Treatment Strategies for the Management of C hronic Illness:
Is Specialization Always Better?
Peter J. Veazie
University of Minnesota
321 19" Ave S
Minneapolis, MN 55455-0430
Phone: (612) 626-8903/ Fax: (208) 988-1797
veaz0005@umn.edu
Paul E. Johnson
University of Minnesota
321 19" Ave S
Minneapolis, MN 55455-0430
Phone: (612) 626-0320/ Fax: (612) 626-1316
pjohnson@ csom.umn.edu
ABSTRACT: Chronic illness cannot be cured, only controlled. In this paper, we describe an
investigation of treatment strategies designed to control the natural progression of type 2
diabetes. We propose that treatment strategies are often specialized to types of patients, and
their performance is sensitive to accurate categorization. We investigate the proposition that
when incorrect categorization occurs, more specialized strategies may perform worse than less
specialized strategies. Following analysis of necessary conditions based on an expected utility
model, we present a dynamical systems model of patient care and define two measures of control
based on the trajectory of patient health states. The first measure characterizes the accumulated
level of control (the extent that health goals are maintained); the second measure characterizes
dynamic structure (the time dependencies among health states). Computer simulation is used to
analyze the effect of incorrect categorization on these measures.
KEY WORDS: Chronic illness, Treatment strategies, Dynamical systems, Computer simulation.
1 Introduction
Much of medical care is designed to cure illness and mend injuries; the duration of care is
typically short-term and success is defined by specific results. This type of patient-care
endeavors to return deviant physiological systems to normal functioning. Chronic illness is an
exception to this model because the underlying adverse physiological conditions are not curable;
instead, the treatment of chronic illness focuses on controlling health effects throughout the life
of the patient. Consequently, the duration of care is long-term and success is defined in terms of
control.
The state of health for a chronically ill patient is dynamic and often unstable. When left
unattended, the nature of disease governs the trajectories of patient health in the space of possible
health states. The patient's state of health eventually deteriorates in normal functioning and,
depending on the illness, can lead to death. The problem faced by physicians who manage
chronic illness is one of developing treatment strategies that control the disease process by
directing the trajectory of patient health-states over time. More successful strategies typically
maintain health-state trajectories in acceptable regions of the space of possible health states.
From the perspective of the healthcare provider, strategies of patient care (hereafter,
treatment strategies) are often tailored to patient categories. Physicians and other healthcare
providers use categorization to reduce the complexity of the information and knowledge they
process. Decision strategies tailored to these categories further reduce cognitive effort
(Gigerenzer et al., 1999). Useful categories embed information that provides associated treatment
strategies an advantage over more general strategies (i.e. strategies that are not tailored to
specific patient categories). Experience leads to more informative categories and better
performing treatment strategies by refining the structure of knowledge and defining better
decision heuristics.
For curable illnesses and injuries, patient categories are primarily based on medical
diagnosis. For chronic illness, patient categories must also account for factors that influence a
patient’s future health state; consequently, factors that govem physician and patient behavior can
be important contributors to patient categorization. Researchers have proposed that physicians
use psychosocial dimensions (e.g. compliance behavior) as well as disease states to develop a
patient category structure having finer granularity than medical diagnosis alone (Johnson et al.,
2001; O'Connor et al., 1997). Treatment strategies based on these additional dimensions reflect
relevant variation across categories and may outperform more general strategies.
Treatment strategies can be characterized along three dimensions: (1) structure, (2)
performance, and (3) specificity. We hypothesize two types of structure for the treatment
strategies employed to manage chronic illness. The first is one in which physicians make
decisions and choose clinical moves based on predictions of the future patient states. We term
this a feedforward treatment strategy (Brehmer, 1990). Feedforward treatment strategies depend
on mental models that include dynamic (time dependent) information regarding (1) patient
disease processes, (2) the consequences of past and present courses of action, including patient
compliance, and (3) knowledge of the way patients move through the clinical care system
(Freyd, 1987)). A feedforward treatment strategy is supported by systems of care in which
physicians follow individual patients over time and by clinics in which patients are tracked and
monitored so that information about patient state (including past compliance) is available at the
time of the patient encounter (Brehmer, 1990; Brehmer, 1992). The second type of structure for
treatment strategies is based on the concept of a feedback (as opposed to feedforward) process
controll. In this type of strategy, physicians make decisions and choose clinical moves using
information about the patient’s current state as evident in the immediate context of care. A
feedback strategy would be expected in a system in which patients are not typically followed by
specific physicians, but receive care based on whichever provider is available when the need for
care arises. A feedback strategy presumes a mental model that is simpler and makes fewer
cognitive and organizational resource demands than a feedforward strategy (Brehmer and Allard,
1991).
Performance and specificity characterize attributes particular to individual treatment
strategies. Performance provides a normative measure of how well a strategy controls patient
health trajectories (depicted as the height of the bars in Figure 1). Specificity reflects the
difference in performance between the category to which a strategy is tailored and other
categories. Figure 1 shows a measure of specificity as the differences denoted by S, and Sz.
Dividing S, and Sp by each respective strategy’s highest performance gives a measure of relative
specificity; in a following section we use a related concept, relative generality, equal to one
minus relative specificity. Categorization generates a relationship between performance and
specificity whereby better performance is achieved through greater specificity.
We compare two kinds of strategies: those that are applied without regard to patient
categories (e.g. clinical guidelines) and those that are tailored to patient categories. We label the
former, general strategies, and the latter, specialized strategies; both kinds may vary in
performance and specificity. Specialized strategies would be expected to outperform general
strategies; strategies with higher specificity should outperform those with lower specificity.
However, levels of performance and specificity can exist such that treatment strategies with
lower specificity outperform those with higher specificity.
Higher specificity implies greater variation in performance across classes. This is
depicted in Figure 1 as the greater difference in expected outcomes for high specificity strategies
than low specificity strategies (i.e. Sa and Sg are larger in the high specificity graph of Figure 1).
Consequently, applying a treatment strategy tailored for one category to patients in another
category can decrease treatment effectiveness (e. g. in Figure 1, AO, and AOg are larger in the
high specificity graph). This suggests strategies with high specificity may not be preferred if
they are applied to patients in other categories.
2 Necessary Conditions for G eneral Strategies
We use an expected utility model to identify necessary conditions for preferring general
treatment strategies to specialized treatment strategies. We represent patient categories as a
partition 2 on a patient state space S. Each element of a partition comprises the set of patient
states that compose a patient category. Patient categories are related to treatment strategies by a
function g that maps § U 2 onto a subset A of all possible treatment strategies. The elements o
of the partition 2 are indexed by i € {1,...k} where k is the total number of element in the
partition (i.e. number of patient categories). Treatment strategies a € A are indexed by j € {0,
1,...k}. The index j =0 denotes a general treatment strategy (i.e. ay =g(S)); the remaining
indices match the corresponding elements of the partition and indicate specialized strategies (i.e.
aj =g(0;) for all j =i). We denote strategy performance by uj: the utility associated with
applying treatment strategy aj € A to a patient in category o;€2. The following matrix
represents the performance (utility) structure relative to 0; and aj:
Physician Strategies
ries
E
The matrix labeled U’, containing elements {uj: i,j =1}, comprises the utilities for specialized
strategies. The diagonal of U" contains utilities associated with the patient categories for which
the specialized strategies are tailored (i.e., the diagonal of U" contains states g’(a)) for all j =1).
The column vector labeled U°, containing elements {uj:j = 0}, comprises the utilities of the
general strategy across patient categories.
We analyze three models, each constrained by two assumptions: First, we assume for
each treatment strategy, aj, the utility associated with the patient category o; = g"'(a)) is greater
than or equal to the utility of that strategy applied to other categories (i.e. Umm 2Uim for each
m21). This assumption embodies the proposition that treatment strategies are tailored to
specific patient categories by virtue of improved utility. Second, we assume the distribution of
patients across categories is independent of the treatment strategy decision (although the
distribution may be a function of past decisions). This assumption reflects a common
ontological commitment regarding the temporal order of causation: causes precede effects.
Expected utility theory implies the general treatment strategy ao is preferred to the set of
specialized strategies {a1,... ax} if the expected utility associated with the use of ap exceeds
the expected utility of using the set {a1,... ax}:
E(uy| ao) > E(uy|{ai,...ax}) . (2)
The expected utility of the general strategy (the left side of equation 2) is
E(uy |@o) =¥ uo p(G,)) (3)
T
where p denotes a probability mass function. We apply two additional assumptions to the first
model: (1) for each patient category 0; the utility of the general strategy is proportional by a
constant factor B to the optimal utility associated with the specialized strategy aj = g(o;); and, (2)
the probability of correctly categorizing a patient is the same across categories. The first
assumption implies
Wo0=B ° Ui (4)
for all i, where B is the proportionality factor representing the relative utility of the general
strategy. This assumption requires that we restrict our analysis to specialized strategies with
non-zero utilities on the diagonal of U*. From equation 3, the general strategy’s expected utility
is rewritten as
E(u, |p) =B-¥ (u, -po,)). (5)
Denoting the expected utility for i = j, with respect to the marginal distribution p(o), as U_,,
equation 5 becomes
E(u; |a)) =B-t.\. (6)
The expected utility of specialized strategies (the right side of equation 2) is
E(u, |{a,..-a,.}) = & uy - pla; [o,,m)- p(m|o,)- p(o,). (7)
Sm
In this formulation m is a dichotomous variable for which m = 0 represents correct categorization
of a patient, and m = 1 represents incorrect categorization of a patient. Denoting the probability
of correct categorization as Tt (i.e. 1m =p(m =0|0;) for all i categories), equation 7 can be written
as
E(u, [4@,...a}) =n -F(u, - p(o,)) +00) “ZHpto.) Ely - pla, |o,,m -0F (8)
fa
or simply
E(u, | {a,,...a,}) = -G_, +(1 1) -a,,, Vaal. (9)
Substituting equations 6 and 9 into equation 2 and solving for B gives the necessary condition for
preferring the general strategy to the specialized strategies in this model:
B >n +(1-1m)-—. (10)
S|
i =
We define the ratio ty as the relative generality y of the strategies {a1,...ax}. Relative
generality is one minus relative specificity. These concepts are mirror images of each other;
hence, either can be used in this analysis without loss of clarity.
We assume U,,, <w;_,, hence relative generality is less than or equal to 1 (i.e, ys1). As
i ej =
Y approaches 1, the specialized strategies perform equally well across patient categories;
consequently, there is no cost associated with incorrect categorization and B must exceed 1 to
prefer the general strategy. Similarly, as m approaches 1, the chance of incurring a cost due to
incorrect categorization is diminished, and again, B must exceed 1 to prefer the general strategy.
As yapproaches 0, the specialized strategies do not function with the patient categories where i +
j, implying a greater cost associated with incorrect categorization. In the last case, the second
term on the right side of equation 10 is 0, implying the relative performance of the general
strategy must exceed the probability of misclassification.
Figure 2 shows a contour plot of B on the parameter space defined by m and y. The
maximum of mt and y defines the lower bound of 8 for preferring the general strategy; the lower
bound is achieved if minimum of nm and y is equal to 0. For example, a physician using
specialized strategies with y near 0 and a 0.8 probability of correctly categorizing patients must
have a general strategy that performs better than 80% of the specialized strategies’ performance
if the general strategy is to be preferred. As y increases, the lower bound of the general
strategy’s performance is higher. This result conforms to intuition: as generality increases, the
cost of incorrect categorization is diminished and a competing general strategy must increase
performance to remain the preferred strategy. Similarly, for a given level of generality, as the
probability of correctly categorizing patients increases, the probability of incurring a loss due to
incorrect categorization decreases; again, a general strategy would require better performance to
compete with the increased accuracy with which the specialized strategies are applied.
The preceding model assumes f is constant across patient categories. A model in which B
varies across categories can be analyzed by expressing the utilities of the general strategy as a
proportion of the mean utility across uy fori = j of the specialized strategies’. Substituting
uo =B, Th (11)
into equation 3 gives
E(u, |ao) =U, °(B, -p(@,)). (12)
Following the reasoning used in the preceding analysis, the necessary condition for preferring the
general strategy in this case is
B>n+0—n) 22), (13)
U
isj
HereB denotes the mean relative performance of the general strategy representing the
summation in equation 12. Figure 2 and the conclusions of the preceding analysis apply to this
model as well, but we use the mean relative performance B of the general strategy in place of B.
Both analyses presented thus far assume the probability of correct categorization is
independent of patient categories. Without this assumption, we can state a more general criterion
for the preference of the general strategy. From equations 2 and 12 the necessary condition is
ax — E(uy | fa,--.ax})
B> (14)
a,
The numerator in the right side term of the inequality is equal to or less than the
denominator, the equality holding only when the probability of correct classification is 1. A
general strategy must therefore perform better on average across patient categories than
specialized strategies relative to the optimal performance of the specialized strategies (i.e.
relative to the expected utility of correctly applied specialized strategies). Factoring equation 14
and rewriting gives
a) Adi) (a, « pla joum =D
gopno tt wae fa (15)
This formulation reveals two category-specific components: (1) the decrease in
performance due to the probability of not applying specialized strategies to the targeted category
(represented by the first term in the parenthesis on the right side of the inequality); and, (2) the
modifying affect of a non-zero utility associated with misapplying treatment strategies
(represented by the second term on the right side of the inequality). The import of these
components is based on the probability of correct categorization mi, and the utilities uj. If patients
are always correctly categorized (i.e. m; = 1 for all i), or if each specialized strategy performs
equally well across categories (i.e. uj =uj for all j =1), then the first term equals 1 and the
second term equals 0. In this case, the general strategy must outperform the optimal application
of the specialized strategies, which would contradict the assumption that specialization is driven
by improved utility. If patients are always incorrectly categorized (nj; = 0 for all i), then the first
term is equal to 0 and the general strategy must outperform the consistent misapplication of the
specialized strategies. If the utilities uj equal 0 for all i #j (i.e. the off-diagonal elements of U),
the second term is 0 and B is bound solely by the relative performance of the diagonal elements
of U’.
In this section we have identified necessary conditions for the preference of general
strategies based on the model presented in matrix 1. We compared the use of general strategies
versus a number of specialized strategies considered as a set. Alternatively, general strategies
can be compared with each specialized strategy individually. Results are the same as those
presented here, only they apply separately to each column of U*.
We next use computer simulation to investigate the relationship between general and
specialized strategies in the context of dynamic interactions between treatment strategies and the
health states of patients with type 2 diabetes.
3 Dynamical Systems Model of Patient Care
We represent patients as a map Gp: from the space of possible treatments Sp, to the
space of possible patient health states Sp;: patients respond, via changes in state, to the moves
generated by treatment strategies. We represent treatment strategies as a map Gry» from the
patient state space Sp; to the treatment space Spx: treatment strategies generate moves in response
to patient state information.
The symbols 6 and @ denote parameters specifying patients and treatment strategies
respectively. For 6 and @ ranging over a set of patient categories, the map Gp.e represents patient
types and Grx» represents treatment strategies of varying specificity (we denote general
strategies as @ = -). There are two general cases: (1) @is the same as 9, and (2) @is not the same
as @. The first case corresponds to the application of a category-specific treatment strategy to a
patient in the same category; the second case corresponds to the application of a
category-specific treatment strategy to a patient of another category.
The interaction between patients and physicians (i.e. the compositions of the patient and
treatment maps) generate trajectories of patient health states and treatment moves:
Foye) =Grre° Garg: Spr Spr (16)
and
Frx(0q =Grxo° Gree Sr Sax. (17)
The pairs [Br Fev] and [Srx, Free@L] are discrete dynamical systems. We focus on the
dynamical system [Sp: Fpyg@[](i.e. the effects of treatment strategies). Specifically, we analyze
patient state trajectories generated by Fp; :
Xn = Fxg (Xo) forn 0, 1, 2,...) (18)
where F py9¢)'(Xo) is the n° iteration of the dynamic function on initial patient state x, € Sp.
We use computer simulation to operationalize the preceding dynamical system and
investigate the relative effects of general and specialized treatment strategies. Both the patient
and treatment strategies, Gp,e and Gry» can be encoded as computer programs; their interaction
(i.e. the composition Fpyg) in equation 16) generates trajectories in the patient state space.
3.1 Patient model
Figure 3 depicts the structure of the patient model that encodes the map Gpre. The
dashed box encloses the patient model; the components labeled medication effort and
psychosocial effort are inputs from the treatment strategy based on patients’ glycosylated
hemoglobin levels (HbA ic). The model updates patient HbA :c in response to inputs generated by
the treatment strategy.
We use HbAic as the outcome variable based on evidence that it corresponds with the
physiological health status of patients with Type 2 diabetes mellitus (United Kingdom
Prospective Diabetes Group, 1998), plus the fact that HbA ic is an important health indicator used
in the management of patients with type 2 diabetes (American Diabetes Association, 1999).
Higher HbA :c levels indicate worse patient health.
The patient model in Figure 3 comprises six concepts:
1) HbAic level— the patient health state variable.
2) Adherence—the extent to which a patient complies with the prescribed treatment
regimens.
3) Side effects— the adverse physical manifestations associated with medications.
4) Stress— the adverse psychophysical response to the psychosocial environment.
5) Medication effort—a representation of the amount and number of medications
prescribed by a given treatment strategy.
6) Psychosocial effort—the psychological pressure, education, and motivation applied
by the health care system to a patient regarding self-care behavior.
Ten relationships integrate these concepts:
1) Increased medication effort decreases HbAc levels, thereby improving patient health.
2) Increased medication effort increases side effects.
3) Increased side effects decreases patient adherence with prescribed medication
regimens.
4) Increased adherence increases side effects by virtue of increasing the effective
medication dose (e.g., a patient that does not take his medicine does not experience
medication induced side effects).
5) Increased psychosocial effort either increases or decreases adherence depending on
the level of psychosocial effort and specific patient parameterization used in the
definition of patient categories.
6) Increased adherence decreases HbA :c by increasing the effective medication effort.
7) Increased psychosocial effort increases stress.
8) Increased side effects increases stress.
9) Increased stress increases HbA ;c (Daniel et al., 1999).
10) Increased Hbic increases adherence. This relation is derived from the link between
HbAic, comorbidities, and motivation: high HbAic corresponds to more
comorbidities, and more comorbidities imply more manifest health consequences,
thereby motivating greater compliance with treatment regimens.
We operationalize the model by (1) defining a discrete-time update function for HbAic
(labeled h in the following equations) and (2) expanding each term to integrate model
relationships. For a given patient i at time t, we assume change in HbAic levels is effected by
two independent factors: (1) a disease effect 6, that increases h and, (2) a medication effect DEi;
that decreases h. Patient HbA1c is updated at time t according to the function?
Hiter = hit + Gt —DEie. (19)
For each patient i at a given time t we assume the disease effect 6 has a truncated
normal distribution
G1 ~ Aut Or | &: 20). (20)
The parameter [;,, is the mean of the corresponding non-truncated distribution and comprises a
patient specific time invariant characteristic m; and a positive perturbation s;, generated by the
patient’s current level of stress
Hi, =, +8, (21)
The time invariant characteristic mj is set for each patient by a draw from a log-normal
distribution
m, ~ LogNormal (-1.830664, 0.8671747) . (22)
This specification represents the distribution of the average 6-week HbAic positive change
among 6,768 patients with type 2 diabetes. The data were obtained from encounter records of a
Minnesota staff-model HMO; all patients with type 2 diabetes that had HbAc tests during the
years 1994 to 1998 are included.
The stress effect s;,: in equation 21 is modeled as
ard Hie)
1
§, =1—-(1-0.73)"" (23)
In this formulation d is the current medication effort, a is the current patient adherence, F is the
current psychosocial effort, and the constant 0.73 is the standard deviation of m; across the 6768
patient’s in the empirical data. Equation 23 is an arbitrary functional specification selected to
satisfy the assumption that increased stress increases the mean disease effect by an amount in
some interval (0, 1). We select an upper threshold t = 0.73 to bound the effect of stress within
one expected deviation. The effect of stress is maximal (equal to 0.73) when adherence,
medication and psychosocial efforts are each equal to 1. The effect of stress is minimal (equal to
zero) when either (1) adherence or medication effort is equal to 0, or (2) psychosocial effort is
equal to 0.
The oj; parameter in equation 20 (i.e. the standard deviation of the non-truncated normal
distribution) is determined as a function of the mean mj for each patient
; = .0104857 + .5189586-m; + .5775702-m;* —.0532329-m;*. (24)
The constants in this equation are estimated by a regression of patient’s standard deviation in
6-week HbA ic positive changes on a third-order polynomial of patient’s average 6-week HbA ic
positive changes.
The effect of medication effort DE in equation 19 is modeled as a proportion of the
current A 1c level
DEit = Pirie . (25)
The proportionality factor P\; is calculated as
Dia
Boe = fare. (26)
Here Dj = aitdi: is the effective medication effort (reflecting the attenuating effect of adherence
on medication effort) and g(D) is the marginal effect of D on P (i.e. dP/dD). Integrating g over
its domain (i. over the interval [0,1]) gives the maximum effect of medication effort
(MaxE ffect). We assume g is monotonically decreasing. In our analysis g(D) is a linear function
of D such that g(0) = 2-MaxEffect and g(1) =0:
g(D) = 2-MaxEffect-(1-D). (27)
Hence, as shown in Figure 4, the effect of changing medication effort from Dj,.1 to Di: can be
calculated as the area of a trapezoid:
Pit= %42-MaxEffect-(1—Dj,)+ 2-MaxEffect-(1—Djt1))-(Dit—Dit.). (28)
Adherence aj; (embedded in the definition of D) is calculated as the product of two
effects. The First effect, S, captures (1) the negative influence of medication side effects
(SideE ffect), (2) the positive influence of Alc level as an indicator of potential comorbidities
(higher HbA ic levels correspond to more comorbidities and worse manifest health conditions),
and (3) the positive influence of the HbA:c gradient with respect to the previous change in
adherence representing improved self-efficacy (Bandura, 1977; Bandura, 1997; Bandura, 2001;
Schwarzer and Fuchs, 1995). The second effect, B, captures the effect of psychosocial effort
(PsychosocialEffort) as a unimodal function representing increased adherence for levels of
psychosocial effort below a threshold and decreased adherence above the threshold.
aits1 = S(SideEffects, hj: Ah; /Aaj1)-B(PsychosocialE ffort, a, B). (29)
The function S, is arbitrarily specified to meet two conditions: (1) adherence must remain
in the interval [0,1], and (2), as a function of the variables x = (Side Effects, hi, Ah /Aai,)', S
must be able to generate both positive and negative deviations from a patient specific base level
of adherence aj. We use the specification
S, =n oar “FARE (30)
where y is a vector of weight parameters specifying the relative importance and direction of
effect for each component in x. In equation 29, B denotes a beta function with parameters a and
B. B is scaled to equal 1 at its mode. Numerous linear and nonlinear functions can be achieved
via the parameterization (a) of B.
As shown in Figure 5, adherence can either increase or decrease in response to
psychosocial effort depending on the level of psychosocial effort and the patient
parameterization of the Beta function. The functions denoted as (S =0.5, a=1, B=5) and
(S =0.8, a =5, B =1) in Figure 6 represent patients that monotonically respond to increased
psychosocial effort; the first reacts by decreasing adherence, the second by increasing adherence.
The functions denoted as (S = 1, a = 2, B =5) and (S =0.6, a =5, B =2) represent patients that
respond well to increased psychosocial up to a threshold point and then decrease adherence after
psychosocial effort exceeds the threshold. The scale S of the Beta function is set by equation 30;
higher values of S imply greater possible adherence.
Medication side effects (SideEffects in equation 29) enter the model as a proportion v of
the effective medication effort
SideEffects;; = v-Dit. (31)
Two patient categories based on adherence behavior are defined for the purpose of this
analysis: Category 1, composed of patients with Beta parameters a < B, and category 2,
composed of patients with Beta parameters a =B. Category 1 represents patients that respond
well to low levels of adherence but react negatively to high levels of adherence. Category 2
represents patients that respond well to higher levels of adherence and react negatively only to
the highest levels of adherence. Specialized treatment strategies will be defined for each
category of patient, and a general treatment strategy will be defined for all patients without
regard to categorization.
3.2. Treatment Strategies
We use goal-directed machine learning to capture the knowledge structure of treatment
strategies. More specifically, we use artificial neural networks to represent the treatment
strategies Gx,» Networks representing specialized strategies of each patient category are trained
using populations of category-specific simulated patients.
The parameter @ of the treatment strategy is an index of patient category. Levels of
specificity are achieved by varying training experience: for each category, three networks are
trained at different experience levels. Networks representing general strategies are trained using
a population composed of simulated patients from a sample of two categories. A total of 7
networks are trained (i.e. [2 categories x3 experience levels] + 1 general strategy).
An online reinforcement-learning algorithm is used to train the neural networks. The
reinforcement function embeds (1) physician goals regarding patient health states and (2)
constraints on treatment. For example, goals derived from the clinical practice literature are the
reduction of HbAic levels, the reduction of LDL levels, and the reduction of Bp level.
Treatment constraints include minimizing the number of drugs given at any one time, and
minimizing drug dosage early in treatment.
Interaction between patient and treatment strategy is achieved by inputting patient state
information to the neural network and passing the resulting treatment moves from the network to
a simulated patients. The simulated patient then generates a new patient state in response to the
treatment move. Recursive operation on resultant patient states produces a trajectory in the
patient state space.
4 Analysis
Strategies at different levels of specificity are compared using functions of patient
health-state trajectories (denoted as O(9¢). We consider two functions of HbAc trajectories: one
summarizes the overall control achieved by treatment strategies; the other summarizes the
constraint imposed by treatment strategies on patient variation across time. The first function is
the sum of state values along the trajectory of HbA: states:
Oo.0)(HbDA\C,) => Fri ee) (HbA,C,) . (32)
This outcome measure associates higher values with trajectories containing poor health states
and lower values with those containing good health states. The second function, which
characterizes the dynamic structure imposed on the trajectory by the treatment strategy, is
defined as the determinant of the autocorrelation matrix of the trajectory. This function returns
smaller values for strategies that better control variation across time in the patient's health state.
The expected outcome of either function for a set of strategies and a probability of
applying a category-specific treatment strategy to a patient of another category p(M) =m is
E(O | p(M) =m) = IZ 20 ¢0)(%) p(@ |8, x, m) p(O |x) f(x)dx. (33)
The function f(x) is the probability density of the patient state space, p(®|x) is the conditional
probability mass function of category membership (the patients actual category), and
p(@| 9, x, m) is the conditional probability mass function of categorization (the category index of
the treatment strategy applied to the patient). Neither f(x) nor p(@|x) are conditioned on m
because both are independent of categorization.
We specify the conditional probability of categorization p(@ | ©, x, m) as
(1 —m) if 9=0
. 34
p(8 |p #8,x)-m ifg +d (34)
p(@|9,m,x) +t
The probability of correct categorization (i.e. @= 8) is one minus the probability of incorrect
categorization (i.e. @ #8). The probability of incorrect categorization is distributed across the
remaining categories according to the distribution of those categories for a given health state.
When p(M) = 0 (ie. category-specific treatment strategies are always applied to
appropriate patients), the conditional probability of categorization becomes
if p =0
= . 35
p(@|8,m,x) fh ifo#8 (35)
The expected outcome then reduces to
E(O | p(M) =0) =F Oo)(x) p(x) E(w)dx. (36)
When the specificity is zero, as is the case of a general strategy (p=), the expected
outcome is
£(0 [@ =) =[$ 0,0)(x) 6 |x) f(x)dx. (37)
The difference between equations 36 and 37 is generated solely by the effect of specificity on the
outcome function.
Results presented at the conference will be in three parts: First, we characterize expected
outcomes as a function of the probability of applying category-specific treatment strategies to
other categories of patients for various levels of specificity (see Figure 6). Second, we identify
the probability of incorrect categorization for which equivalent expected outcomes are achieved,
such as the points a and b in Figure 6. Finally, we determine the probability of incorrect
categorization (such as point d in Figure 6) at which a general strategy achieves the expected
outcome of specialized strategies.
5 Conclusions
Chronic illness requires a distinct mode of care, one for which the goal is system control
rather than system repair. Successful control entails continual management of a patient's health
state, even when physical maladies are not evident. Management is achieved through treatment
strategies often tailored to specific patient categories. However, as we have shown, the
necessary conditions for strategy selection based on performance under optimal conditions may
not be preferred due to the effects of specificity and faulty categorization.
In the context of type 2 diabetes, when category-specific treatment strategies are
appropriately applied, the expected outcome using more specific strategies should exceed the
expected outcome using less specific strategies. This result implies tailoring treatment strategies
to patient categories based on psychosocial attributes, such as compliance with drug regimens,
will be successful if patient categorization is accurate. However, as the probability of applying a
category-specific strategy to patients of another category increases, the advantage of treatment
specificity is diminished.
When the probability of appropriate patient categorization is less than 1, the use of
treatment strategies with lower specificity may be preferred. In circumstances with this result, it
is not necessary that the strategy with lower specificity have a high level of performance; the
only requirement is that the lower specificity strategy performs better than the misapplication of
the strategy with high specificity. This can result in a low level of performance.
If treatment strategies are too specific and there is a possibility of incorrect
categorization, policies that strive to capitalize on tailoring strategies to manage the dynamics of
chronic illness may fail. The benefits of tailoring strategies to patient categories should be
considered in light of the expected accuracy of the categorization process.
FIGURES
Low Specificity
BG N
: dof NY sof On {Sa -\\ Sp
N rp
ae
N
Yd
A | 8 A a:
Category Indicators Category Indicators
NS — Strategy 1 targeted to category A
— Strategy 2 targeted to category B
Figure 1. Treatment strategy performance and specificity
05
09
8 n
Figure 2: Contour map of a general strategy’s relative
performance B on the space defined by relative
generality y and the probability of correct patient
categorization Tl.
Medication
Effort \
< Side Effects
)
Adherence
. +H awk
Psychosocial 4.—-—"
Effort
Figure 3: Patient Model
g(D) =2-MaxEffect:(1-D)
Area of shaded trapezoid is
equal to proportionality factor P
due to a change from Di, to
=
0 Dia
Effective Medication Effort (D)
Figure 4: The effect of changing medication effort
from D,,; to Dy.
r (S =1, a =2, B =5)
(S =0.8, a =5, B =2)
oat
(S =0.6, a =5, B =2
g 06+
: (S =0.5, a =1, B =5)
4p
op
0 0.1 02 03 04 05 06 07 08 d9
Psychosocial Effort
Figure 5: Adherence Functions
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NOTES
' The use of U,_, is a convenience for analysis but it is not an assumption: The model is based on a number system
with an algebraic structure such that V(a,bGR) A(cGR)(a-c=b); therefore, the ratio of any two non-zero numbers
exists and need not be assumed.
? At this level of description, equation 19 appears as a linear discrete-time system with control feedback DE (Sontag,
1990, p. 36); however, the definitions of 6 and DE embed the non-linearity of the system.
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