(CANCER RESEARCH 49. 3290-3296, June 15, 1989)
Effect of Dose, Molecular Size, Affinity, and Protein Binding on Tumor Uptake of
Antibody or Ligand: A Biomathematical Model1
Gillian D. Thomas,2 Michael J. Chappell, Peter W. Dykes, David B. Ramsden, Keith R. Godfrey, John R. M. Ellis,
and Arthur R. Bradwell
Immunodiagnostic Research Laboratory, Department of Immunology, University of Birmingham Medical School, Birmingham BIS 277 ¡G.D. T., P. W. D,, J. R. M. E.,
A. R. B.]; University Department of Medicine, Queen Elizabeth Hospital, Birmingham BIS 2TH [D. B. R.]; and Department of Engineering, University of Warwick,
Coventry CV4 7AL[M. J. C., K. R. G.J, England
constant of 10 3 s ' one-half the bound antibody should have
ABSTRACT
A mathematical model has been developed to determine the best
approach to improving tumor targeting with antibody. The amount of
antibody in the tumor (tumor content) and the tumor:normal tissue
antibody concentration ratio (uptake ratio) were calculated over 12 days
from injection, using the computer program FACSIMILE to solve the
stiff nonlinear differential equations describing the system. Results indi
cate that success requires an optimal combination of dose, size, and
binding affinity of antibody. Increasing the dose to 100 times that
presently used for scanning increased both the percentage of injected
antibody in the tumor and the uptake ratio by up to 2 orders of magnitude
to maximal values determined by affinity. This result could be achieved
by coinjecting unlabeled antibody. Increasing affinity from A',,,= 10'' to
IO" M' ' increased the uptake ratio from 5 to 100 for whole antibody and
to 550 for a small ligand, at the calculated optimal dose, but had no effect
at the current scanning dose. With decreasing molecular size at average
affinity, the same maximum tumor content and uptake ratio were achieved
but progressively earlier. At high affinity there was a substantial advan
tage for a small ligand compared with whole antibody in terms of uptake
ratio (550 versus 100) and minor:normal tissue integral dose ratio (330
versus 60). The uptake of a small ligand was not increased by binding to
plasma protein but with increasing time the tumor content was higher
than without protein binding.
INTRODUCTION
Effective targeting of radioisotopes or toxins to cancer cells
with antitumor antibodies has not yet been achieved (1-3).
Human studies consistently report a maximum TC3 of only
around 0.005% of the injected dose/g since the tumorbackground UR remains less than 5 (1) despite a variety
of experimental strategies (4). The UR must be increased 10fold if antibody scans are to compete with computerized axial
tomography for detection of deep-seated tumors (4) and further
still if curative therapy is to be delivered safely (1). The aim of
this study was to establish how much improvement could the
oretically be achieved by manipulating individual antibody char
acteristics and to find which of these would be of most impor
tance.
Of the many factors which contribute to and limit the final
UR (4), the antigen-binding reaction is fundamental, yet ade
quate binding affinity has not been defined and in one study
low affinity was claimed to be advantageous (5). Affinity con
stants are not quoted in studies of antibody localization as a
rule, and standard low-temperature measurements of affinity
for soluble antigen are likely to differ from true affinity at the
cell surface in vivo (6). Also, for a typical dissociation rate
Received 10/6/88; revised 3/21/89; accepted 3/22/89.
The costs of publication of this article were defrayed in part by the payment
of page charges. This article must therefore be hereby marked advertisement in
accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
1G. D. T. and J. R. M. E. were supported by the Cancer Research Campaign
(Grant CEF 412) and M. J. C. by the Science and Engineering Research Council
(Grant GR/D90642).
2 To whom requests for reprints should be addressed.
3The abbreviations used are: TC, tumor content; UR, uptake ratio; ECF,
extracellular fluid; CEA, carcinoembryonic antigen.
dissociated after 12 min (6), but in practice the tumor residence
time is in days. The extent to which antigen-binding affinity
affects tumor uptake in vivo is thus unknown.
Molecular size introduces a variable time factor to the binding
kinetics by determining the rates of extravasation and excretion
of antibodies or their fragments. Intact antibodies with their
long plasma half-life (7) and slow diffusion in ECF (8) give low
URs due to high background concentration. Comparison of
fragments with IgG in animals shows slightly higher URs but
lower TCs for the fragments (9, 10) suggesting that a major
improvement might be achieved by delaying their urinary ex
cretion, perhaps by engineering binding of small fragments to
plasma proteins. Binding with low affinity to plasma protein
and high affinity to tumor might lead to increased TC with
time, and background clearance could be achieved at any time
by selective competitive displacement from the protein only.
Theoretical analyses of binding systems (antibody, enzyme,
or hormone) usually assume equilibrium or a steady state (1113), inappropriate to the process of tumor uptake. Calculation
of the extent of tumor binding in vivo with time requires a
computer program capable of handling simultaneous nonlinear
differential equations with widely varying rate constants and
has not previously been done. We therefore constructed a
compartmental model to investigate the effect of affinity, mo
lecular size, and injected dose on UR, TC, and tumornormal
tissue integral dose ratio.
MATERIALS AND METHODS
Development of the Model
The model (Fig. 1) contains two central compartments, plasma and
ECF, with entry of antibody, fragment, or ligand by i.v. bolus injection
and exit via plasma into urine. When binding occurs to cell surface
antigens or receptors bathed in ECF, the substance enters a "bound
compartment" from which it returns to the "free in ECF" compartment
on dissociation. For small molecules, compartments can be added
representing binding to proteins (such as albumin) and these can be
entered from plasma and/or ECF. Disposition of the injected dose
depends on molecular size and on affinities for the various binding sites
as well as the concentration of antigen or receptor on tumor and other
cell surfaces. No absolutely tumor-specific cell component has been
identified to date; therefore the model has separate tumor-bound and
other-tissue-bound compartments, the latter postulated to exist homo
geneously throughout ECF. TC equals the total number of mol of
injected substance present in the tumor-bound compartment at any
instant and may be expressed as a percentage of the injected dose. UR
is calculated as the ratio of the concentration in tumor to the mean
concentration over all other compartments, plus I since they must be
at least equal. Total tumor and rest-of-body doses are calculated by
integration and compared on a per-volume basis, assuming uniform
distribution through nontumorous tissue.
Selection of Appropriate Input Variables
Dose. Initially a dose of 6 x 10"I0 mol was used, based on the
quantity normally given for antibody scanning (100 ¿<n).
The effect of
3290
Downloaded from cancerres.aacrjournals.org on June 17, 2017. © 1989 American Association for Cancer Research.
OPTIMAL PROPERTIES
1,2
OF TARGETED MOLECULES
3,4
describing rates of movement between plasma, ECF, and urine. Exper
imentally determined plasma clearance values allow calculation of
linear rate constants for molecules of different sizes, the rate constant
having units of time"' and being equal to clearance (i.e., fraction of
plasma volume/time) divided by plasma volume. Thus
3,5
(Rate constant) x (concentration of solute) x (volume of solution)
*PLASMA**
= Amount of solute transferring into ECF/unit of time
ECF*
For a given molecular size the same rate constant applied to renal
excretion gives renal clearance values consistent with published data
(22, 23). Constants of 10~3to 10~5s~' were found to represent a range
3,6
Fig. 1. Model of the disposition of antibody (or smaller molecule) shown free
in plasma (/) and ECF (3), reversibly bound to identical normal tissue (3,5) and
tumor (3,6) receptors and (small size) to soluble protein (1,2 and 3,4). Urinary
excretion is also indicated (7).
of sizes from a few hundred daltons to the size of an antibody (19, 2528). The rate constant for transfer from ECF to plasma was set at 20%
of that for plasma to ECF in accordance with the ECF:plasma volume
ratio.
Animal studies suggest that tumor capillaries may be up to 10 times
more permeable than those in normal tissues (Table 1) and this was
incorporated into the model by allowing tumor antigens/receptors to
interact with antibodies/ligands from a 50-ml volume of ECF although
the tumor volume was set at 10 ml.
Antigen-binding Affinity. It is usual to describe antibody affinity in
terms of the equilibrium constant A"eq,defined as the ratio of the
association and dissociation rate constants Aaand k¿.According to the
law of mass action (35, 36)
»«IO'
A + Bz=ìAB
*d
This gives rise to the simultaneous nonlinear differential equations
ÃŽ ,-B
Ê10
- ([A] or [A]) = ka(AB] - k,(A](B]
Albumin
Molecular
lg G
Radius
(n m)
Fig. 2. Relationship between molecular radius and permeability/surface
product for intact capillaries (Refs. 17-20).
area
and
Jt([AB\)
increasing dose on tumor uptake was then observed for different mo
lecular sizes and affinities, with a fixed tumor antigen quantity of 2 x
10~'°mol based on the range of CEA molecules per colonie tumor cell
(14) and assuming 10' antigen-expressing cells in the tumor. Tissue
receptor concentration was set at 100 times less based on the range of
CEA expression in noncancerous tissue (15, 16). With a chosen tumor
volume of 10 ml to simulate a barely detectable lesion and receptorbearing normal tissue volume of 121, equal to ECF volume, there was
thus 12 times more antigen/receptor outside the tumor than in it (10
ml x 100 versus 121 x 1). Plasma volume (2.51) was used in preference
to whole blood volume.
Molecular Size. The permeability of continuous capillary endothelia
to intravascular solutes of different molecular sizes is illustrated in Fig.
2 (17-20). The position of whole antibodies (IgG) (M, 150,000) is well
down the curve and they are therefore very slowly extravasated; a
molecule much smaller than Fab' (M, 50,000) would be necessary to
equilibrate quickly with ECF.
The following rough calculation indicates the likely importance of
antibody size, given a scanning (100 ¿ig)dose of immunoglobulin
injected i.V.: plasma volume, 2.5 liters; molecular weight of IgG,
150,000; Avogadro's number, 6.02 x 10" molecules/mol, thus 100 Mg
IgG = 1.61 x 10" molecules/ml of plasma; clearance rate of IgG into
ECF, 0.0036 ml/min/100 g tissue (18, 24), 2.52 ml/min/70-kg man,
and 4.05 x 10" molecules/min; total cells in body, IO14approximately
k,[A](B] - kd[AB]
Typically antibodies have very large /cas (IO5 to IO8 M ' s ') and
lower, more variable k¿s(10~5 - IO3s~') (37) with ^eq for antibodies of
interest being 10s to IO12M~' (38). Values of A. between IO5 and IO9
M ' s ' and of k¿between 10 ' and 10 5 s ' were used in the model,
with tumor and normal tissue binding represented by the same constants
in every case.
Protein Binding. Since small molecules are rapidly lost in urine, the
model examined the effect of increasing retention times by simulating
plasma protein binding for small molecules which also bind to tumors.
Given a lower affinity for plasma protein than for tumor, transfer from
protein to tumor-binding sites might occur with time and result in
greater TC. High URs might then be achieved when TC reached a
maximum by injecting a competitor for the protein-binding site, to
Table 1 Experimentally determined tumor:normal tissue permeability ratios from
animal studies of the extravasation of test solutes, blue dyes (<I nm), albumin
(3.5 nm), IgG (5.5 nm), and fluorescent dextrans (up to 8.2 nm)
All tumors were from rats except for the VX2 carcinoma (rabbit).
(21). It is likely that at least 1% of cells are accessible to solutes in
ECF; i.e., the IgG is distributed among at least IO'2 cells.
normal
ra
TumorWalkerCarcinosarcomaChondrosarcomaSarcoma
tissueMuscleMuscleLiverKidneyConnective
dius
(nm)3.53.53.53.5<13.55.53.55.5<18.2Tumor:
permeability7.221.11.4123.73.2101.88Ref.2
Rd/3FibrosarcomaSarcomaWalkerCarcinosarcomaVX2
Thus at most 0.4 molecule is extravasated per min per cell, or 1 IgG
molecule might reach each cell every 2.5 min, if none is metabolized or
excreted.
Alteration in molecular size is simulated in the model by varying the
rate constants A in the linear equations of the form
tissueIntestineIntestineMuscleMuscleLiverMature
carcinomaNormal
granulationtissueSolute
3291
Downloaded from cancerres.aacrjournals.org on June 17, 2017. © 1989 American Association for Cancer Research.
OPTIMAL PROPERTIES
OF TARGETED MOLECULES
displace non-tumor-bound molecules and allow them to be excreted
rapidly in urine while briefly increasing the amount available to tumor.
Initially binding constants and plasma protein concentrations for
thyroxine binding to thyroxine-binding globulin and albumin were used
(39) with ECF concentration set at two-thirds of that in plasma. The
effect of varying affinity and concentration was then assessed. Compet
itive displacement of ligand from protein was simulated by reducing
the numbers of protein-binding sites at the time of maximum TC.
Computer Simulation. The model has up to six compartments with
transfer of the injected substance between them depending on both first
and second order reactions (representing rates of movement and re
versible binding, respectively). This is a complex nonlinear system
involving many simultaneous differential rate equations (see "Appen
dix"), and the wide variation between their rate constants makes it stiff
40.
ligand
30.
20 _
Jt
(0
(40) because rapid events require integrations to be made at extremely
short intervals which become inappropriate where change is gradual.
FACSIMILE (41) adjusts the integration interval according to the rate
of change within the system so that accurate simulation over long
periods takes a few seconds. Data were generated over IO6 s (11.57
io'9
io'8
10
10
Total
Dose
io10
io"7
io"6
days) from injection and graphical results obtained using the SAS
package (42).
RESULTS
Dose. With the usual antibody scanning dose of 6 x 10~10
mol and typical antigen-binding affinity Keq of IO10M~', pre
dicted peak TC and UR values were 0.02% of the injected dose
and 3.6, respectively, for all molecules although timing of peaks
varied with size (Fig. 5). With increasing dose, peak TC and
UR increased until saturation of tumor receptors occurred and
the substance was present in normal tissues in excess. For a
large molecule such as antibody, this excess was only slowly
excreted and lowered the UR substantially, whereas for a small
ligand, rapid excretion of non-tumor-bound substance meant
that the UR reached a plateau (Fig. 3a). At the lowest (antibody
scanning) dose level, tumor receptors were far from saturated,
but the rate of binding to these was very low in proportion to
the ECF concentration, and with increasing dose a greater
proportion bound to tumor due to mass action (Fig. 3¿>)
until a
maximum was reached beyond which receptors were saturated.
If the labeled dose was assumed constant at 6 x 10~10mol and
0.1 -
-10
10
Tumor- and other-tissue-binding compartments were incorpo
rated and binding affinities for both were: K«,= 10'°M"' (ka =
IO7 M~' s~', ki = 10~3 s~'). The curves are similar in shape to
others seen with different affinities. Peak values did not differ
significantly between sizes but were delayed by increasing mo
lecular size. For the largest molecule the UR was still rising
after 12 days.
The effect of molecular size on TC is shown in Fig. Si/i.
Again peak values were delayed with increasing size, but the
peak TC occurred earlier than the peak UR for a given size.
-7
10
10
(moles)
Fig. 3. Effect of injected dose on (a) peak tumorbackground uptake ratio UR
for whole antibody (IgG) and a small tumor-binding ligand and (A) peak tumor
content TC as percentage of injected dose for all sizes. /Tcq= 10'°M"'. Arrows,
standard dose for antibody scanning.
in
10
-10
-
total
the dose increased by adding unlabeled molecules, the absolute
amount of labeled substance in the tumor was shown to rise
and fall with increasing dose. Thus adding the correct propor
tion of unlabeled molecules increased the absolute TC of the
label (Fig. 4).
Optimal UR and TC values for our given number of tumor
receptors were obtained with a dose of 6 x 10"8 mol and this
dose was used to study the effects of other variables.
Molecular Size. Whole-body retention curves are shown (Fig.
5/) for three simulated molecular sizes ranging from intact IgG
to a putative small tumor-binding molecule or ligand (up to
several hundred daltons). Whole-body half-times with no tu
mor, tissue, or protein binding were 36, 4.7, and 1.4 h, respec
tively.
The effect of molecular size on UR is shown in Fig. 5//'.
-8
ep
0
-10
10
-8
10
10
Total
Dose
-6
-7
10
10
(moles)
Fig. 4. Effect of injected dose on peak TC as mol of injected substance (all
molecular sizes). Curves represent total and labeled molecules bound to tumor
given a fixed number of labeled molecules (6 x IO '" mol) at all dose levels.
Arrow, standard dose for antibody scanning.
For the smallest molecule, peak values of TC and UR occurred
early and almost simultaneously, giving a small integral tumor
dose. For the largest molecule (whole IgG size), at the time of
maximum TC, the UR was still low and rising and the large
area under the TC curve represented a greater integral tumor
dose than for the small molecule. Calculation of tumornormal
tissue integral dose ratios showed an advantage for IgG at
average affinities but a much larger advantage for a small ligand
at high affinity (Table 2).
Antigen-binding Affinity. With increasing affinity, peak TC,
UR, and the integral dose ratio increased markedly (Fig. 6;
Table 2). Lowering the ka at a given Ke<¡
increased the integral
3292
Downloaded from cancerres.aacrjournals.org on June 17, 2017. © 1989 American Association for Cancer Research.
OPTIMAL PROPERTIES
OF TARGETED MOLECULES
0. 33
O.JO
0.25
0.20
0.15
O. IO
0 . O5
0.00
O . 36 i
o. ao
0 . 23 •¿
o. to
)•¿
o.is
40
O. IO
JO
O . 05
O . OO
20
Fig. 6. Effect of binding affinity /f„(= kjk¿)on TC of a small ligand (bottom)
and a Fab' fragment (top, with corresponding Curves a to </). For solid lines, k¿
= IO'3 s'1 and A. is increasing to give AT«,
= 10' (d); 10'°(c); and IO12(*) NT1.
Dotted lines (a) show Afeq= IO12M~' but with k¿= IO"5 s"' (TC as percentage of
IO
O
O
1
2
3
4
injected dose).
Doy.
small
ligand
300
o 0.10
O
a
01 200
IO
6(7
11
12
P«y»
Fig. 5. Effect of molecular size on (i) whole body retention as percentage of
injected dose; (iï)UR; (HI) TC as percentage of injected dose. Curves, whole
antibody (a), a Fab' fragment (b) and a small ligand (<•).
Tumor and other-tissuebinding affinity K«,= 10'°NT1.
100
Table 2 Effect ofK,, on peak UR and TC and integral dose ratio for the smallest
(ligand) and largest (IgG) molecules simulated
Injected dose, 6 x 10 * mol or 100 x normal scanning dose.
URK«o
TC (% of
dose)Ligand0.003
injected
Peak
oLigand1.5
108
tissue integral
ratioLigand0.4
dose
1010
1011
1012
1013
Fig. 7. Effect of AT,,on integral dose ratio for whole antibody (IgG) and a
small ligand.
M ' was equivalent to increasing affinity from 10s to IO13M '
5°31'
0.029
3.6
3.8
0.029
0.157
0.162
16.1
23.8
81"
0.292
0.301
30.3
51.2
99*99°Peak
0.329
0.329
86.7
58.1
429553IgG1.4
0.333IgG0.003
0.333Tumornormal
332.3IgG0.458.9
" Still rising after 11 days.
633172
109
dose significantly for a small ligand, but for the larger sizes the
difference between increasing /caand decreasing k¿was insignif
icant (Fig. 6). At intermediate affinities the integral dose ratio
was greatest for IgG but then reached a plateau, whereas at very
high affinities there was an increasing advantage for small size
(Fig. 7).
Using the scanning dose of 6 x 10 10mol, an increase in Aeq
from 10'°to IO13M~' increased peak TC from 0.018 to 0.019%
using the higher dose.
Protein Binding. Introduction of protein binding of the small
molecule reduced the peak values of TC and UR in direct
proportion to the amount and/or the affinity of the binding
protein due to increasing competition for injected molecules by
protein-binding sites. Total body retention time was also pro
portionately increased and after a short time this resulted in
greater TC in the presence of the carrier protein (Fig. 8). The
anticipated transfer of molecules from protein to tumor, with
corresponding improvement in peak TC, was not seen despite
large differences (up to IO8M"') between tumor and protein A«,
values.
Competitive displacement of molecules from protein-binding
sites at the time of maximum TC caused a transient rise in free
of the dose and peak UR from 3.6 to 3.7 for all sizes. Increasing
concentration, with a rise in TC, and then an increased excre
the dose from 6 x 10"'°to 6 x 10~8 mol at an affinity of IO13
tion rate with a rise in UR. Maximal TC and UR values
3293
Downloaded from cancerres.aacrjournals.org on June 17, 2017. © 1989 American Association for Cancer Research.
OPTIMAL PROPERTIES
£
OF TARGETED MOLECULES
0.0«
7
•¿
•¿
Fig. 8. TC as percentage of injected dose for a small ligand with (PROT) and
without (No PROT) protein binding showing the effect of displacement from
protein (DISP).
approached those reached without any protein binding but in
no case were these exceeded (Fig. 8).
DISCUSSION
This model is unique in achieving the complex mathematical
simulation of the binding kinetics of antitumor antibody in vivo.
For appropriate input data the predicted TC, UR, and tumornormal tissue integral dose ratio correspond closely to
reported values (1) (Table 2). The model demonstrates how
these indices can be maximized and predicts that substantial
improvements in tumor uptake are possible. The model focuses
on the absolute minimum requirements for antibody localiza
tion, disregarding any other factors such as antigenic modula
tion (43), circulating antigen (44), and dissociation of radiolabel
(45). These would reduce TC and UR, as does binding to normal
tissue. This is the model variable subject to the most error, and
the chosen value probably compensates for other uptake-reduc
ing factors not included. The numerical predictions will not be
precise but demonstrate the interplay of dose, size, and affinity
and the magnitude of their effects.
An increased scanning dose of antibody appears essential
since the standard 100 fig is too low to allow the large effect of
increasing affinity to be apparent. Varying the dose in vivo has
produced conflicting results, with increased, decreased, and
unchanged uptake described (7,25,46,47). The model provides
a simple explanation in terms of mass action: the injected dose
must be related to the quantity of antigen present. If this can
be measured or estimated, the model may predict the critical
dose range in a given patient. This could be tested experimen
tally.
The model gave appropriate times to peak tumor uptake for
a range of molecular sizes from antibody to a small ligand (25,
48) (Fig. 5). It is initially surprising that reducing molecular
size at average affinity does not improve TC or UR (Table 2),
but this can be explained on the basis of the short residence
time of small molecules in the tumor (Fig. 5). Large molecules
remain longer in the tumor area despite identical dissociation
constants because of physical trapping in the tissues. This effect
may be enhanced in vivo (49). For diagnostic purposes, small
molecules are clearly preferable, giving much lower whole-body
radiation for the same TC and UR. The extremely narrow peak
would need careful definition, inasmuch as the optimal time for
scanning could easily be missed at low affinity. It follows that
a greater affinity is necessary for a larger molecule than a small
one to achieve adequate TC and UR values together, since their
peaks, although equivalent, are widely separated in time (Fig.
5). For therapy the ideal tumor-binding substance is very small
with very high affinity (Fig. 7; Table 2), properties which may
exist for endocrine-tumor-binding hormone analogues (48) but
which might be impossible to achieve by fragmenting antibod
ies.
Tumor uptake cannot be increased by prolonging the wholebody retention of small molecules with carrier protein. In this
situation tumor accumulation can occur only if a concentration
gradient is maintained at the tumor surface to cause proteinbinding sites to empty locally, and with the simple cell surface
binding in our model this gradient falls as binding sites fill.
Protein lowers the ECF concentration of free ligand; displace
ment restores this transiently but the maximum TC and UR
remain dependent on affinity.
Despite the widely held view that affinity is central to im
proving uptake ratios, the only study to address the question
showed no effect of affinity on tumor localization (39). The
model confirms this finding at current injected doses but pre
dicts a huge effect of affinity if the correct dose is chosen. The
measurement of affinity should accurately reflect the binding
conditions in vivo. In the study quoted (39), the antigen studied
(CEA) is known to be heterogeneous (50) and affinity studies
were not performed using CEA from the tumors themselves or
at body temperature so that, as the authors point out, the true
affinities of the localizing antibodies were unknown. Very few
antibody affinities have been determined at 37°Calthough
dissociation constants may be increased by an order of magni
tude relative to those measured at 20°C(6) and increasing
temperature can also prolong binding (51 ) so that it is not valid
to compare antibodies at the same low temperature.
Since ka can vary widely for a given K^ it has been suggested
that Keq is not a useful predictor of reaction kinetics (52) but
the model shows that it is always A"eqwhich determines peak
TC and UR values, and the integral dose is hardly affected by
ki at a given Keqexcept in the case of very small molecules (Fig.
6).
The model has defined the reasons for the unsatisfactory
results currently being obtained in tumor targeting and predicts
the conditions required for highly specific localization. Accurate
measurements of affinity and antigen quantity are necessary to
ensure that the best possible results are achieved with available
targeting molecules.
ACKNOWLEDGMENTS
The authors wish to thank Frances D. Halstead, Consultant at the
Computing Services Centre, University of Warwick, for operating the
SAS graphics package to produce Figs. 5, 6 and 8, and staff at the
Computing Laboratory, University of Newcastle-Upon-Tyne, for access
to FACSIMILE via their MTS system.
APPENDIX
The compartmental model describing the system is shown in the
"Appendix" figure.'.The injected substance enters into plasma (Com
partment 1). It may then bind to protein in plasma, pass into ECF, or
be excreted in the urine to the environment. Once in ECF the substance
may bind to soluble protein, tissue, or tumor receptors or if still free
may return to plasma.
As the system is not of uniform volume it is more meaningful to
consider the instantaneous quantity of substance in each compartment
as opposed to concentration. The principle of mass balance and stand
ard kinetic theory, ([39], [40]), yield the following set of nonlinear
3294
Downloaded from cancerres.aacrjournals.org on June 17, 2017. © 1989 American Association for Cancer Research.
OPTIMAL PROPERTIES OF TARGETED MOLECULES
with initial conditions
?,(0+) = D
<7u(0+)= 0
93«n = 0
f3.4(0+) = 0
</«(0-) = 0
«/««n
=o
Environment
i.e., the injected dose of size D moles is entered into Compartment 1
at time t = 0.
In each equation we encounter terms involving products of quantities;
for example in Equation A we have the term
denotes linear transition
-k,q,(PP.n
Such expressions
of the substance
term in Equation
of concentrations
denotes chemical reaction ( I.e.A*BüAB)
Compartments:
1 = Free in plasma
Unoccupied protein-binding sites in plasma
2
1,2
Bound to plasma protein
Free in extracellular fluid
3
Unoccupied protein-binding sites in ECF
4
3,4
Bound to ECF protein
Unoccupied specific binding sites on normal tissues
5
3,5
Bound to specific binding sites on normal tissues
Unoccupied specific binding sites on tumor
6
Bound to specific binding sites on tumor.
3.6
mol
Amounts free in plasma, bound to plasma protein, etc.
liters
Volumes of plasma, ECF, tumor
Total plasma protein binding sites in mol/liter of plasma
Total ECF protein binding sites in mol/liter of ECF
Total specific binding sites on normal tissue in mol/liter of
ECF
= Total specific binding sites on tumor in mol/liter of tumor
volume
-k,q,(PP.y\
= k,q, -
-9u)
(A)
(B)
dt
4?3 - kl93(PE.Y3
- 93.4)
- k¡q}(R.Y3 -93.5)
(Q
- 93.4) -
= ksq}(R.Y3 q,(S.Y3,6 -
PP
-
principle due to the binding
(see main text). The latter
from the fact that in terms
law
X,.2
clearly also applies to Compartments 4, 5, and 6 where similar conser
vation laws exist.
It is assumed that protein and tissue in ECF attract substance in a
uniform manner from all that which is possibly available, whereas the
tumor only attracts substance from a small fraction of that available in
ECF, namely that in its immediate vicinity. We have estimated this
region to encompass approximately S times the tumor volume to make
allowance for increased tumor capillary permeability. This accounts for
the ratios 5.F3,6/K3 occurring in Equations C and F.
Solving this set of differential equations using the computer package
FACSIMILE enables us to obtain instantaneous uptake ratios which
are calculated by the expression
UR = (i3.6/K3,6)/((sum - ÕM)/70)+ l
(H)
where
Sum = q, + q,,2 + q¡+ q3.4+ q3,s + q¡,6
(l)
and where the addition of 1 takes into account the initial state of the
system before the substance is injected.
simultaneous differential equations describing the system:
at
=
(G)
where the number of receptors per cell is assumed to be unitary. In this
manner we are able to eliminate the variable A-:(or qi) essentially
making Compartment 2 a "dummy" or redundant compartment; this
=
=
=
V
Yl, F3, V3,6 =
=
PP
=
PE
=
R
i/i, </!..-
arise from the mass action
to cell surface receptors,
G, (PP. V\ - g,.2), arises
we have the conservation
*2
Rate Constants:
to protein; M ' s '
fti Association
Dissociation from protein; s~'
k, Transfer
plasma to ECF; s"1
k, Transfer from
from ECF to plasma; s"'
k, Association
to specific binding site; NT1s"'
k,
Dissociation from specific binding site; s '
Transfer from plasma to urine; s"'.
Terms:
q
-
(D)
(E)
(F)
REFERENCES
1. Dykes, P. W., Bradwell, A. R., Chapman, C. E., and Vaughan, A. T. M.
Radii limimi not licnipy of cancer: clinical studies and limiting factors. Cancer
Treat. Rev., 14:87-106, 1987.
2. Epenetos, A. A., Snook, D., Durbin, H., Johnson, P. M., and TaylorPapadimitriou, J. J. Limitations of radiolabelled monoclonal antibodies for
localization of human neoplasms. Cancer Res., 46: 3183-3191, 1986.
3. Ford, C. H. J., and Casson, A. G. Antibody-mediated targeting in the
treatment and diagnosis of cancer: an overview. Cancer Chemother. 1'har
macol., 17: 197-208, 1986.
4. Bradwell, A. R., Fairweather, D. S., Dykes, P. W., Keeling, A., Vaughan, A.,
and Taylor, J. Limiting factors in the localization of tumors with radiolabeled
antibodies. Immunol. Today, 6: 163-170, 1985.
5. Weinstein, J. N., Covell, D. G., Barbet, J., Eger, R. R., Holton, O. D., Talley,
M. J., Parker, R. J., and Black, C. D. V. Local and cellular factors in the
pharmacology of monoclonal antibodies. In: B. Bonavida and R. J. Collier
(eds.), Membrane Mediated Cytotoxicity, pp. 279-289. New York: Alan R.
Liss, Inc., 1987.
6. Mason, D. W., and Williams, A. F. Kinetics of antibody reactions and
analysis of cell surface antigens. In: D. M. Weir (ed.), Handbook of Experi
mental Immunology, Ed. 4, Vol. 4, pp. 38.1-38.17. London: Blackwell
Seien title Publications, 1986.
7. Carrasquillo, J. A., Sugarbaker, P., Colcher, D., Reynolds, J. C., Esteban, J„
Bryant, G., Keenan, A. M., Perentis, P., Yokoyama, K., Simpson, D. E.,
Ferroni, P., Farkas, R., Schlom, J., and Larson, S. M. Radioscintigraphy of
3295
Downloaded from cancerres.aacrjournals.org on June 17, 2017. © 1989 American Association for Cancer Research.
OPTIMAL PROPERTIES OF TARGETED MOLECULES
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
colon cancer with iodine-131-labeled B72.3 monoclonal antibodies. J. NucÃ-.
Med., 29: 1022-1030, 1988.
Swabb, E. A., Wei, J., and Cullino, P. M. Diffusion and convection in normal
and neoplastic tissues. Cancer Res., 34: 2814-2822, 1974.
Searle, F., Adam, T., and Boden, J. A. Distribution of intact and Fab2
fragments of anti-human chorionic gonadotrophin antibodies in nude mice
bearing human choriocarcinoma xenografts. Cancer Immunol. Immunother.,
21: 205-208, 1986.
Brown, B. A.. Comeau, R. D., Jones, P. L., Liberatore, F. A., Neacy, W. P.,
Sands, H., and Gallagher, B. M. Pharmacokinetics of the monoclonal anti
body B72.3 and its fragments labeled with either '"I or '"In. Cancer Res.,
47:1149-1154, 1987.
Steward, M. W., and Steensgaard, J. Antibody Affinity: Thermodynamic
Aspects and Biological Significance. Boca Raton, FL: CRC Press, Inc., 1983.
Ishikawa, H., Maeda, T., Hikita, H., and Miyatake, K. The computerised
derivation of rate equations for enzyme reactions on the basis of the pseudosteady-state assumption and the rapid-equilibrium assumption. Biochem. J.,
251: 175-181, 1988.
DiStefano, J. J., and Chang, R. F. Computer simulation of thyroid hormone
binding, distribution, and disposal dynamics in man. Am. J. Physiol., 221:
1529-1544, 1971.
Shi, Z. R., Tsao, D., and Kim, Y. S. Subcellular distribution, synthesis, and
release of carcinoembryonic antigen in cultured human colon adenoma cell
lines. Cancer Res., 43: 4045-4049, 1983.
Pusztaszeri, G., and Mach, J.-P. CEA in non-digestive cancerous and normal
tissues. Immunochem., 10: 197-204, 1973.
Martin, F., and Martin, M. S. Demonstration of antigens related to colonie
cancer in the human digestive system. Int. J. Cancer, 6: 352-360, 1970.
Arfors, K.-E., Rutili, G., and Svensjo, E. Microvascular transport of macromolecules in normal and inflammatory conditions. Acta. Physiol. Scand.
Suppl., 463:93-103, 1979.
Garlick, D. G., and Renkin, E. M. Transport of large molecules from plasma
to interstitial fluid and lymph in dogs. Am. J. Physiol., 219: 1595-1605,
1970.
Taylor, A. E., and Granger, D. N. Exchange of macromolecules across the
microcirculation. In: Handbook of Physiology, Ed. 4, Sect. 2, Vol. 4, Part 1,
pp. 467-520. Baltimore: Williams & Wilkins, 1984.
Pilz, I., Kratky, O., Licht, A., and Sela, M. Shape and volume of fragments
Fab' and F(ab')2 of anti-poly(D-alanyl) antibodies in the presence and absence
of tetra-D-alanine as determined by small-angle X-ray scattering. Biochem
istry. 14: 1326-1333, 1975.
Macknight, A. D. C. Introduction to physiology. In: 3. J. Bray, P. A. Cragg,
A. D. C. Macknight, R. G. Mills, and D. W. Taylor (eds.), Lecture Notes on
Human Physiology, p. 1. London: Blackwell Scientific Publications, 1986.
Brenner, B. M., Hostetter, T. H., and Humes, H. D. The molecular and
structural basis of glomerular filtration. N. Engl. J. Med., 298: 826-833,
1978.
Hutt, M. P. Proteinuria and the nephrotic syndrome. In: R. W. Schreier
(ed.). Renal and Electrolyte Disorders, Ed. 3, pp. 565-589. Boston: Little,
Brown* Co., 1986.
Covell, D. G., Barbet, J., Holton, O. D., Black, C. D. V., Parker, R. J., and
Weinstein, J. N. Pharmacokinetics of monoclonal immunoglobulin Gì,
F(ab')2, and Fab' in mice. Cancer Res., 46: 3969-3978, 1986.
25. Koizumi, K., DeNardo, G. L., DeNardo, S. J., Hays, M. T., Hiñes,H. H.,
Scheibe, P. O., Peng, J.-S., Macey, D.-J., Tonami, N., and Hisada, K.
Multicompartmental analysis of the kinetics of radioiodinated monoclonal
antibody in patients with cancer. J. NucÃ-.Med., 27: 1243-1254, 1986.
26. Palmer, J. G., Persson, B. R. R., Strand, S.-E., and Naversten, Y. Computer
assisted biokinetic studies of radiopharmaceuticals in animals and humans:
1. A method for on-line biokinetic studies with computerised scintillation
camera system. //;. Proceedings of the Third International Radiopharmaceutical Dosimetry Symposium, Health and Human Services Publication FDA
81-8166, pp. 157-163. Rockville, MD, United States Government Printing
Office, 1981.
27. Crone, C., and Garlick, D. The penetration of inulin, sucrose, mannitol and
tritiated water from the interstitial space in muscle into the vascular system.
J. Physiol., 210: 387-404, 1970.
28. Weselcouch, E. O., and Gosselin, R. E. Capillary permeability in the isolated
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
rabbit heart as measured by local tissue clearance. Microvasc. Res., 27:175188, 1984.
Heuser, L. S., and Miller, F. N. Differential macromolecular leakage from
the vasculature of tumors. Cancer (Phila.), 57: 461-464, 1986.
Underwood, J. C. E., and Carr, I. The ultrastructure and permeability
characteristics of a transplantable rat sarcoma. J. Pathol., 107: 157-166,
1972.
Papadimitriou, J. M., and Woods, A. E. Structural and functional character
istics of the microcirculation in neoplasms. J. Pathol., 116:65-72, 1975.
Peterson, H.-L, Appelgren, L., Lundborg, G., and Rosengren, B. Capillary
permeability of two transplantable rat tumors as compared with various
normal organs of the rat. Bibl. Anat., 12: 511-518, 1973.
Ackerman, N. B., and Hechmer, P. A. Studies on the capillary permeability
of experimental liver métastases.Surg. Gynecol. Obstet., 146:884-888,1978.
Gerlowski, L. E., and Jain, R. K. Microvascular permeability of normal and
neoplastic tissues. Microvasc. Res., 31: 288-305, 1986.
Rubinow, S. I. Mathematical problems in the biological sciences. In: Regional
Conference Series in Applied Mathematics, Vol. 10, p. 38. Philadelphia:
SIAM, 1973.
Jacquez, J. A. Compartmental Analysis in Biology and Medicine. Ann Arbor,
MI: University of Michigan Press, 1985.
Karush, F. The affinity of antibody: range, variability, and the role of
multivalance. In: G. W. Litman and R. A. Good (eds.), Comprehensive
Immunology, Vol. 5, pp. 85-116. New York: Plenum Publishing Corp.,
1978.
Sharkey, R. M., Primus, F. J., Shochat, D., and Goldenberg, D. M. Com
parison of tumor targeting of mouse monoclonal and goat polyclonal anti
bodies to carcinoembryonic antigen in the GW-39 human tumor-hamster
host model. Cancer Res., 48: 1823-1828, 1988.
Robbins, J., and Rail, J. E. The iodine-containing hormones. In: C. H. Gray
and V. H. T. James (eds.). Hormones in Blood, Ed. 3, Vol. 1, pp. 592-688.
New York: Academic Press, 1979.
Shampine, L. F., and Gear, C. W. A user's view of solving stiff differential
equations. SIAM Rev., 21: 1-17, 1979.
Curtis, A. R., Sweetenham, W.P., FACSIMILE/CHEKMAT
Users' Manual.
In: Harwell Laboratory Report, AERE 12805, 1987.
42. SAS/GRAPH Version 5 Users' Manual. Gary, NC: SAS Institute, Inc., 1985.
43. Chatenoud, L., and Bach, J.-F. Antigenic modulation—a major mechanism
of antibody action. Immunol. Today, 5: 20-25, 1984.
44. Nadler, L. M., Stashenko, P., Hardy, R., Kaplan, W. D., Button, L. N., Kufe,
D. W., Amman, K. H., and Schlossman, S. F. Serotherapy of a patient with
a monoclonal antibody directed against a human lymphoma-associated an
tigen. Cancer Res., 40: 3147-3154, 1980.
45. Anderson, W. T., and Strand, M. Radiolabeled antibody: iodine versus
radiometal chelates. Nati. Cancer Inst. Monogr., 3: 149-151, 1987.
46. Rogers, G. T., Harwood, P. J., Pedley, R. B., Boden, J., and Bagshawe, K.
D. Dose-dependent localisation and potential for therapy of F(ab')2 fragments
against CEA studied in a human tumor xenograft model. Br. J. Cancer, 54:
341-344, 1986.
47. Carrasquillo, J. A., Abrams, P. G., Schroff, R. W., Reynolds, J. C., Woodhouse, C. S., Morgan, A. C., Keenan, A. M., Foon, K. A., Perentis, P.,
Marshall, S., Horowitz, M., Szymendera, J., Englert, J., Oldham, R. K., and
Larson, S. M. Effect of antibody dose on the imaging and biodistribution of
indium-Ill 9.2.27 anti-melanoma monoclonal antibody. J. NucÃ-.Med., 29:
39-47, 1988.
48. Krenning, E. P., Bakker, W. H., Breeman, W. A. P., Koper, J. W., Krooij,
P. P. M., Ausema, L., Lameris, J. S., Reubi, J. C., and Lamberts, S. W. J.
Localisation of endocrine-related tumors with radioiodinated analogue of
somatostatin. Lancet, 1: 242-244, 1989.
49. Jain, R. K. Transport of molecules in the tumor interstitium: a review. Cancer
Res., ¿7:3039-3051, 1987.
50. Olson, W. C., Spitznagel, T. M., and Yarmush, M. L. Dissociation kinetics
of antigen-antibody interactions: studies on a panel of anti-albumin mono
clonal antibodies. Mol. Immunol., 26:129-136, 1989.
51. Wang, A.-C., Banjo, C., Fuks, A., Sinister. J., and Gold, P. Heterogeneity of
the protein moiety of carcinoembryonic antigens. Immunol. Commun., 5:
205-210, 1976.
52. Absolom, D. R., and Van Oss, C. J. The nature of the antigen-antibody bond
and the factors affecting its association and dissociation. CRC Crit. Rev.
Immunol., a: 1-46, 1986.
3296
Downloaded from cancerres.aacrjournals.org on June 17, 2017. © 1989 American Association for Cancer Research.
Effect of Dose, Molecular Size, Affinity, and Protein Binding on
Tumor Uptake of Antibody or Ligand: A Biomathematical Model
Gillian D. Thomas, Michael J. Chappell, Peter W. Dykes, et al.
Cancer Res 1989;49:3290-3296.
Updated version
E-mail alerts
Reprints and
Subscriptions
Permissions
Access the most recent version of this article at:
http://cancerres.aacrjournals.org/content/49/12/3290
Sign up to receive free email-alerts related to this article or journal.
To order reprints of this article or to subscribe to the journal, contact the AACR Publications
Department at [email protected].
To request permission to re-use all or part of this article, contact the AACR Publications
Department at [email protected].
Downloaded from cancerres.aacrjournals.org on June 17, 2017. © 1989 American Association for Cancer Research.