Age- and sex-related regional compressive strength characteristics of human lumbar vertebrae in osteoporosis

Figures & data

Figure 1 Typical load-displacement and stress–strain diagrams with the related strain energy densities of vertebrae. A) load-displacement diagrams with linearization. B) linearized stress–strain diagrams with absorbed and recovered energy densities, the latter calculated at ultimate strains and proportional stresses.

Table 1 Comparing age, density, geometry, and strength data of vertebrae L1–L2

Age, density, geometry		L1		L2
		Mean, SD	rage	Mean, SD	rage
Mean age	years	71.5 ± 13.7	−	71.7 ± 13.0	−
Bone mineral density, BMD	g/cm^2	0.374 ± 0.085	−0.59	0.380 ± 0.095	−0.69
T-score	−	−5.27 ± 0.66	−0.52	−5.97 ± 0.79	−0.47
Superior cross–sectional area, Asup	mm^2	1361 ± 259	+0.20	1389 ± 262	−0.20
Central cross–sectional area, Acentr	mm^2	1061 ± 205	−0.03	1086 ± 244	+0.09
Inferior cross–sectional area, Ainf	mm^2	1398 ± 187	+0.08	1425 ± 261	+0.22
Height without end–plates, h	mm	19.1 ± 2.8	−0.13	21.0 ± 3.3	−0.14
Volume of specimen, V	mm^3	23421 ± 6226	−0.05	26548 ± 8381	+0.05
Ultimate load, Fu	N	3161 ± 1937	−0.63	3109 ± 1898	−0.58
Central ultimate stress, σu, centr	MPa	3.01 ± 1.73	−0.61	2.93 ± 1.74	−0.57
Ultimate strain, ɛu	%	5.11 ± 1.16	−0.06	4.99 ± 1.43	−0.11

Table 2 Age, density and geometry of vertebrae L1–L2 with age-correlation by distinguishing the sexes

Age, density, geometry		L1–L2
		Male (n = 16)		Female (n = 38)		Total (n = 54)
		Mean, SD	rage	Mean, SD	rage	Mean, SD	rage
Mean age	years	65.6 ± 12.6	−	74.2 ± 12.9	−	71.6 ± 13.3	−
Bone mineral density, BMD	g/cm^2	0.446 ± 0.088	−0.82	0.347 ± 0.072	−0.47	0.377 ± 0.089	−0.63
T-score	−	−5.41 ± 0.84	−0.71	−5.63 ± 0.77	−0.31	−5.57 ± 0.79	−0.45
Z-score	−	−4.72 ± 0.64	−0.42	−3.75 ± 0.63	0.48	−4.05 ± 0.77	0.26
Superior cross-sectional area, Asup	mm^2	1591 ± 210	0.33	1272 ± 212	0.50	1366 ± 256	0.19
Central cross-sectional area, Acentr	mm^2	1272 ± 154	0.27	990 ± 189	0.26	1073 ± 221	0.03
Inferior cross-sectional area, Ainf	mm^2	1569 ± 154	0.17	1329 ± 210	0.37	1400 ± 223	0.12
Height without end-plates, h	mm	21.9 ± 1.5	−	19.0 ± 3.3	−	19.9 ± 3.1	−
Volume of specimen, V	mm^3	31212 ± 3588	−	22033 ± 6770	−	24753 ± 7317	−

Table 3 Proportional and ultimate loads and stresses with Young’s moduli of lumbar vertebrae L1–L2 correlation with age and BMD by distinguishing the sexes

Compressive loads and stresses		L1–L2
		Male (n = 16)				Female (n = 38)				Total (n = 54)
		Mean	rage	rBMD	rTscore	Mean	rage	rBMD	rTscore	Mean	rage	rBMD	rTscore
Proportional load, Fp	N	3212	−0.60	0.83	0.76	1582	−0.43	0.38	0.38	2065	−0.55	0.69	0.51
Ultimate load, Fu	N	4636	−0.66	0.90	0.75	2509	−0.52	0.47	0.40	3139	−0.61	0.75	0.52
Proportional size-corrected load, Fp/V	N/mm^3	0.106	−0.61	0.84	0.81	0.082	−0.37	0.41	0.45	0.089	−0.46	0.56	0.57
Ultimate size-corrected load, Fu/V	N/mm^3	0.153	−0.67	0.91	0.82	0.128	−0.44	0.48	0.46	0.135	−0.52	0.60	0.58
Superior proportional stress, σp,sup	MPa	2.09	−0.61	0.84	0.77	1.31	−0.50	0.41	0.40	1.54	−0.58	0.64	0.54
Central proportional stress, σp, centr	MPa	2.58	−0.62	0.85	0.78	1.68	−0.42	0.42	0.42	1.95	−0.53	0.64	0.55
Inferior proportional stress σp,inf	MPa	2.09	−0.60	0.84	0.77	1.25	−0.47	0.41	0.41	1.50	−0.56	0.65	0.55
Superior ultimate stress, σu,sup	MPa	3.00	−0.68	0.91	0.78	2.07	−0.59	0.48	0.41	2.35	−0.65	0.69	0.55
Central ultimate stress, σu, centr	MPa	3.71	−0.69	0.91	0.79	2.66	−0.50	0.50	0.43	2.97	−0.59	0.69	0.56
Inferior ultimate stress, σu,inf	MPa	3.00	−0.66	0.91	0.78	1.97	−0.55	0.49	0.43	2.27	−0.62	0.71	0.57
Young’s modulus, E	MPa	95.6	−0.55	0.79	0.71	67.4	−0.52	0.36	0.32	75.8	−0.56	0.57	0.48
Vertebral proportional stiffness, Kp	N/mm	4888	−0.38	0.53	0.40	3824	−0.23	0.32	0.36	4140	−0.32	0.44	0.39
Vertebral ultimate stiffness, Ku	N/mm	4315	−0.58	0.73	0.59	2925	−0.35	0.34	0.33	3337	−0.47	0.55	0.43

Table 4 Strains, ductility, recoverable, and absorbed energy densities of lumbar vertebrae L1–L2 correlation with age and bone mineral density by distinguishing the sexes

Compressive strains and energy densities		L1–L2
		Male (n =16)				Female (n = 38)				Total (n = 54)
		Mean	rage	rBMD	rTscore	Mean	rage	rBMD	rTscore	Mean	rage	rBMD	rTscore
Proportional strain, ɛp	%	3.22	−0.31	0.38	0.51	2.37	−0.17	0.14	0.22	2.62	−0.28	0.34	0.32
Yield strain, ɛy	%	4.13	0.08	0.12	0.36	3.99	0.00	0.18	0.22	4.03	0.00	0.16	0.24
Ultimate strain, ɛu	%	5.14	−0.06	0.27	0.31	5.02	−0.07	0.29	0.26	5.06	−0.08	0.26	0.28
Maximal (break) strain, ɛmax	%	5.68	−0.10	0.42	0.27	6.20	0.09	0.15	0.26	6.04	0.09	0.10	0.23
Ductility: maximal plastic deformability, ɛd	%	1.55	−0.16	0.22	−0.05	2.21	0.12	−0.20	0.03	2.01	0.11	−0.22	−0.04
Superior ult. absorbed energy density, Wabs, sup^d	mJ/mm^3	0.054	−0.40	0.68	0.59	0.045	−0.54	0.60	0.50	0.048	−0.50	0.61	0.54
Central ult. absorbed energy density, Wabs,centr^u	mJ/mm^3	0.060	−0.37	0.64	0.55	0.052	−0.47	0.57	0.45	0.054	−0.44	0.56	0.49
Inferior ult. absorbed energy density, Wabs, inf^u	mJ/mm^3	0.054	−0.41	0.68	0.58	0.043	−0.49	0.60	0.52	0.046	−0.47	0.63	0.55
Superior duct. absorbed energy density, Wabs, sup^d	mJ/mm^3	0.075	−0.40	0.74	0.49	0.069	−0.39	0.46	0.46	0.071	−0.38	0.52	0.47
Central duct. absorbed energy density, Wabs,centr^d	mJ/mm^3	0.085	−0.38	0.72	0.47	0.084	−0.29	0.42	0.41	0.085	−0.30	0.47	0.43
Inferior duct. absorbed energy density. Wabs, inf^d	mJ/mm^3	0.075	−0.40	0.73	0.48	0.067	−0.33	0.45	0.46	0.069	−0.36	0.53	0.47

Figure 2 Regional cross sectional areas versus aging and sex. A) Regional CSAs. B) Superior CSA vs aging and sex. C) Yearly increase of regional CSAs vs sex. D)Total increase of regional CSAs vs sex.

Abbreviation: CSA, cross-sectional area.

Figure 3 Decrease of ultimate load and load/volume of vertebrae L1–L2 vs aging and sex. A) Decrease of ultimate load vs aging and sex. B) Decrease of ultimate load/volume vs aging and sex.

Figure 4 Decrease of regional ultimate stresses of vertebrae L1–L2 vs aging and sex. A) Decrease of central ultimate stresses vs aging and sex. B) Yearly decrease trends of regional ultimate stresses vs aging and sex. C)Total decrease % of regional ultimate stresses during 50 years between 43–93 years.

Figure 5 Decrease of Young’s modulus and ultimate stiffness of vertebrae L1–L2 vs aging and sex. A) Decrease of Young’s modulus vs aging and sex. B) Decrease of ultimate stiffness vs aging and sex.

Figure 6 Age- and sex-related decrease of energy absorption capacity of vertebrae L1–L2. A, B) Age-related decline of superior ultimate and ductile energy density absorption versus sexes. C) Yearly loss of regional ultimate and ductile energy absorption capacity. D) Total regional loss of ultimate and ductile energy absorption capacity of sexes during 50 years between 43–93 years.

Figure 7 Compressive strength versus BMD. A, B, C, D) Increase of ultimate load, central ultimate stress, Young’s modulus, and ultimate stiffness vs BMD, respectively. Abbreviation: BMD, bone mineral density.

Figure 8 Bone mineral density (BMD) versus aging and sex. A) linear decline of BMD with aging and sex. B) quadratic approximation for women with yearly decline trends at age 50 and 70 years.

Figure 9 Age-related nonlinear approximation of strength parameters for women. A, B, C, D) Quadratic approximation of age-related decline function of ultimate load, central ultimate stress, Young’s modulus, and ultimate stiffness of women, respectively.