Journal of Pharmacokinetics and Biopharmaeeutics, Vol. 8, No. 3, 1980

Heuristic Modeling of Drug Delivery to Malignant Brain Tumors Victor A. Levln, 1'z Clifford S. Patlak, 3 and Herbert D. Laudahl 4 Received August 31, 1979--Final February 28, 1980 It is apparent that chemotherapy against malignant brain tumors is generally ineffective. While some agents are more effective than others, none appreciably alters the clinical course of and the poor prognosis for patients with brain tumors. Even though new and more effective agents are being or will be developed, chemotherapy depends as much on the delivery of drug as it does on the drug used, Therefore, we have defined factors that we believe are of primary importance in drug delivery to brain tumors, and, using computer simulation, we have modeled the effects of these factors. In this article we discuss (a) the extent of the "breakdown" in the blood-brain barrier (BBB) that accompanies the development of malignant tumors in the brain, (b) factors that influence drug transport from tumor capillar&s to tumor ceils at varying distances from the capillaries, (c) the problems inherent in drug delivery from a well-vascularized tumor outward to normal brain tissue that might harbor malignant cells but that does not have leaky vessels (i.e., normaI B B B ), and (d) the difficulties in drug delivery from a well-perfused, highly permeable outer tumor shell to a central, poorly perfused tumor core.

KEY WORDS: chemotherapy; pharmacokinetics; brain tumors; modeling; solid tumors.

INTRODUCTION It is well known that the blood-brain barrier (BBB) is defective in many malignant brain tumors and that the extent of BBB "leakiness" is related to the malignancy of glial and metastatic tumors. This leakiness results from gaps between adjacent endothelial cells and fenestrae in endothelial walls. This work was supported by American Cancer Society Grant CH-75 and NIH Program Project Grant CA-13525. V. A. L. is the recipient of an American Cancer Society Faculty Research Award (FRA- 155). 1 Brain Tumor Research Center, Department of Neurological Surgery, and the Departments of Neurology and Pharmaceutical Chemistry, Schools of Medicine and Pharmacy, University of California, San Francisco, California, 94143. 2Address all correspondence to V. A. L. 3Theoretical Statistics and Mathematics Branch, National Institute of Mental Health, National Institutes of Health, Bethesda, Maryland 20205. 4Department of Biophysics and Biomathematics, School of Medicine, University of California, San Francisco, California 94143. 257 0090-466X/80/0600-0257503.00/0 9 1980 Plenum Publishing Corporation

258

Levin, Patlak, and Landahl

Physiologically, these discontinuities in an otherwise restrictive brain capillary network allow large drug molecules to enter the extravascular extracellular fluid spaces (ECF) of the tumor and, by diffusion, the surrounding normal brain. Interestingly, even though malignant brain tumors have "leaky" capillaries capable of allowing entry of even the largest anticancer agents, drugs that can cross the normal BBB seem to be more effective against intracerebral (i.c.) tumor models and human malignant brain tumors (1-3) than drugs that do not readily cross the BBB. This is not to say that drugs that do not cross the BBB are ineffective, but points to the fact that differences in activity seem to correlate with the ability to cioss the BBB. In this article we discuss computer-simulated drug delivery models to help explain this empirical observation. In order for a drug to be effective against a tumor, the drug not only must be capable of killing the tumor cell when it reaches the site in the cell where it exerts its effect but also must be able to reach this site. For brain tumors, this implies that the drug must cross tumor (and some brain) capillaries, diffuse to the cell, and cross the cell membrane if it is to react within the cell. The drug must not be completely metabolized before it reaches the cell, and the drug blood plasma levels must be at a high enough concentration for a sufficient length of time to achieve some critical cytotoxic level. EXPERIMENTAL

METHODS

Diffusion Coefficients

Diffusion coefficients were determined in 2% agar by a modification (4,5) of the method of Schantz and Lauffer (6). Values cited here were determined either in the laboratory of J. D. Fenstermacher (Section on Membrane Physiology, Division of Cancer Treatment, National Cancer Institute, Bethesda, Maryland) or in our laboratory (V. A. L.). In both cases, the method was the same. BCNU was measured using selective ion monitoring chemical ionization mass spectrometry (7). Diatrizoate meglumine (Conray 60, a contrast dye) was measured using iodine content. All other compounds studied were radiolabeled, and radioactivity in 2% agar was determined using a scintillation spectrometer. Table I lists representative compounds referred to in this article or of value to the reader. Permeability Measurements

The technique for measuring capillary permeability in rat brain and i.e. 9L tumor has been reported (10,11). Adult male Fischer 344 rats weighing

Heuristic Modeling of Drug Delivery to Malignant Brain Tumors

299

Table I. Diffusion Coefficientsin 2% Agar

Compounds

MW

D (cm2/sec • 106)

Source

[14C]Urea [11~C]Hydroxyurea [ C]Creatinine 5-Fluorouracil Dianhydrogalactitol [3HJGalactitol (mannitol) BCNU Dibromodulcitol [14C]Sucrose [3H]Methotrexate [99mTc]DPTA [3H]Epipodophyllotoxin Diatrizonate meglumine [3H]Vincristine Bleomycin [14C]Inulin

60 76 113 130 150 182 214 308 342 454 492 657 809 825 1400 5500

15 12.6 11.3 11.2 10.7 8.8 4.5 7.6 6.9 5.3 4.0 3.0 2.0 1.3 1.1 2.5

Ref. 5 Ref. 8 a b b a c b Ref. 9 Ref. 8 c c c b c Ref. 9

aj. D. Fenstermacher (personal communication, 1979). Computed from least-squares fit of D vs. (MW)- 1 / 2 for measured agar values. CDetermined by V. A. L. for this study.

b

1 6 0 - 2 2 0 g were anesthetized with ether, and the saphenous vein and femoral artery were cannulated with PE50 tubing; the femoral artery catheter was advanced to the descending aorta. Isotopes were injected intravenously and blood samples were taken from the femoral artery. [3H]Dextran or [99roTe]albumin was injected 15 rain before sacrifice, and the terminal tissue/plasma levels were used to measure regional tissue plasma volume (PV). Other drugs or isotopes were measured in plasma at 20-sec intervals up to 6 or 10 rain to generate A U C (plasma drug integral). For m e a s u r e m e n t s of i.e. t u m o r permeability, rats were sacrificed by decapitation, the heads were immersed in liquid nitrogen for 45 sec, and the frozen heads were cut with a Stryker saw. Sections of t u m o r tissue were separated from normal brain. Subcutaneous t u m o r was r e m o v e d after the t u m o r had been clamped at its pedicle. Cubes of tumor, either 2 by 2 by 1 m m or 0.75 by 0.75 by 1 ram, were cut with a specially designed wire apparatus. Tissue and plasma samples were placed in tared scintillation vials, reweighed, and digested with tissue solubilizer, and a toluene base fluor was added. Radioactivity was measured in a B e c k m a n LS250 scintillation spectrometer. Bleomycin tissue plasma samples were shipped frozen to Bristol Laboratories, where they were measured with a radioimmunoassay technique. T h e formula for the permeability coefficient (10), P in cm/sec, is e = ki (0.28)(ICD)/~/B--V

(t)

260

Levin, Patlak, and Landahl

where the intercapillary distance (ICD) in cm and the fractional brain blood volume (BV) in ml/g are constants determined previously (10). The rate of distribution over time ki (sec -1) was computed after a single bolus injection of isotope. Under these circumstances, ki, which is equivalent to the permeability coefficient times the capillary surface area S, is (11) ke = P S = 0.93 C(1 - P W ) / A U C

(2)

where C is d p m / g tissue, PW is fractionated plasma water volume in ml/g, and A U C is the area under the plasma curve during the experimental period, dpm 9 min/g plasma. Computation of CellularlExtracellular f Ratios

The f ratios (the cellular/extracellular partition ratio) were computed from tumor/plasma drug levels at equilibrium or from recent in vitro cell uptake experiments (V. A. Levin and C. Yorke, unpublished data, 1979). For in vivo equilibrium values, the following formula was used: f = (R - 0.27)/0.27

(3)

where R is the ratio at equilibrium of tumor drug/plasma drug and 0.27 is the extracellular fluid (ECF) of the subcutaneous tumor. It was assumed in the derivation of equation 3 that the drug passively equilibrated between E C F and plasma. Representative values are listed in Table II. THE T U M O R A N D ITS MILIEU

Arteriographs and computerized tomographic (CT) scans of tumors cannot be obtained definitively in organs in which there are similarities between tumor and surrounding organ capillary permeability or overlying permeable tissues. Fortunately, there are large differences between tumor and brain permeability to most hydrophilic molecules, which make it possible to distinguish the vascular and permeability features of malignant brain tumors from normal brain. Figures 1 and 2 are the radionuclide and Table IL f Ratios

Drug BCNU 5-Fluorouracil Dianhydrogalactitol Adriamycin Vincristine Ara-C

T u m o r / p l a s m a Tumor/media 0.8 1.10 1.08 60

aV. A. Levin and C. Yorke (unpublishedresults, 1979).

5 7.5

f

Reference

2 3 3 60 5 7.5

12 13 14 15 a 16

Fig. 1, [ 9 9 " T c ] D P T A brain scan of a patient with glioblastoma multiforme. T h e scan on the left is a vertex projection a n d the o n e on the right is a right lateral projection, t~

=.

~t

~

262

Levin, Patlak, and Landahl

Fig. 2. Computerized tomographic scan, with contrast enhancement, of the patient in Fig. 1. CT scans of a patient with a typical malignant tumor (glioblastoma multiforme) within the parenchyma of the brain. The tumor has a hypoxic core exhibiting a variable amount of necrosis, and an oxic, well-perfused "leading edge." Characteristically, radionuclides leak across tumor capillaries to demarcate the margins of the tumor's perfused permeable regions. Generally, two tumor types must be considered: The small, homogeneously permeable tumor, and the large tumor with a poorly perfused core. Figure 3 is a CT scan of a patient whose tumor (anaplastic astrocytoma) has homogeneous distribution of contrast, while Fig. 2 is an example of a CT scan of a patient's tumor (glioblastoma multiforme) that has a poorly

Heuristic Modeling of Drug Delivery to Malignant Brain Tumors

263

Fig. 3. Computerizedtomographicscan, with contrast enhancement,of a patient with a well-perfused anaplasticastrocytoma. vascularized tumor core. The first tumor type presumably has a more uniform growth fraction (proliferating cells/all cells) distribution compared with the second tumor type, where the hypoxic-necrotic core is composed of a population of marginally viable cells that rest in Go phase. Although they are not active, these cells nevertheless have the potential for division should they survive; because many of these cells are nonrepticative in the Go phase, they are protected from the effects of many cell cycle phase specific (CCS) and some cell cycle phase nonspecific (CCNS) drugs. In addition, the outer well-perfused region may be more replicative--have a higher growth fract i o n - t h a n the tumor shown in Fig. 3.

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Levin, Patlak, and Landahl

The infiltrative character of many of these tumors is another consideration. Most gliomatous tumors infiltrate normal brain parenchyma as they grow. Frequently, abnormal or potentially tumorous cells infiltrate in advance of neovascularization of highly permeable vessels. Because of local peritumoral edema and associated hydrostatic pressure, the infiltrative brain tissue around the bulk of the tumor may, in addition, be less permeable to water soluble substances than normal brain tissue (17). Figure 4 is a drawing of the simplified tumor that will be considered in later modeling. It is composed of a central, poorly perfused core of low permeability and a more permeable, well-perfused outer shell. In a macroscopic and microscopic sense, these two components are universal constituents of solid tumors. The outermost shell may be unique to CNS tumors. The outermost shell is a product of the compressive effects of the tumor on the surrounding brain and the infiltration of tumor cells without highly permeable tumor vessels. Characteristically, this region, which we have called the brain adjacent to tumor (BAT), has low capillary permeability and reduced blood flow compared with the outer tumor shell, and, at times, with surrounding brain (17, 18).

Extent of Tumor Capillary Defect Before considering the various models pertinent to the problem of drug delivery to brain tumors, let us first define the extent of defects in the tumor capillary barrier. The frequency and size of these leaks that could account for a positive radionuclide brain scan can be estimated using data obtained CENTRAL

CORE

-OUTER

SHELL

~OUTERMOST SHELL Fig. 4. Drawing of tumor with three regions: the poorly perfused low permeability tumor core, the well-perfused high-permeability outer tumor shell, and the lesspermeable, less-well-perfused, outermost tumor shell.

Heuristic Modeling of Drug Delivery to Malignant Brain Tumors

265

with the i.c. 9L brain tumor. For simplicity, we assume that the breaks are circles. T o compute the fraction of area resulting from breaks of the wall of a right circular cylinder, we used the following terms and algorithm: q l D Ap r a

= = = = = =

radius of breaks, 1-6 x 10 .6 cm (19) length (thickness) of breaks, 1.3-1.5 x 10 .5 cm (19) diffusion coefficient in water, cm2/sec permeability coefficient in t u m o r due to breaks, c m / s e c capillary radius, 9.5 x 10 .4 cm (10) fraction of capillary wall area associated with breaks

Because the permeability of a segment of water of thickness I is simply

D/l, therefore zXP = aD/l

(4)

2~P/D = all

(5)

or

The above formulas assume no steric hinderance and do not consider the case of restricted diffusion. Ap is calculated as the difference between the permeability coefficient in the t u m o r and that in normal brain. F r o m Table III, the average AP/D for the 1975-1976 i.c. 9L t u m o r was found to be 0.9 • 0.4 (SD) cm -1 (10) and for the 1977-1978 i.c. 9L tumor 5.1 • 1.4 (SD) cm -1 (V. A. Levin, unpublished observations, 1977-1979). Therefore, in the earlier 9L t u m o r line only 0.0013% of the capillary surface area was open or leaky, while for the later 9L tumor 0.007% was open. Figure 5 is a graphic representation of the extent of the capillary " b r e a k " ; the dot within the center is the " b r e a k " in a hypothetical capillary c o m p a r e d to the surface area of normal brain capillary enclosed by the black line. While in the above estimates restricted diffusion was not considered, we can briefly consider the effect of bulk flow on 2xP/D. If one can assume Poiseuille flow, then the right side of equation 4 would have the term aqZAp/8~Tl added, where zXp is hydrostatic pressure across the capillary m e m b r a n e and ~ is the plasma viscosity. If Ap is taken to be 20 m m Hg (2.6 x 104 d y n e s / c m 2) and the viscosity is taken to be 0.02 poise, we can then recalculate the average value of (all) together with the standard deviation values for different values of q. For q = 0, 2, 3, 4, and 5 in units of 10 .6 cm, the values of (a/l) and sD/(a/l) for the 1977-1978 9L data are 5.1, 0.27; 4.6, 0.27; 4.1, 0.30; 3.7, 0.34; 3.2, 0.39. F r o m these calculations it can be seen that the value of a is not greatly altered by the value of q chosen and that standard deviation relative to the m e a n is not appreciably changed as long as

266

Levin, Patlak, and Landahl

Table IIL Comparative 9L Tumor and Normal Rat Brain Permeability Relationships Compound

Galactitol (n = 6) a Dibromodulcitol (n = 6) Dianhydrogalactitol (n = 3) 5-Fluorouracil (n = 4) Urea (n = 8)

Ap ~

O b

( • 10 6 c m / s e c )

( • 10 6 cm2/sec)

A P / D (cm -1)

1975-1976 9L studies c 3.8 8.8 8.1 7.6 7.5 10.7 15 11.2 16 15

0.4 1.1 0.7 1.3 1.1 = 0.9•

(SD)

1977-1978 9L studies e Inulin (n = 5) Sucrose (n = 4) Galactitol (n = 12) Creatinine (n = 4) Urea (n = 15)

15.3 27.4 30.6 56.1 100.0

2.5 6.9 8.8 11.3 15

6.1 4.0 3.5 5.0 6.7 = 5.1 • 1.4 (SD)

~Ap = Permeability coefficient of tumor minus permeability coefficient of brain, bD = Free water diffusion coefficient (see Table I). CRef 10. an = Number of animals/determination. eV. A. Levin (unpublished observations, 1977-1978); technique same as ref. 10.

the diameter does not exceed about 4 • 10 -6 cm. The corresponding calculation for the 1975-1976 9L data gives quite similar results for the corresponding value of q: 0.92, 0.39; 0.87, 0.39; 0.81, 0.39; 0.74, 0.40; 0.66, 0.4O. We can calculate the number of breaks, N, one would expect to observe in an electron microscopic section of a vessel end-on for a complete circumference of a capillary for the two area leakage fractions given above.

Fig. 5. The small dark box within the bordered square represents the area of capillary "leak" compared to a unit area of brain capillary surface (from equation 5).

Heuristic Modeling of Drug Delivery to Malignant Brain Tumors

267

Using a depth of visualization through an electron microscope of d = 800 ~ , an average radius of break, q, of 3 x 10 -6 cm (19), and a capillary radius of r = 9.5/xm (10), we have in the observed field a wall area of 2~rrd = 5 x l 0 - S c m 2. For a = 0 . 0 0 0 0 1 3 (1975-1976 i.e. 9L tumor), the total " b r e a k " area would be 21rrda = 6.5 x 10 -13 cm 2, which would be due to N breaks of area r 2 = 28 x 10 -12 cm 2 each. Thus N = 2 1 r r d a / c r q 2 = 0.02 would be the number of breaks expected to be observed in an electron microscopic section, which starts at the inner surface of the capillary wall. For the 1977-1978 i.e. 9L tumor, the corresponding number would be 0.1. These measurements must remain as approximations until an electron microscopist has the time and inclination to follow typical tumor capillaries around their circumference to come up with a "real" number. Nonetheless, these approximations allow us to conceptualize the extent of the "breakdown" in the tumor BBB. MODELING

In sections that follow we will evaluate (a) capillary-to-tumor-cell diffusion within a well-vascularized tumor, (2) diffusion for a well-vascularized tumor outward to brain regions where tumor cells infiltrate, and (3) diffusion of drug from a well-vascularized outer margin inward to a poorly perfused (i.e., hypoxic) core. Considered in our models is the fact that the vascular tumor milieu differs appreciably from norma! brain and other body tissues. Vessels are irregular in size and vary greatly in capillary permeability. Compared with an equal volume of brain cortex, in some tumor regions average blood volume is twice as great, capillary radius is nearly 2.5 times greater, yet blood flow can be as low as 0.3% of the normal cortex (10,18,20). This implies a drop in blood velocity and possible shunting of blood within tumor capillaries a n d / o r closure of some capillaries. In addition, because of the high permeability of tumor capillaries, blood drug levels will fall appreciably between the arteriolar and venular ends of a capillary. Clearly, cells distant from the venular end of capillaries (the "lethal corner," C* of Appendix I) may " s e e " less drug than cells nearer the arteriolar end. Capillary-to.Cell Diffusion

Regardless of whether or not a drug crosses a normal or leaky capillary, sufficient drug must reach tumor cells at varying distances from capillaries to be effective. To treat this problem reasonably, we will only consider the total exposure dose (CT) that a cell receives based on linear models. While it is

268

Levin, Patlak, and Landahl

accepted that drug transport is not always linear at all dose levels, it is reasonable to assume linearity within the small "therapeutic" drug level range. For this reason and to form a basis for future work, we will assume linear pharmacokinetics in all models that follow. Because the exact geometry of the capillaries within the brain is not necessarily a simple parallel array of cylinders, the two models chosen to represent capillary geometry within the brain are the extreme cases. The first is the Krogh cylinder model, in which it is assumed that a cell is subserved only by the two parallel capillaries that it lies between (21). For this model, we will consider only the total exposure dose to those cells that lie at the venous end and midway between the capillaries, that is, the "lethal corner." The exposure dose to these cells will be the minimum exposure dose that any cells in this or any other reasonable model receive. The second model will consider the case of complete mixing in the extracellular fluid around the capillaries (22). In this model it is obvious that the total exposure dose will be greater than the minimum total exposure dose for any other reasonable model. Thus, if for some critical total exposure dose of a drug required to kill cells, this level can be reached in the first model, it implies that this level will be reached in the real situation. On the other hand, if this value is not achieved for the second model, this implies that it will not be achieved in the real situation. Obviously, for drugs that achieve the desired level in the second model but not in the first, it cannot be stated what will happen in the real situation. In our modeling we assume that in the first model the capillaries can be depicted by the Krogh cylinder model (21); that there is radial mixing of drug within the capillary; that within a finite tumor region all capillaries are relatively homogeneous in their "leakiness"; that capillary permeability can be defined; that there is consumption of drug in the ECF and intracellular fluid (ICF) or at the cell surface (i.e., biotransformation, spontaneous hydrolysis, binding) that is expressed as half-life (tl/2) values (tl/2 = In 2/k); and that when the drug is in equilibrium with the cell cytosol the partition value (f ratio) expresses the ICF amount/ECF amount, which includes the limiting case where the drug does not enter the cell (f ratio = 0). The equations and their solutions that describe these models are discussed in Appendix I. As shown there, the total exposure dose for any linear model will be proportional to the total exposure dose of the blood plasma. Figure 6a-f shows the relationship of capillary permeability (P), intracellular/extracellular drug (f ratio), intracellular drug half-life (ICF tl/2), and tumor blood flow (F) to the steady-state drug integrated exposure (actually the percentage of the capillary integral in Fig. 6) for the two extreme models. The values of De, Ve, and r are typical values for a variety of brain tumor models.

I00

ua

f= I

80

// /""

w

/'/'/

o

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_...................

40

~>~_

/

20

,

!

......................... "

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~a)

BLOOD

FLOW (ml/,g/min)

o

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.2 (b)

.4

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BLOOD FLOW (ml/g/rnin)

Fig. 6. These graphs show the total exposure dose as a function of t u m o r blood flow (F) for f ' s of 1 and 6. T h e plasma drug integral is set to 100 u n i t s / m i n / m l for simplicity. T h e worst case total exposure dose ( . . . . . ) was computed for the K r o g h model, equation AI-18, while the best case total exposure dose ( ) was c o m p u t e d for the well-mixed model, equation AI-24. In all cases, S and F were a s s u m e d to be linearly related, the ratio S/F having the value of 750 m i n / c m . T h e t u m o r was a s s u m e d to have an average capillary radius of 9.5 fzm (10), an average extracellular c o m p a r t m e n t Ve = 0.27, an extracellular h/2 of 100 rain (~here is no appreciable difference for tl/e's of 50 to 400 rain), a diffusion coefficient of 3 • blood flow of 0 - i m l / g / m i n , and a capillary permeability coefficient of 1 x 10 -4 cm/sec. T h e plots shown in a were based on an effective intracellular h/2 of 5 min, while those in b, ICF tl/z = 50 rain. c and d are similar to a and b except that the capillary permeability coefficient is 1 x 10 -5 cm/sec, e and f are also similar, except that they were based on P = 1 • 10 -6 cm/sec.

270

Levin, Patlak, and Landahl 100 f =1

BO

o a L~

g

60

Q.

40

0

20

0

(c) BLOOD

FLOW ( m l / g / m i n )

f = I

I00

8O

6o o 4O

20

.2

(d)

,4 BLOOD

.6

,8

I

FLOW ( m l / g / m i n )

Fig. 6. Continued. The constants used in these approximations are listed in the legend to Fig. 6a-6f. Because the ECF is relatively protein free, the ECF drug tl/2 is approximated by the tl/2 of the drug in buffer. For many anticancer drugs (at least those discussed here), the tl/2 in the buffer is 50 to ->400min. Although not illustrated here, the total exposure dose is relatively indepen-

Heuristic Modeling

of

Drug Delivery to Malignant Brain Tumors

271

~00

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~CF

~0

P

tl/2,

5 r n~n

9 I x IO 6 cm/sec

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(e)

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Continued.

dent of E C F 4/2 values greater than 50 min when I C F 4/a values are less than 50 min. Unfortunately, less is known about the rates of drug cell surface binding or intracellular biotransformation or binding. Available information suggests that for many drugs, binding or biotransformation is rapid (for some drugs less than 1 min).

272

Levin, Patlak, and Landahl

Tumor blood flow varies as a function of the type, location, and age of the tumor. Gullimo and Grantham found hepatoma, fibrosarcoma, and Walker carcinoma 256 blood flows to vary between 0.02 and 0.42 ml/g/min, depending on the site of implantation (23). Siracka et al. measured blood flow in 6C3HED lymphosarcoma tumors at 9 and 14 days using xenon-133 and found blood flows of 0.09 and 0.05 ml/g/min, respectively (24). Regional differences in i.c. tumors of 0.1-0.9 ml/g/min have been measured (18) and variations in subcutaneous tumors of 0.004-0.70 ml/g/min have been observed with antipyrine (20). Intracerebral tumor blood flows in humans of 0.3-0.5 ml/g/min have also been observed by external collimation counting methods utilizing xenon-133 (25). Some of our recent studies using rapid sequence CT scanning in brain tumor patients suggest that blood flow may fall to less than 0.1 ml/g/min at times (26). For simplicity, we shall assume that this variation of blood flow is due primarily to the number of capillaries per gram of tumor. That is, we assume that within any tumor the velocity of the blood within a capillary is roughly constant, which in turn implies that the ratio of capillary surface area to blood flow (S/F) will be constant. Because the average value for 9L capillary surface area is 232cm2/g (10), we used the ratio (232 cm2/g)/(0.3 ml/g/min) = 750 min/cm for S / F for our calculations in Fig. 6a-f. Figure 6a,b is based on P = 1 x 10 -4 cm/sec, which is an extremely high value that approximates P values for molecules such as 1,3-bis(2-chloroethyl)-l-nitrosourea (BCNU) and 1-(2-chloroethyl)-3-cyclohexyl-1nitrosourea (CCNU), molecules with "infinite" permeability. For these drugs, characterized by small size and high lipophilicity, the effect of increasing F--which is really an increase in capillary density and therefore also in surface area--is quite dramatic. If the ICF tl/2 is large (Fig. 6b), these effects are less marked. The reasons for these observations are that for both the Krogh and well-mixed models, with increasing f ratios more drug enters cells and, therefore, less drug remains in the ECF. Similarly, because of cellular consumption or binding, less drug is present in the ECF. In Fig. 6c,d, with P set at 1 • 10 -5 cm/sec, the effect of F is decreased. In addition, the level of the total exposure dose is considerably lower because of the lower permeability. If P = 1 • 10 -6 cm/sec, the effect of F is even less marked and the level of the total exposure dose (CT) is even lower (Fig. 6e,f). To analyze specific drugs, we first computed the total exposure dose for BCNU (14 mg/kg) based on pharmacokinetic data obtained in rats. BCNU is a small lipophilic nitrosourea with good activity against i.c. animal tumor models (2, 3) and human brain tumors (1). Because of its high P (1 • 10 -4 cm/sec), it diffuses through cellular membranes as well as within the

Heuristic Modeling of Drug Delivery to Malignant Brain Tumors

273

ECF. The constants used in this computation are listed in the legend to Fig. 7. The formulas in Appendix I (AI-18, AI-24) were used to generate the curves. The right-hand ordinate axis represents the log cell kill of 9L cells in culture (27). With an average blood flow in the tumor of 0.3 ml/g/min, the in vivo total exposure dose to a hypothetical i.c. 9L tumor could produce a m a x i m u m 4 . 1 l o g c e l l kill if t h e I C F t~/2 = 5 0 m i n , b u t o n l y a 3 . 4 l o g c e l l kill if t h e I C F q / 2 = 5 rain. A 3 - 4 l o g c e l l kill w i t h B C N U h a s b e e n o b s e r v e d u s i n g

a colony formation efficiency assay in i.c. 9L tumors following an LDlo dose of 14 mg/kg

(30), the dose on which Fig. 7 was based.

Dianhydrogalactitol (DAG) is an anticancer agent that crosses the BBB but has variable antitumor activity against different i.c. tumor models (2,14) and little activity against human gliomas (31,32). To evaluate a potential reason for its poor antitumor activity against i.c. 9L rat tumors, we computed 4OO

5

BCNU ICF t~/2 -5Omin E

:520

4

I

E

/,1"

o.O-. . . . . . . . . . . . . . . . . . . ~. . . . . . . . . . . . . . . . . .

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tl/z

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240

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P

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o

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P

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.6

.8

(ml/g/min)

Fig. 7. The computed in vivo 9L rat tumor integrated exposure for BCNU (14 mg/kg) as a function of blood flow. The worst case total exposure dose (. . . . ) was computed for the Krogh mode/, equation AI-I8, while the best case total exposure dose ( ) was computed for the well-mixed model, equation AI-24. S and F were assumed to be linearly related, with S / F = 750 min/cm. The computation is based on the pharmacokinetics of BCNU in rat plasma (11). The right ordinate axis is the in vitro log cell kill observed by Wheeler et al. (27) in 9L cells. The constants used were De = 4.5 x i 0 - 4 cm2/sec, P = 1 x 10 -4 cm/sec (10), Ve = 0.27, f = 2(12), capillary radius = 9.5/xm (10), ECF = q / a 5 0 - 4 0 0 rain, and ICF tl/2 = 5 or 50 min. The ICF tl/2 = 50 rain is reasonably well supported by recent measurements of BCNU in tumor tissue in vitro and extrapolation from survival data (28, 29). The plasma drug concentration CvL (/zg/ml) = 7.33 e -~ + 5.75 e-~176 (t in rain) (28).

274

Levin, Patlak, and Landahl

the integrated exposure (Cr) for a dose of 6 mg/kg as the blood flow changes. The constants used are listed in the legend to Fig. 8. When the corresponding log cell kill for i.c. 9L cells exposed in vitro to D A G (31) is plotted, it can be seen that less than a 1 log cell kill would be attained even for the maximum blood flow. In vivo, D A G shows virtually no antitumor activity against 9L tumors, while in vitro a 4 log cell kill is attainable (14,31). Because 6 mg/kg is the LD10 for DAG, the dose cannot be increased; therapy failure is related, therefore, both to inherent biochemical insensitivity of 9L cells to D A G and to drug delivery. Diffusion from a Well-Vascularized Tumor Outward Radionuclide brain scans utilize radionuclide-bound proteins and chelates effectively to determine the size and location of malignant brain tumors. While the size of the scan image bears a direct relationship to the 400[

DIANH YDROGALACTITOL

1.0 qCF f ~/Z - 5 0 r a i n

.......-"

~

ICg

tl/2

9 5rain

160

80

0

I--

//"

.2

.4 .6 BLOOD FLOW(ml/g/rnin)

.8

Fig. 8. Computed in vivo 9L rat tumor integrated exposure for dianhydrogalaetitol (6 m g / k g ) as a function of blood flow. The worst case total exposure dose ( . . . . ) was computed for the Krogh model, equation AI-18, while the best case total exposure dose ( ) was computed for the well-mixed model, equation AI-24. S and F were assumed to be linearly related, with S / F = 750 m i n / c m . The right ordinate axis is the in vitro log cell kill observed by Levin et al. (31) in 9L cells. The constants used were De=4.5xl0-6cm2/sec (42% of 2% agar diffusion coefficient), P = 1 • 10 -5 c m / s e c (10), Ve = 0.27, .1"= 3 (16), capillary radius = 9.5/~m (12), E C F t t / 2 = 5 0 - 4 0 0 min (D. Munger, personal communication, 1976), and ICF tl/2 = 5 or 50 rain. The plasma drug concentration CpL (/~g/ml)= 9.9e -~

+ 3 . 0 e -~176 (t in min) (16).

Heuristic Modeling of Drug Delivery to Malignant Brain Tumors

275

degree and extent of the permeability defect in the tumor, some physicians believe that the scan may overestimate the size of the tumor as a result of diffusion of radionuclide from the tumor to the BAT. A number of factors militate against this, however. Generally, compounds that do not cross normal brain capillaries leak across tumor capillaries and diffuse down a concentration gradient from tumor into BAT and into normal brain tissue (17,33). Compounds can further diffuse from normal brain into the cerebrospinal fluid (CSF) and then be carried by bulk flow out the arachnoid granulations into the systemic venous circulation. Obviously, if a molecule were incorporated into cells as it diffused from the tumor, the ultimate distance the molecule would be able to move from the tumor would be far less than if the molecule was not incorporated. In order to calculate the maximum outward diffusion from the tumor in our model, plasma drug level was chosen to be constant. From Appendix II, the maximum concentration in the region outside of the tumor is given by equation AII-4. To estimate the maximum distances that the material diffuses, we used the fact that the conventional RN scanning time is less than 2 hr and the "free water" diffusion coefficient of the radionuclide 99"Tc-DPTA is 4 x 10 .6 cm2/sec. (For iodinated contrast agents, the diffusion coefficient in 2% agar is approximately 2 x 10 -6 cm2/sec.) Inserting these values into equation AII-4, assuming an intercellular tortuosity factor of 0.40, and choosin_p_gthe distance from the surface of the tumor as 2 mm, we find that 8/2~/Dt= 0.83. Figure 9 shows that the concentration at this point will be less than 20% of the level of the radionuclide in the tumor. Therefore, conventional radionuclide scanning at 1.5-2 hr visualizes tumor almost exclusively and not tumor plus BAT, because BAT in humans is considerably more than 2 mm. When the brain is examined at autopsy, however, it is frequently found that the tumor is much larger (by more than a centimeter in some cases) than that seen on the CT scan. From the above discussion, this discrepancy is most likely due to the fact that the outermost sheli of tumor (BAT) are less permeable than the outer shell. This reduced permeability will be found for contrast agents, radionuclides, and a great many chemotherapeutic drugs. For the reasons discussed in the previous section, drugs may not be able to diffuse to the tumor cells from the plasma in sufficient concentration to be effective. In this instance it is useful to consider diffusion from the more permeable outer shell of the tumor outward to the outermost shell. Because diffusion is so slow, a priori one would not expect that diffusion could account for drug delivery from the outer tumor shell to the outermost shell (Fig. 4) sufficient to overcome low drug levels in the outermost shell resulting from low drug permeability and reduced blood flow. To quantify

276

Levin, Patlak, and Landahl 1.0

'0.8

0.6

0.4

0.2

0 ,4

.8

1.2

1.6

2

6 / 2-,/~t )

Fig. 9. Plot of eric vs. 8 / 2 q ' ~ , where eric is the complementary error function, D is the diffusion coefficient, t is time, and 8 is the distance from the surface of the sphere.

1.0

.8

O r~ w

.6

o 0. x U..l .4

0

.2

0

,

i

2

4

6

8

I0

Ka

Fig. 10. Integrated exposure dose in outer shell/integrated exposure dose in central core ratio vs, K6. Computations based on equation AII-8.

Heuristic Modeling of Drug Delivery to Malignant Brain Tumors

277

the contribution of diffusion and to test the correctness of this assumption, we will use extreme values such that there is relatively little loss of drug by biotransformation (e.g., the ECF and ICF h/2-~400 min). We set f = 1, V, = 0.27, and set PS to a very small number, and allow De to be as large as 3 x 10 -6 cm2/sec. Inserting these values into equation AII-6, we calculate that K = 4.4 cm -1. Figure 10 shows that for the total exposure dose to be greater than 25% of the total exposure dose of the outer shell, K&, must be less than 1.4. Hence 8, the distance from the well-perfused tumor shell surface, must be less than 1.4/4.4=0.3 cm. For &=5 mm, the totat exposure dose is approximately 11% of the central shell total exposure dose. If capillary permeability is not zero or if there is more than the minimum loss of drug due to metabolism and/or binding, or if De is smaller than the value used, the total exposure dose coming from the more permeable outer tumor shell will be smaller. Alternatively, the effect of drug biotransformation on the half-distance for drug penetration at equilibrium can be considered. Figure 11 is such an approximation based on the range of antitumor drug diffusion coefficients and a model that assumes that diffusion takes place perpendicular to an infinite plane. With this assumption, sufficient drug will not be delivered to the outermost edge of the tumor shell unless the total exposure dose of the outer shell is very large. If drug delivery at a distance greater than 5 mm from a leaky tumor is important for brain tumor therapy, then to deliver drugs impermeant to normal brain capillaries it will be necessary to maintain _plasma drug levels for hours to days depending on the size and distribution of the-drugs in question. A large molecule that binds to proteins and cell surfaces might require 24 hr; a smaller molecule that binds less might require only 10 hr for sufficient drug diffusion to take place from tumor to BAT. Based on physical characteristics of the drugs in vilro and some biological data in small animals, the time required to achieve adequate drug levels at varying 2.(3

Fig. 11. How far a drug penetrates tissue in which drug is biotransformed. The distance from a perfused region at which the integrated exposure (CT) is 50% of that of the perfused region is plotted as a function of drug tl/2. Diffusion was computed for an infinite plane where diffusion takes place perpendicular to the plane and where no back-diffusion occurs. The diffusion coefficient, D, between 1 and 5 • 10 .6 cm2/sec includes the range of D for most anticancer agents (see equation AII-

9).

1.6 +P

1.2 -q. ~ .8

D=$x'lO'6 cram2D+++r

J

f f J

,r =

;

.4 ______a_ 0

40

...... ~ 80

, 120

HALF-LIFE (rain)

~

;

1 0

2 0

278

Levin, Patlak, and Landahi

distances from the tumor edge can be determined readily for most chemotherapeutic drugs and for most clinical situations. Unlike malignant gliomas, metaststic brain tumors are usually circumscribed and have relatively few infiltrative tumor cells; however, from experimental studies using i.c. Walker 256 and 9L tumors, both of which are transplantable (metastatic), the leading edge of the tumor and the brain immediately adjacent to the tumor showed a decrease in permeability to water-soluble compounds (17). If this observation can be extrapolated to humans, then whether a tumor is metastatic or primary it would be important for drugs impermeant to normal brain capillaries to be maintained in the plasma for a sufficient time to allow diffusion to these regions. While the tumors in i.e. animal models are relatively small, with correspondingly short drug diffusion distances required to fully saturate the leading edge of the tumor or the brain immediately adjacent to it, brain tumors in humans can be upward of 100 g in size and, correspondingly, the diffusion distances will be measured in centimeters, not millimeters. Diffusion from a Well-Perfused to a Poorly Perfused Tumor Region Inward

One of the more important physiological (structural) features of malignant tumors that CT scans have revealed is the extent of poorly perfused or "low-density" centers. In addition, it may be possible to partially quantitate blood distribution in CNS tumors by rapid CT scanning (26). Figure 12 shows rapid sequence CT scans from one of these studies taken within seconds after a bolus injection of 15 g of Conray into a patient with a malignant glioma of the thalamus. The outer margin of the tumor is high density (enhanced), whereas the inner portions are low density. Static scans taken at 3 min and 1 hr after injection showed that the center portion of this tumor failed to fill with contrast material. From studies similar to these, we concluded that the poorly perfused center of many malignant human glial tumors has a lower blood flow than that of the normal brain, and considerably less than that of the periphery of the tumor. In this patient, the ratio of flow in the periphery to flow in the center of the tumor was between 5 and 10. In addition to decreased perfusion, the central tumor core would also be expected to have reduced capillary permeability. Even though an individual capillary might be abnormally leaky, intercapillary distances are greater in the center than at the periphery of tumors; the net effect would be an actual decrease in the capillary surface and hence a decrease in permeability times surface area (PS) in the center compared to the periphery of the tumor. Measurement of PS in subcutaneous 9L tumors showed a 100- to 200-fold difference in PS values for hydrophilic molecules in different regions of the tumor (20).

Fig. 12. Rapid-sequence contrast-enhanced CT scans in a patient with a malignant glioma who received a bolus of 50 ml of Renografin 76, A rim of contrast enhancement is seen early after injection that does not, at 3 min, enter the lowodensity tumor core~ Even by 20 min, this low'density core still had not filled with contrast material. (Figure courtesy of Dr. David Norman.)

O e~

280

Levin, Patlak, and Landahl

The therapeutic importance of the poorly perfused, variably permeable central tumor core is the fact that while it contains solid and liquid necrotic debris, it also contains a variable number of intact cells. Clearly the majority of the cells in this region will be dead or dying from hypoxia and will have limited access to substrates, but a portion of the cells will be in Go phase and capable of cell division if their nutritional status should improve. Recruitment of Go cells has been experimentally verified in subcutaneous tumors that develop a poorly perfused central core. It has been shown that drug delivery is impaired (34) and that a significant percent of these cells escape cytoxic drug exposure and remain potentially viable (35). We have also found recently that in vitro plating efficiency (colony formation) of 1-2 mm cubes of subcutaneous 9L tumors is independent of in situ perfusion (20; D. C. Wright, unpublished observations, 1979). The low-density central tumor core in Figs. 2 and 12 has a volume of 33-50 g and, therefore, approximately 102~cells. If only 10% of this volume contains Go cells with the potential for later cell division, 109 cells with malignant potential would exist in this low-density center. The potential threat of these remaining cells to the host is quite significant; if the tumor had a doubling time of 7 days, it would attain a cell burden of 102~cells within a month. The pharmacokinetics of drugs as they diffuse from the outer well-perfused portion of the tumor inward toward a poorly perfused tumor center is, therefore, of more than theoretical interest. For our modeling, the tumor is considered to be a sphere with a well-perfused outer shell and a poorly perfused inner core region (Fig. 4). Only diffusion that occurs from the outside is considered. In addition, because the outer shell drug levels are assumed to be constant, the model overestimates the actual drug level in the outer tumor shell and inner core. For simplicity it was assumed further that there is complete mixing around each capillary (effective D = oo), but there is a finite D for large diffusion distances. Last, the extreme case assumptions made in the previous section apply here: capillary permeability is 0, ICF and ECF h/2's are --400 min, and De = 1 x 10 .6 cm2/sec. The formulas, additional assumptions, and constants used for the computation of Fig. 13 are given in Appendix II. As before, the value of K from equation AII-6 for the extreme case parameters will be 4.4 cm -~. Thus for the total exposure dose at the center of the tumor core to be at least 25% of the total exposure dose in the surrounding shell, Kd ~ 3.2, and d -< 7 ram. As shown in Fig. 11, drug biotransformation will similarly reduce effective diffusion from the permeable well-perfused outer tumor shell into the less-well-perfused, low-permeability tumor core. As in the case of diffusion out of the central core into a poorly perfused shell, diffusion into a poorly perfused core from a well-perfused shell may not be a useful method

Heuristic Modeling of Drug Delivery to Malignant Brain Tumors

281

B.0

.8

o

a. N ~d pO ,2

0

2

4

6

8

mO

Kd

Fig. 13, Integrated exposure dose in the center of the central core/ integrated exposure dose in outer shell ratio vs. Kd. Computations based on equation AII-11. of getting drugs to the cells in the center of the core unless the outer shel! attains a very high total exposure dose. DISCUSSION

Drugs that readily cross the BBB are more effective against CNS tumors than drugs that do not cross the BBB because of their size and lipophiticity. As a generalization, drugs that cross the BBB (i.e., produce appreciable brain drug levels) are either hydrophilic with M W -< 160 or hydrophobic with MW-< 400 (36); they cross membranes easily and diffuse readily, and their major transport and distribution, depending on M W and lipophilicity, depend on blood flow. Theoretically, these drugs can diffuse to all areas of tumor and brain, and reach tumor cells infiltrating BAT, the well-perfused tumor, and even some regions of poorly perfused tumor. Unfortunately, because of rapid extra- and intracellular biotransformation, high cellular binding and distribution, and frequently low blood flow and capillary density (which produces low capillary surface area and low effective capillary permeability), these drugs do not achieve uniform and adequate drug penetration. It is clear that even in highly permeable tumors such as the i.c. 9L tumor, only 0.007% of the capillary surface area needs to "break down" to

282

Levin, Patlak, and Landahl

produce a formidable increase in permeability (> tenfold). If tumors were well circumscribed, noninfiltrative, and homogeneously permeable to drugs, drug delivery to tumor cells would be described by the models derived in Appendix I. As shown in Fig. 6a-f, CT for tumor cells is a function of many factors, and a large CT would be favored by a low f ratio, a slow rate of ECF drug breakdown, a large PS, high blood flow rates, high diffusion constants, and moderate to slow rates for cellular biotransformation. Considering drug delivery only, the best tumor to treat would be a small, well-perfused, noninfiltrative metastatic tumor. Unfortunately, metastic tumors in the brain are not vascularized until they are approximately 200-400 ~m in diameter (10,37). Further complicating therapy is the fact that most metastatic and primary tumors, when clinically apparent, have diameters greater than 2 cm (4 g or 4 • 109 ceils) and are infiltrative and/or developing hypoxic, poorly perfused central regions by the time of diagnosis. When chemotherapy is begun, the distribution of capillaries in most tumors is heterogeneous. Few drugs will reach all cells equally well. BCNU (f ratio = 2) is one of the few drugs that can reach cells 200/zm from capillaries with less than a 50% drop in CT. Drugs such as epipodophyllotoxin, adriamycin, bleomycin, vincristine, MTX, DAG, and probably the purine and pyrimidine analogues, which enter cells but are incapable of rapid diffusion through cells because of binding and/or rapid metabolism, will have a rapid drop as they diffuse in the ECF further and further from the capillary. From the data in Table II, it is apparent that most of these drugs have high f ratios (5-FU and most purine and pyrimidine analogues have an f ratio of 3-8, vincristine of = 5, and adriamycin of -> 60, depending on the tissue) that, because of cellular entry, reduce the amount of drug diffusing from the capillary to cells far from the capillary. In reality, CCS drugs, which are active against cycling cells, need not diffuse very far from the capillary for maximum activity because only those cells near capillaries will be proliferative and responsive to CCS agents. The caveat is that less proliferative cells farther from the capillary may be exposed to less of the CCS drug, which may enhance the selection of resistant cell clones. When these cells are recruited into the proliferative pool, they will be insensitive to that particular CCS drug. To minimize this possibility, some form of cyclic therapy with different CCS agents should be employed throughout the contemplated treatment period. CCNS drugs other than nitrosoureas should have low f ratios and high ICF tl/2's (25-75 min), if the goal is to deliver high CT to cells at distances of 100-200 ~m from the capillary. In this instance, suboptimal therapy also may be counterproductive if low levels of drug are delivered and cells becomes resistant to the drug.

Heuristic Modeling of Drug Delivery to Malignant Brain T u m o r s

283

In addition to factors affecting Capillary-to-tumor-cell transport for infiltrative tumors, drug transport to more external regions, such as infiltrative tumor cell regions or edematous, less permeable regions, must be considered. Tumor cell CT for these regions is dependent on the persistence of a diffusion gradient from permeable tumor to less permeable regions external to the main tumor mass, unless the drug crosses the BBB with little hindrance, in which case this model will have little relevance. A rather simple way of looking at drug delivery from a well-perfused tumor shell both outward into infiltrative tumor in normal brain and inward toward the poorly perfused tumor core is to assume that in both cases diffusion distances are great and the speed of tissue penetration can be approximated by assuming that diffusion takes place perpendicular to an infinite plane. In Fig. 14, the relationship between the speed of drug penetration and diffusion coefficient is shown for drugs that are not metabolized in tissue, which is the best possible case for drug delivery.

24HR~

10.00 ~ 5.00

1.00~

S

.

.50

.10 .05

fAdr

bleo~

1 x 3 x I.$ x 5 x

VM-26 8CNU MTX

.oi

,

2

L-

6

.

,

10 ..6 10 .`6 ! 0 .6 10 "6

~

0.4 1.3 1.0 io2

x 10 "G ~c 1G"6 x 10 "~ x 10 ~ J

lO

14

D(x I0-6 cm2,ser

Fig. 14. H o w fast drug penetrates tissue that does not metabolize drug for the case of a constant drug level maintained for 1 rain to 24 hr. T h e distance at which the concentration was 50% of the outside (perfused) drug concentration (or CT) was c o m p u t e d as a function of drug diffusion coefficient (D). Diffusion was c o m p u t e d for an infinite plane where diffusion takes place perpendicular to the plaTl.~_and where no back-diffusion occurs. 8 = 0.954~'Dt (p. 60, ref. 38), where t = sec.

284

Levin, Patlak, and Landahl

Table I lists the diffusion coefficients for some of the commonly used anticancer agents. It is apparent from this table and Figs. 10, 11, 13 and 14 that drugs normally impermanent to the BBB will travel only short distances from the more permeable tumor unless drug levels are maintained for long periods of time. Unfortunately, systemic toxicity often makes this therapeutic approach impossible. For a tumor with a poorly perfused central core, drug delivery of a blood-flow-limited drug (such as nitrosoureas, some procarbazine metabolites, thiotepa, and others) will be less than that received by the betterperfused outer shell. Furthermore, such factors as the f ratio and ICF tl/2 will influence delivery to the central core. High f ratios and short tl/2 will reduce CT in the core. Thus, if a tumor is quite sensitive to a drug of this type, the drug might be used against a large tumor with a poorly perfused central core, but it would make more sense to use a drug that is less permeable (MTX, cis-platinum, adriamycin, bleomycin, and others) that can be perfused intravenously over a long period of time. Generally, the inner tumor core capillary density is low, and, therefore, core capillaries are not as permeable--a reduced capillary surface and not ki account for the decrease in P - - a s the outer shell capillaries and drug transport to the tumor core will be directly related to its diffusion coefficient, capillary permeability coefficient, rate of breakdown in the tumor, levels in the tumor periphery, and the length of time the outside drug level can be maintained. Again, the problems of drug instability and systemic toxicity limit the attainment of therapeutic drug levels. Even though the pharmacokinetic models used in this article can only approximate drug delivery to solid tumors, they allow us to conceptualize the various factors that hinder the delivery of drugs to neoplastic ceils. While drug delivery is by no means the only factor affecting the success or failure of chemotherapy, attaining adequate drug levels at the tumor cell is of primary importance because inadequate tumor cell drug levels will lead to low cell kill and to the potential for the early development of biochemical resistance to a drug. Obviously, systemic toxicity is closely coupled to tumor drug delivery. Sustained drug levels necessary to achieve therapeutic levels for tumor cells distant from capillaries in normal brain or poorly perfused tumor may produce intolerable systemic toxicity. Reduction of systemic toxicity may, of course, be modified by rescue or pretreatment with agents that might block drug induced toxicity. These approaches, currently appealing for lack of any alternative, may not be practical because of the inherent inability to counteract toxicity. Even if myelotoxicity could be blocked, secondary toxicities would probably be dose limiting. Despite our pessimism about their possible success, these approaches must be investigated.

Heuristic Modeling of Drug Delivery to Malignant Brain Tumors

285

Successful chemotherapeutic intervention is also limited by the fact that no tumor is truly homogeneous, truly monoclonal in all biologically important features. Even if a rare tumor is monoclonal at the start of chemotherapy, after a small number of drug treatments the tumor may manifest acquired drug resistance. Whatever its mechanism, development of drug resistance by tumor cells is a leading cause of the failure of chemotherapy. Nonchemotherapeutic (e.g., noncytotoxic) methods of enhancing tumor cell sensitivity might be productive, if the required therapeutic Cr could be reduced by a factor of 10 or more. Alternative approaches would be to cycle or alternate non-crossresistant drugs or drug combinations of equal efficacy. Unless there were at least three or four drugs or combinations, however, such an approach would be doomed to failure because resistance would only be delayed, not prevented. Moreover, if the drugs or combinations are toxic to the same end organs (eg., myelotoxicity), dose reduction during subsequent regimens and therapy courses will reduce their effectiveness. Further compounding these restraints is the cell cycle specificity of drugs and the capriciousness of cell cycle perturbations of tumor cells during therapy. Most solid tumors--especially in the CNS--have low growth fractions (10-30%) (39). It is atso clear from the work of Tannock that growth fraction of mouse mammary tumor cells is a function of 'Moseness" to capillaries (37). Proliferative capacity and function decrease radially from the capillary. Hence low growth fraction tumors have either quantitatively reduced perfusion or large intercapillary distances, or both. While nothing can be said in absolute terms about the potential for proliferation in the remaining cells (many of which are resting Go cells), studies in animals have shown that cells from these regions are capable of neoplastic growth (35, D. C. Wright, unpublished observations, 1979). Small tumors in animals and human tumors that respond to therapy characteristically have an increase in the percentage of cells in the proliferative phase of the cell cycle, which is manifested as an increase in either the labeling index or the growth fraction. Because of this fact, drug must be administered at shorter and shorter intervals to assure that the tumor does not repopulate faster than cells are killed. Given the enormous constraints and complexities of brain tumor (and other solid tumor) chemotherapy, our clinical successes have been limited to short-term palliation. Unfprtunately, future optimism seems unwarranted. Rational design of chemotherapeutic regimens is currently limited by the crudeness of the available drug armamentarium. Any future success with clinical chemotherapy will come only after greater sophistication in drug design has been achieved and means to modify or minimize systemic toxity have been developed.

286

Levin, Paflak, and Landahl

A P P E N D I X I: DISTRIBUTION OF D R U G CONCENTRATION IN A M O D E L T U M O R SYSTEM

We will consider two "extreme" models for the capillary system, the Krogh model and the completely mixed model. For both models, we will be interested only in the total concentration of the drug at any point over all time, i.e., the integrated exposure, called C x T in the pharmacological literature, which we denote by use of the subscript T. Thus, for example, co

CT = J0 C(t) dt

(AI-1)

In order to evaluate CT, use will be made of the following well-known theorem (40). If Co~ is the steady-state concentration of C for a constant input of size unity, and if C,(t) is the plasma concentration, then I"

oo

CT = Coo Jo Cp(t) at = Co~CpT

(AI-2)

Thus only the steady-state equations need be solved, which simplifies the mathematics considerably. Krogh Model

For this case, we will approximate a tissue as predominantly a parallel orientation of capillaries considered as uniform cylinders of length L, radius r, and permeability P, surrounded by a cylinder of radius R. This model yields the smallest minimum value of CT for any model of uniform capillary distribution. While this case has been treated previously (41), the solution is so complex that very little insight may be gained by inspection of the solution. For this reason, we shall attempt to obtain an approximate, but simple, expression for the steady-state distribution of drug concentration around capillaries. We introduce the following notations: r = capillary radius (cm) R = intercapillary distance (ICD)/2 = model cylinder radius (cm) v = capillary blood velocity (cm/sec) L = capillary length (cm) n --number of capillaries per ml of tissue = 1/~rR2L P = capillary permeability (cm/sec) S -- capillary surface area = 2r/(R 2 - r 2) = 2 r / R 2 (cm2/cc tissue) k = rate constant per cm a tissue due to intracellular (kc) and extracellular (ke) metabolism (sec -1) C~ = concentration in capillary

Heuristic Modeling of Drug Delivery to Malignant Brain Tumors

287

Ce = Ce = Cr = Ve = Vc = A~ = A=

concentration in extracellular fluid (ECF) concentration in intracellular fluid (ICF) integrated exposure = integral of concentration over time E C F space fraction ICF space fraction cell surface area per unit volume (cm -1) partition coeffcient between ICF and E C F = C~/C~ at equilibrium f = (amount in ! C F ) / ( a m o u n t in ECF) in any given volume of tissue = v c } t / v~

F = blood flow/cm 3 tissue = r = r2v/R2L (sec -1) D~ = diffusion coefficient in E C F (cm2/sec) FD = rate constant due to intercapillarflkdiffusion

= 4,/RrD~ Ve/(R - r ) 2 (R +r)( sec -1) PD = effective permeability including influence of diffusion;

PD = P/(1 +PS/FD) Pc = cell permeability (cm/sec) Note that S / F = 2L/rv is independent of n and is just twice the average capillary transit time, tR = L/v, divided by r. The capiliary blood volume fraction is (r/R) 2. It is the purpose here to estimate the concentration distribution in the steady state for unit input concentration. As shown in equation AI-2, these values are proportional to the integrated exposure at each point in the surrounding tissue. Let the blood plasma concentration at a point x along the capillary be Cp(x) with Cp(0)= 1. Assume that Ce(x), the average concentration in the E C F between capillaries, is dependent only on C'e(x), the concentration near the capillary. This is reasonable because capillary length is much greater than R = I C D / 2 and hence diffusion along x will have a negligible effect. This will be true if the longitudinal diffusion rate constant, of the order of magnitude DelL 2, is small compared with all other rate constants. We shall further assume that the drug diffuses solely through the ECF. That is, it does not also diffuse through the cells. We then use the conservation condition that, for each circulatory unit, the net amount of drug entering minus drug leaving the capillary is equal to the flow across the capillary membrane, which is equal to diffusion, which in turn is equal to the amount metabolized. For convenience, define k such that kCe is the amount metabolized per unit volume of tissue, extracellularly and intracellularly, at x:

kCe = keVeCe -~kcVcCc

. (A~-3)

288

Levin, Patlak, and Landahl

where Ve is the E C F fraction and Vc is the ICF fraction. But the last term is also equal to the flux into the cells, AcPe ( C e - dc/1), where Ac is the cell surface area per unit volume, Pc is the cell permeability, and A is the partition coefficient. From this equality we can solve for the intracellular concentration Co(x), from which we obtain

k~Vcd~ = Vefkcde/(1 + Vekcf/AePc)

(AI-4)

f being the ratio of intracellular to extracellular amounts for any given volume of tissue in the steady state in the absence of metabolism. Introducing (AI-4) into (AI-3), the net metabolic rate constant per unit volume is

k -~-(ke q-fktc)We

(AI-5)

k'e = k J ( 1 + V~kd/Ace~)

(AI-6)

with

The quantity k'~ is an effective intracellular metabolic rate that approaches k~ for large enough Pc. If, however, Pc is small enough, k'e = A~P~/Vef. Then k = keVe +A~P~. For the drugs studied in this article E C F tl/z = (ln 2)/ke and !CF h/2 = (ln 2)/k'c are the given values. If we equate the loss in capillary concentration between x and x + dx to loss through the capillary wall, and this to the diffusion away from the surface, and this, in turn, to the amount metabolized, we have for each capillary, on dividing by dx,

dC. , - - D, Ve - rfr2v ~ = 2~rrP(Cp - CCe) 2 7 r ~ / R r ~ (Cte -:

de)

= 7r(R2-r2)kd~

(AI-7)

where C'e is the concentration just outside the capillary at p__o_intx and where in the third equality we have used the geometric mean ~/Rr.Rr, instead of the arithmetic mean (R + r)/2 for the cross-sectional area, in order to give a more appropriate weight to the restricting effect on diffusion of the small cross sectional area near the capillary. Equating the second and third expressions and assuming that R2>> r 2, we can obtain C" and thus also

Cp--C'e: Cp

- -

C ; = (Cp - Ce)/(1 +PS/FD)

(AI-8)

where

FD = 4 D e V e ~ r / [ ( R - r) 2 (R + r)]

(AI-9)

FD being a rate constant due to diffusion. If we multiply the first two expressions in equation AI-7 by n = 1/~rRZL, we convert from fluxes per

Heuristic Modeling of Drug Delivery to Malignant Brain Tumors

289

capillary to fluxes per cm 3 of tissue. Thus this equality can be written (42) FdG/dx

= P S ( C p - C'e)/L = - PDS(C~ - C'e)/L

(AI-10)

where PD = P/(1 + PS/FD)

(AI-11)

an effective permeability as modified by diffusion. If we equate the second expression to the last in equation AI-7 and eliminate C'e using equation AI-8, we can solve for C~, the average concentration at a point x along the capillary: C~(x) = Cp(x)/(i + k/PDS)

(AI-12)

Introducing Ce into equation AI-10 and integrating, with Cp(0)= 1, we obtain Cp(x) = e -Ax/L

(AI-13)

A = ( k / F ) / [ 1 + (k/PDS)]

(AI-14)

where Thus we have from equations AI-8 and AI-12, using equation AI-13, C" (x) = [(1 + k/FD) e-a*/L3/(1 + k/PDS)

(AI-15)

de(x) = e-AX/L/(1 + k/PDS)

(AI-16)

Consider for the moment that k/FD is greater than !. If a linear approximation were used, then the concentration at R would be negative. If it is assumed, as a rough approximation, that Ce decays exponentially with distance perpendicular to the capillary, the value of Ce/C'e at R would be the square of the fraction at the midpoint and C* would be (2"/(t + k/FD). Because the concentration must have a zero derivative at R, this latter expression, which is still decreasing, might be expected to be too targe. We find that by introducing an empirical factor of 0.2 in front of k, an estimate of C* can be obtained that is quite satisfactory over the experimental range of blood volumes (0.003-0.09). Thus we write C** (x) = e-a~/L[(1 + 0.2k/FD)~ I + K/_ DS)]

(AI- t7)

for an estimate of the concentration midway between capillaries at a point x. The concentration in the region most poorly perfused (the lethal corner) is then C* (L), given by C* (L) = e-a/[(1 + 0 2k/FD)((J- + k/PDS)]

(AI-18)

If cells are extremely impermeable, k = Veke. But it is then possible that the capillary surface in very close contact with cells may be blocked and

290

Levin, Patlak, and Landahi

hence only a fraction of the capillary surface, of the order of Ve, would be functionally permeable. In this case, one must consider carefully the operational definition of the permeability coefficient P. However, if Pc is very large, if the intracellular diffusion coefficient is not much smaller than De, and if the intracellular metabolic rate is not too much greater than in ECF, then transverse diffusion through the cells cannot be neglected compared with extracellular diffusion, even as an approximation. If the flow rate is high enough (A << 1), the plasma concentration will change negligibly along the capillary length. In this case an exact solution can be obtained. The case with P-> co was used to test the approximate expressions in which blood volumes varied from 0.003 to 0.09, a range of values that covers most experimental situations. The result of these tests indicates that the concentration (~e is within 0.017 of the exact solution at (R + r)/2 ( ~- R/2), the plasma concentration being 1. As long as the value of Ce exceeded 0.06, the relative error was within 8% (e.g., 0.100 vs. 0.108). For values of the concentration farthest from the capillaries, C*, the maximum difference was 0.023, the relative error being less than 19%, but generally much less. In both cases, the relative error may become large when the concentration is about 0.02. Below 0.01, the estimate may no longer be of the correct order of magnitude, the estimated value being too high. However, because C* is already quite small in comparison to Cp, the error is of no practical consequence. Expressions AI-13 to AI-18 were also tested using the results given graphically by Blum (27) in which the permeability coefficients of capillaries ranged from 10 -3 to 10 -2 cm/sec. The approximate expressions for the concentration at (R + r)/2 and at R were satisfactory, being within about 0.02 of the graphical values that were based on a plasma concentration of 1. The average concentrations were all within 0.002 of the exact values. However, the concentration at the capillary surface was overestimated (0.61 instead of 0.52 and 0.34 instead of 0.26) for the two cases at the largest metabolism (ref. 27, Fig. 2c). But note that the concentrations at (R + r)/2 were respectively about 0.02 and 0.01 and the half-penetration depth (8) is only about 3.5/zm in both cases. The plasma concentrations given by equations AI-13 were all within about 0.02 of the exact value, except for these same two cases. Let the plasma concentration C~(t) be approximated by a sum of N exponentials: N

C.(t) = E B~e -*~' i=1

with ~bi ~ 0.

(AI-19)

Heuristic Modeling of D r u g D e l i v e r y to Malignant Brain Tumors

291

Then N

(AI-20)

G r = 2 B,/cb, i=1

Thus, for this case, N

C*r(L) = e-A/[(1 + k/PoS)(1 + 0.2k/FD)] E Bi/&~

(AI-21)

i=1

Completely Mixed Model

For this case, we assume that there is complete mixing within the ECF. For this model, the concentration in the ECF will be the largest minimum concentration for any model of capillary distribution. The steady-state differential equations are (AI-22)

- V d C p / d x = (PS/L) (Cp - Ce) PL

(AI-23)

P S / L Jo (Cp - Ce) dx = ICe

with Cp(O) = 1, where equation AI-5 from the Krogh model still applies to this case. Solving these equations, we have N

C~r = (IF(1 - e-PS/F)~/[F(1 -- e-eS/F) + k]) 2 B,/4i

(AI-24)

/=1

If De and F are allowed to become large, equations AI-21 and AI-24 reduce to the same functions, as would be expected from the assumptions of the two models. APPENDIX Ih D R U G CONCENTRATION IN A TISSUE SUPPLIED ONLY BY DIFFUSION Case A: Time Dependence in a Region with No Metabolism

We shall consider a sphere of radius d whose boundary has concentration CB(t), and we will attempt to calculate the concentration, C(p, t), in the region outside of the sphere with negligible perfusion. The partial differential equation then becomes, where p is the distance from the center of the sphere, Of/at

C(p, o) = 0 C(d, t)= C,~(t)

= D e (O2C/Op 2 +

2/p x Of/lap)

(AIL 1)

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Levin, Patlak, and Landahl

The solution of this equation (38) is

C(p, t)

d

(p-d)

=p 2(,n.De) 1/2

Sot

CB(T) e_~O_d~/4D~,_~~d~ (t_~.)3/-------~a

(AII-2)

Let 8 be the distance from the surface of the sphere, i.e., 8=p- d

(AII-3)

For any value of (p - d), C(p, t) is a maximum for the d/p term in equation (AII-2) equal to 1. That is, as is physically obvious, diffusion will be maximum for a flat boundary. Further, if CB (t) is a monotonically increasing function with a final value of CBo,then C(p, t) will be less than or equal to the value given by equation (AII-2) with Cs (z) replaced by CBo.The integral of equation AII-2 may then be evaluated for this case to yield

C(p, t) <- Cuo erfc (8/2x/D-t)

(AII-4)

Case B: Integrated Exposure in a Region with Metabolism, as a Function of Distance trom the Boundary

If there is a boundary between a well-perfused region and a poorly perfused region, the tissue of the latter region near the boundary can be expected to have an increased concentration due to diffusion from the former, the amount depending on the diffusion coefficient as well as on the metabolic and circulation parameters. We wish to estimate how far this effect extends from the boundary. Because the system is assumed to be linear, the principle of superposition applies. That is, the amount of drug that passes from the wellperfused region into the poorly perfused region will be independent of the amount of drug that enters the poorly perfused region through the capillaries. Because this latter amount has been analyzed in Appendix I, we shall evaluate the amount that passes from one to the other region by assuming that the blood plasma concentration of the drug in the poorly perfused region is 0. We shall assume that the concentration in the well-perfused region is uniform, i.e., we shall ignore the possible decrease in concentration at the boundary of a well-perfused region due to flow into the other region. Further, we shall assume that one of the regions is a sphere and the other region is an infinite shell surrounding the sphere. Because we are interested only in the integrated exposure, we consider only the steady-state diffusion from one macroscopic region into another macroscopic region, and we shall treat the capillaries plus their surrounding ECF and ICF in the poorly perfused region as a homogeneous volume insofar as diffusion from one region to the other is concerned. Further, we

Heuristic Modeling of Drug Delivery to Malignant Brain Tumors

293

shall assume the second model of Appendix I in the poorly perfused regions, i.e., complete mixing within the ECF around each capillary. Thus equation AI-22 with the boundary condition that Cp(0)= 0 applies. The expression for the loss, per unit volume of tissue, into the ECF from the plasma is given by the left-hand side of equation AI-23. These relationships may be readily solved to show that the expression for the loss, per unit volume of tissue, from the ECF into plasma is F(1 - e-eS/F)C,. Thus, including the loss due to metabolism, kCe, the differential equation for the steady-state diffusion of drug into the poorly perfused region is given by

De V~ (d2Celdp 2 + 21p x dC,/dp)

= F[(1 - e-PS/F) + k ]C, (AII-5)

with the boundary condition that Ce(d) = 1 and the parameters applying to the poorly perfused region. Let K 2 = [F(1 - e-eS/F) + k ] / D , V ,

(AII-6)

In order to evaluate C,(p), equations AI-1 and AI-2 are used, where Cpr in equation AI-2 is replaced by Cer(d). Equation AII-3 may be readily solved. Thus, for the situation of diffusion from a well-perfused sphere out to its surrounding poorly perfused shell, the solution is

CeT(p)/ CeT(d) = (d/p) e -K(o-d)

(AII-7)

Cer (p ) / Cer (d) <- e-K8

(AII- 8)

Because p >- d,

The equality applies when the diameter of the tumor shell becomes quite large, that is, for planar diffusion. For P = 0 , V, = 1, and C,r(p)= 0.5C, T(d), this reduces to 6 = 6.45 4~D-tl/2

(AII-9)

where 8 is the half-penetration depth in cm and tl/2 is in minutes. Similarly, the solution for diffusion from a well-perfused shell into the poorly perfused sphere is

C,T(p)/ C,r(d) = d/p x sinh (Kp)/sinh (Kd)

(AII-IO)

The minimum value of C,r, C,T(O), is then given by

C,r (0)/C,T (d) = Kd/sinh (Kd) K being defined by AII-6.

(AII- 11)

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ACKNOWLEDGMENT We acknowledge the collaboration of Neil Buckley for editorial development of the manuscript.

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I8. R. G. Blasberg, C. S. Patlak, W. R. Shapiro, and J. D. Fenstermacher. Metastatic brain tumors: Local blood flow and capillary permeability. Neurology (Minneapolis) 29:547 (1979) (abstr.). 19. N. A. Vick and D. Bigner. Microvascular abnormalities in virally induced canine brain tumors. J. Neurol. Sci. 17:29-39 (1972). 20. V.A. Levin, D. C. Wright, H. D. Landahl, C. S. Patlak, and J. Csejtey. In situ drug delivery. Br. J. Cancer 41:74-78 (1980) (Suppl. IV). 21. A. Krogh. The Anatomy and Physiology of Capillaries, Yale University Press, New Haven, 1929. 22. D. G. Levitt. Theoretical model of capillary exchange incorporating interaction between capillaries. Am. J. Physiol. 220:250-255 (1972). 23. P. M. Gullimo and F. M. Grantham. Studies on the exchange of fluids between host and tumor. II. Blood flow studies of hepatomas and other tumors in rats and mice. J. Natl. Cancer. Inst. 27:1465-1491 (1960). 24. E. Siracka, N. Poppova, V. Pipa, and J. Durkovsky. Changes in blood flow of growing experimental tumor determined by the clearance of aa3Xe. Neoplasma 26:173-t77 (1979). 25. J.D. Weinstein, F. J. Toy, M. E. Jaffe, and H. I. Goldgerg. The effects of dexamethasone on brain edema in patients with metastatic brain tumors. Neurology (Minneapolis) 23:t21129 (1973). 26. D. Norman, W. Berninger, D. Boyd, V. A. Levin, and T. H. Newton. Dynamic computed tomography. Presented at the XIth Symposium Neuroradiologicum, 4-10 June, 1978, Wiesbaden, West Germany (abstr.). 27. K. T. Wheeler, E. F. Tel, M. E. Williams, S. Sheppard, V. A. Levin, and P. M. Kabra. Factors influencing the survival of rat brain tumor cells after in vitro treatment with 1,3-bis(2-chloroethyl)-l-nitrosourea. Cancer Res. 35:1464-1469 (1975). 28. V. A. Levin, J. Stearns, A. Byrd, A. Finn, and R. J. Weinkam. The effect of phenobarbital pretreatment on the antitumor activity of 1,3-bis(2-chloroethyl)-l-nitrosourea (BCNU), 1-(2-chloroethyl)-3-cyclohexyl-l-nitrosourea (CCNU), and 1-(2-chloroethyl)-3-(2,6dioxo-3-piperidyl)-l-nitrosourea (PCNU), and on the plasma pharmacokinetics and biotransformation of BCNU. J. Pharmacol. Exp. Ther. 208:1-7 (1979). 29. R. J. Weinkam, T.-Y. Liu, and H.-S. Lin. Protein mediated chemical reactions of chloroethylnitrosoureas. Chem. Biol. Interact. (in press). 30. M. R. Rosenblum, K. T. Wheeler, C, B. Wilson, M. Barker, and K. D. Knebel. In vitro evaluation of in vivo brain tumor chemotherapy with 1,3-bis(2-chloroethyl)-tnitrosourea. Cancer Res. 35:1387-1391 (1975). 31. V. A. Levin, K. T, Wheeler, and C. B. Wilson. Chemotherapeutic approaches to brain tumors: Clinical and experimental observations with dianhydrogalactitol and dibromodulcitol. Cancer Treat. Rep. (in press). 32. P. Espana, P. H. Wiernik, and M. D. Walker, Phase II study of dianhydrogalactitol in malignant glioma. Cancer Chemother. Rep, 62:1199-1200 (1978). 33. J. L Ausman, V. A. Levin, W. E. Brown, D. P. Rall, and J. Do Fenstermacher. Brain tumor chemotherapy: Pharmacologic principles derived from a monkey tumor model. J. Neurosurg. 46:155-164 (1977). 34. M. G. Donelli, M. Broggini, T. Colombo, and S. Garattini. Importance of the presence of necrosis in studying drug distribution within a tumor tissue. Eur. Y. Drug Metabol. Pharmacokin. 2:63-67 (1977). 35. R.J. Goldacre, Viable tumor regions accessible to chemotherapeutic agents and a possible new strategy for inactivating them. Br. J. Cancer 36:406 (1977). 36. V. A. Levin. Relationship of octanal/watcr partition coefficients and molecular weight to rat brain capillary permeability. J. Med. Chem. 23:682-684 (1980). 37. I. F. Tannock. The relation between cell proliferation and the vascular system in a transplanted mouse mammary tumor. Br. J. Cancer 22:258-273 (1968). 38. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd ed, Oxford University Press, New York, 1959, pp. 63, 247.

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39. T. Hoshino, C. B. Wilson, M. L. Rosenblum, and M. Barker. Chemotherapeutic implication of growth fraction and cell cycle time in glioblastoma. J. Neurosurg. 43:127-137 (1975). 40. N. A. Lassen and W. A. Perl. Tracer Kinetic Methods in Medical Physiology. Raven Press, New York, 1979. 41. J.J. Blum. Concentration profiles in and around capillaries. Am. J. Physiol. 198:991-998 (1960). 42. S. Kety. The theory and application of the exchange of inert gas at the lungs and tissues. Pharmacol. Rev. 3:1--41 (1951).