## Abstract

The pipette aspiration testing technique is considered, and the elastic half-space model, which was originally introduced in the isotropic incompressible case, is revisited and generalized for the case of transverse isotropy. Asymptotic solutions are obtained in the two limiting cases of a wide and a narrow pipette.

## 1. Introduction

In recent years, the micropipette aspiration technique was widely used for testing living cells and measuring their mechanical properties [1–4]. In particular, the cell response is quantified in terms of the aspiration length, which measures projection of the cell into the micropipette under the applied negative pressure.

Being originally developed to handle cells under the microscope and later to investigate their mechanical properties, the pipette aspiration technique is nowadays successfully applied for non-invasive measuring of the *in vivo* mechanical behaviour of soft human tissues [5,6]. Also, the pipette aspiration method was used for the local stiffness measurement of soft materials [7,8].

Recently, the pipette aspiration technique was applied to characterization of non-homogeneous (fibre reinforced), transversely isotropic materials [9]. Comparison of the pipette aspiration testing method with other mechanical testing techniques used for biomaterials (including macroscale testing and nanoindentation) was performed very recently in [10].

The effective application of the pipette aspiration testing technique requires a relationship between the aspirated length and the applied pressure difference. Theret *et al.* [11] developed the so-called half-space model under the assumptions that the tested specimen is idealized to be an incompressible, homogeneous and isotropic elastic medium undergoing small strains.

In their work, the following analytical relationship between the applied pressure difference, Δ*P*, and the resulting aspirated length, *L*, has been established as
*b* is the inner pipette radius, *E* is Young’s elastic modulus, *Φ*(*η*) is a dimensionless factor, called the pipette wall function, and *η* is the ratio of the pipette thickness to the inner pipette radius.

Theret *et al.* [11] provided two analytical approaches to evaluate the pipette wall function. The first method requires solving an integral equation of the punch problem, while the second approach is based on the local force balance and approximately satisfies the non-penetration contact condition. The corresponding solutions are denoted by *Φ*_{P}(*η*) and *Φ*_{F}(*η*), respectively. The advantage of the force approximate model is that the corresponding wall function is given in an explicit form as
**E**(*k*) is the complete elliptic integral of the second kind. From figure 1, it may be seen that the two approaches yield fairly close values of *Φ*(*η*) for values of *η* near 0.2.

Further development of the theory was associated with application of numerical methods to evaluate the pipette wall function for a layered elastic medium [7]. It was also shown by Aoki *et al.* [7] that the friction between the pipette and the specimen surface does not play any significant role. Later, a boundary integral equation model was developed by Haider & Guilak [1] to account for the spherical geometry under the same constitutive assumptions as in the half-space model [11]. The influence of sample thickness in the micropipette aspiration was studied by Alexopoulos *et al.* [12], Sakamoto & Kobayashi [13] and Boudou *et al.* [14].

The half-space model [11] has been most widely adopted for extracting the stiffness characteristics via pipette aspiration [3,10,14,15], which is however applicable provided the tested specimen can be regarded as a relatively large piece of elastic continuum compared with the inner pipette diameter.

However, there is still uncertainty about the significant difference between *Φ*_{P}(*η*) and *Φ*_{F}(*η*) in a wide range of values of *η*. The present paper addresses this issue, and, moreover, in the limit cases of wide or narrow pipettes, asymptotic solutions for *Φ*_{P}(*η*) are presented, which are accurate enough to deal with a complete range of the wall parameter *η*.

Furthermore, it represents a practical interest to extend the half-space model for the case of a transversely isotropic material. Under the assumption of frictionless contact between the pipette and the surface of specimen, this extension is made via the reduction of the pipette punch problem to the corresponding contact problem. In this case, formula (1.1) can be generalized as follows:
*M*_{3} is the so-called indentation elastic modulus [16]. In the case of transversely isotropic material, it can be expressed in terms of the material stiffnesses in the following form [17]:
*A*_{11}, *A*_{13}, *A*_{33} and *A*_{44} are four independent elastic constants, which are related to the engineering elastic constants (*E*, *E*^{′}, *ν*, *ν*^{′} and *G*^{′}) by the formulae *A*_{44}=*G*^{′} and

In the isotropic case, we have
*ν* is Poisson’s ratio. Note that for incompressible isotropic materials, when

The rest of the paper is organized as follows. In §2, the pipette aspiration problem for a transversely isotropic half-space specimen is formulated and asymptotic conditions at infinity are imposed, which distinguish two different cases of a fixed specimen (see §2a) and a free-standing specimen (see §2b). In §3, the pipette aspiration problem is reduced to a contact problem for an annular indenter with a non-flat base. We note here that different aspects of the contact problem for an annular indenter were studied in a large number of papers (e.g. [18–23]), and recently it has attracted considerable attention in the case of adhesive contact [24–26]. The indentation problem for an annular indenter is solved by asymptotic methods in §4. In §5, an asymptotic solution of the pipette punch problem is obtained in the case of a thick pipette. The case of a narrow pipette is treated by another asymptotic method in §6. Finally, in §7, the discussion of obtained results is outlined.

## 2. Pipette aspiration problem formulation

Let us assume that a cylindrical flat-ended pipette with inner radius *b* and outer pipette radius *a* is used to aspirate a circular portion, 0≤*r*<*b*, of the surface of a transversely isotropic elastic half-space, *z*>0, where (*r*,*z*) are cylindrical coordinates (figure 2).

The wall parameter, which is defined as the ratio of the wall thickness, 2*h*=*a*−*b*, to the inside radius *b*, i.e.

It is assumed that the stress imposed by the pipette on the half-space surface (due to the pressure difference Δ*P*) is constant over the inner circular region, while the outside region is stress-free, that is

Furthermore, the shear stress is assumed to vanish on the entire surface of the elastic half-space, while within the annular contact region, *b*≤*r*≤*a*, the normal displacement of the half-space surface is equal to zero (any deformation of the pipette is neglected), i.e.

It is of practical interest, to measure the aspiration length
*P*.

Now, we distinguish two different ways of fixing the tested specimen (figure 3).

### (a) Pipette aspiration of a fixed specimen

If the half-space specimen is assumed to be fixed at infinity (figure 3*a*), we impose the following asymptotic condition for the displacement vector:

In other words, the displacements should vanish at infinity.

### (b) Pipette aspiration of a free-standing specimen

If the half-space specimen is allowed to freely move upward due to aspiration (figure 3*b*), we impose the following asymptotic condition:
**e**_{z} is the unit vector in the downward direction, *δ*_{0} is an unknown vertical displacement of the specimen. The constant *δ*_{0} should be determined from the equilibrium equation
*p*(*r*) is the contact pressure distribution under the pipette, i.e.

Note that in light of (2.7), the following asymptotic condition at infinity should hold:
*σ*_{z}(*r*,*z*)=*O*((*r*^{2}+*z*^{2})^{−1}), when

## 3. Reduction of pipette aspiration problem to a contact problem

In the absence of the pipette, according to the known solution of Love’s circular patch problem [28–30], the vertical surface displacement profile of the elastic half-space loaded by negative uniform pressure, −Δ*P*, acting over a circular region of radius *b* is given by
*M*_{3} is the indentation modulus of the elastic half-space (see equation (1.4)), **K**(*k*) and **E**(*k*) are, respectively, the complete elliptic integrals of the first and second kind of the modulus *k*∈[0,1).

Therefore, the contact pressure, *p*(*r*), under the pipette punch should satisfy the following integral equation:
*δ*_{0} is equal to zero for a fixed tested specimen (figure 3*a*), or should be determined from the equilibrium equation (2.7) in the case of a free standing specimen (figure 3*b*).

According to the principle of superposition, the aspirated length is evaluated as the sum

In the case of a fixed specimen, when *δ*_{0}=0, formula (3.4) reduces to

Therefore, in view of (3.2) and (3.4), formulae (3.3) and (3.5) can be represented in the following form (which generalizes equation (1.1) introduced in [11]):
*Φ*_{P}(*η*) are the so-called pipette wall functions (for a free-standing and a fixed specimen, respectively); they depend solely on the wall parameter *η*, which was introduced by (2.1). Thus, the pipette aspiration problem (2.2), (2.3) is reduced to solving the integral equation (3.2) of contact problem for a frictionless annular punch.

## 4. Indentation of an elastic half-space by an annular punch

### (a) Governing integral equation of the contact problem and the surface deflection for a flat-ended annular punch

We consider the frictionless indentation of an elastic half-space *z*>0 by a rigid axisymmetric punch with a ring-shaped contact region *b*<*r*<*a*. This problem was studied in detail in a number of papers [18,19,21,22]. In particular, the contact pressure *p*(*r*) under a flat-ended punch satisfies the integral equation
*M*_{3} is the indentation elastic modulus, *a* and *b* are the outer and the inner radii of the contact area, respectively, and *δ*_{0} is the punch displacement.

In the equilibrium state, the contact force is given by

Following Gubenko & Mossakovskii [18] and applying the method developed by Collins [31], the solution of equation (4.1) is reduced to the Fredholm integral equation of the second kind

Accordingly, the contact pressure is given by
*ρ*=*r*/*b*, so that *ρ*>1 under the punch.

At the same time, the surface deflection inside the flat-ended annular punch is evaluated as

Note finally [31] that while equation (4.3) holds for all values of *a* and *b*, approximate solutions can be only readily be obtained when *b*≪*a* (i.e. for a wide punch). In particular, the asymptotic solution of equation (4.3) can be obtained by the so-called method of large λ, which in the contact problems for anisotropic materials was recently extended in [32–34].

### (b) Indentation by a wide flat-ended annular punch

Let the contact radii ratio
*O*(λ^{6}) agree with those given by Collins [31].

Now, by making use of equations (4.2), (4.4) and the asymptotic expansion (4.7), the following approximation for the contact force can be established [31,35]:

Finally, according to (4.5) and (4.7), the surface deflection at the origin of coordinates can be evaluated as
*u*_{3}(0)→*δ*_{0} as λ→0.

### (c) Indentation by a narrow flat-ended annular punch

Let us introduce the notation

Now, we assume that the parameter *ε*=(1−λ)(1+λ), where λ is the contact radii ratio (4.6), is small, and thus, λ is close to 1.

In this case, the solution of equation (4.1) can be represented in the form

Now, applying the asymptotic method developed by Grinberg & Kuritsyn [36] (see also [20]), we obtain

Further, from (4.12) and (4.13) it follows that

Finally, the surface deflection at the origin of coordinates can be evaluated as
*Λ* becomes a large parameter when *ε*→0.

### (d) Harmonic capacity and the centre deflection for an annular punch

For a flat-ended punch, the force–displacement relationship can be represented in the form
**c** is the harmonic capacity of the contact area (see, in particular, [37–39]).

In the case of an annular contact area with the inner, *b*, and the outer, *a*, radii, according to equations (4.8), (4.14) and (4.17), the following asymptotic approximations hold:
*ε*≪1. Here, *R* and *Λ* are given by (4.10) and (4.16), respectively.

It is interesting to observe that the simple asymptotic formula (4.19), although being derived for small *ε* only, works well in the whole range of the parameter *ε*∈(0,1) (figure 4). Indeed, the difference between the predictions of the asymptotic formulae (4.18) and (4.19) is less than 0.05% in the range of small λ.

Further, the vertical (normal) displacement of the surface of an elastic half-space at the centre of a flat-ended annular punch can be represented in the form
*ψ* is a dimensionless factor.

According to equation (4.20) and the asymptotic expansions (4.9) and (4.15), the following approximations hold:
*ε*≪1.

Formulae (4.18), (4.21) and (4.19), (4.22) should be used in the cases of wide and narrow annular punches, respectively. Figure 4 shows that the obtained asymptotic solutions cover the whole range of the parameter λ∈(0,1).

## 5. Solution of the pipette punch problem for a thick pipette

### (a) Governing integral equation in the case of a fixed half-space specimen

Following Theret *et al.* [11], the pipette punch problem (3.2) with *δ*_{0}=0 can be reduced to the problem of determining a certain function *χ*(*r*,*z*), harmonic in the half-space *z*>0, vanishing at infinity and satisfying the boundary conditions

According to the method developed in [18,31], this harmonic potential appears as follows [11]:
*j*(*t*) is a real-valued function given by
*g*(*t*) is a real *odd* function, which satisfies the linear integral equation
*t*≤*b*.

At the same time, the surface profile is given by
*r*∈(*b*,*a*).

According to (5.3), the aspiration length is calculated as follows:

Since *g*(*t*) is known from equation (5.2), *j*(*t*) can be evaluated according to (5.1), and therefore the right-hand sides of (5.3) and (5.4) are known.

### (b) Asymptotic solution of the aspiration length for a thick pipette

By introducing new unknown functions

While equation (5.7) holds for all values of λ∈(0,1), approximate solutions can only readily be derived for small values of λ. Indeed, assuming that λ≪1 and applying a perturbation technique, we obtain its approximate solution in the form

Now, in view of (5.6) and (5.8), equations (2.4) and (5.5) yield the aspiration length

Further, in view of (5.6), formula (5.4) takes the form

Integrating the contact pressure density over the contact area, we evaluate the contact force

Thus, the substitution of the asymptotic expansion (5.8) into equation (5.11) results in the following expansion:

Finally, in view of (3.7) and (5.9), the pipette wall function is given by
*η* by formulae
*η*=(1−λ)/λ and *η*.

### (c) Solution in the case of a free-standing half-space specimen

It is easily seen from formula (5.12) that the equilibrium equation (2.7) is not satisfied, because the contact force *F* is not equal to the total aspiration load *πb*^{2}Δ*P*. Now, using the solution of contact problem for a flat-ended punch (see §5b), we readily obtain the following formula for the aspiration length, *L*_{0}, of a free-standing half-space specimen:
*L* is the aspiration length for a fixed specimen given by (5.9), **c** is the harmonic capacity of flat-ended punch given by (4.18), *ψ*(λ) is the normalized surface deflection at the centre of annular punch (figure 5) given by (4.9), while the contact force *F* is given by formula (5.12).

So, making use of the asymptotic expansions (4.9), (4.18) and (5.12), we can evaluate the pipette wall function

Finally, the aspiration displacement, according to (5.14), can be represented as

The application of the asymptotic expansions (4.18) and (5.12) in (5.16) allows us to derive the following approximation:
*δ*_{0}<0, meaning that the rigid displacement of the elastic specimen is directed towards the pipette.

## 6. Solution of the pipette aspiration problem for a narrow pipette

### (a) Aspirated length and aspiration approach

On the basis of the analysis performed in §3 (see equations (3.3) and (3.4))), in the case of a free-standing half-space specimen, the aspirated length can be represented as
*p*_{−1}(*r*) satisfies the integral equation

Thus, taking into account the relation
*ψ* is the centre surface deflection factor (see equation (4.20)), we reduce formula (6.1) to the following one:

Recall that the constant *δ*_{0} is equal to zero for a fixed tested specimen, so that the corresponding aspirated length is given by
*L*_{0} and *L* are related as follows:

Further, recall that in the case of a free-standing specimen *δ*_{0} should be determined from the equilibrium equation
*p*(*r*) is the contact pressure density satisfying the integral equation (3.2) that is

Now, first by application of Mossakovskii’s reciprocal theorem [40], we transform equation (6.6) as

Second, taking into account the relation
**c** of the annular punch, we resolve equation (6.7) for *δ*_{0} as follows:

Finally note that the integrals in (6.3) and (6.8) contain the function *r*∈(*b*,*a*) is given by the second formula (3.1).

### (b) Asymptotic solution for the aspirated length and aspiration approach

Making use of the variable change

Observe that as *ε*→0, the modulus *k*(*ε*) and the complementary modulus *k*^{′}(*ε*) behave as *k*(*ε*)=1+*O*(*ε*) and *k*^{′}(*ε*)=*O*(*ε*^{1/2}). Therefore, the following asymptotic series can be used [42]:

In view of (6.10) and (6.13), we readily get

Let us introduce the notation
*v*_{−1}(*ε*,*α*) determines the solution *p*_{−1}(*r*) of equation (6.2).

By applying the asymptotic method of Grinberg & Kuritsyn [36], we find

Further, in light of (4.19), we put

Thus, in view of (6.8), (6.15) and (6.16), the aspiration displacement can be represented as
*δ*_{0}<0, since *Λ* is large for small *ε*. Figure 6 illustrates the variation of *δ*_{0} in the whole range of the pipette wall radii ratio λ; the difference between the asymptotic approximations (5.17) and (6.22) at the middle value λ=0.65 is less than 0.15%.

### (c) Pipette wall functions for a narrow pipette

Using the above introduced notation (6.15)–(6.17), (6.19) and formulae (6.4) and (6.5), the pipette wall function in the case of a fixed specimen, *Φ*_{P}(*η*), and the pipette wall function in the case of a free-standing specimen, *Φ*^{0}_{P}(*η*), can be represented as follows:
*η* is the wall parameter, which is related to the wall thickness parameter *ε* by the formulae

Thus, making use of the asymptotic expansions (6.11)–(6.14), we obtain from equations (6.23) and (6.24) the following approximations for the narrow pipette wall functions:
*η*=2*ε*/(1−*ε*), the asymptotic expansions (6.25) and (6.26), which are valid as *ε*→0, hold only for small values of *η*.

Figure 7 shows that the obtained asymptotic approximations allow one to evaluate the wall functions in the whole range of the pipette wall parameter λ; the difference between the predictions of the asymptotic approximations (5.13), (6.25) for *Φ*_{P}(*η*) and (5.15), (6.26) for *Φ*^{0}_{P}(*η*) in vicinities of the middle values λ=0.65 and λ=0.70 is less than 0.12% and 0.14%, respectively.

## 7. Discussion and conclusion

The objective of this study was to develop analytical tools to evaluate the local indentation stiffness of a soft tissue by means of pipette aspiration technique. The tissue was assumed to be transversely isotropic, macroscopically homogeneous and linearly elastic, whereas the specimen surface was taken to be frictionless and flat near the tested point. In this regard, we developed asymptotic approximations in the half-space model framework, which was originally introduced by Theret *et al.* [11] and represents the deformation of a tested specimen by the deformation response of an elastic half-space, so that the specimen geometry and its characteristic size do not enter the resulting relation (1.3) for the aspirated length. A unique aspect of this study was the application of the generalized Mossakovskii’s reciprocal theorem for evaluating the central surface deflection in the contact problem for an annular indenter.

Our analysis shows that it is of paramount importance to specify the boundary conditions of clamping for the tested specimen. In the framework of the half-space model, they are accounted through the asymptotic conditions at infinity (see (2.6) and (2.7)). In particular, the following two configurations were considered: (i) half-space specimen is fixed at infinity (configuration implicitly assumed in the punch model [11]); (ii) free-standing half-space specimen (configuration explicitly assumed in the FEM model [7]).

Figure 8 presents a direct comparison of the obtained asymptotic solutions with the results of FEM numerical simulations carried out by Aoki *et al.* [7] in the isotropic case for a nearly incompressible material, when *ν*=0.49. Note that a different normalization for the pipette wall function was adopted in [7]. Since the free-standing specimen configuration was used in [7], the numerical results (dotted line) fall onto the curve corresponding to *Φ*^{0}_{P}(*η*).

To our knowledge, this study is the first report with explanations of the marked differences between predictions of the punch model *Φ*_{P}(*η*) and the force model *Φ*_{F}(*η*), both of which were introduced by Theret *et al.* [11]. Our findings (figure 9) imply that, from one side, the pipette wall function *Φ*_{F}(*η*) of the force model approaches the pipette wall function *η*→0, that is in the narrow pipette limit. From the other side, as is expected, the wall function *Φ*^{0}_{P}(*η*) goes close to the pipette wall function *Φ*_{P}(*η*) of the punch model [11] when *Φ*_{P}(*η*) and *Φ*^{0}_{P}(*η*) may be significant, especially in the case of a narrow pipette, when the wall parameter *η* is small.

## Authors' contributions

I.A. and G.M. worked together on each part of this paper.

## Competing interests

We declare we have no competing interests.

## Funding

The authors acknowledge support from EU project HORIZON2020 RISE Marie Sklodowska Curie grant no. MATRIXASSAY No 644175.

- Received July 12, 2016.
- Accepted September 1, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.