Appendix A. Supplementary data

Appendix A. Supplementary data Appendix A.1: Coordinates of the via attachments Table 1: Coordinates of the via attachments for the muscle elements with a curved lines-of-action. The coordinates are given with respect to the global reference frame. Muscle

Iliocostalis Iliocostalis Iliocostalis Iliocostalis Iliocostalis Iliocostalis Iliocostalis Iliocostalis Iliocostalis

# Via Bone

cervicis cervicis cervicis cervicis cervicis cervicis cervicis cervicis cervicis

Position (m) x

y

z

1 1 1 1 2 2 2 3 3

1 2 3 4 1 2 3 1 2

R1 R2 R3 R4 R1 R2 R3 R1 R2

-0.0316 -0.0405 -0.0505 -0.0573 -0.0322 -0.0413 -0.0501 -0.0327 -0.0383

-0.1504 -0.1248 -0.1008 -0.0811 -0.1520 -0.1222 -0.1015 -0.1511 -0.1235

1.1040 1.0969 1.0770 1.0560 1.1039 1.0941 1.0779 1.1033 1.0951

Iliocostalis thoracis Iliocostalis thoracis Iliocostalist thoracis Iliocostalis thoracis Iliocostalis thoracis Iliocostalis thoracis Iliocostalis thoracis Iliocostalis thoracis Iliocostalis thoracis Iliocostalis thoracis Iliocostalis thoracis Iliocostalis thoracis Iliocostalis thoracis Iliocostalis thoracis Iliocostalis thoracis

1 1 1 1 1 2 2 2 2 2 3 3 3 3 3

1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

R6 R7 R8 R9 R10 R5 R6 R7 R8 R9 R4 R5 R6 R7 R8

-0.0670 -0.0679 -0.0698 -0.0690 -0.0677 -0.0612 -0.0632 -0.0649 -0.0677 -0.0668 -0.0579 -0.0611 -0.0644 -0.0661 -0.0698

-0.0595 -0.0535 -0.0477 -0.0452 -0.0475 -0.0684 -0.0599 -0.0528 -0.0474 -0.0445 -0.0818 -0.0678 -0.0596 -0.0529 -0.0477

1.0148 0.9909 0.9628 0.9308 0.8994 1.0378 1.0160 0.9899 0.9642 0.9330 1.0569 1.0356 1.0150 0.9898 0.9628

Omohyoid

1 1

T1

-0.0411 -0.1799 1.1004

T3 T4 T5 T6 T7 T8 T2 T3 T4 T5 T6 T7 T1 T2 T3 T4 T5 T6

-0.0084 -0.0141 -0.0185 -0.0191 -0.0199 -0.0275 -0.0045 -0.0084 -0.0184 -0.0200 -0.0209 -0.0280 0.0027 -0.0052 -0.0113 -0.0186 -0.0217 -0.0238

Semispinalis thoracis 1 Semispinalis thoracis 1 Semispinalis thoracis 1 Semispinalis thoracis 1 Semispinalis thoracis 1 Semispinalis thoracis 1 Semispinalis thoracis 2 Semispinalis thoracis 2 Semispinalis thoracis 2 Semispinalis thoracis 2 Semispinalis thoracis 2 Semispinalis thoracis 2 Semispinalis thoracis 3 Semispinalis thoracis 3 Semispinalis thoracis 3 Semispinalis thoracis 3 Semispinalis thoracis 3 Semispinalis thoracis 3 Continued on next page

1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6

1

-0.0859 -0.0784 -0.0673 -0.0640 -0.0578 -0.0487 -0.1044 -0.0998 -0.0812 -0.0684 -0.0624 -0.0524 -0.1160 -0.1120 -0.0954 -0.0824 -0.0656 -0.0634

1.0671 1.0503 1.0269 1.0103 0.9851 0.9556 1.0888 1.0746 1.0508 1.0247 1.0075 0.9834 1.1031 1.0900 1.0721 1.0541 1.0178 1.0096

Table 1: Continued from previous page Muscle

# Via Bone

Semispinalis Semispinalis Semispinalis Semispinalis Semispinalis Semispinalis Spinalis Spinalis Spinalis Spinalis Spinalis Spinalis Spinalis Spinalis Spinalis Spinalis Spinalis Spinalis Spinalis Spinalis Spinalis Spinalis Spinalis Spinalis Spinalis Spinalis Spinalis Spinalis Spinalis Spinalis Spinalis Spinalis Spinalis Spinalis Spinalis Spinalis Spinalis Spinalis Spinalis Spinalis Spinalis Spinalis Spinalis Spinalis Spinalis Spinalis

thoracis thoracis thoracis thoracis thoracis thoracis

thoracis thoracis thoracis thoracis thoracis thoracis thoracis thoracis thoracis thoracis thoracis thoracis thoracis thoracis thoracis thoracis thoracis thoracis thoracis thoracis thoracis thoracis thoracis thoracis thoracis thoracis thoracis thoracis thoracis thoracis thoracis thoracis thoracis thoracis thoracis thoracis thoracis thoracis thoracis thoracis

Position (m) x

y

z

4 4 4 4 4 4

1 2 3 4 5 6

C7 T1 T2 T3 T4 T5

0.0099 0.0055 -0.0053 -0.0149 -0.0188 -0.0248

-0.1345 -0.1306 -0.1125 -0.0974 -0.0850 -0.0745

1.1154 1.1100 1.0907 1.0709 1.0542 1.0376

1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 6 6 6 6

1 2 3 4 5 1 2 3 4 5 6 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 9 1 2 3 4

T8 T9 T10 T11 T12 T7 T8 T9 T10 T11 T12 T5 T6 T7 T8 T9 T10 T11 T12 T5 T6 T7 T8 T9 T10 T11 T12 T4 T5 T6 T7 T8 T9 T10 T11 T12 T9 T10 T11 T12

-0.0118 -0.0134 -0.0131 -0.0129 -0.0109 -0.0114 -0.0136 -0.0126 -0.0125 -0.0119 -0.0104 -0.0156 -0.0223 -0.0203 -0.0228 -0.0210 -0.0180 -0.0175 -0.0133 -0.0158 -0.0223 -0.0203 -0.0228 -0.0210 -0.0180 -0.0175 -0.0133 -0.0147 -0.0223 -0.0203 -0.0228 -0.0219 -0.0210 -0.0180 -0.0175 -0.0133 -0.0102 -0.0131 -0.0129 -0.0109

-0.0396 -0.0382 -0.0389 -0.0366 -0.0402 -0.0452 -0.0410 -0.0403 -0.0393 -0.0392 -0.0415 -0.0592 -0.0580 -0.0520 -0.0440 -0.0450 -0.0450 -0.0439 -0.0444 -0.0620 -0.0570 -0.0510 -0.0430 -0.0450 -0.0463 -0.0439 -0.0444 -0.0620 -0.0580 -0.0512 -0.0441 -0.0447 -0.0474 -0.0453 -0.0439 -0.0444 -0.0389 -0.0389 -0.0366 -0.0402

0.9503 0.9257 0.8958 0.8794 0.8538 0.9833 0.9610 0.9277 0.9038 0.8766 0.8524 1.0169 1.0022 0.9785 0.9559 0.9201 0.9058 0.8816 0.8455 1.0227 1.0022 0.9785 0.9559 0.9201 0.9058 0.8816 0.8455 1.0311 1.0022 0.9785 0.9559 0.9380 0.9201 0.9058 0.8816 0.8455 0.9179 0.8958 0.8794 0.8538

Appendix A.2: Mathematical definitions of the wrapping surfaces

2

3

Form

Region

0.0400 0.0120

r

ry

rz

0.1370 0.1100 0.2200

rx

ps y z

x

pe y z

-0.0300 -0.0800 1.0300 0.0166 -0.1732 1.1077 -0.0850 -0.0370 1.0400 -0.0486 -0.1179 1.1209

x

pc y z

x

nx y

z

x

ny y

z

x

nz y

z

0.0000 -0.1320 0.8970 0.9945 0.0000 0.1045 0.0000 1.0000 0.0000 -0.1045 0.0000 0.9945

x

All values are in meters. r is the radius of the cylinder, rx, ry, and rz are the radii of the ellipsoid in x-, y-, and z- directions, ps and pe are the start and end points of the cylinder, pc is the center of the ellipsoid, nx, ny and nz are the axes of the ellipsoid in x, y, and z directions. c The wrapping surface specified for the trapezius muscle is only applicable to its first six elements.

b

a

Serratus anterior Ellipsoid Thorax Serratus posterior superior Cylinder Thorax c Trapezius Cylinder Scapula

Muscle

Table 2: Mathematical descriptions of the geometrical forms of the wrapping surfacesa,b .

Appendix A.3: Architectural tendon parameters Table 3: Architectural tendon parameters for the deeper back musclesa . Muscle Levatores Levatores Levatores Levatores Levatores Levatores Levatores Levatores Levatores Levatores Levatores Levatores Levatores

# Length Mass Area (mm) (g) (cm2 ) costarum costarum costarum costarum costarum costarum costarum costarum costarum costarum costarum costarum costarum

Multifidus Multifidus Multifidus Multifidus Multifidus Multifidus Multifidus Multifidus Multifidus Multifidus Multifidus Multifidus Multifidus Multifidus Multifidus Multifidus Multifidus Multifidus Multifidus Multifidus Multifidus Multifidus Multifidus Multifidus Multifidus Multifidus Multifidus Multifidus Multifidus Multifidus Multifidus Multifidus Multifidus Multifidus Multifidus Continued on next

1 2 3 4 5 6 7 8 9 10 11 12 13

28.0 49.0 29.0 29.0 26.0 33.0 39.0 35.0 44.0 59.0 42.0 72.0 40.0

19 95.0 20 113.5 21 139.0 22 41.0 23 53.0 24 102.0 25 79.0 26 47.0 27 57.0 28 84.5 29 39.5 30 61.5 31 35.5 32 47.0 33 66.0 34 65.5 35 93.0 36 70.0 37 37.0 38 44.0 39 44.0 40 105.0 41 67.0 42 44.0 43 85.0 44 100.5 45 48.5 46 76.5 47 79.0 48 101.0 49 76.5 50 49.0 51 45.0 52 61.0 53 37.5 page

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0.06 0.03 0.06 0.07 0.02 0.05 0.03 0.06 0.05 0.16 0.07 0.08 0.04

0.018 0.005 0.017 0.020 0.006 0.013 0.006 0.014 0.009 0.023 0.014 0.009 0.008

0.09 0.37 0.26 0.02 0.02 0.07 0.04 0.09 0.02 0.08 0.08 0.04 0.01 0.01 0.16 0.07 0.08 0.09 0.03 0.09 0.06 0.17 0.18 0.03 0.03 0.05 0.05 0.15 0.06 0.04 0.11 0.04 0.06 0.06 0.02

0.008 0.027 0.016 0.004 0.003 0.006 0.004 0.016 0.003 0.008 0.017 0.005 0.002 0.002 0.020 0.009 0.007 0.011 0.007 0.017 0.011 0.013 0.022 0.006 0.003 0.004 0.009 0.016 0.006 0.003 0.012 0.007 0.011 0.008 0.004

Table 3: Continued from previous page Muscle

# Length Mass CSA (mm) (g) (cm2 )

Multifidus Multifidus Multifidus Multifidus Multifidus Multifidus Multifidus Multifidus Multifidus Multifidus Multifidus

54 55 56 57 58 59 60 61 62 63 64

66.0 43.0 47.0 39.5 34.0 48.0 44.5 33.0 39.0 36.5 39.0

0.15 0.01 0.02 0.04 0.21 0.04 0.21 0.13 0.1 0.06 0.01

0.019 0.002 0.004 0.008 0.051 0.007 0.039 0.033 0.021 0.014 0.002

Rotatores Rotatores Rotatores Rotatores Rotatores Rotatores Rotatores Rotatores Rotatores Rotatores Rotatores Rotatores Rotatores Rotatores Rotatores Rotatores Rotatores Rotatores Rotatores Rotatores Rotatores Rotatores Rotatores Rotatores Rotatores Rotatores

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

30.0 30.0 26.0 23.0 18.0 20.0 26.0 13.0 15.0 8.0 24.0 8.0 26.5 9.0 38.0 9.0 27.5 7.0 26.5 31.5 27.0 19.0 30.5 30.5 21.0 27.5

0.02 0.02 0.09 0.03 0.01 0.01 0.01 0.02 0.01 0.03 0.03 0.02 0.02 0.03 0.03 0.02 0.02 0.01 0.01 0.02 0.02 0.01 0.08 0.08 0.04 0.03

0.006 0.006 0.029 0.011 0.005 0.004 0.003 0.013 0.006 0.031 0.010 0.021 0.006 0.028 0.007 0.018 0.006 0.012 0.003 0.005 0.006 0.004 0.022 0.022 0.016 0.009

Spinalis Spinalis Spinalis Spinalis Spinalis Spinalis

1 2 3 4 5 6

110.0 140.0 186.0 219.5 224.0 101.5

0.07 0.07 0.15 0.15 0.15 0.07

0.005 0.004 0.007 0.006 0.006 0.006

thoracis thoracis thoracis thoracis thoracis thoracis

a

The elements 16-18 of the multifidus muscle had the similar muscle architecture as the lumbar elements of this muscle (tendon-muscle-tendon architecture). Therefore, we did not measure mass and cross-sectional area of their tendons.

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Appendix A.4: Abbreviations used for the muscles Table 4: Abbreviations used for the muscles illustrated in Fig. 4. Muscle name

Abbreviation

Iliocostalis cervicis Iliocostalis thoracis Intercostales externi Intercostales interni Levator scapulae Levatores costarum Longissimus capitis Longisssimus cervicis Longus capitis Multifidus thoracic Obliquus capitis inferior Obliquus capitis superior Omohyoid Rectus capitis anterior Rectus capitis lateralis Rectus capitis posterior major Rectus capitis posterior minor Rhomboideus (major and minor) Rotatores Scalenus anterior Scalenus medius Scalenus posterior Semispinalis capitis Semispinalis cervicis Semispinalis thoracis Serratus anterior Serratus posterior superior Spinalis thoracis Splenius capitis Splenius cervicis Sternocleidomastoid Sternohyoid Sternothyroid Subclavious Subcostales Thyrohyoid Transversus thoracis Trapezius

ICE IT IE II LS LEC LCA LCE LUCA MFT OCI OCS OH RCA RCL RCPMA RCPMI RM ROT SAN SM SP SSCA SSCE SST SA SPS ST SUCA SCE STCM STH STT SCL SCO TH TT TP

Iliocostalis lumborum Latissimus dorsi Longissimus thoracis Multifidus lumbar Obliquus externus abdominis Obliquus internus abdominis Psoas major Quadratus lumborum Rectus abdominis Serratus posterior inferior Transversus abdominis

IL LD LT MFL OEA OIA PM QL RA SPI TA

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