How to Evaluate Three Dimensional Angle Error From Plain Radiographs
Chen-Kun Liaw, MD, PhD
a,b,c,d, Tai-Yin Wu, MD
e, Sheng-Mou Hou, MD, PhD, MPH
a,b, Rong-Sen Yang, MD, PhD
b, Chiou-Shann Fuh, PhD
faDepartment of Orthopaedics, Shin Kong Wu Ho-Su Memorial Hospital and Health System, Taipei city 11101, Taiwan
bDepartment of Orthopaedics, College of Medicine, National Taiwan University & Hospital, Taipei city 10002, Taiwan
cDepartment of Healthcare Information and Management, School of Health Technology, Ming Chuan University, TaoYuan city 33348, Taiwan
dCollege of Medicine, Fu Jen Catholic University, New Taipei city 24205, Taiwan
eTaipei City Hospital, Renai Branch, Taipei city 10629, Taiwan
fDepartment of Computer Science and Information Engineering National Taiwan University, Taipei city 10617, Taiwan
a b s t r a c t a r t i c l e i n f o
Article history:
Received 23 January 2013 Accepted 20 May 2013
Keywords:
three dimensional angle two dimensional angle plain radiograph
Evaluating three-dimensional angle error is necessary because we cannot get every patient's CT or MRI at all times. Creating a method that can calculate angle error from plain radiographs is therefore important. Using vector and trigonometric mathematics, we gradually deduct our formula which can calculate angle error from goal angles (the angles we plan to achieve before operation) to result angles (the angles we get after operation) by two perpendicular radiographs. We also encode it into Micorsoft Excel (Redmond Campus, Redmond, Washington, U.S.) so that it becomes more user-friendly. We hope this tool can be used when evaluating TKR, corrective osteotomy, fracturefixation, and so on.
© 2013 Elsevier Inc. All rights reserved.
In the past two decades, the discipline of orthopaedics has changed from“free hand” to “mechanical aided”, or more precisely “computer aided”. Meanwhile, evaluation tools have evolved from plain radiograph to three dimensional CT, or MRI.
Two dimensional radiographs have a non-replaceable position due to the following reasons:
1. Some patients do not have CT or MRI. For example, in retrospective studies, the patients may not have CT or MRI during the study period.
2. Some patients may refuse or are not eligible to receive CT or MRI examinations. This may be related to the costs or radiation exposure.
3. Plain radiographs have better resolution than CT or MRI.
Without three dimensional examinations, current studies use two perpendicular radiographs to demonstrate the results separately, such as antero-posterior (AP) view and lateral view (LAT)[1–5]. This may cause another problem: if we do not integrate data from AP view with LAT view, we may not see the truth. For example, we got results with four patients as shown inFig. 1.
The four patients belong to two groups, group #1 (A, B), and group
#2 (C, D). If we calculate the difference from AP view alone, there is no
difference between group #1 and group #2; so is the result from LAT view alone. However, group #1 has one good result (A) and one poor result (B), and group #2 has two median results (C, D). These two groups do have different clinical meanings.
The purpose of this study was to develop a mathematic method to calculate three-dimensional angle difference from AP and LAT view radiographs, without using three dimensional data from CT or MRI.
We can, for example, use this method to calculate the error of tibia component implantation during total knee arthroplasty (TKR).
Methods
We hypothesized that every measurement on AP and LAT view is well done. No technical problem which warrants consideration occurred during measurements.
Wefirst define the three dimensional Cartesian coordinate system.
The X-axis runs from central to lateral. The Y-axis runs from distal to proximal. The Z-axis is from posterior to anterior.
If we want to evaluate the error in implanting tibia component during TKR, wefirst define the unit vector with specific posterior slope angle ps and valgus angle vg.
We deducted our formula from the definitions of trigonometric function, as illustrated inFig. 2:
Vector OD is the normal vector of tibia component.
The X-axis is parallel to line OA.
The Y-axis is parallel to line OB.
The Z-axis is parallel to line GO.
The posterior slope angle means the angle formed by the projection of the normal vector to YZ plane, which is shown as∠BOC.
The Journal of Arthroplasty 28 (2013) 1788–1790
All authors have no conflict of interest of this issue.
This study contains human data from de-link database. No IRB approval is needed.
The Conflict of Interest statement associated with this article can be found athttp://
dx.doi.org/10.1016/j.arth.2013.05.023.
Reprint requests: Chen-Kun Liaw, MD, PhD, Department of Orthopaedics, Shin Kong Wu Ho-Su Memorial Hospital and Health System, No. 95, Wen Chang Road, Shih Lin District, Taipei City, Taiwan.
0883-5403/2810-0019$36.00/0– see front matter © 2013 Elsevier Inc. All rights reserved.
http://dx.doi.org/10.1016/j.arth.2013.05.023
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The valgus angle means the angle formed by the projection of the normal vector to XY plane, which is shown as∠BOF.
Because we wanted to get the unit vector, we assigned the length of the normal vector 1 unit.
length of OD = 1
The cuboid:
length FD = length BC = length OG
length FO = length OB/cos(∠BOF)
∠DOF = tan−1(length FD/length FO) = tan−(length BC/length OB/cos(∠BOF))
= tan−1(tan(∠BOC)*cos(∠BOF)) = tan−1(tan(ps)*cos(vg)) length OF = length OD*cos(∠DOF)
vector OD = (length OA, length OB,−length OG) = (length OA, length OB,−length FD)
length OA = length OF*sin(∠OFA) = length OF*sin(∠BOF)
= length OF*sin(vg) = length OD*cos(∠DOF)*sin(vg)
= cos(tan−1(tan(ps)*cos(vg)))*sin(vg)
length OB = length AF = length OF*cos(∠OFA) = length OF*cos(∠BOF)
= length OF*cos(vg) = length OD*cos(∠DOF)*cos(vg)
= cos(tan−1(tan(ps)*cos(vg)))*cos(vg)
length FD = length OD*sin(∠DOF) = sin(tan− 1(tan(ps)
*cos(vg)))
Thus, we define:
unit vector (ps, vg) = (cos(tan−1(tan(ps)*cos(vg)))*sin(vg), cos(tan− 1 (tan(ps)*cos(vg)))*cos(vg), − sin(tan− 1 (tan(ps)
*cos(vg))))
A similar formula has been used and published in a mathematic journal[6].
Now we can calculate the angle between goal (the angles we plan to achieve before operation) and result (the angles we get after operation) by calculating the angle between the two vectors with replacement of the parameters ps and vg.
For example, our goal is ps = 3° and vg = 0°.
goal unit vector = (cos(tan− 1(tan(3°)*cos(0°)))*sin(0°), cos(tan− 1 (tan(3°)*cos(0°)))*cos(0°), − sin(tan− 1 (tan(3°)
*cos(0°)))) = (0, 0.99863,−0.05234)
InFig. 3, we measure the postoperative X-rays of a TKR patient, and the results are ps = 6.51° and vg =−7.31°.
Fig. 2. Illustration of how to get the normal vector from posterior slope angle ps and valgus angle vg. Vector OD is the normal vector of tibia component. The X-axis is parallel to line OA. The Y-axis is parallel to line OB. The Z-axis is parallel to line GO. The posterior slope angle means the angle formed by the projection of the normal vector to YZ plane, which is shown as∠BOC. The valgus angle means the angle formed by the projection of the normal vector to XY plane, which is shown as∠BOF.
Fig. 3. (A) The TKR postoperative AP view. The measured tibia plate is varus 7.31°, which means valgus−7.31°. (B) The TKR postoperative LAT view. The measured posterior slope is 6.51° (90–83.49 = 6.51).
Fig. 1. An example of angle measurements of 4 patients.
C.-K. Liaw et al. / The Journal of Arthroplasty 28 (2013) 1788–1790 1789
result unit vector = (cos(tan−1(tan(6.51°)*cos(−7.31°)))*sin(−
7.31°), cos(tan−1 (tan(6.51°)*cos(−7.31°)))*cos(−7.31°), −sin(- tan−1 (tan(6.51°)*cos(−7.31°)))) = (−0.12643, 0.985579, − 0.11247)
Angle between two unit vectors = cos−1(vector 1 dot vector 2) The angle between (0, 0.99863, −0.05234) and (−0.12643, 0.985579,−0.11247) equals to 8.062913218°.
This complex formula was combined into a simplified Excel program, as shown in Fig. 4. Because ps is only used in tibia component, we change it toflexion fl so that it can be used in other cases. For example, we can use it to evaluate reduction of distal radius fracture. In this case, we can input volar tilt asfl, inclination as vg.
Fig. 5shows the formula encoded into cell E2 of a Microsoft Excel spreadsheet. The users can input the formula by themselves or from the following TEXT:
ACOS(COS(ATAN(TAN(B2*PI()/180)*COS(A2*PI()/180)))
*SIN(A2*PI()/180)*COS(ATAN(TAN(D2*PI()/180)*COS(C2*PI()/180)))
*SIN(C2*PI()/180)+COS(ATAN(TAN(B2*PI()/180)*COS(A2*PI()/
1 8 0 ) ) ) * C O S ( A 2 * P I ( ) / 1 8 0 ) * C O S ( A T A N ( T A N ( D 2 * P I ( ) / 1 8 0 )
*COS(C2*PI()/180)))*COS(C2*PI()/180)+SIN(ATAN(TAN(B2*PI()/
180)*COS(A2*PI()/180)))*SIN(ATAN(TAN(D2*PI()/180)*COS(C2*PI()/
180))))*180/PI() Acknowledgments
The authors thank United Orthopedic Corporation, Hsinchu, Taiwan for providing us mathematical supports.
This study was supported by the grant of National Science Council, NSC98-2314-B-087-001-MY2, Taiwan, ROC.
References
1.Nam D, Jerabek SA, Haughom B, et al. Radiographic analysis of a hand-held surgical navigation system for tibial resection in total knee arthroplasty. J Arthroplasty 2011;26(8):1527.
2.Nam D, Nawabi DH, Cross MB, et al. Accelerometer-based computer navigation for performing the distal femoral resection in total knee arthroplasty. J Arthroplasty 2012;27(9):1717.
3.Nam D, Weeks KD, Reinhardt KR, et al. Accelerometer-Based, Portable Navigation vs Imageless, Large-Console Computer-Assisted Navigation in Total Knee Arthroplasty:
A Comparison of Radiographic Results. J Arthroplasty 2012.
4.Nam D, Cross M, Deshmane P, et al. Radiographic results of an accelerometer-based, handheld surgical navigation system for the tibial resection in total knee arthroplasty. Orthopedics 2011;34(10):e615.
5.Mullaji AB, Sharma A, Marawar S. Unicompartmental knee arthroplasty: functional recovery and radiographic results with a minimally invasive technique.
J Arthroplasty 2007;22(4 Suppl 1):7.
6.Liaw CK, Yang RS, Hou SM, et al. A simple mathematical standardized measurement of acetabulum anteversion after total hip arthroplasty. Comput Math Methods Med 2008;9(2):105.
Fig. 4. The formula is encoded in Excel. The user can just input goal and result and then get the operational error.
Fig. 5. The entire formula is encoded in cell E2 of an Excel spreadsheet.
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