. (1)
3.3. Three-tissue head model
We investigated the influence of the CSF structure by setting its optical properties to low absorption and scattering values: μa = 0.04cm−1 and μ′s = 0.1cm−1. The two hemispheres (left and right parietal) of the newborn were studied, because the right hemisphere possessed a thick “pocket” of CSF located underneath the optical probe (white arrow on the axial slice of the second row in ), while the left hemisphere possessed a “normal” amount of fluid.
Fig. 8. Brain absorption coefficients μa,brain estimated across subject age (row-wise) and for two values of superficial absorption of μa = [0.1,0.2]cm−1 (column-wise) when the CSF tissue is taken into account. For each sub-figure, solid (more ...)
shows retrieved μa,brain with respect to true values for the 7 groups of SDs. The first two columns correspond to μa,superf set to 0.1 and 0.2cm−1, respectively. The third column shows an axial slice of the left and right newborn (NB) hemispheres. Each graph was created in the same way as in . Distortion of the value of μa,brain by the superficial layer is similar to the effect observed when ignoring CSF, with the difference that the low absorption of CSF pulls the retrieved μa,brain to lower values than in the previous case.
For the newborn, the estimations computed at shorter SDs in the left hemisphere are less distorted than those estimated in the right hemisphere. This is due to the large volume of CSF in the vicinity of the optical probe. The clear pocket of CSF contributes to modification of the signal through the combined effects of very low absorption and scattering. In general, short SDs groups, from 5–20 to 15–30 mm, are poorly estimated for all head models.
shows the averages of the relative errors and standard deviations in percentage [%] of μa,brain for all 52 combinations of optical properties of brain and superficial tissues. The error rises to a maximum of 51% for the newborn right hemisphere, mostly due to the effect of the CSF pocket. For the newborn left hemisphere, the optimal set of SDs is 20–35 mm and the error is 19±5%. Decreasing error with increasing SDs is still true for the adult model to a slightly higher extent. The distinction between 6 and 12 month old infants is more visible than in the case without CSF. The 12 month old infant shows less accurate estimations for almost all SDs because of the thicker superficial layer. In both cases, optimal SDs are between 20–35 and 30–45 mm, where the errors is less than 16±10%. For the adult, the longest SDs group (35–50 mm) is needed to minimize the error to a value of 23 ± 15%.
Fig. 9. Three-tissue head model: averages and standard deviations of the relative error on retrieved μa,brain over 52 combinations of μa,superf = [0.1,0.2]cm−1 and μa,brain = [0.05,0.06, . . . ,0.29,0.3]cm−1 for the 7 sets (more ...)
The influence of the CSF was also studied for a superficial scattering coefficient of 5 cm−1. For all head models, the error at 5–20 mm increases. For the newborn, both hemispheres show lower errors (ranging from 7 to 45%) with the lower scattering coefficient (except for 25–40 mm in the right hemisphere: error increased to 13%). In the case of the 12 mo. old infant, the error increases from 15 to 61% (except for 10–25 and 15–30 mm: down to 30 and 16%, respectively) while decreases to a minimum of 6 and 25% in the 6 mo. old infant and adult, respectively (data not shown).