From CT Scan to Plantar Pressure Map Distribution of a 3D Anatomic Human Foot

Excerpt from the Proceedings of the COMSOL Conference 2010 Paris

P. Franciosa, S. Gerbino* University of Molise, School of Engineering *Corresponding author: Via Duca degli Abruzzi - 86039 Termoli (CB) – Italy [email protected]

Abstract: Understanding the stress-strain behav- methods, based on FE modeling, have also been ior of human foot tissues and pressure map dis- adopted. FE models of human foot have been tributions at the plantar interface is of interest developed under certain simplifications and as- into biomechanical investigations. In particular, sumptions (see [7] and [8]) such as: (i) simplified monitoring plantar pressure maps is crucial to or partial foot shape, (ii) assumptions of non- establishing the perceived human comfort of linear hyper-elastic material law, (iii) ligaments shoe insoles. and plantar fascia modeled as equivalent forces In this contest, a 3D anatomical detailed FE hu- or elastic beams/bars, (iiii) no friction or thermal man foot model was created, starting from CT effect, at plantar foot interface, accounted. (Computer Tomography) scans of a 29 years old Based on these assumptions, valuable 3D FE male. Anatomical domains related to bones and models were presented in the literature [9], [10]. soft tissues were generated, using segmentation In this contest, the geometrical complexity of techniques, and then processed into a CAD envi- foot structure was captured through reverse en- ronment to model 3D domains by using gineering methodologies, based on CT (Com- loft/sweep and boolean operations. Finally, FE puter Tomography) or MR (Magnetic Reso- model was generated into Comsol Multiphys- nance) scans. Starting from high resolution ics® 3.5a, by assigning contact pairs and defin- medical images, suitable FE models were gener- ing non-linear material laws. Particular attention ated by using segmentation procedures. was posed on fine tuning the calculation strongly In the present paper, how to face out the 3D dependent on the convergence parameters influ- geometry reconstruction of the human foot into enced by the contacts and hyper-elastic material Comsol Multiphisics® 3.5a is investigated. properties. Comsol Multiphisics® offers a direct link with two of the most powerful commercial tools able Keywords: CT scan, CAD foot model, hyper- to generate 3D tetrahedral mesh models starting elastic material, contact analysis, plantar pressure from stack-up images: ScanFE® (by Simple- map. ware®) and Mimics® (by Materialise®) [11]. However, as reported and experimented yet in 1. Introduction [12], so-imported mesh models cannot be edited in Comsol Multiphisics®. This program, when It was reported that the perceived comfort working on tessellated geometries, creates ra- related to the human foot can be associated to the tional Bezier patches, defining boundary or do- contact pressure generated at the plantar - insole main equations. Manipulating or editing those interface [1]. High values of pressure can gener- boundaries/domains, which requires geometry ate pain or pathologies due to the obstruction of decomposition analysis, is not allowed. blood circulation in areas with peak values of To avoid all these crucial issues, one should pressure. Experimental studies have pointed out create B-rep CAD geometry starting from seg- a relationship between perceived comfort and mented surfaces, extracted from stack-up images. plantar pressures [2], [3]. Over years, many re- If, on one hand, this stage may be time consum- searchers have addressed their attention on ing, on the other hand a suitable CAD geometry minimizing the plantar pressure, by varying in- model is finally available and ready to be post- sole footwear materials, thickness and anatomi- processed into any multiphysic simulation. cal shape [4], [5], [6]. The above approach was adopted, for exam- Recently, in order to give a valuable support ple, into [13], where, starting from MR scans, the to experimental investigations, computational foot geometry was created and then assembled into SolidWorks® CAD system. CAD model, the FE model was created directly In the present paper, the development of the into Comsol Multiphisics® for the non-linear detailed 3D anatomic FE model of a human foot analysis, by using the live-link between both is described. The balanced standing condition environments. Boundary conditions, contact and was simulated: the foot was supposed to be identity pairs were introduced. Domain equations touching an ideal-rigid ground. Hyper-elastic were set to simulate the non-linear hyper-elastic material law was assigned to soft tissues. Contact behavior of the foot soft tissue. pressures were calculated by adopting a non- linear static simulation. How to speed-up the 3. Foot Modeling solution convergence by fine-tuning non-linear tolerance, penalty factor and scaling value, was A CT (Toshiba® Aquilion 4 equipment) scan also highlighted. was performed on a 29 years old male (65 Kg). 345 slices were captured with a slice distance of 2. Methodology Overview 1.0 mm (see Figure 2.a). Scans were made for both feet in their neutral Figure 1 depicts the general work-flow posture in which there is the least tension or adopted in the present research. pressure on tendons, muscles and bones.

Foot CT scan images .DICOM format (Toshiba® Aquilion 4)

3D Image reconstruction .STL format

(ScanIP® module 3.2) Figure 2.a. DICOM image

CAD modeling live-link (SolidWorks® 2010)

FE modeling (Comsol Multiphysics® 3.5a)

2D segmented regions 3D reconstructed domains Figure 2.b. Segmented regions and 3D reconstruction Figure 1. Work-flow methodology Medical images were, then, exported into A 29 years old male was submitted to a CT standard .DICOM format (image resolution scan. Medical images of both feet were imported 512x512 pixels) and processed by using into ScanIP® module of Simplewere® software, ScanIP® tools. Generally speaking, DICOM where bones and soft tissues were extracted by images are gray-based images. Different gray- adopting density-based segmentation procedures. levels correspond to different density materials At this stage boundary surfaces, into the tessel- (Figure 2.b). Looking at DICOM images the lated format, were generated. Those surfaces following regions can be distinguished: soft tis- were, then, exported into .STL format and im- sue, bone, ligaments, plantar fascia and carti- ported into SolidWorks® CAD system. Here, lages. In the present paper, only bones and soft solid domains were generated by adopting tissue were segmented. Ligaments and plantar loft/sweep and boolean operations. From this fascia were added at FEM stage as equivalent load conditions (see also Section 4); instead, lated surfaces), lofting/sweeping/filling surfaces cartilages were modeled separately at CAD were generated as also described in [14]. stage. In order to reduce the reconstruction ef- Figure 4 shows the typical work-flow (first fort, slice pixels related to left foot were re- metatarsal bone used as example) adopted to moved. Thus, in the following only the right foot generate solid domains, starting from cross- will be analyzed. section curves. Figures 3.a and 3.b show the assembled bone CAD procedure is based on the following structure and the soft tissue domain as seen into criteria: ScanIP® environment, respectively. Bone struc- • minimum number of surface patches; and, ture was composed of 19 bones: tibia, fibula, • surface patches large as much as possible. talus, calcaneus, cuboid, navicular, 3 cuneiforms All this assures more flexibility in handling 3D (bones of the metatarsus), 5 metatarsals (bones mesh and boundary conditions at FEM stage. of the metatarsus) and 5 components of the phalanges (bones of the toes). Phalange bones (a proximal and a distal phalanx for the great toe; proximal, middle and distal phalanges for the second to fifth toes) were fused together since their relative motion do not affect plantar pressures.

a. Bone structure b. Soft tissue Figure 5. 3D CAD domains generated into SolidWorks®

Figure 5 depicts the so-reconstructed CAD geometry. Cartilages that were not extracted into a. Bone structure b. Soft tissue the segmentation phase were then modeled into Figure 3. Tessellated surfaces generated the CAD system in order to joint bones and fill into ScanIP® environment the cartilaginous space (see Figure 5.a). Boolean operations were used to ensure congruence One should note that the so-generated do- among the related interfacial surfaces. For in- mains are geometrically described by the tessel- stance, the whole bone structure was subtracted lated boundary surfaces. from the soft tissue. Finally, toes at soft tissue These tessellated models were then imported level were merged together (Figure 5.b). into SolidWorks® CAD system by using the SCANto3D® add-in module to process and fixed boundary (UFS) manage imported tessellated surfaces.

y x

z a b c Figure 4. Generation of CAD model. a: tessellated geometry; b: cross-section curves; c: 3D CAD model ground Starting from cross-sections (created as inter- section curves between cut planes and the tessel- Figure 6. FE model as seen into Comsol Multiphisics® 4. FE Modeling Constant Value Ebone (MPa) 7300.0 The foot CAD model was thus imported into Ecartilage (MPa) 10.0 ν Comsol Multiphisics® 3.5a, by using the live- bone 0.30 ν link connection with SolidWorks®. cartilage 0.40 Here, domain equations, boundary conditions Table 2. Material constants for bones and cartilages [7] and solver settings were provided.

In the present paper the balanced standing 4.2 Boundary Conditions condition was simulated. The foot was supposed to touch an ideal-rigid flat ground. For this pur- During the balanced standing condition, a pose the “ground” domain was added to the FE vertical force, corresponding to one half of the model (see Figure 6). body weight (F =650/2 N), is transferred from w the body to the foot and then to the ground. 4.1 Domain Equations The plantar fascia stabilizes the longitudinal

arch of the foot and supports the longitudinal Materials were assumed linear and isotropic, forces during the weight application phase [8]. while the soft tissue was modeled as a non-linear As discussed in Section 3, plantar fascia was hyper-elastic material. As reported into [7], the geometrically simplified and modeled at FEM material law, which best represents the com- stage by defining equivalent longitudinal forces. pressibility and the highly non-linear behavior of soft tissue, is a polynomial law, as stated into F relationship (1), where C (N/m2) and d (m2/N) w ij k T are the material constants. II1, II2 and J are the first, the second and the third modified strain Fw invariants, respectively. W is the strain energy per unit of volume. T Fa ( ) ( ) ( ) +−⋅+−⋅= H1 21 110 201 ...3IIC3IICJ,II,IIW Fr ()()()2 +−⋅−⋅+−⋅+ ...3II3IIC3IIC... 120 111 2 (1) ()2 1 ()2 1 ()−⋅+−⋅+−⋅+ 4 IC... 202 3I 1J 1J Ff H3 Fr D1 D2 plantar fascia H 2 Material constants, reported in Table 1, were Figure 7. 2D model for the estimation of plantar fascia force derived from in vivo tests in [7]. Equation (1) can H1=58.74 mm, H2=140.49 mm, H3=32.67 mm be easily implemented into Comsol Mul- tiphisics® as a domain equation. A 2D equivalent model was adopted, as suggested also in [8], to estimate those forces. Look- Constant Value 2 ing at Figure 7, the longitudinal force T may be C10 (N/m ) 85550.0 2 evaluated once Fr (force exerted at rear-foot) or C01 (N/m ) -58400.0 2 Ff (force exerted at fore-foot) is known. From C20 (N/m ) 38920.0 2 simple considerations about the 2D static equi- C11 (N/m ) -23100.0 2 librium, one can write: C02 (N/m ) 8484.0 2 D1 (m /N) 0.4370e-5 D (m2/N) 0.6811e-6  ⋅ ( H-HF ) 2 F = 32w ⋅ () Table 1. Material constants for r =→ F H-H 32w ⋅ H3  H 2 T human soft tissue [7] H H (2)  ⋅=⋅ HFHT 2 1  3r1 ⋅= Table 2 reports material constants for bone and F0.4268T w cartilage domains (“E” is the Young’s modulus, while “ν” is the Poisson’s ratio). Equation (2) says that about 42% of the applied weight, Fw, is supported by the plantar fascia. Assuming that the force T is equally supported ing the right Ac value is not a trivial task. Several by all metatarsal bones, then, the net traction experiments (with a “trial and error” approach) force to be applied to every metatarsal bone were conducted to carry out the optimal scaling equals about 9% of the applied weight. factor, allowing to reach a valid solution problem Furthermore, as suggested into [10] and [13], into a reasonable time. the Achilles’ tendon force, Fa, may be assumed equal to about 75% of the applied weight. Finally, to the upper surfaces (UFS into Figure 6) of soft tissue, tibia and fibula fixed constraints were applied.

Figure 9. Definition of contact pairs between colored surfaces

The penalty factor, pn, may aid the solver to get started and to speed the convergence up.

Figure 8. Definition of identity pairs Generally speaking, pn depends on several fac- on the colored surfaces tors, such as mesh size and material law parameters [15]. In the present paper, pn was chosen Physical interaction among bones-soft tissue equal to pn=sc·Ee/hmesh, where hmesh is the mean and bones-cartilages was modeled by defining mesh size, Ee is an equivalent elastic modulus identity pairs among interfacial surfaces. All this and sc is a correction factor. Ee modulus was set assures that the displacement fields at interfacial equal to Ee=(C10+C20)/2, where C10 and C20 are surfaces are identical each-other. Figure 8 de- the first and the third constant of the chosen hy- picts the so-defined identity pairs. per-elastic material (see relationship (1)), respectively. The sc factor was assumed equal to 10.0. 4.2 Contact Pair Modeling One should note that if, on one hand, a high penalty factor speeds the convergence process up, on Since plantar pressures are also of interest, the other hand the global stiffness matrix may be contact pairs among plantar surfaces and ground- ill-conditioned. Thus, choosing the optimal pen- surface were defined (Figure 9). Moreover, no alty factor is a crucial task to be achieved when friction was accounted in the analysis. facing out contact problems. Foot and ground were supposed to be not in Another feature to be considered, when man- contact initially: a small gap was left among con- aging and solving contact problems, is the contact surfaces. To face out this kind of problem, straint status of each domain. Authors have ex- the initial contact pressure, the scaling factor and perimented that “lack of constraints” causes the penalty factor have to be set properly. These problems with the convergence process into factors strongly influence the convergence proc- Comsol Multiphysics®. Therefore, is it impor- ess and its speed. tant that all domains are constrained in all direc- Since master and slave surfaces are initially tions, so that there is only one possible solution not in contact, zero initial contact pressure was for each convergence iteration. For this reason, assumed. The scaling factor, sf, related to the the ground domain was supposed to rigidly move contact variable was chosen equal to sf=Fw/Ac, only along the global Z direction (see Figure 6) where Fw and Ac are the applied weight and the starting from a distance of about 1 mm from the estimated final contact area, respectively. Choos- foot. In this way, all domains are properly constrained, and unwanted rigid motions are map may become a fundamental issue to be used avoided. when investigating perceived comfort at plantar- insole interface. 5. FE Solution and Results Future improvements of the present research will focus on the experimental validation of nu- FE model was resolved by adopting a non- merical results, by using high resolution insole linear solver ("PARDISO out of core" was used sensors. as linear system solver). To aid convergence process when calculating 6. Final Remarks contact variables, a parametric solver was also adopted. As discussed above, the ground dis- The present paper focuses on the simulation placement along Z direction was parameterized of the balanced standing condition of a human to linearly change into the range [0, wmax]. The foot model. wmax value was chosen so that the reaction force Once the 3D geometry of the human foot was calculated at UFS was greater or equal to the generated from CT images, the non-linear FE applied weight, Fw. So, wmax=8.0 mm was a good model was built up. In particular, hyper-elastic guess. material law and contact pairs were assigned in When facing out contact problems, also the order to well capture the bio-mechanical behav- tolerance (Ntol) for non-linear solver has to be ior of the human foot when touching a rigid flat properly set. As suggested into [16], the follow- surface. ing condition should be satisfied: This research has highlighted some critical issues related to the analysis of a highly non- ⋅ linear FE model in which complex geometry, Ns augf N ≤ (3) hyper-elastic material and contacts are involved. tol ⋅ pu nmax The suggested parameters related to the best choice of the penalty factor as well as the set- where Naug and umax are the tolerance for the tings of the solver come from some general rules augmented lagrangian solver (by default along with a trial-and-error approach. This is true -3 Naug=10 ) and the maximum displacement, re- for several FEM-based software. All these fine- spectively. For the present application, one may tuning operations are very time-consuming and assume umax=wmax. there no guarantee of successful. So, some im- Mesh was generated by fine-controlling ele- provements and suggested best practices to re- ment size (about 3mm) at plantar surfaces. Look- solve these kinds of problems are expected in the ing at Figure 6, one should note that the ground next future. domain was meshed with hexahedral elements, while free tetrahedral elements were adopted for the foot geometry. Final mesh was made of 7. References 25812 nodes and 87149 elements. In order to reduce the computational effort, linear shape 1. Jordan C., Barlett R., Pressure Distribution functions were adopted. Thus, the number of and Perceived Comfort in Casual Footwear, Gait degrees of freedom of the FE model was 81428. & Posture, Vol. 3, Issue 4, pp. 215-220 (1995). Numerical simulations were accomplished in 2. 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a. b.

Figure 10. Domain contour plot for total displacement [mm]. a) undeformed geometry; b) deformed geometry, showed magnified 1x.

Figure 11. Boundary contour plot for pressure map [MPa] 4. Lee Yung-Hui, Hong Wei-Hsien, Effects of eration for Realistic Simulation of the Tran- Shoe Inserts and Heel Height on Foot Pressure, Cranical Current Stimulation, Proc. of COMSOL Impact Force, and Perceived Comfort during Conference 2008, Boston (2008). Walking, Applied Ergonomics, Vol. 36, pp. 355– 12. Franciosa P., Gerbino S., Handling Tessel- 362 (2005). lated Free Shape Objects with a Morphing Mesh 5. Jason TM Cheung, Ming-Zhang, Aaron Kam- Procedure in Comsol Multiphysics®, Proc. of Lun Leung, Yu-Bo Fan, Three-dimensional Fi- COMSOL Conference 2009, Milano (Italy), Oc- nite Element Analysis of the Foot during Stand- tober 14-16 (2009). ing: a Material Sensitivity Study, Journal of 13. Jason TM, Cheung, Ming-Zhang, Finite Biomechanics, Vol. 38, pp. 1045–1054 (2005). Element Modeling of the Human Foot and Foot- 6. Jason TM Cheung, Ming-Zhang, Parametric wear, Proc. of ABAQUS Conference 2006, Cam- Design of Pressure-Relieving Foot Orthosis us- bridge-USA (2006). ing Statistics-based Finite Element Method, 14. Franciosa P., Martorelli M., Ausiello P., 3D Medical Engineering & Physics, Vol. 30, pp. Visualization and Fatigue Simulation in Restored 269–277 (2008). Human Teeth by Using Micro-CT Data, Proc. of 7. Lemmon D., Shiang TY., Hashmi, A., Ul- the Int. ADM-INGEGRAF, Lugo (Spain), June brecht, JS., Cavanagh, PR., The Effect of Shoe 10-12, (2009). Insoles in Therapeutic Footwear: A Finite Ele- 15. Wriggers P., Computational Contact Me- ment Approach, Journal of Biomechanics, Vol. chanics, Wiley, New York (2002). 30, pp. 615-620 (1997). 16. Comsol Multiphysisc® 3.5a User Guide 8. Wright DG, Rennels DC, A Study of the Elas- (Structural Mechanics Module – User’s Guide). tic Properties of Plantar Fascia, Journal of Bone Joint Surg Am. Vol. 46, pp. 482-492 (1964) 8. Acknowledgements 9. Antunes PJ., Dias GR., Coelho AT., Reselo F., Pereira T., Non-Linear Finite Element Modelling Authors wish to express a warm thanks to dr V. of Anatomically Detailed 3D Foot Model, tech- Rufrano, from Univ. of Naples Federico II (IT), nical report (2008). who accepted to undergo CT scan. 10. Jason TM Cheung, Ming-Zhang, A 3D Finite Many thanks also to dr. D. Speranza, from Univ. Element Model of the Human Foot and Ankle of Cassino (IT), for allowing us to use the Sim- for Insole Design, Archives of Physical Medicine plewere® package, and dr. F. Pepe and dr. G. and Rehabilitation, Vol. 86, pp. 353-358, (2005). Severino for their technical support. 11. Said R., Cotton R., Young P., Datta A., El- wassif M., Bikson M., Image Based-Mesh Gen-