2.1.1.

Monte Carlo method

To model the expected output of the practical diffuse reflectance $R_d$ of the wound, we modified the $mcxyz$²² (version July 22, 2019, downloaded on October 6, 2020) light transport modeling program to observe photon propagation in customized heterogeneous tissue. Figure 1(a) illustrates a typical photon path. Individual photons set off at the tissue surface with the same initial weight and direction. The tissue scattering properties cause a photon to potentially change propagation angle at each step while the absorption along the movement path reduces its weight. A sinusoidal illumination pattern is generated by the injected photon density following a sinusoidal probability distribution function along the $y$ axis, such that the stripes are orthogonal to the wound long axis as shown in Fig. 1(b). When a photon escapes from the top surface, i.e., back scattered light, it is collected by the detector ending its journey as shown in Fig. 1. The minimum three-dimensional (3D) detection unit in simulation has a photon collection bin size of 0.02 mm in three dimensions. The illumination wavelength is originally set at 617 nm to match the experiment described in Sec. 2.2. This wavelength is selected to provide a suitable balance between scattering and depth penetration to epidermis and dermis.

Fig. 1

(a) The geometry of tracing the photon within the tissue. The example of the nondestructive detector at depth $z_i$ is shown. The direction of the photon going up and down is judged by the direction cosine between the direction vector and $z$ axis. (b) The 3D structure of the surgical site wound. The collimated sinusoidal pattern (red colored) is projected to the tissue surface. (c) The $xz$ section of the tissue.

To further monitor the photon migration in the tissue, we insert nondestructive detector planes at regular intervals 0.02 mm, to record the depth–dependent weight ${\text{Weight}}_{\text{down}}(x,y,z)$ within the tissue, as shown in Fig. 1, for the photon moving down through the tissue. When the photon moves to a new voxel, the weight of the photon will be updated and subsequently recorded by the detector planes. The upper facing detector will record the current weight of the photon travelling down through the tissue.

2.1.2.

Surgical wound model

During invasive surgery, the surgeon makes an incision through the patient’s skin to access the operation site. After the procedure, the incision will be sealed and form a wound with a vertical structure. We therefore built up a 3D tissue structure to mimic this as shown in Fig. 1(b) with its $xz$ section in Fig. 1(c), where the wound is embedded in skin center with a cubic block structure and vertical boundary. During the wound healing process, we assume any wound infection or possible change only happens within the wound block. The wound width can change during the healing process, but the wound depth remains the constant. The skin remains homogeneous and uniform in optical properties.

As shown in Fig. 1, the tissue size is designed with tissue lengths of $l_x=28\mathrm{mm}$ (to eliminate the boundary effect), $l_y=20\mathrm{mm}$ (to include at least two periods of the sine pattern projection), and $l_z=16\mathrm{mm}$ while the depth of wound is 10 mm (to create a semi-infinite wound and skin structure²³). We used 84,000,000 photons, the number being validated during preliminary simulations, to produce statistically repeatable results.

The absorption and scattering can both change during the wound healing process. The structural change typically results in scattering coefficient changes while absorption coefficient only changes the weight of the photon and is included via with Beer–Lambert law. As we are more interested in structural change, we assume the absorption coefficient of the wound remains constant and the scattering coefficient was varied for possible wound states. As the SFDI method can make assessment of scattering and absorption separately, we assign the wound and healthy skin a specific absorption coefficient ${\mu }_a$ and reduced scattering coefficient ${\mu }_s^{\prime }$. For healthy skin, we use typical human skin parameters²⁴ at 617 nm, where the reduced scattering coefficient is $1.42{\mathrm{mm}}^{-1}$ and the absorption coefficient is $0.023{\mathrm{mm}}^{-1}$.²⁵ The anisotropy, $g$, is fixed at 0.9 for both skin and wound.

2.1.3.

One-dimensional profile for wound monitoring

To simplify the observation of changes in wound width and the intensity of the AC image, here we define a one-dimensional (1D) AC profile. When looking at a small wound site area in our model, we can assume that the wound and skin region are each locally uniform with respect to their own optical properties. Therefore, we determine $I_{\text{curve},\mathrm{AC}}$ by averaging the intensity of the AC image $I_{\mathrm{AC}}$ along the $y$ direction as in Eq. (2), where $N_y$ is the number of pixels in $y$ direction. This helps to reduce the number of the photons that are launched into the original model as well as eliminating random noise in the Monte Carlo method.

3.1.1.

Accuracy in wound width determination

Here, we investigate the dependence of wound monitoring varying the wound reduced scattering coefficient ${\mu }_s^{\prime }$, wound width and illumination spatial frequency $f_x$. Five wound reduced scattering coefficients are considered based on the previous works^9,11,12 (listed in Table 1), while ${\mu }_a$ is kept constant at $0.023{\mathrm{mm}}^{-1}$ and the anisotropy, $g$, is 0.9. The width values used were 0.5, 1, 1.5, 2, 4, and 6 mm and spatial frequencies $f_x$ of 0.1, 0.2, and $0.3{\mathrm{mm}}^{-1}$.

Table 1

Optical properties of five types of wound.

The simulations are run for three phases of the light pattern and the AC images are calculated with Eq. (1). A typical simulation result is shown in Fig. 4(a), where the AC image result is color coded for intensity. The wound area, with its lower reduced scattering coefficient therefore lower returned intensity, can clearly be seen as the blue stripe in the center. Measuring the width of the blue region with ImageJ, we obtain the width value 2.85 mm in Fig. 4(a) demonstrating the partial volume effect.

Fig. 4

(a) The $I_{\mathrm{AC}}$ image of 2 mm width wound model with wound ${\mu }_s^{\prime }=0.71{\mathrm{mm}}^{-1}$. (b) The $I_{\text{curve},\mathrm{AC}}$ of the wound model in panel (a). The green line marks the skin intensity level and the black line mark the half maximum of the wound area. The vertical dashed line is the real wound area according to the wound model. (c)–(e) The measured wound width versus the real wound width. (f)– (h) The corresponding error of the wound width measurement for the data in panels (c)– (e). The dashed line is $y=0$.

Figure 4(b) illustrates the average intensity profile $I_{\text{curve},\mathrm{AC}}$ through the wound. The edge response in the $I_{\text{curve},\mathrm{AC}}$ spills out the shape of the wound. Thus, we used the full width half maximum value, which has the overall best estimation to determine the wound width from the simulations. Figures 4(c)–4(e) show the measured wound width versus the true wound width and Figs. 4(f)–4(h) plot the error of measurements there using $\text{error}=\text{result}-\text{truth}$. For the narrow wounds ($\le 1.5\mathrm{mm}$), the width is significantly overestimated. The accuracy of the measurement increases with wider wound width, higher scattering coefficient, and higher spatial frequency (this matches the conclusion from Bassi et al.³³). The wound width results from 1 to 2 mm wound model have unexpectedly low error in higher scattering wound models. The half maximum here is excellent for the wound width estimation.

To explore whether edge response “spill out” affecting the wound area intensity, in Fig. 5, we plot the calculated wound width measurement error at the full width of 90% maximum with $\text{error}=\text{result}-\text{truth}$. SFDI results exaggerate the wound width as all the error values are positive. The edge response reduces with the higher scattering, higher spatial frequency, and wider width wound, leading to better accuracy in wound structure matching Figs. 4(c)–4(e). We further investigate this with the photon behavior around the skin wound interface in Sec. 3.2.1.

Fig. 5

(a)–(c) The error of the $I_{\text{curve},\mathrm{AC}}$ measured at full width 90% maximum.

3.2.1.

Edge response

From the simulation and phantom experiments, we learn the spatial frequency, reduced scattering and wound width contribute to the SFDI lateral resolution. Here, we look closely into the photon behavior at the vertical interface between the healthy tissue and wound to understand influence on the transition to the images and width estimation. Four signature wound models with two wound widths 0.5 and 2 mm are selected with wound reduced scattering ${\mu }_s^{\prime }=0.473{\mathrm{mm}}^{-1}$ and ${\mu }_s^{\prime }=4.73{\mathrm{mm}}^{-1}$ from the simulation. The spatial frequency used is $f_x=0.1{\mathrm{mm}}^{-1}$ for four models. This serial of wound model combines the narrow and wide wound with the low and high scattering wound. We slice the ${\text{Weight}}_{\text{down}}$ through $xz$ section at $y=0$ [see Fig. 7(e)] to view how the edge response contributes to the diffuse reflectance. In the intensity plots, we only note photons once they have undergone one scattering event.

Fig. 7

(a)–(d) The 3D edge response observed from the $xz$ section of ${\text{Weight}}_{\text{down}}$ profile for the wound width 0.5 and 2 mm wound at ${\mu }_{s,\text{wound}}^{\prime }=0.473{\mathrm{mm}}^{-1}$ and ${\mu }_{s,\text{wound}}^{\prime }=4.73{\mathrm{mm}}^{-1}$. (e) The slicing method through $xz$ section at $y=0\mathrm{mm}$. (g)–(i) The selected area in plots (a)–(d) shown with the dashed yellow rectangle. The red dashed contour line locates where the intensity is 70% of the maximum intensity within the whole wound area. The ideal 70% intensity contour line assuming the wound is infinite homogeneous and with same reduced scattering value is indicated with a black dashed line.

In the low scattering wounds, as shown in Figs. 7(a) and 7(c), the brown arrows point out the first scattering event occurrences at a significant depth into the tissue. In the skin, the first scattering event occurrence is closer to the top surface noted by the white arrows. For the high scattering wounds, the first scattering event occurs at a shallower depth than in the relatively lower scattering healthy skin, as illustrated by the grey arrows in Figs. 7(b) and 7(d). As expected, photons penetrate more deeply in the lower scattering tissue. However, there are clear contributions to the profile from the tissue adjacent to the wound, which have different scattering values.

To investigate this edge response more closely, we plot the contour line of 70% maximum intensity. If we assume there is an infinite homogeneous wound area, any intensity contour line should be flat. However, all of the contour lines here show a U-shape indicating the influence of the tissue transition at the edge to the photon propagation. For the low scattering wound model as shown in Figs. 7(f) and 7(g), the photons propagate down further in the 2 mm wound than the 0.5 mm wound as the two contour lines are both deeper for 2 mm wound. Though the absorption reduces the “weight” of the photons when they travel further down, the wider wound still has significantly more photons at a greater depth ($z\ge 1.5\mathrm{mm}$). The photons between the real and ideal contour line as illustrated by the pink arrow in Figs. 7(f) and 7(g) are the photons entering the wound from the skin. They have less chance of being scattered back to the skin so generally travel within the wound resulting in the upper U-shape 70% contour line. There is decreased ${\text{Weight}}_{\text{down}}$ intensity in the healthy tissue adjacent to the wound, where the photons have been lost. The photons scattered into the wound from skin have already travelled longer distance from the skin area to the wound area. The photons moving from skin to wound have less “weight” than the “local” photons originally launched into the wound. They are not able to travel downward as far as the “local” photon, forming the lower U-shape 70% contour curve.

Inversely, in the high scattering wound [see Figs. 7(h) and 7(i)], photon’s first scattering event happens near the surface leading to a nearly flat 70% contour curve approximately at the surface for all the wound widths. Photons leave from wound to skin leading to the higher intensity in the skin area where adjacent to the wound. Similarly, the photons entering the healthy tissue, relative lower reduced scattering media, are less likely to be scattered back. Therefore the intensity is very low between the real and ideal 70% contour line as indicated by the pink arrow in Figs. 7(h) and 7(i), where photons are lost from wound to the healthy skin. Both pair of narrow and wide wound with same wound scattering properties show the influence of edge response is greater in narrower wounds. The high-scattering narrow wound area demonstrates lower intensity in diffuse reflectance than the wider wound, whereas the low-scattering narrow wound shows greater intensity.

To verify what we find from the ${\text{Weight}}_{\text{down}}$, we align the $I_{\text{curve},\mathrm{AC}}$ (spatial frequency used is $0.1{\mathrm{mm}}^{-1}$) to the full width 90% maximum to compare the transition around the wound-skin boundary area. From Fig. 8, the 0.5 and 1 mm wound curves are separated from other curves due to the more significant edge response and photons staying in the lower scattering media. The 2 and 4 mm wound curves almost overlap at the transition area. Thus, the relative contribution of the edge response effect decreases with the wound width increase matching the ${\text{Weight}}_{\text{down}}$ profiles.

Fig. 8

(a)–(d) Four wound width 0.5, 1, 2, and 4 mm $I_{\text{curve},\mathrm{AC}}$ are plotted particularly at the range where the curve approaching from the skin level to the centre of the wound area. The $I_{\text{curve},\mathrm{AC}}$ are aligned at full width 90% maximum by the 0.5 mm wound at right side.