NL2032347B1 - Intervertebral cage - Google Patents

Abstract

A computer implemented method (100) oomprising the step of synthesizing the topology, the shape, and the size of an intervertebral cage for replacing an intervertebral disc in a patient, wherein the step of synthesizing is based on topology 5 optimization. An example of an intervertebral cage for replacing an intervertebral disc in a patient, wherein the intervertebral cage has a topology, a shape and/or a size, synthesized with topology optimization is shown in Figure 7 and Figure 9. 10 Figure 7

Description

INTERVERTEBRAL CAGE

FIELD OF THE INVENTION

The invention relates to an intervertebral cage. The invention further relates to a method for synthesizing an intervertebral cage. The invention also relates to an embodiment of an intervertebral cage. The invention also relates to a data processing system and a computer-readable storage medium for synthesizing an intervertebral cage.

BACKGROUND OF THE INVENTION

An intervertebral disc sits between two vertebras. Where the vertebras provide a stiff structure to the spine of a human, the intervertebral disc provides flexibility to the two adjoining vertebras. Due to the flexibility of the intervertebral disc the two adjoining vertebras may move to some extend relative to each.

A common result of the continues flexing of, as well as the stress exerted on, the intervertebral disc overtime is that the intervertebral disc may degrade to such an extent that it needs to be replaced. The known intervertebral disc replacements or cages are fixed cages. Further, new intervertebral replacements or cages are known having at least one kinematic pair, such as a revolute joint. A disadvantage of the application of a kinematic pair in an intervertebral disc replacement is that these replacements wear out, specifically the joints wear out resulting in another replacement surgery or even permanent loss of flexibility between the vertebras adjoining the intervertebral disc replacement of the patient.

SUMMARY OF THE INVENTION

An object of the invention is to overcome one or more of the disadvantages mentioned above.

According to a first aspect of the invention, a computer implemented method comprising the step of synthesizing the topology, the shape, and the size of an intervertebral cage for replacing an intervertebral disc in a patient, wherein the step of synthesizing is based on topology optimization.

A flexure is typically a monolithic compliant element that connects two or more (assumed) rigid links, allowing for selectively chosen movements. A flexure may comprise one or more compliant joints. In contrast to conventional hinges, flexures achieve their range of motion through elastic deformation. The finite dimension and operation below a critical stress limit the attainable range of motion. Due to the monolithic nature, flexures hardly require maintenance and have a long lifetime if used within the intended range of motion. Due to the lack of friction and backlash, flexures have high repeatability in use. Hence, the intervertebral cage provides the technical effect of extending the lifetime and/or reliability of the intervertebral cage according to the claimed method. This technical effect is obtained while typically sustaining the range of motion of the patient.

The topology optimization is typically focused on synthesizing so-called short-stroke flexures, for which — in contrast to large-stroke flexures — the assumptions of a linear stress—strain and a linear strain—displacement relationship suffices. The topology optimization typically results in a description of a topology, a shape, and/or a size, preferably a topology, a shape, and a size. The combination of a topology, a shape, and a size may be labelled as the geometry of the intervertebral cage. The intervertebral cage is an intervertebral disc replacement. The synthesized intervertebral cage is a type of flexure. The synthesized intervertebral cage is typically a non-fusion solution for the patient providing a higher likelihood for maintaining the natural range of motion.

The topology optimization method involves energy-based performance measures that are obtained under prescribed displacement conditions, and the method is unique in the way these response functions are combined to form the optimization problem formulation. The resulting advantageous properties of the optimization problem formulation pose benefits in terms of simplicity, versatility and computational efficiency.

According to another aspect of the invention, an intervertebral cage comprising a topology, a shape and/or a size synthesized according to any of the embodiments involving topology optimization. The intervertebral cage provides the advantages also provided by the computer implemented method or forthcoming from the computer implemented method.

According to another aspect of the invention, a method for manufacturing comprising the steps of: receiving a data structure comprising information for manufacturing an intervertebral cage for replacing an intervertebral disc in a patient, wherein the intervertebral cage has a topology, a shape and/or a size synthesized according to any of the embodiments of the computer implemented method; and producing the intervertebral cage. The method provides the advantages also provided by the computer implemented method.

According to another aspect of the invention, a data structure comprising information for manufacturing the synthesized intervertebral cage according to any of the embodiments of the computer implemented method, or any of the embodiments detailing a specific embodiment of the intervertebral cage in relation to manufacturing and/or the data structure. The data structure provides the advantages also provided by the computer implemented method. The data structure may be formatted as a STL file,

CAD file, or any other suitable format for manufacturing.

According to another aspect of the invention, an intervertebral cage for replacing an intervertebral disc in a patient, wherein the intervertebral cage has a topology, a shape and/or a size, according to any of the specific embodiments, such as

Figure 7, more specific Figure 7 and Figure 9. The intervertebral cage provides the advantages also provided by the computer implemented method.

According to another aspect of the invention, a method for manufacturing an intervertebral cage is according to any of the specific embodiments, such as Figure 7, more specific Figure 7 and Figure 9, wherein the method uses additive manufacturing.

The method for manufacturing provides the advantageous also provided by the computer implemented method. Furthermore, the intervertebral cage is typically monolithic, advantageously simplifying the additive manufacturing. In a specific embodiment, the topology optimization may consider restrictions and/or limitations of additive manufacturing, to account for these restrictions and/or limitations the synthesis may comprise steps of applying erosion and dilation effects, for advantageously optimizing the embodiment of the intervertebral cage for additive manufacturing.

Furthermore, in a specific embodiment of the intervertebral cage, considerations of additive manufacturing, such as overhanging parts during manufacturing, are advantageously considered during optimizing the embodiment of the intervertebral cage.

According to another aspect of the invention, a data processing system comprising means for carrying out the steps of any of the computer implemented method mentioned in the description, the steps of the method for manufacturing mentioned, or the data structure mentioned. The data processing system provides the advantages also mentioned for the other aspects of the invention, specifically those provided by the computer implemented method.

According to another aspect of the invention, a computer-readable storage medium comprising instructions which, when executed by a computer, cause the computer to carry out the steps of any of the computer implemented methods mentioned, or the steps of the method for manufacturing mentioned. The computer- readable storage medium provides the advantages also mentioned for the other aspects of the invention, specifically provided by the computer implemented method.

According to another aspect of the invention, a computer implemented method comprising the step of synthesizing the topology, the shape, and/or the size of a flexure, wherein the step of synthesizing is based on topology optimization. The flexure may be applied in or is part of a device. Such a device may be a human implant, such as a non-fusion spinal implant, for example an intervertebral cage for replacing an intervertebral disc in a patient. Another example of a device, typically a non-fusion implant, may be any other joint in the human body, preferably a spinal joint, a knee, a hip, an ancle, a shoulder joint, an elbow, a wrist, a finger joint, or a jaw joint.

Even another example of a device comprising the flexure may be applications in industry, such as X, Y, and Z motion tables applied in for example electron beam microscopes and lithography wafer steppers. The flexure may combine the course and fine motion, typically in one or more directions. Still another example of a device comprising the flexure may be applications in industry, such as precision alignment for space applications e.g., instruments in satellites, providing among others the advantages of highly reliable and/or highly repeatable precision alignment. The instruments may involve optomechanical alignment internal to the instrument or relative to an external reference point. The computer implemented method for the flexure provides the advantages also mentioned for the other aspects of the invention, specifically provided by the computer implemented method for the intervertebral cage.

DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS

In an embodiment of the computer implemented method, the intervertebral cage comprises an element, which is a monolithic element. The monolithic element, not having any material junction, extends the lifetime and/or reliability. Further, the monolithic element may simplify manufacturing of the intervertebral cage.

In an embodiment of the computer implemented method, the intervertebral cage comprises a/the element, which is a compliant element. The compliant element typically comprises a plurality of compliant joints. Compliant joints advantageously increase the lifetime and/or reliability.

In an embodiment of the computer implemented method, the topology optimization is initiated with a build volume having: a length in a lateral direction (X) in 5 the range of 30 mm to 50 mm, preferably 35 mm to 45 mm, more preferably 37 mm to 43 mm, most preferably substantially 40 mm; a length in a dorsal and/or a ventral direction (Y) in the range of 20 mm to 40 mm, preferably 25 mm to 35 mm, more preferably 27 mm to 33 mm, most preferably substantially 30 mm; and/or a length in a cranial and/or a caudal direction (Z) in the range of 8 mm to 20 mm, preferably 10 mm to 18 mm, more preferably 12 mm to 16 mm, even more preferably 13 mm to 15 mm, most preferably substantially 14 mm. The build volume is limited due to that it should fit into the space previously occupied by the removed intervertebral disc. The build volume ranges advantageously allow for adaptation to human bodies of different sizes.

Especially the caudal direction is typically critical as that it should not be too large in this direction. In a further embodiment, the method comprises the step of determining the build volume, specifically the height of the intervertebral cage in the caudal direction, based on the to be replaced intervertebral disc. The step of determining may be based on the shape and size of the to be replaced intervertebral disc before it at least partly lost its function.

In an embodiment of the computer implemented method, the topology optimization applies a maximum predefined mass value requirement for limiting the mass of the vertebral cage. The maximum predefined mass value typically stimulates the removal of non-functional protrusions. This embodiment may advantageously be applied for removing floating mass in the final design. A floating mass, next to being not producible, also causes unnecessary material use as well as is not functional or may even limit a motion over the full range typically in an unpredictable way.

In an embodiment of the computer implemented method, the six degrees of freedom are split in a degrees of freedom subset having a low stiffness, and a degrees of constraint subset having a high stiffness; and the topology optimization is performed based on the degrees of freedom subset and/or the degrees of constraint subset such that the intervertebral cage substantially provides comparable stiffness as the intervertebral disc. The intervertebral disc in the human body typically provides high stiffness in multiple directions of the six degrees of freedom and low stiffness in the other direction(s) of the six degrees of freedom. Mimicking these stiffnesses advantageously provides a comparable freedom of movement to the human body, specifically the spine, after replacing the intervertebral disc with an intervertebral cage, thereby increasing the comfort or quality of life of the patient compared to other replacements and/or before replacement. Separating the degrees of freedom in low and high stiffness provides simple requirements and thereby simplifies the set-up of the topology optimization problem formulation. In rigid body mechanisms, movements are typically restricted or not (binary). However, in compliant mechanisms even the

DOF have stiffness, and the "restricted" movements are not infinite stiff. Thus, a strict definition of DOF may not be well defined. In practice one speaks off a DOF when the stiffness of a motion (e.g. rotation around an axis) is substantially (typically 10 to 100 times) smaller than the stiffness of the restricted motions. The six degrees of freedom are typical for a rigid body, but may be applied also for compliant mechanisms.

Mentioned in the text is that the six degrees of freedom may be generalized to motion patterns, as specified in the description.

In an embodiment of the computer implemented method, the degrees of constraint subset comprises the translations along a lateral axis (TX), and a dorsal and/or a ventral (TY) axis; and/or the degrees of freedom subset comprises the translation along the cranial and/or the caudal axis (TZ), a rotation around a lateral axis (RX), a dorsal and/or a ventral axis (RY), and a cranial and/or a caudal axis (RZ).

The selection of the degrees of constraint subset is advantageously for mimicking the stiffnesses of an intervertebral disc, specifically a healthy intervertebral disc, that needs to be replaced due to that the intervertebral disc lost it at least partly its intended function.

In a further embodiment of the computer implemented method, the subset of the six degrees of freedom relating to rotation comprise: a maximum rotation around a lateral axis (RX) in the range of 2 degrees to 10 degrees, preferably 3 degrees to 8 degrees, more preferably 4 to 6 degrees, most preferably substantially 5 degrees; a maximum rotation around a dorsal and/or a ventral axis (RY) in the range of 2 degrees to 10 degrees, preferably 3 degrees to 8 degrees, more preferably 4 to 6 degrees, most preferably substantially 5 degrees; and/or a maximum rotation around a cranial and/or a caudal axis (RZ) in the range of 2 degrees to 10 degrees, preferably 3 degrees to 8 degrees, more preferably 4 to 6 degrees, most preferably substantially 5 degrees. This embodiment advantageously specifies the range of motion such that the topology optimization method may incorporate and iterate to the appropriate stiffnesses. This embodiment further advantageously allows to introduce abutments in the intervertebral cage. This embodiment further advantageously allows to introduce increased stiffness in the intervertebral cage when close to or exceeding the range specified.

In an embodiment of the computer implemented method, the subset of the six degrees of freedom relating to translation comprises: a maximum translation along a lateral axis (TX) is below 2 mm, preferably 1,5 mm, more preferably 1 mm, most preferably substantially prevented; a maximum translation along a dorsal and/or a ventral axis (TY) is below 2 mm, preferably 1,5 mm, more preferably 1 mm, most preferably substantially prevented; a maximum extension along a cranial and/or a caudal axis (TZ) is below 1 mm, preferably 0,5 mm, more preferably 0,25 mm, most preferably substantially prevented; and/or a maximum compression along a cranial and/or a caudal axis (TZ) divided by the unloaded length in the cranial and/or the caudal direction is in range of 0% to 60%, preferably 10% to 50%, more preferably 20% to 30%, most preferably substantially 30%. This embodiment advantageously specifies the range of motion such that the topology optimization method may incorporate and iterate to the appropriate stiffnesses. This embodiment further advantageously allows to introduce abutments in the intervertebral cage. This embodiment further advantageously allows to introduce increased stiffness in the intervertebral cage when close to or exceeding the range specified.

In an embodiment of the computer implemented method, the topology optimization method comprises defining a constrained nonlinear optimization problem formulation, preferably gradient based, for synthesizing the intervertebral cage; the constrained nonlinear optimization problem P is formulated as: minimize x f[U;[x]], i € €

P= [ec to Ux]<0, jeF ;and xe XN x is the vector of N design variables; X the set defines as {xin R | Xmin < X < Xpnaxh the dimensionless objective f € R* is a function of energy measure U; € R* such as strain energy, work, compliance, or energy derived measure such as stiffness; 7, €

R* is a maximum allowable energy or stiffness; U, is the measure of energy or stiffness of predefined motion patterns in the set € ; U, is the measure of energy or stiffness of a predefined motion patterns in the set F; and C and F are complementary. The constrained nonlinear optimization problem P is advantageously simplified compared to other topology optimization design methods. This simplification reduces the computational time, thus also the energy consumption for coming to a design. The optimization problem is typically solved for an undeformed configuration. The product is synthesized for specific stiffnesses related to a specific motion pattern, such as a rotation and/or translation, in the undeformed configuration. In general, the product has varying stiffness depending on the deformation. U typically represents strain energy, external work, or compliance in Joules, and/or stiffness in Newton per meter or Newton meter per radian. Although in the description strain energy is often selected, the formulas may be generalized, applied and/or altered for fitting other representations of

U.

In an embodiment of the computer implemented method, the translational motions have a stiffness: between 200 N/mm and 20.000 N/mm in undeformed configuration along a medial/lateral axis (TX); between 200 N/mm and 20.000 N/mm in undeformed configuration for along a posterior/anterior axis (TY); and/or between 300

N/mm and 3000 N/mm in undeformed configuration along a cranial and/or a caudal axis (TZ). This embodiment advantageously specifies the ranges of motion of the intervertebral cage. The optimization problem is typically solved with the stiffness from the undeformed configuration. The product is synthesized for specific stiffnesses related to a specific motion pattern, such as a rotation and/or translation, starting out from the undeformed configuration. In general, the product has varying stiffness depending on the deformation.

In an embodiment of the computer implemented method, the rotational motions have a stiffness: between 1Nm/deg and 100 Nm/deg in undeformed configuration around a medial/lateral axis (RX); between 1Nm/deg and 100 Nm/deg in undeformed configuration around a posterior/anterior axis (RY); and/or between 10

Nm/degree and 1000 Nm/deg in undeformed configuration around a cranial and/or a caudal axis (RZ). This embodiment advantageously specifies the ranges of motion of the intervertebral cage. The optimization problem is typically solved with the stiffness from the undeformed configuration. The product is synthesized for specific stiffnesses related to a specific motion pattern, such as a rotation and/or translation, starting out from the undeformed configuration. In general, the product has varying stiffness depending on the deformation, such as increasing stiffness in a direction depending on the distance from the undeformed condition to the deformed condition.

In a further embodiment of the computer implemented method, the topology optimization problem formulation can more detailed be written as mame lx] and yy [x] subject to i, [x] S Kiz subject to Kp. [x] < Kx subject to K;z[x] <¥K subject to Kk, [x] <%, subjectto vl[al <7

Kz , Kz , Kz , and Kz are the maximum allowable stiffness corresponding to respective degrees of freedom; and 7 is the maximum material usage relative to a/the build volume. This embodiment advantageously defines the stiffnesses for the degrees of freedom in a mathematical formulation allowing for relatively simple implementation of the topology optimization method.

In an embodiment of the computer implemented method, a motion pattern is preferably defined as a translation along an axis or a rotation around an axis; preferably the axis are selected as a lateral axis (X), a dorsal and/or a ventral axis (Y), and/or a cranial and/or a caudal axis (Z); and the topology optimization is performed such that stiffness is maximized for preselected motion patterns, and/or for the other preselected motion patterns stiffness is limited to a predefined stiffness value. In this embodiment the alignment of the X, Y, and Z axis with the axis in the human body, specifically the spine, is advantageously detailed.

In an embodiment of the computer implemented method, a/the motion pattern is defined as a translation along an axis or a rotation around an axis; the topology optimization comprises defining a constrained nonlinear optimization problem | preferably gradient based, for synthesizing the intervertebral cage; and the constrained nonlinear optimization problem is formulated as: minimize x —fle:x]], IEC

P= [i to Elst, jeF ;and xe XN the dimensionless objective f € R* is a monotonically increasing function of strain energies €; € R* and , € R* is a maximum allowable strain energy of motion pattern j. The topology optimization problem may advantageously be generalized to other or derived energy measures, such as compliance, stiffness, or external work. Maximizing strain energies for restricted motion patterns translates to restricted motion for these motion patterns.

In an embodiment of the computer implemented method, the topology optimization is based on a topology defined by N continuous differentiable design variables x with its components in X:={x € R| 0 < x < 1}; each element í in the topology optimization has a stiffness, preferably a Young modulus, E;; and E; is a nonlinear function of design variable x;. Furthermore, the range allows to easily round up to 1 or down to 0 as part of ending/concluding the optimization method, this step may be typed as a post-processing step included in the embodiment. Further, the different functions being continuous differentiable also allows for the use of gradient information which aids in finding an improved simplified implementation and calculation of iterations.

In an embodiment of the computer implemented method, the Young's modulus is a function of the filtered design variable bin) =&+(1-8R[x]

E is the material Young's modulus; wherein ¥ is the filtered design variable x; € is the relative stiffness between solid and void; and R the material interpolation function. A relatively simple first order filter is used for advantageously simplifying the implementation and calculation.

In an embodiment of the computer implemented method, the material interpolation function is typically chosen as R[x] = xP ; wherein p € R* ; and preferably p is in the range of 2 to 4, more preferably substantially 3. This interpolation function increases the probability to obtain a 0/1 solution. Typically, the interpolation function allows for stimulates lower values to O and higher values to 1.

In a further embodiment of the computer implemented method, a strain energy of motion pattern i in a discretized setting is defined as: €;[x] = TT Kl[xlu; ;

K[x]u; € R™*" is the design dependent symmetric stiffness matrix; and u; € R° comprises a nodal displacement of motion pattern i. Here n is the number of structural degrees of freedom or unknown displacements in the analysis. This embodiment advantageously provides the discretised strain energy of a motion pattern i, preferably all motion patterns.

In an embodiment of the computer implemented method, the nodal displacement is calculable by solving n linear governing equations K[x]u, = f‚ Vie

M; M is the set of all motion patterns; and f; € R™ are the nodal loads of motion pattern i. Solving of linear governing equations advantageously simplifies the implementation and/or execution of the topology optimization method. This embodiment is advantageously combined with a definition of the direction of the axis in relation to the human body.

In an embodiment of the computer implemented method, the nodal displacement may be partitioned as i” x] |] = [7]: u; are the free nodal displacements; u, are the prescribed nodal displacements; f; are the applied nodal loads; and f,; are the nodal reaction loads of motion pattern i. The motion patterns are advantageously defined purely in terms of prescribed nodal displacements at the interfaces, without additional applied loads.

In an embodiment of the computer implemented method, the applied loads f; = 0in all cases; and the step of synthesizing comprises the step of solving for the nodal displacement Ku; = Kp, Vi € M. The applied loads f, = 0 in all cases advantageously simplifies the notation and/or set-up of problem formulation of the topology optimization method, more specific solving the nodal displacement formula mentioned above.

In an embodiment of the computer implemented method, the topology optimization problem is a gradient-based inequality-constrained nonlinear optimization problem advantageously simplifying the implementation and/or execution of the topology optimization method.

In an embodiment of the computer implemented method, the topology optimization comprises the step of: filtering a design variable field; and differentiable

Heaviside projecting using a high projection threshold for erosion and/or a low projection threshold for dilation. This embodiment advantageously improves the sensitivity to manufacturing tolerances and thereto manufacturability of the intervertebral cage. This embodiment advantageously limits the minimum feature size of the intervertebral cage designed by the topology optimization method.

In an embodiment of the computer implemented method, minimize flu. [xe]], Pec

Pp, = x —~ ; and m7" )subjectto Ula <U, i€F’ xe XN

U[x®] and U[x?] are strain energies based on the eroded and dilated fields, respectively, advantageously simplifying the implementation and/or execution of the topology optimization method.

In an embodiment of the computer implemented method, me flu), iec

Fo = fo to res ; c : ‚ and xe XN o; are the elemental stresses obtained by the predefined motion patterns i for a predefined magnitude; and wherein & is a maximum allowable stress value, advantageously simplifying the implementation and/or execution of the topology optimization method.

In an embodiment of the computer implemented method, the topology optimization is for short stroke flexures. Intervertebral cages are a typical example of short stroke flexures. In an embodiment, the computer implemented method comprises the step of manufacturing the intervertebral cage. In an embodiment, the method of manufacturing comprises additive manufacturing. Additive manufacturing may be 3D printing.

In an embodiment, the computer implemented method comprises the step of providing a data structure comprising information for manufacturing the synthesized intervertebral cage for advantageously allowing storage and/or communication of the design of the intervertebral cage. In a further embodiment of the computer implemented method, the information for manufacturing comprises the density as a function of the location for the intervertebral cage, preferably for each location in a/the build volume of the intervertebral cage. In a further embodiment of the computer implemented method, the information for manufacturing comprises the topology, the shape and the size of the intervertebral cage, preferably for each location in a/the build volume of the intervertebral cage.

In an embodiment of the intervertebral cage, the intervertebral cage comprises a monolithic compliant element; and the monolithic compliant element comprises the topology, the shape and the size. This embodiment advantageously provides the advantages as mentioned for the features of the computer implemented method.

In an embodiment of the intervertebral cage, the intervertebral cage comprises a monolithic compliant element; and the monolithic compliant element comprises the density as a function of the location for the intervertebral cage, preferably for each location in a/the build volume of the intervertebral cage. This embodiment advantageously provides the advantages as mentioned for the features of the computer implemented method. In an embodiment of the data structure, the data structure is in a computer readable format for advantageously communicating the intervertebral cage design.

In an embodiment of the intervertebral cage, the shape and/or the size are further detailed in Figure 10(a) — 10(i). In an embodiment of the intervertebral cage, the intervertebral cage is manufactured using additive manufacturing. In an embodiment of the intervertebral cage, the intervertebral cage comprises compliant joints, and at least the compliant joints are composed of a single composition. In an embodiment of the intervertebral cage, the intervertebral cage comprises a flexure, typically comprising a plurality of compliant joints. The advantages of the preceding embodiments are mentioned throughout the description, specifically in relation to features of the computer implemented method.

In an embodiment of the method for manufacturing, the intervertebral cage comprises compliant joints, and at least the compliant joints are composed of a single composition. In an embodiment of the method for manufacturing, the method is combined with any of the features of the embodiments of the computer implemented method for achieving the same or comparable advantages.

In an embodiment of the method, the method is combined with any of the features introduced in any of the other embodiments, particularly the embodiments of the system, for obtaining the same advantages as mentioned for that particular embodiment or enhancing the advantages mentioned for the individual embodiments.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention will be apparent from and elucidated further with reference to the embodiments described by way of example in the following description and with reference to the accompanying drawings, in which:

Figure 1(a) schematically shows a computational design of a multi-axis flexure;

Figure 1(b) shows an image of a prototyped computational design of a multi-axis flexure of Fig. 1(a);

Figure 2(a) schematically shows two rigid links (gray) connected via a flexure (white);

Figure 2(b) schematically shows a translation along the x-axis, motion pattern tx;

Figure 2(c) schematically shows a translation along the y-axis, motion pattern ty;

Figure 2(d) schematically shows a rotation about the z-axis, motion pattern rz;

Figure 3(a-f) schematically shows generated designs for various varieties of problem formulations based on P;

Figure 4 schematically shows a characteristic convergence history of problem formulation P;

Figure 5(a-i) schematically show generated high-resolution 3D flexure designs;

Figure 6(a) schematically shows a design obtained by solving F;;

Figure 6(b) schematically shows a design obtained by solving P,;

Figure 7 shows an image of a prototyped computational designed intervertebral cage;

Figure 8 schematically shows the intervertebral cage arranged between two vertebras and the definition the axis, translations and rotations;

Figure © schematically shows a computational design of a prototyped computational designed intervertebral cage;

Figures 10(a-i) schematically show a computational design of a prototyped computational designed intervertebral cage;

Figure 11 schematically shows an embodiment of a computer program product, computer readable medium and/or non-transitory computer readable storage medium according to the invention; and

Figures 12(a-n) schematically show the intervertebral cage in several deformed conditions.

The figures are purely diagrammatic and not drawn to scale. In the figures, elements which correspond to elements already described may have the same reference numerals.

LIST OF REFERENCE NUMERALS translation along Z axis rotation about X axis rotation about Y axis rotation about Z axis 1000 computer program product 1010 computer readable medium 1020 computer readable code

DETAILED DESCRIPTION OF THE FIGURES

The following figures may detail different embodiments. Embodiments can be combined to reach an enhanced or improved technical effect. These combined embodiments may be mentioned explicitly throughout the text, may be hint upon in the text or may be implicit. 1. Introduction

A flexure is a monolithic compliant element that connects two or more (assumed) rigid links, allowing for selectively chosen movements. In contrast to conventional hinges, flexures achieve their range of motion through elastic deformation. The finite dimension and operation below a critical stress limit the attainable range of motion. Due to the monolithic nature, flexures hardly require maintenance and have a long lifetime if used within the intended range of motion. Due to the lack of friction and backlash, flexures have high repeatability in use. Given these advantages, flexures may be applied in precision applications such as positioning stages and optical mounts. The present work is focused on so-called short-stroke flexures, for which — in contrast to large-stroke flexures — the assumptions of a linear stress—strain and a linear strain—displacement relationship suffice. Flexures are engineered to have desired characteristic stiffness for specific relative rigid link movements. These movement are hereafter denoted by Motion Patterns (MPs). The mechanism Degrees of Freedom (DOFs) are the number of MPs with relatively low characteristic stiffness; that are the free motion patterns. In contrast, the mechanism

Degrees of Constraint (DOCs) are the number of MPs with a much higher characteristic stiffness. An example of a complex three-dimensional flexure with two mechanism DOFs and four mechanism DOCs is shown in Fig. 1. Fig. 1. Shows a computational design of a multi-axis flexure in Fig.1(a) and its prototyped counterpart in Fig. 1(b). The flexure is compliant in the rotations about the x and y-axis while stiff in the x, y and z translations as well as rotation about the z-axis.

Common single-axis flexures are (i) compliant revolute joints, such as notch hinges that allow relative rotation about a single axis, and (ii) compliant prismatic joints, such as a pair of parallel leaf-springs, that allow relative motion along a single axis.

Revolute joints may also be called (flexural) hinge, flexure bearing or flexure pivot.

Prismatic joints may also be called translational (flexure) hinges. Common multi-axis flexures are, e.g., compliant cylindrical, universal, spherical and planar joints. Complex flexures typically combine multiple primitive flexures as building blocks, thus enabling more complex kinematics. Also, flexures can be classified by the degree of localization of the deformation, ranging between lumped (i.e. highly localized) and distributed compliance.

The primary design requirement of a short-stroke flexure may be the relative stiffness between the mechanism DOFs and DOCs. Secondary considerations may be range of motion, axis drift, deformation and stress, fatigue, volume and mass, as well as the sensitivity of those aspects to, e.g., manufacturing errors. The synthesis methods often used for rigid-body mechanisms, cannot straightforwardly be applied to compliant mechanisms. There is always mechanical stress involved in any motion, and the behavior is dependent on the loading condition. This implies that kinematics (motion) and kinetics (load case) must be treated simultaneously. As a result, the concept of mechanism DOFs fades in compliant mechanisms, because they behave differently for any loading conditions. Furthermore, the complex deformation and motion behavior of compliant mechanisms complicates both their accurate analytical modeling as well as purposeful design. Hence, the synthesis process is iterative, and often time-consuming.

Systematic flexure synthesis methods rely on kinematic or building block approaches, such as rigid-body replacement techniques or the ‘freedom and constraint topology’ method. However, these approaches do not exploit the full range of design possibilities. The use of gradient-based structural optimization techniques to design flexures has gained increasing interest because of the possibility to design optimized flexures, satisfying application-specific requirements. Topology Optimization (TO) in particular, allows for maximum design freedom, while requiring minimal designer input regarding the flexure concept.

Owing to the potential benefits of TO, academics, engineers and designers could benefit from a versatile, simple, easy to implement and use as well as computationally efficient TO method for short-stroke flexure design. Multiple different

TO problem formulations are previously proposed. Section 2 provides a comparison of previously proposed TO problem formulations for flexure design and addresses the remaining challenges in the field with respect to simplicity, versatility and computational effort.

To address the challenges, in Section 3 we propose a novel and intuitive topology optimization formulation to design flexures. The basic idea is to maximize the stiffness of a priori defined constrained MPs, whilst imposing an upper bound on the stiffness of a priori defined free MPs. Motion pattern stiffnesses are evaluated via strain energies under prescribed movements of the rigid links. The contribution of this work is thus on the use of energy-based response functions under prescribed displacement conditions and the manner in which these response functions are combined to form the optimization problem formulation. The contribution, advantageous properties of the problem formulation and consequential simplicity, versatility and computational efficiency is elaborated on in Section 3.2.

The basic formulation proposed in this work focuses on the primary design requirement, that is maximization of the relative stiffness between the free and constraint MPs. Implementation specific considerations are discussed in Section 4, followed by numerical examples in Section 5. We will additionally demonstrate the ease and influence of taking into account stress considerations as well as manufacturing robustness in Section 5, all within the limits of linear elasticity theory.

Section 6 outlines the code associated with this paper, with which the 2D results can be reproduced easily. The application is completed with stating the limitations of the proposed formulation, providing recommendations for future work and concluding remarks.

Table 1 below shows topology optimization problem formulations for flexure design versus quantifiable measures of the quality criteria. The disclosures are ordered by year, ending with the present contribution. Versatility is expressed by the demonstrated range of applications (types of joints, single or multi-axis) and dimensionality (2D or 3D). The Parameters column denotes the minimum number and type of parameters required to set up the formulation.

Implementation includes notable features, such as the type of analyses and responses.

The computational effort of a single design iteration is dominated by the effort of finite element and sensitivity analyses.

The last column indicates, subsequently the number of (i) preconditioning/factorization steps, (ii) physical loads, and (iii) additional adjoint loads per design iteration.

The sum of the loads indicates the number of iterative solves/back-substitutions required. For fair comparison all listed numbers of parameters and loads are for a single-axis flexure formulation.

A. Hasse, et. al., Smart Mater. 2D Any single-axis joint | Max volume Static condensation 1,2,0

Struct. 18 (11) (2009) 115016 Eigenmode Orthogonalization

Eigensystem analysis

M.Y. Wang, J. Mech. Robot. 1 (2) | 2D Any single or Max volume Static condensation 1,2,0 aon | rasan come egen

B. Zhu, et. al. Sci. China Technol. | 2D Prismatic and Max volume Non-design domain 1,2, 1

Sci. 57 (3) (2014) 560-567 evolute joint Non-design domain Exotic responses size Additional spring

Max axis drift

Spring stiffness

J. Pinskier, et. al. Precis. Eng. 55 | 3D Leaf flexure Max volume Strain energy based 1,2,0 woos || men

J. Pinskier, et. al. Mech. Mach. 2D Revolute joint Max volume Non-design domain | Non-design domain 1,3,3

Theory 150 (2020) 103874 size Exotic responses

Max displacement

Present 2D, Any single or Max strain energy Strain energy based 1,2,0 oe ee TTT TT

Table 1

2. Comparison of existing formulations

Currently, a good comparison between different TO formulations to synthesize flexures is absent. To compare different formulations, we define the following three quality criteria: + simplicity, + versatility, and « computational effort.

We define simplicity as the ease of understanding, implementation and use of the formulation. This includes the number of parameters required to define the optimization problem and the ease of assigning an appropriate value to those parameters. Versatility is the applicability of the method to a wide range of uses e.g. planar to three-dimensional or single-axis to multi-axis flexures. The total computational effort to obtain an optimized design in a nested analysis and design process depends on the number of design iterations and the effort per design iteration.

The number of design iterations is highly dependent on the ease of solving the resulting optimization problem and, thereto, the complexity (from an optimization point of view) of the optimization problem formulation. The main contribution to the computational effort per design iteration is the number of analyses and their expense, such as a Finite Element Analysis (FEA). The effort of an analysis can be predominantly separated in the effort of the preconditioning/factorization (most expensive) and the iterative solve(s)/back-substitution(s), for iterative and direct solution approaches, respectively. By focusing on the chosen quality criteria, relevant criteria such as the control of range of motion, feature size, compliance distribution, stress levels and parasitic motion (e.g. change of rotational center), although relevant, are not considered in this comparison.

Below different approaches to flexure design using TO are discussed from the perspective of the aforementioned quality criteria. The aim of the discussion is to provide a concise overview of the field and build the argumentation for the present work. Thereto, in-depth review and/or comparison is out of the scope of this work. For a detailed description of the formulations the reader is referred to the relevant contributions, as presented in the first column of Table 1. This table presents quantifiable measures of the quality criteria for a set of distinct topology optimization problem formulations. The following discussion adopts a categorization in kinetostatic and kinetoelastic formulations as known in the art.

Kinetostatic formulations

Naturally, TO problem formulations for flexure design find their origin in the field of compliant mechanism design. Kinetostatics deals with the determination of forces that act upon the elements of a mechanism, given the mechanical system acts as a static construction. The so-called kinetostatic formulations, in one form or another, simultaneously aim to maximize the energy transmission between the input and output ports and mechanism’s structural stiffness. The mechanism performance is generally quantified using the concepts of mechanical advantage, geometric advantage, mechanical efficiency, flexibility-stiffness or mutual potential energy. Although there is no single universally accepted formulation, it has been shown that these formulations produce almost similar topologies for the optimized compliant mechanisms, viz. these topologies tend to emulate their rigid-body counterpart.

Derived from these kinetostatic formulations for compliant mechanism design, in the art straightforward approaches are proposed for designing planar single- axis (prismatic and/or revolute) joints, taking into account axis drift. In the art, the formulation has been extended to account for geometric nonlinearity, stress constraints, stress and compliance distribution, and prescribed stiffness characteristics. The formulation is utilized in the art for the topology optimization of application specific flexures.

In the art is additionally proposed a simple and intuitive TO formulation aimed at the synthesis of leaf-springs using only strain-based measures. As a result, the formulation is simple and computationally efficient.

In the art it is common to derive the topology optimization response functions from design requirements of their application, such as the design of a structural flexure for force sensing in a wind tunnel balance, the design of flexures for mounting of mirrors, and the redesign of flexural hinges for compliant mechanisms.

Kinetoelastic formulations

As opposed to kinetostatic formulations, kinetoelastic formulations consider the mechanism’s kinematic functions as an integral part of the elastic properties of the continuum structure and seek to find compliant mechanisms with desirable intrinsic properties. This is, thus far, accomplished by shaping the mechanism stiffness matrix entries. The mechanism stiffness matrix is obtained by static condensation of the global stiffness matrix to a small set of nodal displacements that can describe the MPs.

The formulation may be effectively applied to the design of planar prismatic joints, and revolute joints.

From a shape-morphing design philosophy, in the art, a kinetoelastic formulation is proposed to design compliant mechanisms with selective compliance by shaping the modal properties of the mechanism stiffness matrix (i.e. eigenmodes and eigenvalues). Compliant mechanisms with selective compliance combine the advantages of both lumped and distributed compliance, that is reduced stress concentrations and a distributed deformation pattern, while preserving defined kinematics. The method has been improved upon, and extended to multiple mechanism DOFs in the art.

The kinetoelastic formulations in the art use static condensation to obtain the mechanism stiffness matrix. This procedure requires an expensive analysis which scales with the number of nodal displacements required to describe the MPs. Thereto, this is highly efficient for problems like single-input—single-output compliant mechanisms, for which the MP can, generally, be described using only two nodal displacements. However, for the aforementioned problem formulations, a vast number of nodal displacements are required to describe the MPs. As a result, applying static condensation (without further adaptation) to flexure design would generally require substantial high computational effort.

Concluding remarks

Despite the attention devoted to TO of flexures, the previously proposed formulations have disadvantages and pose challenges, see also Table 1. The kinetostatic formulations are straightforward but tend to be specific for a small set of flexures. In contrast, the kinetoelastic formulations are versatile, however are generally more complex to implement. Several formulations include responses that depend highly nonlinear on the nodal displacements or make use of artificial stiffness and additional user-defined parameters, that can make application difficult. Some show inferior convergence properties (many iterations or oscillatory behavior) and/or deliver non-binary (and hence non-manufacturable) topologies due to absence of conflicting requirements [11,13]. Finally, some formulations require substantial computational effort, which makes application of the method unpracticable. To conclude; none of the previously proposed formulations is simple to understand, implement and use as well as versatile and computationally efficient. 3. Method

Consider a structure within a bounded domain £2, made of an isotropic linear elastic material. For simplicity of explanation, we discretize the domain in a structured grid of N finite elements (nelx x nely) with a total of n nodal displacements, as sketched in Fig. 2(a). Let us define a set M', consisting of unique free and constrained

MPs. For example, consider the set M = {tx, ty, rz}, in line with the 2D problem depicted in Fig. 2(b), Fig. 2(c) and Fig. 2(d), respectively. These MPs define prescribed nodal displacements at the interfaces between rigid link and flexure {e.g. a unit displacement in x-direction between top and bottom interfaces for mechanism degree tx). The assumption that these interfaces are rigid is valid if the links can be considered much stiffer compared to the flexure. As such, the MPs correspond to the relative rigid body motions of the interfaces.

Fig. 2a shows two rigid links (gray) connected via a flexure (white). The flexure geometry is discretized using nelx x nely finite elements. The interface nodes used to prescribe the MPs are here denoted by circles (¢). Those nodal displacements are used to prescribe different MPs. Fig.2(b), (¢) and (d) show three different MPs commonly used in 2D flexure design; Fig.2(b) shows relative translation along the x- axis (tx). Fig.2(c) shows relative translation along y-axis (ty). Fig.2(d) shows relative rotation about the z-axis (rz).

We define subset C c M that contains the constrained MPs, and subset F =

M\C that contains the free MPs. In line with the primary design requirement for short- stroke flexures, we aim to maximize the stiffness of the MPs in C, while constraining the maximum stiffness of the MPs in F.

The present work uses the strain energy of the MPs as a measure for stiffness. In contrast to the traditional compliance minimization under applied loads, minimization of strain energy under prescribed displacements results in minimization of corresponding stiffness. That is, stiffness maximization under applied loads equates to minimization of corresponding displacements, whereas under prescribed displacements this equates to maximization of corresponding reaction loads. 3.1. Optimization problem formulation

The proposed constrained nonlinear optimization problem formulation for flexure synthesis now simply reads: minimize x f|E [x]], {eC

P= [a to &lx]<E, jeF ;and xe XN where the dimensionless objective f € R* is a monotonically increasing function of strain energies £‚; € R* and , € R* is the maximum allowable strain energy of MP j.

The topology is described by N continuous differentiable design variables x with its components in X:={x e R| 0 <x < 1}.

The strain energy of MP i in a discretized setting is defined as

Ex] = yu, Kix, where K[x]u; € R™" is the design dependent symmetric stiffness matrix and u; € R" contains the nodal displacements of MP i. These nodal displacements are obtained by analysis of the structural behavior, described by n linear governing equations

Klxlu;,=f,vieM where f; € R” are the nodal loads of MP i. To calculate u;, we partition this equation as w= 17

Kpp Kppl [Up foi where u; are the free nodal displacements, u, the prescribed nodal displacements, fr: the applied nodal loads and f,, ; the nodal reaction loads of MP i. As mentioned, the MPs are defined purely in terms of prescribed nodal displacements at the interfaces, without additional applied loads. Hence, the applied loads f; = 0 in all cases. The solutions to the equation above, u;; can be obtained by solving the system of linear equations

Ky, = Kppu, Vi € M

The allowable strain energies of the free MPs are the primary design requirement. These energies are generally known from system requirements, either directly as energy term or indirectly as stiffness. The maximum allowable strain energy can be approximated by the desired free MP stiffness via simple one-dimensional equivalent models. For example, consider the MP ty from Fig. 2 with a known desirable stiffness k and a prescribed relative displacement zu between the interfaces. Then, the maximum allowable strain energy may be approximated by for ~ ~ku?. The desired stiffness can, if unknown, be derived from the required stroke for a given maximum actuation force or vice versa from the required actuation force for a given stroke.

Sensitivity analysis

TO generally requires the consecutive calculation of structural responses (objectives or constraints) and their sensitivity to the design variables. Both generally involve one or multiple computationally expensive FEA. For specific optimization responses — for example strain-energy — the problem becomes so-called ‘self-adjoint’.

In self-adjoint problems, the loading terms of the analyses required to obtain the structural response and sensitivity information are linearly dependent. As a result, the computational cost of the sensitivity analysis reduces dramatically. Note that this advantage is only applicable to the linear case, which is the focus of this study. All responses in P are self-adjoint. As a result, the sensitivities can be calculated based on available information. In addition, the sensitivities are separable, i.e. each design variable contributes solely via its elemental strain energy. Thereto, one may write = = ij [ole with €; ; € R* and y; ; € R the elemental strain energy and multiplication factor of element j due to degree i. The interpretation and derivation of y, ; is further explained in Section 4.

As a consequence of the prescribed displacements scenarios — in contrast to ‘classic’ compliance minimization under applied loads — the sensitivities of strain energy, and thus the constraints, have a strictly positive sign. Intuitively this can be understood by the increase of an elastic body's strain energy under prescribed deformation upon increase of the Young's modulus. The sensitivities of the objective have a strictly negative sign due to the reformulation of a maximization problem to a minimization problem (max f equates to min -f). 3.2. Formulation properties

The contribution of this work is on the use of energy-based response functions under prescribed displacement conditions and the manner in which these response functions are combined to form the optimization problem formulation. The resulting advantageous properties of the optimization problem formulation pose benefits in terms of simplicity, versatility and computational efficiency.

Simplicity

The simplicity and effectiveness of the formulation is directly related to the similarity with the primary used ‘compliance minimization’ TO problem formulation in the art. The objective and constraints are monotonic functions with strictly opposite sign of design sensitivities, which proves a well-defined optimization problem. This results in a, relative to the state-of-the-art, easy to solve optimization problem with good convergence properties if a standard optimization is applied. The constraint(s) take over the ‘role’ of the volume constraint known in the art to provide auto- penalization of design variables with intermediate values, which is evidenced in binary topologies.

The formulation requires a minimal number of independent parameters to define the optimization problem (only maximum strain energies of the free MPs), simplifying its use and circumventing the common ‘trial-and-error’ approach towards parameter value selection.

The formulation is uniquely based on strain energy measures. This makes implementation in/in combination with commercial FEA software packages simple, as such packages generally make this data accessible for the user. Since element strain energies (or elemental stiffness matrices in combination with the nodal displacements) are common output data in commercial finite element analysis software, also the sensitivity analysis is straightforward to implement, even when using software packages that do not already provide sensitivity information.

Most of the existing formulations share one of the above advantageous properties, see Table 1. However, none of the formulations simultaneously show ease of implementation and use, a minimal number of parameters, and a well-defined and easy to solve optimization problem (demonstrated by fast and smooth convergence).

Versatility

All response functions in the problem formulation are of the same form; that is strain energy measured under prescribed displacement conditions. Note that, due to this generality of the method, the problem formulation can include one or multiple constrained MPs in the objective while constraining the stiffness of one or multiple free

MPs. As such, the formulation can be used to design both single-axis as well as multi- axis flexures.

What is more, the MPs are defined by the user, and are not restricted to specific geometries, design domains or applications. Although the proposed optimization problem formulation is relatively simple and only involves strain energy contributions from the considered MPs, it is thus effective in generating many types of flexures, as will be shown in Section 5.

Computational effort

Independent of the number of MPs, the formulation requires a single factorization/preconditioning step plus one back-substitution/iterative solve per MP.

Thus the majority of the computational effort does not scale with the number of MPs.

What is more, the optimization problem is self-adjoint and obtaining the sensitivity information requires negligible computational effort. Note that multiple formulations share this computational efficiency, as shown by the comparison in Table 1. 4. Implementation

Independent of the problem formulation as presented, the user has to consider, select and implement a variety of methods to effectively use the formulation in a TO setting. Without loss of generality, the following aids in the consideration and implementation of design parametrization, filtering, material interpolation, response formulation and gradient-based optimization. All numerical examples employ the implementation choices described here. The default constants used in the examples, as implemented in the attached code, are listed in Table 2 here below.

Symbol Description ~~ Value e Stiffnessrato =~ 10°

Vv Poisson ratio 0.3 p SIMP penalty 3.0 r Filter radius (no. elements) 2.0 € Maximum design change 1073 x° Homogeneous initial design 0.5 ~~ Table 2: Constant parameters and assigned values.

For the Finite Element Analysis (FEA), we opt for standard 4-node quadrilateral (2D) and 8-node hexahedral elements (3D) in structured meshes. The design domain is parametrized by assignment of a design variable x; € y to each finite element i, which allows for local control of the material properties.

It is generally recognized that both final design and performance are sensitive to the initial design x. This is especially the case for compliant mechanisms, and thus also for flexure optimization. We consider this influence out of the scope of this paper and thereto opt for the in the art known homogeneous initial design.

To eliminate modeling artifacts, the design variable field is generally blurred as to obtain the filtered field ¥ € zy using a linear filtering operation K[x]: y¥ — x" with relative filter radius r € R*. This operation is also accounted for in the sensitivity calculation, as described in the cited reference.

Asymmetric topologies resulting from problems with symmetric boundary conditions is, although not often explicitly reported, common and inevitable. The gradient-based optimizer solves many independent convex problems until a finite convergence criterion is met. As a result, round-off errors are inevitable. This leads, in most cases, to divergence from the symmetric local optimum. One may easily enforce symmetry by linking design variables over one or multiple axes; either by creating a dependency or by averaging.

The Young's modulus of an element is related to the filtered design variable via an element-wise composite rule, that is

Ex] = € + (1-— S)RIZ]

E with E the material Young's modulus, ¢ the relative stiffness between solid and void and R the material interpolation function. We apply modified Solid Isotropic

Material with Penalization (SIMP) interpolation function as may be known in the art, that is

R[x] = x? with p € R* a user definable parameter. It may be known in the art that this interpolation function increases the probability to obtain a 0/1 solution of a strain-based optimization problem. Note that, as a result, the elemental multiplication factor is simply obtained via vi; i= (1 - 5

It may be beneficial to scale the objective such that it holds a reasonable value (as compared to the constraints). We opt to normalize the strain energy of degree i to its strain energy at the first optimization iteration, that is ait &; with a the relative strain energy and k the optimization iteration counter.

Note that, as a result of the above equation, the normalized strain energy « is a dimensionless positive scalar value by definition and a” = 1 for all i.

In order to simultaneously maximize the stiffness of multiple [DOCs ] MPs, the corresponding normalized strain energies are combined in a monotonically increasing function. We opt here for a simple summation, that is the objective at iteration count k yields ici file] = > alle] i

With |C| the DOCs. One might, in addition, add weight factors to the individual strain energy measures to control relative importance or opt for a smooth minimum function. Note that, as a consequence of the second above equation, the magnitude of the prescribed displacements for different constrained MPs become irrelevant.

The gradient-based inequality-constrained nonlinear optimization problem

Pis solved in a nested analysis and design setting. The design variables are iteratively updated by a sequential approximate optimization scheme, as may be known in the art of the topology optimization field. The Method of Moving Asymptotes (MMA) as may be known in the art is used. The resulting convex sub-problems are solved using a primal-dual interior point method. The optimization is terminated when the maximum design change is smaller than €.

This work may include a MATLAB code to design 2D single and multi-axis flexures, which is discussed in more detail in Section 6. We provide a briefly introduction here to allow the reader to understand how to replicate the results in upcoming sections. The code is an extension of the top/1.m code by Andreassen et al. known in the art, and can be called using a similar syntax, that is flexure(nelx, nely, doc, dof, emax) where doc and dof are lists of strings of the names of the desired constrained and free MPs, respectively. Parameter emax is a list of maximum allowable strain energies corresponding to the free MPs in dof. 5. Numerical examples

In order to demonstrate the method proposed in Section 3, we apply it to common problems for which results have been reported in literature. Thereto, we introduce a set D{ tx, ty, rz } consisting of the three rigid body MPs of the rigid links; two relative translations and rotation around the center of the flexure, see Fig. 2. A sketch of the deformed structure resulting from the prescribed MPs for x!® are shown in Fig. 2.

Note that, without adjusting the formulation, any other set of unique MPs may be used.

Fig. 3. Generated design for various varieties of problem formulations based on P. This includes both single and multi-axis mechanisms for 2D topologies. These designs are generated by flexure(200,200,doc,dof,emax), with doc and dof given in the subcaptions and the maximum strain energies in emax equal for all cases. Note that we have omitted the string signature (e.g. "tx") here for simplicity.

Fig. 3 show the resulting topologies for a variety of planar design cases.

Primitive topologies for the design of a compliant prismatic and revolute joint are shown in Fig. 3(a) and Fig. 3(c) respectively. The results are as expected and fully in accordance to the results obtained by both synthesis methods and topology optimization formulations.

Complex topologies appear for different less common design cases, such as those shown in Figs. 3(d)-3(f). These results, to the best knowledge of the authors, have not been reported in literature. Increasing the DOCs and/or DOFs generally results in more complex (number of bodies and rotation points) and innovative topologies, see Figs. 3(e) and 3(f). The convergence history of the responses for a planar optimization problem with representative set of input parameters is shown in

Fig. 4. For feasible input parameters, the convergence history of problem P is characterized by a quickly active and satisfied constraints and a steadily and smoothly increasing performance, converging within a limited number of iterations.

Fig. 4. Characteristic convergence history of problem formulation P: objective f and constraint g as a function of design iteration k. Note the objective is relative with respect to the first iteration, i.e. f{% = 1. This specific convergence plot is generated by flexure(100,100,tx,ty, 1.2).

Fig. 5. High-resolution 3D flexure designs generated using a C++ implementation. The number of MPs allows for a high number of variations. From left to right: the topology as a result of solving P for the given set of constrained and free

MPs, and corresponding deformed topology for a free and constrained MP. Note that the deformations are highly scaled for visualization purposes.

High-resolution 3D topologies are presented in Fig. 5. Those topologies are examples of the high variety of designs that can be obtained based on the set of rigid body MPs in a 3D space.

The resulting topologies validate the correct working principle of the proposed formulation. In addition, it shows optimized flexures have relatively small features with highly lumped strain energy, see e.g. Fig. 3(c). As a result, those flexures have a small range of motion limited by the critical stress and their performance is sensitive to manufacturing errors.

In order to practically use the resulting designs, the maximum allowable stress as well as manufacturing uncertainties should be taken into consideration. In the following we will show the possibilities of limiting stress levels and/or introducing manufacturing robustness in the formulation, without aiming to provide a thorough investigation of design parameters. To this end, we use the resulting design from Fig. 3(d) as a reference. We denote its objective by f° and corresponding maximum stress by 7°.

Fault-tolerant design

The desired kinematics of a flexure are sensitive to both uniform and spatially varying geometric deviations. However, in classical deterministic topology optimization, the effect of such uncertain parameters on the performance of the structure is not taken into account. This may lead to a design that is very sensitive to manufacturing errors. As a consequence, the performance of the actual structure may be far from optimal. In the art a robust approach to topology optimization where the effect of uniform manufacturing errors may be taken into account. Uniform erosion and dilation effects, from here on denoted by superscripts (e) and (d), are simulated by means of a projection method: the filtering of the design variable field is followed by a differentiable Heaviside projection using a high projection threshold n° = 7 + An to simulate an erosion and a low projection threshold 17% = 7 — 47 to simulate a dilation.

An additional advantage of the robust formulation is the direct control of the minimum feature size of both solid and void.

For the robust design of flexures, only a slight difference of the earlier equation is required, that is minimize x —fle:xe]], {eC

Pp, = [rai to Ei [x4] < €, ieF

AEX

With £[x¢] and €[x?] strain energies based on the eroded and dilated fields, respectively.

Since the eroded and dilated designs will always hold the maximum and minimum strain energies, respectively, the intermediate design can be excluded from the optimization formulation without compromising robustness in terms of length scale control. This allows to, partially, reduce the added cost of the robust formulation. Note that all responses still only involve self-adjoint strain energy terms.

Fig. 6. Resulting design of the topology optimization problem from Fig. 3(d) extended with (a) the robust formulation, and (b) stress constraints.

Fig. 6(a) Design obtained by solving F,, see the equation above. The eroded and dilated designs are indicated by, respectively, green and red contour lines.

The design is robust with respect to uniform manufacturing errors and satisfies a minimum feature size (both solid and void). However, the DOC stiffness is decreased with 68% as compared to the design in Fig. 3a.

Fig. 6(b) Design obtained by solving P;, see the equation below. Major change in topology can be observed as compared to Fig 3a. As a result of the applied stress constraints the range of motion is extended, with just 8% decrease of DOC stiffness as compared to the design in Fig 3a

Fig. 6(a) shows the resulting designs of an optimization problem with filter radius r = 4 finite elements, 7 = 0.5 and An = 0.2. The robustness poses a heavy restriction on the achievable performance, as can be observed by the decrease in performance, i.e. f = 0.32 x f° The hinges are clearly lengthened, thus distributing the strain energy over larger areas of the topology. In line with this observation, it may be shown that it is possible to indirectly achieve stress-constrained topological design via length scale control. Note the non-intuitive presence of protrusions along the center horizontal axis. Upon further investigation, it is observed that those do not add stiffness to the free MPs, whilst contributing some (although little) stiffness to the constrained

MPs. Considering this lack of sensitivity, those are expected to be removed first upon, for example, introduction of a volume constraint.

Stress-based design

In order to limit the maximum stress for a given range of motion, or similarly extend the range of motion for a given maximum stress, one can simply extend the problem formulation P with stress constraints on the free MPs, which yields = x f|E, [x]], ieC p — /subjectto Ex] <E, [EF gs; lo]] <5, i€F x €X where a; are the elemental stresses obtained by prescribing free MP i, and 5 the maximum allowable stress, based on some theory of failure. To evaluate stress constraints, elemental strain energies are no longer usable. Many different formulations of g, are known in the art. Without loss of generality, we use a unified aggregation and relaxation approach.

Fig. 6(b) shows the resulting design of an optimization problems with = 0.4 x 3° The stress constraints are satisfied by introduction of (more) hinges with a more distributed deformation energy. Although the maximum stress is drastically reduced, the introduction of stress constraints has a relative limited impact on the performance decreases, namely f = 0.92 x f° This demonstrates that the proposed formulation can effectively be extended with stress constraints, yet a thorough investigation thereof is considered out of the scope of this work. 6. Replication of results

The supplementary material includes a MATLAB function (.m-file) that is provided to replicate the 2D single and multi-axis flexure designs from Fig. 3 and to use as a basis for further research. The function file flexure.m is based on the top71.m by Andreassen et al. and can be called using flexure(nelx,nely,doc, dof, emax).

Here nelx and nely define the dimensions of the rectangular design domain in terms of number of elements in x and y-direction. Both doc and dof are arrays containing the strings of names of the constrained and free MPs, respectively.

Parameter emax is an array of maximum allowable strain energies corresponding to the MPs in dof. Further explanation on the code can be found in Appendix A.

To further encourage use of the proposed method and to show its versatility and ease of implementation, the material in addition contains both 2D and 3D implementations of some more unusual geometries in commercial FEA package

COMSOL Multiphysics (.mph-file).

Computer Aided Design (CAD) models of the 3D designs presented in Fig. 5 (and some more) are available in STL format (.stl-file) for the purpose of additive manufacturing. 7. Discussion

Before concluding, a reflection on the formulation in light of existing methods, the limitations and related future work and potential applications is in order.

Although dissimilar in formulation and implementation, in the art implementation may be found sharing the same kinetoelastic design philosophy, resulting in designs with selective compliance. Thereto, the current invention can be considered a simplification.

The introduction of independent MPs allows to easily perform design variations such as multi-axis flexures (e.g. as demonstrated in Figs. 3 and 5) and straightforward adaptation to specific applications.

While clear advantages can be identified, the proposed formulation is not without limitations. As presented here, it is intended for and limited to the design of short-stroke flexures, i.e. satisfying linear strain-displacement and stress-strain relationships. However, the prototyped samples indicate many of the resulting topologies can be used effectively in a finite range of motion. In the art an analysis is made for the finite range of motion behaviour for a subset of the prototyped flexures using a novel nonlinear eigenmode analysis technique. Results indeed indicate the flexures retain their predicted properties at least for a small finite range. As expected, for a large range of motion, stiffness properties deviate substantially. Considering both the need for and interest in large-range compliant joints, e.g. compliant implants, the authors pursue a study to extend the proposed formulation to design for long-stroke flexures.

This application does not comprise all possible variations building on this formulation, e.g. change of geometry, MPs or objective function. To that, we stress that, in our experience, the formulation does not pose limitations for shape-morphing or compliant mechanism design. As such, design studies and variations of the formulation to applications mentioned above are considered comprised in the claims.

The ease of implementation and use, versatility and modest computational effort of the proposed optimization problem formulation poses a high potential of the formulation to be used for designing an intervertebral cage as claimed. 8. Conclusion

Despite the extensive efforts devoted to research in the field of TO and the need for an effective tool to synthesize flexures, such as for an intervertebral cage as claimed, a generally accepted TO problem formulation for short-stroke flexure synthesis has been absent so far. Motivated by this, we propose a simple, versatile and computationally efficient topology optimization problem formulation. Using motion patterns this strain energy based formulation simplifies understanding and implementation of TO synthesis of flexures, and features low computation cost, smooth convergence to well-defined designs, and a minimum of tuning parameters. The base formulation is easily extended with additional design requirements and maintains its favorable properties. Although designed for short-stroke applications, the resulting 3D designs prove practically useful within a small finite range of motion. With source code provided to replicate the demonstrated results, this formulation is ready to be further explored and applied in academia and industry.

Degree ~~ No. da ~~ ~*~ 1 + 0 ty 2 0 1 rz 3 1 1 - 2(x/nelx)

Table A.3: Motion pattern numbering and prescribed nodal displacements of the top interface.

Elaboration on code

This section elaborates on the function file flexure.m. The code is separated in multiple sections, starting with a section title (e.g. %% SENSITIVITY ANALYSIS) followed by contiguous lines of code. As introduced in Sections 4 and 6, the function can be called via: flexure(nelx, nely, doc, dof, emax)

Some possible variations are: flexure(100, 100, "tx", "ty", 1) flexure(200, 100, ["tx","rz"], "ty", 1) flexure(100, 120, "rz", ["tx","ty"], [1,0.1])

Section PREPROCESSING collects and converts the user input. MPs tx, ty and rz are related to displacement fields 1, 2 and 3, respectively. The section asserts the doc and dof are appropriate {no overlap, at least one constrained and one free

MP).

The generation of mesh (PREPARE MESH), preparation of FEA (PREPARE FEA) and preparation of density filter (PREPARE FILTER) are equivalent to the top71.m code.

Section BOUNDARY CONDITILNS apply the BC (prescribed displacements) for the three MPs. Instead of prescribing the nodal displacements of top and bottom nodes, we have opted to fix the nodal displacements of bottom nodes, while prescribing the nodal displacements of top nodes. In line with Fig. 2 the displacements in x-direction (u) and y-direction (v) are found in Table A.3. Note that v of degree rz is a linear decaying function of the location in x-direction (zero in the middle). The nodal displacements are separated in fixed and free for the purpose of partitioning (see the equation above).

The design variable field is symmetrized in Section FORCE SYMMETRY to eliminate any round off errors that might, ultimately, lead to asymmetric designs. Upon commenting of the aforementioned section and using a random initial design one can explore synthesis of asymmetric designs.

The filtering of design variables (DENSITY FILTER), interpolation of material Young's Modulus (MATERIAL INTERPOLATION) and stiffness matrix assembly routine (STIFFNESS MATRIX ASSEMBLY) are equivalent to the top71.m code. Note the stiffness matrix is symmetrized (eliminate any round-off errors) to ensure the preferred solver (Cholesky factorization) is used.

Section SOLVE SYSTEM OF EQUATIONS contains solving the system of equations and calculation of MP strains in line with the equations above. To limit computational cost, the calculations are performed only on active MPs, i.e. if a MP is not in doc or dof the corresponding nodal displacement field is not solved for.

The objective and constraint(s) are calculated in Section RESPONSES.

Note the objective is normalized with respect to its value in first iteration, see equation above, and hence is dimensionless. Corresponding sensitivities (SENSITIVITY

ANALYSIS) are in good accordance with the code in top71.m. If required/preferred a volume constraint can be added to the optimization problem formulation by uncommenting Section VOLUME CONSTRAINT.

The optimization routine provided in the code (DESIGN OPTIMIZATION

STEP) consists of three subsequent steps, that is « determination of the variable bounds via movelimit, » generation of an approximated subproblem using the supplied approx.m file, and + solving the strictly convex approximated subproblem using MATLAB’s fmincon interior-point algorithm.

The movelimit strategy detects oscillations of design variables and increases/decreases the movelimit accordingly. The behavior of the movelimit strategy can be adapted by changing mlinit, mlincr and mldecr.

For computational efficiency and improved convergence behavior it is recommended to substitute the provided optimization routine with the original MMA by

Svanberg. For simplicity the function call is added as a comment on Section

ORIGINAL MMA. Section TERMINATION CRITERIA calculates the norm of the KKT conditions as well as mean variable change, followed by Section VARIABLE UPDATE handling history information.

Finally, in Section PRINT RESULTS and PLOT DESIGN the performance measures (objective, constraints, mean variable change, KKT norm and strain energies) are printed to command window.

Section TERMINATION checks constraint feasibility as well as design variable and KKT norm change with respect to user defined tolerances. If satisfied, the optimization problem is terminated.

Figure 7 shows an image of a prototyped computational designed intervertebral cage. When the topology optimization is executed with a different parameter, and/or a different boundary condition the shape or form of the designed intervertebral cage may be different or even radically different, while still providing an intervertebral cage complying to the general requirements as set out in this description.

Thus, not only the shown intervertebral cage is claimed, but also all variations resulting from the topology optimization method resulting in an intervertebral cage complying to the requirements set out.

Figure 8 schematically shows the intervertebral cage arranged between two vertebras and the definition the axis, translations and rotations. The X, Y and Z axes are shown. The translation along the different axes are indicated with arrows on the end of the axes and the respective indications TX, TY, and TZ. The rotations about the different axes are indicated with arrows around the axes and the respective indications

RX, RY, and RZ.

Figure 9 schematically shows a computational design of a prototyped computational designed intervertebral cage. The intervertebral cage is shown in a perspective view. The intervertebral cage shown in Figure 9 is also shown in Figure 7.

Figures 10(a-i) schematically show a computational design of a prototyped computational designed intervertebral cage. The designed intervertebral cage is symmetric to the X plane, Y plane and Z plane. For this reason, only one-eighth of the intervertebral cage is shown in the Figures 10(a-c). Figure 10(d) shows a perspective view of the intervertebral cage along the X-axis. Figure 10(e) shows a cross-section view of the intervertebral cage halfway along and perpendicular to the X-axis. Figure 10(f) shows a perspective view of the intervertebral cage along the Y-axis. Figure 10(g) shows a cross-section view of the intervertebral cage halfway along and perpendicular to the Y-axis. Figure 10(h) shows a cross-section view of the intervertebral cage halfway along and perpendicular to the Z-axis. Figure 10(i) shows a perspective view of the intervertebral cage along the Z-axis.

Figure 11 schematically shows an embodiment of a computer program product 1000, computer readable medium 1010 and/or non-transitory computer readable storage medium according to the invention comprising computer readable code 1020. The compounding system typically comprises a controller arranged for executing one or more of the methods as specified throughout the description and claims as typically coded in software.

Figures 12(a-n) schematically show the intervertebral cage in several deformed conditions. The intervertebral cage in Figures 12(a-n) in deformed condition is also shown in Figures 10(a-i), Figure 9, Figure 8, and Figure 7.

Figures 12(a) schematically shows the intervertebral cage stretched along the Z axis and viewed along the Y-axis.

Figures 12(b) schematically shows the intervertebral cage stretched along the Z axis and viewed along the X-axis.

Figures 12(c) schematically shows the intervertebral cage compressed along the Z axis and viewed along the Y-axis.

Figures 12(d) schematically shows the intervertebral cage compressed along the Z axis and viewed along the X-axis.

Figures 12(e) schematically shows the intervertebral cage rotated about the

X-axis and viewed along the X-axis.

Figures 12(f) schematically shows the intervertebral cage rotated about the

X-axis and viewed along the Y-axis.

Figures 12(g) schematically shows the intervertebral cage rotated about the

Y-axis and viewed along the Y-axis.

Figures 12(h) schematically shows the intervertebral cage rotated about the

Y-axis and viewed along the X-axis.

Figures 12(i) schematically shows the intervertebral cage rotated about the

Z-axis and viewed along the Z-axis.

Figures 12(j) schematically shows the intervertebral cage rotated about the

Z-axis and viewed along the X-axis.

Figures 12(k) schematically shows the intervertebral cage translated along the X-axis and viewed along the Y-axis.

Figures 12(l) schematically shows the intervertebral cage translated along the X-axis and viewed along the X-axis.

Figures 12(m) schematically shows the intervertebral cage translated along the Y-axis and viewed along the X-axis.

Figures 12(n) schematically shows the intervertebral cage translated along the Y-axis and viewed along the Y-axis.

It will also be clear that the above description and drawings are included to illustrate some embodiments of the invention, and not to limit the scope of protection.

Starting from this disclosure, many more embodiments will be evident to a skilled person without departing from the scope of the invention as set forth in the appended claims. These embodiments are within the scope of protection and the essence of this invention and are obvious combinations of prior art techniques and the disclosure of this patent. Devices functionally forming separate devices may be integrated in a single physical device.

The term “substantially” herein, such as in “substantially all emission” or in “substantially consists”, will be understood by the person skilled in the art. The term “substantially” may also include embodiments with “entirely”, “completely”, “all”, etc.

Hence, in embodiments the adjective substantially may also be removed. Where applicable, the term “substantially” may also relate to 90% or higher, such as 95% or higher, especially 99% or higher, even more especially 99.5% or higher, including 100%. The term “comprise” also includes embodiments wherein the term “comprises” means “consists of”.

The term "functionally" will be understood by, and be clear to, a person skilled in the art. The term “substantially” as well as “functionally” may also include embodiments with “entirely”, “completely”, “all”, etc. Hence, in embodiments the adjective functionally may also be removed. When used, for instance in “functionally parallel’, a skilled person will understand that the adjective “functionally” includes the term substantially as explained above. Functionally in particular is to be understood to include a configuration of features that allows these features to function as if the adjective “functionally” was not present. The term “functionally” is intended to cover variations in the feature to which it refers, and which variations are such that in the functional use of the feature, possibly in combination with other features it relates to in the invention, that combination of features is able to operate or function. For instance, if an antenna is functionally coupled or functionally connected to a communication device, received electromagnetic signals that are receives by the antenna can be used by the communication device. The word “functionally” as for instance used in “functionally parallel” is used to cover exactly parallel, but also the embodiments that are covered by the word “substantially” explained above. For instance, “functionally parallel” relates to embodiments that in operation function as if the parts are for instance parallel. This covers embodiments for which it is clear to a skilled person that it operates within its intended field of use as if it were parallel.

Furthermore, the terms first, second, third and the like in the description and in the claims, are used for distinguishing between similar elements and not necessarily for describing a sequential or chronological order. It is to be understood that the terms so used are interchangeable under appropriate circumstances and that the embodiments of the invention described herein are capable of operation in other sequences than described or illustrated herein. Thus, these terms are not necessarily intended to indicate temporal or other prioritization of such elements.

The devices or apparatus herein are amongst others described during operation. As will be clear to the person skilled in the art, the invention is not limited to methods of operation or devices in operation.

It should be noted that the above-mentioned embodiments illustrate rather than limit the invention, and that those skilled in the art will be able to design many alternative embodiments without departing from the scope of the appended claims. In the claims, any reference signs placed between parentheses shall not be construed as limiting the claim. Use of the verb "to comprise" and “to include”, and its conjugations does not exclude the presence of elements or steps other than those stated in a claim.

Also, the use of introductory phrases such as “at least one” and “one or more” in the claims should not be construed to imply that the introduction of another claim element by the indefinite articles "a" or "an" limits any particular claim containing such introduced claim element to inventions containing only one such element, even when the same claim includes the introductory phrases "one or more" or "at least one" and indefinite articles such as "a" or "an." The article "a" or "an" preceding an element does not exclude the presence of a plurality of such elements.

The invention may be implemented by means of hardware comprising several distinct elements, and by means of a suitably programmed computer. In the device or apparatus claims enumerating several means, several of these means may be embodied by one and the same item of hardware. The mere fact that certain measures are recited in mutually different dependent claims does not indicate that a combination of these measures cannot be used to advantage.

The invention further applies to an apparatus or device comprising one or more of the characterising features described in the description and/or shown in the attached drawings. The invention further pertains to a method or process comprising one or more of the characterising features described in the description and/or shown in the attached drawings.

It will be appreciated that the invention also applies to computer programs, particularly computer programs on or in a carrier, adapted to put the invention into practice. The program may be in the form of a source code, a code intermediate source and an object code such as in a partially compiled form, or in any other form suitable for use in the implementation of the method according to the invention. It will also be appreciated that such a program may have many different architectural designs. For example, a program code implementing the functionality of the method or system according to the invention may be sub-divided into one or more sub-routines.

Many different ways of distributing the functionality among these sub-routines will be apparent to the skilled person. The sub-routines may be stored together in one executable file to form a self-contained program. Such an executable file may comprise computer-executable instructions, for example, processor instructions and/or interpreter instructions (e.g. Java interpreter instructions). Alternatively, one or more or all of the sub-routines may be stored in at least one external library file and linked with a main program either statically or dynamically, e.g. at run-time. The main program contains at least one call to at least one of the sub-routines. The sub-routines may also comprise function calls to each other. An embodiment relating to a computer program product comprises computer-executable instructions corresponding to each processing stage of at least one of the methods set forth herein. These instructions may be sub- divided into sub-routines and/or stored in one or more files that may be linked statically or dynamically. Another embodiment relating to a computer program product comprises computer-executable instructions corresponding to each means of at least one of the systems and/or products set forth herein. These instructions may be sub- divided into sub-routines and/or stored in one or more files that may be linked statically or dynamically.

The carrier of a computer program may be any entity or device capable of carrying the program. For example, the carrier may include a data storage, such as a

ROM, for example, a CD ROM or a semiconductor ROM, or a magnetic recording medium, for example, a hard disk. Furthermore, the carrier may be a transmissible carrier such as an electric or optical signal, which may be conveyed via electric or optical cable or by radio or other means. When the program is embodied in such a signal, the carrier may be constituted by such a cable or other device or means.

Alternatively, the carrier may be an integrated circuit in which the program is embedded, the integrated circuit being adapted to perform, or used in the performance of, the relevant method.

The various aspects discussed in this patent can be combined in order to provide additional advantages. The mere fact that certain measures are recited in mutually different claims does not indicate that a combination of these measures cannot be used to advantage. Furthermore, some of the features can form the basis for one or more divisional applications

EMBODIMENTS

1. Computer implemented method (100) comprising the step of synthesizing the topology, the shape, and the size of an intervertebral cage for replacing an intervertebral disc in a patient, wherein the step of synthesizing is based on topology optimization. 2. Computer implemented method according to the preceding claim, wherein the intervertebral cage comprises an element, which is a monolithic element. 3. Computer implemented method according to any of the preceding claims, wherein the intervertebral cage comprises a/the element, which is a compliant element. 4. Computer implemented method according to any of the preceding claims, wherein the topology optimization is initiated with a build volume having: - a length in a lateral direction (X) in the range of 30 mm to 50 mm, preferably 35 mm to 45 mm, more preferably 37 mm to 43 mm, most preferably substantially 40 mm; - a length in a dorsal and/or a ventral direction (Y) in the range of 20 mm to 40 mm, preferably 25 mm to 35 mm, more preferably 27 mm to 33 mm, most preferably substantially 30 mm; and/or - a length in a cranial and/or a caudal direction (Z) in the range of 8 mm to 20 mm, preferably 10 mm to 18 mm, more preferably 12 mm to 16 mm, even more preferably 13 mm to 15 mm, most preferably substantially 14 mm. 5. Computer implemented method according to any of the preceding claims, wherein the topology optimization during iterating applies a maximum predefined mass value requirement for limiting the mass of the vertebral cage. 6. Computer implemented method according to any of the preceding claims, wherein the six degrees of freedom are split in a degrees of freedom subset having a low stiffness, and a degrees of constraint subset having a high stiffness; and wherein the topology optimization is performed based on the degrees of freedom subset and/or the degrees of constraint subset such that the intervertebral cage substantially provides comparable stiffness as the intervertebral disc. 7. Computer implemented method according to the preceding claim,

wherein the degrees of constraint subset comprises the translations along a lateral axis (TX), and a dorsal and/or a ventral axis (TY); and/or wherein the degrees of freedom subset comprises the translation along the cranial and/or the caudal axis (TZ), and a rotation around a lateral axis (RX), a dorsal and/or a ventral axis (RY), and a cranial and/or a caudal axis (RZ).

8. Computer implemented method according to any of the preceding claims 6-7, wherein the subset of the six degrees of freedom relating to rotation comprise:

- a maximum rotation around a lateral axis (RX) in the range of 2 degrees to 10 degrees, preferably 3 degrees to 8 degrees, more preferably 4 to 6 degrees, most preferably substantially 5 degrees;

- a maximum rotation around a dorsal and/or a ventral axis (RY) in the range of 2 degrees to 10 degrees, preferably 3 degrees to 8 degrees, more preferably 4 to 6 degrees, most preferably substantially 5 degrees; and/or

- a maximum rotation around a cranial and/or a caudal axis (RZ) in the range of

2 degrees to 10 degrees, preferably 3 degrees to 8 degrees, more preferably 4 to 6 degrees, most preferably substantially 5 degrees.

9. Computer implemented method according to any of the preceding claims 6-8, wherein the subset of the six degrees of freedom relating to translation comprises:

- a maximum translation along a lateral axis (TX) is below 2 mm, preferably 1,5 mm, more preferably 1 mm, most preferably substantially prevented;

- a maximum translation along a dorsal and/or a ventral axis (TY) is below 2 mm, preferably 1,5 mm, more preferably 1 mm, most preferably substantially prevented,

- a maximum extension along a cranial and/or a caudal axis (TZ) is below 1 mm, preferably 0,5 mm, more preferably 0,25 mm, most preferably substantially prevented, and/or

- a maximum compression along a cranial and/or a caudal axis (TZ) divided by the unloaded length in the cranial and/or the caudal direction is in range of 0% to 60%,

preferably 10% to 50%, more preferably 20% to 30%, most preferably substantially 30%. 10. Computer implemented method according to any of the preceding claims, wherein the topology optimization method comprises defining a constrained nonlinear optimization problem formulation, preferably gradient based, for synthesizing the intervertebral cage; wherein the constrained nonlinear optimization problem P is formulated as: minimize x f[U;[x]], ie €

P= aje to Ula SU, jeF ;and xE XN wherein x is the vector of N design variables; wherein X the set defines as {xin R | Xin < X < Xmac}: wherein the dimensionless objective f € R* is a function of energy measure

U; € R* such as strain energy, work, compliance, or energy derived measure such as stiffness; wherein U, € R* is a maximum allowable energy or stiffness; wherein U; is the measure of energy or stiffness of predefined motion patterns in the set C ; wherein U; is the measure of energy or stiffness of a predefined motion patterns in the set F; and wherein C and F are complementary. 11. Computer implemented method according to any of the preceding claims, wherein the translational motions have a stiffness: - between 200 N/mm and 20,000 N/mm in undeformed configuration along a medial/lateral axis (TX); - between 200 N/mm and 20,000 N/mm in undeformed configuration for along a posterior/anterior axis (TY); - between 300 N/mm and 3,000 N/mm in undeformed configuration along a cranial and/or a caudal axis (TZ). 12. Computer implemented method according to any of the preceding claims, wherein the rotational motions have a stiffness:

between 1 Nm/deg and 100 Nm/deg in undeformed configuration around a medial/lateral axis (RX); between 1 Nm/deg and 100 Nm/deg in undeformed configuration around a posterior/anterior axis (RY); and/or between 10 Nm/degree and 1000 Nm/deg in undeformed configuration around a cranial and/or a caudal axis (RZ) - between 1Nm/deg and 100 Nm/deg in undeformed configuration around a medial/lateral axis (RX); - between 1Nm/deg and 100 Nm/deg in undeformed configuration around a posterior/anterior axis (RY); - between 10 Nm/degree and 1000 Nm/deg in undeformed configuration around a cranial and/or a caudal axis (RZ). 13. Computer implemented method according to the preceding claims 11 and 12,

MAXIE Kela] and ky [x] subject to i, [x] < Ez wherein subject to Kp. [x] < Krx subject to kK, [x] S Ky subject to k,,[x] S Krz subjectto v[x]< © wherein Kz , Kz , Kz , and x, are the maximum allowable stiffness corresponding to respective degrees of freedom; and wherein 7 is the maximum material usage relative to a/the build volume. 14. Computer implemented method according to any of the preceding claims, wherein a motion pattern is preferably defined as a translation along an axis or a rotation around an axis; wherein preferably the axis are selected as a lateral axis (X), a dorsal and/or a ventral axis (Y), and/or a cranial and/or a caudal axis (Z); and wherein the topology optimization is performed such that stiffness is maximized for preselected motion patterns, and/or for the other preselected motion patterns stiffness is limited to a predefined stiffness value. 15. Computer implemented method according to any of the preceding claims, wherein a/the motion pattern is defined as a translation along an axis or a rotation around an axis;

wherein the topology optimization comprises defining a constrained nonlinear optimization problem formulation, preferably gradient based, for synthesizing the intervertebral cage; wherein the constrained nonlinear optimization problem is formulated as: minimizex —f[&;[x]], i eC

P= [i to Elst, jeF ;and xe XN wherein the dimensionless objective f € R* is a monotonically increasing function of strain energies €; € R* and €, € R* is a maximum allowable strain energy of motion pattern j. 16. Computer implemented method according to the preceding claim, wherein the topology optimization is based on a topology defined as N continuous differentiable design variables x with its components in X:={x € R| 0 < x < 1}; wherein each element i in the topology optimization has a stiffness, preferably a

Young modulus, E;; and wherein E; is a nonlinear function of design variable x;. 17. Computer implemented method according to the preceding claim, wherein

Ex] =E€+ (1-E)R[X]

E wherein E is the material Young's modulus; wherein X is the filtered design variable x; wherein £ is the relative stiffness between solid and void; and wherein R the material interpolation function. 18. Computer implemented method according to the preceding claim, wherein R[x] = x? ; wherein p € R* ; and wherein preferably p is in the range of 2 to 4, more preferably substantially 3.

19. Computer implemented method according to any of the preceding claims 15-18, wherein a strain energy of motion pattern i in a discretized setting is defined as: glx] = su; Kle] ; wherein K[x]u; € R™*" is the design dependent symmetric stiffness matrix; and wherein u; € R* comprises a nodal displacement of motion pattern i. 20. Computer implemented method according to the preceding claim, wherein the nodal displacement is defined as n linear governing equations

Klx]u, =f, vieM; wherein M is the set of all motion patterns; and wherein f; € R™ are the nodal loads of motion pattern i. 21. Computer implemented method according to the preceding claim,

K K Uys; wherein the nodal displacement may be partitioned as bi | | = ze] fp pp pt foi wherein u;,; are the free nodal displacements; wherein u,,; are the prescribed nodal displacements; wherein f; are the applied nodal loads; and wherein f,,; are the nodal reaction loads of motion pattern i. 22. Computer implemented method according to the preceding claim, wherein the applied loads f, = 0 in all cases; and wherein the step of synthesizing comprises the step of solving for the nodal displacement Kg; = Kppyu,;, Vi € M. 23. Computer implemented method according to any of the preceding claims, wherein the topology optimization problem is a gradient-based inequality-constrained nonlinear optimization problem. 24. Computer implemented method according to any of the preceding claims, wherein the topology optimization comprises the step of: - filtering a design variable field; and

- differentiable Heaviside projecting using a high projection threshold for erosion and/or a low projection threshold for dilation. 25. Computer implemented method according to any of the preceding claims, minimize flu. x1], ie wherein B, = x —~ ; and no )subjectto Ux] <U, i€F’ xe XN wherein U[x¢] and U[x“] are strain energies based on the eroded and dilated fields, respectively. 26. Computer implemented method according to any of the preceding claims, mr flu), tec wherein P, = Ux] <0, i€F :and subject to go lo] <3, ieF xe XN wherein g; are the elemental stresses obtained by the predefined motion patterns i for a predefined magnitude; and wherein 7 is a maximum allowable stress value. 27. Computer implemented method according to any of the preceding claims, wherein the topology optimization is for short stroke flexures. 28. Computer implemented method according to any of the preceding claims, comprising the step of manufacturing the intervertebral cage.

29. Computer implemented method according to the preceding claim, wherein manufacturing comprises additive manufacturing. 30. Computer implemented method according to any of the preceding claims, comprising the step of providing a data structure comprising information for manufacturing the synthesized intervertebral cage. 31. Computer implemented method according to the preceding claim, wherein the information for manufacturing comprises the density as a function of the location for the intervertebral cage, preferably for each location in a/the build volume of the intervertebral cage. 32. Computer implemented method according to the preceding claim, wherein the information for manufacturing comprises the topology, the shape and the size of the intervertebral cage, preferably for each location in a/the build volume of the intervertebral cage. 33. Intervertebral cage comprising a topology, a shape and/or a size synthesized according to claim 1-32. 34. Intervertebral cage according to the preceding claim, wherein the intervertebral cage comprises a monolithic compliant element; and wherein the monolithic compliant element comprises the topology, the shape andthe size. 35. Intervertebral cage according to any of the preceding claims 33-34, wherein the intervertebral cage comprises a monolithic compliant element; and wherein the monolithic compliant element comprises the density as a function of the location for the intervertebral cage, preferably for each location in a/the build volume of the intervertebral cage. 36. Method for manufacturing comprising the steps of: - receiving a data structure comprising information for manufacturing an intervertebral cage for replacing an intervertebral disc in a patient, wherein the intervertebral cage has a topology, a shape and/or a size synthesized according to any of the claims 1-32; and - producing the intervertebral cage. 37. Data structure comprising information for manufacturing the synthesized intervertebral cage according to any of the claims 1-32, preferably claims 29-32. 38. Data structure according to the preceding claim, wherein the data structure is in a computer readable format and/or on a computer readable medium.

39. Intervertebral cage for replacing an intervertebral disc in a patient, wherein the intervertebral cage has a topology, a shape and/or a size, according to Figure 7 and/or

Figure 9. 40. Intervertebral cage according to the preceding claim, wherein the topology, the shape and/or the size are further detailed in Figure 10(a) — 10(i). 41. Intervertebral cage according to any of the preceding claims 39-40, wherein the intervertebral cage is manufactured using additive manufacturing. 42. Intervertebral cage according to any of the preceding claims 39-41, wherein the intervertebral cage comprises compliant joints, and wherein at least the compliant joints are composed of a single composition. 43. Intervertebral cage according to the preceding claim, wherein the joints are compliant joints. 44. Method for manufacturing an intervertebral cage according to any of the claims 39-43, wherein the method uses additive manufacturing. 45. Method for manufacturing according to the preceding claim, wherein the intervertebral cage comprises compliant joints, and wherein at least the compliant joints are composed of a single composition. 46. Method for manufacturing according to the preceding claim, wherein the method is combined with any of the features of the claims 1-32. 47. Data processing system comprising means for carrying out the steps of any of the claims 1-32, the steps of claim 36, or the data structure of the claims 37-38. 48. Computer-readable storage medium comprising instructions which, when executed by a computer, cause the computer to carry out the steps of any of the claims 1-32, the steps of claim 36, or the steps of the claims 44-46.

Claims (48)

CONCLUSIONS A computer-implemented method (100) comprising the step of synthesizing the topology, shape and size of an intervertebral cage for intervertebral disc replacement in a patient, wherein the synthesizing step is based on topology optimization. 2. Computer-implemented method according to the preceding claim, wherein the intervertebral cage comprises an element which is a monolithic element. 3. Computer-implemented method according to any of the preceding claims, wherein the intervertebral cage comprises an element that is a yielding element. Computer-implemented method according to any of the preceding claims, wherein the topology optimization is started with a building volume with: - a length in a lateral direction (X) in the range from 30 mm to 50 mm, preferably 35 mm to 45 mm, further preferably 37 mm to 43 mm, most preferably substantially 40 mm; - a length in dorsal and/or ventral direction (Y) in the range from 20 mm to 40 mm, preferably 25 mm to 35 mm, further preferably 27 mm to 33 mm, most preferably substantially 30 mm; and/or - a length in cranial and/or caudal direction (Z) in the range from 8 mm to 20 mm, preferably 10 mm to 18 mm, further preferably 12 mm to 16 mm, even more preferably 13 mm to 15 mm, most preferably substantially 14 mm. 5. Computer-implemented method according to any of the preceding claims, wherein the topology optimization applies during iteration a maximum predefined mass value requirement for limiting the mass of the vortex cage. 6. Computer-implemented method according to any of the preceding claims, wherein the six degrees of freedom are divided into a subset of degrees of freedom with a low stiffness and a subset of degrees of constraint with a high stiffness; and wherein the topology optimization is performed based on the subset of degrees of freedom and/or the subset of the degrees of constraint, such that the intervertebral cage provides substantially comparable stiffness to the intervertebral disc. A computer-implemented method according to the preceding claim, wherein the subset of constraint degrees comprises the translations along a lateral axis (TX) and a dorsal and/or ventral axis (TY); and/or where the subset of degrees of freedom is translation along the cranial and/or caudal axis (TZ) and rotation about a lateral axis (RX), a dorsal and/or ventral axis (RY) and a cranial and/or a caudal axis (RZ). 8. Computer-implemented method according to any of the preceding claims 6-7, wherein the subset of the six degrees of freedom with respect to rotation comprises: - a maximum rotation about a lateral axis (RX) in the range from 2 degrees to 10 degrees, at preferably 3 degrees to 8 degrees, further preferably 4 to 6 degrees, most preferably substantially 5 degrees; - a maximum rotation about a dorsal and/or ventral axis (RY) in the range of 2 degrees to 10 degrees, preferably 3 degrees to 8 degrees, further preferably 4 to 6 degrees, most preferably substantially 5 degrees, and /or - a maximum rotation around a cranial and/or caudal axis (RZ) in the range of 2 degrees to 10 degrees, preferably 3 degrees to 8 degrees, further preferably 4 to 6 degrees, most preferably substantially 5 degrees . 9. Computer-implemented method according to any of the preceding claims 6-8, wherein the subset of the six degrees of freedom with respect to translation comprises: - a maximum translation along a lateral axis (TX) is less than 2 mm, preferably 1.5 mm, further preferably 1 mm, most preferably substantially; - a maximum translation along a dorsal and/or ventral axis (TY) is smaller than 2 mm, preferably 1.5 mm, further preferably 1 mm, most preferably substantially prevented; - a maximum extension along a cranial and/or a caudal axis (TZ) is less than 1 mm, preferably 0.5 mm, further preferably 0.25 mm, most preferably substantially absent; and/or - a maximum compression along a cranial and/or caudal axis (TZ) divided by the unloaded length in the cranial and/or caudal direction is in the range of 0% to 60%, preferably 10% to 50%, further preferably 20% to 30%, most preferably substantially 30%. A computer-implemented method according to any one of the preceding claims, wherein the topology optimization method comprises defining a constrained non-linear optimization problem formulation, preferably slope based, for synthesizing the intervertebral cage; where the bounded non-linear optimization problem P is formulated as: minimize x flu: [x]], ot P= [nian to Ux] <U, jE€F ;and x € XV where x is the vector of N design variables; where X defines the set as {x in R | Xmin < X < Xmach where the dimensionless objective f € R* is a function of energy measure U; € R* such as strain energy, work, compliance or energy-derived quantity such as stiffness; where U, € R* is a maximum allowable energy or stiffness; where U; is the measure of energy or stiffness of predefined movement patterns in the set C; where U; is the measure of energy or stiffness of a predefined movement pattern in the set F; and where C and F are complementary. Computer-implemented method according to any of the preceding claims, wherein the translational movements have a stiffness: - between 200 N/mm and 20,000 N/mm in undeformed configuration along a medial/lateral axis (TX); - between 200 N/mm and 20,000 N/mm in undeformed configuration for along a posterior/anterior axis (TY); - between 300 N/mm and 3000 N/mm in undeformed configuration along a cranial and/or caudal axis (TZ). A computer-implemented method according to any one of the preceding claims, wherein the rotational movements have a stiffness: between 1 Nm/deg and 100 Nm/deg in undeformed configuration about a medial/lateral axis (RX); between 1 Nm/deg and 100 Nm/deg in undeformed configuration about a posterior/anterior axis (RY); and/or between 10 Nm/deg and 1000 Nm/deg in undeformed configuration about a cranial and/or caudal axis (RZ) - between 1 Nm/deg and 100 Nm/deg in undeformed configuration about a medial/lateral axis (RX) ; - between 1 Nm/deg and 100 Nm/deg in undeformed configuration about a posterior/anterior axis (RY); - between 10 Nm/deg and 1000 Nm/deg in undeformed configuration around a cranial and/or caudal axis (RZ). 13. Computer-implemented method according to the preceding claims 11 and 12, maximize xc Sy subject to k,,[x] < Kz subject to vx] SD where Kz, Kz . Ktz €nk;, are the maximum allowable stiffness corresponding to the respective degrees of freedom; and where 7 is the maximum material use with respect to a/ the building volume. A computer-implemented method according to any one of the preceding claims, wherein a movement pattern is preferably defined as a translation along an axis or a rotation around an axis; wherein the axis is preferably chosen as a lateral axis (X), a dorsal and/or a ventral axis (Y), and/or a cranial and/or a caudal axis (2); and wherein the topology optimization is performed such that the stiffness is maximized for pre-selected motion patterns, and/or for the other pre-selected motion patterns the stiffness is limited to a pre-defined stiffness value. A computer-implemented method according to any one of the preceding claims, wherein a/the movement pattern is defined as a translation along an axis or a rotation about an axis; wherein the topology optimization includes defining a bounded non-linear optimization problem formulation, preferably slope based, for synthesizing the intervertebral cage; where the restricted nonlinear optimization problem is formulated as: minimize x f[e[x]], [EC P= foie aan [x] <E, jeF and x € XN where the dimensionless objective f € R* is a monotonically increasing function of the load energies €; € R* and £ € Rt is a maximum allowable stretch energy of the movement pattern j. A computer-implemented method according to the preceding claim, wherein the topology optimization is based on a topology defined as N continuously differentiable design variables x with its components in X ={x ER[O < x < 1}; where each element j in the topology optimization has a stiffness, preferably a Young's modulus, E;; and where E; is a non-linear function of design variable x;. A computer-implemented method according to the preceding claim, wherein Ex] = E+ (1-8R[X] E where E is the material Young's modulus; where ¥ is the filtered design variable x; where € is the relative stiffness between solid and empty; and where R is the material interpolation function. A computer-implemented method according to the preceding claim, wherein R[x] = 27; wherep € R*; and where p is preferably in the range of 2 to 4, more preferably substantially 3. 19. Computer-implemented method according to any of the preceding claims 15-18, wherein a strain energy of movement pattern i in a discretized environment is defined as: Eil] = Zu; Klxlu;; where K[x]u; € R™" is the design-dependent symmetric stiffness matrix; and where u; € R™ includes a nodal displacement of motion pattern i. A computer-implemented method according to the preceding claim, wherein the nodal displacement is defined as n linear governing equations K[x]u; = f‚Vi € M; where M is the set of all movement patterns; and where f; € R” are the nodal loads of movement pattern i. A computer-implemented method according to the preceding claim, wherein the node movement can be partitioned as [ | [oe] _ [" “J. Kp Kp] lpi foil where u; are the free nodal displacements; where u,; are the prescribed nodal displacements; where f;; are the applied nodal loads; and where f,,; are the nodal reaction loads of movement pattern i are. 22. Computer-implemented method according to the preceding claim, wherein the applied loads f, = 0 in all cases; and wherein the synthesizing step is the step of solving the nodal displacement K;fU;,; = Kp; Vi € M includes. A computer-implemented method according to any one of the preceding claims, wherein the topology optimization problem is a slope-based inequality bounded non-linear optimization problem. A computer-implemented method according to any one of the preceding claims, wherein the topology optimization comprises the step of: - filtering a design variable field; and - differentiable Heaviside projection using a high projection threshold for erosion and/or a low projection threshold for dilation. 25. Computer-implemented method according to any one of the preceding claims, minimizing flux), ieC where Pp = no to Ux! <0, ier ©" xe 26. Computer-implemented method according to any one of the preceding claims, me flu, iec where Ps = here to Owl] < 0. ; Chen Jo;loil <6, i€F x € are the elementary stresses obtained by the predefined motion patterns i for a predefined magnitude; and where & is a maximum allowable voltage value. A computer-implemented method according to any one of the preceding claims, wherein the topology optimization is for short-stroke bends. 28. Computer-implemented method according to any of the preceding claims, comprising the step of manufacturing the intervertebral cage. A computer-implemented method according to the preceding claim, wherein the manufacturing comprises additive manufacturing. A computer-implemented method according to any one of the preceding claims, including the step of providing a data structure containing information for manufacturing the synthesized intervertebral cage. A computer-implemented method according to the preceding claim, wherein the information for manufacturing includes the density as a function of the location of the intervertebral cage, preferably for each location in a/the building volume of the intervertebral cage. A computer-implemented method according to the preceding claim, wherein the information for manufacturing includes the topology, shape and size of the intervertebral cage, preferably for each location in a building volume of the intervertebral cage. An intervertebral cage comprising a topology, a shape and a size synthesized according to claims 1-32. 34. Intervertebral cage according to the preceding claim, wherein the intervertebral cage comprises a monolithic compliant element; and wherein the monolithic compliant element includes the topology, the shape and the size. 35. Intervertebral cage according to any of the preceding claims 33-34, wherein the intervertebral cage comprises a monolithic yielding element; and wherein the monolithic compliant element comprises the density as a function of the location for the intervertebral cage, preferably for each location in an intervertebral cage building volume. 36. Method for manufacturing, comprising the following steps: - receiving a data structure containing information for manufacturing an intervertebral cage for replacing an intervertebral disc in a patient, wherein the intervertebral cage has a topology, a shape and/or a size has been synthesized according to any of claims 1-32; and - making the intervertebral cage. 37. Data structure comprising information for manufacturing the synthesized intervertebral cage according to any one of claims 1-32, preferably claims 29-32. 38. Data structure according to the preceding claim, wherein the data structure is in a computer-readable format and/or on a computer-readable medium. 39. Intervertebral cage for replacing an intervertebral disc in a patient, wherein the intervertebral cage has a topology, a shape and/or a size according to figure 7 and/or figure 9. 40. Intervertebral cage according to the preceding claim, wherein the topology, shape and/or size is further detailed in Figures 10(a) - 10(i). 41. Intervertebral cage according to any of the preceding claims 39-40, wherein the intervertebral cage is manufactured using additive manufacturing. 42. Intervertebral cage according to any one of the preceding claims 39-41, wherein the intervertebral cage comprises flexible joints, and wherein at least the flexible joints are composed of a single composition. An intervertebral cage according to the preceding claim, wherein the joints are flexible joints. 44. Method for manufacturing an intervertebral cage according to any one of claims 39-43, wherein the method uses additive manufacturing. 45. Method of manufacturing according to the preceding claim, wherein the intervertebral cage comprises compliant joints, and wherein at least the compliant joints are composed of a single composition. 46. Method for manufacturing according to the preceding claim, wherein the method is combined with one of the features of claims 1-32. A data processing system comprising means for performing the steps of any one of claims 1-32, the steps of claim 36, or the data structure of claims 37-38. A computer-readable storage medium comprising instructions that, when executed by a computer, causes the computer to perform the steps of any one of claims 1-32, the steps of claim 36 or the steps of claims 44-46.