The problem of predicting the consequences of a change applied to a variable in a set of constraints too complex for explicit solution has traditionally been solved by simulating the constraint equations. The prime example of that strategy in the cardiovascular domain is Guyton's cardiovascular simulation[2]. That program consists of a large number of equations relating the physiological parameters. To simulate the effect of some intervention, all parameters are set to a consistent initial state, the values of the intervention variable are set to correspond to the desired change, and the model is run forward in time by determining next values for the parameters from the equations at appropriate time increments. There are three problems with this strategy for our purposes.
Work on qualitative reasoning has produced a second kind of simulation strategy partially addressing these problems. The qualitative simulation approach applied to the medical field is exemplified by Kuipers work with Kassirer on qualitative simulation of fluid balance[3]. Using this technique, only the direction (monotonicity) of the relationship between two parameters is used to determine the changes. However, when the models become complicated, the amount of ambiguity in the solution increases. In a model as complicated as the essential relationships in the cardiovascular system, the predictive capability of this methodology is severely limited.
The approach we have developed for predicting changes is based on signal flow analysis. Signal flow analysis takes a set of linear equations representing the gains between elements in a circuit and computes the overall gain from input to output if the circuit is stable. To apply this approach in a medical domain, the technique must be extended to consider all parameters as outputs, to do the computation in such a way that the paths of influence are preserved, and to handle quantities that are integrations of other quantities. In addition, it is necessary to make some simplifying assumptions to apply the approach to our domain.
The computational algorithm used in signal flow analysis is Mason's general gain formula[5]. To keep track of the mechanisms of influence and to determine the change in all parameters, we have derived a different version of this algorithm that computes the gain incrementally from parameter to parameter correcting for feedback each time a new feedback path is encountered. In effect the algorithm traces each path from the changed parameter to all other parameters, computing the cumulative gain at each point along the way. The change to a parameter is determined by summing the gains on paths through that parameter. Because the paths are explicitly computed, it is possible to compare the gains across different paths and identify the mechanisms with the most influence on any parameter change.
The problem of integration parameters is made tractable by the assumption of signal flow analysis that the system is in a steady state. For a medical domain, that means that the analysis is answering the question: If the system is in a steady state and a change is applied to the system, what will be the steady state after transients have died down? The assumption that the system goes from steady state to steady state can be relaxed somewhat by taking advantage of the widely differing times required by different mechanisms in the body to change, from seconds for shifts of fluid from pulmonary to systemic circulation, to years for ventricular hypertrophy. By assuming that for the time period of concern, mechanisms that react more rapidly have reached equilibrium and for slower mechanisms no significant change has taken place, it is possible to reason in terms of quasi-steady states. Overall, the assumption of steady states is justified by the stability of the cardiovascular system[4]. The only possible problems would arise if transient states could occur that would change relationships between parameters sufficiently to change the next steady state.
Integrations can be handled because the derivative of the integrated parameter is zero in steady state. For example, the shift of blood in and out of the pulmonary circuit rapidly reaches a state in which both ventricles have the same output. The requirement for a zero derivative yields a set of linear equations in the integrated parameters analogous to the original problem that can be solved for the changes to those parameters by the same algorithm. The changes to the integrated parameters in steady state then become additional inputs to the system of the remaining parameters. Thus, the system can be solved in a two step process. 1) Use the change to determine how the integrated parameters must have changed for their derivatives to be zero. 2) Use these changes to determine the state of the rest of the parameters.
The requirement of linear equations implies that non-linear systems must be approximated by piece-wise linear equations. Many of the relationships between physiologic parameters are non-linear, from the Frank-Starling curve between end diastolic pressure and cardiac output to relationships that are multiplicative. Since the system is dealing with the change from one state to another, the initial gain across a link is the partial derivative with respect to the parameter at the head of the link. For example, the equation determining blood pressure is:
This approximation is adequate for small changes, but for large changes it is necessary to step through the segments of the change until the complete change is modeled or find values of gain that better approximate the average gain for the change. In effect, this means the change is being applied slowly so that the system is never very far away from a quasi-steady state. In our experience thus far, these approximations seem to be reasonable ones for the domain. Thus, we have a methodology for predicting the effects in a model for any change applied to it.