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ILAR Journal V38(2) 1997
The Role of Computational Models in Animal Research
David M. Foster and Ray C. Boston
| David M. Foster, Ph.D., is with the Center for Bioengineering, University of Washington, Seattle. Ray C. Boston, Ph.D., is with the Department of Clinical Studies, School of Veterinary Medicine, University of Pennsylvania. |
INTRODUCTION
Kinetic analysis and integrated systems modeling have contributed significantly to our understanding of the physiology and pathophysiology of metabolic systems in humans and animals. With the advent of personal computers and scientific software packages, a growing number of experimental biologists are using mathematical modeling techniques in their experimental design. At the same time many of these users recognize that the discipline of kinetic analysis requires its own expertise in order to develop and maintain state-of-the-art modeling tools in such a way that the power of modeling is maximized.
With the development of sophisticated systems models, questions are arising about the boundary between the use of these models and laboratory experimentation. This is particularly important when an experimental design calls for laboratory animals as models for human physiology and pathophysiology. These questions arise primarily because the development, testing, and application of mathematical models are poorly appreciated. In particular, robust mathematical models obtained by using the power of mathematics to bind together diverse pieces of physiological information lead to the formulation of specific hypotheses that can only be tested through further experimentation. Model development and laboratory experimentation is thus a cyclical process.
Against this background, the purpose of this article is to describe the modeling process and the purposes for which models can be used. Examples will be given to illustrate specific situations. In this light, the limitations of a model' s utilitarian values can be discussed, and the boundary between simulating and performing experiments will become clearer. We will illustrate 2 case studies in which laboratory experimentation and modeling are used together: one uses an animal model for a human disease (atherosclerosis), and the other uses an animal model to understand the physiology of that particular species (water metabolism).
MATHEMATICAL MODELS OF BIOLOGICAL SYSTEMS
Mathematical modeling of biological systems is becoming increasingly important in experimental design. To understand why this is the case, we must first define what we mean by a model and a mathematical model and what can be expected from such models. In this section, we draw much of the material from Berman (1963a; 1963b; 1964).
Models and Mathematical Models
A model is a construct invented as an aid to understand a specific system under study, and possibly the relationships of that system with other systems. It is a formal statement of assumptions, conceptualizations, and experimental design.
Consider, for example, a model for facilitated diffusion. Facilitated diffusion is a non-energy-requiring transport process whereby a substance, in order to enter or leave a cell, must interact first with a carrier protein. Once attached to the carrier protein, movement into or out from the cell is "facilitated'' by the carrier protein. Facilitated diffusion is the mechanism by which glucose, for example, enters red blood cells.
A model of this system can be given diagrammatically in Figure 1. The model is called a 2-state model because the substance can be transported in both directions by binding either to the carrier protein on the outside or inside of the cell membrane. In the figure, the carrier protein C is denoted by the triangular shape spanning both sides of the cell membrane (denoted by the filled bar line); this carrier protein can move its active site to either the inside or outside of the membrane as denoted by the solid, 2-headed arrow. The substance of interest, S, can interact with the carrier protein at either surface; this is indicated by the dashed, 2-headed arrow. In this way, we have schematized the movement, which is possible in both directions, of the substance once it has associated with the carrier membrane. It is a model for facilitated diffusion since it represents schematically the processes involved.
A mathematical model is a model that can be described by a set of equations. Clearly there is a difference between a conceptual model and its mathematical expression. A conceptual model is simply a schematic of how a system might work. A mathematical model takes such a schematic and tries to describe the various processes by equations. The mathematical model thus marries the precision of mathematics with the vagaries of biology. We will have more to say about this in a moment.
Suppose one were to describe the processes illustrated in Figure 1 in mathematical terms. Then using binding kinetic notations, one could redraw the figure as illustrated in Figure 2.
Notice in moving from Figure 1 to Figure 2 we have put more details on the movements that are actually occurring. For example, the movement of free carrier protein between both sides of the membrane, denoted Co, and Ci respectively for the protein at the outer and inner face, is diagrammed by the arrows labeled k2 and k-2. The binding of the substance outside the cell, So, to the carrier on the outside surface, Co, forms the carrier protein-substance complex, CSo and is represented by the arrow k1. That is, the dotted line labeled k1 indicates the reaction So + Co ® CSo.
Using standard binding kinetic equations, one can write a system of differential equations describing the movements depicted in Figure 2. These are:
The terms on the left-hand side of the equations, for example dCo/dt describe the rate of change with respect to time (in this case of Co,) Co and Ci are the concentration of free carrier at the outer and inner surface of the membrane, So and Si are the concentrations of the substance outside and inside the cell, [CS]o and [CS]i are the concentrations of the carrier-substance complex at the outer and inner surface of the membrane, and T = Co + Ci + [ CS]o + [ CS]i is the total carrier concentration.
At this point, we have taken the model shown in Figure 1 and, using Figure 2, have translated the model into a mathematical model. The question is: for a specific experiment on a specific system, will this model describe the data? Since Co, Ci, So,, Si, [CS]o, and [CS] represent what can be measured experimentally, this question can be restated: can estimates of the parameters ki and k-i be obtained so that the solution of the differential equations given above will predict the data?
No matter how appealing this model is, experimental data is needed to verify that it can describe the system under study. More specifically, no matter how a mathematical model is formulated, it is characterized by unknown parameters. Numerical values for these parameters must be provided before the model's equations can be solved. There are 2 ways to estimate these parameter values. One way is to estimate them arbitrarily; in this case, one does not know whether they bear any resemblance to "reality" or not. The other way is to estimate them from data. This requires, as noted above, an experiment, and in the end is the only way to validate the physiological meaning of the parameter estimates.
What, then, is the purpose of a mathematical model? This depends upon an individual investigator's needs and how modeling is used in experimental design. Among the purposes, we list the following.
To describe a set of experimental data and estimate desired parameters. One purpose of a mathematical model is to describe a set of data. That is, one seeks a particular set of values for the parameters characterizing a model, which will produce model-calculated values that agree closely with the experimental data. These parameters can then be used to calculate other metabolic parameters of interest such as volumes, production rates, and clearance rates. When a model is fitted to data, error estimates for parameter values can be obtained along with their correlations. This enables statistical comparisons between different study groups.
Many experiments are designed as single input-single output studies. That is, there is a single input into the system (such as a drug administration) and a single measurement (such as plasma concentration of the drug). It is known that there are a standard set of methods available to estimate many parameters of interest. These methods have been validated using a combination of mathematical modeling and experimentation.
To characterize more fully the system under study. On the other hand, a richer design can call for a single input-multiple output, or multiple input-multiple output study; both yield more than one set of data since there is more than one sampling site. Such studies yield rich data sets that normally require an integrated model to interpret. Unfortunately many investigators do not realize this potential, and so what we call "free information in the data" is disregarded. Thus one of the purposes of mathematical models is to characterize more fully the system under study by developing a model that simultaneously describes all the data and concomitantly incorporates known physiology.
To simulate experiments. Mathematical models can be used to simulate experiments. This is, in fact, one of the most important yet poorly understood function of such models.
If one wishes to test a hypothesis about how some aspect of a system functions, and if one already has a mathematical model describing this system, one can simulate the proposed experiment to test a priori if the data generated will be sufficient to test the hypothesis. One can modify the data collection times, the duration of the experiment, and inputs and outputs that are necessary to enhance success.
This can and should be done prior to performing a pilot or series of pilot studies. The combination of simulating experiments and performing pilot studies that can be analyzed using the model greatly enhances the probability of success and often saves time and money.
It should be emphasized, however, that simulating experiments can never take the place of actually performing them.
How does one choose a particular model or modeling methodology? The choice depends upon several factors.
The information desired from the experimental data. When an experiment is performed to test a hypothesis, certain kinds of information are required from the data in order to test the hypothesis. Sometimes this information consists of standard parameters that can be calculated easily; an example would be the clearance of a drug from the body. However, each of the formulas used to calculate these parameters is based upon a set of assumptions, and the system under study must satisfy these assumptions in order for each calculated parameter value to be valid. Complex models can be used to justify the application of simpler formulas, but these too require experimental validation.
The experimental design. Biological systems are very complex, and no one experiment can be designed to generate data detailing every facet of a system. The reason is that there is a wide range of times for metabolic events and transport that occur in the body.
For example, suppose one wanted to study glucose metabolism. The catabolism of glucose, from the biochemical point of view, is via glycolysis. The first step is phosphorylation of glucose, which can be reversed in the liver by hydrolysis of glucose-6-phosphate. One can certainly study glucose metabolism at this level.
On the other hand, suppose one wanted to study glucose production and catabolism in more general terms. In this case, a study would need to be designed in which some of the finer details of glucose metabolism are lumped together. Some specifics would be lost without affecting the estimates of the desired parameters.
One way to address this problem is to use tracer-labeled glucose. One can, for example, purchase glucose that has had one of its carbon or hydrogen atoms exchanged with radioactive 14C or 3H. When a small amount of this is injected into plasma, serial plasma samples can be taken and the amount of radioactivity quantitated. Modeling can be used to interpret these data (Figure 4 below).
Therefore experimental design plays a role in terms of the postulation of a model structure and the kind of information one can obtain from the data as interpreted by a model.
The complexity of the system under study. While an experiment is usually designed to estimate certain parameters of interest, the formulas used depend themselves upon a model. Very simple formulas are often derived from very simple models. A common parameter estimated from certain kinds of data is the fractional catabolic rate; this parameter estimates the fraction of a substance irreversibly removed from the system per unit time. Very simple formulas exist to estimate this parameter. However, in more complex systems, the assumption upon which the formula is based may not be satisfied. One needs, therefore, to be aware of the complexity of the system under study, and incorporate this complexity in the model structure.
A benefit of this exercise can be to justify the use of the simple formulas. That is, most of the common metabolic parameters can be estimated from a model of any complexity. In some cases, they can be shown to coincide with the parameters estimated from simple formulas.
Theoretical considerations. One can postulate a model structure and the equations describing it in very complex terms. However, it must be possible to solve the equations. In some cases, particularly when a model is specified using partial differential equations with a rich parameterization, the solutions are difficult if not impossible. It is therefore necessary because of theoretical considerations to use a simpler model formulation. This area of modeling is normally reserved for those with expertise in mathematics or statistics.
Suppose one is studying a system for which no mathematical model exists. What are the steps involved in the model development and testing process? The answer to this question is very complex and in fact is a discipline in many bioengineering or biomathematics programs at universities. Below we list some of the steps.
Formulate a model structure. Formulating a model structure is not unlike the process we went through in describing the model for facilitated diffusion in Figure 1. One first identifies the components of the system and then specifies their interconnections. The interconnections usually represent biochemical transformation or physical transport. One needs to incorporate as much known physiology and biochemistry as possible in this structure recognizing there may be a need to combine some of the processes.
Choose the equations and specify the parameters. Two sets of equations are usually required. First, there are the equations that characterize the components and their interconnections, and hence will be determined by the nature of the system being studied. They will be based on known or hypothesized physical chemistry and will be characterized by values of parameters that must be supplied in order to obtain a solution.
It is possible that a set of equations may be specified for which there is no solution (in contrast to a system that is difficult to solve); this can happen for a variety of reasons most of which are theoretical. In this case, simplification must be made. However, it must be recognized that simplifications are in a sense hypotheses that must ultimately be tested.
A second set of equations is required when experimental data are involved. These are the measurement equations. As the link between the model equations and the data, they ensure that the units between the two are consistent. Normally parameters such as volumes or masses are required for these equations, with the parameters characterizing the model equations.
Solve the equations (simulation). When the model is fully specified, that is, when the equations have been written and parameter values assigned, the equations can be solved. This is called a simulation. One can then examine the predictions of the model and compare them with experimental data.
The best one can hope for is compatibility between the model predictions and the data within some statistical tolerance (Berman 1963a, 1963b). Compatibility, however, does not mean the model is correct. It does mean that it can be used to estimate parameters of interest such as production and catabolic rates. "Correctness" comes through the validation process as described below, and one quickly learns that no model is ever totally "correct" because experimental methodology will eventually produce knowledge that will force a change in the model structure.
Incompatibility between the model predictions and the data means the model is not correct. More specifically, it means there is one or more incorrect components or inter-connections. Since the postulation of a model is equivalent to the postulation of a hypothesis about how the system works, this is valuable information since it will force a change in the model formulation. In fact it is during this part of the model development process that the modeler learns the most about the physiology of the system under study. In the model development process, one easily learns from the errors.
Identify and estimate parameters. As noted, a model is specified by a set of parameters. The question arises as to whether the parameters can be estimated with a predefined degree of statistical precision from a particular set of experimental data. The problem can be posed in two ways.
A priori identification addresses the issue as to whether the parameters describing the model can be estimated from a set of ideal data where ideal data are continuous, noise-free time measurements. If they cannot, there is no way they can be estimated from a set of real data. A priori identifiability is a very difficult problem to solve in the general case (Carson and others 1983).
A posteriori identification addresses the question as to whether the parameters can be estimated from a set of experimental data. Normally one uses a computer software package that contains at least one optimization routine (see SAAM 1995). Since most biological models are nonlinear, that is, parameters characterizing them are nonlinear, the optimization itself is nonlinear and thus is an approximation (see Bates and Watts [1988]).
When one successfully optimizes the model parameters to obtain a best fit to a set of data (that is, the optimizer in the software package being used moves the parameter values from their initial estimated values to a set of values so that a "best fit" to the data is obtained), statistical information is returned to the user. This contains information on the estimated standard deviations of the parameters and the correlations among them. From this knowledge, one can assess the a posteriori identifiability of the model by using preset criteria for acceptance of the estimated standard deviations and correlations.
It is important to point out that optimizers in different software packages can act differently. More particularly, if a model has more parameters than can be estimated from the data, some software packages will return no statistical information. The reason why this happens depends both upon the particular optimization algorithm chosen and how it is implemented numerically. The crucial point is the following: When a model is overparameterized and statistical information is returned, the researcher can tell by examining the coefficients of variations of the parameters and the correlation coefficients which parameters cannot be estimated. If this problem is discovered early enough, the experimental design can be modified so that sufficient data are collected; mathematical tools such as optimal sampling design (Carson and others 1983) are available to help. In the event that no such modification is possible, the model structure may have to be changed, or specific constraints will have to be imposed on some parameters.
Search for other compatible structures (model uniqueness). Just because one has postulated a model structure that is compatible with the data does not mean this is the only such model. It is incumbent upon the investigator to search for other model structures that are also compatible. If more than one can be found, then an experiment must be designed to distinguish between the different structures. An example of this situation can be found in Foster and others (1986) where 3 different models describing low density lipoprotein metabolism were formulated, all of which had different physiological meanings.
Simplify (minimal models). What is a "minimal" model? Mathematical models are characterized by unknown parameters. By a minimal model, we mean a mathematical model with the fewest parameters needed to describe the data, that is, the simplest structure. Determining this number requires statistical knowledge in terms of testing for goodness-of-fit and model order (the number of parameters in a model).
Clearly the notions of model uniqueness and minimal models go hand-in-hand. It is very rare that one can postulate a model structure that has all the characteristics of the data on the first try. Thus one starts with a simple structure and adds complexity based upon known physiology and biochemistry until such a structure is found. The parameters of the model can either be estimated with acceptable precision or they cannot. If they can, then as noted above one must explore other plausible model structures to see if they can describe the data equally well. If there are none, then one can accept the model as a minimal model .of the system. If, on the other hand, there are other model structures of the same model order that describe the data equally well, experiments must be designed to distinguish among the acceptable structures.
If the parameters cannot be estimated with precision and further experiments are not possible, then the model structure must be simplified again in accordance with known physiology and biochemistry. This may mean some processes are lumped together because in the time frame of the experiment they cannot be separated. In the process of model reduction, when an acceptable model is found, then the steps described above must be followed before concluding a model is a minimal model.
Validate the model (testing). When a model has been postulated that is compatible with a set of data, it can then be validated. Validation usually involves a perturbation to the system under study, that is, performing a different experiment on the system in a different physical state to see if the model can predict the outcome from this new experiment (see Carson and others [1983]). Validation in this sense refers to a model's ability to predict the behavior of a system under conditions different from those from which it was derived.
Examples of such perturbations include pathophysiological conditions (that is, comparing a normal situation with a group having a specific disease) or intervention studies (such as administering a drug or dietary therapy to control a pathophysiological condition).
Examples
Many different mathematical modeling methodologies have been used in biomedical research. One that is commonly used and that has produced many useful results over the years is compartmental modeling. Compartmental models, technically, are schematics representing systems of ordinary differential equations (Jacquez 1985, 1993). This, however, is not the point in the present discussion. What we wish to do is give some background into compartmental models and what, from a modeling perspective, they mean. We will then give some actual examples.
The kinetics of a substance refers to its temporal and spatial distribution in a system such as the human body. However, it is impossible to follow every molecule of a substance all of the time. It is therefore necessary to discretize the system, that is, to lump various parts of the system together into a finite number of entities. Compartments and compartmental models accomplish this.
A compartment is an amount of material that is kinetically homogeneous, well-mixed, and distinct from other material in the system. The easiest compartment to think about is plasma; other examples could be red blood cells or liver cells. Glucose and lactate could be two compartments, both of which are in plasma. This lumping process will reduce the system to a finite number of compartments. Material can then flow into or out from a compartment, or be exchanged between compartments. We will see in a moment, however, that compartments are mathematical constructs, and equating a compartment with a physical (anatomical) or chemical space requires great care.
A compartmental model is a collection of interconnected compartments with a specified set of inputs and outputs. Mathematically, it represents a set of differential equations. More importantly, however, it provides a mechanistic framework for direct physiological interpretation of data. We reiterate that great care must be exercised in equating a compartment, which actually represents a differential equation, with a physiological volume.
Figure 3 is an example of a 3-compartment model. Each of the compartments, labeled 1, 2, and 3, represents a differential equation, that is, an expression of the form dqi/dt = where qi is the mass of a substance in compartment i at time t. What comes after the equals sign is determined by the arrows in the model. The arrows labeled kij represent the fractional transfer of material from compartment j to compartment i. For example, k12 is the fraction of mass in compartment 2 transferred to compartment 1 per unit time. The arrows k01 and k02 are the fractional losses from compartments 1 and 2 respectively, and the bold arrow U1 denotes exogenous mass input per unit time into the system.
A model such as this is frequently used to described glucose kinetics in the whole body (see Ferrannini and Cobelli [1987]. It is assumed that compartment 1 is plasma, and that this compartment can be sampled; this is indicated by the dotted line with the bullet. The two loss pathways are hypothesized to be insulin-dependent and insulin-independent glucose disposal. While compartment 1 can describe the plasma data, this model can predict the kinetics of glucose in compartments 2 and 3, two compartments in the body that are not accessible to measurement and thus represent hypotheses as to the kinetics of glucose in these nonaccessible "compartments''. One can hypothesize that compartment 2 represents glucose in the brain and that k02 is the insulin-independent glucose utilization in that organ. However, this is just a hypothesis, and if one wanted to test that hypothesis, one would have to design an experiment in which brain glucose was available for measurement; these data could then be compared with the model-predicted value for compartment 2.
Figure 3 is an example of how a model can be used to predict the behavior of a system. Again, however, we are operating at the boundary between experiments and mathematical models. These model predictions can only be tested through experimentation.
The plasma lipoproteins are aggregates of fat and protein molecules that allow the transport of fats in the aqueous environment of plasma. They are subdivided into a number of specific classes using density value, the most common one of which is buoyant density. The proteins of lipoproteins are called apolipoproteins (apo). Much is known about the metabolism of the plasma lipoproteins (see, for example Dam-merman and Breslow [ 1995]).
In the following example, we will concentrate on the metabolism of the apoB-containing lipoproteins. These are the very low (VLDL; d < 1.006g/ml), intermediate (IDL; 1.006 < d < 1.019g/ml) and low (LDL; 1.019 < d < 1.083g/ ml) density lipoproteins. LDL is primarily responsible for transporting cholesterol to cells in the periphery. LDL is positively correlated with cardiovascular disease, and hence studies dealing with its physiology and pathophysiology are very important (Kwiterovich and others 1993).
The relationship among VLDL, IDL, and LDL can be summarized in Figure 4. A number of experimental techniques can be used to quantify the transport of apoB from VLDL to LDL, taking advantage of the fact that the apoB once secreted on a VLDL particle remains as the particle is converted through IDL, or remnant particles, to LDL.
The LDL particles were once considered to be a homogeneous collection of particles. However, it is now known this is not the case (see Krauss and Burke [1982]). This makes the design and interpretation of metabolic studies much more difficult (see, for example, Foster and others [1986]). The problem is trying to understand the balance between LDL production and catabolism in different subclasses of particles since these are the two metabolic processes responsible for determining an individual's plasma LDL level.
In addition, it has long been argued that there are two sources for LDL. One represents the conversion from the less dense VLDL particles, and the other is de novo LDL production. Both are assumed to occur in the liver (Dammerman and Breslow 1995). However, one cannot distinguish between de novo LDL production in the liver and the synthesis of a small class of VLDL particles in the liver that are rapidly converted to LDL.
The reason why the African green monkey was chosen as an animal model in this particular study was that previously it had been shown that VLDL isolated from an isolated per-fused African green monkey liver was rapidly converted to LDL (Marzetta and others 1989). Such a study, which provided the first clue that such a small class of particle might exist, is not possible in humans. The study described in Murthy and others (1990), which built upon this information, was designed to assess the contribution of this small class of particles to plasma LDL levels in vivo. This can be regarded as a first step towards understanding how such a class could contribute similarly in humans. The long-term goal, of course, is a better understanding of human lipoprotein metabolism and the discovery of interventions to correct pathophysiological conditions.
The study design was a multiple input-multiple output tracer kinetic study that again would not have been possible in humans. To determine if hepatic VLDL could contribute to plasma LDL levels (that is, if there was a precursor pool to LDL that never appeared in plasma but was incorporated directly into LDL in the liver), hepatic VLDL was isolated from perfused livers and tracer-labeled with tritium. At the same time, plasma LDL was isolated and radioiodinated. The 2 tracer-labeled particles were injected simultaneously, and serial plasma samples were taken. The samples were subfractionated into different lipoprotein subfractions (including 2 LDL subfractions), and quantitated for radioactivity. The set of data was rich in information and required a model to interpret. Basically, a compartmental model structure was derived to create a mathematical model of the conceptual model shown in Figure 4. The final structure, which was derived and tested using the principles described earlier, is shown in Figure 5 below.
In this paper, additional compartmental modeling techniques were used to test a variety of hypotheses using the data that would not have been possible using conventional methods. The main conclusion was that plasma LDL could be derived from various sources including both hepatic VLDL and LDL, and each gives rise to a metabolically distinct subfraction of LDL.
This is a very important observation in the study of atherogeneity of LDL in humans since it can provide clues in humans as to why certain subfractions of LDL are more atherogenic than others, what the origins of these subfractions may be, and what kind of a therapeutic intervention might be designed to alter the pathophysiological condition.
Many people think that animal research, and the mathematical models that have been developed based upon that research, deal only with human needs. While we have given examples of this in the past section, there are many instances when studies are undertaken because of a need to understand a metabolic system in a particular animal species. We give an example below of water metabolism, and go into more details of the modeling so the interested reader can appreciate by example some of the points we have been making.
Background of the study. Water is the ultimate substrate; its role in digestion and metabolism is ubiquitous and pivotal. Animals have evolved ingenious procedures to deal with environmental factors impacting their water management; these embrace, for example, facilities for limiting losses due to sweating and other evaporative cooling processes. A question naturally arises as whether the animal has compromised its other functions as this adaptation has occurred. The project discussed here describes how an investigation using compartmental models led to the identification of a model and the rate constants describing water metabolism in sheep; this can be used as a baseline to study other issues related to water metabolism (Faichney and Boston 1985).
The experimental data. We postulate that 3 body pools are associated with water movement in the sheep: the gut (dominated by the rumen), the blood and extracellular space, and the intracellular space. In essence water enters the sheep's body via drinking episodes. It is then transported into the extracellular space, and then into the intracellular space. Whereas these processes are bidirectional, their susceptibility to osmotic pressures in either direction dictates that a characterization of water transport will need to be with reference to a specific metabolic state of the animal. Thus our study will focus on sheep in a steady state in regard to water balance (see Figure 6).
Water leaves the sheep's body largely in the form of urine. However, salivary, evaporative, and fecal water losses also take place. Thus to quantitate water transport we need access to kinetic data describing the movement of water between extracellular space and gut, and, in a reciprocal fashion, between gut and extracellular space. The intracellular space is inaccessible for measurement and hence its role will be predicted from variations needed to a simple exchanging 2-pool system to describe the kinetic transport data.
Figure 6 shows a multicompartmental model of water metabolism reflecting the points above.
Experimental data to test the model. After a period of metabolic stabilization, tritiated water was injected either into the rumen or plasma and serial samples were taken from both sites (in each sheep) for 8 days. The bolus applications into plasma or rumen were separated by 8 days. Figure 7 shows the model depicted in Figure 6 with these 2 experimental protocols. This served as the basis for the data to test the model; representative sets are shown in Figure 8. In addition, the steady-state inputs and losses of water were monitored, that is, metabolic water intake, feed water, drinking water, fecal output, and urinary output were measured. The balance of water loss was presumed to coincide with evaporative processes.
The important point to notice here is that in attempting to describe water transport processes in the sheep we have incorporated information and data from as many sources as we have at our disposal. We will now use these data to determine if the model shown in Figure 6 can adequately describe the metabolic processes.
We encountered irreconcilable inconsistencies between our (tracer) data and our model-based predictions when we tried to fit the model shown in Figure 6 to the data. If the plasma-predicted activities agreed with the data then we found that the rumen-predicted activity levels fell far too quickly when compared with the data. Furthermore, preserving the rumen-prediction consistency caused the plasma predictions to be inconsistent with the plasma data.
To address this deficiency, the model required the inclusion of a delayed or prolonged entry of water into the plasma-gut (compartments I and 3) subsystem of the model (see Figure 9). This could be achieved by including a lower gut pool with absorption and secretion processes in the same manner as for the rumen pool. This would make sense from a physiological standpoint as well.
The original model was not compatible with the data and had to be modified. That is, the hypotheses built into this model needed modification. The model was accordingly changed (see Figure 9) to accommodate a lower gut pool and to allow for exchange of the contents of this pool with the blood pool. Another compartment could be resolved from the data together with appropriate connections with other compartments.
This is an example, then, of how one can start with a simple model structure, discover that it cannot predict experimental data, and add additional structure to correct the problem. How that structure is added is dictated both by the nature of the data themselves and by known physiology.
This revised model was found to describe the data quite well; representative fits are shown in Figure 10.
With a model that can predict the experimental data, what kind of information is available? Table 1 provides a summary of the kind of information that can be obtained. These numbers can serve as baseline numbers to understand water metabolism in sheep and to study sheep with specific water-based diseases. In particular, the power of the model shown in Figure 9 is that in pathophysiological states, the information summarized in Table 1 can be compared with the baseline state, and specific defective pathways can be identified and potentially corrected.
To understand how complex this problem is in terms of estimating the parameters and how the respective parameters are specifically identified, we present the following. There are in fact 1 3 parameters in the model to be determined (the rate constants k12, k21, k13, k31, k01, k43, k04; the compartmental masses M1, M2, M3, M4; and the "de novo" water inputs U2 and U3). We see, for example, from Table 1, that from the plasma-specific activity data from the plasma injection, k31, k21, k01, and M1 are either explicitly, or in combination, identified. The rate constant k01 will be resolved in conjunction with the steady-state water balance data, k31 will be apparent through the early rumen appearance data, and k21 will represent the balance of the fractional outflow from plasma necessary to explain the early disappearance there.
A similar picture to that regarding use of the plasma injection data also unfolds for the rumen injection data.
The discussion of the water model illustrates several points we have discussed previously. The model describes the experimental data and concomitantly will permit us to estimate parameters of interest, primarily in terms of water movement. The model also characterizes the system more fully since it is able to describe the multiple input-multiple output study. We also described how the model structure was put together, both initially and then modified. We discussed the parameter identification and estimation issues. To finish the discussion, we would have to deal with issues related to model uniqueness and validation, which is beyond the scope of this article.
Experiments and Mathematical Models: the Boundary
It should be clear now how mathematical modeling and experimental design interact. In fact, that interaction can take place in two ways. In the first, diagrammed in Figure 11, the modeler and experimentalist are working on the same system, and they both have access to the same literature. However, the modeler does not need the experimentalist to postulate and test a model; neither does the experimentalist need the modeler to design an experiment. However, since both have access to the same literature, what is missed by not having an interaction is the expertise that both bring to an understanding of the system. Because of the nature of their training, each will bring a different interpretation of the same literature. Therefore Figure 12 shows a paradigm that can work more effectively.
The dotted lines indicate how the interaction can work. Both can participate in the initial design of the physical model and experiment. Using the initial model, the experimental design can be altered. At the same time, pilot studies can result in changes to the model structure. The figure also indicates where the expertise of both lies. For the modeler, it lies in the mathematical formulation of the model, and for the experimentalist it lies in the execution of the experiment with the resulting data.
It is at the comparison level that one probes more deeply into the system being studied. Either the comparison produces compatibility between the model predictions and the data, or it does not. If there is not compatibility, then both the model and the experimental design may have to change. The model structure will have to change to accommodate characteristics in the data, and the experimental design may have to change to generate more data to estimate the parameters in the model.
If the model is compatible, then there will be a number of hypotheses built into the model structure that need to be tested by further experimentation. The cycle thus starts over again, but with the model, the formulation of new hypotheses can be stated with far greater precision. This is a major advantage to operating under the scheme shown in Figure 12.
We have spoken throughout about the boundary between mathematical models and experiments. This boundary becomes apparent when one is dealing with a model that is compatible with a set of data and known physiology. Such a model has predictive value for conditions in the system that are different from the conditions under which the model was derived. However, these predictions have no real value without confirmation; and confirmation can come only from experiment data.
The importance of this observation is clear when one thinks of trying to formulate therapeutic interventions in specific pathophysiological conditions. One may have a model of a system in the normal state, but without knowing the specifics of how an intervention affects the system, one does not know if the intervention will actually have the desired result.
For example, returning to the situation with LDL being an atherogenic lipoprotein, as noted earlier it is now known that one of the subclasses of LDL is far more atherogenic than the others. Suppose one were to prescribe a drug that lowered plasma LDL levels, but actually raised the levels of the atherogenic subclass. By looking at only one part of the puzzle, one would think the drug had the desired effect while in fact it did not. Only a combination of modeling and experimentation can unravel these kinds of situations.
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TABLE 1 A summary of information that can be obtained from a model that can predict experimental data
| Data Form Identified | Observation Point | Features of Model |
| Steady state observations | Intake | |
| Metabolic water | U2 (ml d-1) | |
| Fecal water intake + drinking water | U3 (ml d-1) | |
| Outflow | ||
| Fecal water | R04 (ml d-1) | |
| Water balance | R01 (ml d-1) | |
| Flow | ||
| Rumen outflow | R43 (ml d-1) | |
| Pool size | ||
| Rumen water | M3 (ml) | |
| Lower gut water space | M4 (ml) | |
| Total body water | M1+M2+M3+M4 (ml) | |
| Loss | ||
| Total fractional water loss | (U2+U 3)/(M1+M2+M3+M4)(d-1) | |
| Specific activity observations | ||
| Plasma injection | Plasma observations | 1k31, k21, k011 (d-1) M1 (ml) |
| Rumen observations | k31 (d-1) | |
| Rumen injection | Rumen observations | 1k13, k43 (d-1) |
| Plasma observations | M3 (ml) k13(d -1) |
1 Without evidence to the contrary the rate constants k14, and k41 were respectively assigned the same values as k13, and k31. In each case this is the combination of parameters whose 'sum' is best identified from the tracer study.
FIGURE 1 A 2-state model for facilitated diffusion of a substance into and out from a cell.
FIGURE 2 The 2-state model shown in Figure 1 redrawn as a first step towards a mathematical formulation of the model.
FIGURE 3 An example of a multicompartmental model.
FIGURE 4 The metabolic relationship among VLDL, IDL, and LDL apoB. The light arrows represent conversion, which is one-way, from VLDL to IDL and LDL, or irreversible losses. The bold arrows represent de novo synthetic pathways for VLDL and LDL.
FIGURE 5 A schematic of the model derived in Murthy and Foster (1990) to describe the metabolism of apoB through VLDL + IDL (combined because of the low apoB levels in monkeys), and the less and more dense subfractions of LDL, LDL-I, and LDL-2. The radioactivity in VLDL + IDL, LDL-1, and LDL-2 are respectively the sums of the radioactivity in compartments 10, 1, and 11, 9, 2 and 12, and 8 and 3. Compartment 13 is an extravascular compartment which exchanges with compartment 3.
FIGURE 6 A 3-compartment model for water metabolism. Drinking water enters the system into the "gastro-intestinal" compartment GIT. Water can leave via fecal water. Water in the GIT compartment can exchange with the plasma-extracellular space compartment; water can leave this compartment via urine and evaporative loss. Water in the plasma compartment can exchange with water in the intracellular space compartment, IC Space. Water that results from metabolic processes can enter the lC Space compartment.
FIGURE 7 The 3-compartment model shown in Figure 6 with the two experimental protocols superimposed. The dotted line with the bullet indicates sampling sites, i.e. the GIT and Plasma and EC Space compartments. The top figure shows the bolus (solid arrow with asterisk) into the plasma compartment; the lower figure shows the bolus into the GIT compartment.
FIGURE 8 Representative sets of data from the 2 bolus injection studies. The top panel shows the rumen and blood specific activities following a bolus injection into plasma; the bottom panel the same for the bolus injection into the rumen. Specific activity is calculated as the quotient of dpm (disintegrations per minute) and water mass in each sample.
FIGURE 9 An expanded model accommodating the lower gut. Compartment 1 is the plasma and extracellular space compartment, compartment 2 is the intracellular compartment, compartment 3 is the rumen compartment, and compartment 4 is the lower gut compartment. The two experimental designs are superimposed upon the model. U3 and U2 represent drinking water and metabolic intake respectively.
FIGURE 10 Representative fits of the data shown in Figure 8 by the model shown in Figure 9. The top panel is for the bolus injection into plasma; the lower panel for the bolus injection into the rumen.
FIGURE 11 Schematic for the role of mathematical modeling and experimental design where there is no interaction between the modeler and experimentalist.
FIGURE 12 Schematic for the role of mathematical modeling and experimental design where there is interaction between the modeler and experimentalist.
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