sensitivity of tissue differentiation and bone healing predictions to tissue properties

8/3/2019 Sensitivity of Tissue Differentiation and Bone Healing Predictions to Tissue Properties

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Sensitivity of tissue differentiation and bone healing predictions to

tissue properties

Hanna Isaksson a,b,c,, Corrinus C van Donkelaar b, Keita Ito a,b

aAO Research Institute, AO Foundation, Clavadelerstrasse 8, 7270 Davos, Switzerlandb Department of Biomedical Engineering, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlandsc Department of Physics, University of Kuopio, PO Box 1627, 70211 Kuopio, Finland

a r t i c l e i n f o

Article history:

Accepted 2 January 2009

Keywords:

Fracture healing

Mechanobiology

Material properties

Fractional factorial design

Design of experiments

Orthogonal array

a b s t r a c t

Computational models are employed as tools to investigate possible mechano-regulation pathways for

tissue differentiation and bone healing. However, current models do not account for the uncertainty in

input parameters, and often include assumptions about parameter values that are not yet established.

The aim was to clarify the importance of the assumed tissue material properties in a computational

model of tissue differentiation during bone healing. An established mechano-biological model was

employed together with a statistical approach. The model included an adaptive 2D finite element model

of a fractured long bone. Four outcome criteria were quantified: (1) ability to predict sequential healing

events, (2) amount of bone formation at specific time points, (3) total time until healing, and (4)

mechanical stability at specific time points. Statistical analysis based on fractional factorial designs first

involved a screening experiment to identify the most significant tissue material properties. These seven

properties were studied further with response surface methodology in a three-level BoxBehnken

design. Generally, the sequential events were not significantly influenced by any properties, whereas

rate-dependent outcome criteria and mechanical stability were significantly influenced by Youngs

modulus and permeability. Poissons ratio and porosity had minor effects. The amount of bone

formation at early, mid and late phases of healing, the time until complete healing and the mechanicalstability were all mostly dependent on three material properties; permeability of granulation tissue,

Youngs modulus of cartilage and permeability of immature bone. The consistency between effects of

the most influential parameters was high. To increase accuracy and predictive capacity of computational

models of bone healing, the most influential tissue mechanical properties should be accurately

quantified.

& 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Fracture healing mainly aims to restore bones load-bearing

function. It involves sequential differentiation of cells and tissues,

which is influenced by the local mechanical environment

(Einhorn, 1998; Gerstenfeld et al., 2003). Computational modelsof tissue differentiation during bone healing are frequently used

to study possible mechano-regulation pathways. Increasing

biological knowledge and computational power have pushed

recent developments towards focusing on biological aspects

of tissue differentiation, such as how to better describe cell

processes (Gomez-Benito et al., 2005; Isaksson et al., 2008a), cell

dispersal (Perez and Prendergast, 2007), inclusion of growth

factors and angiogenesis (Bailon-Plaza and van der Meulen, 2001;

Geris et al., 2008a). In contrast to the wealth of literature on

mechanical behavior of fracture callus (Claes et al., 1999;

Kenwright and Goodship, 1989; Richardson et al., 1994), insuffi-

cient data is available on the tissue material properties of the

sequentially developing callus tissues. Therefore during computa-

tional modeling, callus tissue material properties are oftenestimated based on material properties of similar tissue types,

but obtained from mature tissue or as educated guesses when no

literature data is available. Hence, the accuracy of the assumed

material properties may become the limiting factor in the

precision of the simulations.

Most studies of tissue differentiation today assume identical

tissue mechanical properties (Table 1) (Andreykiv et al., 2008;

Epari et al., 2006b; Geris et al., 2004, 2008b; Isaksson et al.,

2006b, 2008a; Kelly and Prendergast, 2005; Lacroix and

Prendergast, 2002; Perez and Prendergast, 2007). Unfortunately,

many of these properties are not well established. Lacroix

introduced this set of material properties and showed in a

parametric study by varying-one-parameter-at-the-time that

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journal homepage: www.elsevier.com/locate/jbiomechwww.JBiomech.com

Journal of Biomechanics

0021-9290/$ - see front matter & 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.jbiomech.2009.01.001

Corresponding author at: Department of Physics, University of Kuopio, PO Box

1627, 70211 Kuopio, Finland. Tel.: +358 17162341; fax: +3581716 3032.

E-mail address: [email protected] (H. Isaksson).

Journal of Biomechanics 42 (2009) 555564
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sequential events during bone healing were not altered by

changes in material properties, as long as the tissues were

sequentially stiffer (Lacroix, 2001). However, current models aim

to evaluate rates of healing, as well as effects of biological and

mechanical interventions and potential non-union treatment

strategies. For such purposes, qualitative descriptions of sequen-

tial spatial events during normal healing are no longer sufficient.

The importance of the material properties on the quantitativeresponse needs further clarification.

A design of experiments (DOE) approach based on fractional

factorial designs was used to computationally evaluate the

influence of each assumed tissue property involved during bone

regeneration in a poroelastic FE model of tissue differentiation.

Different from varying-one-parameter-at-a time, the DOE ap-

proach does not need a baseline model, and can reach a more

reliable conclusion about factor effects with fewer simulations

(Funkenbusch, 2005; Phadke, 1989). The outcome was assessed as

sequential spatial and temporal tissue differentiation events, bone

formation rate, time until complete healing and mechanical

stability. The investigated properties were Youngs modulus,

Poissons ratio, permeability and porosity in each of the tissue

types; bone marrow, granulation tissue, fibrous tissue, cartilage,immature bone and mature bone. The objective was to determine

which material parameters are of the greatest influence to each of

the major processes during tissue differentiation and to the

bone healing capacity. We hypothesize that material properties

would influence both the spatial and the temporal progression

of sequential tissue transformation during bone healing, since

material properties in combination with loading govern the

mechano-regulation algorithm.

2. Methods

2.1. Adaptive tissue differentiation model

The computational mechano-regulatory model was developed to describe

the temporal and spatial distributions of fibrous tissue, cartilage and bone,

regulated through cellular activity (Isaksson et al., 2008a). Dependent on

mechanical stimulation, mesenchymal stem cells, fibroblasts, chondrocytes and

osteoblasts responded by proliferation, differentiation, migration and/or apoptosis.

Additionally, the cells could produce or degrade extracellular matrix for their

respective tissue type (Isaksson et al., 2008a).

An axisymmetric FE model of an ovine tibia was adopted from a previous

fracture healing study (Isaksson et al., 2006b). The geometry represented a 3 mm

transverse fracture gap and an external callus (Fig.1a). A 1 Hz cyclic load of 300 N

was applied proximally on the cortical bone. The magnitudes of deviatoric shear

strain and fluid velocity were calculated at the peak load (v 6.5 ABAQUS, Simulia,

Dassault Systems) and used to predict cell and tissue differentiation behavior(Prendergast et al., 1997). Parameter values for all cell processes remained constant

(Table 2).

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Table 1

Tissue material properties.

Commonly used

properties

Additional literature

review

Factor levels

High Low

Cortical bone Youngs modulus (MPa) 15750a Not included

Cortical bone Permeability (m4/N s) 1.0E17d Not included

Cortical bone Poissons ratio 0.325b Not included

Cortical bone Porosity 0.04c Not included

Granulation tissue Youngs modulus (MPa) 1 0.99l 1.5 0.5

Gran ulation tis su e P ermeability (m4/N s) 1.0E14 1.5E14 5.0E15

Granulation tissue Poissons ratio 0.167 0.2004 0.1336

Granulation tissue Porosity 0.8 0.96 0.64

Fibrous tissue Youngs modulus (MPa) 2e 1.9e; 7.8m 3 1

Fibrous tissue Permeability (m4/N s) 1.0E14e 1.5E14 5.0E15

Fibrous tissue Poissons ratio 0.167 0.19m 0.2004 0.1336

Fibrous tissue Porosity 0.8 0.70m 0.96 0.64

Cartilage Youngs modulus (MPa) 10g 3.10l; 14n; 5.3o ; 7p;

5.8q; 4.511.8r ; 10s15 5

Cartilage Permeability (m4/N s) 5.0E15f 4.7E15f; 2.0E15t;

1.9E157.0E15u;

2.3E15s

7.5E15 2.5E15

Cartilage Poissons ratio 0.167h 0.1740.185h; 0.19u; 0.17v 0.2004 0.1336Cartilage Porosity 0.8k 0.79k; 0.73t; 0.76v 0.96 0.64

Immature bone Youngs modulus (MPa) 1000 201l; 2250x; 2139y; 540z 1500 500

Immature bone Permeability (m4/N s) 1.0E13 10E13ab; 4.7E13y;

0.8E1310E13aa,ae,af1.5E13 5.0E14

Immature bone Poissons ratio 0.325 0.32x; 0.23z; 0.24aa 0.39 0.26

Immature bone Porosity 0.8 0.79x,z; 0.77ac;

0.750.80ag,ah,ai0.96 0.64

Mature bone Youngs modulus (MPa) 6000i 8300z; 13000aa 9000 3000

Mature bone Permeability (m4/N s) 3.7E13j 10E13ab; 4.7E13y;

0.8E1310E13aa,ae,af5.55E13 1.85E13

Mature bone Poissons ratio 0.325 0.32x; 0.23z; 0.24aa 0.39 0.26

Mature bone Porosity 0.8 0.79x,z; 0.77ac;

0.750.80ag,ah,ai0.96 0.64

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2.2. Literature review of tissue material properties

An extensive literature review was conducted to determine how well each

tissue property is defined. We focused on finding soft tissue properties and on

determining the variability in reported literature properties for well-characterized

tissues. All tissues were assumed linear poroelastic and required a Youngs

modulus, Poissons ratio, permeability and porosity. We assumed that granulation

tissue, fibrous tissue, cartilage, immature and mature bone are involved during

sequential tissue differentiation. Also bone marrow in the intramedullary canal

and cortical bone were included. The initial tissue material properties were taken

identical to those commonly used during computational analyses of tissuedifferentiation (Table 1) (Isaksson et al., 2006b; Lacroix and Prendergast, 2002).

An extensive literature review was conducted to find additional experimental

references for characterization of the tissues involved as well as the variability of

the reported tissue properties (Table 1).

Cortical bone and bone marrow do not undergo tissue differentiation during

bone healing. The material properties of cortical bone are well established and

much stiffer than the other tissues. Therefore, it was assumed that variation would

not have a significant effect and it was excluded from the parametric study. For

bone marrow, mechanical properties are less well known. They were assumed to

be potentially important during the early phases of healing, and were therefore

included. Despite variations in constitution between yellow and red bone marrow

(Blebea et al., 2007; Hartsock et al.,1965) and with site, age, and species ( Meunier

et al., 1971; Schnitzler and Mesquita, 1998), the only material properties for

bone marrow found are those by Hosokawa and Otani (1997, 1998), who used

ultrasound to quantify the modulus to be 2 MPa.

Immature bone is less mineralized than mature bone and was therefore

assigned a lower Youngs modulus. Recently, a nanoindentation study on fracture

callus tissue reported variations in bone modulus between 271010MPa,depending on degree of mineralization (Leong and Morgan, 2008) the low values

being in the range of those measured for early embryonic mineralized bone (Tanck

et al., 2004). Similarly, the reported values for permeability of human cancellous

bones range over two orders of magnitude (10141012 m4/Ns) and depend

strongly on porosity and anatomical site (Arramon and Nauman, 2001; Grimm and

Williams, 1997; Lim and Hong, 2000; Nauman et al., 1999; Pakula et al., 2008).

Porosity of human and bovine cancellous bone ranges from 7095% depending on

the anatomical site and bone status (Table 1) (Chaffai et al., 2000; Fellah et al.,

2004; Hosokawa and Otani, 1997, 1998; Kohles and Roberts, 2002; Lundeen et al.,

2000; Pakula et al., 2008; Salome et al., 1999; Wear et al., 2005).

Literature values for Youngs modulus of cartilage vary greatly, partly because

different types of moduli are reported. We collected studies that measured

instantaneous modulus, since our mechanical model is evaluated during a load

cycle of 1 s. Generally this parameters is reported to be 35 MPa under

compression, with variations up to 10MPa (Elliott et al., 1999; Korhonen et al.,

2002; Laasanen et al., 2003; Setton et al., 1993, 1997; Wei et al., 1998). One

study assessed the cartilage modulus within a rat fracture callus to be 3.10MPa

(Leong and Morgan, 2008). Permeability, Poissons ratio and porosity are well

characterized in young, normal, and aged cartilage, and reported with fairly high

consistency (Table 1).

Fibrous tissue in ligaments and tendons are well characterized under tension

(Anaguchi et al., 2005). However, this tissue in its native environment is

vastly different from the quickly formed fibrous tissue during repair. Hory and

Lewis determined fibrous tissue modulus during repair under compression at a

bonecement interface after total joint replacement in a canine model to be

1.9MPa (Hori and Lewis, 1982). The formed tissue was described as consisting of

heavy collagen fibers with fibrocytes interspersed throughout the tissue matrix

(Hori and Lewis,1982). Hence, it is a fair assumption that it is similar to the tissue

that develops temporarily during bone healing. Granulation tissue, formed shortly

after the trauma, is assumed the softest and least organized tissue. It is also theleast characterized tissue. Recently, its modulus was quantified for the first time in

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Table 1 (continued )

Commonly used

properties

Additional literature

review

Factor levels

High Low

Marrow Youngs modulus (MPa) 2 2x,ad 3 1

Marrow Permeability (m4/N s) 1.0E14 1.5E14 5.0E15

Marrow Poissons ratio 0.167 0.2004 0.1336

Marrow Porosity 0.8 0.96 0.64

Youngs modulus and permeability were chosen 750%, and Poissons ratio and porosity were chosen 720% of those commonly used.a Smit et al. (2002).b Cowin (1999).c Schaffler and Burr (1988).d Johnson et al. (1982).e Hori and Lewis (1982).f Armstrong and Mow (1982).g Lacroix and Prendergast (2002).h Jurvelin et al. (1997).i Claes and Heigele (1999).

j Ochoa and Hillberry (1992).k Mow et al. (1980).l Leong and Morgan (2008).m Moussa et al. (2008).n Wei et al. (1998).o

Korhonen et al. (2002).p Laasanen et al. (2003).q Akizuki et al. (1986).r Shepherd and Seedhom (1999).s Setton et al. (1997).t Wayne et al. (2003).u Julkunen et al. (2007)v Julkunen et al. (2008).x Hosokawa and Otani (1997).y Kohles and Roberts (2002).z Wear et al. (2005).aa Pakula et al. (2008).ab Arramon and Nauman (2001).ac Fellah et al. (2004).ad Hosokawa and Otani (1998).ae Grimm and Williams (1997).af Nauman et al. (1999).ag

Chaffai et al. (2000).ah Lundeen et al. (2000).ai Salome et al. (1999).

H. Isaksson et al. / Journal of Biomechanics 42 (2009) 555564 557


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a rat fracture callus, where it was determined to be 1 MPa (Leong and Morgan,

2008).

2.3. Design of experiment approach to study bone healing

A two-step parametric analysis was conducted, similar as the study by

Isaksson et al. (2008b). All the material properties were investigated and their

influences were determined using analysis of variance (ANOVA). The investigated

material properties were Youngs modulus, Poissons ratio, permeability and

porosity for granulation tissue, fibrous tissue, cartilage, immature bone and

mature bone, respectively. The chosen parameter space for Youngs modulus and

permeability were 50% and 150%, and for Poissons ratio and porosity it was 80%

and 120% of the commonly assumed properties for each tissue type. These

parameter spaces covered most reported properties in literature (Table 1). First, a

two-level screening experiment was used to identify the most important factors

(material properties). It investigated all material properties at two levels, high and

low (Table 1), using a L64 resolution IV array (Funkenbusch, 2005; Phadke, 1989),with 24 control factors (material properties) and a total of 64 treatment conditions

(simulations with different factor level combinations). The screening experiment

assumed approximately linear factor influence (Isaksson et al., 2008b; Phadke,

1989). Thereafter, a more detailed examination was carried out using the response

surface methodology on the identified most important factors to further evaluate

curvature and interactions. A BoxBehnken design was used with 7 factors, each

with 3 equally spaced levels, by adding a mid level to the high and low levels from

the screening experiment. This design resulted in 62 treatment conditions, and

allowed us to independently estimate all factors, quadratic factors and two factor

interactions. The arrays were generated and analyzed using JMP software (7.0.1.,

SAS Institute, Inc., NC).

To assess the results obtained from the parametric study, four criteria that

characterize the performance of the system for each treatment condition were

determined. The first criterion assessed the ability to predict sequential spatial

events observed during normal fracture healing, independent of time. Each event

received a score of 0 for non-physiological and 1 for normal event. The events

were: (1) fibrous tissue formation in the gap, (2) initial periosteal-bone formation,

(3) growing periosteal callus including endochondral ossification, (4) fibrous/

cartilage formation in the gap, (5) external bony bridging, (6) bone creepingsubstitution, and (7) complete callus filled with bone. The second criterion

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Fig.1. (a) Geometric FE model. Poroelastic axisymetric FE model (left) used for all analyses. The initial conditions include concentrations of mesenchymal stem cells at the

periosteum, at the marrow interface, at the outer boundary, and randomly in the callus tissue. All other cell types and tissue types have zero concentrations initially and the

tissue material parameters of 100% granulation tissue. (b) Sketch of the adaptive tissue differentiation model including the cell processes involved ( Isaksson et al., 2008a).

Table 2

Normalized cell parameter data that was used for all treatment conditions.

Cell Initial cell density TransportD (mm2day1) ProliferationfPR (day1) DifferentiationfD (day1) Apo ptos isfAP (day1)

Periost Marrow Outer/ external Callus

MSC 0.5 0.30 0.05 0.005 0.65 0.60 0.30 0.05

FB 0.0 0.0 0.0 0.0 0.50 0.55 0.20 0.05

CC 0.0 0.0 0.0 0.0 0.0 0.20 0.10 0.10

OB 0.0 0.0 0.0 0.0 0.20 0.30 0.15 0.15

Matrix Initial conc. ProductionfPM (day1) DegradationfDM (day1)

FT 0.0 0.20 0.05

C 0.0 0.05 0.05

B 0.0 0.10 0.05

Parameter values were calculated based on the literature review in Isaksson et al. (2008a). The rates of all processes for mesencymal stem cells (MSC), fibroblasts (FB),

chondrocytes (CC), osteoblasts (OB), fibrous tissue (FT), cartilage (C), and bone (B) were constant throughout the parametric study.



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measured the progression of bone healing, based on the amount of bone formation

in regions of interest (Fig. 2), during early (day 10), mid (day 25), and late (day 50)

phases of healing. The third criterion measured the total time required until

complete fracture healing as the number of days until the whole callus was

predicted to be filled with bone, i.e. when each element contained over 75% bone

matrix. These three criteria originated from our previous study ( Isaksson et al.,

2008b). The fourth criterion was added based on mechanical stability assessed by

interfragmentary movement and axial stiffness at early and mid phases of healing.

ANOVA was used to investigate the significance and contribution of each factor.

The percentage of total sum of square (%TSS) was calculated as the ratio of the sum

of square of deviation about the mean for each factor divided by the total sum

of square of deviation about the mean (Funkenbusch, 2005). %TSS for each of the

outcome criteria were used to determine the contribution of each factor to the

variance (Dar et al., 2002).

3. Results

3.1. Screening experiment

Former predictions of bone healing were used as the baseline

for evaluation of normal healing (Isaksson et al., 2008a). The

expected sequential events during normal bone healing were not

affected by the material properties. All simulations scored high,

and the contribution to the variance was not informative. In

contrast, the amount of bone formation, the time until complete

healing and the mechanical stability were significantly affected by

the tissue properties. In general, Youngs modulus and perme-

ability had high influence, whereas Poissons ratios and porosity

had little influence (Table 3). All outcome criteria were mostly

influenced by three parameters; the permeability of granulationtissue, the Youngs modulus of cartilage and the permeability of

immature bone. Outcome criteria evaluated during early phases

of healing were most dependent on modulus and permeability

of granulation tissue and modulus of cartilage, and outcome

criteria assessed during later phases of healing were more highly

dependent on the modulus of cartilage and permeability of

immature bone (Table 3). From the results of the screening

experiment, the most contributing factors were collected for the

BoxBehnken design (Table 3). These were Youngs modulus ofbone marrow, granulation tissue, cartilage, immature and mature

bone as well as permeability of granulation tissue and immature

bone.

3.2. BoxBehnken experiment

All expected sequential events scored high, and the contribu-

tion to the variance was not informative. For amount of bone

formation, the time until complete healing and mechanical

stability, the properties that were of highest importance con-

curred with those identified in the screening experiment (Table 4).

The amount of bone formation during the early stages of healing

was most influenced by the permeability of granulation tissue

(20%), whereas at mid and late stages of healing it was mostsensitive to the Youngs modulus of cartilage (mid 39%, late 20%)

and permeability of granulation tissue (mid 28%, late 23%). Time

to complete healing was substantially influenced by parameters

related to immature bone (permeability 38%, modulus 15%).

Mechanical stability during the early phases was most influenced

by permeability of granulation tissue (interfragmentary move-

ment 35%, stiffness 44%), followed by modulus of cartilage and

interaction between modulus and permeability of granulation

tissue. During later time points, the mechanical stability was more

influenced by modulus of cartilage (interfragmentary movement

41%), and permeability of immature bone (stiffness 33%) (Table 4).

Most material properties had an approximately linear influence

on the outcome criteria (Figs. 3 and 4). The moduli of cartilage and

immature bone were the only parameters which showed non-linear responses for bone formation during the late phases of

healing (Fig. 3b). Response surface analysis combined with the

ANOVA showed that most interactions were minor. In contrast,

few outcome criteria showed significant interactions, exemplified

by the amount of bone formation at late phases where the

interaction between modulus of cartilage and permeability of

granulation tissue was the second most important parameter

(Fig. 4). However, significant interactions were always related to

already identified parameters of high importance for that out-

come criterion (Table 4).

Finally, the results of the statistical model were confirmed by

running single simulations with the most beneficial material

properties for amount of bone formation and time until complete

healing criteria. It confirmed that those simulations resulted in

the highest amount of bone formation (15% more than average),

the shortest time until complete healing (16 days shorter than

average) as well as the lowest interfragmentary movement and

highest stiffness (80% lower and 280% higher) (Fig. 5).

4. Discussion

This study was motivated by the importance of callus tissue

material properties on the mechanical behavior of the fracture

callus, and thereby the predictions of tissue differentiation during

healing. Similar to what was suggested by Lacroix (2001), material

properties did not have a significant effect on the sequence

of predicted events during bone healing. However, they did

influence the rates of healing and the mechanical stability(Tables 3 and 4). Time to complete healing, amount of bone

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Fig. 2. Regions of interests (ROI) that were used for the amount of bone formation

outcome criteria. During early stage (day 10) of healing, the amount of bone in

periosteal reaction and callus formation were measured and averaged. During mid-

phase (day 25) of healing, the amount of bone in the endosteal (intramedullary

canal) callus and the bridging regions were assessed. To assess the amount of bone

formation during the late stage, the bridging and gap (complete healing) regions

were evaluated.



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formation at specific time points and interfragmentary movement

and stiffness were mainly affected by permeability of granulation

tissue, Youngs modulus of cartilage, and permeability of im-

mature bone.

Clinical and experimental evidence exists for the importance of

two of these factors. The character and magnitude of initialmechanical stability is important for success-rate of healing

(Lienau et al., 2005; Schell et al., 2005). Therefore, it could be

anticipated that the properties of the haematoma and initial

granulation tissue are important. When the relative parameter

space was identical, the permeability was more influential than

the Youngs modulus. We speculate that this is because the

modulus is too low to largely affect the outcome. However, boththese tissue properties are today inadequately characterized.

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Table 3

ANOVA of each of the outcome criteria for the L64 screening experiment.

ANOVA, %TSS Time to

complete

healing

Amount of bone formation Mechanical characterization Average

Factors Early

phase

Mid

phase

Late

phase

IFM early

phase

IFM mid

phase

Stiffness

early phase

Stiffness

mid phase

X1 Youngs modulus Marrow 2.7 9.4 3.8 1.6 11.1 5.9 1.8 1.7 4.7

X2 Youngs modulus GT 4.6 12.1 0.7 1.8 15.9 1.4 12.7 0.4 6.2X3 Youngs modulus FT 3.3 4.3 5.2 0.0 3.5 7.3 1.3 4.5 3.7

X4 Youngs modulus C 15.8 5.9 19.8 18.9 11.5 13.7 10.5 16.9 14.1

X5 Youngs modulus IMB 9.7 1.4 3.5 5.4 4.0 4.5 5.1 6.4 5.0

X6 Youngs modulus MB 1.9 8.4 0.5 11.4 0.1 1.2 0.9 2.5 3.4

X7 Permeability Marrow 0.4 0.0 1.3 0.9 2.1 0.0 0.3 0.7 0.7

X8 Permeability GT 16.6 14.5 28.5 7.3 20.1 17.3 25.5 17.2 18.4

X9 Permeability FT 1.5 3.5 0.0 0.0 0.3 0.7 0.2 0.5 0.8

X10 Permeability C 1.1 0.1 0.2 1.6 0.2 1.4 0.0 0.7 0.7

X11 Permeability IMB 14.1 9.8 4.2 16.4 0.1 0.1 7.8 15.5 8.5

X12 Permeability MB 0.5 0.4 3.2 0.0 1.6 0.7 0.1 2.8 1.2

X13 Poisson ratio Marrow 0.1 0.0 1.5 1.3 0.4 0.2 0.1 0.8 0.6

X14 Poisson ratio GT 0.1 0.0 0.4 0.0 0.1 1.0 0.0 0.3 0.2

X15 Poisson ratio FT 1.0 2.0 0.8 0.0 1.1 0.4 0.0 0.5 0.7

X16 Poisson ratio C 0.0 2.5 0.8 0.0 0.6 1.1 0.6 0.4 0.7

X17 Poisson ratio IMB 2.2 0.9 1.6 1.6 0.8 1.6 0.4 1.5 1.3

X18 Poisson ratio MB 1.5 1.0 0.2 2.4 0.5 0.7 0.3 0.9 0.9

X19 Porosity Marrow 0.4 0.0 2.0 1.3 1.4 0.5 0.0 1.2 0.9X20 Porosity GT 0.5 0.0 0.0 2.0 0.2 0.6 0.3 0.0 0.5

X21 Porosity FT 0.4 0.0 0.3 0.0 0.5 2.2 0.1 0.2 0.5

X22 Porosity C 1.5 0.5 0.2 1.9 0.7 1.2 0.0 0.0 0.8

X23 Porosity IMB 0.6 0.6 0.4 1.4 0.4 0.3 1.0 0.5 0.6

X24 Porosity MB 0.7 0.0 0.6 0.0 1.6 0.5 0.0 0.1 0.4

The percentages of the total sum of squares (%TSS) are listed. The most influential parameters are highlighted. The total influence and average were used to determine the

factors in the higher level design. Abbreviations: GTgranulation tissue, FTfibrous tissue, Ccartilage, IMBimmature bone, MBmature bone.

Table 4

ANOVA of each of the outcome criteria for the BoxBehnken response surface array.

ANOVA, %TSS Time to complete

healing

Amount of bone formation Mechanical characterization Average

Main factor effects Early

phase

Mid

phase

Late

phase

IFM early

phase

IFM mid

phase

Stiffness early

phase

Stiffness mid

phase

X1 Youngs

modulus

Marrow 2.2 6.6 5.7 0.1 14.3 4.4 2.7 3.1 4.9

X2 Youngs

modulus

GT 0.8 3.0 1.8 0.1 7.0 2.4 7.0 1.4 2.9

X3 Youngs

modulus

C 10.9 7.5 38.6 19.8 23.9 41.1 2.2 19.7 20.5

X4 Youngs

modulus

IMB 15.4 0.0 6.3 5.1 6.8 1.3 4.0 9.9 6.1

X5 Youngs

modulus

MB 0.7 0.0 0.0 0.1 0.0 0.0 0.0 0.6 0.2

X6 Permeability GT 13.8 20.3 27.7 22.6 34.9 10.5 44.2 12.3 23.3

X7 Permeability IMB 37.7 11.7 6.9 0.9 1.4 1.5 3.5 32.7 12.0

Significant interactions

X3 X4 Modulus C Modulus

IMB

0.5 0.3 0.3 9.7 0.7 0.4 0.0 0.0 1.5

X2 X6 Modulus

GT Permeability GT

0.2 18.7 0.4 0.0 0.2 0.7 10.0 1.0 3.9

X3 X6 Modulus C Permeability

GT

1.2 2.6 1.1 21.5 0.4 9.7 0.7 0.7 4.7

All main factor effects are given together with three significant two-factor interactions as percentages of the total sum of squares (%TSS). The most influential parameters

are highlighted. Abbreviations: GTgranulation tissue, FTfibrous tissue, Ccartilage, IMBimmature bone, MBmature bone.



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Despite differences in basic biological assumptions between

mechano-biological models, the assumed equations for convert-

ing tissue partition into mechanical properties are similar, and

since most current studies assume a similar set of material

properties the identified most important parameters are likely

to be the same. Early developments of computational models

of fracture repair focused on improving mechanical models

of biological tissues. Today poroelastic mechanical models are

standard and recent developments have focused on implementingbiological aspects of healing. These characteristics are crucial for

bone healing models. However, until the assumptions and

descriptions of tissue mechanical properties are better validated,

the predictive capacity of these models remains qualitative.

The benefits and limitations of fractional factorial analysis

have been discussed extensively in Isaksson et al. (2008b). When

matrix production occurs, several tissue properties will be

affected simultaneously. In the current study a 3 level BoxBehn-

ken design was used to additionally be able to evaluate these

interactions between parameters. The results indicate that certain

interactions are prominent such as combinations of modulus and

permeability of cartilage and granulation tissue (Table 4).

However, from all two-variable interactions, only three interac-

tions were within the 3 most influential parameters for any of theoutcome analyses. Those 3 interactions all involved the para-

meters that also had the highest main factor influences. Porosity

and Poissons ratio were given a relatively smaller parameter

space compared to Youngs modulus and permeability. This was

motivated by the higher consistency in literature for these

parameters, and by the physical limitations for Poissons ratio

and porosity (Table 1). Youngs modulus and permeability of both

tissues that are well characterize and those without literature

references, were given identical relative parameter spaces to avoid

bias related to the parameter space in the statistical model(Isaksson et al., 2008b). Due to the difficulty in finding one

parameter which can be used to characterize the progression

of fracture healing, we chose to use several outcome criteria.

Three of them were used before (Isaksson et al., 2008b). Since

this study is focusing on material properties, we also quantified

the mechanical stability based on interfragmentary movement

and stiffness. All together, these four criteria are believed to

characterize the system well.

For the first time, this study provides a systematic approach to

evaluate the sensitivity of the assumed tissue material properties

during computational modeling of bone healing and showed that

material properties, especially permeability of granulation tissue,

Youngs modulus of cartilage and permeability of immature bone

needs better characterization before the full potential of compu-tational mechano-biological models can be achieved.

ARTICLE IN PRESS

Fig. 4. Surface contour plots of the most influential parameters for amount of bone formation during early and late phases of healing and the mechanical stability during

early and mid phases of healing. The interactions between parameters were generally of minor importance, but for the amount of bone at late phases of healing, the

interactions between permeability of granulation tissue and modulus of cartilage were second most influential parameter. The contribution of the material parameters

were calculated at high (1), mid (0) and low (+1) levels.



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Conflict of interest statement

None of the authors have any conflicts of interest.

Acknowledgements

We acknowledge CSC, the Finnish IT Center for Science for

computational tools, and the European Commission for funding

(BONEQUAL-219980).

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Fig. 5. The statistical model was confirmed by running single simulations with the

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interfragmentary movement and 280% higher stiffness compared to the average

response.



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