John H. Frenster, M.D., Assistant Professor
Rockefeller Institute, New York, N.Y. 10021 

In many clinical disease states, a body process may be performing at a rate many times greater than normal, but because the process is markedly overloaded, it fails to deal successfully with all of its imposed load. A residual fraction of the imposed load accumulates before such a process, and this fraction is the direct stigmata of process failure and disease. This medical paradox of high-output failure provided the early stimulus for an analysis of the processes of human systems as throughputs (1).

In further studies, it has become possible to classify the action elements within each process (2), to analyze the interaction of such elements (2), and to begin the measurement of the effectiveness, control, renewal, and cost efficiency of such systems. From these studies has emerged the concept that the health of a body system is represented by its load-tolerance (3).

This portrays health as much more than the mere absence of disease. The range of load-tolerance in a tested process can be measured among normal subjects and in patients with disease, especially during the stages of a disease and its therapy (4), providing a quantitative estimate of health or disease within the process (3).

Human Systems as Throughputs:

Throughputs are fluxes of substrate units passing through transforming processes which are arrayed serially within systems (5). A wide variety of human processes of clinical interest can be analyzed by the techniques of Queueing Theory as components of throughput systems. Such processes include the propulsion of blood by the heart, the exchange of respiratory gases by the lungs, the absorption of molecules across the intestinal mucosa, the excretion of bilirubin by the liver, the clearance of urea by the kidneys, the uptake of iodide by the thyroid, and the transport of glucose across cell membranes.

Queueing Theory in System Kinetics:

Queueing Theory considers the waiting lines formed when throughput units, arriving at some process channel

Fig. 1.

(Fig. 1), cannot be served without causing some of them to wait from time to time (5). In each human throughput system, the length of the waiting line (L) that is accumulated before a process service channel, starting from an unloaded state, is a function of the rate of arrival (A) and of the rate of departure (D) of throughput units from the process:

Eq. 1.

D, the output rate, is a function of the interaction of the waiting line length (L), the maximum service rate capacity (Cmax), and the resistance (R) opposing the exit of output (2, 3). This interaction within a throughput system or component process is described by a generalization of the Michaelis-Menton equation for saturation kinetics:

Eq. 2.

where the basal state values Lo and Ro and the constants k₁, k₂, K_L, and K_R characterize the particular system. 
Starting with the unloaded process, (L = 0), an increase in input rate (A) results in a non-linear increase in output rate (D), with saturation of service capacity ultimately limiting further increases of output rate at levels that are a function of the capacity (Cmax) and the resistance (R) to output (Fig. 2):

Fig. 2.

Input rate (A) is a function of the ouput rate of the preceeding process in the system, while resistance (R) is a function of the output rates of the process studied and of the following process in the system. Rate control of a throughput process is most often achieved by the adjustment of the Cmax rates of both the process itself and of other processes in the system (5).

Activation and Hypertrophy of Systems:

In most human throughput systems, Cmax rate is variable in two stages, rapid increase resulting from activation of existing idle service channels, and more gradual increases resulting from net formation of new service channels, as in hypertrophy states. Such variation of Cmax rates is actuated through superimposed control systems (Fig. 3):

Fig. 3.

which sample the waiting line and/or the exiting output rate of the process and of other processes in the system. Sampling of exiting output rates provides an exterior criterion of the adequacy of process or system output, while sampling of process waiting lines provides an interior criterion of the balance between process arrival and departure rates. Such superimposed control systems permit an optimal adjustment of Cmax rates to provide both an adequate output rate and a balance to the existing imposed load and opposed resistance within the system. Frequently these control systems are themselves variable, the values of their component parameters being functions of sampled waiting lines and output rates, as mediated via proportional, differential, or integral responses within adaptive control circuits (Fig. 3).

Measuring System Effectiveness:

In clinical medicine it is of great interest to measure two features of throughput systems, the load-tolerance of the system, and the Cmax after activation of the system. If a test load (A)_test is administered of size and rate just sufficient to lengthen the existing waiting line (L)_exist to a known pathologic length (L)_path, then:

Eq. 3.

where the magnitude of (A)_test is a quantitative estimate of the load-tolerance of the system, and is determined by by the excess of available capacity over that required for existing input load and opposed resistance (3, 4). In latent disease, a decrease in capacity or an increase in load or resistance results in a decrease in the load-tolerance of the system, while in overt disease, load-tolerance declines even further to zero or frankly negative. A state of health is characterized by a high load-tolerance within a tested system, and by its measurement the degree of health or disease of a throughput system can be quantitatively estimated (3).

The Cmax rate after activation of a throughput system may be estimated after the pulse administration of a large input load to the system while in the basal state, or by a graphical analysis obtained by plotting the reciprocals of input rate and output rate for various tested input rates. Knowledge of the Cmax rate of a system will permit those disease mechanisms operating to decrease capacity to be distinguished from those which increase input load or resistance to output of the system (1).

System Cost-Efficiency:

The occurence of reversible hypertrophy and atrophy states illustrates the operation of cost-efficiency criteria in achieving an optimal matching of capacity with the levels of load and resistance usually encountered (1). To achieve a given output rate, a large number of combinations of input load and service capacity are possible in any given system (Fig. 4):

Fig. 4.

Combinations employing large loads and minimal capacity are wasteful of throughput units, but minimize those resources of renewal, design, and control used in the formation of service channel capacity. The optimal combination of load and capacity for a given output is that one which entails the minimal total resource cost for load and capacity. The efficiency with which human systems can achieve such an optimal natural selection is variable (C₁ vs. C₂ vs. C₃), and is often reduced in disease states (Fig. 4).

Summary:

Many human processes of clinical interest can be represented mathematically as loads of throughput units flowing through service channels and opposed by resistances to output. Each human throughput system normally presents a waiting line of throughput units before each service channel. Disease states, whether caused by decreases in capacity or by increases in load or resistance, are manifest as a lengthening of the waiting line to pathologic values. The ability to service additional loads without an increase in the waiting line to pathologic values is a measure of the load-tolerance and health of the process. Human throughput systems analysis permits measurement of system effectiveness, control, renewal, and cost-efficiency.

Support:

Supported by NIH Grant No. 5-K3-CA-17857-02.

References:

1. Frenster JH, "Limits to Functional Hypertrophy in High-Output Failure", Annals of Internal Medicine 53, 647-655 (1960).


2. Frenster JH, "Interaction of Load, Capacity, and Resistance in Body Processes", Perspectives in Biology and Medicine 4, 152-158 (1961).


3. Frenster JH, "Load Tolerance as a Quantitative Estimate of Health", Annals of Internal Medicine 57, 788-794 (1962).


4. Frenster JH, "The Magnitude of Disease as Measured by Tolerance Tests", Journal of Theoretical Biology 2, 159-164 (1962).


5. Frenster JH, "Throughput Interaction within General Systems", General Systems, (In Preparation).

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1. Editorial: "Medical Feudalism", J. Am. Med. Assoc. 184, 1039 (June 29, 1963).

1. Frenster JH, "Analysis of Queueing and Renewal within Human Systems", Nature 207, 1139-1140 (1965).

2. Frenster JH, "Medicine 275: Systems Analysis of Latent Disease".