Trapezium Bone

The derivation of the true colour of natural waters is based on the calculation of the Tristimulus values that are the three primaries (X, Y, Z) that specify a colour stimulus of the human eye [6 ,14 ]. Suppose the radiation spectrum that comes from the water is given by I that is a function of wavelength (λ), then the tristimulus values are given by:
$X={\displaystyle \int I\left(\lambda \right)\overline{x}(\lambda )d}\lambda$
$Y={\displaystyle \int I\left(\lambda \right)\overline{y}(\lambda )d}\lambda$
$Z={\displaystyle \int I\left(\lambda \right)\overline{z}(\lambda )d}\lambda$
The CIE 1931 standard colourimetric two degree Colour Matching Functions (CMFs) are presented by x (red), y (green) and z (blue). These serve as weighting functions for the determination of the tristimulus values. The intensity I can be replaced by I = E × R, the product of the illumination E times the remote sensing reflectance of water (R) [10]. For notation purposes we introduce the symbol T that represent the three tristimulus values (X, Y, Z) and $(\overline{t) }$ that represents the three CMFs:
$T={\displaystyle \int E(\lambda )R_{\text{RS}}\left(\lambda \right)\overline{t}(\lambda )d}\lambda$
To further simplify the calculations the illumination E is taken as a constant, independent of wavelength, and the remote sensing reflectance is assumed to be corrected for the surface effects (Fresnel reflectance, foam, capillary waves). We refer to the standard books on water remote sensing by Mobley [11 ] and Kirk [10 ]. Thus R_RS (λ) describes the intrinsic colour of the water, independent of air-water interface effects or illumination effects. Finally, because the integrals cannot be solved analytically, T can be written as the summation:
$T=E{\displaystyle {\sum }_{i=400}^{710}{R}_{\text{RS}}(\lambda )\overline{t}(\lambda )\Delta \lambda }\text{ or }T={\displaystyle {\sum }_{i=400}^{710}y(\lambda )\Delta \lambda }\text{ or }T={\displaystyle {\sum }_{i=400}^{710}\Delta T}$
Note that the summation is taken between 400 and 710 nm. This will be discussed in more detail below. Also E is taken as unity and y is the product of the remote sensing reflectance times the CMF weighting functions:
$y(\lambda )=R_{\text{RS}}(\lambda )\overline{t(\lambda )}$
Because ocean colour satellites do not provide full-spectral coverage, the y-spectrum must be first reconstructed by linear interpolation, based on the remote sensing reflection measured at the spectral bands (b). The contribution to T of a small interval of the spectrum between wavelengths L1 and L2 can be approximated by the trapezium rule (Figure 1):

To calculate y (Equation (4)), R(λ) = R_RS(λ) must be retrieved from the values at the satellite bands b1 and b2. This can be done by linear interpolation at wavelength L1 and L2:

If Equations (4)–(6) are combined we find:

with:

Rewriting Equation (7) to an expression that is linear in the satellite bands R_b1 and R_b2 we find:

This implies that if we have the measured R values at b1 and b2, we can estimate ΔT between those bands (Equations (3), (8) and (9)) as a linear combination of those two, because for every wavelength interval we can calculate A, B and C and know $(\overline{t)}$ at wavelengths L1 and L2 from [6 ]. Once the tristimulus values T (X, Y, Z) have been calculated, the three values are normalized and the colour is expressed in the coordinates:
$x=\frac{X}{X+Y+Z} y=\frac{Y}{X+Y+Z}$
The white point has the coordinates x = y = 1/3. In the (x, y) chromaticity plane, the coordinates are transformed to polar coordinates with respect to the white point and the hue angle is derived. The hue angle (α) lies between the vector to a point with coordinates (x − x_w, y − y_w) and the positive x-axis (at y − y_w = 0), giving higher angles in an anti-clockwise direction (see Figure 2).
$\alpha =\arctan \left(y-y_W,x-x_W\right)\text{ modulus 2}\pi$
All calculations in this paper were made with the ATAN2 function (four-quadrant inverse tangent) and the derived angles (in radians) are multiplied by 180/π to get the angles in degrees. In the development of the water hue angle concept, Wernand et al. [12 (link)] used (α_M) for the hue angle derived with the FUME algorithm for MERIS, while Novoa et al. [15 (link)] introduced (α_W) for the hue angle of water. In this manuscript we will refrain from indices and use (α) as the hue angle that represents the “true” or “intrinsic” colour of a natural water, which can be approximated by satellite remote sensing reflectance measurements.

The operation was performed under brachial plexus block anesthesia using an upper arm tourniquet. A straight incision was made in the distal forearm between the distal portion of the flexor carpi radialis (FCR) and the radial artery and was carried across the distal wrist crease using a hockey-stick incision that angles toward the base of the thumb. The FCR tendon was retracted ulnarly, and the radial artery was retracted radially. The wrist capsule was entered through a longitudinal incision from the volar lip of the radius to the proximal tubercle of the trapezium. The capsule and intracapsular ligaments were carefully divided and reflected sharply off the scaphoid with a scalpel. Fibrous tissue or pseudarthrosis of opposing bone surfaces of the proximal and distal fragments were thoroughly debrided by using a high-speed burr with constant irrigation, leaving a shell of intact cartilage (Figure 2A). We used Kirschner wires as joysticks to distract across the nonunion site. A cancellous graft obtained from the patient’s iliac crest was impacted into the shell, and a wedge-shaped cortico-cancellous graft was shaped to fit the gap between distal and proximal fragments and then inserted into the defect to maintain the reduction (Figure 2B).
The first guide wire was inserted along the long axis and was approximately 2 mm distant from the medial cortex of the scaphoid. Usually, two or three attempts were needed for guide wire insertion until the position of the first guide wire was ideal. Then, we inserted the second guide wire adjacent to the first, but separated in the radioulnar plane by the distance of the diameter of the planned screws. Usually, one or two attempts were needed for second screw guide wire insertion.
After the positions of the guide wires were verified by the C-arm, two cannulated screws (one Acutrak Mini and one Acutrack Mirco Hillsboro, OR, USA) were inserted into the scaphoid by the palmar approach. For most cases, we inserted the 2.5 mm-diameter screw through the radial guide wire first and then the 3.5 mm-diameter screw through the ulnar guide because the bone quality of the ulnar side of the scaphoid waist was the highest. Only in one female patient, we used two 2.5 mm-diameter screws, while in two male patients, we inserted two 3.5 mm-diameter screws into the scaphoid. The length and position of the screws were confirmed by C-arm (Figures 2C,D). Finally, the wires were removed, and the capsule and radioscaphocapitate ligament were repaired. All procedures were performed by one senior surgeon.

Rice cake in a container was pressed downward with a cylindrical press jig (20 mm diameter) from the surface of the rice cake to 6 mm at a test speed of 60 mm min⁻¹, and the stress was measured using a tabletop universal tensile tester (Trapezium X software version 1.5.3, EZ-test; Shimadzu, Kyoto, Japan).