Effect of carrier size and mechanical properties on adhesive unit stability for inhalation: A numerical study

TheDiscreteElement Method (DEM)wasusedtoinvestigatethe micromechanics ofadhesive units,thebuilding blocksofadhesivemixtures,comprisingacarrierparticlewithattachedmicronised ﬁ neparticles.Binaryhead-on collisionsbetweenadhesiveunitswasperformedinthehandlinganddispersionvelocityregimes,withvariations incarriermechanicalproperties(mannitolor lactose)and size(50, 100, 200 μ m). The systemwasmodeledwith sphericalparticleswithinteractionsrepresentedbyHertz-MindlinandJohnson-Kendall-Robertscontactmodels. The results were analysed in terms of stability ratios (retention, loss and transfer) and mechanical properties (restitutioncoef ﬁ cient).Itwasobservedthat mannitolparticlesaremorestablethan lactosewith similar disper-sionperformanceandthatthe200 μ mcarrieristhemoststableamongtheparticlesinvestigated.Theresultspro-vide a parametric knowledge about the performance of carrier particles and could be used for development of macro models for particle engineering and optimization of formulations. © 2021 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).


Introduction
Pulmonary drug delivery is becoming an important aspect of local and systemic drug delivery [1] owing to its targeted action and release.Drug delivery through dry powder inhalers (DPIs) provides a varied set of advantages over traditional inhalers [2].DPIs use a special kind of ordered mixture known as adhesive mixture [3,4].Adhesive mixtures consist of micronised particles (< 5 μm aerodynamic diameter) attached to considerably larger carrier particles (50 μm -300 μm) forming adhesive units.The fine particles being smaller in size have a high surface to volume ratio and are highly cohesive in nature.In order to prevent agglomeration and to improve powder flowability, adhesive units with carrier particles are formed.
The stability and integrity of adhesive units depend upon the chemical and morphological structure of the carrier and fine particles [5,6].Previous studies have shown the influence of fine particle size [7], surface property of carrier [8,9] and carrier sizes [10] to name a few.Lactose has been the preferred choice of carrier particle used for inhalation owing to its stability and inert nature [11].Although versatile, lactose could react with some fine particles (comprising the active pharmaceutical ingredient, API) [12].Recent experimental studies conducted by Hertel et al. [13] analysed the aerosolisation performance and explored the possibility of using mannitol as carrier particle.There are numerous studies focusing on the effect of carrier particle size during aerosolisation using model silica particles [10] or traditional lactose particles [13][14][15].Studies focusing on the variability of size of carrier particle have reported varied and inconclusive results.This calls for a study of carrier mechanical properties and size during handling of the adhesive mixtures.
Adhesive mixtures for inhalation are complex and poorly understood [16,17].A wide range of interdependent parameters are critical to formulation, handling and dispersion.Even simplified systems consisting of lactose carriers and micronised fines are complex and incompletely understood.Numerical simulations, which enable welldefined parametric studies and detailed characterisation of the obtained results, constitute one possible ways to tackle these challenges.
Numerical simulations to study the micromechanics [18][19][20], dynamics within the inhaler [21] and the manufacturing process [22] have gained impetus in the last decade.Computational fluid dynamics (CFD) and the Discrete Element Method (DEM) are the two most commonly used tools to study the dynamics of particles.Carrier shape [23] and surface asperities have received considerable attention.Both numerical [24][25][26][27] and experimental studies [28] have focused on agglomerate breakage and impact.In order to improve the understanding of the stability of adhesive units our previous study [29] focused on the simulation of adhesive units of different surface coverage ratio (SCR) and different surface energy of interaction and spherical carrier particles of 100 μm diameter.In a subsequent study, the effect of the fine particle shape on the stability and performance of adhesive mixtures was addressed [30].Related experimental attempts to understand the effect of the nature [13,14] and size of carrier particles [10] on the aerosolisation performance have been made.
The current study is the first attempt to systematically characterise the parameters governing the stability of adhesive units during handling and packaging [31][32][33] using numerical methods in an attempt to minimise segregation.To this end, the DEM was used to study two different aspects of the selection of carriers.First, the effect of replacing lactose as a model carrier with mannitol was studied.Second, the effect of changing carrier particle size in a system with lactose fines and lactose or mannitol carrier was addressed.The simulations were performed using spherical carrier and fine particles.Hence the effect of shape of particles was disregarded in order to enable a parametric study of effect of carrier size and mechanical property.Likewise, additional factors such as the surface energy and moisture content were either kept fixed or disregarded.In order to develop a holistic map, different randomised adhesive units with varying SCR were taken into consideration.An understanding of the effect of using mannitol carrier particles or lactose carriers of different size was obtained and the impact of the selection of carrier on the stability and integrity of the adhesive units was established.The present studies are performed in the handling velocity regime (0.04-1.7 m/s) for carrier size and in both the handling and the dispersion velocity regime (3.5-62 m/s) for carrier properties.In this article for the sake of brevity, adhesive units comprising of lactose carrier and lactose fines will henceforth be referred to as LL and adhesive units with mannitol carrier and lactose fines will be referred to as LM.

Numerical framework
The DEM is used to simulate the interaction between adhesive units.It is based on Newtonian mechanics with forces between particles inferred from binary interactions.Newton's law governing the translation motion of the target particle i is defined as, where j is a neighbouring particle, m i is the mass, and x i is the position of particle i, g is acceleration due to gravity, f ij nc and f ij tc are normal and tangential forces acting on the particle i due to the contact with particle j and f fluid i is the force in particle i resulting from fluid interaction.Simulations are performed at low handling velocity regime, which results in negligible drag forces.Rotational motion on the particle is governed by, where I is the inertial tensor, M is the applied torque and ω is the angular velocity about the center of mass of the particle.Contact interaction between the two particles is governed by the Hertz-Mindlin model, as described in detail in previous works [29], and summarised below: • The effective radius (R * ) and effective mass (M * ) of the two particles in contact are defined as the harmonic mean of the their radii and masses, respectively.• The effective Young's modulus (E * ) and shear modulus (G * ) are defined as, where E i is Young's modulus for particle i, G i is the shear modulus for particle i and ν i is the Poisson's ratio for particle i (same holds for particle j).
• The Tabor parameter μ [34] is a dimensionless parameter whose value is used for choosing contact interaction: the DMT (Derjaguin-Muller-Toporov) [35] model for μ < 1 and the JKR (Johnson-Kendall-Roberts) [36] model for μ > 1.For the system under consideration, μ was found to be greater than 1 for all interactions, which corresponds to the JKR model.• Normal forces are described by two components, the normal elastic force (F ne ) and the normal damping force F nd .The normal elastic force is represented as, where F c is the pull off force between the particles, a is the contact area at a given time and a 0 is the finite contact area due to adhesion/cohesion in absence of external load.The pull off force between the particles is defined as, where Γ is the surface energy of interaction.The normal damping force is defined as, where S n is the normal stiffness, β is a dimensionless parameter which is a function of coefficient of restitution (e) and v n is the normal relative velocity between the two particles.
• The tangential force acting on the particle is the sum of two components, the tangential elastic force (F te ) and the tangential damping force (F td ).The tangential elastic force is defined as, where S t is tangential stiffness and δ t is the tangential overlap.The tangential damping force is defined as, where S t is tangential stiffness v t is the tangential relative velocity between the particles.3. Methods

Particle properties and formation of adhesive units
The fine particles and carrier particles were modeled as elastic spheres with contact damping.The diameter (Ferret) of the fine particles was fixed at 3 μm and property mimicked lactose fines.In the first part, addressing the effect of changing the nature of the carrier particle from standard lactose to mannitol, the size of the carrier particle was fixed at 100 μm.Adhesive units comprising lactose fines on lactose carriers (LL) and lactose fines on mannitol carriers (LM) were formed, using the parameters found in Table .1. Lactose fines were used to mimic the drug particles.In the second part, addressing the effect of varying carrier size, both the LL and the LM system was considered, with three different carrier sizes.As illustrated in Fig. 1, these were selected as 50 μm, 100 μm and 200 μm diameter.Again, the remaining parameters are stated in Table .1.
Adhesive units were created by psuedorandomising fine particles on the surface of the carrier particle.This was achieved by formation of a uniform cloud of particles around the carrier.Each fine particle was given a uniform random velocity followed by a velocity directed to the centre of the carrier.Further details can be found in [29].Based on the above description three different number densities or surface coverage ratios (SCRs: 0.5, 0.7 and 1) of the carrier particle were created.The SCR is defined as the ratio between the surface area of the carrier covered by the fines to the total surface area of the carrier.The total number of fines required to achieve a certain SCR is stated in Table .2. In the table, the volumetric ratio of the fine particles is also provided, as an indication of the volume fraction of drug in an inhalation powder.The surface energy of interaction between carrier-carrier, carrier-fine and fine-fine was fixed at 0.03 J/m^2 for both the LM and the LL system [19,21,22].

Simulation setup and simulations
Binary head on collision between adhesive units was performed in the handling and the dispersion velocity regime, taken to correspond to the velocity range of 0.04 m/s to 1.7 m/s [31-33] and 3.5 m/s to 62 m/s, respectively.The dispersion velocity corresponds to the pressure drop in a Next Generation Impactor (NGI) which mimics the inhalation pattern.The handling velocity regime corresponds to flow of particles through a hopper and packaging into a capsule, canister or inhalation device, etcetera.Similar units in terms of carrier material and size were allowed to collide with each other.Each of the adhesive units were created in triplicates with three different initial spatial distributions (using different randomisation) of fines on the surface of the carrier particle.Based on the above parametric variation over 1000 independent simulations were performed.
Results were quantified based on the transfer, retention and loss ratio.The transfer ratio represents the fraction of fines transferred between the two carrier particles, the retention ratio is given by the fraction of fine particles retained on the surface of the carrier particles after they have undergone collision and the loss ratio represents the total fraction of particles lost from the binary system under consideration.The restitution coefficient (e) was used to quantify the effective mechanical properties of the adhesive units.It is given as, where v after is the relative velocity between adhesive units after collision and v before is the relative velocity between adhesive units before collision.Note that the square of the restitution coefficient thus represents the fraction of the kinetic energy mv 2 /2 retained after collision.The simulations were conducted using Altair EDEM™ 2020 DEM simulation software with necessary custom plugins.The Verlet integration scheme was used and the time step was fixed at 0.2 ns to ensure a numerically stable system.

Statistical analysis
Averages of the restitution coefficient and loss ratio over the velocity range 1-1.7 m/s were calculated for each particular adhesive unit (randomisation seed value).The thus derived variable (the average) constitutes a summary measure of the results obtained in the selected velocity range amenable to straightforward statistical analysis.A collision velocity greater than 1 m/s was chosen to avoid the stochastic variability observed at lower velocities.A t-test was conducted to compare the LL and LM systems and ANOVA (Analysis of Variance) and Tukey HSD (Honestly Significant Difference) was conducted to compare results obtained for different carrier sizes.A significance level of 0.05 was used and p values were reported.2a shows a qualitative picture of the recoil after impact of adhesive units with lactose fines and mannitol carrier (LM) and Fig. 2b shows the collision of the adhesive units with lactose fines and lactose carrier (LL).It could be observed that the LL system shows a higher dispersion of fines than the LM system for the same SCR, surface energy of interaction and collision velocity.

Handling velocity regime
Fig. 3 summarises the quantitative results for mechanical properties and stability ratios for the LL and LM systems.The surface energy is fixed at 0.03 J/m 2 for all particles (lactose or mannitol carriers and lactose fines).Both LL and LM adhesive units undergo binary head on collisions with the relative collision velocity varying from 0.04 m/s to 1.7 m/s.4.1.2.1.SCR 0.5.Fig. 3a and b summarise the result of binary collision of LL adhesive units and LM adhesive units with each other with an SCR of 0.5.It can be clearly seen that the restitution coefficient (Fig. 3a) is higher for the LL than for the LM system (p < 0.05) and the curves for the restitution coefficient approach a plateau at higher velocity in both cases.Fig. 3b summarises the stability of the adhesive units in terms of the transfer, retention and loss ratio.Retention is higher for the LM system (p < 0.05) with minimal losses and the LL system undergoes higher losses as the restitution coefficient plateaus out.At low velocities both the systems are comparable to each other.
4.1.2.2.SCR 0.7.The results obtained for an SCR of 0.7 are summarised in Fig. 3c and d.On increasing the SCR to 0.7, the restitution coefficient is similar for the LL and LM systems at a higher velocity (larger than about 1 m/s, Fig. 3c).However, some indication of an increased tendency for the LM system to stick to each other after collision may be seen for velocities around 0.5 m/s.Fig. 3d summarises the stability ratios.The LM system has a higher retention ratio and a lower loss ratio compared to that of the LL system (p < 0.05) and the LM system is thus more stable than the LL system.
4.1.2.3.SCR 1. Fig. 3e and f summarise the result for the binary collision of adhesive units with an SCR of 1.The restitution coefficients (Fig. 3e) are similar for the LL and LM systems.The stability ratios (Fig. 3f) show that both the LL and LM systems retain a similar number of fines on the carrier surface and the loss ratio is comparable for both systems.
Moreover, there is a tendency for the loss for the LL system to decrease with increasing SCR.An increase in the number of fines acts as a shield on the surface of the carrier resulting in a net interaction of lactose fines whereby leading to similar stability for both the LM and LL systems.

Dispersion velocity regime
The results in the following section summarises the binary collision of LL and LM adhesive units undergoing head on collision in the velocity range of 3.5 m/s to 62 m/s representative of dispersion velocity inside an inhaler.The surface energy of interaction is fixed at 0.03 J/m 2 for all particles (lactose or mannitol carriers and lactose fines).
4.1.3.1.SCR 0.5.Fig. 4a and b summarise the collision of adhesive units with an SCR of 0.5.In Fig. 4a it could be observed that the coefficient of restitution for LM plateaus to an effective value dominated by the mannitol carrier particle.For LL there is an initial plateau observed but with increase in impact velocity the coefficient decreases to a lower value.The results from Fig. 4 show that all the fine particles are dispersed from the carrier surface at high impact velocity (≳30 m/s).At lower impact velocity, adhesive units with mannitol carriers are more stable than those with lactose carriers.
4.1.3.2.SCR 0.7.Fig. 4c shows the plot of restitution vs. impact velocity in the dispersion regime for an SCR of 0.7.With increasing impact velocity, the restitution coefficient of LM plateaus at an effective value which is slightly lower the one obtained for LM at an SCR of 0.5.The restitution coefficient for the LL system follows a similar trend as that for an SCR of 0.5 (Fig. 4a).It could be observed in Fig. 4d that the retention ratio for LM is higher than that of LL for lower velocities (≲30 m/s) and plateaus at same value for higher velocities, indicating dispersion of all fine particles.4.1.3.3.SCR 1. Fig. 4e and f summarises the result of collision of adhesive unit with an SCR of 1.The restitution coefficient for LM plateaus at a value higher than that of LL for the same impact velocity.The LM system is more stable at lower impact velocities compared to LL.For higher impact velocities, virtually all the fines are dispersed from the adhesive units.

Impact of carrier size
Fig. 5 (LL) and Fig. 6 (LM) illustrate the results of the simulation of binary collision of adhesive units with different carrier sizes (50 μm, 100 μm and 200 μm).The surface energy of interaction between carrier-carrier, fines-fines and carrier-fines is kept constant at 0.03 J/ m 2 and SCR is varied as 0.5, 0.7 and 1 for each carrier size.

LL system
The results in the following section summarise the binary head on collision of adhesive units with lactose carrier and lactose fines.4.2.1.1.SCR 0.5.Fig. 5a and b summarise the results from a binary collision of adhesive units with an SCR of 0.5.As seen in Fig. 5a, with an increase in velocity, the restitution coefficient plateaus at a similar value for all three sizes of carrier particles, a value which is representative of a carrier-carrier collision.Stability ratios in terms of transfer, retention and loss are summarised in Fig. 5b.The retention and loss ratio are similar for the 50 μm and 100 μm carriers but the 200 μm carrier has a lower loss (p < 0.05) and a higher retention ratio and thus is more stable.4.2.1.2.SCR 0.7.Fig. 5c shows the plot of restitution vs. impact velocity for an SCR of 0.7.It could be observed from the figure that the restitution coefficient is similar for all the three particle sizes.The results obtained (Fig. 5d) indicate that the 200 μm carrier particle is most stable with a higher retention ratio and low loss ratio followed by the 50 μm and 100 μm carrier particle (p < 0.05).4.2.1.3.SCR 1. Fig. 5e and f summarise the collision of adhesive units of different sizes with an SCR of 1.The restitution coefficient varies across the velocity range but no consistent difference between the three sizes is observed.The 200 μm carrier is more stable (p < 0.05) compared to the 50 μm and 100 μm ones but the difference is less than 3%.There are no statistical differences between the 50 μm and 100 μm carriers.With an increase in SCR, the effect of the carrier particle size thus decreases.

LM system
The results in the following section summarise the binary head on collision of adhesive unit with mannitol carrier and lactose fines in the handling velocity regime.4.2.2.1.SCR 0.5.Fig. 6a and b summarise the collision of adhesive units with an SCR of 0.5.It is observed in Fig. 6a that the restitution coefficient of particle size 100 μm and 200 μm plateau at similar values which are representative of carrier-carrier collision.The 50 μm carrier plateaus at an effective value of carrier and fine collision.The retention and loss ratios are similar for 100 μm and 200 μm but 50 μm has a higher loss (p < 0.05) and thus is least stable.4.2.2.2.SCR 0.7.Fig. 6c and d summarise the result of collision of adhesive unit with an SCR of 0.7.It could be observed that the restitution coefficient is similar for all three particle sizes, as was previously observed for the LL system.The stability ratios show that the 100 μm and 200 μm particles have similar retention and loss ratio which are higher than the ones for the 50 μm particles which is least (p < 0.05) stable among the three.
4.2.2.3.SCR 1. Fig. 6e shows the coefficient of restitution vs. impact velocity plot for adhesive units undergoing collision with an SCR of 1.It could be observed that the coefficients of restitution are similar to each other and are yet to reach a plateau.The results in Fig. 6f shows that 200 μm is most stable followed by 100 μm and 50 μm.
An overall comparison between the results presented in Fig. 5 and Fig. 6 reveals that the losses for the mannitol carrier are lower than those for the lactose carrier for the same systems.

Stochastic behaviour at low velocity
Energy exchange between two impacting adhesive units depends on the initial spatial distribution of the fines on the carriers.This is prominent at low velocities, for which an impact could result in total or partial loss of energy, corresponding to zero (sticking) to low restitution coefficients.Since different random initial spatial arrangements of fines were used in the simulations, this phenomena leads to stochastic results at low velocities (Figs. 3, 5 and 6 a, c and e).This behaviour has been observed and explained in our previous study [29].

Impact of carrier properties
The nature of the carrier particle plays a critical role in the aerosolisation performance during inhalation.An understanding of the properties is a crucial step in optimising the overall formulation performance.Previous studies involving mannitol as a carrier focused on aerosolisation performance with minimal control on other physical parameters such as shape, size and interaction energy.Controlled simulations using mannitol as a carrier were performed to pinpoint the effect of the nature of the carrier particle while keeping other parameters constant.A comparison between the mechanical properties provided in Table 1 reveals that the major difference between mannitol and lactose carriers is that the former has a considerably lower Young's modulus.Low Young's modulus leads to a higher flexibility which is manifested in a less dramatic acceleration of the particles upon impact, as illustrated in Fig. 7.The lower acceleration is expected to result in lower inertial forces on the fines and hence increased stability of the adhesive units.This feature is prominent for adhesive units with low surface coverage such as SCRs of 0.5 and 0.7 (Fig. 3b and d).On increasing the SCR of the adhesive units, the surface density of fines increases, resulting in a shielding effect for the carrier particle.As a result, the interaction is dominated by lactose fines, i.e. lactose-lactose interaction for both the LM and LL adhesive units (Fig. 3f).A similar trend is observed for LL and LM adhesive units undergoing collision in the dispersion velocity regime as seen in Fig. 4. LM adhesive units are more stable than LL units for moderate impact velocities (≲30 m/s) whereas virtually all lactose fines are dispersed from both lactose and mannitol carriers at higher velocities owing to the high inertial forces.A comparison between Figs. 5 and 6 indicates that similar results hold for the 50 μm and 200 μm carrier sizes, at least in the handling velocity regime.
In conclusion, the effect of changing the nature of the carrier particle is prominent in the handling regime whereas the LM and LL systems have similar retention ratio (dispersibility) for high impact velocity during dispersion.Based on the above observation it could be inferred that the mannitol carrier could perform better than lactose from a stability aspect owing to its higher stability during handling and similar dispersion for higher velocity impacts as would occur within the inhaler.Mannitol could thus further be looked into as a viable alternative to lactose.

Impact of carrier size
The size of the carrier particle plays a significant role in the stability, flowability and aerosolisation performance of the adhesive mixture inside the inhaler.Previous studies with respect to carrier particle size Fig. 5. Restitution coefficient (left) and stability ratio (right) vs. impact velocity for lactose carrier particles of size 50 μm, 100 μm and 200 μm diameter for different surface coverage ratios (SCR) and surface energy of interaction of 0.03 J/m 2 .The data are grouped according to SCRs: a) and b) SCR 0.5, c) and d) SCR 0.7, e) and f) SCR 1.Each data point represents the mean obtained from three independent simulations of different initial seed values used during creation of adhesive units and the error bars represent the standard deviation among the three seed values.
have been inconclusive and varied.One notable experiment focusing on the performance of size of carrier particle was based on model silica particles [10] which focused on the aerosolisation performance and its dependence on particle size maintaining the same drug load on the mixtures.Understanding the stability of adhesive units during handling is an important step in selecting the right carrier size to optimise formulation performance.
As for the comparison between different carrier sizes, it is instructive to look at the acceleration experienced by the carrier particles upon impact in the absence of fines.It could be observed from Fig. 8 that the maximal acceleration experienced by the carriers decreases with increasing carrier size, being inversely proportional to the carrier diameter.Thus the 200 μm carrier particles experience the lowest acceleration followed by the 100 μm and 50 μm particles.As a results, the inertial forces on the fines are expected to decrease with increasing carrier size, leading to an increased stability of the adhesive units.A notable deviation from the expected trend for LL adhesive unit is observed in Fig. 5d where 50 μm shows a higher retention compared to 100 μm; one possible explanation for this could be higher cohesiveness of 50 μm particle at intermediate SCR.It is indeed observed from the simulations that the 200 μm carrier particle has the highest retention ratio among the three particle sizes.Higher energy is required to aerosolise adhesive units with 200 μm carriers and very small carrier particles have a low flowability owing to high cohesive forces between them.This points to an intermediate carrier size (100 μm based on the studied sizes) as an optimal choice.5.4.Pull-off force, fine-particle collision and press-on forces From Eq. ( 6) it could be seen that the pull off force (F c ) is directly proportional to the effective radius (R * ) of the particles for particles with same surface energy.Hence the pull off forces increases with the carrier particle size, so that more energy is required to remove fines from the surface of a large carrier.This effect is modest, however, Fig. 6.Restitution coefficient (left) and stability ratio (right) vs. impact velocity for mannitol carrier particles of size 50 μm, 100 μm and 200 μm diameter for different surface coverage ratios (SCR) and surface energy of interaction of 0.03 J/m 2 .The data are grouped according to SCRs: a) and b) SCR 0.5, c) and d) SCR 0.7, e) and f) SCR 1.Each data point represents the mean obtained from three independent simulations of different initial seed values used during creation of adhesive units and the error bars represent the standard deviation among the three seed values.since R * , being dominated by the small particle size, exhibits a modest increase from 1.42 to 1.48 μm when the particle size increases from 50 to 200 μm.
Table .2 shows the number of fine particles for the same SCR value.Naturally, larger carrier particles require a larger number of fine particles to achieve the same SCR, something which potentially could lead to more collisions among the fines and higher energy loss as a result of which the retention ratio is higher.
A larger carrier diameter during mixing could lead to higher presson forces [37,38] and higher stability and poor dispersion of the fine particles.The model used in the current study does not consider press-on forces during mixing which could be investigated in future studies.
Previous experiments [10] have shown that high energy forces is the major contributor to the detachment of fines from the carrier surface during aerosolisation inside the inhaler.Larger carrier particles (200 μm) are more stable during handling but require higher forces for fines to be released during aersolisation.Smaller carrier particles (50 μm) are more cohesive owing to the surface to volume ratio and would have low flowability which can lead to bridging and clogging in the hopper during handling.Among the three carrier sizes studied, 100 μm appears to be optimal as it has better flowability and requires less energy for fines to be dislodged from the carrier surface.

Cross comparison between property and size
On analysing the Hertz model [39] in absence of damping and interfacial interaction, the maximum deceleration between the particles after collision a max can be represented as, where ρ is the density of the particle, v 0 is the relative collision velocity, R * effective radius between the particles and E * is the effective Young's modulus.It could be inferred from Eq. ( 11) that the maximum deceleration between the bare carrier particles is inversely proportional to effective radius and directly proportional to effective Young's modulus to the power 2/5 as seen in Fig. 8 and Fig. 7, respectively.Overall, the manifestation of the carrier material and size in the form of the maximal acceleration controls the magnitude of the inertial forces experienced by the fines upon impact.It could be observed from Fig. 7 that the acceleration of a 100 μm mannitol carrier after collision is lower than that of the 200 μm lactose carrier, Fig. 8.In Fig. 3 and Fig. 5 it is observed that at low SCR, LM with 100 μm carrier particle has a higher retention ratio compared to the LL adhesive unit with 200 μm diameter.An increase in SCR leads to similar retention as the fine particles mask the surface of the carrier particle resulting in a net interaction between fine particles.
In this study, the carrier and fine particles were considered to be spherical to minimise the interference of shape effects.It should also be noted that the dispersion of particles inside an inhaler depends on the inhaler design and other competing mechanisms such as collision with the wall and drag forces on the surface of the carrier particle which could result in dispersion of fines alongside head-on collisions.Other parameters such as friction, shape of fine particles, hollow carrier particles could potentially effect the results but have been excluded in the study in and attempt to focus on the effect of the carrier size and mechanical properties.

Conclusion
Two different aspects of the selection of carriers in adhesive mixtures intended for inhalation have been systematically studied using DEM simulations.
First, the effect of using mannitol rather than lactose was assessed, using a design of the adhesive units that ensured that the all other factors except the mechanical properties of the carrier particle remained constant.Binary collision of adhesive units with varying SCR was considered in a handling and dispersion velocity regime.The results were analysed based on mechanical properties and stability ratios for the adhesive units.It was observed that adhesive units with mannitol as carrier are more stable compared to those with lactose.This finding is attributed to the higher flexibility (lower Young's modulus) of mannitol resulting in lower deceleration upon impact and hence lower losses for the same impact velocity.However it is crucial to note that an improved stability may lead to a reduced dispersion inside an inhaler as could be noted from the dispersion curves.A careful balance between stability, dispersibility and inhaler type is needed for improved performance.This leads to the conclusion that mannitol could be considered a viable alternative to lactose in terms of mechanical stability provided that sufficient energy is imparted to the mixture during administration to accomplish an adequate dispersion.
Second, the effect of the size of the carrier particle was studied, using lactose and mannitol carriers with sizes 50 μm, 100 μm and 200 μm.Again, binary collision between the adhesive units was carried out for different SCRs and impact velocity in the handling regime.Mechanical stability of the units after collision was recorded and analysed.It was observed that the 200 μm carrier particle imparts a larger stability to the adhesive units than the 50 μm and 100 μm particles.This could be attributed to larger inertia of the 200 μm particles, resulting in a lower deceleration upon impact.Other contributing factors are a higher pull off force and more intra fine collisions.It has previously been observed that more energy is required to aerosolise an adhesive unit with 200 μm large particles compared to smaller ones.However, the flowability

Fig. 1 .
Fig. 1.Illustration of the different sizes of carrier particles.

Fig. 2 .
Fig. 2. Illustration of recoil after impact of adhesive units with a surface coverage ratio (SCR) of 0.5, surface energy (Γ) fixed at 0.03 J/m 2 and relative collision velocity of 1.7 m/s.(a) Adhesive unit with lactose fines and mannitol carrier, (b) adhesive units with lactose fines and lactose carrier.

Fig. 3 .
Fig.3.Restitution coefficient (left) and stability ratio (right) vs. impact velocity for LL (lactose fines and lactose carrier) and LM (lactose fines and mannitol carrier) for different surface coverage ratios (SCR) and surface energy of interaction of 0.03 J/m 2 in the handling velocity regime.The data are grouped according to SCRs: a) and b) SCR 0.5, c) and d) SCR 0.7, e) and f) SCR 1.Each data point represents the mean obtained from three independent simulations of different initial seed values resulting in different spatial distribution used during creation of adhesive units.

Fig. 4 .
Fig.4.Restitution coefficient (left) and stability ratio (right) vs. impact velocity for LL (lactose fines and lactose carrier) and LM (lactose fines and mannitol carrier) for different surface coverage ratios (SCR) and surface energy of interaction of 0.03 J/m 2 in the dispersion velocity regime.The data are grouped according to SCRs: a) and b) SCR 0.5, c) and d) SCR 0.7, e) and f) SCR 1.Each data point represents the mean obtained from three independent simulations of different initial seed values resulting in different spatial distribution used during creation of adhesive units.

Fig. 7 .
Fig. 7. Maximum acceleration vs. impact velocity during head on collision of 100 μm large carrier particles of different materials.

Fig. 8 .
Fig. 8. Maximum acceleration vs. impact velocity during head on collision of lactose carrier particles of different size.

Table 2
Number of fine particles required to achieve the indicated surface coverage ratios (SCRs) for different sizes of carrier particles.The ratio between the total volume of fines and the volume of the carrier is also provided.