Mathematical Modeling in Surgical Research

ed knowledge about an object that can be embedded into a static model is frequently used in bedside surgical care. For example, water in the adult human is commonly modeled to occupy two compartments, an intracellular space and an extracellular space, where the extracellular space itself consists of two compartments, an interstitial space and an intravascular space. Clinical estimates of the magnitudes of fluid and electrolyte deficits rely on such a static model. Dynamic Models The passenger in 15-C is likely less concerned with dimension than with a safe and swift journey. The journey depends on engineering, and the passenger in 15-C is reassured that a professional team has designed systems and subsystems to reliably interact in highly specific and predictable ways. The key phrase is “designed...to interact”. Biomedical engineering excepted, the surgical investigator does not participate in the design of the object under study. In most surgical research projects, the goal is to elucidate the design. The key tool is controlled perturbation of the study object followed by sequential measurement of object parameters. From the measurements—whether the data describe gene expression, bulk flow of blood through the heart, or spread of a particular bacterium through an intensive care unit—surgical investigators make inferences about the relevance of a particular process. The inferences become hypotheses that are experimentally probed, most often by comparing objects that differ in a single feature: the knockout mouse versus its parent; flow at a hematocrit of 20 versus a hematocrit of 40; use of water-based handwashing versus alcohol foam degerming. Mathematical Modeling Page 4 Data accumulate much faster than knowledge. The classical, reductionist approach to scientific inquiry requires a full factorial experimental design such that each relevant process ought to be tested across the full range of expected performance in order to understand the effect of that individual process upon the whole experimental system . Organ physiologists a generation ago often performed such systematic studies. Their detailed experiments became the basis for clinically essential models such as cardiac performance as a function of preload, afterload and contractility. Such experiments on the microscale of cells and molecules and on the macroscale of large populations are difficult to design and even more difficult to perform. The usual approach is that a relatively few observations made under arbitrary but strictly controlled conditions in which the object under study has been intentionally “isolated from confounding influences” are extrapolated to more general, analytically more complicated situations. The potential for error is obvious, the realization all too frequent. The Hidden Hypothesis The passenger in 15-C is flying in an airplane whose behavior over time was predicted on the basis of an explicit design. The surgical investigator pursues the “inverse problem”. The design of the object under study is to be extracted from its behavior over time subject to a host of noisome experimental constraints. We have already alluded to the limit of the number of data that may be collected. Biologic objects also limit the types of data that can be collected. The precision of the data obtained from biological objects is typically less than that obtained from physical objects. And so on. The extrapolation to the more general situation and to behaviors over time is an hypothesis in its own right, an hypothesis that is subject to verification by experiment. Testing this “extrapolation hypothesis” drives modeling such that behaviors are predicted and then experimentally tested. As stated in the introductory paragraphs, models themselves are merely collections of hypotheses regarding the mechanisms and magnitudes of processes that influence the object under study. 1 What is (and what is not) a mathematical model? 1 As a collection of hypotheses which itself is an hypothesis, mo dels can never be proven “correct”. Their greatest investigational value lies in illuminating what is “missing” or “wrong”. Mathematical Modeling Page 5 A mathematical model is a tool with that an investigator encapsulates hypotheses concerning the mechanisms and relationships that underlie the behavior of a system over time. Equations are used to describe the relationships. It is important that surgeons understand that once the relationships are described as mathematical equations, solutions to those equations are often readily obtained by desktop computers. Mathematical models are ubiquitous in surgical care. Some are expressed as informal "rules", such as the "three-for-one rule" (which states that three volumes of a balanced salt infusion are required to compensate for each volume of acute blood loss). The 3:1 rule originates from experiments showing that water and small ions readily equilibrate across blood vessel walls into the interstitial compartment, and a model that envisions the interstitial compartment to be twice as large as the intravascular compartment. Other mathematical models are more formal, such as the pharmacokinetic models that guide administration of aminoglycoside antibiotics. The nomograms that surgical residents use to make dose adjustments are simply graphic representations of models of the aqueous compartments and the predicted clearance rate of the drug. Each patient is viewed as an individual experiment, with the model offering continuous predictions about plasma concentrations. Measuring the patient's plasma level of the drug at a particular time is a test of the model, not of the patient. An accurate prediction merely indicated that the dose may be left unchanged. An inaccurate prediction does something more--it not only indicates to the surgeon that the dose must be changed but also suggests that the model contains relations that are inaccurate or incomplete. Indeed, unexpectedly high levels may suggest that there is incipient renal insufficiency whereas unexpectedly low levels may suggest that the patient has a larger-thannormal volume of distribution. However useful they may be, memorable "rules" and nomograms are no more than representations of someone else's model. The surgical investigator must ultimately venture into building his own model if he is to make and test hypotheses concerning the design of the object being studied. He must ultimately propose relationships that govern the measurable parameters, make predictions, perturb the object, and observe the fidelity with which his model describes the behavior of his system. Simply stating the anticipated Mathematical Modeling Page 6 change in a parameter ("I predict drug D will cause parameter P to decrease") is not a model. It may well be an event predicted by a model, but the prediction is not the model. Model Building: An Example To illustrate one way that modeling illuminates a problem to focus attention on particular aspects of that problem, consider this familiar and vexing scenario. Review during rounds of a postoperative patient shows two abnormalities. First, the urine output is decreasing. Second, the serum creatinine concentration is rising. The patient has received appropriate volumes of fluid. The inescapable conclusion is that the patient has acute renal insufficiency. The apparent cause of the kidney failure is identified and reversed. The next day, the serum creatinine level has climbed again. Has the true cause of the renal insufficiency been identified? Why has the serum creatinine level risen? Is there another cause for the problem? When will the creatinine concentration peak and begin returning towards normal? These gnawing questions have cost every surgeon anxious moments. To apply mathematical modeling to this (or any other) problem, the universe of the problem must be explicitly defined along with the hypothesized relationships among the components of the experimental system. In the case of the patient with renal insufficiency, it is enough to define the universe to include a source of creatinine (muscle breakdown), a reservoir in which the creatinine is accumulated ( in total body water), and sinks into which the creatinine flows (urine). total creatinine in total body water muscle breakdown ~ tubular secretion glomerular filtration Mathematical Modeling Page 7 This graphic representation encapsulates not only the universe but also the relationships represented in what will become a "conservation of mass"relationship. The graphic emphasizes that we are not particularly interested in the exact source of the creatinine, only that the source continues to pour creatinine into the reservoir by the process of myolysis. The graphic also recognizes that the kidney has two distinct mechanisms by which it removes creatinine from plasma (and, by extension, from total body water): filtration by the glomerulus and secretion into the renal tubule. Although both mechanisms deliver creatinine into the urine, we can and will treat them as distinct processes. The conservation of mass relationship can be "read" as follows. "The rate of change in the total amount of creatinine (where the total amount is equal to the concentration of creatinine multiplied by the volume in which the creatinine is distributed) must equal the difference in the rates at which creatinine is being delivered and creatinine is being disposed. Creatinine is delivered by a single process (muscle breakdown). Creatinine is disposed by two processes, tubular secretion and glomerular filtration. The rate of glomerular filtration depends on the local creatinine concentration." A conservation of mass equation containing these relationships might read: ]) [ ( ) ] ([ Cr g S R dt V Cr d Cr & & & + − = ∗ (1) where [Cr] is the concentration of creatinine in body water, VCr is the volume of that body water, R& is the rate of creatinine released by muscle breakdown, S& is the rate at which creatinine is secreted by the renal tubules, and g& is the glomerular filtration rate. The instantaneous rate of change is denoted by the derivative, d/dt. Two data series are immediately available to the clinician at the bedside. One is the series of concentrations of creatinine [Cr]. Surgeons mentally calculate ∆[Cr] as the data are being examined ("The creatinine went up 1 mg/dl since yesterday!") The other series, typically ignored on patient rounds, is the series of time intervals (∆t) at which the [Cr] determinations Figure 1. Schematic representation of production and elimination of creatinine. Tubular secretion and glomerular filtration are independent processes which occur in the kidneys and deliver creatinine into the urine. Mathematical Modeling Page 8 were made. What (if anything) can be inferred from relationships between the incremental change in creatinine, ∆[Cr]/∆t and the average value of [Cr] during the change? The following simple expansion comes from elementary calculus. If m and n are both functions of the variable t, then Rearrangement of terms in equation (1) yields Inspection shows that so long as VCr, R& and S& are constant, the slope of a d[Cr]/dt vs. [Cr] plot will be a linear function of g& . 2 In other words, the slope of the d[Cr]/dt vs. [Cr] plot--which some refer to as a phase plot or phase portrait--represents the glomerular filtration rate as long as the volume of distribution is more or less constant. Few of us—surgeons or mathematicians—have the intuition or experience to relate clinical data to this rather unfriendly-looking equation. Fortunately, neither are necessary. Desktop microcomputers with appropriate modeling software substitute nicely. The next several sections illustrate how equation (2) might be analyzed using a couple of popular modeling software packages. 2 Mathematically inclined readers may wish to examine this equation in several special cases. First, if coefficients are constant then an analytic solution is possible. In this case, ∆[Cr]/∆t can be calculated precisely, and the difference between the measurable and the infinitessimal d[Cr]/dt can be estimated. Second, behaviors during an acute change in g (step, ramp and so on) display characteristic plots of d[Cr]/dt vs. [Cr]. Third, and perhaps most important, the effect of sequential acute changes in g (two steps) give characteristic behaviors in the plot. ) 2 ( ] )[ ( 1 ) ( 1 ] [ Cr dt dV g V S R V dt Cr d Cr Cr Cr + − − = & & & dt dm n dt dn m dt mn d + = ) ( Mathematical Modeling Page 9