- Subject clinical information
After the parents and institution’s Ethics Committee approval, a pediatric DDH patient was randomly selected as the subject for this study in our hospital, who was a female of 16 months of age, diagnosed with unilateral DDH of the right hip. The patient is 80cm tall, weighs 12.6kg. Physical examinations showed that she had a Trendelenburg gait and limb length discrepancy of 1 centimeter shorter on the right side and positive Allis sign. Radiographs showed large acetabular index (AI) (38.80°) and was classified as grade IV of IHDI on the right side; The AI of the left side was 22.67° (Fig. 1).
- Closed reduction
CR under fluoroscopic guidance was performed under general anesthesia. The hip was reduced by placing it in flexion nearly 100 degrees and gradually abducting it on the position of stability. Then a hip spica cast was fixed in a human position with a gentle posterior mold.
- Imaging
To create patient-specific anatomic models, MRI was used to capture nonbony geometry. On a Siemens 3.0T symphony MR scanner, we obtained coronal 3D gradient-echo images of the pelvis and proximal femur, slice thickness 2.6 mm, matrix 640*640, and 10 min scan time.
- Model assembly
First, the DICOM data were imported into the Mimics software (17.0, Materialise). Next, different masks were set up by the processes of thresholding, region growing, manual editing, Boolean operation, etc. Finally, the 3D models of four bones and three cartilages were established based on these masks.
- FEA modeling
The CAD models were imported into the software, Hyper-mesh (13.0 Altair), and the FEA models were established by element size and local refinement based on the requirement of mechanical analysis, and the acetabulum was divided into six regions. (AL: Anterolateral; AM: Anteromedial; SL: Superolateral; SM: Superomedial; PL: Posterolateral; PM: Posteromedial.) (Supplementary fig.1) The type of tetrahedral solid element was used in the analysis, and there are 228,000 tetrahedral elements and 49,000 nodes in the FEA model. In order to improve the convergence of contact calculation, the elements on the contact surfaces of the femoral head and acetabulum were refined. Moreover, the selective reduction integral with node rotation was adopted for tetrahedral elements to improve the convergence of the solution preferably (Fig. 2).
A. Mechanical properties of materials
There were two materials, namely bone and cartilage, in the FEA model [7]. The young’s modulus was defined as 17GPa, and the Poisson ratio was 0.3 for the cortical bone. The cartilage was a viscoelasticity material that its mechanical features were influenced by different sclerostin and loading rate. Moreover, according to different bone age, properties of cartilage are also not exactly identical from surface to inside. Thus, the cartilage was described as an inhomogeneous material. It was difficult to describe the mechanical properties of cartilage accurately by using a mathematical curve. The stress-strain curve was defined approximately according to the previous research [8] (Fig.3).
B. Definition of boundary conditions
When a relaxed muscle is stretched beyond its resting length, it behaves as a deformable body: it deforms and provides passive resistance to the stretch, and the passive response is characterized as hyperplastic. Under the conditions of leg abduction and flexion, the adductor muscles are the main resistance. The points of attachment and physiological cross-sectional area (PCSA) were determined for the seven muscles (Pectinus, Adductor Magnus 1, Adductor brevis, Adductor longus, Adductor Magnus 2, Adductor Magnus 3, Gracilis), respectively, according to the reference [9]. Given the individual difference of the PCSA among patients, the PCSA of a muscle in a subject was defined as a ratio value of one to the whole, according to described in literature [10].
Step 1, Based on the fact that the muscle’s force is proportional to its PCSA. The relationship between force and PCSA is F=k*A 1. The k is an unknow constant.
Step 2, Given the difference of the PCSA among children and adults, the PCSA of a muscle in a subject was defined as a ratio value of know PCSA described in literature [10]. A=PCSA*α 2. (Table 1) The α is another unknow constant.
Step 3, Expert radiologist judgment and assessment of a 3D FEA model were employed to finalize attachment locations of each muscles. Then, we calculated the length of each muscles (L) according to the points of attachment.
Step 4, Supposedly muscles were in the natural state when the thigh was in the state of 90-degree flexion and 0-degree abduction. At this position, the length of the muscle was defined as La1, and the PCSA was Aa1.
Combined with formula 12.
F1a=k*α*PCSA1a
…
F7a=k*α*PCSA2a
We assumed that the total muscle force equals 1.
Then FTa=F1a+F2a+…+F7a=1
F1a/FTa= PCSA1a / (PCSA1a+ PCSA2a+…+ PCSA7a)=0.083
…
F7a/FTa=0.035
Step 5, The total volume remains constant when the muscle abducts from angle a to angle b.
L1a*A1a=L1b*A1b 3
Combined with formula 123.
F1b=k*A1b=k*A1a*L1a/L1b=k*a*PCSA1a* L1a/L1b
F2b= k*α*PCSA2a* L2a/L2b
…
F7b= k*α*PCSA7a* L6a/L7b
We assumed that the total muscle force equals 1.
Then FTb=F1b+F2b+…+F7b=1
F1b/FTb= PCSA1a* L1a/L1b / (PCSA1a* L1a/L1b+ PCSA2a* L2a/L2b+…+ PCSA7a* L7a/L7b)=0.070 (For example b=45°).
…
F7b/FTb=0.038 (Table 2)
Step 6, The abduction force was defined approximately as 1/5 of the baby weight (24N) [11]. Based on the FEA model, the value of the force for each muscle can be solved by force analysis according to the force ratio of each muscle when the muscle forces and external applied loads staying in the equilibrium state.
Table 1 The relationship of length and PCSA under different flexion angle for different muscle
Number
|
Muscles
|
Friederich PCSA
(mm2)
|
FA 0°
PCSA(mm2)
|
FA 0°
ML
(mm)
|
FA 45°
ML
(mm)
|
FA 65°
ML
(mm)
|
FA 80°
ML
(mm)
|
1
|
Pectinus
|
903
|
A1=903*α
|
31.5
|
49.1
|
55.7
|
59.3
|
2
|
Adductor Magnus
(minimus)
|
2552
|
A2=2552*α
|
63.3
|
88.5
|
96.0
|
100.2
|
3
|
Adductor brevis
|
1152
|
A3=1152*α
|
72.4
|
97.2
|
104.5
|
108.7
|
4
|
Adductor longus
|
2273
|
A4=2273*α
|
93.5
|
121.5
|
130.8
|
135.4
|
5
|
Adductor Magnus (middle)
|
1835
|
A5=1835*α
|
111.3
|
134.9
|
141.6
|
146.1
|
6
|
Adductor Magnus (posterior)
|
1695
|
A6=1695*α
|
134.8
|
161.7
|
170.4
|
175.8
|
7
|
Gracilis
|
373
|
A7=373*α
|
137.4
|
164.0
|
172.9
|
177.9
|
PCSA: physiologic cross sectional area [12], FA: flexion angle, ML: muscle length, a: a unknown coefficient, which can be used to predict the new PCSA, according to the Friederich PCSA.
Table 2 The ratio of muscle force under different abdution angle for different muscle
Number
|
Muscles
|
AA 0°
TRMF
|
AA 45°
TRMF
|
AA 65°
TRMF
|
AA 80°
TRMF
|
1
|
Pectinus
|
0.083
|
0.07
|
0.066
|
0.065
|
2
|
Adductor Magnus
(minimus)
|
0.237
|
0.221
|
0.219
|
0.217
|
3
|
Adductor brevis
|
0.107
|
0.104
|
0.104
|
0.103
|
4
|
Adductor longus
|
0.211
|
0.212
|
0.211
|
0.212
|
5
|
Adductor Magnus (middle)
|
0.170
|
0.183
|
0.187
|
0.189
|
6
|
Adductor Magnus (posterior)
|
0.157
|
0.171
|
0.174
|
0.175
|
7
|
Gracilis
|
0.035
|
0.038
|
0.039
|
0.039
|
Total
|
1
|
1
|
1
|
1
|
1
|
AA: abduction angle, TRMF: The ratios of muscle forces
C. Mechanical analysis
To simulate the action of CR, the abduction force, F, was applied on the point of knee, with the force direction being vertical to the axis of the thigh. The fixed constraint was applied to the surrounding of the femur and to the rotation of thigh axis to simulate the restriction of surrounding tissues. A muscle produces two kinds of forces, active and passive, which sum to compose muscle total force. Under the condition of leg abduction and flexion, the adductor muscles stretched beyond its resting length are the main resistance of the passive response. Thus, the active force is neglected. The abduction force was defined approximately as 1/5 of the baby weight (24N) [11]. Based on the FEA model, the value of the force for each muscle can be solved according to the force ratio of each muscle when the muscle forces and external applied loads (24N) staying in the equilibrium state. Moreover, the distribution of acting force and stress between the femoral head and the acetabulum also could be solved (Fig.4).