BACKGROUND
Pressure produced on the residual limb by the socket during gait can be beneficial or harmful to the amputee; it can produce benefits when it enhances control, stability, and proprioception or produce costs if it creates discomfort and leads to tissue trauma. The goal of this research was to determine whether linear multiple regression models that estimate pressures on the residual limb can be developed using data from a load cell attached to the base of a socket. The ability to model the relationships between the kinetics of gait and pressure effects on the residual limb would provide a powerful tool for research and help clinicians diagnose socket discomfort problems. It would facilitate research on socket interface designs and materials and assist clinicians in prescribing interventions for amputees with sensitive residual limbs. It might also make it possible to examine the relationship between intrasocket pressures and the gait characteristics of amputees or prosthetic foot and socket interface design characteristics.

Linear multiple regression analysis offers a means for examining and possibly simplifying potentially complex cause-and-effect relationships involving multiple variables. It is important to determine how the magnitude of an outcome variable, such as intrasocket pressure, changes with the magnitude of the independent causal variables, such as the forces and moments occurring at the base of the socket. Tests of significance alone do not indicate how cause and effect covary, but regression analysis reveals the strength of the contribution of each independent variable to the dependent variable. There may be multiple causal relationships between intrasocket pressure and independent variables that correlate with pressure. Besides the forces and moments occurring at the base of a socket, other variables that could influence pressure include foot type, liner type, residual limb morphology, and socket interface design, but these were not examined. It can be difficult to isolate or control many important pressure-producing variables in an experimental setting. During gait, the forces and moments at the base of a socket vary together over stance in a nonexperimental manner, and it would be difficult to design an experiment which isolated each while an amputee walked. Regression is useful in nonexperimental settings where both dependent and independent variables change in a complex manner over time.

Linear multiple regression is a category of linear multivariable models that assumes that the effects of multiple independent variables add together to produce an outcome where the magnitude of the contribution of each independent variable to the outcome is proportional to the magnitude of the independent variable. It is capable of retaining variables that have a significant impact on the outcome and eliminating variables that have an insignificant effect on the outcome. When certain assumptions are met, linear multiple regression models are able to isolate the effects of individual independent variables on the outcome.

Using strain gauges glued onto the pylon to estimate force and moment, an early pilot study found a high correlation between the forces and moments occurring in the pylon at the base of the socket and the pressures on the patellar tendon and gastrocnemius regions of the residual limb (R ^{2} = 0.877 and 0.960, respectively).^{1} The authors proposed the matrix formulation shown below to model socket pressures as a linear function of the forces and moments. The proposed linear model hypothesized that as measured force and moment vary during stance, pressure at specific regions within the socket will change proportionally, with the total pressure being the sum of the pressure caused by the force and the pressure caused by the moment. The cells of the matrix [W (γ )] indicate the proportional relationship between the measured forces and moments and pressure.

where

σ _{p} (γ , t ) = pressure on the patellar tendon region, which varies during stance (t );
σ _{g} (γ , t ) = pressure on the gastrocnemius region, which varies during stance (t );
W (γ ) = matrix of coefficients that relate pylon forces and moments to socket pressures, which does not vary during stance;
F_{Z} (γ, t ) = axial force in the pylon, which varies during stance (t );
M_{y} (γ, t ) = flexion-extension moment in the pylon, which varies during stance (t);
γ = socket flexion-extension alignment; and
t = time during stance.
A modern load cell attached to the base of a socket can be used to measure F_{z} and M_{y} instead of using strain gauges glued to the pylon.^{2–11} The research question was how accurately intrasocket pressures can be estimated using a linear multiple regression modeling approach and force-moment data obtained from a load cell.

In the current study, the theory of the early study was extended to multiple regions in a socket by adding rows to the matrices for [σ ] and [W (γ )] to include the popliteal and distal tibia regions. Stepwise multiple linear regression was used to fit data to the model and estimate the matrix of coefficients for a fixed alignment as the β values from the regression equations, which is discussed below. The β terms represent the proportional relationship between load cell measurements and pressure. A load cell was used to measure the forces and moments, which made it possible to include force in the anterior-posterior X direction, F_{x} , as one of the causal variables, which was not done in the earlier study. The robustness of the models was examined using load cell and pressure data produced by making small alignment perturbations to the prostheses used by the subjects.

METHODS
A convenience sample of four unilateral transtibial amputee subjects were recruited, S1, S2, S3, and S4, but data from only S2 and S3 are presented because of pressure measurement technical problems that occurred with S1 and S4 (Table 1 ). Inclusion criteria included functioning at a K3 or K4 level; ability to undertake walking without a loss of balance or an unsteady gait, which might imply a proneness to falling; and good socket fit. New amputees, preparatory prostheses, residual limb infections, open sores, and pain were grounds for exclusion. All subjects read and signed a Letter of Informed Consent describing the protocol that had been approved by the institutional review boards of the US Army Medical Research and Materiel Command and the University of Nevada, LV. The subjects supplied their own sockets, which were spare sockets that fit well in both cases. The brand of gel liner was not recorded.

Table 1: Subject characteristics

A commercially produced load cell was used to measure forces and moments at the base of the socket^{2–11} (model 45E15A41000N125; JR3, Woodlands, CA), and a commercially produced system using force sensing resistors (FSRs) (F-Socket9811E; Tekscan, South Boston, MA) was used to measure pressure between the socket and the gel liner covering the residual limb.^{12–17} One sensor array was placed on the inside of the anterior wall of the socket between the wall and the liner used by the subject to measure pressures at the patella tendon and distal tibia regions, and one array was placed on the inside of the posterior wall of the socket to measure pressures at the popliteal and gastrocnemius regions (Figures 1 B, C). The sensors cells windowed for each socket region are shown in Figure 2 . The technical specifications for the load cell and pressure measurement systems are shown in Table 2 . Single-point calibration of the sensors was undertaken following the recommendations of the manufacturer. Complete pressure data were obtained for only two subjects, S2 and S3. The wire leads from the middle columns of the anterior force sensor for S1 were destroyed by the socket brim, leading to a loss of pressure data over the patella tendon and distal tibia, and the cable connecting the anterior sensor handle to the data logger malfunctioned for S4, which also led to a loss of pressure data.

Table 2: Technical specifications of the load cell and Tekscan F-Socket used in the research

Figure 1: (A) Load cell and axes. The positive x axis direction of the load cell is toward the anterior of the foot, the positive y axis direction of the load cell is toward the left, and the positive z axis direction of the load cell is parallel to the pylon and vertical. Counterclockwise moments are positive when looking down an axis. (B) Tekscan FSR sensor array and approximate positioning for the anterior regions of the socket. (C) Setup of load cell and pressure sensors with cables leading to the data logger.

Figure 2: F-socket pressure windows.

The data from the load cell were recorded and reported in the frame of reference of the load cell as forces directed toward the residual limb and moments as they would be experienced on the residual limb (Figure 1 A). Only forces and moments in the sagittal plane were used in the analysis because the socket regions for which pressure measurements were taken were in the frontal plane. F_{z} _{pos} was directed upward toward the residual limb and parallel to the pylon, F_{x} _{pos} was directed anterior and perpendicular to the pylon, and F_{x} _{neg} was directed posterior and perpendicular to the pylon. The y axis pointed to the left; M_{y} _{pos} was counterclockwise about the y axis as viewed from the left, representing the couple produced by the sagittal plane force components F_{z} _{pos} and F_{x} _{pos} that began with heel contact. M_{y} _{neg} was clockwise about the y axis, representing the couple produced by the force components F_{z} _{pos} and F_{x} _{neg} that ended with toe-off.

Two new solid ankle cushioned heel (SACH) feet of identical design (Ohio Willow Wood, Mt Sterling, OH) were obtained for the two subjects. Subject 3 had no previous experience with the SACH foot, and S2 had very limited or no experience. Subject time limitations precluded allowing an adaption period for the feet, but this was expected to have no influence on the goal of examining the validity of a linear modeling approach. Use of SACH feet was intended to help control for variation due to type of foot.

Three sets of data were obtained for each subject. One set of pressure data, for the initial acceptable alignment, was used to calibrate the regression models, and the other two sets, for the +5-mm perturbations, were used to examine the accuracy of predictions when forces and moments changed because of alignment perturbations. Alignment perturbations were carried out using a Spectrum Alignment System (Hosmer, Campbell, CA) attached distal to the load cell, which produced 1 mm of anterior-posterior movement of the foot for each revolution of an adjusting screw. The sleeve of the alignment system, into which the pylon was inserted and secured, was perpendicular to the x and y axes and parallel to the z axis of the load cell, which ensured that the longitudinal pylon axis would be parallel to the z axis of the load cell and perpendicular to its x and y axes.

A certified prosthetist replaced each subject’s original pylon with a pylon shortened to allow the load cell and the spectrum alignment system to be attached and produced an alignment that matched the original alignment as closely as possible. Using a standard question about the acceptability of the alignment, both subjects reported that an acceptable alignment had been produced. Subjects were asked to support their weight entirely on the prosthesis to obtain a measurement of body mass supported by the load cell during single-limb stance. This measured force was used to normalize the data from the load cell. The protocol called for each subject to take approximately 15 steps at a self-selected comfortable walking speed on a level surface during a trial, which would allow several nonrepresentative steps at the beginning and end of the trial to be discarded as well as other nonrepresentative steps that might occur in the middle of a trial, for example, if the subject had to turn around. After the initial walking trial (0 mm of perturbation), the foot was moved anterior 5 mm (+5) and the walking trial was repeated, and then the foot was moved posterior 5 mm (−5) and the walking trial was repeated. Subjects were blinded to the initial alignment as well as to the direction of movement for each perturbation. Experiments were conducted in a hallway, which was sufficiently long to allow approximately 15 steps to be taken on the prosthetic side without having to turn around.

A graphic user interface (GUI) was developed to break the stream of data from the load cell and pressure sensors down into 50 consecutive stance phase intervals of equal 2% duration.^{18} Ten similar consecutive steps occurring after gait initiation were selected for analysis. The same points in time used to define stance based on F_{z} _{pos} were used to define the beginning and ending of stance for the remaining five load cell variables. The mean values of the load cell variables for each interval were exported as Excel-ready matrices having 50 rows and 10 columns (intervals by steps). The values of F_{x} and M_{y} variables, which changed sign during stance, were further processed to produce four separate variables, with two representing the positive values of the variables and the other two representing the negative values. This was necessary to maintain internal validity within the regression model, and if it had not been done, the least-squares algorithm would have attempted to compute only one regression coefficient for the independent variables that changed sign even though pressures were always positive. F_{x} _{neg} was created as a separate variable and assigned a positive value equal to the force when F_{x} was less than zero and was assigned a value of zero elsewhere, and F_{x} _{pos} was created as a variable and assigned a positive value equal to the force when F_{x} was greater than zero and was assigned a value of zero elsewhere. Load cell data for M_{y} were processed in a similar manner to produce M_{y} _{pos} and M_{y} _{neg} as independent variables.

Pressure data (FSR Pressure) were processed using the GUI in a manner similar to the load cell data. Socket regions were windowed (Tekscan F-Scan Mobile Research Installation 6.31 software) and exported as files containing the average pressures in each of the socket regions as a function of time. Each of the two sensor arrays first was windowed in its entirety to produce a movie that was used to detect when pressure began to build up after heel contact and when it decreased after toe-off. The windows for the socket regions were identified by playing the Tekscan movies and noting which cells showed the greatest increase in pressure (Figure 2 ). The GUI processed the pressure data for each window in the same way the load cell data were processed and exported one 50 × 10 Excel-compatible matrix for each windowed region of the socket. Because the GUI divided both the load cell data and pressure data into 50 intervals of equal duration having matched beginning and ending times, it was possible to merge the two data sets for regression modeling.

Means were computed for each of the 50 intervals over the 10 steps and used for fitting of the models. Pearson product-moment coefficients of correlation were computed between the resultant sagittal plane force, FR_{S} = (F_{z} ^{2} + F_{x} ^{2} )^{0.5} , and resultant moment about the y axis, MR_{Y} = (M_{y} ^{2} )^{0.5} , to determine the amount of colinearity between the independent variables.

Stepwise regressions were fitted to the processed data for each of the pressure regions using all five sagittal plane variables and the following theoretical linear model (SPSS Statistics 17.0):

where

(t ) is the time interval, 1…50
P _{meas} (t ) is the average socket region pressure measured in kPa during interval t ;
α and β are the true regression coefficients to be estimated using least-squares methods and represent the cells of [W ] in the model proposed earlier;
F_{z} (t), F_{x} _{pos} (t) and F_{x} _{neg} (t ) are forces measured in N during interval t ; and
M_{y} _{pos} (t) and M_{y} _{neg} (t) are moments measured in Nm during interval t .
This resulted in fitted equations of the form

where

P _{est} (t ) is the pressure estimated by the model during interval t ; and
A and B are the calibrated estimates of the true regression coefficients.
Each of the B coefficients could be interpreted to have units that make the equation dimensionally correct. If interpreted in this manner, the B s associated with forces would have units in terms of “units of pressure per unit of force,” the B s associated with moments would have units in terms of “units of pressure per unit of moment,” and A would have units in terms of “pressure.” If the stepwise algorithm did not result in the inclusion of an independent variable in the model, B was zero.

To aid in the evaluation of the linear models, the terms in the equations representing forces were summed to create a force component, PF, and the terms representing moments were summed to create a moment component, PM. This allowed the Equation 3 model to be represented as

where

P _{est} (t ) = estimated socket region average pressure in kPa during interval t ;

A = estimate of α ;

PF(t ) = B_{Fz} × [F_{z} (t )] + B_{Fx} _{pos} × [F_{x} _{pos} (t)] + B_{Fx} _{neg} × [F_{x} _{neg} (t )] is the component of estimated pressure due to force during interval t ; and

PM(t ) = B_{My} _{pos} × [M_{y} _{pos} (t )] + B_{My} _{neg} × [M_{y} _{neg} (t )] is the component of estimated pressure due to moment during interval t .

Stepwise regression creates linear models in a series of steps using algorithms that make decisions on which variables to include and exclude. At each step, the variable that most improves the coefficient of multiple correlation is entered into the model, and variables previously entered into the model are tested to determine if they should remain in the model. If two or more variables are strongly colinear, only one should be allowed to remain in the final model. To help achieve this, a significance of p < 0.005 was required for entry of a load cell variable into the stepwise regressions, and if the significance of its contribution to P _{est} became p < 0.10 when a new variable was entered, the variable was removed. This ensured that a variable had to produce a highly significant contribution to R ^{2} to enter a model and allowed it to be removed if its contribution to R ^{2} declined substantially when another colinear variable was introduced to the model.

After fitting the regression equations for the initial alignment, the values of the load cell variables were entered back into the calibrated models to estimate the contributions of the force components PF(t ) and moment components PM(t ) to total estimated pressure, P _{est} (t ). To determine if the models were alignment specific, the models obtained for the initial alignment were applied using the load cell and pressure data for each of the perturbations (+5, −5) to produce pressure estimates. Root-mean-square error (RMSE) was computed for each alignment and each region of the socket as

To facilitate interpretation of the results, the ratio of RMSE to the peak pressure estimated from the alignment model was computed as PRSME:

RESULTS
The values of the five independent variables measured by the load cell, which were tested by the stepwise algorithm for entry into the models, are plotted in Figure 3 . Normalized resultant sagittal plane force FR_{S} versus normalized moment in the sagittal plane about the y axis MR_{Y} is plotted in Figure 4 . The figures reveal that patterns of loading were different between subjects for SACH feet of the same design. For both subjects, FR_{S} exhibited a double peak similar to vertical ground reaction force. S3 had a first peak for FR_{S} that occurred after the moment MR_{Y} had passed through zero, which suggests that the loading of the first force peak was along a line of action anterior to the pylon. S2 exhibited force peaking when the line of action of the resultant was posterior to the pylon. Thus, first peak force loading may have occurred for S2 as the heel was loaded, and for S3, it may have occurred as the forefoot began to be loaded. The normalized moment during the first peak resultant force was higher for S3 than for S2. S2 generated a greater normalized moment than S3 at the second peak, implying that peak force may have occurred when its line of action was further toward on the toe. However, both subjects exhibited approximately equal second normalized peak resultant forces. The data suggest that S2 may have been using a greater proportion of the keel length of the SACH foot between the two resultant force peaks than S3 was.

Figure 3: Sagittal plane load cell measurements.

Figure 4: Normalized sagittal plane resultant force versus normalized resultant moment M_{y} . As shown by the arrows, stance progresses to the right along the bottom portion from the origin (heel contact) to the first peak of resultant force, vertical to the second peak of resultant force, and left along the top portion to toe-off at the origin. The distance between pairs of markers represents 2% of stance. The relative angles of lines drawn from the origin to the markers indicate the relative perpendicular distances from the load cell to the line of action of the resultant force.

The regression coefficients and values of R ^{2} for the linear models that resulted are shown in Table 3 and the relative contributions of PF and PM are shown in Figure 5 . The stepwise regression algorithm selected independent variables M_{y} _{neg} and F_{x} _{neg} for every model. Moment M_{y} _{neg} was the major source of peak pressure at the patella tendon during late stance. Variable F_{z} , the force parallel to the pylon, appeared in seven of eight models and dominated pressures generated at the distal tibia, popliteal, and gastrocnemius regions throughout stance. F_{x} _{pos} and M_{y} _{pos} appeared in only two models each, suggesting that they played a less pervasive role in pressure generation.

Table 3: Pressure model regression coefficients and goodness of fit

Figure 5: Estimated versus measured pressure. PF, force component; PM, moment component; P _{est} , pressure estimated by the model during interval t ; P _{meas} , average socket region pressure measured in kPa during interval t .

The accuracy of the linear models with respect to the measured pressure data is shown in Figure 5 and Table 4 . The contributions of the force and moment components to estimated pressure, PF and PM, are shown in Figure 5 . The calibrated regression models exhibited coefficients of multiple correlation very close to 1.00, a value that would indicate no error in the estimation of pressure. Values ranged from 0.996 (S3 gastrocnemius) to 0.929 (S3 patella tendon).

Table 4: RMSE for initial alignments and perturbations

Figure 5 indicates that measured pressures exhibited different patterns for the two subjects. S3 exhibited higher pressures than S2 did for all regions except the patella tendon, where S2 exhibited higher pressures. The linear relationship that is hypothesized to underlie linear regression analysis seemed to exist at all socket sites for most of stance with several exceptions. Possible reasons for nonlinearity are proposed in the Discussion section below. At the gastrocnemius regions for both subjects and at the popliteal and distal tibia regions for S2, pressures estimated by the models were nearly identical to measured pressures throughout stance. Residual limb tissues at the gastrocnemius region are homogeneous and the hypothesis of a linear relationship seemed to be supported. At the patella tendon region of both subjects and at the distal tibia and popliteal regions of S3, the measured pressures exhibited peaking that was not captured by the linear models, although estimated pressures were very close to estimated pressures. The sharp peaking of P _{meas} in the patella tendon region suggests a slightly nonlinear relationship, possibly because of changes in the stiffness of the underlying tissues, as discussed below. An interesting finding was that at the distal tibia during the second peak of pressure in late stance, the models predicted for both subjects that the moment component reduced pressure created by the force component; for S3, the amount of the reduction was very large. The model for S3 also predicted that the force component reduced pressure noticeably in the patellar tendon region during the latter 75% of stance.

The Pearson product-moment correlations between FR_{S} and MR_{Y} are shown in Table 4 for all alignments and indicate that between 26.4% and 50.6% of their variance was common. Colinearity between the force and the moment produced by it was present. The magnitude of the correlation was numerically similar within each subject as alignment varied but was different between the two subjects. This was further evidence that response to feet of the same design varied between subjects. Figure 4 , which graphs the resultant force versus resultant moment for both subjects, indicates that strong colinearity occurred wherever the lines were relatively straight and sloped. Colinearity was evident before the first peak of resultant force, as shown along the bottom leg of each triangle and after the second peak of the resultant force in the upper right hand corner of each triangle as stance progressed toward toe-off in the lower left corners of the plots. Where the lines tended to run vertically between the first and second force peaks, there appeared to be a low amount of colinearity.

Root-mean-square error expressed as a percentage of peak estimated pressure, PRMSE, was lowest when the models were run for the initial alignment data used to calibrate them but averaged 1.6 to 4.6 times greater when data from the alignment perturbations were used in the models (Table 4 ). This suggests that the models were alignment specific and lost accuracy when variations occurred in the values of the independent variables because of small alignment perturbations. This could be the result of colinearity in combination with the use of a mathematical algorithm to minimize the sum of the square of the error term during regression model calibration. The increase in error at the patella tendon was high for S2, with perturbation in either the anterior or posterior direction and for both subjects at the gastrocnemius with anterior translation of the foot. The condition for which error increased the least was at the distal tibia for S3.

DISCUSSION
THEORETICAL IMPLICATIONS
The linear multiple regression modeling approach is based on a hypothesis of residual limb loading that is conceptually appealing. Linear models are widely used to quantify many types of relationships. The least-squares fitting criterion estimated peak pressures with low error for the different subjects and socket regions of importance to prosthetic fitting. However, the models seemed to provide good estimates of pressure for only a single alignment and subject and lacked the universality needed for scientific generalizations.

The rationale for proposing that a linear relationship exists between force/moment at the base of a socket and pressure on the residual limb is based on concepts from engineering mechanics. However, several key requirements must be met for a linear model of socket pressure to be theoretically correct. The primary requirements are that the pressure measurements at the wall of the socket must be linearly proportional to the force and moment measurements at the base of the socket and the pressure effects of the force and the moment must add together.

The force acting on the center of mass of the body is transmitted down through the hip and femur to the knee and then to the tibia of the residual limb, and the tibia transmits force through the soft tissues of the residual limb to the wall of the socket, where it produces pressure. A linear relationship can be hypothesized between the pressure on the wall of the socket and the load caused by the mass of the body if it is assumed that the transtibial residual limb is shaped like an inverted truncated cone fitted into a socket to which it conforms. A proportional relationship can be hypothesized to exist between the pressure on the walls of the socket and the load measured distal to the socket. This follows from the static analysis of force versus pressure relationships for a wedge.^{19} Although the tissues of the residual limb are viscoelastic and may absorb energy and exhibit quasilinear behavior,^{20} a load cell attached to the base of the socket will measure only the load that is transmitted to the inside walls of the socket subsequent to energy-absorbing phenomenon related to residual limb tissue response.

A linear relationship also can be expected between pressure and moment. The distance from the load cell to a point on the interior socket wall such as the center of the popliteal region remains constant, so the force exerted at this point in the socket due to the moment measured at the base of the socket should increase linearly with the measured moment. Strongly linear relationships were observed for the gastrocnemius regions of both subjects and the popliteal and distal tibia regions for S2.

However, for the linear relationship to hold over a range of loading magnitudes, the distribution of force over an area of the socket must not vary with the magnitude of the force. This assumption could be more difficult to meet in some socket locations, such as the patella tendon, where nonlinear relationships may have existed during socket loading. The force measured at the load cell is not generated at a single point on the socket but is the result of pressure generated over a region on the inner wall of the socket. If the proportion of the force generated by a fixed socket region such as the patella tendon does not remain constant relative to other regions of the socket as the force measured at the base increases, then the relationship between pressure and force in a specific socket region may be nonlinear. For example, as the knee joint flexes and extends during stance, the prominence and pressure generating role of the stiffer tissue of the patella tendon may vary. If stiff patella tendon tissue emerges within softer adjacent tissue as stance progresses, the stiffer tissue may generate a higher force per unit of socket area than the softer adjacent tissue. Greater amounts of force may be distributed over an increasingly smaller area in the socket as the underlying tissue stiffness increases and becomes more prominent over a smaller region of the socket. If this happens, the rate of pressure increase may exhibit a nonlinear response as force increases, and pressure could peak more sharply than would occur from a linear relationship where pressure increases in a constant proportion to force. This peaking effect on pressure was observed for both subjects at the patella tendon during the occurrence of peak force and may have also occurred at the distal tibia of S3.

Concerning the need to have the effects of force and moment be additive, it is a basic principle of Newtonian mechanics that forces add together as vectors. Thus, the pressure effects of the force occurring at the base of the socket and the pressure effects of the additional force produced by the moment will add together.

A related theoretical requirement of linear multiple regression models is that the independent variables should not exhibit colinearity or correlation. This requirement was not fully met because the moment at the base of the socket is dependent on the force that occurs there during stance; MR_{Y} = r _{S} × FR_{S} , where r _{S} is the perpendicular distance from the point at which the moment is measured to the line of action of the resultant force. When colinearity is present, least-squares methods will not be able to isolate completely the effects of force and moment on pressure, and the result will be a degree of indeterminacy in the regression coefficients. However, Figure 4 reveals that as stance progresses, colinearity is strongest before the first peak force and subsequent to the second peak, where the curves are nearly straight lines that slope upward and to the right from the origin. These strongly linear portions of the curves represent approximately 40% of stance and comprise the forces and moments of lowest magnitude. Thus, the effects of colinearity on least-square estimates of the regression coefficients will be somewhat muted by the strongly non-colinear relationships and high magnitudes associated with the portions of the curves between the first and second peaks, where the length of the moment arm increases as the line of action of FR_{S} progresses from the heel toward the toe. Visual inspection of Figure 5 facilitates interpretation of the validity of the models and is discussed below in the section on clinical implications.

Colinearity may be an issue that prevents the development of general regression models of intrasocket pressure. The probabilities used in the stepwise regression to allow independent variables to enter and be removed from the model can minimize the colinearity problem to some extent but cannot completely eliminate it. Regression models were run for the perturbations and revealed changes in the magnitudes of the regression coefficients. In some cases, the independent variables that entered into the models and were removed from them changed as the least-squares algorithm was applied. However, the R ^{2} values for the models of the perturbations were as high as those for the initial alignment. Results also were dependent on the quality of pressure measurement, and FSR sensors have shortcomings related to drift, hysteresis, rate of loading, lag, and calibration.^{21–25}

CLINICAL IMPLICATIONS
The interpretation of the results shown in Figure 5 has relevance to clinical problems that involve the diagnosis of socket fit and discomfort. The regression approach used in the study was able to very accurately estimate pressure for an individual patient as a function of the magnitudes and timing of the forces and moments at the base of the socket, and Figure 5 indicates the relative contribution of the forces and moments to the timing and magnitude of pressure. If one of the modeled regions of an amputee’s residual limb developed a pressure-related problem, a depiction similar to Figure 5 for the amputee would allow the clinician to determine the relative contribution of the force and moment components to peak pressure and the time during stance when the peak occurred. This could help identify a strategy to ameliorate the problem. A relevant question is “What factors influence the magnitudes and timing of the forces and moments?” They could be related to the type of activity, foot design, alignment, socket interface design, or the gait habits of the individual.^{26–28}

Figure 5 suggests that for self-selected comfortable speed level walking, pressure complaints or pressure-related tissue problems in the gastrocnemius, distal tibia, and popliteal regions are more likely to be a result of the force component than the moment component. Pressure problems in these regions may be a function of the peak force generated during weight acceptance in early stance and the propulsion phase in late stance. With respect to weight acceptance, the pressure effects of the moment component M_{y} _{pos} peaked before the first peak of the force component and were small for both subjects. Thus, M_{y} _{pos} , which would normally be associated with the heel loading moment, does not seem to play a major role in generating high pressure at any socket region in the models during weight acceptance.

Figure 5 also suggests that during the propulsion phase of stance, the moment component becomes an increasingly likely candidate as a source of pressure, particularly in the patella tendon region, where it is the dominant pressure-producing component. In the popliteal and gastrocnemius regions during propulsion, the forefoot loading moment plays a smaller role than force but is still a significant component of pressure. Figure 5 suggests that the moment component may actually reduce pressure at the distal tibia during the propulsion phase. This seems logical because the loading of the forefoot is associated with a moment that tends to rotate the socket toward the patella tendon and away from the distal tibia.

An unexpected result of the regression model for S3 at the patella tendon was that during forefoot loading, the force component seemed to reduce the pressure-producing effect of the moment component because the force component was subtractive. Reasons for the subtractive effect of FR_{S} can only be conjectured. Figure 5 suggests that it could be a consequence of the attempt to model nonlinear behavior in the patella tendon region using a least-squares algorithm and a linear model. Additional research would be necessary to determine if nonlinear modeling approaches would improve peak pressure estimation at the patella tendon.

Additional research also would be necessary to determine scientifically the most cost-effective type of ameliorative action for pressure-related problems in the socket—a change in socket interface design, foot design, alignment, or gait modification. Based on Figure 5 , it can be conjectured that if residual limb pathology or sensitivity is the underlying reason for pressure problems, then the only prosthetic options that will reduce the problems might involve reducing the peak force associated with weight acceptance and propulsion. This could call for having the patient produce a less forceful gait, selecting a foot design that reduces the force component, adding a shock absorber, or using a socket interface design that reduces pressure. Reduction of the peak moment component through foot design and alignment perturbation may not alleviate the problem. The forefoot loading moment component is a function of foot design and alignment,^{26,27} and Figure 5 suggests that forefoot moment would have its greatest impact on pressure at the patella tendon during propulsion. The moment component during propulsion would have less effect than the force component at other locations in the socket. Although alignment perturbation may decrease the force and moment at the base of the socket during one phase of stance, it may increase the force and moment during another phase of stance, and this could increase pressure in a different region of the socket.^{27}

Additional research would need to be conducted to confirm whether the patterns of the two subjects are the norm for most transtibial amputees and to determine if a general model could be developed that would allow load cell data from any amputee to be plugged into it to estimate pressures. As mentioned above, one of the confounding influences could be the design and fit of the socket, which is a function of individual practitioner habits and skills. Sockets produced by different practitioners along with differences in socket fit could cause large variations in pressure profiles, as evidenced by the differences between S2 and S3 in Figure 5 . The regression models indicate that the peak weight acceptance and propulsive forces produced by an amputee are major contributors to intrasocket pressures, but these might be related to gait habits that are difficult for the amputee to modify. The extent to which gait habits of the amputee influence pressures, and vice versa, also may be causally related to model regression coefficients. The influence of socket interface design^{28} and gait habits on the regression coefficients and their relative contribution to the force and moment components could be a focus of future research.

CONCLUSION
Additional research is necessary to determine if a multivariable modeling approach is useful to researchers and practitioners. There is a clinical need for models that estimate pressures on the residual limb for purposes of finding ways to control and reduce unwanted pressure. This study demonstrated that linear regression models fitted to load cell data collected from individual amputees can predict pressures very accurately for those amputees and can explain the relative contribution of force and moment to pressure in regions of the socket where tissue composition is homogeneous or stiffness remains consistent during gait. This includes the gastrocnemius, popliteal, and distal tibia regions. The regression models indicate how pressure magnitudes change as a linear function of changes in the magnitudes of the forces and moments at the base of a socket. At the patella tendon, a nonlinear modeling approach may be needed to improve accuracy.

However, colinearity issues and the apparent uniqueness of the fitted models to individual amputees, alignments, and possibly socket interface designs may limit their use for scientific research. If there is a need within a scientific experiment to maintain strict independence between the forces and moments produced at the base of the socket, then a regression model of the type developed in this study may not meet completely the theoretical requirements of the experiment. If the scientific hypotheses to be tested do not seek to recognize or account for the apparent uniqueness of each amputee, the variance of the regression coefficients among research subjects could be problematic.

Research on the use of the regression approach needs to be conducted using a greater number of subjects, including those who function at K3 or K2 levels, to determine the extent to which generalizations can be made. Additional research might lead to improved paradigms that could assist clinicians diagnose discomfort problems.

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