Biomechanical Modeling of Brachialis-to-Wrist Extensor Muscle Transfer Function for Daily Activities in Tetraplegia

Background:

We recently reported a novel case demonstrating the feasibility of a brachialis (BRA)-to-extensor carpi radialis brevis (ECRB) tendon transfer, but it is not yet known whether this transfer provides robust functional results across activities. The purpose of this study was to use biomechanical modeling to define the functional capacity of the BRA-to-ECRB tendon transfer in terms of enabling the performance of several activities of daily living.

Methods:

A model of the transferred BRA-ECRB muscle-tendon unit was developed to calculate isometric elbow and wrist joint torque as a function of elbow and wrist angles resulting from different BRA reattachment locations from 50 to 80 mm proximal to the wrist joint crease. Using this model, mathematical optimization predicted the optimal location for BRA reattachment in order to perform each of a number of important upper extremity tasks as well as to calculate a global optimum for performing all of the tasks.

Results:

Analysis of active joint torque showed that the entire elbow torque-angle curve surface shifted “diagonally” toward elbow flexion and wrist extension as the attachment location approached the wrist joint; peak wrist torque was produced at extended wrist angles. Our model predicted that the optimal attachment location for each different task ranged from 54.3 to 74.6 mm proximal to the wrist joint, which is feasible given the anatomy of the muscle-tendon unit. The attachment location to optimize performing all tasks was calculated as 63.5 mm proximal to the wrist joint.

Conclusions:

This study clearly demonstrates that the BRA, which is underused as a donor in tetraplegia surgery, is an excellent donor muscle to provide wrist extension. Biomechanical simulation further highlighted the need to consider not only donor-muscle appropriateness but the patient’s desired function when planning surgical tendon transfers.

Clinical Relevance:

Quantitative evaluation of the way that surgery affects daily tasks rather than simply matching muscle properties may be a more appropriate approach for surgeons to use when choosing and tensioning donor muscles.

Tendon transfers are routinely used to improve hand function in individuals with muscle paralysis¹. The classic criteria used to choose donor muscles relate to donor availability, donor site morbidity, direction of transfer, the one muscle-one function concept, matching of muscle architectural properties, and functional synergism^2-4. In spite of the numerous transfers that have been proposed and implemented, there is not universal agreement among surgeons on the exact criteria used to choose a donor muscle, and appropriate donor muscles are not always available.

Transfer of the brachioradialis to the extensor carpi radialis brevis (ECRB) has been most commonly used to restore wrist extension in patients with tetraplegia, which is needed for maneuvering an electric wheelchair, feeding, gripping, drinking, and donning pants⁵. However, a subgroup of patients with a cervical spinal cord injury has nonfunctional or very weak brachioradialis muscles (group 0 according to the International Classification for Surgery of the Hand in Tetraplegia [ICSHT]) and are not considered eligible for surgical reconstruction of wrist extension.

Although anatomically well described⁶, the brachialis (BRA) is an underappreciated donor muscle that can provide a valuable salvage procedure to correct wrist extensor weakness not only after brachial plexus injury⁷ but also after high-level cervical spinal cord injury. We reported what we believe to be the first successful completion and 3-year follow-up of a BRA-to-ECRB tendon transfer, thus demonstrating the feasibility of this transfer⁸. However, practical considerations did not permit us to broadly define the extent to which this transfer provides good functional results across a number of patients in a full clinical trial. Addressing these issues in humans typically requires biomechanical modeling⁹ as the first phase of laboratory validation.

Most previous quantitative and semiquantitative approaches to determine muscle-donor suitability compared the force-generating properties of the transferred muscle after surgery to the muscle’s original function before surgery^10-13. However, the patient’s desired function should be a primary driver for a tendon transfer. Thus, quantitative evaluation of the way in which the surgery affects tasks to be performed after the surgery, rather than matching muscle properties, may provide more appropriate information to a surgeon choosing and tensioning donor muscles.

The purpose of this study was to use biomechanical modeling to define the functional capacity of the BRA-to-ECRB tendon transfer in terms of enabling the performance of several activities of daily living. These data highlight the need to consider not only donor-muscle appropriateness but also the patient’s desired functions when planning surgical tendon transfers. The results clearly reveal that the BRA is an excellent choice as a donor muscle for wrist extension in light of the many activities of daily living that can be accomplished after this transfer.

Materials and Methods

Biomechanical Model Details

To predict maximum isometric elbow flexion and wrist extension torque produced by the BRA and ECRB, respectively, we created a modified Hill muscle model in MATLAB (R2018b, The MathWorks; code available on request at rlieber@sralab.org) and used OpenSim v3.3¹⁴ to define muscle-tendon unit (MTU) length given elbow and wrist kinematics based on the MoBL_ARMS_module2_4_allmuscles model¹⁵. We used default MTU paths for the BRA and ECRB to calculate the MTU lengths and moment arms before the BRA-to-ECRB tendon transfer surgery (Fig. 1-A) and then modified these values (Table I) based on estimated moment arms measured 1 year after the surgery in 2 different elbow positions (full extension and 90° of flexion)⁸ (Fig. 1-B). In the neutral wrist position (no pronation or supination and no radial or ulnar deviation), the MTU length and moment arm were calculated as a function of both the elbow angle (from full extension [0°] to 120° of flexion) and the wrist angle (from 60° of extension to 60° of flexion).

Fig. 1.

Figs. 1-A and 1-B Graphical representations of the muscle-tendon unit (MTU) model used to simulate the brachialis-to-extensor carpi radialis brevis (BRA-to-ECRB) tendon transfer. Fig. 1-A Presurgery parameters for the BRA and ECRB. Fig. 1-B One-year post-surgery parameters for the BRA-ECRB MTU model based on the estimated moment arm measured by ultrasound at 2 different elbow positions of a single patient (red dots). Note that the BRA-ECRB after surgery is biarticular and thus its length varies as a function of both the elbow and the wrist angle, achieving its longest length with the elbow extended (0°) and wrist flexed (60°).

TABLE I.

Musculotendon Path Attachments Defined for BRA-ECRB After Transfer

Attachment Type	X Coordinate	Y Coordinate	Z Coordinate	Body Segment	Coordinate System	Min. Angle (deg)	Max. Angle (deg)
fixed	0.007	−0.174	−0.004	humerus	—	—	—
fixed	0.01	−0.23	−0.003	humerus	—	—	—
via	0.024	−0.281	0.004	humerus	elbow_flexion	0	10
via	0.019	−0.001	0.018	ulna	elbow_flexion	0	20
via	0.019	−0.01	0.019	ulna	elbow_flexion	0	30
via	0.017	−0.018	0.02	ulna	elbow_flexion	0	40
fixed	0.013	−0.027	0.022	ulna	—	—	—
fixed	0.024	−0.071	0.013	radius	—	—	—
fixed	0.029	−0.131	0.024	radius	—	—	—
fixed	0.035	−0.228	0.039	radius	—	—	—
fixed	0.005	0.028	0.001	third metacarpal	—	—	—

To predict the muscle force generated by the BRA and ECRB before surgery and by the transferred BRA-ECRB after surgery, the Hill model used 2 inputs (muscle activation and MTU length) and several muscle-specific parameters (Table II). Muscle activation was set to 0.01 for the passive state and 1.0 for the active state. The active fiber force-length relationship was represented as a Gaussian function¹⁶: ${\overline{F}}_A^F(a,L^F)=a{e}^{-{(L^F/L_o^F)}^2/\gamma }$ where ${\overline{F}}_A^F$ is active fiber force normalized to maximum isometric force ($F^M$),$\hspace{0.17em}a$ is activation level, $L^F$ is absolute fiber length, $L_o^F$ is optimal fiber length, and $\gamma$ is the shape factor. $F^M$ was calculated based on a constant specific tension (22.5 N/cm²)^17,18 and the muscle physiological cross-sectional area (PCSA; Table II). The value for $\gamma$ was set to 0.15 to approximate the normalized force-length relationship of human muscle sarcomeres^19,20 (see Appendix Fig. S1 for the muscle-specific comparison between the BRA and ECRB). The passive fiber force-length relationship was represented by the natural exponential function²¹ measured for a whole human gracilis muscle: ${\overline{F}}_P^F\left(L^F\right)=0.005786e^{7.519(L^F/L_o^F-1)}-1$ where ${\overline{F}}_P^F$ is the normalized passive fiber force. The total normalized fiber force was then calculated as the sum of the active and passive fiber forces: ${\overline{F}}^F\left(a,L^F\right)= {\overline{F}}_A^F\left(a,L^F\right)+{\overline{F}}_P^F\left(L^F\right)$ where ${\overline{F}}^F$ is the total normalized fiber force. The tendon force-strain relationship was determined by fitting a natural exponential function to experimental data previously reported for the major forearm muscles by our laboratory²²: ${\overline{F}}^T\left(L^T\right)=a{e}^{b(L^T/L_S^T-1)}-1$ where ${\overline{F}}^T$ is normalized tendon force, $L^T$ is tendon length, $a$ and $b$ are fitting coefficients, and $L_S^T$ is tendon slack length. (See Appendix Figs. S2 through S13 for sensitivity analysis of the effects of different tendon curve assumptions on the simulation outcomes. In general, the values selected for these tendon parameters do not have a large influence on the results presented here.)

TABLE II.

Muscle Model Parameters Used in Tendon Transfer Simulation

Parameter*	Parameter Value
	BRA	ECRB	BRA-ECRB
$L_o^F$ (mm)	85.815,31	58.515,32	85.8 (same as BRA)
$L_S^T$ (mm)	53.515	222.315	Varied
$\alpha$ (deg)	031	8.932	0 (same as BRA)
PCSA (cm2)	5.431	2.732	5.4 (same as BRA)
$F^M$† (N)	121.5	61.4	121.5 (same as BRA)

^*See Materials and Methods for explanation of parameter abbreviations.

^†Peak muscle force was predicted from the muscle physiological cross-sectional area (PCSA) and a muscle-specific tension of 22.5 N/cm².

To calculate the MTU force for a set of muscle activations (i.e., 0.01 for passive or 1.0 for active) and MTU lengths, fiber and tendon lengths must meet the constraint: $L^{MT}=L^T+L^F\cos \alpha$ where $L^{MT}$ is MTU length and $\alpha$ is muscle fiber pennation angle. Once fiber length is given, tendon length is calculated as $L^T=L^{MT}-L^F\cos \alpha$. Then the force developed by the muscle is calculated and the corresponding tendon length is determined. Here, fiber and tendon force must satisfy the relationship: ${\overline{F}}^T={\overline{F}}^F\cos \alpha =\left({\overline{F}}_A^F+{\overline{F}}_P^F\right)\cos \alpha .$The final normalized force is then multiplied by the specific tension and muscle moment arm to calculate joint torque for a given combination of muscle activation and elbow and wrist angles.

Optimization of the Tendon Graft Attachment Location for a Given Functional Task

The surgical procedure was modeled as shown schematically in Figure 2. Briefly, a 180-mm tendon graft (nominal length of the tibialis anterior [TA] tendon harvested from the leg) connects the distal BRA tendon to the proximal ECRB insertion tendon with 30 mm of overlap between the donor and recipient tendon at each side, resulting in the effective tendon graft length of 120 mm. For the given 180-mm tendon graft, the tendon graft attachment location was defined relative to the wrist joint and then varied systematically from 80 mm proximal to the wrist crease to a more distal location 50 mm proximal to the wrist crease (suture positions closer to the wrist joint correspond to shorter resting MTU length). We then optimized the surgical attachment location for many important activities of daily living (Table III) by calculating the attachment location that produced the maximum average torque over the selected range of motion. These upper extremity functional tasks were chosen based on a standard set of such tasks created to define functional outcomes needed for patients with tetraplegia²³, which in turn were based on the range-of-motion requirements for upper extremity activities of daily living^24,25.

Fig. 2.

Schematic representation of the brachialis-to-extensor carpi radialis brevis (BRA-to-ECRB) tendon transfer surgery. A tendon graft connects the BRA distal tendon to the ECRB insertion tendon. This insertion site position (black dots) is varied based on the specific simulation. Note that the passive and active BRA-ECRB force depends greatly on the attachment location relative to the wrist joint. TA = tibialis anterior.

TABLE III.

Range of Motion of Elbow and Wrist Joints Required for Specific Upper Extremity Tasks

Task*	Elbow Flexion (deg)	Elbow Extension (deg)	Wrist Flexion (deg)	Wrist Extension (deg)
Drinking (sitting position)24	120	80	5	−5
Removing object from shelf25	120	10	11	−32
Applying deodorant25	104	25	11	−27
Moving hand to back pocket25	101	20	28	−15
Donning and zipping pants25	98	20	38	−40

^*The drinking task was performed by hemiplegic subjects while other tasks were performed by neurologically intact individuals.

To calculate the optimal surgical decision for a given functional activity, we used an “objective function” to define the best outcome of the surgery—i.e., the one that maximized the average joint torque across the range of elbow and wrist joints, expressed as:

$$\max \frac{{{\displaystyle \sum }}_{{\theta }_{e1}}^{{\theta }_{e2}}{{\displaystyle \sum }}_{{\theta }_{w1}}^{{\theta }_{w2}}\left(w_e{\overline{T}}_e+w_w{\overline{T}}_w\right)}{\left({\theta }_{e2}-{\theta }_{e1}+1\right)\left({\theta }_{w2}-{\theta }_{w1}+1\right)}$$

where ${\theta }_{e1}$ is the elbow extension angle, ${\theta }_{e2}$ is the elbow flexion angle, ${\theta }_{w1}$ is the wrist extension angle, ${\theta }_{w2}$ is the wrist flexion angle, $w_e$ is the weight factor for the elbow joint torque normalized to peak elbow joint torque (${\overline{T}}_e$), and $w_w$ is the weight factor for the wrist joint torque normalized to peak wrist joint torque (${\overline{T}}_w$), with angles represented as integers in degrees. The weight factors were designed to allow the surgeon to adjust the relative importance of the elbow joint compared with the wrist joint during functional tasks—both were set to 1 in this study (see Appendix Figs. S14 and S15, which illustrate the sensitivity of the effects of different weight factors on each simulation outcome).

Finally, to facilitate comparison across these many conditions, we defined a “performance index” to summarize the way in which a surgical technique optimized for one task performed for each of the other tasks. The performance index was defined as the average normalized torque-generating capacity within the joint range for each functional task divided by the average normalized torque-generating capacity within the joint range of the optimized task.

Source of Funding

Research support was provided by Department of Veterans Affairs Grant I01 RX002462 and salary support for Dr. Lieber was provided by Grant IK6 RX003351. Research support was provided by the Swiss Paraplegic Centre and the Swiss Paraplegic Foundation to Dr. Fridén.

Results

As expected, the model demonstrated that passive joint torque during elbow flexion and wrist extension increased exponentially as the attachment location approached the wrist joint (smaller attachment locations are closer to the wrist; Fig. 3). Regardless of the attachment location, peak passive torque always occurred at full elbow extension (i.e., 0°) and 60° of wrist flexion. However, torque magnitude varied, with a peak elbow passive torque of 0.21 N m, 0.64 N m, and 2.00 N m with attachment positions of 80 mm, 65 mm, and 50 mm, respectively (Figs. 3-A, 3-B, and 3-C). Similarly, peak wrist passive torque was 0.13 N m, 0.39 N m, and 1.21 N m with attachment positions of 80 mm, 65 mm, and 50 mm, respectively (Figs. 3-D, 3-E, and 3-F).

Fig. 3.

Passive torque-angle characteristics after surgery for different tendon graft attachment locations. As the attachment location approaches the wrist joint (from 80 to 50 mm), elbow flexion torque (Figs. 3-A, 3-B, and 3-C) and wrist extension torque (Figs. 3-D, 3-E, and 3-F) increase exponentially.

Our simulation also demonstrated that the active torque-angle surface was significantly affected by the attachment location (Fig. 4). The attachment location of 80 mm yielded a transfer that produced peak active elbow flexion torque at 60° of elbow flexion and 60° of wrist flexion (Fig. 4-A). However, as the attachment location approached the wrist joint (decreasing the resting BRA-ECRB MTU length), the entire torque-angle curve surface shifted “diagonally” toward elbow flexion and wrist extension (Figs. 4-B and 4-C). With regard to wrist extension torque, the active peak torque was produced at 60° of wrist extension regardless of the attachment location. However, the elbow joint angle at which peak active torque was produced shifted from elbow extension to elbow flexion as the attachment location approached the wrist joint (Figs. 4-D, 4-E, and 4-F).

Fig. 4.

Active torque-angle characteristics after surgery for different tendon graft attachment locations. Note that attachment location shifts the peak torque of the active torque-angle shape “diagonally” on this surface for both elbow flexion torque (Figs. 4-A, 4-B, and 4-C) and wrist extension torque (Figs. 4-D, 4-E, and 4-F).

Our simulation predicted that the attachment locations providing optimal function for each task ranged from 54.3 to 74.6 mm proximal to the wrist joint (Fig. 5). For instance, an attachment location of 54.3 mm was optimal for the “drinking” task while an attachment location of ∼74 mm was more appropriate for “moving hand to back pocket” and “donning and zipping pants.” The tasks of “applying deodorant” and “removing an object from shelf” required an attachment location of ∼68 mm for optimal function. The optimal attachment location that maximized function across all tasks was calculated as 63.5 mm, which represented a tradeoff between the “applying deodorant” and “drinking” tasks, while the optimal attachment location across the entire range of both elbow and wrist joint angles (i.e., not task-specific) was calculated as 71.7 mm, which was between the “applying deodorant” and “moving hand to back pocket” tasks.

Fig. 5.

Tendon graft attachment locations optimized for specific tasks. All distances (in mm) are given relative to the wrist joint crease, and all passive forces (in N) are estimated at 20° of elbow flexion and neutral wrist and forearm positions. ECRB = extensor carpi radialis brevis and TA = tibialis anterior.

It is clearly seen that the curve surface for the “drinking” task (Fig. 6-A) is the most different compared with the other tasks (Figs. 6-B through 6-F), indicating that “drinking” is the functional outlier among the selected tasks. However, further analysis demonstrated that any surgical approach provides a performance index of >70% relative to optimal for any task (Fig. 7). This reveals the very robust nature of this muscle transfer combination in that almost any insertion produces a high performance. Interestingly, our simulation suggested that a transfer optimized for all tasks achieved ≥93% of the optimal outcomes for all tasks while a transfer optimized for the entire range of elbow and wrist joint angles achieved a performance index of ≥99% for all tasks except “drinking,” which had a performance index of 77%.

Fig. 6.

Active force-angle characteristics after surgery for different tendon graft attachment locations optimized for specific tasks. Note that the peak force of the active force-angle shape based on an attachment location optimized for the “drinking” task (Fig. 6-A) occurs at a more flexed elbow position compared with the other tasks (Figs. 6-B through 6-G).

Fig. 7.

Performance index calculated for each task when the simulation was optimized for each of the other tasks. Note that there is a clear difference between “drinking” and all other tasks, but any surgical approach provides a performance index of >70% of optimal for any given task. See the Materials and Methods for the definition of the performance index.

Discussion

The purpose of this study was to use mathematical optimization in combination with musculoskeletal modeling to predict the functional outcomes of a recently-reported novel tendon transfer in reconstructive tetraplegia hand surgery—i.e., the BRA-to-ECRB transfer⁸. Based on the almost perfect muscle architectural match between the BRA and ECRB muscles, the main result of our simulation is that the BRA is an excellent choice as a donor to restore wrist extension. The 2 most important muscle functional properties, excursion and force production, are fairly similar between the 2 muscles, with the ECRB predicted to generate approximately 60 N of force over 6 cm of length while the BRA can easily match those properties with a predicted force of 120 N over 8 cm of length (Table II). The increased force and excursion of the BRA are quite functional as the new BRA-ECRB MTU is biarticular and thus, with the elbow extended and the wrist flexed, experiences a greater length range than either the BRA or the ECRB alone. The BRA is monoarticular and, while the ECRB is biarticular anatomically, the relatively small elbow flexion moment arm²⁶ results in most of ECRB excursion resulting from wrist extension. These data thus provide strong theoretical support for using this transfer and are consistent with our case presentation in which high active torque and low passive torque occurred with the wrist and elbow extended (Fig. 4 of reference 8).

The present study extends our previous finding of a nearly perfect match between BRA and ECRB muscle properties by showing how effective the new BRA-ECRB MTU would be at enabling the performance of a number of common tasks (Fig. 7). Intraoperatively, the surgeon has the choice to set the new BRA-ECRB MTU length (by “tensioning” the transfer) based on how far along the ECRB tendon the graft is placed. In our coordinate system, this ranges from 80 to 50 mm proximal to the wrist crease (Fig. 2). We found that suturing the tendon 63.5 mm proximal to the wrist crease (Fig. 5) would provide >90% of optimal function for removing objects off a shelf, applying deodorant, placing a hand in the back pocket, and donning and zipping pants (Fig. 7). Clinically, the C5-C6-innervated synergistic biceps muscle will be intact in patients undergoing the BRA-ECRB transfer. Therefore, the relative contribution of the BRA to elbow flexion strength is less important and the transfer should instead focus on optimizing wrist extension torque and stability—i.e., by using the more distal insertion for the donor. This analysis provides a novel and strong rationale for the use of this transfer. We suggest that future theoretical studies of tendon transfers include these types of specific patient functional criteria and that the patient’s desired function be the focus of the preoperative and intraoperative decision-making process. That approach is more challenging than a “simple” architectural-biomechanical match because the relevant moment arms pre- and post-transfer as well as the kinematics of all relevant movements must be defined. We suggest that creating a “library” of such parameters be a goal for our surgical specialty to enable reconstructive upper extremity surgery to function more like the “personalized medicine” approach seen in other specialties, most notably oncology²⁷.

Our modeling approach has a number of limitations that must be considered. First, we assumed uniform and maximal activation of the BRA after surgery, which, if not present, would decrease the specific tension from the 22.5-N/cm² value used here; however, there would be no change in the shapes of the curves (Figs. 3 and 4) or in the performance index values (Fig. 7). Second, the attachment site was considered fixed once and for all, but it is known that insertion sites of tendon transfers can be subject to slippage²⁸. Third, our numbers are based on single-sized subject moment arm values obtained from the OpenSim software. In addition, because only one BRA-to-ECRB transfer has been reported⁸, to our knowledge, we have no other clinical data to use for comparison or to assess variability of BRA-to-ECRB transfer results. Finally, chronic use of the BRA as a biarticular wrist extensor and elbow flexor over a range that is much larger than what the BRA normally experiences could result in serial sarcomere number adjustment by the BRA, as is observed in other mammalian systems^29,30. Future experiments that address the limitations of the current study for the BRA-to-ECRB and related transfers will enhance the scientific credibility of tendon transfer surgery.

Appendix

Supporting material provided by the authors is posted with the online version of this article as a data supplement at jbjs.org (http://links.lww.com/JBJSOA/A403).