Particle deposition in the respiratory tract

Chapter 7

Particle deposition in the respiratory tract From discussions in earlier chapters, we already know that particle size plays an important role in determining where an inhaled pharmaceutical aerosol particle will deposit in the respiratory tract. However, several other factors, such as inhalation flow rate, affect particle deposition as well. By combining simplified lung geometries like those described in Chapter 5 with some basic fluid dynamics, it is possible to develop simple deposition models that quantitatively describe how these different factors affect particle deposition in the lung. Such models are useful in guiding the design of aerosol delivery devices, and their basis will be described in this chapter. However, because of the dramatic simplifications in lung geometry and fluid mechanics that are needed to make these models tractable, there are a number of aspects that these models do not capture (e.g., details of localized deposition hot spots within lung generations, effects of lung remodeling, and variations in regional ventilation due to disease). While experimental data can help illuminate some of these aspects, computational fluid dynamics simulations of aerosol motion in portions of the respiratory tract can also be used to provide a detailed understanding of certain effects (see, e.g., Koullapis et al., 2018; Longest and Holbrook, 2012; Kleinstreuer and Zhang, 2010). However, the simple models that are the focus of this chapter likely provide sufficient accuracy and understanding for many inhaled pharmaceutical aerosol applications. Bear in mind though that because the actual geometry of the respiratory tract is so complicated and because predicting particle trajectories throughout an entire lung is beyond prediction or measurement with current methods, our understanding of particle deposition in the respiratory tract is far from complete and remains a topic of current research. Despite this, a reasonable understanding can be achieved by considering several simplified problems to which we now turn.

7.1 Sedimentation of particles in inclined circular tubes The effect of gravity on particles inhaled into the respiratory tract can be understood to a certain extent by examining the deposition of particles in inclined The Mechanics of Inhaled Pharmaceutical Aerosols. https://doi.org/10.1016/B978-0-08-102749-3.00007-5 © 2019 Elsevier Ltd. All rights reserved.

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D = 2R

vfluid

q

L

FIG. 7.1 Fluid flow in a circular tube of diameter D and length L with the tube axis at an angle θ from the horizontal is a simple approximation for estimating particle sedimentation in the respiratory tract.

circular tubes in which there is a one-dimensional, laminar (i.e., nonturbulent) air flow. Although this is a simplification of what actually occurs in the respiratory tract, it provides a starting point in understanding sedimentation of particles in the lung. The basic geometry is shown in Fig. 7.1. What we want to determine is the fraction of particles of a given size that will deposit in a length L of a circular tube if they enter the tube uniformly distributed across the entrance to the tube. To solve this problem, we must first know the fluid velocity field in the tube. A closed form solution for the velocity field in a circular tube, known as Poiseuille flow (named after the French physician Poiseuille who performed experiments on flow in tubes in the mid1800s), is obtained by solving the Navier-Stokes equations and is given by
 (7.1) vfluid ¼ 2U 1  r 2 =R2 where U is the average velocity across the tube, r is radial distance from the tube centerline, and R ¼ D/2 is the radius of the tube. Poiseuille flow is valid only if the velocity of the fluid in the tube is steady, has only one component, and is independent of distance along the tube. These conditions will only be satisfied for laminar flow in straight tubes at distances, x, downstream from the inlet that satisfy the “fully developed” condition that x/D > 0.06Re, where Re is the Reynolds number Re ¼ ρUD/μ (White, 2016). Upstream of these locations, the velocity field is not well represented by Poiseuille flow. We saw in Chapter 6 that Re 1 for most of the conducting airways, so that we can expect Poiseuille flow to be a poor approximation to the velocity field in most of these airways. Thus, only in the smallest conducting airways, and possibly more distal regions, do we expect the flow to be similar to Poiseuille flow. Note also that deep in the lung, where sedimentation is most important, airways are covered with alveoli, making the airways much different from circular tubes. Because such alveolated ducts lack the large surface area of containing walls that would normally result in slower velocities near the duct walls, a uniform velocity field vfluid ¼ U (called “plug” flow), directed along the tube axis, may be a better approximation to the actual velocity field than Poiseuille flow in such regions when estimating sedimentation.

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Once we have specified a velocity field in the tube, then, to determine what fraction of particles will deposit in the tube, we need to determine the trajectories of particles entering the tube at all points in the tube cross section and see which particles deposit on the tube walls before they manage to exit the tube with the fluid. To determine particle trajectories, we can solve the equation of motion for each particle, which is given in Chapter 3 as
   v  vfluid νsettling d v ^ ¼ g (7.2) Stk ∗ dt U U U where we are using the average fluid velocity U as the fluid velocity scale and Stk is the Stokes number: Stk ¼ Uρparticle d2 Cc =18μD

(7.3)

νsettling ¼ Cc ρparticle gd2 =18μ

(7.4)

and

is the settling velocity, t* ¼ t/(D/U) is a dimensionless time, v is particle velocity, and ^ g ¼ g/g is a unit vector in the direction of gravity. The term in Eq. (7.2) with the Stokes number in front is responsible for deposition of particles by inertial impaction. However, from our discussions in Chapter 3, we know that sedimentation is an important deposition mechanism only in the more distal parts of the lung, where we know impaction is not an important deposition mechanism. So in terms of predicting amounts of particles sedimenting in the lung, as a first approximation, we can neglect the impaction term in Eq. (7.2). In this case, Eq. (7.2) reduces to the following equation for particle velocity: g v ¼ vfluid + νsettling ^

(7.5)

Because we are assuming vfluid is parallel to the tube for both Poiseuille flow and plug flow, Eq. (7.5) predicts that all particles in a monodisperse aerosol will settle in the vertical direction at the same speed. Thus, neglecting particleparticle interactions (which in Chapter 3 we already decided was reasonable for many pharmaceutical aerosols), no particle can overtake another particle in a monodisperse aerosol. For this reason, we can draw a line in the cross section of the tube entrance that divides those particles that will deposit in the tube from those particles that will travel through the tube to exit without depositing. Let us refer to this line as the “sedimentation line.” Particles on the sedimentation line are said to follow “limiting trajectories.” Once we determine the sedimentation line, we can determine the fraction of the particles entering the tube that deposit in the tube by calculating the mass flow rate over the two sections of the tube on either side of the sedimentation line.

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7.1.1 Poiseuille flow Using Eq. (7.5), the limiting trajectories and sedimentation line have been determined for horizontal tubes by Pich (1972) and for arbitrarily oriented tubes by Wang (1975) with the Poiseuille flow velocity. For Poiseuille flow, the fraction Ps of particles depositing in the tube is given by (Wang, 1975): Ps ¼ 1  E  Ω

(7.6)

where E is the fraction of particles escaping the tube without depositing, 8 hqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffii 2 > > for  90°  θ  0° ði:e:, uphill flowÞ γ ð1  γ Þð1  2γ Þ + arcsin 1γ > > π > > > pffiffiffiffiffiffiffiffiffiffiffiffi >     qffiffiffiffiffiffiffiffiffiffiffiffi > <2 3νsettling L νsettling 1  η2  cos θ  2 + sin θ η arcsin 1  η2   E¼ νsettling >π UD U > > sin θ π 1+ > > U > > > > : for 0° < θ  90° ði:e:, downhill flowÞ (7.7)

and Ω is the fraction of particles retreating out of the tube due to gravitational settling (and thus not depositing): 8 " rffiffiffiffiffiffiffiffiffiffiffi# > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi > < 1  1 3 sð1  sÞ + arcsin 1  s +
1  9s2  arcsin 1  s for  90°  θ  0° π 1 + 3s Ω¼ > > : 0 for 0° < θ  90°

(7.8)

Recall θ is the angle of the tube from the horizontal as shown in Fig. 7.1. The parameters appearing in Eqs. (7.7), (7.8) are  2=3 3νsettling L cos θ 4UD γ¼ (7.9) νsettling 1 sin θ 2U 0 131=3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    2 νsettling @ L νsettling L 6 cos 2 θA7 6 cos θ + 4 sin 3 θ + 36 7 6 U D D U 7 6 7 η¼6 7 6 16 7 6 5 4 2

  22=3

νsettling

2=3

sin θ U sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi!1=3    2 νsettling L L 6 cos θ + 4 sin 3 θ + 36 cos 2 θ D D U

(7.10)

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νsettling

s¼

sinθ 6U νsettling 1 sin θ 2U

(7.11)

The parameter η given in Eq. (7.10) is the solution to a third-order polynomial equation given in Wang (1975), for which Wang gives an approximate solution but for which an exact solution can be obtained as given in Eq. (7.10). Eqs. (7.6)–(7.11) are rather cumbersome. Thus, simplifications to these equations are useful. Under the condition that νsettling sin θ ≪ U, Heyder and Gebhart (1977) show that Eqs. (7.6)–(7.11) reduce to  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 (7.12) Ps ¼ 2κ ð1  κ2=3 Þ  κ 1=3 1  κ2=3 + arcsin κ 1=3 π where κ¼

3 νsettling L cos θ 4 U D

(7.13)

and use has been made of the result pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 1  arcsin 1  κ 2=3 ¼ arcsin κ 1=3 π π

(7.14)

Eq. (7.12) is symmetrical about θ ¼ 0, so that deposition with this equation is independent of whether the flow is uphill or downhill. The condition νsettling sin θ ≪ U, which is required for Eq. (7.12) to be valid, can be written as a restriction on particle size using the definition of settling velocity in Eq. (7.4) and average flow velocities in the simplified lung geometry presented in Chapter 5, yielding the result that Eq. (7.12) is a good approximation to Eqs. (7.6)–(7.11) for particles of diameter d satisfying sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 72Qμ (7.15) d≪ds where ds ¼ n 2 sin θρp gπD2n where Q is the flow rate at the trachea, n is the generation number (n ¼ 0 in the trachea), and Dn is the diameter of the nth generation airway. Examining ds vs generation number with θ ¼ 38.24° (which is a commonly used tube orientation in lung model sedimentation, as we will see shortly), we find that only in the last few alveolar generations of the lung do we expect there to be any difficulty in satisfying Eq. (7.15) for typical inhaled pharmaceutical aerosols (which have particle diameters normally between 1 and 10 μm or so). However, we have already suggested that Poiseuille flow is probably not a particularly good approximation to the flow field deep in the lung anyway, so that Eq. (7.12) is reasonable for the small conducting airways where we expect the flow to be similar to Poiseuille flow.

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Note that Eq. (7.12) gives complex numbers for the deposition fraction when κ > 1, so it is usual to set Ps ¼ 1 if κ > 1, since when κ > 3/4 the time needed for a particle to move one tube diameter perpendicular to the flow streamlines (because of sedimentation) is less than the time it takes for the average flow velocity to travel the length of the tube.

7.1.2 Laminar plug flow For plug flow, simple geometric consideration of the area between overlapping ellipses is all that is needed to separate depositing from nondepositing particles and determine the sedimentation line (since the fluid velocity is the same everywhere in the tube and the particles occupy an elliptical region that settles at constant velocity inside the vertical cross section of the tube, which is also an ellipse). These considerations are given in Heyder (1975). The fraction of particles depositing, Ps, in a circular tube in this case is given by 2 3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    2ffi 2 4 4 4 Ps ¼ 1  4 arccos κ  κ 1  κ 5 (7.16) π 3 3 3 where κ is given in Eq. (7.13). Note that Ps is only a real number for 4κ/3  1, so if κ 3/4, it is usual to set Ps ¼ 1, since if κ > 3/4 then, as mentioned in Section 7.1.1, the time for a particle to move one tube diameter perpendicular to the flow streamlines is less than the time it takes for the average flow velocity to travel the length of the tube.

7.1.3 Well-mixed plug flow The velocity field in the central airways is probably not well approximated by simple flow fields such as Poiseuille flow or plug flow. This is because of secondary flows associated with inertial effects in the curved regions of bifurcations of the larger airways as discussed in Chapter 6. The development of exact models would require simulation of the Navier-Stokes equations to predict sedimentation in these regions rigorously. However, an approximation for sedimentation in these regions can be made by assuming that the effect of the secondary flows is to produce a well-mixed aerosol in the tube cross section. (This is in contrast to the sedimentation results above for Poiseuille and plug flow where the entire aerosol settles with a well-defined upper boundary and no mixing occurs between the aerosol-free and aerosol-containing regions.) With a well-mixed aerosol, the problem then reduces to estimating sedimentation in a plug flow where the aerosol is assumed to have a uniform number density in the tube cross section. The rate of deposition in such a flow can be obtained from steady mass conservation: ð Mv  dS ¼ 0 (7.17) S

Particle deposition in the respiratory tract Chapter

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where M is the mass of aerosol per unit volume and S is the surface bounding the volume containing the aerosol under consideration. This equation is simply a statement of the fact that the rates at which aerosol mass enters or leaves the tube along its cylindrical sides or ends must sum to zero because of mass conservation. Since we are assuming a well-mixed aerosol at each tube cross section, M varies only with distance, x, along the tube. The mass flux of aerosol through the tube entrance is ð
 D2 (7.18) Mv  dS ¼ MðxÞ U + νsettling sin θ π 4 entrance

and the mass flux of aerosol exiting through the tube exit for an infinitesimally short length of tube, dx, is ð


 D2 Mv  dS ¼ Mðx + dxÞ U + νsettling sin θ π 4

(7.19)

exit

Realizing that no aerosol deposits on the upper side of the tube due to sedimentation, the mass flux of aerosol depositing on the sides of the tube, DE, is ð Mv  dS (7.20) DE ¼ bottom half of tube

With the tube axis oriented at an angle θ downhill from the horizontal, geometric considerations yield
 (7.21) v  dS ¼ νsettling cos θ ðRdϕsin ϕÞdx where ϕ denotes angular distance around a circular cross section of the tube. Eq. (7.20) can thus be written as 

 ðπ dx DE ¼ M x + νsettling cos θdx sin ϕRdϕ 2

(7.22)

0

Integrating yields



 dx νsettling cos θDdx DE ¼ M x + 2

(7.23)

Putting Eqs. (7.18), (7.19), and (7.23) into Eq. (7.17) yields  
 πD2 dx ¼ M x + ½Mðx + dxÞ  MðxÞ U + νsettling sin θ νsettling cos θDdx 4 2 (7.24)

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Expanding M(x + dx) and M(x + dx/2) in Taylor series about x, dividing through by dx, and taking the limit as dx ! 0, we finally obtain 4νsettling cos θ dM  ¼ M
dx U + νsettling sin θ πD

(7.25)

Integrating Eq. (7.25), we obtain the mass of aerosol per unit volume as a function of distance x along the tube: " # 4νsettling cos θ  x MðxÞ ¼ M0 exp 
(7.26) U + νsettling sin θ πD where M0 is the aerosol mass per unit volume at the tube entrance. The fraction of mass depositing in a tube of length L is given by Ps ¼

M0  ML M0

(7.27)

where ML ¼ M(x ¼ L). Using Eq. (7.26) to evaluate ML, Eq. (7.27) gives us our final result for the fraction of aerosol depositing in an inclined tube assuming the aerosol concentration remains uniform over the tube cross section: " # 4 νsettling cos θ L  Ps ¼ 1  exp 
(7.28) π U + νsettling sin θ D In the central and upper airways (where we expect secondary flows to give well-mixed aerosols), we have already seen that Eq. (7.15) implies νsettling sin θ ≪ U, so that Eq. (7.28) can be well approximated in these regions by   16 (7.29) Ps ¼ 1  exp  κ 3π which is a result given by previous authors for horizontal tubes (Morton, 1935; Fuchs, 1964). Here, κ is given in Eq. (7.13). The fraction of particles Ps depositing with the different types of flow we have considered is shown in Fig. 7.2 at various κ. It can be seen from Fig. 7.2 that the assumed velocity field in the tube affects deposition fractions. Each of the three approximations shown in Fig. 7.2 might be reasonable approximations for different parts of the respiratory tract, with Eq. (7.29) (well-mixed plug flow) probably the most reasonable of the three in the central conducting airways, Eq. (7.12) (Poiseuille flow) applying in the small conducting airways, and Eq. (7.16) (plug flow) applying in the alveolated airways, although none of them will exactly duplicate sedimentational deposition in an actual lung geometry since the flow there is neither strictly Poiseuille nor plug flow. Instead, if high-accuracy or detailed three-dimensional knowledge of the deposition pattern within an airway is desired, computational fluid dynamics simulation of the detailed flow and particle motion should be done, as discussed later in this chapter.

Particle deposition in the respiratory tract Chapter

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141

1 0.8 0.6 Ps 0.4 Plug flow Well-mixed plug flow Poiseuille flow

0.2

0.2

0.4

0.6

0.8

1

k FIG. 7.2 The fraction Ps of aerosol sedimenting in a tube is shown for the different velocity and aerosol fields including plug flow Eq. (7.16), well-mixed plug flow Eq. (7.29), and Poiseuille flow Eq. (7.12), as a function of the parameter κ in Eq. (7.13).

7.1.4

Randomly oriented circular tubes

The above equations allow estimation of the fraction of aerosol depositing in a tube at a known angle θ. However, the different airways in the lung are oriented in many different directions. One approach to determining where particles would deposit in the lung due to sedimentation is to track many different individual particles through many different individual paths through the lung (using a Monte Carlo approach to give the orientation of each tube, in which a random number is used to select an orientation from a distribution of tube orientations), using the above equations to approximate the amount depositing in each generation due to sedimentation. This is the approach taken originally by Koblinger and Hofmann (1990) and subsequently by various authors since. An alternative approach to dealing with sedimentation in the many different orientations θ of airways in the lung is to treat the airways as a collection of randomly oriented tubes. The average fraction of aerosol depositing in one of a randomly oriented set of tubes is then given by π=2 ð

Ps f ðθÞdθ Ps ¼

π=2

(7.30)

π=2 ð

f ðθÞdθ π=2

Here, Ps is the fraction of aerosol depositing in a tube at known angle θ and is given by one of the various approximations considered above, while f(θ)dθ is the probability of finding a tube at an angle θ. An expression for f(θ) is obtained by realizing that an infinite number of randomly oriented tubes of length L having one end centered at the origin will fill a sphere of radius L. The fraction

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The Mechanics of Inhaled Pharmaceutical Aerosols

FIG. 7.3 The area of a ring on the surface of a sphere is proportional to the fraction of randomly oriented tubes that are directed at angle θ from the horizontal.

of these tubes that are oriented between angles θ and θ + dθ is then proportional to the surface area of a ring on the sphere between these two angles as shown in Fig. 7.3. The area of this ring is simply the arc length Ldθ multiplied by the circumference of the ring 2πL cos θ, while the total surface area of the sphere is 4πL2, so the fraction of tubes oriented at angles between θ and θ + dθ is then f ðθÞdθ ¼

2πL2 cos θdθ 4πL2

(7.31)

cos θdθ 2

(7.32)

which simplifies to f ðθÞdθ ¼

Putting Eq. (7.32) into Eq. (7.30), we obtain 1 Ps ¼ 2

π=2 ð

Ps cos θdθ

(7.33)

π=2

We can evaluate Ps using Eq. (7.33) with the various different equations we have developed for Ps. However, the only case in which an exact integration of Eq. (7.33) is possible is for well-mixed plug flow (Eq. 7.29), for which we obtain "   #  0 2 4t 0:5 4t0 1  21 F 2 1, Ps ¼ 2 + πI1 , 2 (7.34) 1:5 π π

Particle deposition in the respiratory tract Chapter

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where t0 ¼

νsettling L U D

(7.35)

is the time it takes a particle to travel one tube length divided by the time the fluid takes to settle one tube diameter. Here, I1 is the first-order modified Bessel function of the first kind, while 1F2 is a generalized hypergeometric function. Hypergeometric functions are defined as the solutions of an ordinary differential equation called the hypergeometric equation. A series expansion solution to the hypergeometric equation gives the following series expansion for the hypergeometric function in Eq. (7.34):     X ∞ xk b1 1 ,x ¼ F 2 1, (7.36) 1 b2 ðb1 Þk ðb2 Þk k¼0 where the notation (α)k indicates the following product: ( αðα + 1Þðα + 2Þ⋯ðα + k  1Þ for k  1 ðαÞk  1 for k ¼ 0

(7.37)

For both plug flow Eq. (7.16) and Poiseuille flow Eq. (7.12), numerical integration of Eq. (7.33) is necessary in order to accommodate the definition that Ps ¼ 1 when κ 3/4 (for Eq. 7.16)) or κ 1 (for Eq. 7.12)). Values for the average fraction of aerosol Ps depositing in a randomly oriented circular tube with the various different types of flows using Eq. (7.33) are shown in Fig. 7.4 as a function of the parameter t0 defined in Eq. (7.35). It can be seen from Fig. 7.4 that the different flow fields give reasonably similar values to Ps . This fact combined with the knowledge that all of these flow fields are only approximations to the actual flow field in the airways, and the fact that most inhaled pharmaceutical aerosols are polydisperse (resulting in a wide range of t0 ) reduces the importance of the differences between the different Ps shown in Fig. 7.4. Because it is somewhat cumbersome to use Eq. (7.34) or to integrate Eq. (7.33) numerically for the different sedimentation functions Ps given earlier, approximations to these equations are sometimes used. In particular, for κ ≪ 1, it can be shown (Pich, 1972) that Ps for Poiseuille flow reduces to 4 Ps ¼ κ ðPoiseuille flow with κ≪1Þ π

(7.38)

Substituting Eq. (7.38) into Eq. (7.33) one obtains Ps ¼ κ ðPoiseuille flow with κ≪1Þ

(7.39)

However, Eq. (7.39) can be obtained by observing that coincidentally, for κ ≪ 1, Ps ¼ Ps jθ¼ arccos ðπ=4Þ

(7.40)

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1 0.8 0.6 Ps

0.4

Plug flow Well-mixed plug flow Poiseuille flow

0.2

0.5

1

1.5

2

2.5

3

t′ FIG. 7.4 The average fraction Ps of aerosol sedimenting in a tube with randomly chosen angle from the horizontal is shown as a function of the parameter t0 in Eq. (7.35) by evaluating Eq. (7.33) for the different velocity and aerosol fields, including plug flow Eq. (7.16), well-mixed plug flow Eq. (7.29), and Poiseuille flow Eq. (7.12).

For this reason, a commonly used empirical approximation for Ps for Poiseuille flow is to use Eq. (7.40) for all κ, not just κ ≪ 1 (Heyder and Gebhart, 1977). In this case, Eq. (7.12) is evaluated using a constant value of θ ¼ 38.24° ¼ arccos(π/4) for all airways in the lung. For well-mixed plug flow, a sometimes used empirical approximation for Ps is Ps ¼ Ps jθ¼0 ðhorizontal well  mixed plug flowÞ

(7.41)

which simply assumes that all the airways are horizontally oriented. The two simplified sedimentation equations in Eqs. (7.40), (7.41) are shown in Fig. 7.5 together with the more rigorous sedimentation results obtained by

1 0.8 0.6 Ps

Plug flow Well-mixed plug flow Poiseuille flow Poiseuille flow for q = 38.24° Horizontal well-mixed plug flow

0.4 0.2

0.5

1

1.5

2

2.5

3

t′ FIG. 7.5 The two simplified equations, Eq. (7.40) (Poiseuille flow for θ ¼ 38.24°) and Eq. (7.41) (horizontal well-mixed plug flow), for the average fraction Ps of aerosol sedimenting in a circular tube with randomly chosen angle from the horizontal are shown with the more rigorously derived values shown already in Fig. 7.4.

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integrating Eq. (7.33) numerically for Poiseuille flow (Eq. 7.12), plug flow (Eq. 7.16), and well-mixed plug flow (Eq. 7.29). It can be seen that Eq. (7.40) is a reasonable approximation to the more general Poiseuille flow result for most t0 , while Eq. (7.41) is a reasonable approximation to either of the two more general plug flow results especially for t0 < 0.5. Example 7.1 Estimate the probability that a 3 μm particle of density 1000 kg/m3 entering the 20th generation of the idealized lung geometry given in Chapter 5 will deposit in that generation by sedimentation if the inhalation flow rate is 50 L/min. Solution An average value of this probability can be estimated by assuming this generation is randomly oriented with respect to the horizontal and using one of the equations developed above for randomly oriented tubes and shown in Fig. 7.5. For this purpose, we need to evaluate the parameter t0 given in Eq. (7.35): t0 ¼

νsettling L U D

(7.35)

To do this, we must evaluate the settling velocity, which we know from Chapter 3 is given by νsettling ¼ ρparticle gd 2 =18μ where μ is the viscosity of air (μ ¼ 1.8  105 kg/m s). Thus, we obtain
2
 νsettling ¼ 1000 kg=m3∗ 9:81m=s2∗ 3  106 m = 18∗ 1:8  105 kg=ms ¼ 0:273mm=s Also, we evaluate U from the volume flow rate Q, since Q ¼ U*(cross-sectional area of 20th generation), so that
 U ¼ Q= 220∗ πD 2 =4 But from Chapter 5, we know that generation 20 of our idealized lung model has diameter D ¼ 0.033 cm, so

U ¼ 50L= min ∗ 1000cm3 =L∗ 1 min =60s= 220∗ π ∗ ð0:033cmÞ2 =4 ¼ 0:929cm=s Using the airway length L ¼ 0.068 cm from Chapter 5, we thus obtain t 0 ¼ ð0:0273cm=s=0:929cm=sÞ∗ð0:068cm=0:033cmÞ t 0 ¼ 0:061 If we assume a randomly oriented tube, then from Fig. 7.5, we obtain estimates for the average sedimentation probability as P s ¼ 0.059 for randomly oriented Poiseuille flow P s ¼ 0.059 for well-mixed plug flow P s ¼ 0.061 for plug flow We also obtain P s ¼ 0.075 for horizontal plug flow and for Poiseuille flow oriented at 38.24°.

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We see that the three randomly oriented results differ little, while the two simplified Eqs. (7.40), (7.41) both overestimate deposition by approximately 25%. Such differences are not a major concern given that we are making several dramatic simplifications (e.g., this is actually a three-dimensional geometry where the walls of the duct would be partially covered by alveoli, besides being a bifurcating shape, not cylindrical, not to mention being a time-dependent geometry as the lung inflates or deflates during the inhalation maneuver). If less uncertainty in predicted deposition is desired, computational fluid dynamics simulations should be done, as discussed later in this chapter, in order to remove some of these simplifying assumptions.

7.2 Deposition by impaction in the lung The other main mechanism that causes inhaled pharmaceutical aerosols to deposit in the respiratory tract is inertial impaction. We might at first think that we could proceed as we did with sedimentation and develop relatively simple approximate equations for estimating impaction from theoretical solutions of the governing equations. However, inertial impaction in the lung occurs because particles are unable to follow the curved streamlines that the air follows in passing through bifurcations. Thus, to develop a model that might predict impaction with reasonable accuracy, we must at least have a fluid velocity field that duplicates the curved nature of the streamlines that occurs in the lung. Thus, we cannot use such simple straight tube flows as plug flow or Poiseuille flow as we did in estimating sedimentation. Instead, more complex flows must be considered. However, before considering impaction in such flows, it is worthwhile deciding what parameters will be important in determining impaction. From our discussions in Chapter 3 on similarity of particle motion, we know that the motion of a spherical particle in a given geometry with low particle Reynolds number is affected only by the following parameters: Stokes number Stk, flow Reynolds number Reflow, and nondimensional settling velocity. These parameters are given by Stk ¼ U0 ρparticle d 2 Cc =18μD Reflow ¼ ρfluid U0 D=μ nondimensional settling velocity: νsettling =U0 Here, U0 and D are a characteristic flow velocity and linear dimension, respectively, in the given geometry; d is particle diameter, and Cc is the Cunningham slip correction factor. In addition to these three parameters, geometric parameters (e.g., branching angle and parent/daughter diameter ratio) and the actual geometric shape of an airway can affect impaction. If all these parameters are important in determining deposition, it would be difficult to develop simple

Particle deposition in the respiratory tract Chapter

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formulas for estimating impaction. However, from Chapter 3, we know that if we compare the Stokes number with the nondimensional settling velocity, we obtain an estimate of the importance of impaction versus sedimentation. This comparison is readily done by examining the Froude number Fr, where
 (7.42) Fr ¼ Stk= νsettling =U0 or Fr ¼ U0 2 =ðDgÞ Its value is shown in Fig. 7.6 throughout the idealized lung geometry of Chapter 5 for two different inhalation flow rates. From Fig. 7.6, it is apparent that sedimentation is negligible compared with inertial impaction throughout the conducting airways at an inhalation flow rate of 60 L/min, while at the tidal breathing flow rate of 18 L/min, this is true only in the large airways. In these regions, impaction is the dominant mechanism of deposition, and we expect impaction to be independent of the sedimentation parameter νsettling/U0, so that the only parameters affecting impaction then are the dynamical parameters Stk and Reflow, in addition to nondimensional geometric parameters that characterize the bifurcation geometry. However, numerous experimental and numerical studies have measured deposition of particles in airway replicas and casts, finding that for typical airway geometries in normal lungs and for the Reynolds numbers encountered in the regions where impaction is important (typically Reflow > 1), inertial impaction is only weakly dependent on both Reflow and the various geometric parameters. Thus, we come to the empirical conclusion that for inhaled pharmaceutical aerosols, for which sedimentation and impaction are the dominant deposition mechanisms, deposition

FIG. 7.6 The Froude number Fr from Eq. (7.42), defined here as the ratio of Stokes number to nondimensional settling velocity, is shown as a function of airway generation in the idealized lung geometry of Chapter 5 for two inhalation flow rates. Large values of Fr correspond to regions where sedimentation can be expected to be small compared with impaction, while small values of Fr correspond to regions where impaction can be expected to be small compared with sedimentation.

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by inertial impaction in the airways can be approximated as being only a function of the Stokes number. The result that inertial impaction can be approximated as being dependent on only the Stokes number for normal airway geometries is a major simplification. However, obtaining the functional dependence of impaction on Stokes number requires us to duplicate the curvature of streamlines in the airways that leads to this impaction. A variety of flows have been considered by various authors for this purpose, varying from the simple (e.g., flow in a bent circular tube) to the complex (e.g., simulations of the Navier-Stokes equations in realistic human airways), and also include experiments on the deposition of monodisperse aerosols in models and casts of human airways. These studies give various empirical correlations that approximate the variation of impaction efficiency with Stokes number, with the different correlations giving quite a range of impaction probabilities. However, most of these equations are based on experiments or theory using only a single airway generation, often without cartilaginous rings or without consideration of the effect that upstream generations, including the larynx, have on the fluid dynamics in the generation, leading to inaccurate predictions of impaction deposition in actual airways. Fortunately, the following equation of Chan and Lippmann (1980) Pi ¼ 1:606 Stk + 0:0023

(7.43)

is based on data that include such factors (it is based on experiments in casts of airways), so that this equation represents typical values that would be expected in an actual lung. Indeed, Borojeni et al. (2014) have confirmed the accuracy of this equation using realistic airway replicas of the first few lung generations of five adults, built from high-resolution medical images. Borojeni et al. (2014) also find that Eq. (7.43) accurately predicts deposition in replicas of the airways of children (age 2–8 years). Example 7.2 Use Eq. (7.43) to estimate the probability that a 3 μm diameter particle of density 1000 kg/m3 entering the 10th generation of the idealized lung geometry given in Chapter 5 will deposit in that generation by impaction if the inhalation flow rate is 30, 60, and 90 L/min. Solution Eq. (7.43) is Pi ¼ 1:606 Stk + 0:0023 To use this equation, we must first calculate the Stokes number Stk ¼ U0 ρparticle d 2 Cc =18μD Here, we must evaluate U0 from the volume flow rate Q, since Q ¼ U0*(crosssectional area of 10th generation), so that
 U ¼ Q= 210∗ πD 2 =4

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But from Chapter 5, we know that generation 10 of our idealized lung model has diameter D ¼ 0.162 cm, so
 U0 ¼ Q L= min ∗ 1000cm3 =L∗ 1 min =60s= 210∗ π ∗ 0:1622 cm2 =4 ¼ 0:007896Q m=s, where Q is in L= min Putting this into our definition of Stk and approximating Cc ¼ 1, then
2
 Stk ¼ 0:007896 Q m=s ð1000kg=m3 Þ 3  106 m = 18∗ 1:8  105 kg=ms∗ 0:00162m ¼ 0:000135 Q where Q is in L=min Thus, we have Stk ¼ 0:0041 for Q ¼ 30L= min Stk ¼ 0:008 for Q ¼ 60L= min Stk ¼ 0:012 for Q ¼ 90L= min Putting these into Eq. (7.43), we have the probability of impaction in this generation as Pi ¼ 0:0088 for Q ¼ 30L= min Pi ¼ 0:0153 for Q ¼ 60L= min Pi ¼ 0:0218 for Q ¼ 90L= min We see that the probability of impaction is linearly dependent on flow rate, so that an aerosol consisting of particles of this size would deposit significant amounts in the conducting airways at the higher flow rates but would penetrate well into the alveoli at low flow rates. This is partly why there are no strict criteria as to what is an appropriate particle size for inhaled pharmaceutical aerosols—deposition in the airways is dependent not only on particle size but also on flow rate, which can vary considerably in patients.

7.3 Deposition in cylindrical tubes due to Brownian diffusion We have already seen in Chapter 3 that molecular diffusion (i.e., Brownian motion) can play a role in the deposition of small-diameter inhaled pharmaceutical aerosols in the respiratory tract. To proceed to rigorously estimate diffusion in the respiratory tract would require us to solve the Navier-Stokes equations for the fluid flow and then solve either the equations of motion for a particle moving in this flow field with Brownian motion superposed on the trajectory, or else solve a convection-diffusion equation for the aerosol concentration with this velocity field. With the latter approach, we need to solve a standard convection-diffusion equation for the aerosol concentration (Fuchs, 1964): ∂n + r  ðnvÞ ¼ Dd r2 n ∂t

(7.44)

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subject to the boundary condition that n ¼ 0 at walls and appropriate initial conditions. Here, Dd is the diffusion coefficient, which we gave in Chapter 3 for spherical particles as. Dd ¼

kTCc 3πμd

(7.45)

and v is the bulk velocity field of the particulate phase (usually assumed equal to the fluid velocity for the simplified case here of a uniform aerosol concentration and deposition due to diffusion alone). Because of the complexity of solving the Navier-Stokes equations and Eq. (7.44) in such a complicated geometry as the respiratory tract, approximations are desirable. Two such approximations are obtained by solving Eq. (7.44) with either an assumption of a Poiseuille flow velocity field Eq. (7.1) or a uniform, plug flow velocity field. For Poiseuille flow, several authors have solved Eq. (7.44) with various simplifying asymptotic approximations to the resulting infinite series (Townsend, 1900; Nusselt, 1910; Gormley and Kennedy, 1949; Ingham, 1975) to obtain expressions for the average deposition probability Pd in a cylindrical tube with Poiseuille flow. For example, Ingham (1975) gives Pd ¼ 1  0:819e14:63Δ  0:0967e89:22Δ  0:0325e228Δ  0:0509e125:9Δ 2=3 (7.46) while Gormley and Kennedy (1949) give the commonly used result accurate for Δ < 0.1: Pd ¼ 6:41Δ2=3  4:8Δ  1:123Δ4=3

(7.47)

where in both Eqs. (7.46) and (7.47), Δ¼

kTCc L 1 3πμd U 4R2

(7.48)

and k ¼ 1.38  1023 J/K is Boltzmann’s constant, T is the gas temperature, μ is the gas viscosity, Cc is the Cunningham slip correction factor, d is particle diameter, R is the airway radius, L is airway length, and U is average flow velocity in the tube. If a plug flow velocity field is assumed, Eq. (7.44) reduces to the same equation as that for a stationary aerosol residing in a cylindrical tube for a time t ¼ L/U. This equation is simply the time-dependent diffusion equation, which can be solved analytically by straightforward separation of variables to obtain (Buchwald, 1921; Fuchs, 1964) Pd ¼ 1  4

∞ X 1 4λ2m Δ e 2 λ m¼1 m

(7.49)

where λm is the mth zero of the zero-order Bessel function J0. The sum in Eq. (7.49) converges slowly for small values of Δ, so that a large number of terms in the sum are required for reasonable accuracy when Δ is small.

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FIG. 7.7 Probability of deposition due to diffusion, Pd, is shown for plug flow Eq. (7.60) and Poiseuille flow Eqs. (7.46), (7.47) in a cylindrical tube as a function of the parameter Δ in Eq. (7.48).

A comparison of Pd for Poiseuille flow using Eqs. (7.46), (7.47) and the result for plug flow (Eq. 7.49) are shown in Fig. 7.7. It can be seen that Eqs. (7.46), (7.47) differ negligibly, but plug flow (Eq. 7.49) gives considerably higher deposition probabilities than either of these equations at low values of Δ. Because the flow in the airways and alveolar regions of the lung is neither plug flow nor Poiseuille flow, none of Eqs. (7.46), (7.47), or (7.49) will exactly predict diffusional deposition in the lung airways. For this reason, various authors have examined deposition in geometries more closely resembling airways and developed alternative models for diffusional deposition, some of which account for the nonplug/non-Poiseuille flow nature in the conducting airways and the effect of alveoli. However, much of this work on diffusional deposition is aimed at deposition in the conducting airways or at particles that are much smaller than those occurring in inhaled pharmaceutical aerosols, so that at this point, it is worth examining how important diffusion is expected to be as a deposition mechanism in the different parts of the lung for inhaled pharmaceutical aerosols. We have already examined this in an order of magnitude manner in Chapter 3, but let us reexamine this issue more specifically. In particular, we can use Eqs. (7.46) or (7.49) to compare diffusional deposition probabilities with sedimentation and impaction probabilities from the equations we saw earlier in this chapter. Shown in Figs. 7.8 and 7.9 is the ratio of the probability of diffusional deposition to the probability of deposition by either impaction or sedimentation. In Fig. 7.9, κ  0.3 for the parameter range shown, so that Eq. (7.40) is a reasonable approximation for sedimentation probability in a randomly oriented tube with Poiseuille flow, while Eq. (7.41) is a reasonable

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FIG. 7.8 The ratio of the probability of deposition of a 3.5 μm-diameter particle due to sedimentation versus diffusion (Ps/Pd) and impaction versus diffusion (Pi/Pd) is shown in each generation of the idealized lung geometry of Chapter 5 for an inhalation flow rate of 60 L/min. Plug flow is assumed in the different lung airways in calculating sedimentation and diffusion. Thus, Ps is from Eq. (7.41), and Pd is from Eq. (7.60). Pi is from Eq. (7.43).

FIG. 7.9 Same as Fig. 7.8, but now, for a 2.0 μm-diameter particle, Poiseuille flow is used for the calculation of sedimentation and diffusion probabilities. Thus, Ps is from Eq. (7.40), and Pd is from Eq. (7.46), while Pi is still from Eq. (7.43).

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approximation for sedimentation probability in a randomly oriented tube with plug flow in Fig. 7.8. From Figs. 7.8 and 7.9, we see that during inhalation at 60 L/min, diffusion starts to become nonnegligible only in the alveolar region (generations 15 and higher). This fact makes the empirical correlations developed for diffusional deposition in the conducting airways (e.g., Yu and Cohen, 1994; ICRP, 1994) less useful for inhaled pharmaceutical aerosols than for occupational exposure aerosols, since these equations only apply to the conducting airways (where Figs. 7.8 and 7.9 show that diffusion is unimportant for inhaled pharmaceutical aerosols). Fig. 7.8 also shows that for plug flow, diffusion begins to become nonnegligible in the alveolar region once particles have diameters smaller than approximately 3.5 μm, but Fig. 7.9 shows that for Poiseuille flow, we need particle diameters below approximately 2.0 μm for this to occur. At first sight, this may not be apparent from Figs. 7.8 and 7.9, but once it is realized that the ratios Pi/Pd and Ps/Pd both increase with particle size, it is seen that the particle sizes shown in Figs. 7.8 and 7.9 represent the approximate critical particle sizes where both these ratios together drop below 10 over a significant part of the lung. The difference in critical particle size below which diffusion becomes important (i.e., 3.5 μm for plug flow vs 2.0 μm for Poiseuille flow) is due to the much larger diffusional deposition probabilities that occur with plug flow vs Poiseuille flow (see Fig. 7.7) at small Δ. Recall that in Chapter 3, we roughly determined that diffusion would become important relative to sedimentation for particle sizes on the order of 3.5 μm for an inhalation flow rate of 60 L/min, which agrees well with our result here for plug flow. Note that at lower flow rates, diffusional probabilities decrease in importance relative to sedimentation as discussed in Chapter 3, so that smaller particle sizes are needed for diffusion to become important at lower flow rates (e.g., in Chapter 3 at 18 L/min, we estimated that diffusion was negligible for particles larger than approximately 3 μm in diameter). It should be noted that Figs. 7.8 and 7.9 give the relative importance of diffusion while air is flowing through the lung airways. If there is a breath hold at the end of inhalation, then because the amount of aerosol sedimenting in a given time is approximately proportional to time and because the time occupied by inhalation is often considerably less than a typical breath hold, the amount of aerosol sedimenting during inhalation is much smaller than that sedimenting during the breath hold. In this case, diffusion becomes less important as a deposition mechanism, as we already mentioned in Chapter 3. Indeed, we can calculate probabilities for sedimentation or diffusion during a breath hold by replacing the particle residence time L/U with tbreath hold in the equations we developed above for sedimentation or diffusion probabilities. Doing so, we find that diffusion probabilities with either Eq. (7.46) or (7.49) are <1/3 the sedimentation probabilities in Fig. 7.4 for particles larger than 1 μm with a 10 s breath hold.

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In conclusion, we see that diffusion is not generally a dominant mechanism of deposition for inhaled pharmaceutical aerosols when a breath hold is used, but can become nonnegligible during the act of inhalation itself in the more distal parts of the lung (where impaction probabilities are small). For this reason, diffusion is usually included when modeling the fate of inhaled pharmaceutical aerosols, even though its importance may not be large for many such aerosols, particularly if breath holding is performed. It should be noted that in low-gravity environments where sedimentation becomes negligible, Figs. 7.8 and 7.9 indicate that diffusion is likely to be the dominant mechanism of deposition in the distal parts of the lung.

7.4 Deposition in the mouth and throat Equations for sedimentation, impaction, and diffusion like those given in the previous sections are useful in understanding the behavior of aerosols in the lung. However, a significant portion of an inhaled aerosol may never reach the lung because of the filtering effect of the mouth and throat (the nose is an even better filter, and for this reason, delivery of inhaled pharmaceutical aerosols to the lung normally avoids nasal inhalation, if possible, because of high losses in the nose). To achieve drug delivery to the lungs, it is important to know how much of the aerosol and what particle sizes are able to travel past the mouth and throat. Unfortunately, the geometry of the mouth and throat does not resemble any simple idealized shape and varies considerably between individuals, so that simple analyses (like the straight tube deposition equations we have looked at in modeling deposition within the lung), or experiments in very simple geometries, do not normally provide reasonable accuracy. In addition, the fluid mechanics in the mouth-throat is normally turbulent and dependent on the fluid dynamics created by the inhalation device placed at the mouth (DeHaan and Finlay, 2001). In the case of pressurized metered-dose inhalers (pMDIs), the aerosol enters the mouth as a high-speed jet, and the particle diameters change rapidly while transiting the mouth-throat, further complicating matters. Because of the above complexities, it is common to measure mouth-throat deposition with inhalers experimentally using mouth-throat replicas, rather than trying to predict this deposition. Because of ethical concerns with using a replica of a given subject’s mouth-throat in research aimed at development of a commercial product, idealized mouth-throats are attractive in this regard (Martin and Finlay, 2015). Despite the above-noted preference for experimental measurements to determine mouth-throat deposition for a given inhaler, correlations do exist to predict deposition in the mouth-throat with stable aerosols entering the mouth at the same speed as the inhaled air. Since mouth-throat deposition is largely the result of inertial impaction, it is expected to depend on Stokes number. However, since the flow and turbulent dispersion in the mouth-throat depends

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on the Reynolds number, a Reynolds number dependence is also expected (Grgic et al., 2004a; Nicolaou and Zaki, 2013). As a result, these correlations specify the fraction PMT of particles entering the mouth that deposit in the mouth-throat typically as being of the form a PMT ¼ 1  (7.50) a + b Rec Stkd where Re is a Reynolds number and Stk is a Stokes number. The constants a, b, c, and d are empirical coefficients usually determined by fitting Eq. (7.50) to experimental data obtained in extrathoracic airway replicas for a particular category of subjects (e.g., either adults, children, infants, or neonates) and breathing pattern (e.g., either tidal or constant flow rate). Such correlations can be used to predict average mouth-throat deposition of nebulizer aerosols and dry-powder inhalers, but not pressurized metered-dose inhaler aerosols since the above-noted high speed and highly transient diameter changes with the latter are not currently captured by existing correlations. Values of the fitting constants for a variety of different cases, including modifications to Eq. (7.50) that capture electrostatic charge effects, are summarized in Carrigy et al. (2015). For our purposes, one of the most useful correlations of the form given in Eq. (7.50) is that of Grgic et al. (2004b), developed for constant flow rate inhalation in the range of 30–90 L/min through large diameter (>1 cm) mouthpieces: PMT ¼ 1 

1 1 + 11:5 Re0:707 Stk1:91

(7.51)

where ρfluid UD μ

(7.52)

ρparticle d 2 U 18μD

(7.53)

Re ¼ Stk ¼

Here, the geometric length scale D is an average dimension of the mouth-throat defined by Grgic et al. (2004b) based on mouth-throat centerline length L and volume V: rffiffiffiffiffiffi V (7.54) D¼2 πL Similarly, U is an average velocity in the mouth-throat defined in terms of inhalation flow rate Q and the average area A ¼ πD2/4, that is, U¼

4Q πD2

(7.55)

For the adult population considered by Grgic et al. (2004b), they found an average length scale D ¼ 0.023 m.

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For small-diameter mouthpieces (<1 cm), the flow entering the mouththroat takes on a jet-like behavior that is not considered by Eq. (7.51). However, a mouth-entry jet gives higher impaction at the back of the mouth, so that modern inhaler designs typically avoid small-diameter mouthpieces, although DeHaan and Finlay (2004) give a correlation that includes this effect. While Eq. (7.51) is appropriate for inhalation flow rates seen with dry-powder inhalers, nebulizers are used with tidal breathing at lower flow rates outside the range of validity of this equation. Instead, for tidal breathing, Golshahi et al. (2013) (with coefficients corrected by Yang et al., 2017) give the following correlation: 1 (7.56) PMT ¼ 1  1 + 2:23  105 Re0:206 Stk3:04 where Re and Stk are defined similarly as in Eqs. (7.52)–(7.55) but including the Cunningham slip correction (defined in Chapter 3) and also using a slightly different definition of the length scale D that is instead based on the average local cross-sectional area, yielding the definitions 4ρfluid Q πμD

(7.57)

2ρparticle d 2 QCc 9μπD3

(7.58)

Re ¼ Stk ¼

where Q is the average tidal breathing flow rate. Despite a somewhat different definition of length scale D, the adult subjects considered by Golshahi et al. (2013) had an average D ¼ 0.021 m, very similar to the average length scale in Grgic et al. (2004b). An alternative correlation that has commonly been used for predicting mouththroat deposition with tidal breathing of environmental and occupational aerosols is that of Rudolf et al. (1990), which is a curve fit to in vivo mouth-throat deposition in human subjects inhaling tidally via straight tubes inserted into the mouth. This model gives the impactional deposition in the mouth-throat region as Pi ¼ 1 

1:1  10

4




1

 2 Q0:6 V 0:2 1:4 dae t

(7.59) +1

where dae is the particle’s aerodynamic diameter in μm (defined in Chapter 3), Q is the inhalation flow rate in cm3/s, and Vt is tidal volume in cm3. Eq. (7.59) is used in the ICRP (1994) deposition model for radiological exposure assessment. If impaction in the mouth-throat was solely determined by the particle’s impaction properties, we would expect particle diameter to appear via a term proportional to dae2Q (since this is the form of the Stokes number, which determines impaction). Instead, flow rate appears to the power 0.6 in the term dae2Q0.6. This fact and the appearance of tidal volume in Eq. (7.5) are presumably empirical effects of flow rate and tidal volume changes to the mouth-throat geometry.

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Both Eqs. (7.56) and (7.59) are intended for use with tidal breathing. Nebulizers are one of the few inhaled pharmaceutical aerosol devices that operate with tidal breathing, and both equations have been found to predict average mouth-throat deposition with nebulizers quite well (Yang et al., 2017). The well-known rule of thumb that suggests particles larger than approximately 5 μm in aerodynamic diameter are not suitable for efficient inhalation delivery is derived from the rapid rise in mouth-throat deposition that occurs with increasing particle size in the mouth-throat and seen with the above equations. Note that this rule of thumb does not give allowance to the effect of inhalation flow rate, nor does it account for the differing fluid dynamics that occurs with different inhalers at the mouth, so that this rule of thumb should be used with due caution.

7.4.1

Intersubject variability in mouth-throat deposition

Equations of the form given by Eqs. (7.50), (7.51), and (7.55) use a characteristic mouth-throat diameter D that appears via the Reynolds number and Stokes number. While this diameter will have a different value for each subject, the use of a specific subject’s mouth-throat diameter in these equations is unlikely to allow accurate prediction of deposition in that specific subject, since differences in shape are not accounted for with such an approach (Yang et al., 2017). However, the use of different mouth-throat diameters in these equations can be used to estimate intersubject variability in mouth-throat deposition, which is important with inhaled pharmaceutical aerosols since such variability plays a major role in causing intersubject variability in lung dose (B€orgstrom et al., 2006). In this regard, the approach of Ruzycki et al. (2017) can be used to predict intersubject variability in mouth-throat deposition as follows. To demonstrate the approach, let us consider Eq. (7.56), from which we can predict an average value of mouth-throat deposition PMT for adults that have a population-average tidal breathing flow rate Q and a population-average mouth-throat diameter D. We expect intersubject variability about this mean value of mouth-throat deposition to occur primarily because of three factors: (1) Variations between subjects in flow rate Q (2) Variations between subjects in their mouth-throat diameter D (3) Variations between subjects in the shape of their mouth-throat Variations in mouth-throat deposition due to the first factor can be estimated in a straightforward manner by using Eq. (7.56) to predict mouth-throat deposition when the flow rate is one standard deviation higher than the population-average flow rate Q (using the population-average value of D), giving PMT +. Similarly, we use Eq. (7.56) with a flow rate that is one standard deviation lower than average to find PMT . Then, we can estimate the standard deviation in mouth-throat deposition that is due to intersubject variations in flow rate as SMTQ ¼ ðPMT +  PMT Þ=2

(7.60)

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Similarly, intersubject variability due to the second factor (mouth-throat diameter) can be obtained by evaluating Eq. (7.56) with a mouth-throat diameter D that is one standard deviation larger than the population average and one standard deviation lower than the population average and use an equation just like Eq. (7.60) to obtain an estimate for the standard deviation, sMTD, due to intersubject mouth-throat diameter variations. For tidal breathing adults, Ruzycki et al. (2017) measured a mouth-throat diameter D ¼ 0.0188 0.0027 m and used this approach to find sMTD with Eq. (7.56). Values of standard deviation in mouththroat diameter for various populations are given in Finlay et al. (2019), who give D ¼ 0.023 0.0018 m for the mouth-throat diameter definition (7.54) of Grgic et al. (2004b) used in Eq. (7.51). Variations due to mouth-throat shape, the third factor noted above, are more difficult to obtain. However, since Re and Stk account for all dynamical variations in geometrically similar mouth-throats, any differences between predicted and measured mouth-throat deposition with Eq. (7.56) or (7.51) must be due to intersubject variations in mouth-throat shape (i.e., geometric dissimilarity). Examining these differences, Ruzycki et al. (2017) find that for tidal breathing, the standard deviation in mouth-throat deposition due to geometric dissimilarity, sgd, is 0.05 with Eq. (7.56) and tidal breathing. Assuming independence of the above-noted three factors, variability s due to all three factors is then obtained using the L2-norm: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (7.61) s ¼ s2MTQ + s2MTD + s2gd For a polydisperse aerosol, calculation of mouth-throat deposition must be integrated over the particle size distribution but is otherwise straightforward. Ruzycki et al. (2017) find the above approach used with Eq. (7.56) accurately predicts intersubject variability in mouth-throat deposition with nebulizer delivery measured in vivo by Yang et al. (2017). The ability of this approach to predict intersubject variability in other cases (e.g., for higher flow rates using Eq. 7.51) has not yet been examined. Note that with the above approach, intersubject variability due to geometric dissimilarity (i.e., factor three above) is different for different correlations. Martin et al. (2018) give values for correlations that have been developed for different age groups and breathing patterns, with sgd ¼ 0.089 for Eq. (7.51). Example 7.3 Estimate average mouth-throat deposition and its intersubject variability for a population of adults tidally breathing an aqueous aerosol from a nebulizer with a mass median aerodynamic diameter (MMD) of 4.5 μm and geometric standard deviation of 1.7. Assume an average flow rate of Q ¼19.1 L/min, with an intersubject standard deviation in flow rate of 5.6 L/min. Also assume there are no hygroscopic changes in particle diameter during transit through the mouth-throat, and assume a particle density of 1000 kg/m3. Compare to the case of delivery of the same aerosol with a single-breath dry-powder inhaler (that has a mouthpiece diameter

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of 1.3 cm) using Eq. (7.51) with an adult population that has an average flow rate through the inhaler of 60 20 L/min (mean st. dev.), with this population’s mouth-throat diameter, defined by Eq. (7.54), having a value D ¼ 0.023 0.0018 m (mean st. dev.) as given by Grgic et al. (2004b). Solution First, let us calculate the average mouth-throat deposition for the tidally inhaled nebulizer aerosol using Eq. (7.56) at the given flow rate of 19.1 L/ min ¼ 0.000318 m3/s, air density ρfluid ¼ 1.2 kg/m3, air dynamic viscosity μ ¼ 1.8  105, particle density ρp ¼ 1000 kg/m3, and average mouth-throat diameter D ¼ 0.0188 0.0027 m (mean st. dev.) from Ruzycki et al. (2017). In order to use Eq. (7.56) with a polydisperse aerosol, we must perform an integration over the particle size distribution; that is, we must evaluate ð P MT ¼ PMT ðx Þmnormalized ðx Þdx (7.62) where x is particle diameter, PMT(x) is given by Eq. (7.56) at particle diameter x, and mnormalized(x) is the normalized log-normal mass distribution from Chapter 2, that is, " # 1 ð ln x  ln MMD Þ2 mnormalized ðx Þ ¼ pffiffiffiffiffiffi exp (7.63)
2 x 2π lnσ g 2 lnσ g Substituting (7.63) and (7.56) with the given parameter values into Eq. (7.62) and integrating numerically, we find that average mouth-throat deposition for this aerosol is P MT ¼0.198. We now redo our integration of Eq. (7.62) using a flow rate of 19.1 + 5.6 ¼ 24.7 L/min, finding PMT + ¼ 0.267 and at a flow rate of 19.1– 5.6 ¼ 13.5 L/min, finding PMT  ¼ 0.124. Eq. (7.60) then gives us sMTQ ¼ 0.071. Similarly, using Eq. (7.62) at a flow rate of 19.1 L/min but now with D ¼ 0.0188 + 0.0027 m and also D ¼ 0.0188–0.0027 m, we obtain sMTD ¼ 0.103. Ruzycki et al. (2017) find that Eq. (7.56) has a geometric dissimilarity value of sgd ¼ 0.05, and combining this variability with our calculated values of sMTD ¼ 0.071 and sMTQ ¼ 0.124 via the L2-norm in Eq. (7.61), we find s ¼ 0.135. Thus, we predict that mouth-throat deposition for this aerosol inhaled during tidal breathing in a population of adults will have a mean standard deviation of 0.198 0.135. Repeating the above procedure but instead using PMT from Eq. (7.51) in the integrand of Eq. (7.62), we find average mouth-throat deposition for the single-breath inhaler case at Q ¼ 60 L/min (0.001 m3/s) with D ¼ 0.023 m is now P MT ¼0.305, while at Q ¼ 60 + 20 ¼ 80 L/min Eq. (7.51) gives PMT + ¼ 0.410 and Q ¼ 60– 20 ¼ 40 L/min gives PMT- ¼ 0.183, so we find sMTQ ¼ (0.410–0.183)/2 ¼ 0.114. Similarly, Eq. (7.51) with Q ¼ 60 L/min and D ¼ 0.023 + 0.0018 and D ¼ 0.023– 0.0018 gives sMTD ¼ 0.0662. Combining these flow rate and mouth-throat diameter deposition variability values with the geometric dissimilarity variation sgd ¼ 0.089 that Martin et al. (2018) find for Eq. (7.51), then, Eq. (7.61) gives s ¼ 0.159. Thus, we predict that mouth-throat deposition for this aerosol inhaled during a single constant flow rate breath in the given population of adults will have a mean standard deviation of 0.305 0.159, compared with 0.198 0.135 for the same aerosol delivered with tidal breathing.

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As expected, the higher flow rate of the single-breath inhaler gives higher mouth-throat deposition for the same aerosol size distribution, and along with this comes larger absolute intersubject variability. Note that the problem statement did not mention any variation in the particle size distribution with flow rate, that is, we are using the same MMD and GSD for all inhalation flow rates. This may be a reasonable approximation for nebulizers, but many single-breath dry-powder inhalers produce somewhat different size distributions at different inhalation flow rates, a factor we have not included here. Note also that in both cases, we are not including any suboptimal use of the delivery device (e.g., exhaling into the device instead of inhaling, failure to load the device with drug, and improper positioning of inhaler mouthpiece). Our predicted mean and intersubject variability values here should be viewed with these additional variability issues in mind.

7.5 One-dimensional and empirical deposition models By combining equations and analyses like those presented in the preceding sections of this chapter, it is possible to develop relatively simple models for predicting the amount of an inhaled aerosol that will deposit in the different lung regions. A plethora of such models have been presented in the archival literature for environmental and occupational aerosols (e.g., Heyder and Rudolf, 1984 list 27 models already prior to 1984, while Hofmann, 2011 covers a few more recent models), dating back >65 years to Findeisen’s work (1935). Such simple deposition models can be broadly categorized into three types, based on the approach they take to the fluid and particle motion in the lung: empirical models, Lagrangian dynamical one-dimensional models, and Eulerian dynamical one-dimensional models. In the two types of dynamical models, equations governing the dynamics of the aerosol are solved to predict the amount of aerosol depositing in the different parts of the respiratory tract. In Lagrangian models, the aerosol is examined in a reference frame that moves with the aerosol, while in Eulerian models, the aerosol is examined in a stationary frame. It should be noted that respiratory tract deposition models developed for use with environmental or occupational aerosols often suffer from their inability to model mouth-throat deposition at the higher nontidal inhalation flow rates, or with mouth-entry jets, that occur with many pharmaceutical inhalation devices. However, if mouth-throat deposition is given empirically using more recently developed extrathoracic deposition correlations, like those noted in Section 7.4, such models are capable of modeling intrathoracic (lung) deposition. When such updated models are combined with models that predict airway surface fluid volumes and mucociliary clearance, they can be used to predict not only doses of drugs initially depositing in the different regions of the lungs but also the subsequent time-dependent disposition of drug, that is, its pharmacokinetics (Martin and Finlay, 2018).

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Empirical models are the simplest models. These models give a set of algebraic equations that fits a set of experimental in vivo data. Empirical models do not give explicit consideration to the particle and fluid dynamics and so do not rely on dynamical analyses like those used to develop the equations given earlier in this chapter. Empirical models are usually relatively easy to implement and have low computational demands. However, they do not lend themselves well to extrapolation outside the parameter space of the experimental data nor to inclusion of dynamical effects like hygroscopicity. This limits their applicability in some cases.

7.5.1

Lagrangian dynamical one-dimensional models

The next level of conceptual complexity beyond empirical models is achieved by Lagrangian dynamical models. In current versions of these models that treat the entire respiratory tract, particles are followed in one dimension (i.e., depth into the lung) through an idealized lung geometry in which the fluid flow in each lung generation is specified (usually either plug flow or Poiseuille flow). The particles are simply convected through the idealized lung geometry at the average local flow velocity in each lung generation. The probability of deposition as a particle travels through each generation of this geometry is estimated using equations like those given earlier in this chapter. Symmetrical lung geometries are most commonly used, although asymmetrical geometries can be considered, for example, with the addition of Monte Carlo techniques (Koblinger and Hofmann, 1990). Two-way and one-way coupled hygroscopic effects (e.g., Javaheri et al., 2013; Finlay and Stapleton, 1995; Ferron et al., 1988; Persons et al., 1987) can also be readily incorporated into these models, since the equations governing these effects are written very naturally in a Lagrangian form. The principal limitations of current Lagrangian models stem from their use of only one spatial dimension and the assumed fluid dynamics (although these same features also lead to the attractive simplicity and low computational requirements of these models). These limitations result in the need for additional considerations when simulating the axial dispersion of an aerosol bolus (i.e., a short burst of inhaled aerosol spreads out axially as it travels through the lung—see Park and Wexler, 2007; Hofmann et al., 2008 for approximate approaches for incorporating axial dispersion). Although generalizing these models to more than one spatial dimension would allow circumvention of some of their difficulties, it would also make them far more computationally demanding. A second limitation of Lagrangian models is the difficulty they have in treating flow rates and aerosol concentrations that vary during a breath, a feature that can be important with some inhaled pharmaceutical aerosols, such as dry-powder inhalers.

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7.5.2 Eulerian dynamical one-dimensional models Bolus dispersion and time dependence can be more easily implemented using the third framework mentioned above, the Eulerian approach. The general concept of an Eulerian model is to solve a convection-diffusion equation for the aerosol in an idealized version of the lung geometry, using ideas first developed for modeling gas transport in the lung (Taulbee and Yu, 1975; Taulbee et al., 1978; Egan and Nixon, 1985). The aerosol not only is convected through the lung by the air motion but also diffuses relative to the air due to Brownian motion. As with the Lagrangian models, purely Eulerian deposition models that simultaneously treat the entire respiratory tract are restricted to one spatial dimension due to computational cost, with depth into the lung being the spatial dimension that is used. Such one-dimensional models can be derived by considering the equation for mass conservation of the aerosol particles, which in integral form can be written ð ð ð ∂ ndV + nv  dS ¼ Dd rn  dS (7.64) ∂t V

S

S

where V contains the volume of aerosol under consideration and S bounds this volume. Here, v is the velocity of the aerosol when treated as a continuum, and Dd is the molecular diffusion coefficient given in Eq. (7.45). This equation can be reduced to one dimension as follows. First, introduce a coordinate system (x,y,z) with x representing depth into the lung and y,z representing the cross section of the air-filled portions of all parts of the lung at a depth x. Next, we consider a short section of lung from depth x to depth x + dx. We integrate the first term on the left-hand side over the two spatial dimensions y and z and break up the other two terms into integrations over two types of surfaces: airway surfaces Syz and airway lumen Sx, as shown in Fig. 7.10. In this way, we can rewrite Eq. (7.64) as ∂ ∂t

ð

x +ðdx

ð

AT ndx + nv  dS + x

Sx

ð

ð

nv  dS ¼ Dd rn  dS + Syz

Sx

Dd rn  dS

(7.65)

Syz

where AT is the total cross-sectional area of the airways at depth x and n is the average aerosol number concentration over this cross section. To simplify this equation further, we can write     ð ∂n  ∂n  Dd rn  dS ¼ Dd AA  D A (7.66) d A ∂x x + dx ∂x x Sx

where AA is the total area in the cross section of the airway passages that make up Sx, and we have assumed for simplicity, as is normally done, that the average

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y z x

Syz

x Sx

x+dx

FIG. 7.10 A portion of a lung airway at depth x in the lung is shown. Sx is the airway lumen surface at depth x, and Syz is the airway surface itself.

value of aerosol concentration over AA is the same as over AT. In addition, we can write ð

nv  dS5 ðnAA uÞjx + dx  ðnAA uÞjx + Fdx (7.67) Sx

where u is the average velocity in the x-direction over the area AA, and the term Fdx is a flux correction (White, 2016) that arises because of the nonlinear integrand in Eq. (7.67), so this equation is exact with F ¼ 0 only in the case of plug flow with a uniform aerosol concentration across the airway cross section. In general, F 6¼ 0. In order to evaluate F exactly, we would need to know the value of the aerosol number concentration and velocity across all cross sections Sx. Because obtaining this information defeats the simplicity of a one-dimensional model, it is usual to introduce the following approximation, in which the flux correction F is represented as an effective diffusive flux with diffusion coefficient DF:   ∂ ∂n A A DF (7.68) F¼ ∂x ∂x where DF must be specified in some empirical manner. The difference between AA, appearing in Eqs. (7.66), (7.67), and AT appearing in Eq. (7.65) arises because the terms in Eqs. (7.66), (7.67) represent convective and diffusive transport of aerosol along the x-direction. Because direct connection between successive lung generations occurs through the airway passages, not via alveoli, such transport occurs only across the area AA (the crosssectional area of the airway passages), not across the entire lung cross-sectional area AT (which includes alveoli). In contrast, AT arises because we have taken an average over the entire lung cross section at depth x. At this point, we also need to realize the meaning of the last term on each side of Eq. (7.65). Together, these two terms represent the rate at which the aerosol number concentration changes due to deposition of aerosol at the airway surface walls. Defining L as this deposition rate per unit length, then we can define L such that

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ð Ldx ¼

ð Dd rn  dS 

Syz

nv  dS

(7.69)

Syz

Substituting Eqs. (7.66)–(7.69) into Eq. (7.65), dividing by dx and taking the limit as dx ! 0, we obtain the following equation for the average aerosol concentration, n, at depth x:   ∂ ∂ ∂ ∂n ðAT nÞ + ðAA nuÞ ¼ AA Deff L (7.70) ∂t ∂x ∂x ∂x where Deff is an effective diffusion coefficient given by Deff ¼ Dd + DF

(7.71)

and Dd is the molecular diffusion coefficient from Eq. (7.45), while DF is from Eq. (7.68). Although Eq. (7.70) is exact as written, it cannot be solved without knowing AT, AA, u, L, and DF, which require approximation, as follows. The areas AT and AA can be approximated using an idealized lung geometry like those described in Chapter 5. (Note that the areas AT and AA can be made time-dependent to allow for lung expansion during inhalation, c.f. Taulbee et al., 1978; Egan and Nixon, 1985.) For the chosen idealized lung geometry and a given inhalation flow rate, the velocity u in Eq. (7.70) can be approximated as the mean air velocity of the airway cross section by using simple mass conservation of the air. To obtain the deposition rate L, it is useful to realize that the number of aerosol particles depositing per unit time throughout a given lung generation can be approximated by nQP, where Q is the air flow rate through the lung generation and P is the total deposition probability in a given lung generation due to impaction, sedimentation, and diffusion. The deposition rate L per unit time and length is then given simply by L¼

nQP lm

(7.72)

where lm is the length of an airway generation at depth x in the idealized lung geometry being used. Eq. (7.72) allows L to be estimated using the approximate expressions for the deposition probabilities Pi, Pd, and Ps that we developed earlier in this chapter. One of the most empirical aspects of one-dimensional Eulerian models lies in the specification of the effective diffusion coefficient DF defined in Eqs. (7.68), (7.71). Most authors follow Scherer et al. (1975), who performed experiments on the dispersion of gases in a glass tube model of the first five generations of a Weibel A lung geometry. For dispersion of gases, they found DF had the form DF ¼ αudm

(7.73)

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where α is a factor of order 1 (they found α ¼ 1.08 during inhalation, while α ¼ 0.37 for exhalation, although more recent work by Fresconi and Prasad (2008) suggests using instead α ¼ 0.5 for both inspiration and expiration when the Reynolds number in the given airway is <100). Here, u is the average air velocity in the mth lung generation, and dm is the diameter of this lung generation. For aerosols, various authors have suggested using Eq. (7.73) with various values of α in the range 0.1–1.0, in addition to empirical forms different than Eq. (7.73) (Taulbee and Yu, 1975; Taulbee et al., 1978; Darquenne and Paiva, 1994; Edwards, 1995). Given that different velocity profiles occur in different parts of the lung, from Eqs. (7.67), (7.68), we can see that it is not unreasonable to expect different forms of DF in different parts of the lung (Lee et al., 2000b), and indeed, Li et al. (1998) suggest an alternative equation for DF in the mouth-throat. Based on Scherer et al. (1975) and the simulations of Lee et al. (2000b), Hoffman et al. (2008) use Deff ¼ 0:7ðDd + 1:08 ud m Þ for inspiration

(7.74)

Deff ¼ 0:65ðDd + 0:37 udm Þ for expiration

(7.75)

At present, there is little detailed direct experimental evidence for any of these forms for Deff in aerosol deposition models (evidence is usually given in indirect comparisons of dispersion predictions of respiratory tract models where many other assumptions are made, so that direct scrutiny of the validity is not possible). Further research is needed in this area. One limitation of standard Eulerian models is the difficulty they have in including two-way coupled hygroscopic effects, since a Lagrangian viewpoint is more natural in predicting these effects. Generalizing Eulerian models to more than one spatial dimension is also difficult because inertial impaction is more naturally dealt with by tracking individual particles (using a Lagrangian approach). Instead, in three dimensions, the most common approach is to use an Eulerian approach for the fluid and a Lagrangian approach for the particles that requires using computational fluid dynamics, which we now examine.

7.6 Computational fluid dynamics deposition models For many applications of inhaled pharmaceutical aerosols, the above-noted simple deposition models provide enough information. Indeed, such models can give deposition at each generation of the conducting airways that is surprisingly accurate in comparison with computational fluid dynamics simulations of these airways (Zhang et al., 2009). However, there may be situations where limitations of the simple models are problematic, for example, when information is desired regarding three-dimensional localized deposition within airway generations or the effect of three-dimensional local abnormalities in lung geometry associated with disease. Such information requires more detailed modeling of

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the flow and particle motion in the respiratory tract, which is normally achieved by solving the Navier-Stokes equations and particle trajectories using computational fluid dynamics (CFD) in a portion of the lung airways. Many authors have used such an approach (see, e.g., Koullapis et al., 2018; Longest and Holbrook, 2012; Kleinstreuer and Zhang, 2010 for reviews). In general, well-established CFD software can be used to obtain the fluid flow and particle trajectories in a relatively straightforward manner. However, the following points need to be borne in mind when performing such CFD calculations. First, cartilaginous rings are present on the proximal conducting airways that cause irregularity of the airway surface that enhances deposition compared with a smooth surface (Evans and Castillo, 2016; Zhang and Finlay, 2005). For this reason, it is preferable to use realistic airway geometries, usually reconstructed from high-resolution medical images of the airways of human subjects. Unfortunately, such imaging can currently only resolve the first few airway generations. If the CFD simulation is restricted to the resolved airways, then boundary conditions (e.g., flow rates entering each distal branch) must be supplied at the distal ends of the truncated airways, which is most accurately done using knowledge of regional ventilation in the distal lung regions (Lin et al., 2013). Alternatively, artificially constructed distal airway geometries may be specified. In either case, two points must be realized: (1) the geometries of the lung airways change during inhalation as the lung inflates and (2) regional ventilation within the lung varies as lung volume changes during inhalation, adding spatial inhomogeneity to the unsteadiness of the aforementioned geometry changes throughout the lung. Accurate accounting of such effects can be done by obtaining time-dependent in vivo information on lung geometry and regional ventilation (Burrowes et al., 2017), but this of course adds considerable complexity to any such study. While inclusion of time-dependent regional ventilation and time dependence of conducting airway geometries is desirable to achieve the very highest levels of accuracy, errors in deposition values obtained with CFD in static conducting airways may be acceptable for many inhaled pharmaceutical aerosol applications (Katz et al., 2017; Miyawaki et al., 2016; Longest et al., 2016), especially given the increase in effort associated with a dynamic simulation. In contrast to the conducting airways, CFD simulations in the alveolar regions normally include time dependence to the geometry (Khajeh-Hosseini-Dalasm and Longest, 2015; Sznitman et al., 2009), although research is needed on the importance of including self-consistent fluid-structure interaction effects, where asymmetrical expansion of alveoli has been observed in fluid simulations (Roschenko et al., 2015). Besides the above issues, the usual concerns with CFD must be addressed (Versteeg and Malalasekera, 2007), including demonstration of grid independence (i.e., simulations on grids with differing mesh sizes must be performed to show that the results do not depend on mesh size) and independence of deposition on the number of particles being tracked (i.e., deposition does not change if more particles are tracked). In the mouth-throat, if a Reynolds-averaged

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Navier-Stokes (RANS) approach is used to model the turbulence, corrections that account for near-wall anisotropy of turbulence on particle motion and adequate near-wall resolution should be used (Bass and Longest, 2018), although the former issue can be circumvented if large eddy simulation (LES) or direct numerical simulation is used. While highly detailed information regarding particle and fluid behavior in the respiratory tract can be gained with CFD, the level of complexity required for such simulations to achieve sufficient accuracy remains a topic of ongoing research that is hampered by the difficulty in obtaining detailed in vivo data to compare with.

7.7 Understanding the effect of parameter variations on deposition One of the most useful features of respiratory tract deposition models is their ability to show the effect of how different parameters affect deposition in the lung. Indeed, we can obtain a qualitative understanding of these effects simply by examining the equations we developed earlier in this chapter for deposition in simplified geometries. From these equations, it becomes clear that deposition due to the three principal deposition mechanisms (i.e., impaction, sedimentation, and diffusion) increases monotonically in a given lung generation with three parameters, as follows. From Eq. (7.43), we see that inertial impaction increases monotonically with Stokes number: Stk ¼ Uρparticle d2 Cc =18μD

(7.76)

From Figs. 7.4 and 7.5, we see that the different sedimentation equations all increase monotonically with the parameter. t0 ¼

νsettling L U D

(7.77)

where νsettling ¼ Cc ρparticle gd2 =18μ

(7.78)

Finally, from Fig. 7.7, we see that deposition for the different diffusion equations increases monotonically with the parameter. Δ¼

kTCc L 1 3πμd U 4R2

(7.79)

Thus, even though different deposition models may use different equations governing each of the three principle deposition mechanisms, all these deposition models predict that deposition in a given lung generation will increase if Stk, t0 , or Δ is increased. From this fact, we can draw an understanding of how different variables affect deposition.

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Before doing this, however, let us make the flow rate Q appear explicitly in Eqs. (7.76)–(7.79), by realizing that flow rate is related to flow velocity and airway diameter D by Q ¼ UπD2

(7.80)

We can thus rewrite Eqs. (7.76)–(7.79) as 2Cc ρd 2 Q 9πμ D3

(7.81)

Cc gπ ρd 2 LD 72μ Q

(7.82)

kTCc L 12μ Qd

(7.83)

Stk ¼ t0 ¼

Δ¼

From these equations, we see that particle size (appearing as d2) and airway diameter (appearing as D3) are the only variables that appear to a power other than unity. Thus, of all the variables, these two have the potential to have the largest effect on deposition, since changes in d or D are amplified by being raised to a nonunitary integer power. Since particle size can be controlled when designing a delivery device, while airway dimension cannot, it is for this reason that particle size is the single most important parameter in pharmaceutical aerosol delivery. Eqs. (7.81), (7.82) show that deposition in a given lung generation will increase with particle diameter (since the decrease in diffusional deposition with particle diameter seen in Eq. (7.83) usually only plays a secondary role in deposition of inhaled pharmaceutical aerosols). However, this conclusion is strongly altered by the fact that if more aerosol deposits in one generation, then less aerosol will reach downstream generations. Indeed, because the mouth-throat filters out particles before they reach the tracheobronchial region, while the tracheobronchial region filters out particles before they reach the alveolar region, actual deposition in the tracheobronchial region and alveolar region decreases with particle size for larger particles due to this effect. From Eqs. (7.81)–(7.83), we see that inhalation flow rate increases impactional deposition but decreases sedimentational and diffusional deposition. Thus, tracheobronchial deposition can increase with inhalation flow rate if the flow rate is high enough that impaction is the dominant mechanism there (although again we must be careful because of the filtering effect of upstream regions, since we saw that mouth-throat deposition increases with flow rate as well, so that less aerosol reaches the tracheobronchial region at higher flow rates and it is possible to actually have a decrease in tracheobronchial deposition with increased flow rate due to this effect). The effect of increasing flow rate on alveolar deposition is clearer, since if impaction increases in the mouth-throat and tracheobronchial region due to

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flow rate effects, then less is available to deposit in the alveolar regions, and in addition, Eqs. (7.82), (7.83) show that even less of this available aerosol will deposit since the primary deposition mechanisms in the alveolar region (i.e., sedimentation and diffusion) diminish with increased flow rate. The effect of changes in airway diameter on deposition is dramatic in Eq. (7.81), since any such changes are raised to the third power there (due to the combination of increased velocity that occurs at fixed flow rate in a smaller diameter tube and changes in the ratio of stopping distance to tube diameter). As a result, impaction in the mouth-throat and tracheobronchial airways decreases rapidly if airway dimensions are made larger (e.g., as a person grows through childhood). This allows more aerosol to reach the alveolar region, which acts together with the increase in sedimentation that occurs in Eq. (7.82) as D increases, resulting in dramatic increases in alveolar deposition if airway diameters are made larger. At first sight, the increase in sedimentation with airway diameter in Eq. (7.82) seems counterintuitive, since an increase in tube diameter will increase the distance a particle must settle before depositing (so that one might think that sedimentation should decrease with increased tube diameter). However, this effect is outweighed by the fact that the particle has more time to deposit in the tube because flow velocity varies inversely with the square of tube diameter. Eqs. (7.81)–(7.83) show that increases in airway length cause increases in sedimentational and diffusional deposition, resulting in increased alveolar deposition if the remaining variables are in the usual range where impaction is the dominant deposition mechanism in the conducting airways. Since increases in airway length occur in concert with increases in airway diameter during progression through childhood, in this case, such increases in alveolar deposition with increased airway length would add to the increased alveolar deposition that we have already seen occurs with increases in airway diameter.

7.8 Respiratory tract deposition A principle reason for the existence of respiratory tract deposition models is to provide an understanding of how different factors affect this deposition, as we have just seen. However, such an understanding can also be partly gained by examining experimental data in which deposition of aerosol particles has been measured in human subjects. The largest and most systematic such experimental data sets have been obtained during tidal breathing of subjects breathing monodisperse aerosols delivered via a tube inserted into the mouth, much of which is summarized by Stahlhofen et al. (1989) and shown in Figs. 7.11–7.14. Although mouth-throat deposition with pharmaceutical inhalation devices will in general be different than that given in Fig. 7.11 (due to changes in the fluid flow caused by the presence of the device at the mouth—see Section 7.4), this figure clearly shows the usual increase in oropharyngeal

FIG. 7.11 Mouth-throat (“extrathoracic”) deposition fraction (i.e., fraction of inhaled aerosol depositing in mouth-throat) in human subjects measured during mouth breathing shown as a function of dae2Q, where dae is aerodynamic diameter and Q is inhalation flow rate (Lippmann, 1977; Foord et al., 1978; Emmett et al., 1978; Stahlhofen et al., 1980, 1981, 1983). (Adapted from Stahlhofen, W., Rudolf, G., James, A.C., 1989. Intercomparison of experimental regional aerosol deposition data. J. Aerosol Med. 2, 285–308.)

FIG. 7.12 Fast-cleared deposition fraction (i.e., fraction of inhaled aerosol depositing in the lung that is cleared within a day or so) versus aerodynamic particle diameter, measured in human subjects during tidal mouth breathing at a variety of tidal breathing flow rates and tidal volumes. Tracheobronchial deposition consists of this fast-cleared deposition, plus a small (unknown) portion of the slow-cleared deposition. (Adapted from Stahlhofen, W., Rudolf, G., James, A.C., 1989. Intercomparison of experimental regional aerosol deposition data. J. Aerosol Med. 2, 285–308.)

FIG. 7.13 Slow-cleared deposition efficiency (i.e., fraction of inhaled aerosol depositing in the lung that isn’t cleared within a day or so) versus aerodynamic particle diameter measured during tidal mouth breathing at a variety of tidal volumes and flow rates. The majority of this deposition is alveolar deposition, but a small (unknown) portion is due to tracheobronchial deposition. (Adapted from Stahlhofen, W., Rudolf, G., James, A.C., 1989. Intercomparison of experimental regional aerosol deposition data. J. Aerosol Med. 2, 285–308.)

FIG. 7.14 Total respiratory tract deposition as a function of particles with diameter equivalent to that of spheres having a density of 1000 kg/m3 are shown for several tidal volumes Vt and breathing frequencies f (low corresponds to Vt ¼ 500 mL, f ¼ 15 breaths/min; medium corresponds to Vt ¼ 1000 mL, f ¼ 7.5 breaths/min; high corresponds to Vt ¼ 2000 mL, f ¼ 3.75 breaths/min) at an inhalation flow rate of 250 mL/s, measured in human subjects during tidal mouth breathing. (Adapted from Stahlhofen, W., Rudolf, G., James, A.C., 1989. Intercomparison of experimental regional aerosol deposition data. J. Aerosol Med. 2, 285–308.)

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(mouth-throat) deposition with particle size and inhalation flow rate. Indeed, for particles >10 μm, deposition is on average >90% in this figure at an inhalation rate of 60 L/min and >50% at an inhalation rate of 18 L/min. Figs. 7.12 and 7.13 suggest that for tidal breathing, there is a broad maximum in lung (i.e., intrathoracic) deposition (which is near 6 μm for tracheobronchial deposition but near 3 μm for alveolar deposition at the 30 L/min inhalation flow rate in this figure). These maxima arise because of the combination of effects mentioned in the last section, where increases in deposition with particle size are offset by increased filtering in upstream regions. It is because of these maxima and the dramatic increase in mouth-throat deposition at higher particle sizes that rules of thumb regarding optimal particle sizes for inhalation have arisen, with a commonly used such rule of thumb being that inhaled pharmaceutical aerosols must in the 1–5 μm range to reach the lung. However, because of the flow rate dependence of particle deposition and the lack of abrupt cutoffs in actual deposition curves like those in Figs. 7.12 and 7.13, such rules of thumb should be used with caution. Fig. 7.14 shows total respiratory tract deposition during tidal breathing for a few individuals, which is the sum of the individual regional deposition fractions shown in Figs. 7.11–7.13. The well-known minimum in total deposition seen in this figure for submicron particle sizes occurs because sedimentation and impaction decrease with decreasing particle size, but diffusion increases with decreasing particle size, resulting in a crossover between these different deposition mechanisms where the minimum in total deposition occurs.

7.8.1 Slow-clearance from the tracheobronchial region One of the principal difficulties in understanding experimental data on deposition within the lung (i.e., intrathoracic deposition) in human subjects is the difficulty of obtaining accurate measures of deposition in the different anatomical regions of the lung. This difficulty arises because the resolution of radionuclide imaging methods used in such experiments does not allow accurate discrimination of individual airways within the lung lobes, so that direct quantitative mapping of the radioactivity onto anatomical airway generations, for example, has not been possible. Although progress in these directions has occurred using models that map the different generations of idealized models of the lung (discussed in Chapter 5) to three-dimensional locations in the lung (Fleming et al., 1995; Lee et al., 2000a; Conway et al., 2012), at present, experimental data cannot accurately determine deposition at the generational level throughout the lung. Instead, the most commonly used approach has been to measure the so-called “fast-cleared” and “slow-cleared” fractions of lung deposition like those shown in Figs. 7.12 and 7.13. The rationale for these measurements is based on the presence of cilia in the tracheobronchial airways and the absence of cilia in the alveolar regions. As a result, mucociliary clearance causes clearance of most particles depositing within the tracheobronchial

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region within 1 day or so, while insoluble particles depositing in the alveolated regions are cleared much more slowly. Thus, alveolar deposition has commonly been taken to be equal to the “slow-cleared” fraction, while tracheobronchial deposition is obtained by subtracting this slow-cleared fraction from the initial total lung deposition (giving the “fast-cleared” fraction, which has often been equated with tracheobronchial deposition). Unfortunately, some particles depositing in the tracheobronchial region are actually cleared slowly. While the reasons and mechanisms for this slow clearance remain a topic of research, Sturm and Hofmann (2006) note that they “probably include the transfer of particles through the airway epithelium toward lymph or blood vessels, the uptake of particulate mass by airway macrophages, and the simple storage of particles on the epithelial surface with subsequent retransfer onto the mucus layer.” Fig. 7.15 shows the fraction of insoluble particles depositing in the tracheobronchial region that is cleared slowly in the clearance model proposed by Sturm and Hofmann (2003). Notice the dependence on both airway generation, i, and particle geometric diameter d in Fig. 7.15, with slow clearance reaching 100% for submicron particles in the most distal

20

io n

100

ge n

er at

15

rw ay

60

10

Ai

fs(i,d) (%)

80

40 5 20 0 1

2 3 4 Particle diameter (μm)

5

6

FIG. 7.15 The average fraction fs of tracheobronchial deposition that is “slow-cleared” (i.e., that remains after 24 h) in the clearance model of Sturm and Hofmann (2003) is shown versus particle diameter d and airway generation i. (Adapted from Sturm, R., Hofmann, W., 2003. Mechanistic interpretation of the slow bronchial clearance phase. Radiat. Prot. Dosimetry 105 (1–4), 101–104.)

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conducting airways, although this figure indicates that slow clearance is negligible for much of the conducting airways for larger particles. The model of Sturm and Hofmann (2003) assumes that slow clearance decreases linearly with particle diameter and has a fourth order polynomial monotonic dependence on airway generation number i, as can be seen in Fig. 7.15. Fig. 7.15 suggests that a portion of the “slow-cleared” deposition in Fig. 7.13 in the range of interest for inhaled pharmaceutical aerosols (e.g., 1–5 μm) will actually be due to tracheobronchial deposition, particularly for the smaller particles. However, because of their small mass (so that impaction and sedimentation are small), but not too small diameter (so that diffusion is still small), these smaller particles have relatively low deposition probabilities in the tracheobronchial region compared with the alveolar region. As a result, the actual amount of slow-cleared aerosol from the tracheobronchial region is relatively small compared with alveolar deposition, and the correction that is needed in order to equate slow-cleared fractions with alveolar deposition is relatively minor (ICRP, 1994). The question then remains as to the validity of equating fast-cleared deposition with tracheobronchial deposition, since tracheobronchial deposition may be underestimated due to the existence of a slow-cleared fraction. The error made by incorrectly assigning the slow-cleared tracheobronchial fraction to alveolar deposition will depend not only on particle diameter d but also on what generations i the particles deposit within, due to the dependence of fs on both of d and i (as clearly seen in Fig. 7.15, and also supported by experiments, e.g., Smith et al., 2008). Because inhaled pharmaceutical aerosols typically have larger particle diameters that are inhaled at higher flow rates than ambient or environmental aerosols, slow-cleared tracheobronchial deposition is likely to be less of a concern when using fast clearance as a measure of tracheobronchial deposition with pharmaceutical aerosols. Indeed, Fig. 7.16 shows a comparison of fast-cleared and tracheobronchial deposition for various aerosols (with volume median diameters in the range of 3–6 μm and geometric standard deviation 1.5–1.6) produced by a vibrating mesh nebulizer (see Chapter 8) measured in 11 healthy male subjects by Conway et al. (2012), where they used an assumed idealized lung geometry to map in vivo deposition with single-photon emission computed tomography (SPECT) onto individual lung generations. The radioactivity in their study was contained within the nebulized droplets in submicron particles that, according to Fig. 7.16, would be subject to a significant slow-cleared fraction. However, tracheobronchial deposition among these subjects matches fast-cleared deposition on average to within 1%, although considerable intersubject variability can be seen in this match. A similar study by Fleming et al. (2015) in six asthmatic subjects found that average fast-cleared deposition was 47%, while their measured tracheobronchial deposition was 52%, again indicating minor differences due to the possible presence of slow clearance. However, Fleming et al. (2015) note that their Weibel idealized lung model may not be a good representation of lung geometry in their asthmatic

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0.8

Fast-cleared deposition

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.1

0.2

0.3 0.4 0.5 Tracheo-bronchial deposition

0.6

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FIG. 7.16 Fast-cleared deposition is shown vs tracheobronchial deposition from the in vivo study of Conway et al. (2012) in 11 subjects with several aerosols. The solid line is the line of identity.

subjects, making their mapping of SPECT deposition onto airway generations uncertain. Studies confirming that slow clearance from the tracheobronchial region is similarly minor for other pharmaceutical aerosol formulations, for example, dry powders or pressurized metered-dose inhaler particles, and the effect of disease state on slow-cleared tracheobronchial fractions with such aerosols have not yet been performed.

7.8.2

Intersubject variability

We have already discussed intersubject variability of deposition in the extrathoracic region and its causes and prediction (see Section 7.4.1), and Fig. 7.11 clearly shows this variability for tidal breathing through tubes. Figs. 7.12 and 7.13 show that there is also tremendous variation from individual to individual in the amount of aerosol that will deposit in the different regions within the lung. Although empirical formulas for this variation have been developed for tidal breathing through tubes (Rudolf et al., 1990), Ruzycki et al. (2017) found that these considerably underpredicted extrathoracic intersubject variability with a nebulized aerosol, thereby hampering accurate estimation of intersubject variability in lung deposition since intersubject variability in extrathoracic deposition itself causes variability in the aerosol reaching the lungs (B€ orgstrom et al., 2006). We saw in Section 7.4.1 that including intersubject variations in flow rate and geometry (both size and shape) allowed accurate prediction of intersubject

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variability in extrathoracic deposition. Added to this variability, intersubject variations in breathing pattern and lung geometry (again, both size and shape of the airways) cause further intersubject variation within the lungs, although application of an approach similar to that outlined in Section 7.4.1 has not yet been applied to the lungs, at least in part because detailed characterization of in vivo intersubject variations in airway geometry and shape in all but the most proximal parts of the lung remains out of reach with current imaging methods.

7.9 Targeting deposition at different regions of the respiratory tract By combining observations from experiments like that presented above with deposition model predictions, it is possible to achieve a reasonably good understanding of how different parameters affect respiratory tract deposition. With this understanding, it is possible to have some control over where in the respiratory tract an aerosol will deposit. Such targeting of the respiratory tract can be of importance with therapeutic agents where efficacy is thought to depend in part on where the drug deposits in the lung, such as with drugs intended for systemic uptake into the blood through the alveolar epithelium or with antimicrobial agents delivered to regional sites of infection. It should be noted though that such targeting will not be very precise, for a number of reasons, as follows. First, it is not possible to adequately control the regions to which air carries aerosol to in the lung (i.e., regional ventilation is not controllable in any precise way, due to a combination of the stochastic nature of the lung, dispersion, chaotic mixing, and diffusion). For example, we cannot target only the tracheobronchial region since the pathways to some terminal bronchioles are much shorter than others and will start to fill alveolar regions before the air has even reached the terminal bronchioles in other pathways. We also cannot have air reach only the alveolar region since it must first travel through the tracheobronchial region and the mouth-throat. Second, the factors that affect deposition change gradually from region to region in the lung. Thus, for example, if we choose particles that deposit mainly by sedimentation, such particles will deposit mainly in the alveolar regions. However, as we saw in Fig. 7.7, sedimentation is also operational in the small bronchioles, so that we cannot entirely avoid deposition in the tracheobronchial region in this manner. Finally and perhaps most importantly (Clark and Hartman, 2012), variations between subjects (i.e., intersubject variability) in the parameters that control deposition will also dramatically broaden regional deposition in any attempt at targeting specific lung regions within a population of subjects. Despite these difficulties, broad targeting of aerosol is possible. In particular, it is clear that if we increase inhalation flow rate, mouth-throat deposition will increase according to Fig. 7.11, while the total dose to the lung will

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decrease. In addition, if flow rate is already relatively high (so that Fig. 7.6 indicates impaction is the dominant deposition mechanism in the tracheobronchial region), then further increases in flow rate will increase impaction in the tracheobronchial region so that whatever aerosol does manage to make it past the mouth-throat will tend to deposit in the tracheobronchial region.1 Conversely, reducing inhalation flow rate will shift deposition away from the mouth-throat and into the lung. If inhalation flow rate is very low, then impaction in the larger airways can be mostly avoided, resulting in small airway and/ or alveolar deposition, depending on particle size. It should be realized that it is unrealistic to expect patients to consistently control inhalation flow rate very precisely on their own, so that the use of specific inhalation flow rates for targeting is probably best achieved with a delivery device that either delivers aerosol only when the patient supplies a certain flow rate range or that causes patients to supply such flow rates. Probably the most commonly used approach to targeting aerosol deposition is through particle size. We have seen clearly throughout this chapter how important particle size is in determining deposition, with larger particles impacting and sedimenting more readily than smaller ones, while smaller ones deposit more readily by diffusion. We already saw that increasing particle size causes increased mouth-throat deposition, resulting in less aerosol reaching the lung. Within the lung, Figs. 7.12 and 7.13 show that during tidal breathing at 30 L/min and 1 L tidal volumes, alveolar deposition is maximized at smaller particle sizes (near 3 μm) than is tracheobronchial deposition (near 6 μm). However, the maximum in deposition versus particle diameter is quite broad so that when one of these regional depositions is maximized, the other is still near 50% of its maximum in Figs. 7.12 and 7.13. Thus, although some targeting can be achieved using particle size (and indeed, this is the principal approach used in existing pharmaceutical inhalation devices, particularly when targeting the alveolar region by using particle sizes near 1–3 μm), it should be realized that such targeting is typically very broad. Flow rate and particle size targeting are usually the most commonly thought of means for achieving targeting of inhaled pharmaceutical aerosols. However, inhalation volume can affect deposition, since for smaller volumes, the conducting airways occupy more of the inhaled volume, resulting in a relative shift toward more tracheobronchial than alveolar deposition at low inhalation volumes and an opposite shift at higher lung volumes (particularly with breathholding so that all aerosol reaching the alveolar regions deposits there).

1. At lower flow rates, impaction and sedimentation respond oppositely to flow rate in the tracheobronchial region as seen in eqns. (7.81) and (7.82) since sedimentation decreases with increases in flow rate due to reduced residence times, while impaction increases with flow rate. As a result, flow rate targeting is not as effective at moderate flow rates where neither impaction nor sedimentation is dominant.

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In addition, particle density affects deposition, since impaction and sedimentation are affected by ρd2 in Eqs. (7.81), (7.82), so that aerodynamic particle size, dae, rather than particle size itself governs these mechanisms, as we discussed in Chapter 3. Thus, the use of low-density porous particles in dry-powder inhalers can allow targeting via particle density reductions in a way that parallels particle size targeting. Airway dimensions also strongly affect deposition (as already mentioned in discussing Eqs. 7.81)–7.83), with smaller airway diameters leading to enhanced tracheobronchial deposition due to increased flow rates and impaction, although control over airway dimensions is not a practical approach to targeting. However, alterations in airway dimensions can occur in a given patient due to changes in disease state or age, which can change the intended deposition location of an inhaled aerosol. One of the most helpful uses of aerosol deposition models is in aiding methods for targeting aerosols in the respiratory tract in pharmaceutical aerosol research and development. Indeed, serious attempts at targeting are often guided by deposition models (either simplified models discussed earlier in this chapter or more complex models involving CFD) since these models allow quantitative exploration of the effects of different variables on respiratory tract deposition targeting in a manner that is much more detailed than is possible with the general discussion presented above, although such models do not usually incorporate the broadening of targeting that occurs due to intersubject variability.

7.10 Conclusion By analyzing aerosol deposition in simplified geometries that resemble parts of the respiratory tract, we have seen that relatively simple deposition models can be developed, from which a good understanding of deposition in the respiratory tract can be produced. Such models produce results that are in generally good agreement with experimental and CFD results and can be useful in the development of inhaled pharmaceutical aerosol devices. However, there are a number of simplifying assumptions that go into these models that limit such models in their applicability. CFD simulations can overcome some limitations of the simplified models, although our present inability to measure the geometry of the smaller lung airways and local in vivo deposition in three dimensions hampers such studies. The most severe limitation regarding our knowledge of deposition from the point of view of inhaled pharmaceutical aerosols is probably our inability to rigorously predict deposition in diseased lungs, which remains a topic for future research.

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