Transcript No Slide Title

Distributed Models of Drug Kinetics
Paul F. Morrison Ph.D.
Drug Delivery and Kinetics Resource,
Division of Bioengineering and Physical Science,
NIH
Previous: Whole organ pharmacokinetic models

C(t), Cp(t)
•Multiple organs
•Multiple chemical species
Today: Distributed models of pharmacokinetics

C(x, t)
•Multiple chemical species
Outline
• General principles behind distributed models
– Simple formalism
– Major physiological-physical factors
• Examples where distributed kinetics are important in
drug delivery
–
–
–
–
–
Peritoneal cavity
Intraventricular infusion
Direct interstitial infusion in the CNS
Microdialysis
Delivery of tight-binding antibodies
General Principles
• Central issue: What is the situation which leads to a
spatially-dependent distribution of drug in a tissue, and
how is this distribution described quantitatively?
• Essential Characteristic: Target tissue can not be
approximated as a well-mixed compartment.
Specifically, drug is delivered along a path from the
source along which local clearance or binding causes
the concentration to vary.
Common Situations Leading to Spatial
Distributions
• Delivery of agents from spatially non-uniform sources
– e.g. point source of needle tips
– e.g. surface (planar) sources encountered when drug
solution bathes the surface of a target organ
– e.g. intravenous delivery into tissues with highly
heterogeneous capillary densities
– Certain tumors
• Delivery of very highly binding substances leading to a
“moving front” concentration boundary
– High affinity antibody conjugates
Distributed Capillary Bed
Diffusion-Reaction Formalism
Mass balance for substance in tissue:
2
C
t
=
rate of conc
change in DV
 C
D 2
x
km
C
R
-
net diffusion
metabolism
in DV
in DV
If C p (t)  0,
C(x, 0) = 0,
IC
C(0, t ) = R Cinf , BC
C( , t) = 0,
-

C
pa  - C p 
R

BC
net transport
across microvasculature
c
-D
x x
DV
c
-D
x x + Dx
kmC /R
pa(C/R-Cp)
(Diffusion-reaction Cont’d)
Then C(x, t)
R Cinf
1
 - x 
 x

erfc 
= exp 
- kt 

 k / D 
 4Dt

2
1
 x 
 x

erfc 
+ exp 
+ kt 

 k / D 
 4Dt

2
where k = ( km + pa ) / R
At steady state, this simplifies to just:
C(x)
= R exp[- x k / D ]
Cinf
If no reaction is present, then:
C(x, t)
 x 
= R erfc 
 4Dt 

Cinf
Implications
• Drugs can be delivered to tissue layer near exposed
surface but the thickness of this layer depends
strongly on metabolism of agent
• Delivery of non-metabolized agents across surfaces
for purposes of systemic drug administration (e.g.
I.P.) is dominated by distributed microvascular
uptake in the tissue layer underlying the surface
– And not by the rate of transfer across the peritoneal
membrane per se
• Importantly, local dose-response relationships also
become spatially dependent, e.g.

AUC =  C(t) dt
is NOT applicable,
0
instead

AUC(x) =  C(x, t) dt
0
is applicable.
Diffusion Constants and Reaction Rate
Constants
• Diffusion constants extrapolated from known
values for reference substances
37
2 *
- 0.6
Dtissue = l2a*Daqueous
MW
la(
)
 Dtissue  MWref 0.6

 = 

D
 tissue,ref   MW 
• Reaction rate constants
– Cultured cell metabolic data
– Fittings of compartment models to whole tissue
data
– From autoradiography and fit to exp[-x√(k/D)]
(continued)
•
pa  MW -
0.63
so
 pa  MWref 0.63


pa  = 
 ref   MW 
– Concentration profile insensitive to MW if k= pa/R
(i.e. inulin depth ~ urea)
Example 1
Intraperitoneal Administration of
Agents
Treatment of ovarian cancer patients
The ovarian goals:
• Achieve sufficient penetration of surfaces to treat
resident tumor nodules
– Nodules lie on serosal surface, not invasive
– Not metastatic
– Diameter of <5 mm (in 73% of post surgical cases)
• Complete irrigation of serosal surfaces
– Predicted pharmacokinetic advantage of I.P. over I.V. delivery
– Alberts et al confirmed survival advantage for patients treated
with cisplatin
Predicted Penetration in Peritoneum
• Cisplatin
– Cis-diamminedichloroplatinum (II)
– Slow hydrolysis
– Compute C(x) at steady state
• How good is the model?
– Estimate predictability of diffusion model for surrogate
EDTA
Concentration Profile of EDTA at
Peritoneal Interface of GI Tissue
1
Relative Tissue Concentration
C / Cinf
(C- f C )
e p
= fe exp[ -x k / D ]
(Cinf - C p )
where fe = 0.26, k / D = 0.0078m -
1
0.1
Plasma
0.01
0
200
400
600
800
Distance from Peritoneum (µ)
1000
Example 2
Intraventricular Delivery
-- Attempt to circumvent the BBB
Blood-Brain Barrier Exclusion
of Histamine
Pardridge et al, Ann Int Med 105; 82-95 (1986)
Sucrose Distribution After
Intraventricular Administration
Groothuis et al, J. Neurosurg. 1998
Similar Penetration with
Macromolecules
Penetration depth (d p ) =
ln 2
pa + km
fe De
• For macromolecules (MW > 67kD), D = fe De /R
decreases by >10-fold and effective pa decreases
by a similar amount. Thus, in the capillary
permeation limit
d p  d p (unchanged)
Intraventricular Infusion of BDNF
Yan et al, Exp. Neurol. 127; 23-36 (1994)
Delivery of NGF from Implanted
Polymer
2.5mm
2.5mm
Krewson et al, Brain Res. 680; 196-206 (1995)
Example 3
Direct Interstitial Infusion
(microinfusion)
-- approach to achieve more widespread
distribution
Direct Infusion Cannula in Brain
On a cellular scale:
Slow flow:
0.5 to 10 µl/hour
(Alzet range)
High flow:
0.1 to 4.0 µl/min
(Harvard pump)
Two Techniques for Direct Interstitial
Infusion
Harvard Pump
Alzet pump
250 µl
250 µl
Syringe system
10 µl
Slow Microinfusion
(0.9 µl/hr of cisplatin for 160 hrs)
Applicable mass balance:
km
C
1  2 C
C
= D
r
- pa( - C p ) C
2
t

r

r
R
R
r
changein
net diffusion
total tissue
intoa volume
conc per Dt
element
permeation
irreversible
reaction
Steady state solution for continuous infusion,
after matching mass inflow rate to
diffusive flux at cannula tip, is:
q Cinf
C(r) =
exp(-d r)
4 Dr
where d = ( pa + km )/( DR)
qCinf is mass infusion rate, and D=f De/R.
R= 1 for cisplatin, R=ffor IgG, where f is
the extracellular volume fraction. Time to
steady state at r = 4 mm is 3 hrs for cisplatin.
Concentration Profile of Cisplatin in
Rat Brain
(0.9 µl/hr)
Issues
• Treatment volumes r >1 cm
• If pumping rates taken to maximum range, how
does the response of high-flow delivery compare
with low-flow (diffusional) delivery?
• Simple estimators of profiles and volumes of
distribution
High Flow Infusion Model
 pi
v(r ) = - k
r
Pressure:
Darcy’s Law:
Continuity Equation for Water
Minor deformation
Macromolecular mass balance:
km
1  2 C
1  2
C
C
r v C - pa( - C p ) - C
= D 2 r
2
R
R
t
r
r r
R r r
change
net diffusion
net bulk flow
net mass transfer
in total
into avolume
into a volume
across capillaries
conc in Dt
element
element
metabolic
loss
Simplification of Mass Balance
C
1  2
= r vC -kC
2
t
R r r
where k =(km + pa)/ R and
R = f inabsenceof binding and confinement to ECS
• Concentration Profile:
C(r)/ Cinf
 4 (km + pa) 3

3
= R exp r - ro 

3q

(
)
where q Cinf is the mass infusion rate, ro is the cannula radius
• Penetration depth at S.S. and time to reach it:
rp = 3 2q / [4 (km + pa)]
1.8 cm
3.6 cm
t p = 2 R/[3(km + pa)]
1.2 days
14.0 days
33 hr=> km
infinity => km
REPRESENTATIVE MACROMOLECULAR PARAMETERS[1]
Parameter
Symbol
Value
Source
Tissue hydraulic conductivity
(cm4/dyne/sec)
Capillary permeability (cm/sec)
Capillary area/tissue volume
(cm 2/cm3)
Extracellular fraction
Catheter radius (cm)
Diffusion coefficient (cm2/sec)
Volumetric infusion rate (cm3/sec)
k
0.34x10-8
Morrison et al 1994
p
1.1x10-9
Blasberg et al 1987
a
100
0.2
0.023
Bradbury 1979
Patlak et al 1975
32 gauge
De
q
0.7x10-7
5.0x10-5
Metabolic rate constant (sec-1)
km
5.7x10-6
Tao and Nicholson 1996[2]
Typical 3 µl/min high-flow
infusion rate.
Arbitrary value[3]
f
ro
[1] Typical of a 180 kDa protein. Note that D in other equations is defined as D=f De /R
where R=f for a non-binding molecule confined to the extracellular space. R=1 for
a non-binding molecule that distributes equally between the intra and extracellular spaces.
[2] Albumin De value (gray matter) of these authors scaled by molecular weight to 180 kDa.
[3] This rate corresponds to a half-life of 33 hr and is roughly 5 times the average brain
protein turnover rate
Concentration Profiles in Brain
(3 µl/min)
Relative Interstitial Concentration
C e=C/( f C inf )
1
0.8
0.6
0.4
1 hr
8 hr
steady state
0.2
27 hr
64 hr
0
0
0.5
1
1.5
Radial Dis tance
2
(cm )
2.5
3
Brain Infusion Technology
• Delivery
Predicted distribution of albumin-aCSF in
striatum from flow-diffusion-reaction equation
111-In-DTPA-Transferrin Infusion
MODEL: POST-INFUSION PHASE
•
Pure Diffusional Relaxation of
End-of-Infusion Profile
• Profile:
C(r, tˆ) =
- k tˆ

2
2
e
- (r - r ) /(4Dtˆ )
- (r + r ) /(4Dtˆ )
r dr 
 C(r , t inf ) [ e
-e
]
2r  Dtˆ 0
tˆ > 0
where
D = f De / R, k = ( pa + km ) / R, tˆ = t - t inf
Post-infusional Behavior of
Concentration Profiles
Post-infusional Behavior of
Concentration Profiles
Increased Treatment Volume with High
Flow Infusion
Application: Chemical Pallidotomy
• Parkinson’s disease
• Alternative to thermal ablation
– Problem of optic nerve destruction
• Quinolinic acid
– Parameters from microdialysis
– Small molecular weight
Simulated Quinolinic Acid Profiles in
Gpi Following Direct Interstitial
Infusion
Chemopallidectomy
Chemopallidectomy
Gpi, Intact Side
Gpi, Lesioned Side
Summary
• General principles of distributed
pharmacokinetics
• Examples
-- Peritoneal
-- Intraventricular
-- Interstitial
• Next week: Population pharmacokinetics