Abstract

Introduction

Drug efficacy in brain is believed to be driven by the concentration of free (unbound) drug in brain (C_u,brain) [1]. One approach to the measurement of C_u,brain requires the dual measurement of the ratio of total drug in brain, divided by the total drug in plasma, or K_p,brain, in parallel with determination of the k_p,uu,brain, which is the ratio of C_u,brain divided by the free (unbound) drug in plasma, or C_u,plasma. The C_u,brain and C_u,plasma are measured in vitro by equilibrium dialysis of brain homogenate and plasma, respectively [2, 3].

In vitro in vivo extrapolation assumes the C_u,plasma measured in vitro is a reliable index of the free (bioavailable) drug in vivo within the brain capillary plasma compartment, i.e., that C_{u,plasma,in vivo} is equal to C_{u,plasma,in vitro}. This assumption requires that drug bound to plasma proteins, such as albumin or alpha-1 acid glycoprotein (AGP), is not available for transport in vivo. As reviewed by Zhang et al. [1], work in the 1980s provided evidence for drug delivery to organs in vivo via the plasma protein bound pool, and that this work has been largely forgotten. Recently, there has been renewed interest in drug delivery to tissues in vivo via the plasma protein-bound pool [4–6], a process referred to as plasma protein-mediated uptake or PMU [7].

The purpose of the present work was to develop a physiologically based pharmacokinetic (PBPK) model of brain delivery of drugs that are bound in plasma by albumin and AGP. The partly flow-partly compartmental model is tested under conditions of both the steady state and non-steady state, and is based on 18 parameters governing the kinetics of drug binding to albumin and AGP in plasma, drug binding to tissue proteins in brain, drug influx and efflux across the blood–brain barrier (BBB), drug metabolism, and cerebral blood flow. The PBPK model is examined for 2 drugs, propranolol, which undergoes modest PMU by brain from the AGP-bound pool in plasma [8], and imipramine, which undergoes extensive PMU by brain from both the albumin-bound and AGP-bound pools in plasma [9].

Methods

The partly flow-partly compartmental model of brain drug delivery is outlined in Fig. 1. The PBPK model is comprised of 11 variables, which are also defined in Table ​TableI.I. The concentration of free albumin in the capillary compartment, AF, is not treated as a variable, and is approximated by the total albumin concentration, because the plasma albumin concentration is log orders higher than the total plasma drug concentration. In the non-steady state model, as in the case of oral drug administration, the total plasma drug concentration (LT0) varies with time, and LT0[t] is a 12^th variable. The steady state model and the non-steady state model include 18 and 23 parameters, respectively. The basal values for the model parameters for propranolol and imipramine, along with literature citations, are given in Table ​TableII.II. The non-steady state model was examined only for propranolol, as detailed pharmacokinetic (PK) parameters are not available following oral administration of imipramine.

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Fig. 1

Partly flow-partly compartment PBPK model of brain delivery of drugs bound to plasma proteins such as albumin or AGP. The 11 variables, GL0, GF0, AL0, LF0, GL, GF, AL, LF, LM, PL, and PF, are distributed over 3 compartments: the systemic circulation, the brain capillary plasma volume, and the brain post-vascular volume. Within the systemic circulation, GL0, AL0, LF0, and GF0 is the concentration of globulin-bound drug, albumin-bound drug, free (unbound) drug, and free (unbound) globulin, respectively. Within the brain capillary plasma volume, GL, AL, LF, and GF is the capillary globulin-bound drug, the capillary albumin-bound drug, the capillary free (bioavailable) drug, and the capillary unoccupied globulin, respectively. Within the post-vascular brain compartment, LM, PL, and PF are the free drug in brain, the drug bound to brain tissue proteins, and the unoccupied brain tissue protein pool, respectively. The basal values for the 18 parameters of the model are defined in Table ​TableIIII.

Table II

Model Parameters for Propranolol and Imipramine

Parameter	Parameter name	Propranolol (reference)	Imipramine (reference)
LT0	Total plasma drug concentration	100 nM [10]	100 nM [11]
AF	total systemic and capillary albumin concentration	800,000 nM [12]	800,000 nM [12]
GT0	total systemic globulin concentration	20,000 nM [12]	20,000 nM [12]
KAin vitro	KD of drug binding to albumin in vitro	290,000 nM [8]	42,000 nM [13]
KGin vitro	KD of drug binding to globulin in vitro	3,300 nM [8]	1,200 nM [14]
PT	total brain binding protein concentration	5,000 nM	5,000 nM
VP	brain capillary volume	0.01 L/kg [15]	0.01 L/kg [15]
VT	brain extravascular volume	0.7 L/kg [15]	0.7 L/kg [15]
k1	rate constant of drug dissociation from globulin in brain capillary	1140 min−1 [8]	5,400 min−1 [9]
k2	rate constant of drug association with globulin in brain capillary	0.06 nM−1 min−1 [16] (106 M−1 s−1)	0.06 nM−1 min−1 [16] (106 M−1 s−1)
k3	rate constant of drug influx from plasma to brain across BBB	66 min−1 [8]	150 min−1 [9]
k4	rate constant of drug efflux from brain to plasma across BBB	0.943 min−1 [8]	2.1 min−1 [9]
k5	rate constant of drug association with brain protein binding site	0.006 nM−1 min−1 (105 M−1 s−1)	0.006 nM−1 min−1 (105 M−1 s−1)
k6	rate constant of drug dissociation from brain protein binding site	0.52 min−1 [17]	≤0.5 min−1 [9]
k7	rate constant of drug dissociation from albumin in brain capillary	1740 min−1 [8]	≥6,000 min−1 [9]
k8	rate constant of drug association with albumin in brain capillary	0.006 nM−1 min−1 [18] (105 M−1 s−1)	0.006 nM−1 min−1 [19] (105 M−1 s−1)
k9	rate constant of drug metabolism in brain	0 min−1 [17]	0 min−1 [20]
k10	rate constant of brain capillary plasma flow	60 min−1 [15]	60 min−1 [15]
b	fractional oral bioavailability	0.3 [21]	-
s	drug dose	4,600 nmol/kg, 80 mg [22]	-
k	rate constant drug oral absorption	0.023 min−1 [22]	-
d	rate constant drug systemic elimination	0.0027 min−1 [22]	-
V	drug systemic volume of distribution	5.0 L/kg [21]	-

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Plasma total drug concentration in steady state after intravenous infusion is LT0; plasma total drug concentration in non-steady state after oral administration is determined from the pharmacokinetic parameters (b, s, k, d, and V) as described in Eq. 13. KD=dissociation constant of drug binding to albumin or globulin

Results

The steady state simulations for propranolol delivery to brain are given in Table III for 9 of the 11 variables shown in Table ​TableI.I. The concentration of unbound AGP in the systemic compartment (GF0) or capillary compartment (GF) are not shown, as these approximate the total AGP (GT0), since GT0>>LT0. The basal propranolol parameters in Table ​TableIIII were used to produce the results for simulation 1 (Table III), where the KG^{in vivo} of propranolol binding to AGP is 19 uM, as compared to the KG^{in vitro} of 3.3 uM (Methods). The KG^{in vivo} of 19 uM produces a 102% increase in the capillary free (bioavailable) drug, LF, relative to the free drug in plasma in vitro, LF0. The free drug in brain, LM, is equal to the free drug in the capillary plasma, LF, and is also 102% greater than LF0. Owing to the enhanced dissociation of propranolol from AGP in vivo in the brain capillary, the capillary GL concentration decreases 61% relative to the systemic GL0 (Table ​(TableI,I, simulation 1). As there is no enhanced dissociation of propranolol from albumin, the capillary AL concentration increases 98% relative to the systemic AL0, as the albumin captures drug dissociated from AGP (Table ​(TableI,I, simulation 1). In simulation 2, there is no enhanced dissociation of propranolol from AGP, where KG^{in vivo}=KG^{in vitro}=3.3 uM. In simulation 2, k1=198 min⁻¹, which was computed from k1=k2·KG^{in vitro} given a k2 value of 0.06 nM⁻¹ min⁻¹ (Table ​(TableII).II). In simulation 2, the concentration of free drug in vitro, LF0, the concentration of free drug in the brain capillary, LF, and the concentration of free drug in brain, LM, are identical. In simulation 3, the basal value for KG^{in vivo}, 19 uM, is used (Table ​(TableII),II), and the plasma albumin and AGP concentrations are set at the levels observed in metastatic cancer, AF=600 uM and GT0=70 nM [12]. In this setting, the concentration of free drug in the brain capillary in vivo, LF, and the free drug in brain, LM, are increased 258% relative to the free drug in vitro, LF0 (Table III, simulation 3). In simulation 4, the KG^{in vivo} is set at 19 uM, but the k1 and k2 values are both increased tenfold relative to simulation 1; the k2 value of 0.6 nM⁻¹ min⁻¹ corresponds to an association rate constant of 10⁷ M⁻¹ s⁻¹. In simulation 5, the KG^{in vivo} is set at 19 uM, but the k1 and k2 values are both decreased tenfold relative to simulation 1; the k2 value of 0.006 nM⁻¹ min⁻¹ corresponds to an association rate constant of 10⁵ M⁻¹ s⁻¹. Despite the 100-fold variation in k1 and k2 values, simulations 4 and 5 show increases in LF and LM, relative to LF0, which are comparable to simulation 1. Simulation 6 represents the dissociation-limited model of transport, which posits very high membrane permeability of the drug, relative to rates of drug dissociation from the plasma protein in vivo. The dissociation rate constant for propranolol binding to AGP within the brain capillary in simulation 6, k1=198 min⁻¹, is the same as in simulation 2. Simulation 6 increases membrane permeability, k3 and k4, 1,000-fold. The k3 value of 66,600 min⁻¹ corresponds to a propranolol PS/F ratio of 1,100. However, the results of simulation 6 show that a dissociation-limited model of drug transport does not allow for transport of drug into brain from the AGP or albumin bound pools in plasma, as the LF and LM values are identical to LF0 in either simulation 6 or simulation 2 (Table III). Simulation 7 represents the case of both enhanced dissociation of drug from AGP in the brain capillary compartment, as in the case of simulation 1, but in the presence of an active efflux transport (AET) mechanism that causes a selective tenfold increase in k4, but not k3. In this setting, the concentration of LF is 102% of LF0, but the concentration of LM is reduced 90% relative to LF (simulation 7, Table III). In simulation 8, the only change from the basal state of simulation 1 is an increase in k9 from 0 to 6 min⁻¹, which represents drug metabolism in brain. Metabolism causes a 17% and an 89% decrease in LF and LM concentrations, respectively (simulation 8, Table III), relative to the LF and LM concentrations in simulation 1. In simulation 9, drug is metabolized in brain in parallel with the presence of ischemia, as represented by a tenfold decrease in k10, and these combined conditions cause a 59% decrease in both the LF and LM concentrations, relative to the LF and LM concentrations in simulation 8, where drug is metabolized in the presence of normal cerebral blood flow. Simulation 10 is the case where drug is metabolized in brain, where k9 is increased from 0 to 6 min⁻¹, and the drug is a substrate for an AET system, where k4 is selectively increased tenfold relative to the basal state. The combination of drug metabolism, and AET, produces a decrease in the concentrations of LF and LM of 9% and 94%, respectively, relative to the LF and LM concentrations in simulation 1, where there is no drug metabolism and no AET. In the special case where the drug is not bound by plasma protein, then LF0=LF=LM=LT0 as shown by simulation 30 of Table S1 of the Supplementary Material.

Table III

Propranolol Steady State Model Simulations

No	Parameter changea	Arterial (nM)			Capillary (nM)			Brain (nM)			Kp,brain
		GL0	AL0	LF0	GL	AL	LF	LM	PL	PF
1	basal	61.7	28.2	10.2	23.6	55.8	20.6	20.6	959	4,040	9.8
2	k1=198 min−1	61.7	28.2	10.2	61.7	28.2	10.2	10.2	526	4,473	5.4
3	AF=600 uM; GT0=70 nM	87.4	8.5	4.1	55.7	29.6	14.7	14.7	724	4,276	7.4
4	k1=11,400 min−1; k2=0.6 nM−1 min−1	61.7	28.2	10.2	22.2	56.8	21.0	21.0	973	4,026	9.9
5	k1=114 min−1; k2=0.006 nM−1 min−1	61.7	28.2	10.2	33.6	48.6	17.9	17.9	854	4,145	8.7
6	k1=198 min−1; k3=66,600 min−1; k4=940 min−1	61.7	28.2	10.2	61.7	28.2	10.2	10.2	526	4,473	5.4
7	k4=9.4 min−1	61.7	28.2	10.2	23.6	55.8	20.6	2.1	116	4,883	1.2
8	k9=6 min−1	61.7	28.2	10.2	20.2	46.5	17.1	2.3	130	4,869	1.3
9	k9=6 min−1; k10=6 min−1;	61.7	28.2	10.2	7.61	19.23	6.96	0.95	54	4,946	0.55
10	k4=9.4 min−1; k9=6 min−1	61.7	28.2	10.2	21.9	51.2	18.8	1.2	66	4,934	0.67

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^aBasal parameters are given in Table ​TableII and are used in simulation 1; for simulations 2–10, all parameters are the basal parameters except for the parameters listed. Parameters are defined in Table ​TableI,I, and shown in Fig. 1

The dissociation rate constant of propranolol binding to brain protein, k6 (Fig. 1), has been measured in vivo, and is 0.52 min⁻¹ [17]. The k5·PF product of propranolol binding to brain proteins in vivo is 1.8 min⁻¹ [17]. A k5 value of 0.006 nM⁻¹ min⁻¹ (Table ​(TableII),II), in parallel with a PT value of 5,000 nM (Table ​(TableII),II), produces a K_p,brain of 9.8 for propranolol (simulation 1, Table III), which matches the experimentally observed K_p,brain for propranolol in brain, 9.7, following an IV injection of 7.5 mg/kg [39]. Simulations listed in Table S1 of the Supplementary Material show that reduction of PT to either 1,000 nM, or 500 nM, produces progressively reduced K_p,brain values of 2.1 and 1.2, respectively, which are low compared to observed K_p,brain value for propranolol [39]. Other simulations listed in Table S1 of the Supplementary Material show the use of a k5 value of either 0.06 or 0.0006 nM⁻¹ min⁻¹ produced K_p,brain values that were too high, or too low, respectively, compared to the observed K_p,brain for propranolol [39]. The effect of varying the PT value was evaluated by simulations that used the basal values in Table ​TableII,II, but varied the PT value from 100 nM to 10,000 nM, and the predicted values for LM and PL are given in Fig. 2. Increasing PT in brain resulted in a progressive increase in the drug pool bound to brain proteins, PL, but had no effect on the concentration of free drug in brain, LM (Fig. 2). This result is anticipated by Eq. 5 (Methods), which shows the LM variable is independent of drug tissue binding parameters, PT or KP (k6/k5).

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Fig. 2

The steady state model for propranolol computes the values for free drug in brain, LM, and protein-bound drug in brain, PL, relative to the total brain concentration of drug binding tissue protein, PT, over a PT range of 0.1 to 10 uM.

Imipramine undergoes enhanced dissociation from both albumin and AGP binding sites within the brain capillary [9], as reflected by the KG^{in vivo} of 90 uM [9] and the KA^{in vivo}>1,000 uM [9], as compared to the KG^{in vitro} and KA^{in vivo} values of 1.2 uM [14] and 42 uM [13], respectively. The results for simulations of imipramine distribution in brain using the steady state model equations and the imipramine parameters are given in Table ​TableIV.IV. Using the basal parameters for imipramine listed in Table ​TableII,II, the free (bioavailable) imipramine in brain capillary plasma, LF, and the free imipramine in brain, LM, are 18-fold greater than the free drug in vitro, LF0 (simulation 11, Table ​TableIV).IV). Simulation 12 represents the case where there is no enhanced dissociation of imipramine from either albumin or AGP, and the KG^{in vivo} and the KA^{in vivo} values are fixed at the KG^{in vitro} and KA^{in vitro} values. For simulation 12, k1 is reduced to 72 min⁻¹, and k1 was computed from k1=k2·KG^{in vitro}, where k2=0.06 nM⁻¹ min⁻¹ and KG^{in vitro}=1.2 uM (Table ​(TableII).II). In simulation 12, k7 was reduced to 252 min⁻¹, where k7 was computed from k7=k8·KA^{in vitro}, where k8=0.006 nM⁻¹ min⁻¹ and KA^{in vitro}=42 uM (Table ​(TableII).II). Eliminating enhanced dissociation of imipramine from albumin and AGP lowers the LF and LM concentrations in brain in vivo to the free drug concentration in vitro, LF0 (simulation 12, Table ​TableIV).IV). The K_p,brain value for simulation 11, 19.3, approximates the experimentally observed K_p,brain, 23, for imipramine in vivo in the mouse [40], which is 12-fold higher than the K_p,brain predicted for simulation 12 (Table ​(TableIV).IV). The K_p,brain for imipramine for the rat ranges from 26–30 [41, 42]. In simulation 13, the AF and GT0 concentrations observed in metastatic cancer are used (Table ​(TableIV).IV). These changes in plasma protein concentrations produce a 31-fold increase in LF and LM in vivo, relative to the free drug in vitro, LF0 (simulation 13, Table ​TableIV).IV). In simulations 14 and 15, the rate constants of imipramine dissociation, k1, and association, k2, with AGP are either increased or decreased tenfold, respectively, relative to the basal values in Table ​TableII,II, and this has only a minimal effect on the LF and LM concentrations relative to the values in simulation 11 (Table ​(TableIV).IV). Simulation 16 represents the cases of a dissociation-limited transport mechanism, where the rate constants of imipramine dissociation from AGP, k1, and from albumin, k7, are much lower than the rate constant of drug transport through the BBB, given by k3 and k4. In simulation 16, the k1 and k7 values are the same as in simulation 12, but the k3 and k4 values are increased 1,000-fold to k3=150,000 min⁻¹ and k4=2,100 min⁻¹. However, these very high membrane permeation rates for imipramine used for simulation 16 produced variable concentrations identical to simulation 12, and the in vivo concentrations of LF and LM were equal to the in vitro concentration of free drug, LF0 (Table ​(TableIV).IV). In simulation 17, the k4 of imipramine efflux is selectively increased tenfold to simulate transport via an AET system at the BBB, as there is evidence that imipramine is a substrate for p-glycoprotein [43]. This simulation caused no change in the drug concentration in the brain capillary compartment, LF, but resulted in a 90% reduction in the LM concentration, relative to simulation 11 (Table ​(TableIV).IV). In simulation 18, drug metabolism was modeled as the value for k9 was increased from 0 to 6 min⁻¹. This produced a 48% and 86% decrease in the LF and LM concentrations as compared to simulation 11 (Table ​(TableIV).IV). In simulation 19, drug metabolism was combined with ischemia, as represented by a tenfold decrease in k10, and this resulted in a 90% and 97% decrease in the LF and LM concentrations as compared to simulation 11. In simulation 20, AET and drug metabolism were combined with a tenfold increase in k4 and an increase in k9 from 0 to 6 min⁻¹. This produced a 22% and 94% decrease in the LF and LM concentrations, respectively, compared to simulation 11 (Table ​(TableIV).IV). A summary of all simulations for brain delivery of imipramine is given in Table S2 of the Supplementary Material.

Table IV

Imipramine Steady State Model Simulations

No	Parameter changea	Arterial (nM)			Capillary (nM)			Brain (nM)			Kp,brain
		GL0	AL0	LF0	GL	AL	LF	LM	PL	PF
11	basal	45.3	51.9	2.73	11.3	39.5	49.2	50.2	1,880	3,120	19.3
12	k1=72 min−1 k7=252 min−1	45.3	51.9	2.73	45.3	51.9	2.73	2.78	162	4,838	1.6
13	AF=600 uM; GT0=70 nM	79.2	19.4	1.36	33.1	25.1	41.9	42.7	1,695	3,305	17.4
14	k1=54,000 min−1; k2=0.6 nM−1 min−1	45.3	51.9	2.73	11.0	39.6	49.4	50.3	1,884	3,116	19.3
15	k1=540 min−1; k2=0.006 nM−1 min−1	45.3	51.9	2.73	14.1	38.3	47.6	48.6	1,843	3,157	18.9
16	k1=72 min−1; k3=150,000 min−1; k4=2,100 min−1 k7=252 min−1;	45.3	51.9	2.73	45.3	51.9	2.73	2.77	161	4,839	1.6
17	k4=21 min−1	45.3	51.9	2.73	11.3	39.5	49.2	5.02	284	4,716	2.9
18	k9=6 min−1	45.3	51.9	2.73	6.13	20.8	25.6	6.78	376	4,624	3.8
19	k9=6 min−1; k10=6 min−1	45.3	51.9	2.73	1.13	3.94	4.86	1.29	76	4,924	0.77
20	k4=21 min−1; k9=6 min−1	45.3	51.9	2.73	8.97	31.1	38.6	3.06	177	4,823	1.8

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^aBasal parameters are given in Table ​TableIIII and are used in simulation 11; for simulations 12–20, all parameters are the basal parameters except for the parameters listed

The non-steady state model was used to simulate variable concentrations over 1440 min after a single oral ingestion of 80 mg propranolol in humans. The total plasma concentration of drug, LT0[t], was computed from Eq. 13 using the PK parameters in Table ​TableII,II, and the plasma LT0 concentrations are plotted in Fig. 3a. The simulations were performed with the basal propranolol parameters in Table ​TableII,II, except the AF=600 uM and the GT0=70 nM as in simulation 3 (Table III). The values for globulin bound drug and albumin bound drug in the arterial compartment, GL0 and AL0, and brain capillary compartment, GL and AL, are plotted in Fig. ​Fig.3b.3b. The concentration of free (bioavailable) drug in the brain capillary, LF, the free drug in brain, LM, and the free drug in vitro, LF0, are plotted in Fig. 3c. The sum of the LM and PL concentrations are plotted in Fig. 3a. The (LM+PL)/LT0, or K_p,brain values, are 5.4, 7.2, 8.0, and 9.1, at 2, 4, 6, and 10 h after administration and reached equilibrium K_p,brain values of 10 between 12–24 h after administration. The plasma area under the concentration curve (AUC) for each variable was determined with the trapezoid method, and these values are given in Table ​TableVV for the arterial, brain capillary, and brain compartments. The AUC for the brain capillary unbound (bioavailable) drug, LF, and unbound drug in brain, LM, are 260% greater than the AUC for unbound drug in vitro, LF0 (Table ​(TableV).V). These results on the comparative values of the AUC for LF, LM, and LF0 with the non-steady state model are identical to the comparative values for concentrations for LF, LM, and LF0 with the steady state model (simulation 3, Table III). The ratio of AUC for the LM+PL pools, relative to the LT0 pool, is 7.4. The total propranolol plasma concentration, LT0, used in the steady state model, 100 nM (Table ​(TableII),II), is reached at 360–480 min after PO administration (Fig. 3a). At this time, the concentrations of LF0, LF, and LM (Fig. 3c), approximate the same concentrations generated with the steady state model for these variables (simulation 3, Table III). The AUC values for the globulin-bound and albumin-bound drug in the arterial and brain capillary compartments are inverted, owing to the selective enhanced dissociation of propranolol from AGP, but not from albumin, within the capillary compartment relative to the arterial compartment of brain.

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Fig. 3

The non-steady state model for propranolol computes the concentrations for the total drug in brain (LM+PL) and total drug in plasma (LT0) (panel a), for the concentrations of albumin-bound drug and AGP-bound drug in the systemic compartment (AL0, GL0) and brain capillary compartment (AL, GL) (panel b), and the concentrations of free (bioavailable) drug in the brain capillary (LF), the concentration of free drug in brain (LM), and the concentration of free drug in vitro (LF0) (panel c).

Table V

Propranolol AUC in Arterial, Brain Capillary, and Brain Compartments

Compartment	Variable (nM)	AUC (nmol·min/L)
Arterial	LT0	48,489
	LF0	1,994
	GL0	42,373
	AL0	4,136
Brain capillary	LF	7,122
	GL	27,059
	AL	14,379
Brain	LM	7,211
	LM+PL	359,655
	PL	352,655

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AUC computed from T=0 to T=1,440 min with the trapezoid method based on the concentrations shown in Fig. 3. Variables are defined in Table ​TableII

Discussion

The PBPK model described in this work focuses on the in vivo kinetics of drug binding to plasma proteins, albumin and AGP, and brain tissue proteins, as outlined in Fig. 1. Model parameters include the individual association rate constants (kon) and dissociation rate constants (koff) of drug binding in vivo to albumin and AGP in plasma, and to drug binding proteins in brain (Table ​(TableII).II). The model does not address drug binding to the target receptor, which has been examined in prior work on the relative effects of the kon and koff rates governing the drug-receptor binding reaction [44]. The present model is based on the premise that the dissociation constant, KD^{in vitro}, governing drug binding to the plasma protein in vitro may, or may not, be equal to the KD^{in vivo} that governs drug binding to the plasma protein at the interface of plasma and the endothelial glycocalyx in vivo within the brain capillary compartment. It is possible to determine the KD^{in vivo} within the brain capillary in vivo using carotid arterial injection methods [34]. Such vivo studies of drug binding to plasma proteins within the brain capillary compartment show that the KD^{in vivo} is generally, but not always, greater than the KD^{in vitro}, which is indicative of enhanced dissociation of drug from the plasma protein in vivo, such that the protein-bound drug is available for uptake by brain, without exodus of the plasma protein, per se, from the capillary compartment. Plasma protein mediated drug uptake, or PMU [7], in brain is examined in the present PBPK model. This study models 2 drugs, propranolol and imipramine. Propranolol undergoes a modest transport into brain in vivo from the AGP-bound pool, but not from the albumin-bound pool [8]. In contrast, imipramine undergoes marked transport into brain in vivo from both the AGP-bound and albumin-bound pools [9].

The different models that have been proposed for PMU have been recently reviewed [45], and include a receptor model, a dissociation-limited model, and an enhanced dissociation model. Although both albumin and AGP bind multiple proteins [46, 47], no gene encoding a specific albumin or AGP receptor has been cloned to date. Both in vivo investigations [48], and in vitro studies with isolated brain microvessels [49], show no specific albumin receptor at the BBB. In the dissociation-limited model, it is posited that membrane permeability is actually 2–3 log orders higher than experimentally observed values, and in this setting, drug is rapidly transported into the organ following dissociation from the plasma protein within the capillary compartment, rather than undergo re-association with the plasma protein [50]. While there is no experimental basis for the postulate of extremely high PS products for drug transport at the BBB, the dissociation-limited model is examined in simulations 6 and 16 for propranolol and imipramine, respectively (Tables III-​-IV).IV). These simulations show that there is no transport from the plasma protein-bound pool even if BBB permeability to the drug (k3, k4) was increased 1,000-fold over experimentally observed values.

The observation, from several separate laboratories, is that the KD^{in vivo}, within the brain capillary compartment, is often times much greater than the KD^{in vitro} [34]. The KD is a measure of ligand affinity for the protein, and this affinity is generally dictated by rates of ligand dissociation, which are a function of the conformation of the protein binding site [51, 52]. Both albumin and AGP undergo conformational changes upon contact with biomembrane surfaces [53–56]. Within the brain capillary, the major surface exposed to circulating plasma proteins is the endothelial glycocalyx (EG). The EG is rich in proteoglycans and glycosaminoglycans [57], which bind albumin and AGP [58–60]. The EG is 400 nm thick at the brain microcirculation [61, 62]. Therefore, the thickness of the EG at the brain capillary endothelium is greater than the thickness of the brain endothelial cell, which is about 300 nm [63]. The EG covers about 40% of the surface of the endothelium in the brain, as compared to only 4% of the endothelial surface in the lung [62]. Potential interaction between the EG and plasma proteins is rarely discussed in investigations of brain delivery of drugs bound to plasma proteins.

The investigations demonstrating an increase in the KD^{in vivo}, relative to the KD^{in vitro}, of drug binding to albumin or AGP have been conducted with human albumin and human AGP [34]. There may be important differences in drug binding to plasma proteins from non-human species, particularly rats and mice. The amino acid sequence of rat serum albumin (RSA, AAH85359) or mouse serum albumin (MSA, AAH49971) is 72–73% identical to the sequence of human serum albumin (HSA, AAA98797). Owing to these sequence differences, the diazepam binding site on HSA is not present on RSA [64], and the affinity of RSA for diazepam is 30-fold lower as compared to diazepam binding to HSA [65]. The amino acid sequence of rat AGP (NP_445740) or mouse AGP (NP_032794) is only 49–50% identical to the sequence of human AGP (AAA35515). In addition, the plasma concentration of AGP in the rat [66] or mouse [67] is more than tenfold lower than the concentration of AGP in human plasma, which is about 1 mg/mL [12, 68]. There is little information on the extent to which drugs such as propranolol or imipramine, or other CNS drugs, are bound to rat or mouse albumin or to rat or mouse AGP, as compared to the human proteins [69].

The principal lines of evidence against plasma protein mediated uptake are experiments with cerebral microdialysis, which are generally performed in rats or mice. The unbound drug in the brain microdialysate corresponds to the unbound drug in plasma measured in vitro in the case of brain microdialysis of diazepam [26], imipramine [43], or propranolol [70]. These studies in the rat are assessments of drug binding to rat albumin or rat AGP. Apart from the role of species effects in plasma protein binding of drugs, studies with microdialysis are confounded by the brain penetration injury and brain ischemia induced by the implantation in brain of a dialysis fiber [71–73]. Brain ischemia is induced in the region around the fiber, because the diameter of the dialysis fiber, 200–600 microns, is much greater than the inter-capillary distance in brain, which is about 40 microns [74]. The IV administration of fluorescent 100 nm microspheres shows no perfusion of capillaries in the region contiguous with the dialysis fiber implanted in brain [71]. The fluid within the dialysis fiber implanted in brain may be in equilibrium with the extracellular space (ECS) up to 2,000 microns removed from the dialysis fiber for a poorly diffusible compound confined largely to the ECS [75]. However, for molecules such as lipophilic amine drugs, which are highly diffusible across cell membranes and are sequestered within brain cells, the dialysis fiber may be in equilibrium with brain ECS up to only 100–200 microns removed from the dialysis fiber [75], a region within the ischemic zone surrounding the fiber. Both brain injury and/or brain ischemia cause rapid shedding of the EG at the brain capillary within 60 min of the insult, and this shedding of the EG in brain persists for at least 7 days [76–78]. If the implantation in brain of a dialysis fiber causes shedding of the EG and if PMU is initiated by plasma protein interaction with the EG, then no PMU may be detectable with cerebral microdialysis.

The PBPK model used in this investigation also examines drug binding/sequestration within the brain compartment (Fig. 1). Drug sequestration in brain produces a K_p,brain value>1, which is the case both for propranolol, where the K_p,brain is 10 [39], and imipramine, where the K_p,brain is 23 [40]. The brain sequestration of lipophilic amines, such as propranolol or imipramine, which have high pKa values, and which are protonated at physiologic pH, may be due binding to anionic phospholipids, sequestration within the acidic compartment of the lysosome, which constitutes 1–3% of the cell volume [79], or binding to brain microsomal or mitochondrial proteins. The unbound volume of distribution, V_u, of propranolol and atenolol in brain slices is 112 mL/g and 2.5 mL/g, respectively [80]. Since both propranolol and atenolol have a pKa of 9.5–9.6 [81], the binding of either drug to acidic phospholipids, or lysosomal entrapment, should be comparable. The 45-fold higher V_u for propranolol as compared to atenolol in brain slices [80] suggests the major mechanism for the selective sequestration in brain of propranolol, as compared to atenolol, is binding to brain tissue proteins. The absence of atenolol binding to brain proteins is confirmed by the f_u,brain=1.0 in brain homogenate, and the low K_p,brain of 0.04 for atenolol [82]. In contrast, the f_u,brain for propranolol or imipramine is low 0.01–0.02 in brain homogenate [83], and both propranolol and imipramine are bound, via a saturable mechanism, by tissue microsomal proteins [84]. An early study showed the saturable binding of propranolol to brain tissue proteins was equally distributed to the microsomal and mitochondrial fractions of brain cells [39], and these subcellular compartments comprise 73% of the total cellular protein [85]. Drug sequestration in tissue homogenates in vitro is inversely related to the f_u,tissue, and the sum of f_u,tissue in multiple organs is predictive of the systemic volume of distribution [86, 87].

Drug binding to brain tissue proteins is directly related to K_p,brain, and this is demonstrated in Fig. 2, where the brain concentration of protein-bound drug, PL, is directly related to the total brain tissue protein concentration, PT. The data in Fig. 2 also shows that the concentration of the free drug in brain, LM, or C_u,brain, is independent of the PT concentration. The independence of the free drug in brain on drug binding to tissue proteins is predicted by Eq. 7 (Methods), which shows, in the absence of drug metabolism, where k9=0, that the free drug in brain, LM, is a function only of the bioavailable drug in plasma, LF, and bi-directional BBB transport. Equation 7 shows the LM in brain is proportional to the concentration of the free (bioavailable) drug in plasma, LF. The LM/LF ratio, or C_u,brain/C_u,plasma, constitutes the k_p,uu,brain parameter (Methods), which, similar to C_u,brain, is also independent of drug partitioning in brain, such as binding to tissue proteins. However, the k_p,uu,brain parameter is typically measured in vitro by equilibrium dialysis (ED) of aliquots of plasma, for determination of C_u,plasma, and brain homogenate, for determination of C_u,brain. The C_u,brain measured in vitro by ED of brain homogenate is inversely related to drug binding to brain tissue proteins, but the C_u,brain in vivo is independent of drug binding to brain tissue proteins (Fig. 2). In vitro measurements of C_u,plasma by ED will underestimate the unbound (bioavailable) drug in plasma in brain in vivo, if PMU takes place at the brain endothelial interface.

Conclusion

The PBPK model described in this work examines brain delivery of plasma protein bound drugs by simulating the effects of changes in multiple kinetic parameters on drug binding to albumin and AGP in vivo in the brain capillary compartment, and to tissue proteins in the brain compartment (Fig. 1). The results of this study highlight the limitations of the in vitro measurement of the free drug in plasma or brain by equilibrium dialysis of plasma or brain homogenate, respectively. Equilibrium dialysis of plasma will under-estimate the free drug in brain if there is significant PMU of the plasma protein-bound drug. Equilibrium dialysis of brain homogenate is a function of drug binding to tissue proteins in vitro, whereas the free drug in brain in vivo is independent of drug binding to tissue proteins. The free drug in brain is a function of the free (bioavailable) drug in the brain capillary compartment, bi-directional BBB drug transport, and brain metabolism of the drug.

Supplementary Information

Declarations

Footnotes

References