Modelling of the deformation behaviour of a magnetic hydrogel in a magnetic field gradient

An ink made of alginate and methylcellulose with embedded magnetite microparticles was developed for extrusion printing. Constructs, so-called scaffolds, are colonised with cells which can be activated by mechanical stimulation. In this work, a defined magnetic field gradient is applied to achieve non-contact deformation. However, the deformation behaviour or relevant material parameters of the hybrid material are unknown. While the properties were determined with experiments adapted to hydrogels, a separate experimental set-up for micro-computed tomography, adapting the Maxwell configuration, was developed to investigate the deformation behaviour. These analyses were performed depending on ageing and particle concentration. For these tests, strands were used as bending beams, since these are simple and well known systems. Firstly, a model for the bending curve was erected, which defines a range in which the real bending curve would be expected. It was compared with the measured bending curves. There was very good agreement for the first days. On day 14, the measured bending curves were still within the calculated range, but at the lower limit due to the shortcomings of the model as the violation of the small deformations condition at this point. Secondly, the bending as a function of incubation duration was observed by a series of radiograms when a magnetic field gradient was applied. From this, a functional approach was formulated to describe the system response. Some parameters have already been identified, for others a proposal is given. Thirdly, microscopic analyses were carried out to observe the effects of the field gradient on particle distribution and structure. It was revealed that a homogeneous particle distribution was found even after 2.5 h. Also, in the direction of the field gradient, no chains were formed and no damage of the network could be detected. The obtained results show, that the material is suitable for mechanical stimulation.

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Introduction
In an aging society, the probability increases that an implant will be needed.This can be caused by age itself, e.g.due to degeneration of the joints or due to diseases.On the other hand, accidents can also make the replacement of damaged tissue necessary.Since every person is individual, massproduced parts often cannot be perfectly adapted to the body (Chrcanovic et al 2014).In turn some negative effects are possible such as implant failure e.g.due to inadequate attachment of the implant (Glauser et al 2004), inflammation and pain, which can lead to further surgery (Leonhardt et al 1999, Glauser et al 2004).Bone degeneration (Glauser et al 2004) or abrasion of implants or bone are also possible consequences (Grunder et al 1999).The healing process also varies from patient to patient (Morlock et al 2008).
Therefore, there are numerous groups that are conducting research into flexible implant designs (Ahlfeld et al 2020, Dhawan et al 2019, Kilian et al 2020) and also offer commercial solutions (3D Systems Corporation Press Release 2018).One promising approach is the use of additive manufacturing processes such as extrusion printing of biopastes.A new development in this field is the use of magnetic hydrogels, in this case alginate, which are colonised with stem cells (Lima et al 2015, Czichy et al 2022).By applying an alternating magnetic field with a constant field gradient such a structure can be deformed in a controlled manner (Czichy et al 2022).The idea is to stimulate cells by the deformation to divide and differentiate, so that over time they completely replace the biologically degraded structure (Shahin and Doran 2015, Weber et al 2015, Anderson and Johnstone 2017).In this way, patient-specific implants will be generated from the body's own cells.
The approach of using magnetic hydrogels is widely accepted, since the deformation behaviour can be controlled in a targeted manner and is of medically relevant magnitudes.In addition to medical applications with regard to hyperthermia or thermoregulation (Crippa et al 2017) and the release of active substances and cells (Zhao et al 2011, Cezar et al 2014, Crippa et al 2017), the spectrum ranges from soft robotics (Tang et al 2019, Tognato et al 2019) or actuators (Podstawczyk et al 2020) and biometrics (Tang et al 2019) to shape morphing (Hölzl et al 2016, Tang et al 2019).A typical method to determine the deformation capacity is to compare the heights of a cylinder before and while a magnetic field is applied.The ratio of the change in height compared to the initial height was defined as compression.This method was used for both block material as in Ramanujan and Lao (2006) also in porous scaffolds as in Zhao et al (2011).Likewise, Cezar et al applied to consider the deformation capacity for mono-and biphasic cylinders for microparticles (2, 3, 7, 13 wt%) (Cezar et al 2014).The application of a magnetic field itself also has a stimulating effect on cells (Xu andGu 2014, Huang et al 2019).Magnetic fields have long been an issue in the treatment of e.g.fractures (Hai-Yan and Ning 2014), so that stimulation using magnetic fields represents an important approach for tissue engineering.
In this paper for the material to be characterised alginate with methylcellulose (MC) was used as matrix, as this is a typical printing paste for extrusion printing in the tissue engineering area on the one hand and offers a cell-friendly environment on the other (Nicodemus and Bryant 2008, Andersen et al 2015, Akkineni et al 2016, Ahlfeld et al 2018, Neves et al 2020).Also, successful experiments with cells are already published (Ceccaldi et al 2012, Spangenberg et al 2021, Czichy et al 2022).Alginate is a polysaccharide, extracted from brown algae.It is not cytotoxic and biocompatible as well as biodegradable (Vargas et al 2018).The cross-linking process takes place via a physical linkage in which divalent cations intercalate in the areas of two guluronate blocks.
As magnetic material the iron oxide magnetite was chosen.The advantage over iron is the corrosion resistance of magnetite.In addition magnetite is non-toxic for cells as known from various studies with magnetic nanoparticles (as shown e.g. in (Aurich et al 2007, Mbeh et al 2012, Reddy et al 2012)), and was also tested for magnetic microparticles (Spangenberg et al 2021).
To use a solution based on alginate as a bioink, its viscosity have to be increased by adding 9 wt% autoclaved MC.It is used in a variety of industrial applications (Chevillard and Axelos 1997) and prominent in tissue engineering (Ahlfeld et al 2018).MC has hydrophobic components.This leads to a release of MC over time, which occurs more at lower temperatures (<10 • C) than at room temperature (20 • C) (Li 2002).
Since the material degrades over time, its properties such as the Young's modulus, and thus its deformation behaviour, change drastically (Czichy et al 2020).Furthermore, it is not possible to select any random strain for the deformation, since cells die if the load is too high (Schmelter et al 2006).Therefore, the first step was to characterize the deformation behaviour and to analyse the dependence on various influencing factors.A suitable experimental setup was designed for this purpose, which could be used in micro computed tomography (µCT).This is a suitable measuring method for obtaining threedimensional data sets and comparing different states of the object.Deformations of the system can be visualised in this way.Since it is a non-contact measuring method, there is no influence on the sample or the sample environment.Together with digital image processing (DIP), analysis at macro-and microscopic level are possible (Günther et al 2012, Borbáth et al 2014, Gundermann and Odenbach 2014).On the one hand, the deformation of the overall structure can be analysed (Günther et al 2012).On the other hand, the structure is resolved with regard to the distribution of the magnetic particles (Borbáth et al 2014, Gundermann andOdenbach 2014).
The basics for dimensioning the field have already been described and are implemented in a bioreactor called CyMAD (Czichy et al 2022).In this previous work it was shown that the approach is suitable to process samples with 25 wt% magnetite over 14 d.We now present the development of a model to describe the deformation as a function of sample age and particle concentration and validate it with corresponding measurements.The published equations, including the assumptions and simplifications made for them, are reviewed.The necessary material properties such as the Young's modulus and the magnetisation were determined.Furthermore, the particle distribution on the microscopic scale and, using the pair correlation function (PCF), the structure formation by applying an external magnetic field gradient were investigated.

Materials
The research object is a particle matrix system based on alginate.Alginate powder (alginic acid sodium salt from brown algae, M/G ratio 1:2) and MC powder (molecular weight ≈ 88 000 Da, 4000 cP) were purchased from Sigma, USA.Phosphate buffered saline solution (PBS) was obtained from Thermofisher, USA.Calcium chloride dihydrate (M = 147.02g mol −1 ) was used from Roth, Germany and the magnetite powder (natural) for the magnetic particles was used from Inoxia Ltd, UK.

Sample preparation
For matrix preparation, as previously described (Czichy et al 2020), 3 wt% alginate were dissolved in PBS; which was chosen because of its cell friendly environment due to the pH value of 7.4 and the contained salts.After adding 9 wt% MC to the alginate solution, samples were left to swell in the mixture for 120 min.
Magnetite powder was then added and mixed in homogeneously with a spatula to obtain the final paste.In this study 5, 15, 25 and 35 wt% of magnetite was used for tests regarding time response, bending behaviour and measurands required in the models.For microscopic analysis 25 wt% was chosen, since it is the preferred compromise for the application case (Spangenberg et al 2021).
Cylindrical strands were produced by manual extrusion from an appropriate nozzle, inspired by the extrusion printing of scaffolds as described before (Czichy et al 2020(Czichy et al , 2022)).For tensile tests, strands were made with a diameter of approx.4.8 mm and a length of 25 mm.For the investigations in the µCT regarding bending and microstructure, strands with a diameter of approx.2.3 mm and a length of 16 mm were manufactured.The strands were pressed directly into the cross-linking solution (100 mM CaCl 2 ) to obtain a defined shape and remain there for 20 min.
For the measurements with the vibration sample magnetometer (VSM), to determine the magnetic properties, spherical samples with a volume of around 40 mm 3 were produced as described before (Czichy et al 2022).
All samples were stored in cell culture medium DMEM from fisher with 1% penicillin at 37 • C for 14 d.The medium was renewed twice a week.

Methods
The tests, which concerned the analysis of the deformation behaviour, were carried out on day 0 after sample preparation, after 24 h incubation and after 14 d.The investigations concerning the analysis of the microstructure were carried out after 14 d.

Vibrating sample magnetometer.
For spherical samples the magnetisation M in dependence of the magnetic field strength H was determined with a VSM (LakeShore 7307, Lake Shore Cryotronics, USA) with a maximum field strength of 1500 kA m −1 .
2.3.2.Tensile tests.The Young's Moduli were obtained by tensile tests via the MCR301 from Anton Paar with a speed of 30 µm s −1 as described before (Czichy et al 2020).

Micro computed tomography.
To obtain 3D images as primary input for DIP for insight regarding the particle structure and deformation behaviour, measurements were taken with the µCT system TomoTU (TU Dresden, Chair of Magnetofluiddynamics, Measuring and Automation Technology, Germany).Radiograms were recorded at different angles in the form of grey-value images of 2304 × 2940 px.The tube current was 170 µA and the acceleration voltage was 90 kV.3D images were created from these radiograms using a filtered back projection and FDK algorithm based software package.The resolution is 7.12 µm px −1 in case of macroscopic analysis and 4,996 µm px −1 for the particle analysis.
In order to investigate the effects of a magnetic force on the material, an experimental set-up was constructed, which generates a suitable magnetic field gradient and a defined force in only one direction, shown in figure 1.This is achieved by a Maxwell configuration of neodymium-ironboron magnets (d = 50 mm, h = 10 mm, N52, B R = 1.42-1.47T), shown in figure 2. The Maxwell configuration is a pair of cylindrical magnets of equal strength whose magnetic fields oppose each other.As a result, the magnetic fields in the centre between the magnets cancel each other out and an approximately constant field gradient is formed in axial direction.
Two different magnetic fields are defined for the area, where the samples are placed.When the magnets are in position 1 (figure 2 left), the sample experiences no force because the magnetic fields of the magnets cancel each other out.This is defined as initial state.By moving the lower magnet upwards to position 2 (figure 2 right), a gradient of approx.10.5 kA m −1 mm −1 can be realised.In case of a bending setup this is defined as deformed state.
For the setup, the magnetic field gradient was determined from the simulation of the field with the freeware finite element method magnetics (FEMM, www.femm.info/wiki/HomePage) based on Lua script (Meeker 2020).Simplified layout for a setup with two permanent magnets for the µCT to determine the three-dimensional position of the sample without and with applied magnetic field gradient.

Digital image processing
2.3.4.1.Bending curve.For each state (initial, deformed) a 3D image is taken.The centre of gravity (S x |S y ) is determined for each layer in axial direction y s .That corresponds to the position of the neutral fibre.The difference between the initial and deformed state equals the actual bending curve.The principle is shown in figure 3.
Although the difference between the bending line and the deformed state is small even on day 14 with a network weakened by ageing, the difference between the states is used.This establishes a general procedure that can be applied to other materials.The reason why the difference is so small is that the material hardly shows any deformation due to its own weight, this is even valid for the weakened network on day 14.This in turn is due to the good positioning of the sample and the chosen experimental set-up including the sample design.

Particle analysis.
The reconstructed tomography data was processed in Matlab with the plugin DIPimage (Quantitative Imaging Group 2013).The key element for determining the particle position and thus the particle distribution is the seeded watershed algorithm (Beucher and Lantuéjoul 1979), how it has already been described in other works (Günther et al 2012, Gundermann andOdenbach 2014).Likewise, the particle structure could be examined more closely on the basis of their positions using the PCF as already done for magneto-active elastomers (Schümann et al 2020).The function g(r) can be used to draw conclusions regarding potential formation of chains or agglomerates (Schümann et al 2020).This is possible since g(r) describes the probability g of detecting an object at a distance r in the adjacent space surrounding a reference object.

Time response.
To analyse the time response, a series of images with 37 radiograms per minute was taken during the application of the magnetic field.The bending sample was aligned so that the cylinder axis was perpendicular to the direction of the x-rays.From the middle section of the sample, where the maximum deformation occurs, the centre of the sample is determined for each image.This allows the displacement of the centre to be determined and the observation of the deformation, when a magnetic field is applied and removed after a defined time.

Model
Since the scaffolds used in Tissue Engineering have a complex geometry, which can only be reproduced to a limited extent, a simple, the well-known bending beam was used instead (figure 4).
As bending beams cylindrical strands were used, which were clamped into a special holder and placed centrally between the magnets (figure 5(a)).
The modelling of the gradient dimensioning and the description of the bending curve can be found in detail in a previous paper regarding the bioreactor CyMAD (Czichy et al 2022).In this previous work, samples with 25 wt% magnetite were cyclically deformed at a frequency of 1 Hz for 3 h over 14 d.As this was a medical application, the material was autoclaved.On day 14, it was examined whether material damage had occurred due to the stimulation.No differences were found between the stimulated and control groups, neither on a macroscopic nor microscopic level (Czichy et al 2022).Now the question is whether the equation for describing the bending curve w in dependence of the coordinate y w (y) = q l 4 24 E I x with the simplified Kelvin force actually represents the measurements.Here q is the linear load, which is related to the Kelvin force via q l = F with the sample length l.Furthermore, E is the Young's modulus, I x is the moment of inertia, µ 0 is the magnetic permeability of the vacuum, V is the volume of the sample, M V is the volumetric magnetisation and ∇ ⃗ H the field gradient.The assumptions made regarding the Kelvin force are a homogenous magnetisation over the sample and that there is just a magnetic gradient in z-direction (Czichy et al 2022).
In order to verify the model, some parameters had to be measured for the magnetic hydrogel in dependence on particle concentration and incubation duration.An exception is the field gradient, as this was calculated using the simulation software FEMM 4.2.It was designed as 'magnetic problem', type 'axisymmetric'.For the magnets (r = 25 mm, h = 10 mm, distance 45 mm) the magnetic properties could be obtained from the datasheet for NdFeB N52.The magnetisation curves M V (H) were determined via VSM from LakeShore 7307.An example for 25 wt% magnetite on day 0 is given in figure 6.It can be seen, that the remanence M R is with 2.7 kA m −1 very low compared to the saturation magnetisation M S with 33.3 kA m −1 .The ferrimagnetic magnetite particles embedded into the matrix are thus at a distance from each other in which they significantly increase For calculating volume and moment of inertia the diameter of the sample is extracted from the 3D data of the sample.Also the length of the samples can be gained from it.Thus, the applied force and the individual bending curve can be calculated for each sample.
Taking into account the double standard deviation of the measured values, two lines are defined for each sample as the upper and lower boundary in which the actual measured curve is expected to lie if the model applies.The area spanned by these two lines is defined as the model range MR.

Bending behaviour
The bending behaviour depends on the particle concentrations and the incubation duration.For the relation to be analysed, measurements for the parameter combinations mentioned above were carried out in a µCT.The samples in shape of approx.16 mm long and 2.3 mm thick cylindrical strands were placed centrally in a radial direction in the sample chamber.Initially, the sample is in an area with a field strength of 0 A m −1 (figure 2).This is defined as the initial state, as there is no force applied to the sample.Taking a measurement via µCT, this unloaded state is recorded and the position of the neutral fibre is determined from the data.The magnetic field is then applied.Analogously, a tomogram is taken of the deformed state and the position of the neutral fibre is determined.The actual bending curve results from the difference of the two curves (figure 3).For 25 wt% magnetite on day 0 all curves are displayed in figure 7 as example.
Figure 7 shows that the reproducibility is very good both within one test series (V1-V3) and among different ones.This can be seen in the comparison of the sample from another test series (V0).For this reason, instead of the single curves the mean values of the bending w depending on particle concentration c and incubation duration t are shown in figure 8 as an overview.
The maximum bending in the centre of the samples is summarised in table 2 and the resulting strain is presented in table 3.An increase in bending with increasing particle concentration and incubation duration can be determined.However, the correlation between bending and concentration differs for each measurement day.The reason for this is the interplay between Young's modulus and magnetic force and will be discussed later.
For the comparison of the measured bending curves with the model, the minimum bending w min (solid red line) and the maximum bending w max (dashed red line) were determined for the confidence interval of 95% on the basis of the error calculation.The area covered by the two lines is referred to in the following as the range of the model of the expected bending.For each concentration a representative example is given for day 0 in figure 9, for day 1 in figure 10 and for day 14 in figure 11.

Time response
The time responses are shown exemplarily for samples with a concentration of 25 wt% for different incubation durations in figure 12. First, a rapid increase can be seen as a direct response to the application of the magnetic field.This is followed by a concave section, which, however, does not change into a stationary, but into a linearly increasing area.

Particle analysis
In the following, the results for the particle distribution and the evaluation by PCF regarding the structures before and after application of the magnetic field gradient for 2.5 h are shown.Figure 13 shows the assignment of the axis to the sample and their reference regarding manufacturing and field gradient.The section for analysis is also displayed by a box.The  box goes through the entire sample in the z-direction.In x and in y, however, an area of 350 px was used from the centre of the sample.

Particle distribution.
As can be seen in figure 14, in the x-direction, which runs parallel to the sample axis, a stable frequency level of 2.9% for detecting particles has developed over the observed range.In the y-direction, influenced by the shape of the sample, a slight rounding can be seen.The distributions in the z-direction show a curvature caused by the sample shape.The differences in width are caused by the variation of the sample diameter.
The z-direction is important with regard to the effects of the magnetic field gradient.The difference between the initial state of the individual samples and after application of a magnetic field for 2.5 h, whereby the second measurement was also carried out with the field applied, is shown in figure 15.As a control group, the results in the x-direction are also presented in figure 15.
3.4.2.Particle structure.The PCF g(r) describes the probability of occurrence of a particle in distance r to another particle plotted over the distance.It can be used to evaluate the structure within a sample or to track structure formations.
The PCF results for 14 d old samples containing 25 wt% magnetite are shown in figure 16.The rise of the curves only starts at approx.6.6 µm.This corresponds to the smallest distance between the centres of the particles.
Only 5% of the particles have a longest major axis of less than 20 µm.Because of this, the curve does not rise abruptly but continuously over 50 µm to the value 1, corresponding to the particle size distribution (see (Spangenberg et al 2021)).
All PCFs show an ideal course of a homogeneous particle distribution.There are neither peaks at the beginning nor  overshooting of the curves.Instead, the value 1 was directly hit and maintained by all curves after a slow curve rise.Even after 2.5 h in the magnetic field gradient, no significant changes in the PCF are discernible either in the zdirection or the other directions with respect to the initial state.

Bending behaviour
On day 0, the elastic modulus is almost constant between 5 and 25 wt% (table 1), so that the bending depends only on the force and thus linear on the particle concentration.This is also reflected in the measured bending curves.35 wt% on the other hand exhibit a significantly higher Young's modulus.Since applies, for the comparison of 25 and 35 wt% an increase in Young's modulus of 39.4% compared to a force increase of 40% leads to a theoretically constant bending.This is also supported by the data (figure 8, red box).
On day 1, the influence of the Young's modulus is evident, as the bending has increased to the same extent as the Young's modulus has decreased.The ratios of the Young's modulus for the different concentrations are, apart from 35 wt% with a higher decrease, similar to day 0. The curves relate to each other accordingly.On day 14, there is a linear correlation of the elastic moduli with the volumetric particle concentration.In combination with the force, this means that in the exemplary comparison of 25 wt% with 15 wt%, the curves of the samples containing 25 wt% magnetite are below those of the samples containing 15 wt%, as the difference in force is 40% and the difference in Young's modulus is 48%.This is reflected by the data in figure 8.The smallest deformation for each concentration was measured on day 0. For these samples, the condition of the Euler-Bernoulli beam with regard to small deformations can be considered as fulfilled.There is a very good agreement between the measured bending curves and the MR.It can be said that the model represents the bending behaviour on day 0 very well.A tendency of the measured bending curves towards the upper half of the MR, i.e. towards w min , is recognisable.Since sample preparation took place on this day and the samples show very good reproducibility, the MR is quite narrow due to the low variances of the erroneous of the measured variables compared to the other incubation durations.
Day 1 shows an equivalent pattern compared to day 0: small deformations, bending and strength of the MR increase with concentration and the measured curves tend to be in the upper half of the MR.Thus, the model represents the actual bending behaviour on day 1 very well.
On day 14, the shortcomings of the model are revealed.Firstly, the condition of small deformations is considerably violated.With a bending of approx. 1 mm over a length of 11 mm, these are neither small deformations, nor can it be assumed that the planes are parallel to the z-axis.Secondly, although the curves are still within the confidence interval, and despite the large MR of 806 µm for 5 wt% and 503 µm for 25 wt%, the measured bending curves do run at the limit of the minimum bending w min .
Further reasons for the differences between model and measurements can be found in the interaction of restoring forces, the material stiffening due to the considerable elongation of around 10%, the magneto active effect, which causes a stiffening of the material when a (homogeneous) magnetic field is applied (Stepanov et al 2013, Borin et al 2019, 2020), the influence of the Kelvin force and the incubation duration influence, which depends on the sample composition.The demagnetising field within the samples was also not taken into account in the model.This makes it clear, that the modelling for big deformations is a very complex problem, for which this work offers a fundamental first step towards a solution and creates a starting point for further, adapted models, on which e.g.computer-aided simulations can be developed.

Time response
The curves in figure 12 were fitted with the function depending on the time t and are presented as red lines in figure 17.In addition, the measured data are given in blue.
Besides the complete functional approach, the fit functions without the linear component m t are given in purple.The approach is based on the model of Burgus, which is used to describe the time behaviour of polymers with elastic, viscoelastic and viscous behaviour (Czichos et al 2007, Rösler et al 2019, Eyerer et al 2020).
The first summand of the fit function represents the spontaneous behaviour of the material.Due to the delayed application of the magnetic field, this elastic deformation w 1 does not occur as an offset value for t = 0 s but is also delayed.In order to represent this delay, an exponential factor is used with the constant K 1 .
The second summand is used to model the viscoelastic behaviour, which is described for polymers with a proportional transfer behaviour.This is adopted for this model approach.There are various possibilities regarding the constant K 2 as will be explained later.
The third summand describes the linear increasing part, which is found in the last third of the measured curves.Accordingly, this is formed as the product of the increase m and the time t.
In the case of polymers, the viscous behaviour is represented with linear terms.However, in the present case, there is another possible cause besides retardation.Since a linear magnetic field is applied and the magnetisation depends on the field strength, there is a dependence of the applied magnetic force on the position of the sample within this field and thus its position in space.With increasing bending, a part of the sample shifts to an area with a higher field.This increases the magnetisation corresponding to the new position for this part of the sample, so that the sample is no longer homogeneously magnetised.The greater the deformation, the greater the difference in local magnetisations within the sample.With regard to the time behaviour, the increase in magnetisation means an increase in the local force, which results in further deformation.In the literature, hydrogels as alginate are mostly compared with elastomers, which only show elastic and viscoelastic behaviour (Kaklamani et al 2014, Baruffaldi et al 2021).Thus, the measured deformation is a superposition of the actual time behaviour under the effect of a constant load and the deformation caused by the increase in force.This is expressed in the second and third summands of equation ( 4) while the first summand comes from a spontaneous reaction forced by the application of the field gradient.
For hydrogels, since their properties are considered to be those of elastomers, the Voigt-Kelvin model can be applied, according to which a stationary end value is reached after some time (Czichos et al 2007, Rösler et al 2019, Eyerer et al 2020).Transferred to the material system at hand, this means that the linear component is entirely due to the mentioned external influences.Accordingly, a correction procedure based on the fit functions could be set up by deducting the linear part of the function approach from the determined bending curve.This procedure is to be understood as a first approach, which is only valid if there is no viscosity.
The identification and significance of the fit parameters w 1 , K 1 and m could be determined either from the collected measurement data regarding the time response or through further experiments carried out in the magnetic test setup.
The increase m can be calculated directly from the data of the step test for t = [10; 15] min.Due to the similarity to a PT1 element, the constant K 1 can be defined as the time constant T 1 .However, since no stationary end value is reached for the curves, the data from the step test cannot be used to determine T 1 .Instead, an impulse test was performed in which the magnets were removed as soon as they reached position 2.Although this data cannot be considered as an actual impulse, the test is suitable for replicating the behaviour for the primary section of the step test, as the same ramp is driven in both cases.
With T 1 known, w 1 can be calculated with In contrast, the variables w 2 and K 2 cannot be determined from the measurement data.At first glance, for w 2 the formula seems promising.However, the values calculated in this way do not agree with the values of the fit function.
In the case that the increase in force is only expressed by a linear ramp, K 2 could be described according to the Voigt-Kelvin model in the form of (1 − e −t τ ) via the relaxation time τ (Czichos et al 2007).K 2 is probably a combination of damping d and spring constant c, as described by Eyerer in the form of (1 − e −c d t ) (Eyerer 2020).However, all of the mentioned variables are unknown and cannot be determined from the measurement data.The used test setup is unsuitable for carrying out pure retardation and relaxation experiments, as neither strain nor stress can be kept constant.If these parameters are known through a suitable experiment, the superposition components and thus the influence of the magnetic field could be determined on the basis of the difference to the fit function.
The reactivity of magnetic hydrogels is mostly considered in connection with flow control.This is used to either vary the volume flow or to create waves via applying an alternating field (Satarkar et al 2009, Liu et al 2019).Liu et al have shown the response as a function over time when the field is applied for a cyclical progression.A similar course as the presented curves (figure 12) is shown with a part that initially rises linearly, then flattens out and end into a stationary part.Since these are microfluids and therefore much smaller deformations, the time for one cycle is less than 1 s.In the field of tissue engineering, mostly only the deformation capacity is considered.In the case of cyclic stimulations, the deformation over time is dictated by stamps or direct stretching of the samples, so that the time behaviour is not relevant.The situation is different in the area of actuators and soft robotics, since of interest is the time until a certain deformation is reached or a certain distance is covered.For example, in the field of shape morphing the deformation can be achieved between 1 s and 100 s, depending on different parameters such as the magnetic field strength, the stiffness and the particle concentration (Liu et al 2022).Thus, the actuator designed by Goudu et al manages a rotation of 360 • due to a rolling movement in 7 s (Goudu et al 2020).

Particle analysis
Concerning the evaluated sample region, with an edge length of 350 px, which corresponds to approx.1748.6 µm, the mean value for the frequency of particles is 2.9% for an equal distribution of the particles.This agrees with the value to be read off, so that a homogeneous distribution of the particles can be concluded and it can be observed particularly well in the xdistribution both for the initial state and after a stay of 2.5 h in the magnetic field.In principle, it can be said that the manufacturing process is suitable for producing homogeneous samples.
Between the initial state and the state in the magnetic field there is no discernible difference for any axis direction.This is particularly interesting for the z-direction, as this corresponds to the direction of the gradient.Since there is no difference, it can be concluded, that the weakening of the network due to the loss of the MC and the diffusion of the Ca 2+ into the medium over 14 d is not present to such an extent, that the particles leave their position due to the magnetic field gradient.The network of alginate chains is thus dense enough to bind the particles in place.There are neither peaks nor significant shifts in the distributions in favour of one side.This is particularly important for the distributions in the z-direction, as the force acts in this direction.
Considering the results of the PCF, which show also very good agreement between both states, the matrix withstands the magnetic force.It can be concluded, that a homogeneous distribution and no structures are present even after 2.5 h in the magnetic field.With this, the mentioned prerequisites of a homogeneous, isotropic material needed for the theory of beams of the first order is considered as fulfilled.
Compared to the chain formation processes described by Schümann and Morich for soft magnetic microparticles in a soft elastomer matrix in a homogeneous magnetic field (Morich 2018, Schümann 2020, Schümann et al 2020), these could not be demonstrated for alginate MC with magnetite particles in a constant field gradient.On the one hand, other forces operate here on the particles, and on the other hand, the network structure seems to exert a significant influence.In the present case, the particles are firmly bound in the alginate network.
Since only the centre of gravity, but not the orientation, of the particles is available as reliable data due to their size, no statements can be made about rotation processes.

Conclusion
The aim of the bending model based on the Kelvin force and the theory of beams of the first order is to give a first approach to the description of the bending behaviour and the resulting possibility to predict it.This objective can be considered as fulfilled in the context of alginate MC hydrogels containing magnetite.
The calculated bending curves using the measured elastic moduli and the magnetisations represent the measured bending curves very well using the error calculation with the confidence interval of 95%.Thus, the assumptions made regarding the application and simplification of the Kelvin force, as well as the simplified theoretical consideration of an Euler-Bernoulli beam, are legitimate for the development of this approach.
This means that calculations based on magnetisation and Young's modulus can be used to predict the deformation behaviour and select the matrix material suitable for the task at hand.The corresponding measurements can be carried out with less time and (computational) technical effort compared to using a µCT and the associated data processing and special set up.Especially since access to the devices required is easier achieved.
At microscopic level, there is a homogeneous particle distribution, which is neither influenced by the ageing process nor by the application of a magnetic field over a period of 2.5 h.Therefore, the particles are firmly embedded in the matrix and neither cut through the alginate chains nor slip through the alginate network due to the Kelvin force.
Also, no near orders can be detected, especially in the field direction.Rotation processes or minimal movements are probably still possible, but not to the extent that they influence the homogeneous distribution or a structure formation in the form of chains or agglomerates occurs.
It can be concluded, that the material is suitable for mechanical deformation and the models are applicable for simulations and scaffold architecture.Thus, the next steps are the adaptation of the scaffold structure to the force conditions in the CyMAD (see therefore (Czichy et al 2022)) and the transfer of the findings to other materials or improved matrices such as the use of blood plasma instead of PBS (see therefore (Ahlfeld et al 2020)).

Figure 1 .
Figure 1.Experimental setup (a) and visualisation of the magnetic field (b), as well as the resulting gradient in axial direction (c) in the sample chamber at minimum magnet distance of 43 mm.

Figure 2 .
Figure 2. Simplified layout for a setup with two permanent magnets for the µCT to determine the three-dimensional position of the sample without and with applied magnetic field gradient.

Figure 3 .
Figure 3. Determination of the position of the neutral fibre to calculate the bending curve as the difference between the two states.(a) Bending beam and coordinates (b) start and end of the digital image process (c) position of the neutral fibre of the initial state (lime) and deformed state (dark green) (d) resulting bending curve.

Figure 4 .
Figure 4. Schematic illustration of a magnetic scaffold (left) and the simplified model for determining the bending behaviour of the chosen material for the theoretical and practical approach.

Figure 5 .
Figure 5. (a) Set up for a bending beam, (b) radiogram of the initial state, (c) radiogram of the deformed state in the magnetic field gradient; (d) difference of images [t = 14 d, c = 25 wt%].

Figure 8 .
Figure 8. Bending curves (mean value, n ⩾ 3) in dependence of incubation duration and particle concentration.

Figure 9 .
Figure 9.Comparison of the measured bending curves with the model range, given by the curves w min and wmax, for one example of each concentration on day 0.

Figure 10 .
Figure 10.Comparison of the measured bending curves with the model range, given by the curves w min and wmax, for one example of each concentration on day 1.

Figure 11 .
Figure 11.Comparison of the measured bending curves with the model range, given by the curves w min and wmax, for one example of each concentration on day 14.

Figure 12 .
Figure 12.Time response to applying a magnetic field gradient for samples with 25 wt% magnetite depending on time.

Figure 13 .
Figure 13.Coordinate system and sample section (red box) for particle analysis.

Figure 15 .
Figure 15.Comparison of the particle distribution in x-and in z-direction for the initial state and after 2.5 h in a magnetic field gradient for three different samples [c = 25 wt%, t = 14 d].

Figure 16 .
Figure 16.Pair correlation function for the initial state and after 2.5 h in a magnetic field gradient [c = 25 wt%, t = 14 d, n = 3].

Figure 17 .
Figure 17.Fit function of the time response.

Table 1 .
(Czichy et al 2020) magnetic alginate-MC-hydrogels depending on the particle concentration c and the incubation duration t. overall magnetisation of the hybrid material, but do not interact with each other and thus not alter the paramagnetic behaviour of the overall system.The Young's moduli E(t,c) were determined via tensile test with AntonPaar MCR 301 as described in(Czichy et al 2020)and are summarised in table 1. the

Table 2 .
Maximal deformation wm in the middle of the bending curves depending on incubation duration t and concentration c (n ⩾ 3).

Table 3 .
Maximal strain ε in the middle of the bending curves depending on incubation duration t and concentration c (n ⩾ 3).