Introduction to Bioengineering Lecture 2: Pharmaceutical Aerosols

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Introduction to BioengineeringLecture 2
Pharmaceutical Aerosols

• Taken from
• WH Finlay, The mechanics of inhaled
  Pharmaceutical Aerosols, Academic Press, 2001

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Why Pulmonary Delivery
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Respiratory Tract

• From an engineering point of view the geometry of
  the respiratory tract is not well know
• Geometry contains fine detail
• Geometry is time dependent
• Geometry varies between individuals
• Topologically the lungs simply consist of a
  series of bifurcating pipes
• Three basic regions
• Extrathoracic region (upper air ways)
• Tracheo-bronchial region
• Alveolar region

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Extrathoracic Region

• Extrathoracic region (upper air ways)
• Oral cavity (mouth) transient with variation
  in position of tongue and jaw
• Nasal cavity
• Larynx (constricted entrance to trachea
  containing vocal cords and trap door)
• Pharynx (throat region of between larynx and
  mouth oropharynx and between larynx and nose
  nasopharynx)

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Tracheo-Bronchial Region

• Tracheo-bronchial region (lower airways)
• Airways that conduct air from the larynx to the
  gas exchange regions
• Trachea to bronchi to terminal bronchioles
• Glottis is the opening from the larynx into the
  trachea which changes shape with flow rate
  (larger with higher flow rates)
• Main bronchi is the first generation of branching
  after trachea
• Lobar bronchi branch off the main (second
  generation)
• Two in the left lung
• Three in the right lung
• Lobar bronchi ventilate lobes
• Segmental bronchi branch off the lobar (third
  generation)
• Lobes are subdivided into bronchopulmonary
  segments each ventilated by segmental bronchi

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Tracheo-Bronchial Airways

• Covered with a mucus layer that overlays fine
  hairs (cilia)
• Cilia act to clear the mucus layer to the throat
  (swallowed or expectorated)
• Clearance occurs within 24 hours for particles gt
  1 micron
• Particles lt 1 micron borrow in mucus layer
• Cartilaginous rings in trachea and main bronchi
  causing corrugated inner surface (may effect
  fluid dynamics)
• Extrathoracic and tracheo-bronchial airways are
  termed the conducting airways since the move
  air to the gas-exchange region

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Alveolar Region

• Alveolar region -- aka parenchyma or pulmonary
  region
• Contains all parts of the lung with alveoli
• All the daughter generations from a single
  terminal bronchiole is called acinus
• Respirator Bronchioles
• first generation daughter branching after the
  terminal bronchioles
• Relatively few alveoli
• Subsequent generations will have an increasing
  number of alveoli
• Alveoli ducts
• Entirely covered by alveoli
• Several generations
• Alveoli sacs

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Weibel A Model (1963)

• Assumptions
• generation of the lung branches are symmetrical
  into tw0 identical daughter branches
• Generations 0-16 Tracheo-bronchial region
• Generations 17-19 Respirator Bronchioles
• Number of alveoli on each generation
• G17 5 alveoli
• G18 8 alveoli
• G19 12 alveoli
• Generations 20-23 are alveolar ducts with 20
  alveoli per duct
• Problems
• Not symmetrical
• Estimated volume 6 times to small for an adult
  male (people increase geometry by 6x to account
  for problem)
• Under predicts diameters of the tracheo-bronchial
  airways
• Conduction airways end at generation 14 not 16

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Breath Volumes Flow Rates

• Tidal Volume (Vt)
• Average volume inhaled and exhaled during
  periodic breathing
• Breath Frequency (f)
• The number of tidal breaths per minute
• Approximately 12 breaths per minute for an adult
• Duty Cycle
• Ratio of the inhalation time divided by the
  breathing period
• Inhalation occupies 43.5 of breathing period
• Exhalation occupies 51.5 of breathing period
• 5 pause
• Total Lung Capacity (TLC)
• The total volume of airspace in the lung when
  maximally inflated
• Typically around 6.1 liters for adults

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Definitions Continued

• Functional Residual Capacity (FRC)
• The volume of airspace during tidal breathing at
  start of inhalation
• Typically around 3.1 liters for adults
• Residual Volume (RV)
• The volume of airspace in lung at min.
• Vital Capacity (VC)
• Largest possible volume one can inhale
• Typically 4.1 liters in adults
• Forced Expiratory Volume (FEV)
• Max. volume that can be exhaled in 1 sec.

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• Volumes in adults are functions of
• Age
• Height
• Weight
• Race
• Disease
• Notice large standard deviations (i.e., 50)

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Fluid Dynamics

• To understand the fate of aerosols in the
  respiratory tract we must first understand the
  fluid motion
• We can not define detailed fluid mechanics for
  entire tract (see lung structure/models) but we
  can make a number of informative statements
• Incompressible fluid with constant density

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Governing Equations

• Navier-Stokes Equations (Conservation of
  Momentum)
• Non dimensionalizing the equation leads to
• Reynolds number (Re) is the ratio of inertial
  forces to viscous forces telling us the
  relative importance of viscosity
• Strouhal number (St) is the ratio of inertial
  forces to fluctuation forces -- tells us the
  relative importance of the unsteady term

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Calculating Re and St

• Previously we found
• f 12
• Vt 0.751 liters
• Q 18 liters/min or 300 cc/s
• Consider trachea (from Weibel A Model)
• Diameter (D) 1.539 cm
• Velocity (U) Q/A 161 cm/s
• Characteristic time (T) 5 s
• Calculate Re and St

Turbulent!
????
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Relative Importance of Terms

• The ratio of Reynolds number to Strouhal number
  provides an indication of the relative importance
  of the viscous term and the unsteady term (ratio
  of fluctuating forces to viscous forces)
• Womersley number is typically employed
• Fluctuating terms dominate in this case

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Homework

• Calculate and plot the Reynolds, Strouhal
  Womersley numbers for the larynx plus 23
  generations provided by the Weibel A Model
• Conditions
• characteristic time 5 seconds
• normal adult breathing rate 18 liters per
  minute
• and
• MDI/PDI breathing rate of 60 liters per minute
• Identify regions of turbulence and laminar flow
  (Regt1000 turbulent flow) and those where
  viscous and unsteady forces dominate

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Re St
Turbulence production occur distal to larynx by
shear in the boundary layers. Turbulence will
not exist long enough to be convected into the
next generations (via viscous dissipation).
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Womersley number

• It appears unsteady term can be neglected
  everywhere except for the first few generations
• Actually this term is much larger because flow
  changes direction during a breath

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Effect of Fluctuating Velocity

• Conservation of momentum with the average and
  fluctuating components
• The average moment must be conserved as in
  Navier-Stokes leaving a conservation of
  fluctuating momentum
• Non dimensionalizing the equation leads to

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Effect of Fluctuating Velocity

• Now unsteady effects are important if eRe is
  large
• Note that U must be associated with the U at
  that time.
• Thus e is the largest when we are between the
  exhalation and inhalation.
• Here U0 and U is finite
• e??
• This time over with e is significant is small
  over the breathing cycle (i.e. less than 5)
• Systems continuously supplying an aerosol (e.g.
  nebulizer) this problem is not significant to
  deposition rates
• However, MDI and PDI have a short burst of
  aerosol right at the start of inhalation (a
  trouble zone), unsteady effects are crucial

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Particle Reaction

• Up to now we have only considered the fluid
  motion and not considered the particle equations
  of motion nor the boundary conditions
• The time a particle is in a generation is
• If the fluid motion changes significantly over
  the time the particle is in a generation then
  unsteady effects are important
• This term look a lot like e

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Particle Reaction

• For a typical tidal breathing pattern unsteady
  effects are important even for continuous
  aerosols in the alveolar regions

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Particle Influence

• Particles not only react to fluid motion but
  generate fluid motion around them
• This particles can both increase and decrease
  turbulence
• The volume fraction of aerosol is the key
• Volume fractions lt 10-6 have negligible effect on
  fluid motion/turbulence

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Particle Influence

• Increasing or decreasing turbulence depends on
  particle diameter (d) compared to the eddy length
  scale (l0.5D)
• d/llt0.1 reduced turbulence
• d/lgt0.1 increases turbulence
• Example
• Generation 17 D5mm so l2.5mm
• dlt250mm reduce turbulence, dgt250mm increase
  turbulence
• Most inhaled therapeutics have particle sizes
  less than 100mm so we expect reduced turbulence

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Boundary Conditions

• Air inhaled into the respiratory tract is rapidly
  heated and humidified by heat and water vapor
  transfer from the airway walls
• This occurs within the first few generations
• Mucus layer on the airways will have little
  impact on the fluid motion in healthy patients,
  but in may disease states the mucus layer may
  become so thick it decreases the diameter of the
  generations

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Particle Deposition Models

• Particle size plays an important role in where
  particles will deposit
• Lung geometry and flow rate will also play
  significant roles
• Because our ability to adequately model the lung
  geometry we can only develop simplified
  deposition models based on fluid dynamics
• However a reasonable understanding for the
  dominate mechanisms can be illuminated through
  such simplified models
• Types of depositions
• Sedimentation
• Impaction
• Brownian Motion

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Sedimentation

• The effect of gravity on particles in the
  respirator tract can be understood by examining
  the deposition of particles in inclined circular
  tubes of laminar flow
• What fraction of particles deposit within the
  length of the tube?
• For simplicity we will assume a plug flow.

q
L
U
D
Plug flow
Poiseuille flow
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Settling Velocity

• The terminal velocity/settling velocity of a
  particle determined when the force of gravity on
  the particle is equal to the force of drag on the
  particle.
• A correction must be included to account for the
  failure of the no slip conditions. The Cunninghan
  slip correction factor uses the ratio of particle
  diameter (d) to mean free path of the fluid
  (l0.072mm in air).
• Thus the settling velocity becomes

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Fraction depositing due to settling

• The time allotted in a generation is
• The distance a particle can travel in that time
  via settling is
• Thus particles in the intersection between two
  circles offset by the above distance represent
  those that have not had sufficient time to
  deposit on the walls due to gravity

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Sedimentation effect of size
18 liters per min and 45 degree angle
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Sedimentation effect of body position
1 micron particle at 18 liters per min
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Sedimentation effect of fluid velocity

10 micron particle at 45 degree angle
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Homework 2

• Estimate the probability of a 3 micron particle
  of density 1000kg/m3 entering the 20th generation
  will deposit in that generation by sedimentation.
  Assume the flow rate is 50 liters per minute and
  the body position is at 45 degrees.

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Impaction

• The other main mechanism for particle deposition
  in the lung is by inertial impaction resulting
  from the particles trajectory differing from flow
  streamlines, such as flow around a bend

• By dimensionless analysis the motion of a
  particle is governed by
• geometry
• Reynolds number (Re)
• Stokes number (Stk)
• Non dimensional settling velocity

• Rule of thumb Stklt0.2 will follow streamlines

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Deposition by impaction

• Comparing the Stokes number to the
  non-dimensional settling velocity gives a
  indication of how important impacting is compared
  to sedimentation (Froude number)

Impact
Settling

• Notice impact is more important in the conducting
  airways

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Deposition by impaction

• Particle motion resulting in impaction in the
  conducting airways therefore is governed by
  geometry, Reynolds number and Stokes number
• Experimental results indicate impaction is only
  weakly dependent on geometry and Reynolds number
• Impactionf(Stk)
• The Stokes number is a measure of how important
  inertial effects are in determining particle
  trajectories (aka particle relaxation time, or
  stopping/starting distance)

Non dimensionlize
Notice in a absence of gravity if Stk ? 0 then
the relative velocity must go to zero and the
particle and fluid velocities match. Hence Stkltlt1
will follow streamlines
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Impaction relations based on Stk

• Most relations dont include effects of previous
  generations (i.e., secondary flow) except for
  Chan and Lippman (1980)

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Brownian Motion

• For small particles collisions with randomly
  moving air molecules will cause the particle to
  change direction
• This random walk is called Brownian motions
• The randomness of this motion can not be
  predicted except at in small time frames
• As d decreases Dd increases, thus small particles
  (dltlt1 micron) diffuse readily due to molecular
  collision

Dd is molecular diffusivity
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Brownian Motion vs. Sedimentation

• Importance of Brownian motion only comes into
  play when sedimentation becomes too slow
  (xb/xslt0.1)

Sedimentation
Brownian
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Deposition by Brownian Motion

• Brownian motion not terribly important in any
  region of the lung (except maybe in the alveolar
  region)

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Simultaneous deposition

• The sum of all the probabilities is

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Effect of induced electrical charge

• So far we have discussed deposition by impaction,
  sedimentation and Brownian motion however
  aerosols have a net electrical charge (result of
  formation) thus electrostatic forces can affect
  particle motion
• The force requires a charged particle (q) and an
  external electric field (E)
• A lung does not set up an electric field unless
  charge particles are present when tissue
  molecules orient themselves (dielectric effect)

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Effect of induced electrical charge

• Lung tissue has a dielectric constant (e) of 80
  (same as water), this large value simplifies the
  problem where the lung gains the opposite charge
  of the particle

x
x
Imaginary particle

• The resulting force is a function of particle
  charge, the twice the distance between the
  particle and the lung wall (r), permittivity of
  the free space

Lung wall
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Effect of charge on motion

• Particle motion is there by influenced through an
  additional term
• Non dimensionalizing as done previously leads to

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Importance of electrostatics

• The ratio of Inc to Stk to determine when induced
  charge is more important that inertia (i.e.,
  Inc/Stk gtgt 1)
• For typical breathing patterns ngt43 before
  electrostatics are important (for early airways
  only)
• The ratio of Inc to u to determine when induced
  charge is more important that sedimentation
  (I.e., Inc/u gtgt 1)
• For typical breathing patterns ngt30 before
  electrostatics are important (for early airways
  only)

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Delivery systems

• Currently there a few aerosol delivery/generation
  systems
• Meter Dose Inhaler (MDI)
• Meter Dose Inhaler (MDI) with spacer
• Powder Dose Inhaler (PDI)
• Jet Nebulizer
• Ultrasonic Nebulizer
• How do they work??

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Meter Dose Inhaler /- Spacer
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Powder Dose Inhaler
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Jet Nebulizer
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Ultrasonic Nebulizer










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Homework 3

• Calculate and compare the total, impact,
  sedimentation and Brownian motion deposition for
  a spherical particle of density 1000kg/m3 in
  generations 0, 15 and 23.. Particle sizes will
  range between 0.5 and 5 micron. Airway velocity
  is 60 liters per min. Assume a 45 degree posture.
• In which generations is the aerosol generated by
  the new Turbuhaler effected by electrostatic
  forces. Assume particle density is 1000kg/m3 and
  the average diameter is 5 microns