MMID4 Overview

1.1. What Is MMID4?

MMID4 is an acronym for Multiple path, Multiple tracer, Indicator Dilution, 4 region model. It is a simulation of blood-tissue exchange that can be used to examine the behavior of three types of tracers: one that remains in the vasculature, one that can leave the vasculature but remains extracellular, and a fully permeant tracer that enters cells. The model can be used to study the physiology of the exchange process or as a tool to help analyze actual experimental data. The exchange process may be examined by viewing the instantaneous outflow concentration, the instantaneous extraction, or the amount of tracer remaining in the exchange system (i.e., the residue function).

The model provides ways to examine tracer delay and dispersion between the injection site and the target organ and the effects of flow heterogeneity on exchange within the organ. Fig. 1.1 shows the general layout of the model. Each of the multiple tracers flows through the common arterial nonexchanging vessels, the parallel pathways each containing an arteriole, exchange unit, and venule, and then through the common venous nonexchanging vessels.

Figure 1.1. MMID4 provides up to 20 pathways to account for tissue heterogeneity. Flow enters each pathway after having passed through optional large vessel components. Each pathway includes optional small vessel components and one exchange unit. Large and small vessel components account for delay and dispersion of the input function before it reaches the exchange unit.

In formal notation, the outflow concentration-time curve seen from an exchange unit in response to an impulse input of tracer is denoted as h(t). We use h(t) as a general symbol to denote the normalized (unity area) outflow dilution curve even when the input has forms other than an impulse. A subscript is added to designate specific tracers. Thus, h_D(t) denotes the "permeating" or "diffusible" tracer; h_E(t) signifies the extracellular tracer; and h_R(t) is used to represent the intravascular "reference" tracer.

1.2. Vascular Components

1.2.1. Nonexchanging vessels

The nonexchanging vessel operators

The nonexchanging vessels in MMID4 are modeled by vascular operators (King et al., 1993). Each nonexchanging vessel is completely described by its volume and relative dispersion (RD). If a particular operator is not desired in a given simulation, it can be effectively removed by setting its volume to zero. Although the nonexchanging vessels are normally used to describe vascular delay and dispersion, they can also describe pure delay (no dispersion) by setting RD to zero.

Arterial vessels

Three types of arterial vessels are shown in Fig. 1.1. The first is the input tubing. This models a cannula or catheter through which tracers are injected. While not strictly an arterial vessel, it is upstream of the organ and can be described in the same way as the nonexchanging vessels, by its volume and RD.

The second type of arterial vessel, downstream from the input tubing, is the artery. Tracer flowing out of the artery is distributed to the various flow paths where it enters the third type of arterial vessel, the arterioles, and then the exchange units. The distribution of flow among the pathways is controlled by the flow heterogeneity parameters described in Section 3.3.2.

Venous vessels

Fig. 1.1 shows that the venous vessels in MMID4 are similar to the arterial vessels. Each exchange unit drains into a venule. All venules drain into the common vein, then into the output tubing.

1.2.2. Exchange units

The blood-tissue exchange unit (BTEX) operator

The basic tissue unit in which exchange takes place, shown in Fig. 1.2, consists of four regions: an intravascular space, the endothelium, the interstitium, and the parenchymal cells. Note that the interstitium (ISF) is split into two parts. The first, ISF₁, lies between the endothelial and parenchymal cells. The second, ISF₂, is connected to ISF₁ and the parenchymal cell. This is considered a four region model despite the split of the ISF into two parts.

Figure 1.2. Basic exchange unit in the MMID4 model with four regions in which exchange can take place.

Flow and exchange within the capillary is modeled by a Lagrangian sliding fluid element algorithm, shown in Fig. 1.3. The exchange unit is divided into a series of axially distributed segments. In the first time step, exchange between the capillary and surrounding tissue is calculated. At the beginning of each subsequent time step, the fluid in the capillary is advanced one segment, and exchange is allowed to occur again. The rationale for this approach has been discussed by Bassingthwaighte (1974) and Bassingthwaighte et al. (1989). The model provides for up to 60 axial segments.

Figure 1.3. Lagrangian sliding element algorithm for blood-tissue exchange: t = V_p/ (F_p · Number of segments). With each t, the capillary contents slide instantaneously one segment downstream, after which the radial transmembrane and axial diffusional exchanges, and the consumption, are computed.

Intravascular tracer exchange units

The parameters affecting the intravascular tracer are shown in Fig. 1.4.

Figure 1.4. Parameters affecting the intravascular tracer. In this case, the PS product of the gap (PS_g) and apparent volume of distribution in the interstitial space (V´_isf) are included to simulate receptor binding on the endothelial surface. See text for details.

The volume of distribution within the intravascular region is the plasma volume (V_p). The tracer can undergo axial diffusion (D) and may also be bound to receptors on the endothelial surface. Such binding, if reversible, may be simulated by allowing movement into the interstitium similar to the movement of the extracellular tracer. Hence, PS_g and V´_isf are also included as parameters for the intravascular tracer. The plasma flow, F_p, and plasma volume, V_p, are the same parameter for all three tracers.

Extracellular tracer exchange units

The parameters affecting the extracellular tracer concentrations are shown in Fig. 1.5. The extracellular tracer has an apparent volume of distribution in the interstitial space (V´_isf) and in the plasma (V_p). It can undergo axial diffusion in the intravascular and interstitial regions (D). It can only move between the two regions through the interendothelial cell clefts. Hence, its movement depends on only one PS product.

Figure 1.5. Parameters affecting exchange of the extracellular tracer. See text for details.

Permeant tracer exchange units

The parameters affecting the concentration of the permeant tracer in each region are shown in Fig. 1.6. Because this tracer can move to all regions, it has an apparent volume of distribution (V) in all regions. In the intravascular region, this is the plasma volume (V_p). Also in each region, the tracer has the potential to undergo axial diffusion (D). The tracer may be metabolized or undergo chemical reactions that result in its being cleared from the region. The degree to which this happens is indicated by the intraregion consumption or "gulosity" (G). Movement of the tracer between regions is governed by the permeability-surface area product (PS) of the various barriers. A molecule of tracer may move, for example, between the capillary and the interstitium. Two routes are available; it can either move through the clefts between the endothelial cells, or it can move through the endothelial cell. In the first case, movement depends on one PS product (PS_g). In the second, movement depends on the PS product at the luminal side of the cell (PS_ecl) and the PS product on the abluminal side of the cell (PS_eca).

Figure 1.6. Parameters affecting exchange of the permeant tracer. F_p and V_p are the same for all tracers. See text for details.

1.3. Additional Features

1.3.1. Input function generation

General

MMID4 permits the user to generate an input function in a number of ways. These are discussed briefly below. For a given simulation run, the user selects one of the available methods. Generation of the input function is discussed in detail in Section 3.2 of this manual and is the subject of exercises in Section 2.2.2.

Function generator

MMID4 gives the user access to a generalized function generator. It will generate a variety of functions including several types of pulses and pulse trains and a number of density functions. This generator is named cinput; additional information about it can be found in the UNIX online manual page.

Curves from reference data

If the user specifies that reference data is to be loaded by XSIM, curves from these data can be used as the input function for one or more of the tracers.

Deconvolution

When there are reference data that contain an outflow curve for an intravascular tracer, the user can direct MMID4 to deconvolve those data with the model transfer function and use the result as the input function.

1.3.2. Heterogeneity

General

When the user configures MMID4 to use more than one flow path, the flow paths may have different flows, membrane conductances, and tracer consumptions. These heterogeneities are used to simulate tracer transport and exchange in a whole organ. Heterogeneity is discussed in Section 3.3 of this manual and is the subject of exercises in Section 2.4.3.

Flow

In a multipath configuration of MMID4, the flow for each path must be specified. These flows can be specified by using a probability density function, by a curve in the reference data, or by the user entering the flow for each path.

Membrane conductance

When multiple paths are specified, the user can specify that the conductances of the capillary wall and endothelial cell for each path be constant or proportional to the flow in that path.

Tracer consumption

As with membrane conductances, the user can specify that the consumption of tracer in the endothelial cells and parenchymal cells be proportional to flow.

1.3.3. Parameter tools

Parameter Evaluation

In certain instances, it may be useful to have several parameters have the same numeric value (e.g., the capillary permeability for the three tracers). Parameter expression evaluation can link the value of "slave" parameters to that of a "master" parameter. Changing the value of the master automatically changes the values of the slaves. Note: a parameter, including scalars, may not be slaved onto another parameter that is already slaved to a parameter or constant.

Two types of parameter tools are provided in MMID4: The eval field for each parameter, and a set of general scalars. The former allows a parameter to be set to equal a function of one or more other parameters.

Parameter scalars

While MMID4 has a wide range of input parameters, situations arise in which a user wants to use a parameter that is not provided by the program. General scalars are provided for this purpose. These scalars allow the user to create a new result parameter that is a linear combination of other parameters. The set of scalars are simply a set of parameters that are not tied to the model, but can be set like any other parameter, either by entering a value directly or by slaving the scalar to another parameter in the Eval field.

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