## Abstract

Biological experimentation has many obstacles: resource limitations, unavailability of materials, manufacturing complexities and ethical compliance issues; any approach that resolves all or some of these is of some interest. The aim of this study is applying the recently discovered concept of finite similitude as a novel approach for the design of scaled biomechanical experiments supported with analysis using a commercial finite-element package and validated by means of image correlation software. The study of isotropic scaling of synthetic bones leads to the selection of three-dimensional (3D) printed materials for the trial-space materials. These materials conforming to the theory are analysed in finite-element models of a cylinder and femur geometries undergoing compression, tension, torsion and bending tests to assess the efficacy of the approach using reverse scaling of the approach. The finite-element results show similar strain patterns in the surface for the cylinder with a maximum difference of less than 10% and for the femur with a maximum difference of less than 4% across all tests. Finally, the trial-space, physical-trial experimentation using 3D printed materials for compression and bending testing provides a good agreement in a Bland–Altman statistical analysis, providing good supporting evidence for the practicality of the approach.

## 1. Introduction

There are a variety of problems that arise when considering biological experimentation ranging from financial to ethical concerns. Scaled experimentation potentially provides a solution but presently its application is rather limited, lacking credibility with previous work on the scaling of biological systems having little to no control over material properties and other aspects [1]. A fundamental concern of any scaled experiment is, in what sense is it truly representative of the full-scale system and how can this be gauged? With this in mind, this paper aims to introduce a new theory for scaling experiments and apply this to the design of biological experimentation.

The most common and presently the only plausible scaling approach is dimensional analysis [2–4]. The fundamental concept underpinning this method is that physical properties and their behaviour cannot depend on the choice of units used to measure them. This basic idea leads to the Pi Theorem, which states that it is possible to transform any relation to a dimensionless expression that does not depend on the units of measurement. Essentially, if there are *n* governing parameters and there are *k* independent dimensions, then the expression is reduced to a dimensionless form with *n* − *k* parameters. The dimensionless expression allows the use of certain parameters that are called similarity parameters that aid in the assessment of whether the scaling of a certain experiment is possible. Although a good approach, dimensional analysis has many shortcomings as expertise is generally required for its application. The main issue with dimensional analysis, however, is that it is not principally a scaling theory and only achieves scaling indirectly by insisting that the governing equations in dimensionless form remain invariant at different length scales. This imposition is hardly realistic in scaled experimentation with the matching of similarity parameters being virtually impossible, and the approach offers no solution when matching is not achievable.

A review of the methods used for scaling in biology reveals that it is necessary to appreciate what aspects can be scaled in relation to the study of biological systems [1]. There is a variety of literature on the subject although the majority of papers focus on a particular aspect such as the scaling of sample sizes in a statistical study, or the change in scale of some quantity such as force, size, metabolic rate (i.e. allometry), etc. However, these aspects are not particularly relevant to the design of trial experiments as they provide little insight into the choice of material properties and the matching of the underlying physics involved.

The underlying assumption of finite similitude is that the complete physics of a system is unlikely to scale but conservation laws must undoubtedly hold over a extensive range of length scales [5]. Thus, in continuum bio-mechanics, the laws of nature necessitate that volume, mass, momentum and energy must be conserved along with compliance of non-conservative laws for movement (for the description of displacement, see reference [6]) and entropy. These laws are describable in transport form which can be interpreted as the rate of change of a physical quantity on a control volume being equal to the transfer and flux of the quantity at the boundary. This provides the foundation for the scaling theory presented here, which involves relating the laws in both the physical and trial spaces. For this purpose, a scaling map *χ* is introduced and, interpreting this map as a ‘deformation’ of space, it can then be used to relate the balance laws in their transport form giving rise to similarity equations between the trial and physical spaces. These have to be satisfied for all the transport equations involved to guarantee finite similitude which, in turn, guarantees that the governing physics remains representative. This can be done by obtaining the scaling parameters (as proportional physics is sought) from the different equations for the balance laws of volume, continuity, momentum, movement, energy and entropy (if needed). Constitutive responses are only then considered to provide direct-scaling material properties in the trial space, which can then be used as part of a materials selection procedure to arrive at physical materials for trial experimentation.

The principal concern with scaled biomechanical experimentation (and all scaled experimentation for that matter) is the inherent uncertainty associated with it. The methodology presented here addresses this using mathematical principles that allow the quantification of the mismatch between spaces. Exact similitude is seldom achieved in practice; however, for biomechanical experimentation in particular, localized errors in the finite-element analysis of bones (when measuring strain or stress) can be used to improve the trial experimentation under consideration. The quantification of discrepancies and the element of experimental design (selection of materials) with the novel scaling approach aid in this regard.

The novel scaling theory is presented in §2, where the control volume theory [6] and the balance laws in their transport form are stated to derive the similarity equations which are fundamental in the use of finite similitude and the design of trial experiments. These equations provide the means to compare the physical and trial space as well as obtaining the scaling parameters and material properties in the trial space. The application to biomechanical experimentation using isotropic scaling is considered in §3 with the application of pertinent conservation laws of volume, continuity momentum and angular momentum. The scaling parameters and the properties for the trial-space materials are obtained after considering the constitutive response to biomechanical experiments involving validation of finite-element models by means of image correlation software and the use of synthetic composite bone. The resulting predicted trial-space materials are replicated using three-dimensional (3D) printed materials, which are selected on the basis of properties, cost and the availability of specimens for complex geometries. Printed replica materials do not of course provide a perfect match, but a feature of the new theory is reverse scaling, which facilitates the qualification of any mismatch, which is a feature that is fundamental for gauging the success of a trial-space experiment. Finally, case studies in biomechanics are analysed in §§4 and 4.2 where the first step for assessing the success of the scaling approach is numerical evaluation using finite-element models and the reverse scaling particular to the novel scaling approach, where considered in the first instance is the simple geometry of a cylinder followed by the geometry of a femur undergoing tests of compression, tension, torsion and bending. The physical-trial experimentation undertaken in this study is supported by results obtained in the numerical models and Bland–Altman statistical analysis [7] for tests of compression and bending in a cylinder and subsequently a femur geometry. The classical dimensional-analysis approach is included in appendix A to validate the novel scaling theory and also to highlight the advantages of finite similitude over other scaling approaches.

The overall aim of the study was to assess the applicability of the novel scaling approach in biomechanical experimentation, specifically consisting of the use of synthetic composite bone and validation by means of image correlation software. The results and the feasibility of the application of finite similitude to the pertinent biomechanical experiments is discussed in §5 as well as future work pertaining to finite similitude in biomechanical experimentation. This is the first approach to the design of scaled experimentation in biomechanics which provides a foundation for further study of anisotropic material properties and validation of patient-specific geometries.

## 2. Finite similitude

Finite similitude is concerned with space, which naturally leads to the control volume concept and the definition of a control volume *Ω**_{ps} as opposed to a continuum body, where the subscript ‘ps’ denotes physical space. The motion of the control volume *Ω**_{ps} is formally defined by means of a deformation map *κ*_{ps} : *Ω** → *Ω**_{ps}, where *Ω** is a reference control volume and *κ*_{ps} is the corresponding deformation map. The control-volume velocity is immediately definable as ** v***

_{ps}=

*D**

*κ*

_{ps}/

*D**

*t*, where

*D**/

*D**

*t*= ∂/∂

*t*|

_{x*}is the control-volume derivative, which is analogous to the material derivative.

The same formulation can be invoked for the trial space which will be the scaled space, where *Ω**_{ts} is the scaled control volume and *κ*_{ts} : *Ω** → *Ω**_{ts} is the associated deformation map.

The time parameter in the trial space need not run at the same rate as the physical space; consequently, time in the trial space is identified by the symbol *τ* as opposed to *t* to distinguish this feature. The velocity of a control volume in the trial space makes use of the time within the space and takes the form ** v***

_{ts}=

*D**

*κ*

_{ts}/

*D**

*τ*= ∂/∂

*τ*|

_{s*}. With the definition of the maps

*κ*

_{ps}and

*κ*

_{ts}, deformation gradient tensors immediately follow and take the form

*F*_{ps}= ∂

*κ*

_{ps}/∂

**and**

*x**

*F*_{ts}= ∂

*κ*

_{ts}/∂

**for the physical and trial spaces, respectively.**

*s**The methodology underpinning the scaling concept consists of assuming there exists a differentiable map (diffeomorphism) between the trial and physical spaces *χ* : *Ω**_{ts} → *Ω**_{ps} The map *χ* is not explicitly time-dependent as it relates points from *Ω**_{ts} to points in *Ω**_{ps}, i.e. is of the form *χ* : ** s*** ↦

***. A deformation tensor is assumed to exist for the map**

*x**χ*and takes the form

**= ∂**

*F**χ*/∂

***. The explicit independence of**

*s**χ*on time ensures that the motions of control volumes are synchronized, which transpires to be a necessary requirement for relating the physics in physical and trial spaces. In a similar way, time between the spaces is related by 2.1where the

*h*is a bijection and is positive.

A general balance or conservation law in transport form for the property *Ψ*_{ps} that governs the behaviour of *Ψ*_{ps} over the control volume *Ω*_{ps} is
2.2where is a ‘flux’ term, *b*^{ψ}_{ps} is the source term, ∂*Ω**_{ps} is the orientable boundary for *Ω**_{ps}, *n*_{ps} is an outward pointing unit normal on ∂*Ω**_{ps}, *v*^{m}_{ps} is the material velocity and *ρ*_{ps} is the density.

Equivalently for the trial space 2.3

To relate the conservation laws (2.2) and (2.3), it is necessary to use the tensor ** F** associated with the scaling map along with Nanson's geometric identities and equation (2.1) to obtain
2.4where .

The search for proportional physics leads to the application of scaling parameters *α*^{Ψ} > 0 to scale the conservation laws described by (2.4). Comparing therefore a scaled equation (2.4) with equation (2.3), the two can be made identical if and only if the following equations hold:
2.5*a*
2.5*b*
2.5*c*
2.5*d*where the tensor ** F** is diagonal in this study and is restricted to take the form

**=**

*F**β*

**associated with isotropic scaling, and where**

*I**β*is a scalar and

**is the unit tensor.**

*I*The similarity conditions in (2.5) are fundamental to the process of scaling and the application of finite similitude. It is necessary therefore to invoke these conditions for the physics pertinent to bio-mechanical experimentation.

## 3. Isotropic scaling in biomechanics

For isotropic scaling ** F** =

*β*

**, as a result, it follows**

*I***=**

*J**β*

^{3}and

*F*^{−1}

**=**

*v**β*

^{−1}

**for any vector quantity**

*v***. Thus, equations (2.5) reduce to 3.1**

*v**a*3.1

*b*3.1

*c*3.1

*d*

The inclusion of the conservation laws for continuity, momentum and angular momentum are considered because, in the case of biomechanical experimentation consisting of validating finite-element models with image correlation software, these are the most critical laws as they lead eventually to strain. These balance laws impose constraints on what is permissible, but also the laws serve as an example of how the balance laws relate to each other when considering the scaling methodology. While a more general scaling is possible, as a first approach, only isotropic scaling is considered in this study.

### 3.1. Scaling of volume

In continuum mechanics the conservation of volume is assumed evident because no useful information is obtained; this is not the case for similitude however. In this case, *Ψ*_{ps} = *ρ*^{−1}_{ps} and *Ψ*_{ts} = *ρ*^{−1}_{ts} and patently no flux or sink terms exist therefore the scaling relationships obtained from equation (3.1) are 1 = *α*^{1}*β*^{3} and **v***_{ts} = *hβ*^{−1}**v***_{ps}. As the scalar *α*^{1} is arbitrary, the relationship 1 = *α*^{1}*β*^{3} infers that the space-scalar *β* is also arbitrary although practical limitations may well exist. The relationship **v***_{ts} = *hβ*^{−1}**v***_{ps} synchronizes the movements of the control volumes as is expected for coupled control-volume type analysis in the two spaces.

### 3.2. Scaling of mass

For the balance law of continuity i.e. the conservation of mass, *Ψ*_{ps} = *Ψ*_{ts}≡1, *J*^{Ψ}_{ts} = *J*^{Ψ}_{ps} = 0 because there is no flux term at the boundary and *b*^{Ψ}_{ts} = *b*^{Ψ}_{ps} = 0 as no source (or sink) of mass exists within the volume. Using the similarity equation (3.1a) returns the relationship
3.2which is required to be satisfied for finite similitude and where *α*^{ρ} is the scalar, which can be set in any material selection procedure. The material properties are considered as a spatially piecewise constant function as this is sufficient for the representation of Sawbones composite bone consisting of a cortical exterior and a trabecular interior. In the case of density, (*ρ*_{ps})^{e} is the value for the exterior synthetic bone material and (*ρ*_{ps})^{i} the interior value. The scaling factor in (3.2) can be written as
3.3

From this, once the material for the trial space cortical material is chosen, it follows that the properties of the trabecular material is set. As *α*^{ρ} is constant, it follows that
3.4which constrains the value of the interior density (*ρ*_{ps})^{i} of the composite material.

### 3.3. Scaling of momentum

Given that the density is restricted by equation (3.2) and the balance law for mass also returns the relationship for velocity **v**^{m}_{ts} = *hβ*^{−1}**v**^{m}_{ps}, it is necessary to consider the balance law of momentum which must also provide restrictions on velocity. For this conservation law *Ψ* = *v*^{m} and from equations (3.1a) and using (3.2), it follows that
3.5and taking into account the relationship **v**^{m}_{ts} = *hβ*^{−1}**v**^{m}_{ps}, the relationship
3.6must be satisfied for finite similitude.

Note that equation (3.1) also gives the relationships 3.7and 3.8

The value for *α*^{ν} is determined from the constitutive response of (3.7), which in this study is restricted to a linear-elastic response. Therefore, substitution of Hooke's linear stress–strain relationships into equation (3.7) provides
3.9which, under the assumption that Poisson's ratio is identical for the materials in both spaces (i.e. *ν*_{ts} = *ν*_{ps}), simplifies to
3.10

This assumption reflects the fact that exact similitude seldom exists in practice; however, the new approach caters for inexact similitude due to the focus on space rather than the object. While in practice Poisson's ratio may not be an exact match, the ability to project trial-space reality into the physical space provides evidence on how critical this discrepancy is in the trial experimentation under consideration.

Substitution of equation (3.6) provides
3.11and on rearrangement gives
3.12which confirms in reference to equation (2.1) that *h* is constant and that, in general, timescales in the two spaces will be different.

It is worth re-emphasizing at this point that time scaling is present with the finite-similitude approach, which is attempting to link real physical processes and not some idealization of reality. Thus, for example, with quasi-static processes it might be anticipated that the time-scaling parameter *h* is not present. However, the quasi-static concept is an idealization and the new approach applies to experimental reality, where time is invariably involved. It is possible, however, that changes in *h* have little impact on a particular ‘quasi-static’ process but such a feature is automatically accounted for with finite similitude.

Note that Young's modulus is considered (as the case of density) as a spatially piecewise function, so an equation similar in form to equation (3.4) is obtained;
3.13which not too unexpectedly constrains the value of the interior modulus (*E*_{ps})^{i} of the composite material in order to capture the behaviour of the synthetic bone.

### 3.4. Scaling of angular momentum

The relationships for angular momentum can now be obtained; in this case, *Ψ* = ** r**∧

*v*^{m}and using equations (3.1a) and (3.2), the following expression is obtained: 3.14which on use of equation (3.6) results in

*α*

^{M}=

*α*

^{ν}

*β*

^{−1}.

Substituting this expression in the flux term equation confirms its consistency with equation (3.7).

### 3.5. Reverse scaling

The inverse deformation map can be used to derive the reverse scaling material properties that are useful for the comparison between spaces,
3.15*a*
3.15*b*
3.15*c*
3.15*d*

The consideration of the inverse scaling map can be used to assess the efficacy of the chosen materials in the physical space; this particularly facilitates the assessment of the approach and experimental design in finite-element models. It is important to emphasize that, for a particular practical analysis, direct-scaled material properties may not in fact exist for any known material. Selecting materials for the trial space is informed by the direct scaling but a perfect match in all properties is in general unlikely. The mappings (direct and inverse) relating the spaces can be interpreted as projections of models that occur in reality. These projections appear as virtual models in the trial space (for a direct map) and the physical space (for a inverse map). This feature makes the approach extremely flexible because exact similitude seldom exists but mismatch with inexact similitude can be contrasted in both the trial and physical spaces on comparing virtual models with reality.

The reverse scaling feature thus provides a mechanism for gauging the performance of any scaled model in the unscaled physical space, thus providing strong justification for using scaled experimentation. This is particularly useful when assessing the approach in numerical models instead of moving directly to the physical-trial experimentation.

Finite similitude is a scaling methodology valid for all continuum mechanics and provides the general rules for scaling. The rules remain unchanged when approximating a general problem by some idealization (i.e quasi-static or linearized processes). The scaling relationships automatically account for the complete physics of a problem occurring in reality such as the scaling of time or the conservation of energy; the approach deals with experimental reality and subsequently by selection of the scalars focuses on the particular physics that is dominant.

### 3.6. Three-dimensional printing materials

Synthetic bone plays an important role in the validation of finite-element models because they are considered to be a viable alternative to the cadaveric bone for biomechanical evaluation [8,9]. In this case, fourth-generation composite bones are considered for the validation of the finite-element models present here. The corresponding material properties can be found in the 2016 Sawbones Biomechanical Test Materials Catalog [10]. The choice of material for the trial space is made on consideration of their cost; as such, the materials chosen are plastics that have a lower cost and are easily obtained for 3D printing purposes.

The use of 3D printing facilitates control of material properties and gives prompt access to specimens for the different case studies for use in the scaling approach, as well as future work with validation of finite-element models.

The materials proposed for the exterior material are the thermoplastic polymer acrylonitrile butadiene styrene (ABS) and the polylactic acid (PLA). The most important material property is its elastic modulus as such, the average values of the elastic modulus for these 3D printed materials are used [11]. Sources providing average values for the elastic modulus in terms of the specifications given with 3D printing prove useful in this regard.

The direct scaling gives values for the interior material and since the most important component for tensile tests is Young's modulus, thermoplastic polyurethane (TPU) flexible filaments with Shore hardnesses of 85A, 90A and 93A were chosen for the interior material. Using the relationship between Shore hardness and Young's modulus given in detail by Gent [12], it is possible to obtain an estimate of Young's modulus in MPa using
3.16where *s* is the Shore hardness.

Using reverse scaling yields the reverse-scaled properties of the materials. In this case, three different combinations are assessed: ABS with a 0.3 mm layer thickness with the 85A hardness TPU, ABS with a 0.4 mm layer thickness with the 90A TPU and PLA with a 0.4 mm layer thickness with the 93A TPU. All of the scaling parameters and material properties for the physical and trial space as well as the reverse scaling materials are summarized in table 1.

The material properties and scaling parameters for each of the proposed trial experiments can be compared using the reverse scaling in the physical space to assess the success of the approach in biomechanical experimentation. With this in mind, the comparison in a finite-element model for different geometries is undertaken.

The 3D printed materials were analysed in two geometries, a composite cylinder and a composite femur. The tests under consideration are compression, tension, torsion and bending, which are analysed using finite-element models and the actual trial experimentation when available. The simulations are solved using Abaqus [13] software, and for the experimentation Aramis [14] digital image correlation software is used. The material properties in the models are considered linear homogeneous isotropic, even though the synthetic bone composite materials are reported as transversely isotropic, due to the consideration of an isotropic material constitutive response as a first approach to the scaling.

## 4. Cylinder trial experiment

The composite cylinder used in these tests is 141 mm long, with a radius of 20 mm and a 3 mm thick cortical layer. The different tests are considered quasi-statically loaded for the purpose of finite-element analysis. The strain measured is compared using the reverse scaling of the ABS/TPU 85A and ABS/TPU 90A in the FE (finite-element) model (the PLA/TPU 93A is considered identical to the composite cylinder since the reverse scaling yields almost identical properties). In the case of compression and tension tests the strain component in the *z*-axis is used and for bending and torsion tests the maximum principal strain is used.

The loads are presented in table 2. The compression and tension tests are congruent to the average peak loads for the ‘one-leg standing’ posture [15]; the bending test loads are congruent to the values in the *x*-axis and *y*-axis for the same posture [15]. The torsion values are consistent with the mechanical peak torque a femoral bone can withstand [16].

For the compression test, figure 1*a* shows that the maximum percentage difference for the reverse scaling ABS/TPU 85A is 0.43% and 0.47% for the ABS/TPU 90A. The maximum percentage difference between the composite material in the FE model and the analytical solution is 0.62% for all the materials.

Even though the main interest is the strain in the surface as it can be measured experimentally, the analysis of stress for the trabecular material is revealing and is presented in figure 1*b*, which shows the percentage difference in stress for the interior material using the reverse scaling ABS/TPU 85A is 17.5% and 16.2% for the ABS/TPU 90A. However, the cortical material preserves the same percentage difference in stress as that in the strain analysis 0.43% for the ABS/TPU 85A and 0.47% for the ABS/TPU 90A.

The tension test is shown in figure 2*a*, the similarity between the strain values resembles those for the compression test. The maximum percentage difference between the composite and the reverse scaling of the 85A is 0.43% and for 90A is 0.48%.

For the torsion test, the comparison between the reverse scaling of the different 3D printing materials and the composite materials is shown in figure 2*b*. The graph shows that the maximum percentage difference of the strain values in the surface is 0.17% for the reverse scaling of 85A and 0.19% for the 90A. The test reveals a greater degree of similarity than the previous tests as a consequence of the greater importance of the outer fibres of the cylinder when loaded in torsion.

For the bending test, the comparison is performed node-to-node in the regions shown in figure 3, where the higher values of maximum principal strain are considered.

For the comparison, the difference between each node in the regions is considered and, as a result, the average percentage difference for each value of the force is compared using the reverse scaling materials (figure 4).

The comparison between the reverse scaling (85A and 90A) and the composite material is shown in figure 5*a*,*b*. The graphs show an increasing average percentage difference with the minimum difference being 4% and maximum difference being 10.3% in region I and a minimum difference 0.11% and maximum 7% for region II.

The values in region I show the reverse scaling of the ABS/TPU 90A resembles more closely the synthetic material since the values of the average percentage difference for the strain in the surface are smaller than that of the ABS/TPU 85A. Similarly, for the values in region II except the values differ significantly more than in region I.

The maximum percentage differences recorded in the tests, qualify to a degree, the extent of inexact similitude as a consequence of the materials selected for the trial space tests. Maximum differences differ in the two material combinations selected, highlighting the importance of material choice and depending on the purpose of the scaling experiment whether the errors involved are considered acceptable.

### 4.1. Trial experimentation

The trial experimentation for the 3D printed cylinder was conducted using the scaled specimens of the synthetic composite bone cylinder with a scaling parameter ; the specimens were printed with the two combinations of scaled materials ABS/TPU 85A and PLA/TPU 93A. The cylinder specimens have a height of 106 mm and a radius of 15 mm; the force applied in the trial space was calculated using the scaling parameters, equation (3.7) and the expression for the average normal component of stress in the direction of the *z*-axis, *σ* = *F*_{n}/*A*, where *F*_{n} is the normal component of the force. Thus, the expression for the normal component of force in the trial space follows from equation (3.7) and is
4.1

This expression can also be used to calculate the values for torque for the trial space because ** M** =

**∧**

*r***. In the case of the torsion test considered in the FE model and in accordance with the conservation of angular momentum analysis in §3.4,**

*F*

*M*_{ts}=

*r*_{ts}∧

*F*_{ts}= (

*β*

^{−1}

*r*_{ps})∧(

*α*

^{ν}

*h*

*F*_{ps}) =

*α*

^{ν}

*hβ*

^{−1}

*M*_{ps}. A scalar version of this relationship is applied to relate torques in the two spaces as shown in figure 2

*b*.

#### 4.1.1. Compression

Using the aforementioned expression, the axial force that was applied in the compression test was calculated for the trial space using the respective parameters for the ABS/TPU 85A and the PLA/TPU 93A. As the force values in the physical space range from 1200 N to 1800 N with 200 N increments consistent with the average peak loads in a one-leg standing posture (and taking into account the equipment's accuracy) the trial space force values are calculated which results in the respective values ranging from 71.6 N to 105.4 N with 11.6 N increments for the ABS/TPU 85A and 132.8 N to 199.2 N with 22.1 N increments for the PLA/TPU 93A (figure 6).

The experiment was considered quasi-static the image correlation software records images at each loading stage (one stage-test for each value of force) and consequently calculates strain values in the surface for each value of force. The results were analysed in the statistical data of the software which displays the average strain in the region selected after the data from the computation mask has been processed. To compare the results obtained in each space a Bland–Altman statistical analysis [7] is applied, where measurements of the synthetic composite bone with the image correlation software regarded as the gold standard and the measurement of trial space materials (i.e. 3D printed materials) as the new proposed method.

#### 4.1.2. Bending

As above, on use of expression (4.1) the force applied in the bending test was calculated for the trial space using the respective parameters for ABS/TPU 85A and PLA/TPU 93A. As the force values in the physical space range from 700 N to 900 N with 100 N increments consistent with the average peak loads in a one-leg standing posture on the *x*-axis (and taking into account the equipment's accuracy), the trial space force values are calculated, which results in the respective values ranging from 41 N to 52.6 N with 5.8 N increments for ABS/TPU 85A and 77.5 N to 99.6 N with 11.05 N increments for PLA/TPU 93A (figure 7).

#### 4.1.3. Results

Table 3 shows the repeatability analysis for each method of validation as well as the Bland–Altman analysis for each combination of materials in the trial space. Eight measurements of strain were taken for each of the values of force in each space. For the analysis of repeatability the difference between every measurement for each value of the force is considered, resulting in 108 values. The within-subject repeatability is assumed to be independent because a plot between the standard deviation and the mean shows no tendency to change with the magnitude of measurements. The same is done for the bending test, resulting in 81 values; the within-subject repeatability is assumed to be independent as in the case of the compression test. These repeatability coefficients are satisfactory for the purpose of validation with the image correlation software because the accuracy of the Aramis software is 0.01 in percentage strain (100 micro-strains).

The agreement between the synthetic-composite bone and the trial space materials was analysed. The difference between each measurement from the synthetic-composite cylinder and each measurement of the 3D-printed cylinder is considered (the same is done for the average); as such, the resulting data have 256 values for the compression test and 192 for the bending test. As before the assumption of independence with the magnitude of measurements is made. Table 3 shows the mean difference and standard deviation as well as the limits of agreement which are below the equipment's accuracy. The Bland–Altman plots are included in appendix B in table 7.

### 4.2. Femur trial experiment

There are many advantages to studying the behaviour of a femur in a finite-element analysis such as the study of the role of longitudinal bone curvature in the design of limb bones [17], the optimization of the position of the acetabulum in a periacetabular osteotomy [18] or the study of stress in the femoral head–neck junction after osteochondroplasty [19]. Hence, complex femur geometry was analysed for the different trial space materials to assess the success of the scaling approach in a complex geometry when considering strain in the surface.

In this case, a CT scan of the composite femur was performed and imported to ScanIP segmentation software to create the surface. The solid model was exported to Abaqus FE analysis software to be meshed and analysed.

As the case of the cylinder geometry PLA/TPU 93A is considered an identical match to the synthetic composite femur in the FE models since the reverse scaling materials are almost identical (percentage difference less than 0.00001). The loads and boundary conditions for each test is shown in table 4.

The regions used for node-to-node comparison throughout the tests are shown in table 5; these regions were chosen for their high values of strain since these are the main interest when considering the accuracy of the image correlation software; however, these do not correspond to the regions used in the trial experimentation because the average strain is measured.

#### 4.2.1. Compression

In this case, a compression test was analysed in the FE model; the average percentage difference in region I is 3.4% for 85A and 4.3% for 90A, the average percentage difference for region II is 3.8% for 85A and 4.7% for 90A and the average percentage difference for region III is 0.1% for 85A and less than 0.1% for 90A.

#### 4.2.2. Tension

Similarly, a tension test is analysed in the FE model with a simulated acetabular cup to emulate the conditions present of a femur undergoing tension. The average percentage difference in region I is 1.2% for 85A and 1.3% for 90A, the average percentage difference for region II is 2.6% for 85A and 2.7% for 90A and the average percentage difference for region III is less than 0.1% for both 85A and 90A.

#### 4.2.3. Torsion

The analysis of torsion is done simulating an acetabular cup to emulate the conditions of a femur undergoing torsion. The average percentage difference in region I is less than 0.1% for both 85A and 90A, the average percentage difference for region II is 3.2% for 85A and 3.3% for 90A and the average percentage difference for region III is 1.2% for 85A and 1.3% for 90A.

#### 4.2.4. Bending

For the bending test, the FE model is used to compare the strain values in the surface of the femur for each region. The average percentage difference in region I is 0.01% for 85A and 0.02% for 90A, and the average percentage difference for region II is 0.1% for 85A and 0.1% for 90A. The small values for the difference in the bending test are expected as the trabecular bone in the regions analysed is considerably smaller than the cortical bone and, as such, its contribution to the strain in the surface of the synthetic-composite material is not significant. As in the case of the cylinder geometry the percentage difference involved is a consequence of material combinations selected (i.e. ABS/TPU 85A and ABS/TPU 90A).

### 4.3. Trial experimentation

The trial experimentation was carried out to validate the results of the scaling approach and assessing the success of PLA/TPU 93A when measuring strain on the surface of a femur geometry by means of the image correlation software. The PLA/TPU combination is selected considering it shows the closest agreement in the cylinder trial experiment even though other trial space materials are viable options as noted in the finite-element analysis. This, in turn, provides insight into the extent of the approach in complex geometries that will potentially translate to other experiments consisting of synthetic-composite bone.

The trial experiment was conducted with the 3D printed scaled specimens of the synthetic-composite femur with a scaling parameter *β* = 4/3; the specimens were printed with the combinations of trial space materials PLA 0.4 mm/TPU 93A. The force applied in the trial space was calculated using the same equation as the cylinder experiment for the force (i.e. equation (4.1)) considering that the contact area of the compression test is equivalent in both spaces the change in area given by the scaling map is .

#### 4.3.1. Compression

As in the case of the cylinder trial experiment, the axial force applied in the compression test was calculated for the femur trial space using the respective parameters for the PLA/TPU 93A, the force values in the physical space range from 1200 N to 1800 N with 200 N increments thus the trial space force values range from 132.8 N to 199.2 N with 22.1 N increments. The experiment is considered static loading since the Aramis software records images at each loading stage (one stage tests for each value of the force) and consequently calculates the strain value at the surface for each value of the force. The results are analysed in the statistical data of the software that shows the average strain in the region selected after the data from computation mask has been processed as shown in figure 8.

To compare the results obtained in each space a Bland–Altman statistical analysis is carried out considering the synthetic-composite bone as the gold standard and the proposed trial space materials PLA/TPU 93A as the new method of validation.

It is important to note the regions are chosen differently from the finite-element models because a node-to-node comparison is no longer available, in the case of the image correlation software, only an average of the strain measured in the facets within the region is obtained. As such, the regions are chosen such that similar values of the strain are contained (e.g. positive or negative) in order to guarantee the comparison is representative.

#### 4.3.2. Bending

The final test was a bending test for the femur geometry; on the use of expression (4.1) the axial force applied was calculated for the trial space using the respective parameters for PLA/TPU 93A. As the force values in the physical space range from 700 N to 900 N with 100 N increments consistent with the average peak loads in a one-leg standing posture in the *x*-axis the trial space force values are calculated which results in the respective values ranging from 77.5 N to 99.6 N with 11.05 N increments (figure 9).

#### 4.3.3. Results

Table 6 shows the repeatability analysis for each method of validation as well as the Bland–Altman analysis for each combination of materials in the trial space for the femur geometry. Twelve measurements of the average strain were taken (six for each region) for each value of the force in each space. For the analysis of repeatability the difference between every measurement for each value of the force is considered resulting in 112 values. The within-subject repeatability is assumed to be independent because a plot between the standard deviation and the mean shows no tendency to change with the magnitude of measurements. The same is done for the bending test, resulting in 81 values; the within-subject repeatability is assumed to be independent as in the case of the compression test. These repeatability coefficients are satisfactory for the purpose of validation with the image correlation software because the image correlation software's accuracy is 0.01 in percentage strain.

The agreement between the synthetic-composite bones and the trial space materials was analysed. The difference between each measurement from the synthetic-composite cylinder and each measurement of the 3D-printed cylinder is considered (the same is done for the average) as such the resulting data have 256 values for the compression test and 192 for the bending test. As before the assumption of independence with the magnitude of measurements is made. Table 6 shows the mean difference and the standard deviation as well as the limits of agreement which are below the equipment's accuracy and show good agreement for the purpose of validation by means of image correlation. The Bland–Altman plots are included in appendix B in table 7.

## 5. Discussion

In this work, the application of a novel scaling methodology in bio-mechanical experimentation consisting of validation of finite-element models was analysed for isotropic scaling with the main objectives being the validation of the mathematical theory for scaling and overcoming the difficulties that arise in biological experimentation by means of scaled experiments. The following conclusions can be drawn:

—the scaling theory allows the use of 3D-printed materials which overcome two of the main concerns, the availability and considerable reduction of the cost of materials as well as ethical issues;

—the numerical results for the simple geometry (cylinder) show a small percentage (less than 10%) difference in the surface strain with the synthetic composite bone across all the mechanical tests for the three combinations of materials (ABS/TPU 85A, ABS/TPU 90A and PLA/TPU 93A) and the subsequent trial experimentation on the two combinations (ABS/TPU 85A and PLA/TPU 93A) show a good agreement using Bland–Altman statistical analysis with the closest agreement being the PLA/TPU 93A combination;

—the numerical results for the complex geometry (femur) show a small percentage difference (less than 5%) in the surface strain with the synthetic composite femur across all tests for the three combinations (ABS/TPU 85A, ABS/TPU 90A and PLA/TPU 93A) and the subsequent trial experimentation with the PLA/TPU 93A printed femur shows a good agreement using Bland–Altman statistical analysis; and

—the results confirm the applicability of the scaling theory and the use of 3D-printed materials as a viable alternative to synthetic-composite bone in bio-mechanical experimentation consisting on the validation of models by means of the digital image correlation software. It is important to note that the errors are a consequence of experimental limitation that occur in practice (selection of materials) and not the methodology proposed. The work provides a good foundation for scaled biomechanical experimentation using finite similitude as a framework for the design of trial experiments.

### 5.1. Future work

The study of anisotropic scaling and its transport equation approach, which is one of the characteristics of the approach, warrants further study as well as the application of the scaling theory to anisotropic mechanical properties such as the mechanical properties of synthetic composite bone. The study of strain in trabecular bone and mechanical bone adaptation using scaled experimentation is also of interest in future work. The consideration of these 3D-printed materials for validation of finite-element models in cases where patient specific geometries are analysed also warrants further study.

## Data accessibility

This article has no additional data.

## Competing interests

We declare we have no competing interests.

## Funding

We received no funding for this study.

## Acknowledgements

The authors would like to acknowledge the National Council of Science and Technology of Mexico for providing support for Raul Ochoa-Cabrero to facilitate his research at the University of Manchester.

## Appendix A

The dimensional analysis approach is included here to highlight the differences between the two approaches and give supporting evidence to the strengths of the proposed methodology for scaling. It is demonstrated here how dimensional analysis can be used to obtain equations (3.13) and (4.1). For the biomechanical experimentation under consideration the surface strain is the main concern; thus the deformation in the simple cylinder geometry undergoing compression depends on the following governing parameters: the length of the cylinder *L*; the area where stress is distributed over each material *A*^{i} and *A*^{e}; Young's modulus of each material *E*^{i} and *E*^{e}; and the applied force *F*. The governing equation is of the form
A 1for some function *f*.

The parameters having independent dimension are *L* and *E*^{e} and the rest are dependent on these. On use of the Buckingham Pi theorem, the following Pi groups are obtained:
A 2*a*
A 2*b*
A 2*c*
A 2*d*
A 2eand the expression is simplified to *Π* = *F*(*Π*^{1}, *Π*^{2}, *Π*^{3}, *Π*^{4}).

Using the definition of similarity (*Π*^{m}_{ps} = *Π*^{m}_{ts}, *m* = 1 : 4) and assuming the governing equation is the same for the trial experiment and the physical experiment, the following equations are obtained:
A 3*a*
A 3*b*
A 3*c*
A 3*d*The first two equations refer to the geometrical groups and undoubtedly hold because they are identical to the relationship *A*_{ps} = *β*^{2}*A*_{ts}. The relationship between Young's moduli in equation (3.13) is identical to equation (A 3a). Finally, substitution of equation (3.10) into equation (A 3b) gives *F*_{ts} = *F*_{ps}(*α*^{ν}*hβ*^{2}.)(*β*^{−1})^{2} = *α*^{ν}*hF*_{ps}, which is identical to equation (4.1) on substitution of *A*_{ps} = *β*^{2}*A*_{ts}. Thus, equations (A 3) are consistent with relationships obtained from finite similitude, providing additional evidence of the validity of the approach.

The classical dimensional-analysis theory presented here unlike finite similitude does not:

—integrate with the finite-element method by providing boundary conditions and constitutive equations for exact finite similitude;

—project a virtual-scaled model in the trial space for the design of scaled experimentation;

—project a virtual-reverse model in the physical space for assessing the success of a trial experiment in the physical space;

—evaluate the source and effect of errors;

—accommodate inexact similitude; and

—account for all of continuum mechanics and experimental reality.

## Appendix B

The Bland–Altman plots for each of the physical-trial experimentation tests are included to support the results reported.

- Received April 15, 2018.
- Accepted May 18, 2018.

- © 2018 The Author(s)

Published by the Royal Society. All rights reserved.