Análisis por ítem de choferes que transitan con sus vehículos por el

In order to construct a model, the biology of IgG was reviewed, then mass-balance principles were applied. It was constructed without considering constraints on measur- able variables or availability of known parameters; assumptions will be applied later to produce a more tractable model for simulation. A schematic of the full kinetic model is shown in Figure 5.2.

On first inspection the model is clearly symmetrical as the IgG Kappa (IgGK) and IgG Lambda (IgGL) proteins are treated identically by the metabolic processes. All compartments labelled K represent the quantity of IgG Kappa, whilst L denotes the quantity of IgG Lambda. The model was initially constructed by considering the flow of IgG in plasma. It is known, through radio-nuclide marking [Waldman and Strober, 1969], that IgG moves between plasma and extravascular (EVF) pools. Therefore, two compartments were created for each IgG type (K1,L1 representing plasma and

K2,L2 EVF quantities) with two linear rate constants (k12 and k21 describing the flow

across the plasma membrane. The flow between plasma/EVF is considered to be the same process as EVF and plasma; it is therefore possible to define this exchange by a single parameter, with respect to the volume ratio between the two compartments

Section: 5.2.2 92

Figure 5.2: IgG full model. IgGK (K) and IgGL (L) with the subscripts i, i.e Ki and Li representing the following compartments: 1) Plasma, 2) EVF, 3) Free in the endosome, 4) bound in the endosome, and FcRn) bound to FcRn in the endosome.

from radiolabelling experiments Waldman and Strober [1969] have demonstrated that for IgG the volume of distributions are approximately equal (v1 ≈v2). The synthesis of

intact IgG is controlled by two time-dependent functions: PK(t) (for IgGK) andPL(t)

(for IgGL).

Both IgGK and IgGL are continuously exchanged between plasma and the endo- some through pinocytosis. This is represented by the linear rate constant k31. TheK3

and L3 compartments represents free IgG in the endosome, from which IgG is either

cleared from the system, rate constant k03, or bound to FcRn (bound IgG is repre-

sented by K4 and L4). The quantity of free receptors in the endosome is represented

by the F cRn compartment. FcRn receptors are also catabolised in the endosome; this is represented by the clearance of FcRn (k0F) and synthesis of new receptors by the

input function, PF(t). A concise list of the parameters and model nomenclature can

be seen in Table 5.1.

Through the model the clearance mechanism of IgG can be easily described. The IgG is allow to move freely between plasma and EVF (k12 and k21) but is passed into

Table 5.1: IgG and FcRn competitive binding parameters description.

Variable Description

PK(t),PL(t) and PF(t) Production of IgGK, IgGL and FcRn (mg min−1)

Ki orLi Quantity of Kappa or Lambda in compartment i (mg)

F cRn Quantity of FcRn in the Endosome (mg)

yi Observation of compartment i, ci will be used as obser-

vation gain, in all cases this _v1 i

kij Rate Constant of flow of material from compartment j

to compartment i(min−1₎

k0j Rate Constant removal of material from compartment j

(min−1₎

ka Association rate of IgG and FcRn receptor (mg min−1)

kd Dissociation rate of IgG and FcRn receptor (min−1)

vi Volume of compartment i (1=plasma, 2=extra-vascular

and 3=endosome) (L)

and out of the endosomes via pinocytosis (k31 and k14). If an individual IgG is bound

to an FcRn receptor it is returned to plasma, whilst if it is not bound it is eliminated from the endosome (k03).

The system of equations generated from the model (Figure 5.2) can be seen in equations (5.2). Only the equations for IgGK, with the equation for free FcRn recep- tors, are shown but due to the symmetry in the model, the IgGL variant can be derived by replacing K for L in each case.

˙ K1(t) =−(k21+k31)K1(t) +k12K2(t) +k14K4(t) +PK(t) ˙ K2(t) =k21K1(t)−k12K2(t) ˙ K3(t) =−kbK3(t)F cRn(t) +kdK4(t)−k03K3(t) +k31K1(t) ˙ K4(t) =kbK3(t)F cRn(t)−kdK4(t)−k14K4(t) ˙ F cRn(t) =−kb(K3(t) +L3(t))F cRn(t) +kd(K4(t) +L4(t)) −k0FF cRn(t) +PF(t) (5.2)

Section: 5.3.0 94 The association constant (ka) in equation 5.2 has been replaced by kb, wherekb = ka

v3, to convert the concentrations as shown in equation 5.1 and quantity used above. On clinical presentation, prior to treatment, it is assumed that the patient’s tumour has produced sufficient IgG to saturate all compartments and receptor sites; with the system therefore in steady-state the initial conditions are (once again only IgGK examples):

K1(0) = CK10v1 K2(0) = CK20v2

K3(0) = CK10v3 K4(0) = CK40v3

F cRn(0) = F cRn0 v3

whereCK10represents the concentration of the IgGK in plasma prior to any chemother-

apy treatment. Due to the steady-state assumption the concentration difference be- tween plasma and EVF has equilibrated; in addition, as pinocytosis is a sampling process it can be assumed that IgG free in the endosome is of equal concentration to that in plasma.

Whilst in a restricted laboratory experiment it may be possible to take measure- ments relating to numerous aspects of the above model, the expectation of this mod- elling study is that it is used in a clinical environment and as such the only mea- surements available are those that can be taken in vivo and hence, are restricted to concentrations in plasma allowing the following measurements to be taken:

yK1(t) = c1K1(t),

yL1(t) = c1L1(t) or

yΣ(t) = c1(K1(t) +L1(t)).

(5.3)

wherec1 represents an observation gain. It is possible to observe IgGK (yK1) and IgGL

(yL1) individually, and the total IgG present (yΣ). As with FLC it is assumed that all

IgGK and IgGL are captured by the measurement technique and as such equates c1 to