Intermolecular Forces Model for Lipid Microbubble Shells - Langmuir

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Interface Components: Nanoparticles, Colloids, Emulsions, Surfactants, Proteins, Polymers

An intermolecular forces model for lipid microbubble shells Mark Andrew Borden Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b03641 • Publication Date (Web): 13 Dec 2018 Downloaded from http://pubs.acs.org on December 18, 2018

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Figure 1 157x137mm (300 x 300 DPI)

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An intermolecular forces model for lipid microbubble shells Mark Andrew Borden Mechanical Engineering, University of Colorado, Boulder, CO, USA, 80309-0427

Abstract Lipid-coated microbubbles are currently used clinically as ultrasound contrast agents for echocardiography and radiology, and are being developed for many new diagnostic and therapeutic applications. Accordingly, there is a growing need to engineer specific formulations by employing rational design to guide lipid selection and processing. This approach requires a quantitative relationship between lipid chemistry and interfacial properties of the microbubble shell. Just such a model is proposed here based on lateral coulomb and van der Waals interactions between lipid head and tail groups, using previous coarse graining and force fields developed for molecular dynamics simulations. The model predicts with sufficient accuracy the monolayer permeability, the elasticity as a function of either lipid composition or temperature, and the equilibrium spreading surface tension of the lipid onto an air/water interface. In the future, the intermolecular forces model could be employed to elucidate more complex phenomena and to engineer novel microbubble formulations.

Introduction The biomedical utility of microbubbles was first realized in the late 1960’s with the observation by Gramiak and Shaw of enhanced ultrasound signals from the aortic valve after the injection of dyes.1 They determined that bubbles formed by agitation of the dye solution were 1


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responsible for enhancing the ultrasound echoes. This important discovery opened the possibility of improving the ultrasound scattering contrast between blood and tissue to image cardiac and vascular structures. Research on injectable microbubbles progressed over the ensuing years, and the first ultrasound contrast agent for echocardiography was approved in the 1990’s for clinical use. So-called “first-generation” microbubbles comprising an albumin shell and air core were sufficiently stable and small ( 0] for like charges, 11


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and attractive [𝑤(𝑟) < 0] for opposite charges. The potential energy scales as 1/𝑟, which is the and longest range of the pair-potentials, and the force scales as 1/𝑟2. The dielectric constant 𝜀 of water in the headgroup region weakens the interaction. Polar and Nonpolar Groups. Many of the important molecules involved in microbubble fabrication have polar groups, such as the glycerol backbone of a phospholipid. A group is polar if it forms a permanent dipole. A dipole occurs when there is partial charge separation within the group, owing to asymmetry in the electron distribution between the constituent atoms. The affinity of an atom to the electron pair of a covalent bond is termed the electronegativity. When two atoms of different electronegativity form a bond, there is a natural charge separation with the more electronegative atom gaining a partial negative charge. In the O—H bond of a water molecule, for example, oxygen (electronegativity = 3.44) has a higher affinity for the electron pair than hydrogen (electronegativity = 2.20). The oxygen atom generates a partial negative charge, while the hydrogen assumes a partial positive charge. Asymmetry in the partial charges results in a permanent dipole moment 𝑢, which has both direction and magnitude in units of C m. The convention is for the vector of a dipole to point from the negative to the positive partial charge. The charge-dipole interaction is the second strongest of the pair-potentials. An example is the interaction between a polar water molecule and the charged phosphate group of a lipid in the microbubble shell. The charge-dipole interaction is attractive [𝑤(𝑟) < 0] when the dipole points toward the charge, and it is repulsive [𝑤(𝑟) > 0] when the dipole points away from the charge. The dipole tends to orient toward the charge in such a way as to maximize the attractive force or, equivalently, minimize the free energy. The dipole produces a weaker magnetic field than a charge, and thus the potential energy for a fixed dipole-charge interaction scales as 1/𝑟2, and the force scales as 1/𝑟3. 12


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The dipole-dipole interaction is the next strongest pair potential. The free energy is minimized when the dipoles are oriented with opposite charges in close proximity. In this fixed orientation, the potential energy scales as 1/𝑟3, and the force scales as 1/𝑟4. The charge-dipole and dipole-dipole interactions are weakened in water owing to its large dielectric constant. Nonpolar groups do not ionize and have a symmetric partial charge distribution. These groups interact with each other through fluctuations in their electron states that lead to transient dipole moments. Such charge fluctuations can occur through the interaction with electromagnetic waves, such as light (hence the term “dispersion” interaction). The electron configuration of a nonpolar group can also become polarized by a nearby charged or polar group. The ability of a nonpolar group to become polarized is given by its electric polarizability 𝛼 in units of C2 m2 J-1. The magnitude of the pair-potential depends on the strength of the electric field that polarizes the nonpolar group, and generally follows the trend: charge-nonpolar > dipole-nonpolar > nonpolarnonpolar. The potential energy between two nonpolar molecules scales as 1/𝑟6, which is much shorter range than the other pair-potentials, and the force scales as 1/𝑟7. The dispersion interaction is ubiquitous – it is always present, even between charged and polar groups. Interestingly, freely rotating dipoles also give pair-potentials that scale as 1/𝑟6 18. This happens for both in the dipole-dipole (Keesom) interaction and the dipole-nonpolar (Debye) interaction. In these cases, the rotating dipole produces transient attraction and repulsive forces, thus decreasing the range of the interaction. The attractive force is usually not strong enough to overcome the rotational kinetic energy and fix the dipole, but the interaction does bias the molecular tumbling in such a way as to always produce an attractive force. The interaction weakens as the groups gain kinetic energy with increasing temperature. Conceptually, the rotating dipole is similar to the nonpolar group in that it produces a fluctuating electric field. As a result, 13


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the dipole interactions can be lumped together with the dispersion interaction, since the pairpotentials all scale as 1/𝑟6, to give an overall van der Waals force. Steric Repulsion.

The electrons of each atom are confined to molecular orbitals

surrounding the protons and neutrons of the nucleus. The Pauli Exclusion Principle states that each orbital can contain a maximum of two electrons, each of opposite spin. Therefore, as two atoms approach and their molecular orbitals begin to overlap, there is a very strong hard-sphere repulsion between them. The potential energy for steric repulsion is quite strong and very short range, and is generally written as a power law scaling as, e.g. 1/𝑟12. Such a potential may be differentiated to yield a very short-range force that scales as 1/𝑟13. As with dispersion attraction, the steric repulsion is a ubiquitous interaction that is always present between charged, polar and nonpolar groups. The Lennard-Jones Potential. The van der Waals attraction (freely rotating dipole-dipole, dipole-nonpolar and nonpolar-nonpolar) and steric repulsion interactions are always present between molecular groups. These two fundamental interactions are often lumped together in the Lennard-Jones (LJ) potential: 𝑤𝐿𝐽(𝑟) = 4𝜖

𝜎 12 𝑟

𝜎 6 𝑟

[( ) ― ( ) ]

(4)

where 𝜖 is the depth of the potential well (J) and 𝜎 is the range of the steric interaction. For the interaction of two identical groups, 𝜎 is simply equal to the group diameter. The LJ potential has shown remarkable utility in representing intermolecular interactions in analytical calculations and computer simulations. Charge interactions simply are added onto the LJ pair-potential. The LJ parameters 𝜖 and 𝜎 can be determined from the critical properties of a fluid that obeys the van der Waals equation of state. The van der Waals fluid is the simplest example of an 14


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interacting system of molecules that exhibits a phase transition, in this case a first-order transition between liquid and gas (vapor) phases.41 The ability to extract intermolecular force expressions from thermophysical properties can be quite useful for understanding the behavior of the molecular components when designing a microbubble formulation. The equations linking the LJ parameters to the critical properties for a van der Waals fluid are as follows:41 𝜎= 𝜖=

(

𝑉𝑐 3 2𝑁𝐴

1/3

)

81 2 ∙ 𝑉𝑐𝑘𝑇𝑐 160𝜋𝑁𝐴𝜎3

(5a) (5b)

where 𝑉𝑐 is the critical volume, 𝑇𝑐 is critical temperature, 𝑁𝐴 is Avogadro’s number and 𝑘 is Boltzmann’s constant. Fortunately, these expressions are valid for many of the fluids used to form the microbubble core, such as fluorocarbons.42

Figure 4. Lipid packing in a microbubble shell.

The Lipid Shell of a Microbubble. The lipid shell can be modeled by defining separate headgroup and tail regions and assuming hexagonal packing in each region (Fig. 4). The area per 15


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molecule (𝐴) is related to the separation distance between headgroups (𝑟𝐻) and tail groups (𝑟𝑇) by the following expression: 𝐴=

3 2 2 𝑟𝐻

= 3𝑟𝑇2

(6)

The intermolecular pair potential between two diacyl PC lipids is given by: (7)

𝑤 = 𝑤𝐻 + 𝑤𝑇

where the subscripts H and T refer to the head and tail groups, respectively. The headgroup pair potential is a sum of contributions from the charge-charge and LJ interactions of the choline, phosphate and glycerol groups; the tail group potential is a sum of contributions from the LennardJones interactions of the ester and methylene groups. The force field parameters developed by Orsi and Essex33 are shown in Table 2.

Table 2. ELBA force-field parameters for diacyl PC groups.33 Group

𝑄

𝜖 (kJ/mol)

𝜎 (nm)

Choline

+0.7 e

6.0

0.52

Phosphate

- 0.7 e

6.0

0.52

Glycerol

-

4.0

0.46

Ester

-

4.0

0.46

Methylene

-

3.5

0.45

The ELBA force fields are used with equations (3)-(7) to estimate the pair potentials. For simplicity, only lateral interactions between nearest neighbors are considered. Figure 5a shows the contributions for the head and tail groups, as well as the total intermolecular pair potential 16


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between two DPPC molecules.

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The charges in the headgroups results in repulsion at all

intermolecular separations, whereas the van der Waals attraction and steric repulsion between the tails leads to long-range attraction beyond a molecular area of ~0.38 nm2. Table 3 shows cohesive energies (𝑤𝐸𝐿𝐵𝐴) of the homologous series of PC lipids calculated using the ELBA force field and intermolecular separation distances calculated from the experimental headgroup areas in Table 1.

Table 3. Comparison of intermolecular pair potentials to the work of monolayer penetration. Lipid

𝑤𝐸𝐿𝐵𝐴 × 10 ―20 (J)

Wp × 10 ―20 (J)

DPPC, (16:0)2

1.62

1.7

DSPC, (18:0)2

1.99

2.0

DAPC, (20:0)2

2.37

2.4

DBPC, (22:0)2

2.72

2.8

DLiPC, (24:0)2

3.11

3.2

The intermolecular force between two lipids is given by differentiating the pair potentials and summing the head group and chain contributions: 𝐹 = 𝐹𝐻 + 𝐹𝑇 = ―

𝑑𝑤𝐻 𝑑𝑟𝐻

―

𝑑𝑤𝑇

(8)

𝑑𝑟𝑇

The corresponding coulomb and LJ contributions for the head and tail groups to the overall intermolecular force are therefore: 𝐹𝑐ℎ𝑎𝑟𝑔𝑒 = ― 𝐹𝐿𝐽 = ―

𝑑𝑤𝐿𝐽 𝑑𝑟

= 4𝜖

𝑑𝑤𝑐ℎ𝑎𝑟𝑔𝑒 𝑑𝑟

𝑄1𝑄2

= 4𝜋𝜀 𝜀𝑟2 0

[( ) ― ( )] 12𝜎12 13

𝑟

6𝜎6 𝑟7

(9a) (9b)

17


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The value of the equilibrium area per molecule is found by setting equation (8) equal to zero and solving for the intermolecular separation. The ELBA model parameters give an equilibrium area per molecule (𝐴𝐸𝐿𝐵𝐴) of ~0.45 nm2 for the diacyl PC lipids (Fig. 5b). This value was slightly less than the headgroup area of 0.47 nm2 obtained experimentally for bilayers,31 probably due to the thermal energy neglected in the calculation. The intermolecular pair potential in Figure 5a shows a rather wide and asymmetric energy well, indicating that thermal expansion is significant.

a)

2 Head Tail Total

w (J) x 10-20

1

Repulsion

0 Attraction

-1 -2 -3

0.00

0.35

b)

0.40

0.45

0.50

0.55

0.60

Area per molecule (nm2)

4

Head Tail Total

3

F (N) x 10-10

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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2 Repulsion

1 0

Attraction

-1 -2 0.35

0.00

0.40

0.45

0.50

0.55

0.60

2

Area per molecule (nm )

Figure 5. Interaction between two DPPC molecules calculated using the ELBA force fields. A) Intermolecular pair potentials. B) Intermolecular forces. See text for details on the calculation.

Lipid Shell Properties

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Gas Permeability. The gas permeability is a function of the intermolecular forces derived in the previous section. The topic of gas permeation through lipid monolayers has been studied extensively by luminaries in colloid and surface science, such as Irving Langmuir and VK La Mer, for example as a potential barrier to evaporation that could preserve surface water reservoirs.43 Even though this application ultimately failed owing to surface convection effects, the research produced a treasure trove of experimental data and theory for understanding monolayer permeation. It was shown that owing to the monomolecular nature of the film, gas permeation occurs by an energy barrier mechanism, rather than simple Fickian diffusion. The energy barrier theory predicts that monolayer permeability (𝐾) is proportional to the fraction of permeating molecules having sufficient thermal energy for penetration:44 𝐾 = 𝐾0𝑒 ― 𝑊𝑝/𝑘𝑇

(10)

where 𝐾0 is the permeability of a clean (no film) air/water interface and is estimated to be ~0.8 cm/s for molecular oxygen from statistical thermodynamics45 and 𝑊𝑝 is the work for monolayer penetration.44 The oxygen permeability of microbubble shells comprising PC at different chain lengths has been measured by an ultra-microelectrode experiment.46 Oxygen permeability was observed to scale with microstructure, and extrapolation allowed an estimation of the permeability for a pristine monolayer with zero defect density.47 This data was used with equation (10) to determine the work for monolayer penetration. For these calculations, the area per molecule was set to the experimental values (~0.47 nm2) from Sun et al.31 for bilayers (Table 1). Remarkably, this experimentally determined work for monolayer penetration has very similar values and follows the same trend with lipid acyl chain length to the cohesive energy calculated a priori from the ELBA model (Table 3).

19


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Surface Elasticity. The intermolecular forces analysis can be extended to another important interfacial property that can be independently measured: the surface elasticity. The surface pressure of the monolayer is determined by dividing the intermolecular force over the intermolecular separation: 𝐹𝐻

𝐹𝑇

(11)

𝛱 = 𝛱𝐻 + 𝛱𝑇 = 𝑟𝐻 + 𝑟𝑇

The corresponding coulomb and LJ contributions for the head and tail groups to the overall surface pressure are therefore: 𝛱𝑐ℎ𝑎𝑟𝑔𝑒 = 𝛱𝐿𝐽 =

𝐹𝐿𝐽 𝑟

𝐹𝑐ℎ𝑎𝑟𝑔𝑒 𝑟

= 4𝜖

𝑄1𝑄2

(12a)

= 4𝜋𝜀 𝜀𝑟3 0

[( ) ― ( )] 12𝜎12

6𝜎6

14

(12b)

𝑟8

𝑟

The surface elasticity (𝜒), equivalent to the surface compressibility modulus, is defined as: 𝑑𝛱

(13)

𝜒 = ― 𝑎 𝑑𝑎

where 𝑎 is surface area. Elasticity can therefore be derived from equation (9): 𝜒 = 𝜒𝐻 + 𝜒𝑇 = ―

𝑟𝐻𝑑Π𝐻 2 𝑑𝑟𝐻

―

𝑟𝑇𝑑𝛱𝑇

(14)

2 𝑑𝑟𝑇

where the terms for the coulomb and LJ interactions between the groups are given by: 𝑟 𝑑Π𝑐ℎ𝑎𝑟𝑔𝑒 𝑑𝑟

𝜒𝑐ℎ𝑎𝑟𝑔𝑒 = ― 2 𝑟 𝑑𝛱𝐿𝐽 𝑑𝑟

𝜒𝐿𝐽 = ― 2

= 4𝜖

3𝑄1𝑄2

(15a)

= 8𝜋𝜀 𝜀𝑟3 0

[( ) ― ( )] 84𝜎12 14

𝑟

24𝜎6 𝑟8

(15b)

Using these equations, the surface elasticity can be estimated from the intermolecular surface forces for each lipid species. This elasticity is strictly valid for very small-scale deformations. Table 4 shows calculated elasticity values for the series of PC lipids based on the Orsi and Essex force field parameters, along with experimentally determined values for small-amplitude 20


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oscillations obtained by a photoacoustic technique.48 For these calculations, the area per molecule was again set to the experimental values (~0.47 nm2) from Sun et al.31 for bilayers (Table 1). The intermolecular forces model employed here overestimates the elasticity for DPPC, but it agrees with experiments for DSPC and DAPC. The longest acyl-chain lipid tested in this series, DBPC, gave an anomalously low elasticity in experiments, presumably due to phase-separation of the DBPC lipid into solid domains surrounded by a continuous interdomain region of the emulsifier 1,2-distearoyl-sn-glycero-3-phosphoethanolamine-N-[methoxy(polyethylene

glycol)-2000]

(DSPE-PEG2000), which is a diacyl (18:0)2 lipid.48 This explanation is supported by fluorescence microscopy experiments showing evidence for phase separation,49 and the fact that the elasticity is of similar magnitude to that for DSPC, which like DSPE-PEG2000 is also an (18:0)2 acyl-chain lipid. Overall, the intermolecular forces model appears to provide a fairly accurate estimate of the microbubble shell elasticity at room temperature, and may provide insights into lipid packing and the effects of mixing different lipids.

Table 4. Comparison of elasticity determined by the ELBA for fields and experimental values (mean ± standard deviation) determined by a photo-acoustic technique48 for a series of PC lipids. 𝜒𝐸𝐿𝐵𝐴 (N/m)

𝜒𝑒𝑥𝑝 (N/m)

DPPC, (16:0)2

2.26

1.6 ± 0.2

DSPC, (18:0)2

2.50

2.2 ± 0.2

DAPC, (20:0)2

2.81

2.7 ± 0.6

DBPC, (22:0)2

2.73

2.3 ± 0.3

DLiPC, (24:0)2

3.12

-

Lipid

21


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The effect of temperature on microbubble shell elasticity was recently measured by Lum et al. using a novel photo-acoustic technique.50 Upon rapid heating from room temperature to body temperature (~20 to 37 oC), the microbubble was observed to grow and then shrink, and the surface elasticity and viscosity changed drastically during this process. This result indicated that the lipid film on a microbubble is highly dynamic and experiences interesting interfacial transport phenomena, such as rupture, spreading and re-sealing, that lead to transient mechanical properties. Under slow heating, where the microbubble size was approximately constant, the elasticity was shown to decrease linearly with temperature (Fig. 6). We calculated the temperature effect on microbubble shell elasticity using the ELBA force fields and the expansion coefficient (𝑑𝐴𝑐/𝑑𝑇) given by Sun et al.31 for bilayers. The calculation was made assuming that the area per molecule was twice the chain area, and thus increased with temperature by 5.8 × 10-4 nm2/K (see Table 1). The elasticity calculated by the intermolecular forces model was observed to align very closely with the experimental data. As above, the calculated elasticity was observed to be slightly greater than the experimental value, indicating that perhaps the monolayer shell in these experiments was slightly expanded beyond the equilibrium value.

3.5

Shell Elasticity (N/m)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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3.0 2.5 2.0 1.5 1.0 0.5

20

25

30

35

40

45

50

55

Temperature (°C)

22


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Figure 6. Microbubble shell elasticity as a function of temperature. Data points are measurements (mean ± standard deviation) for microbubbles coated with DSPC:DSPE-PEG2000 = 9:1 taken from Lum et al.50 The line shoes theoretical calculation using ELBA force fields. See text for details on the calculation.

Lipid Domain Sublimation.

Marmottant et al. developed a model for microbubble shell

mechanics during large-scale oscillations, which includes three regimes: buckling, elastic expansion/compression and rupture.51 According to the model, the lipid monolayer stretches and then ruptures during expansion of the gas/water interface of a growing microbubble. Lipid monolayer rupture is supported by several experimental observations, including: (1) ultrasounddriven microbubbles oscillate like free (unshelled) microbubbles at amplitudes necessary to overcome “compression only” behavior (owing to the buckling and elastic regimes);51 (2) the surface tension of expanding microbubbles during gas exchange increases from zero to ~70 mN/m, which is the surface tension of water;52 and (3) surface pressure-area isotherms on the Langmuir trough show significant hysteresis between compression and expansion, the latter showing a steeper slope indicative of fracture.52,53 Upon rupture of the lipid monolayer, one would expect the lipid molecules at the edge of the lipid domains to sublime and diffuse evenly over the bare air/water interface. Detachment of each lipid molecule from the monolayer domain matrix would require overcoming the cohesive forces between the chains of neighboring lipids, and also possibly the free energy loss associated with rehydrating the hydrophobic chains. The process is a two-dimensional analogy to sublimation because the lipids are transforming from a solid phase to a vapor phase.

23


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According to this mechanism, following rupture some lipid molecules break away from the lattice into the two-dimensional vacuum (zero surface pressure) of the bare air/water interface. This process of sublimation occurs until the surface pressure of free lipid molecules is equal to the equilibrium sublimation pressure of the lipid domains. The equilibrium surface sublimation pressure is established when the rate of sublimation of molecules away from the lipid domains is equal to the rate of molecular deposition back onto the lipid domains. The surface sublimation pressure (𝛱𝑆𝑉) for a solid is given by the two-dimensional Clapeyron equation: 𝑑𝛱𝑆𝑉 𝑑𝑇

Δ𝐻𝑆𝑉

(16)

= 𝑇Δ𝐴𝑆𝑉

where Δ𝐻𝑆𝑉 and Δ𝐴𝑆𝑉 are the changes in enthalpy and area per molecule between the twodimensional solid and vapor states, respectively. Assuming that the surface vapor is an ideal gas, then equation (16) reduces to: 𝑑ln 𝛱𝑆𝑉

Δ𝐻𝑆𝑉 = ―𝑘 𝑑(1/𝑇)

(17)

Thus, one can determine the molecular enthalpy change associated with sublimation from lipid monolayer domains by use of measurements of 𝛱𝑆𝑉 with respect to 1/𝑇. Fortunately, such data exists for DMPC, DPPC and DSPC in the form of equilibrium surface tension measurements taken by Lee et al.54 using a micropipette technique. The equilibrium surface tension of the lipid monolayer (𝛾𝑒𝑞) is converted to the equilibrium spreading pressure (𝛱𝑒𝑞) by the classic relation: 𝛱𝑒𝑞 = 𝛾0 ― 𝛾𝑒𝑞

(18)

where 𝛾0 is the surface tension of a clean air/water interface (~72 mN/m). Note that 𝛾0 decreases with temperature, and this decrease must be accounted for when determining 𝛱𝑒𝑞 as a function of 𝑇. It is next assumed that the equilibrium surface pressure is equivalent to the equilibrium surface vapor pressure: 24


Langmuir

(19)

𝛱𝑒𝑞 = 𝛱𝑆𝑉

Thus, using equations (17)-(19), one can determine the enthalpy of surface sublimation for phospholipid monolayers. Figure 7 shows a plot of the natural logarithm of the equilibrium surface vapor pressure (𝛱𝑆𝑉) as a function of the inverse temperature. Linear fits to the data points for DMPC, DPPC and DSPC yielded slopes of 5,576 K, 10,860 K and 15,900 K, respectively. The slopes from Figure 7 were input into equation (17) to give enthalpy of sublimation values of 7.7 x 10-20, 1.5 x 10-19 and 2.2 x 10-19 J/molecule for DMPC, DPPC and DSPC, respectively. These values are plotted in Figure 7, showing a linear trend with respect to acyl chain length.

7 DMPC DPPC DSPC

6

-ln SV (N/m)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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5

4

3 0.0030

0.0032

0.0034

0.0036

1/T(K)

Figure 7. Equilibrium surface vapor pressure versus inverse temperature. Points show data taken from Lee et al.,54 where equilibrium surface tension values were converted to surface vapor pressures as described in the text. Lines are linear least-square fits to the data, where the slopes are 5576 ± 568.7 K, -10860 ± 625.7 K and -15900 ± 698.6 K for DMPC, DPPC and DSPC, respectively.

Values for the enthalpy of sublimation can be estimated from the intermolecular forces model described above and compared to the experimental values derived from Lee et al.54 The 25


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lipids are assumed to be in the equilibrium packing density given by Kučerka et al.55 for DMPC and Sun et al.31 for DPPC and DSPC (Table 1). The value of Δ𝐻𝑆𝑉 is calculated by using the ELBA force fields. The calculation involves taking half of the intermolecular pair potential for the headgroup and adding it to half of the pair potential for the two chains, then multiplying by a factor of 6 to account for nearest neighbors in the hexagonal packing arrangement. Figure 8 shows that the intermolecular forces model provides estimates for DMPC and DPPC that are in excellent agreement with experiments, but underestimates the sublimation enthalpy for DSPC. The latter result suggests that DSPC molecules are slower to spread onto the bare air/water interface than the model predicts. This could perhaps be explained by the experimental methodology, which required lipid to adsorb from lipid bilayer vesicles in the medium to the air/water interface.54 It is possible that this mechanism involves an additional energy barrier, such as lipid hydration, which shifts the equilibrium to lower equilibrium surface tensions than for domain sublimation alone.

4

HSV (J) x 1019

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

3

2

1

0

14

16

18

Acyl Chain Length

Figure 8. Enthalpy of lipid sublimation from domains to the bare air/water interface. Data points show experimental values taken from equilibrium surface tension measurements by Lee et al.54 The line shows prediction from the intermolecular forces model using ELBA force fields. See text for details. 26


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Viscosity and Other Properties. Other properties of lipid-coated microbubbles have been measured, such as surface shear viscosity23 and dilatational viscosity,50 as a function of lipid acylchain length and temperature. Unfortunately, these friction-based properties are difficult to estimate using our simple intermolecular forces model due to the complex molecular mechanism involved. However, the general trends of increasing viscosity with acyl chain length that have been observed experimentally can be explained by the increased cohesive forces predicted by the model. The trends may also explain effects of lipid composition and temperature on lipid shell dilatational mechanics52 and collapse,56 as well as microbubble condensation57 and droplet vaporization.58 These trend predicted by the intermolecular forces model may also explain effects of lipid shell composition on biomedical performance, such as in vivo circulation lifetime59 and corresponding blood-brain barrier opening by focused ultrasound.60

Conclusions and Future Prospects As the number of biomedical applications for lipid-coated microbubbles continues to expand, there is an increasing demand for rational design of microbubble shells guided by quantitative relationships between lipid chemistry and the resulting interfacial properties. A new model is presented here based on lateral coulomb and van der Waals interactions between lipid head and tail groups, using previously force fields developed for molecular dynamics simulations. The model predicts with sufficient accuracy the monolayer permeability, the elasticity as a function of either lipid composition or temperature, and the equilibrium spreading surface tension of the lipid onto an air/water interface. In the future, the intermolecular forces model could be employed to elucidate more complex properties, such as surface shear and dilatational viscosity, 27


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and to explain microbubble dynamic behaviors based on the lipid kinetics. Additionally, the library of chemical groups could be expanded to investigate a broader class of lipid compositions and processing conditions. Ultimately, the approach taken here can be employed to engineer novel microbubble formulations for applications in medicine and beyond.

Acknowledgements The author wishes to thank his current and former graduate students and postdoctoral fellows for the many experiments and conversations that have motivated and informed this review. Funding for this work was provided by NSF grant DMR 1409972 and NIH grant R01 CA195051.

Glossary DPPC, (16:0)2: 1,2-dipalmitoyl-sn-glycero-3-phosphocholine DSPC, (18:0)2: 1,2-distearoyl-sn-glycero-3-phosphocholine DAPC, (20:0)2: 1,2-diarachidoyl-sn-glycero-3-phosphocholine DBPC, (22:0)2: 1,2-dibehenoyl-sn-glycero-3-phosphocholine DLiPC, (24:0)2: 1,2-dilignoceroyl-sn-glycero-3-phosphocholine

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Intermolecular Forces Model for Lipid Microbubble Shells - Langmuir

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