3D Particle-Based Cell Modelling for Haptic Microrobotic Cell Injection Marzieh Asgari1, Ali Ghanbari2, Saeid Nahavandi1 1
Centre for Intelligent Systems Research, Deakin University, Victoria, 3216, Australia 2 Mechanical Engineering Department, University of Canterbury, New Zealand Corresponding:
[email protected]
Abstract – Introducing haptic interface to conduct microrobotic intracellular injection has many beneficial implications. In particular, the haptic device provides force feedback to the bio-operator's hand. This paper introduces a 3D particle-based model to simulate the deformation of the cell membrane and corresponding cellular forces during microrobotic cell injection. The model is based on the kinematic and dynamic of spring – damper multi particle joints considering visco-elastic fluidic properties. It simulates the indentation force feedback as well as cell visual deformation during the microinjection. The model is verified using experimental data of zebrafish embryo microinjection. The results demonstrate that the developed cell model is capable of estimating zebrafish embryo deformation and force feedback accurately. Keywords - cell indentation force and deformation modelling, haptic cell microinjection.
1
INTRODUCTION
Micromanipulation of biological cells is widely undertaken in medical and cell related research [1]. Drug discovery, functional genomics and toxicology are some examples of this regularly-performed cell manipulation [2-4]. Cell injection procedures are traditionally performed manually where the operator relies completely on visual feedback from a microscope. It has been demonstrated that it takes approximately one year for an operator to be adequately trained in the cell injection process and even so, the success rates remain low [5]. The accuracy, speed and trajectory of the micropipette are important metrics relating to the success of the cell injection [6]. Microrobotic biological cell injection can be characterised by cells of varying size, shape and properties as well as contact forces which range widely from μN to mN [7]. The works by [7-10] propose different image processing techniques for the determination of parameters for various cells including suspended cells (e.g., embryos/oocyte) and adherent cells (e.g. HeLa, fibroblasts, and endothelial). Sun and Nelson [6] introduced a system for autonomous injection of individual cells. The work focuses on a mouse embryo and uses image processing to locate the cell nucleus and micropipette tip. In contrast to full automation, the authors are developing a haptically enabled microrobotic system which introduces haptic interaction to enhance human-in-the-loop cell injection in the aim of not losing human expertise. Such an approach would enhance bio-operator control in microinjection process while retaining the ability to exercise their own judgment relating to cell injection [11-14]. Other than that, such an approach would provide the platform to train the operator/ using a haptically enabled virtual environment. Haptic guidance can then be used to train the operator in various aspects of the cell injection process including; trajectory following, speed of insertion, accuracy training and cell force interaction. After sufficient training, the operator/s can move to the physical cell injection system and directly transfer their training experience to real cell injection using an identical mapping framework. Such a framework requires acquiring haptic representation of the cell membrane indentation force to be able to replicate cell injection process in virtual training environment. The existing approaches for mechanical modelling for living cells are categorised in three main groups: continuum approach, micro/nanostructural approach and energetic approach. The former treats the biological cell as a continuum material with certain fluidic, elastic, viscoelastic or solid properties. The model then is derived based on dynamics of fluids. The model parameters are obtained by experimental measurements [15]. The first cell model was introduced by Yeung & Evans [16] considering non-adherent cells as a liquid drop with a viscoelastic cortical shell. Although ignoring the contribution of the extra cellular matrix (ECM) and microstructural events inside the cell to the cell mechanics, the continuum approach is amenable to computational methods developed for fluid and solid mechanics and can provide the distribution of the force on the cell [17]. The micro/nanostructural approach seeks to account the molecular structure inside the cell as the main component that determines the cell mechanics[18]. Some advantages of this method are taking into consideration of the cytoskeleton and ECM to cell mechanical behaviour, describing nonlinear features of cellular mechanics and its discrete network structure [19, 20].However, these models do not take into account the non-elastic and dynamic behaviour of cytoskeleton and are purely mechanical. The last category is employing percolation theory and polymer physics models. For the reason that the model contributes energy budget of the cytoskeleton structure, it is independent of coordinate system and particular details of cytoskeleton
architecture. In addition, it can explain the intracellular mechanical signalling in the cell [21]. However, it requires the assumption of large deformation of the cell and ignores the contribution of the ECM to cellular mechanics. This paper suggests an approach to model the mechanical properties of the biological cells using a hybrid model of multi particles in micro-nano structural approach and continuum approach. The aim is to employ the modelling results to improve a haptically assist microrobotic cell injection process. The paper is organized as follows: in section 2, the developing haptic microrobotic cell injection system is introduced in brief, then the particle-based cell modelling is addressed in section 3. Section 4 simulation of the model is presented and analysed. Finally, the concluding remarks are presented in the last section.
2
HAPTIC MICROROBOTIC CELL INJECTION SYSTEM
Figure 1 demonstrates the microrobotic cell injection system. The MP-285 micromanipulator from Sutter Instruments which provides 3 actuated Degrees of Freedom (DOF) is used for microinjection. Each actuated DOF provides a linear range of 25mm with 0.04µm positioning resolution. A glass micropipette is employed to penetrate the cell. A computer controlled injection trigger provides positive pressure for material deposition (PMI-200, Dagan). The cell holding dish is placed in the view of the microscope lens and a CMOS camera (A601f-2, Basler) is mounted on top of the optical microscope (SZX2ILLB, Olympus) providing visual feedback from the cell injection process. The micromanipulator is interfaced to a PC (Intel Core Duo CPU 2.66GHz, 4GB RAM) using a DAQ card (NI PCI-6259). The PC is utilised as the control and monitoring system. All setup (excluding the PC and the microinjection system) are mounted on an anti-vibration table to minimise vibration.
Fig. 1: Haptic microrobotic cell injection system
The aim of the developing haptically enabled microinjection system is to enhance human-in-the-loop cell injection while retaining the operator’s human-level knowledge, intuition and expertise. Taking advantage of the bilateralism of haptic medium, the introduction of haptic assistance necessitates introduces two major benefits, namely; •
Mapping framework: It delivers the ability for the bio-operator to interact with the micromanipulator using a single-point haptic interface. This introduces an intuitive method to be able to control the motion of the micropipette in a similar fashion to conventional handheld needle insertion [13].
•
Haptic assistance: Virtual fixtures haptically assist the operator in performing the cell injection process by providing haptic feedback [11, 12].
In such a scheme both the operator’s movement of the haptic stylus (i.e. position input) and haptic forces (i.e. haptic force output) occur simultaneously at the haptic device’s haptic interaction point. This bilateralism provides the basis for the operator to receive haptic guidance while retaining ultimate control during the cell injection operation. The ability for the operator to control the microrobot using the following mapping framework also lends itself to training the operator offline (using the addressing cell model). In such a scenario, the operator/s can be trained using the mapping framework within a haptically enabled virtual environment. Haptic assistance can then be implemented to train the operator in various aspects of the cell injection process including interaction with cell. After sufficient training, the bio-operator/s utilises the physical cell injection system and directly applies their training experience to real cell injection using an identical mapping framework. In this perspective, this paper concentrates to address the development of the required cell model in the following sections.
3
3D PARTICLE-BASED CELL MODELLING
Non-adherent cells implement a spherical shape when suspended[22]. They can deform under certain stimulation and retrieve the initial shape upon release [23]. In the proposed model the cell interior is considered as a Voigt liquid-like satisfying hydrodynamic compressibility condition surrounded by a viscoelastic cortical shell. A particle-based model is developed to simulate cell mechanics under manipulation which addresses the mechanical properties of the cell. The model of the cell is obtained via a hybrid approach which contains cell wall model and interior fluid model as following:
3.1 Cell Membrane Model Discrete Element Method (DEM) has been proved to be an effective technique for modelling of a range of cells such as plant cells [24] and red blood cells [25]. In this paper the cell membrane which is assumed to be a viscoelastic layer is modelled using DEM. The spherical shell is divided into a set of discrete surface elements via a meshing algorithm. Then a particle is assigned to the centre of each surface cell element. Each particle is then connected to the neighbour particles by a number of spring-dashpots. They are also connected to the cell centre with an additional spring-dashpot as shown in Figure 2. This assumption aims to bring the lateral and axial movement of each particle due to an external stimulus. Given the particle model for the surface wall at any time step of the modelling, the position of the adjacent cells are updated from updating the spring, damping and hydraulic forces.
3.2 Viscosity and Compressibility inside the Cell The structure inside the cell is considered as a Voigt material, comprising of both viscosity and elasticity properties, which has a small compressibility factor. In favour of considering the contribution of the cell interior, a particle mass is supposed in the centre which is linked to all the membrane particles by a set of spring-dashpots. In view of the fact that, the cell interior is more viscous than elastic, the damping effect of the inner springs is considered much higher than their elasticity effect.
Fig. 2: Cell membrane meshing model. Each solid connected line represents a spring dashpot to the adjacent neighbour particles
3.3 Mathematical Model of the Cell Due to mutual interaction between the adjacent nodes, there is a connection between each particle and its four adjacent particles on membrane as well as the centre particle. These connections model the interaction of adjacent particles considering both elasticity and damping. The position of each particle is derived by applying these interactions and solving the following differential equation in 3D space:
+ B R + F sp = F ext + F comp M m,k R m,k m,k m,k m,k m,k m,k
(1)
sp 3 where Mm,k is the mass of the particle (m,k), Fm , k ∈ R is the tension forces of the adjacent and the centred particles,
Fmext,k ∈ R 3 is the external force applied to the cell in the node (m,k) which can be the needle force as well as holding device
T force, Rm ,k = [xm ,k , ym ,k , zm ,k ] ∈ R 3 is the position vector of each particle, Bm ,k is the damping factor of the cell in each point
comp 3 and Fm , k ∈ R is the hydraulic force exerted to the membrane due to incompressibility condition of the liquid inside the
cell.
1.3.1.
Spring Effects
The spring and damping interaction between the adjacent particles provides lateral and axial movements. As Figure 3 indicates, the model of the mutual interaction is represented by five connecting springs. These springs connect each selected particle to the adjacent particles forming a force arrangement as follows: top bottom + Fmcenter Fmsp,k = Fmleft,k + Fmright , k + Fm , k + Fm , k ,k (2)
where
Fmleft,k = K left ∆Lleft m ,k rˆmleft,k
(3)
∆Lleft m,k = ( x m ,k − x m ,k −1 ) 2 + ( y m ,k − y m ,k −1 ) 2 + ( z m ,k − z m ,k −1 ) 2
(4)
−L
left m ,k
left ˆ left K left , Lleft m , k , ∆Lm , k and rm , k are the spring constant, free length, length change and direction of the particle (m,k) left spring
respectively. Similar to equations (3) and (4) would be considered for other four neighbours to the right, top, bottom and centre. This set of equations gives tension forces imposed to each particle of the model.
1.3.2.
Damping Forces
The damping matrix, Bm, k, should be considered in a way to provide the non-isotropic damping. If the diagonal damping matrix
Bm ,k
Bmx ,k = 0 0
0 Bmy ,k 0
0 0 Bmz ,k
(5)
is considered with equal diagonal elements, then the damping would be isotropic. However, if the diagonal elements are different, the model includes the anisotropic behaviour.
1.3.3.
Hydraulic Force
The whole cell is considered as an incompressible liquid. Therefore, the cell volume should make no considerable changes during the injection process. To achieve it, a small cell incompressibility coefficient β should be taken into account: β=
1 dV V dP
(6)
where V is the cell volume, dV is the volume change and dP is the hydraulic pressure change inside the cell. At each iteration, V is evaluated to determine dV. By considering an appropriate quantity for β, equation (6) gives dP. Now, comp
referring to the relation of force and pressure, the hydraulic force, Fm ,k
would be derived.
3.4 Deformation and Indentation Force Estimation To calculate cell deformation and indentation force at any iteration, the set of differential equations represented by equation (1) is to be solved. The initial condition is considered the static position of the cell where all particles are at rest with initial velocity of zero and cell is fixed on suction surface from bottom. By applying the external forces, the acceleration of particle (m,k) is calculated as follows: = F ext + F comp − B v − F sp / M a m ,k = R m ,k m ,k m ,k m ,k m ,k m ,k m ,k (7)
(
)
where a m ,k is the particle (m,k) acceleration. Subsequently, velocity to calculate the damping effect and position to estimate
the deformation would be derived. Then, the indentation force is calculated based on the estimated acceleration and other forces at the injection point.
4
ANALYSIS AND RESULTS The model parameters identification is conducted using experimental data of a real zebra fish cell microinjection [26].
(a)
(b)
Fig. 3: Cell membrane deformation response. (a) 3D view of deformation (b) deformation projected in the xz-plane over the penetration period
To verify this model, a force is applied to the cell membrane to estimate the indentation force of the cell. Figure 3 indicates cell model deformation response over penetration period. The indentation force profile estimated by the model is shown in Figure 4 which is close to the force profile derived from the real zebrafish cell injection force measurement experiment [26].
Fig. 4: The indentation force derived from the model
5
CONCLUSION
In the aim of providing the bio-operator with the haptic representation of the cell membrane indentation force, a 3D particle-based cell model is proposed. The model considers the inner cell organelles effective to the cell shape, membrane mechanical behaviour, liquid like property of the cell and external imposed forces such as micropipette tip forces. The membrane is considered as a viscoelastic material modelled by DEM. In addition, the cell inner material is assumed a viscoelastic incompressible liquid. The results indicate that the proposed model replicates cell deformation response and the indentation force over the penetration period successfully. The cell indentation forces would then haptically rendered to provide the force feedback to the operator hand during the cell injection process.
6
References
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