**5. Appendix**

### **5.1 FBP algorithm**

The 3D Fourier transform of the 3D volume data generated by the backprojection is based on the following equation (1):

$$F(\alpha\_{x'}, \alpha\_{y'}, \alpha\_z) = \iiint f(\mathbf{x}, y, z) \cdot \exp\left\{-j(\alpha\_x \cdot \mathbf{x} + \alpha\_y \cdot y + \alpha\_z \cdot z)\right\} d\mathbf{x} \cdot d\mathbf{y} \cdot dz \tag{1}$$

where *fxyz* (,,) is the simple backprojection intermediate image, and *x*, *y*, and *z* are real numbers. The meaning of the filtering process performed in 3D Fourier space is described below, and it is mathematically expressed by the following equation (2):

$$\mathcal{F}\mathcal{M}(a\mathbf{o}\_{\mathbf{x}},a\mathbf{o}\_{\mathbf{y}},a\mathbf{o}\_{\mathbf{z}}) = \mathcal{F}(a\mathbf{o}\_{\mathbf{x}},a\mathbf{o}\_{\mathbf{y}},a\mathbf{o}\_{\mathbf{z}}) \cdot \mathcal{M}(a\mathbf{o}\_{\mathbf{x}},a\mathbf{o}\_{\mathbf{y}},a\mathbf{o}\_{\mathbf{z}}) \tag{2}$$

X-Ray Digital Linear Tomosynthesis Imaging of Arthoroplasty 107

*R tem* \_ *<sup>p</sup> W*

\_

2

*spec r inverse R norm*

*ρH*

*π*

0

*R norm*

where *W* is the addition of a direct current component.

*spec r inverse*

*<sup>ρ</sup><sup>H</sup>* is the Nyquist frequency 2

undergone Fourier space low-pass filtering.

*science*, 49, pp.2366-72, ISSN 0018-9499

*CFR WFR* .

frequency <sup>2</sup>

**6. References** 

has a artifact reduction filter characteristic expressed by the following equation (8):

\_

*R* 

\_ *H R inverse*(ω ) ω*R norm*

*R tem* \_ *<sup>p</sup>* value. The characteristics in the negative direction along the horizontal axis are omitted because these are in linear symmetry about the vertical axis with the characteristics in the positive direction. ω (ω ) *H HR spec r inverse* is expressed by the following equation (9):

\_

*πρ <sup>H</sup> <sup>ω</sup> <sup>H</sup> <sup>ω</sup><sup>R</sup> ρ ρ<sup>H</sup>*

*H ω H ωR ρ ρH*

The 3D back Fourier transforms the Fourier space data back to 3D volume data, having

Bleuet P, Guillemaud R, Magin I. Et al. (2002) An adapted fan volume sampling scheme for

Dobbins JT III, Godfrey DJ. (2003) Digital x-ray tomosynthesis: curent state of the art and clinical potential. *Physics in medicine and biology*, 48, R65-106, ISSN 0031-9155 Duryea J, Dobbins JT, Lynch JA. (2003) Digital tomosynthesis of hand joints for arthritis

Feldkamp LA, Davis LC, Kress JW. (1984) Practical cone-beam algorithm. *Journal of the* 

Gomi T, Hirano H, Umeda T. (2009) Evaluation of the X-ray digital linear tomosynthesis

Gomi T, Hirano H. (2008) Clinical potential of digital linear tomosynthesis imaging of total joint arthroplasty. *Journal of Digital Imaging,* 21, pp.312-22, ISSN 0897-1889

reconstruction processing method for metal artifact reduction. *Computerized Medical* 

assessment. *Medical Physics*, 30, pp.325-33, ISSN 0094-2405

*Optical Society of America*, A1, pp.612-619, ISSN 0030-3941

*Imaging and Graphics*, 33, pp.257-274, ISSN 0895-6111

3D algebraic reconstruction in linear tomosynthesis. *IEEE transactions on nuclear* 

\_ max *R temp*

2 2

 *x y*

sin 2

*ρH*

*CFR WFR* and *ρ* is the no-processing region

(8)

*<sup>R</sup>*\_ max is the maximum value of a

(9)

**5.2 Artifact reduction processing** 

( ) *H R inverse* 

where (,,) *FM x <sup>y</sup> <sup>z</sup>* is the filtered 3D Fourier distribution image, and ( , , ) *M x <sup>y</sup> <sup>z</sup>* is a function representing filter characteristics. The filtering process carried out in 3D Fourier space is to weight the 3D Fourier distribution image of complex data with the real-valued filter function *M* dependent on the respective frequency values. The weighting function *M* is compressed in the *<sup>z</sup>* direction. ( , , ) *M x <sup>y</sup> <sup>z</sup>* is expressed by the following equation (3) as a product of three functions representing the filter characteristic:

$$M(o\_{x'}o\_{y'}o\_z) = H\_{pm'}(o\_z) \cdot H\_{spec}(o\_r) \cdot H\_{inverse}(o\kappa \mathbb{R}) \tag{3}$$

( ) *Hprof <sup>z</sup>* has a low-pass filter characteristic, i.e., a Gaussian characteristic, which is expressed by the following equation (4):

$$H\_{proj}(o\_z) = \exp\left(-0.693 \left[\frac{o\_z}{CFD}\right]^2\right) \tag{4}$$

where CFD is the frequency with the Gaussian attenuation halved. ( ) *Hspec r* has a filter characteristic which is expressed by the following equation (5):

$$\begin{cases} H\_{\text{spec}}(\boldsymbol{\alpha}\_{r}) = 1 & \left( \text{case} \quad \boldsymbol{\alpha}\_{r} < \frac{\text{CFR} - \text{WFR}}{2} \right) \\\\ H\_{\text{spec}}(\boldsymbol{\alpha}\_{r}) = \frac{1 - \sin(\boldsymbol{\alpha}\_{r} - \text{CFR}) \cdot \pi \ / \text{WFR}}{2} & \left( \text{case} \quad \frac{\text{CFR} - \text{WFR}}{2} < \boldsymbol{\alpha}\_{r} < \frac{\text{CFR} + \text{WFR}}{2} \right) \\\\ H\_{\text{spec}}(\boldsymbol{\alpha}\_{r}) = 0 & \left( \text{case} \quad \frac{\text{CFR} + \text{WFR}}{2} < \boldsymbol{\alpha}\_{r} \right) \end{cases} (55)$$

However, <sup>222</sup> *r x <sup>y</sup> <sup>z</sup>* . The function has a sine wave form with high-frequency components smoothly attenuated. CFR is the cut-off frequency, and WFR is the total transition frequency width of the filter strength. 2 *CFR WFR* is the Nyquist frequency and 2 *CFR WFR* is the no-processing region frequency. This ( ) *Hspec r* removes high-frequency components from the origin of the 3D Fourier space. ( ) *H R inverse* has a filter characteristic which is expressed by the following equation (6):

$$H\_{inverse}(\text{co }R) = \left| \text{co }R \right|\_{\text{o}} \qquad \text{co }R = \sqrt{\left| \text{co}\_x \right|^2 + \left| \text{co}\_y \right|^2} \tag{6}$$

The 3D back Fourier transforms the Fourier space data back to 3D volume data, having undergone Fourier space low-pass filtering. The 3D back Fourier transform is expressed by the following equation (7):

$$fm(\mathbf{x}, y, z) = \frac{1}{8}\pi^3 \iiint \operatorname{FM}(a o\_x, o\_y, o\_z) \cdot \exp\left\{j(o\_x \ge +o\_y \ge +o\_z \ge z)\right\} d o\_x d o\_y d o\_z \tag{7}$$

where equation (7) is the transformation from the frequency domain to the space domain. It is the inverse of the relation described by equation (1).

### **5.2 Artifact reduction processing**

106 Recent Advances in Arthroplasty

function representing filter characteristics. The filtering process carried out in 3D Fourier space is to weight the 3D Fourier distribution image of complex data with the real-valued filter function *M* dependent on the respective frequency values. The weighting function *M* is

*x y z prof z spec r inverse*

 

( ) exp 0.693 *<sup>z</sup> Hprof z CFD*

has a filter characteristic which is expressed by the following equation (5):

1 sin( ) / ( ) . <sup>2</sup> 2 2

*CFR WFR CFR WFR CFR WFR <sup>H</sup> case*

components smoothly attenuated. CFR is the cut-off frequency, and WFR is the total

The 3D back Fourier transforms the Fourier space data back to 3D volume data, having undergone Fourier space low-pass filtering. The 3D back Fourier transform is expressed by

<sup>1</sup> <sup>3</sup> (,,) ( , , ) exp ( ) <sup>8</sup> *<sup>x</sup> <sup>y</sup> z x <sup>y</sup> z x <sup>y</sup> <sup>z</sup> fm x y z FM*

where equation (7) is the transformation from the frequency domain to the space domain. It

   

*j x y zd d d* (7)

*<sup>y</sup> <sup>z</sup>* is the filtered 3D Fourier distribution image, and ( , , ) *M*

 

*<sup>z</sup>* has a low-pass filter characteristic, i.e., a Gaussian characteristic, which is

*x <sup>y</sup> <sup>z</sup>* is a

(4)

(5)

 

*CFR WFR* is the Nyquist frequency and

removes high-frequency

has a filter characteristic

 

*<sup>y</sup> <sup>z</sup>* is expressed by the following equation (3) as

 *H H HR* (3)

2

 

*<sup>y</sup> <sup>z</sup>* . The function has a sine wave form with high-frequency

2 2 , *H RR R inverse*(<sup>ω</sup> ) ω ω ωω *<sup>x</sup> <sup>y</sup>* (6)

> 

 

where (,,) *FM x* 

compressed in the

( ) *Hprof* 

( ) *Hspec r* 

2

However, <sup>222</sup> 

the following equation (7):

*r x* 

expressed by the following equation (4):

*<sup>z</sup>* direction. ( , , ) *M x* 

(, ,) () () ( ) *M*

()1 <sup>2</sup>

*CFR WFR <sup>H</sup> case*

*<sup>r</sup> spec r <sup>r</sup>*

*spec r r*

()0 <sup>2</sup>

*CFR WFR* is the no-processing region frequency. This ( ) *Hspec r*

components from the origin of the 3D Fourier space. ( ) *H R inverse*

*CFR WFR <sup>H</sup> case*

where CFD is the frequency with the Gaussian attenuation halved.

*spec r r*

transition frequency width of the filter strength. 2

which is expressed by the following equation (6):

is the inverse of the relation described by equation (1).

 

a product of three functions representing the filter characteristic:

( ) *H R inverse* has a artifact reduction filter characteristic expressed by the following equation (8):

$$o o\_{\mathbb{R}\\_temp} = \mathbb{W} \quad + \quad \sqrt{o o\_x^2 + o\_y^2}$$

$$o o\_{\mathbb{R}\\_norm} = \frac{o o\_{\mathbb{R}\\_temp}}{o o\_{\mathbb{R}\\_max}} \tag{8}$$

$$H\_{inner}(o \, R) = \left| o\_{\mathbb{R}\\_norm} \right| $$

where *W* is the addition of a direct current component. *<sup>R</sup>*\_ max is the maximum value of a *R tem* \_ *<sup>p</sup>* value. The characteristics in the negative direction along the horizontal axis are omitted because these are in linear symmetry about the vertical axis with the characteristics in the positive direction. ω (ω ) *H HR spec r inverse* is expressed by the following equation (9):

$$\begin{cases} H\_{\text{spec}}\left(\text{o}\_{r}\right) \cdot H\_{\text{inverse}}\left(\text{o}\mathbb{R}\right) = \frac{2}{\pi} \left| \text{o}\_{\text{R\\_norm}}\right| + \sin\left(\frac{\pi \rho}{2\_{\rho H}}\right) & \left(|\rho| \le \rho H\right) \\\\ H\_{\text{spec}}\left(\text{o}\_{r}\right) \cdot H\_{\text{inverse}}\left(\text{o}\mathbb{R}\right) = 0 & \left(|\rho| > \rho H\right) \end{cases} \tag{9}$$

*<sup>ρ</sup><sup>H</sup>* is the Nyquist frequency 2 *CFR WFR* and *ρ* is the no-processing region

frequency <sup>2</sup> *CFR WFR* .

The 3D back Fourier transforms the Fourier space data back to 3D volume data, having undergone Fourier space low-pass filtering.

### **6. References**


**7** 

*3ICEMEQ Institut* 

*Spain* 

**Blood Transfusion in Knee Arthroplasty** 

Josep Maria Segur1, Francisco Macule1 and Santiago Suso1,3

*1Orthopedic Department, Knee Unit, Hospital Clinic Barcelona* 

*2Anesthesioly Department, Hospital Clinic Barcelona* 

Oscar Ares1, Montserrat Tio2, Juan Carlos Martinez Pastor1, Luis Lozano1,

Orthopedic surgery is one of most blood-consuming surgical specialties since it is associated with a significant preoperative hemorrhage requiring frequent allogeneic blood transfusions. A special mention needs to be done to hip and knee arthroplasty, complex rachis arthrodesis and tumor-pathology removal. The intervention on older and higher-risk patients has raised the demand on allogeneic blood to such levels that even Blood Banks are unable to attend. Besides the high cost, using allogeneic blood has its risks, such as immunosuppression, patient's wrong identification, transfusion reactions or the possibility of infectious disease transmission. This imbalance between blood demand and availability, together with the awareness about potential risks of blood transfusions and the continuous advances both in technology and pharmaceutics, should lead us to extreme changes in transfusion politics; developing a series of therapeutic measures to reduce blood transfusion to minimum, leaving its use only when it is strictly necessary, especially in scheduled


**1. Introduction** 

surgery.

**PREOPERATIVE STRATEGIES**


> Intraoperative blood savers Normovolemic hemodilution


Table 1. Blood-saving techniques during perioperative time.

**POSTOPERATIVE STRATEGIES**

and folic acid




