A method to predict location of non-coup brain injuries
Rajesh Kumar, Md Zubair, Sudipto Mukherjee, Jacobo Antona Makoshi
AA method to predict location of non-coup braininjuries
Rajesh Kumar a , Md Zubair a , Sudipto Mukherjee a and Jacobo Antona Makoshi ba Department of Mechanical Engineering Indian Institute of Technology Delhi, India; b Japan Automobile Research Institute, Ibaraki, Japan*[email protected]
Brain injuries are a major reason for mortality and mor-bidity following trauma in sports, work and traffic. Apartfrom the trauma at the site of impact (coup injury), otherregions of the brain remote from the impact locations (non-coup) are commonly affected. We show that a screw theory-based method can be used to account for the combined effectof head rotational and linear accelerations in causing braininjuries. A scalar measure obtained from the inner prod-uct of the motion screw and the impact screw is shown tobe a predictor of the severity and the location of non-coupbrain injuries under an impact. The predictions are con-sistent with head impact experiments conducted with non-human primates. The methodology is proved using finite ele-ment simulations and already published experimental results
Keywords : Non Coup Injuries, Screw Theory, Injury func-tion, Primate Injuries
Traumatic brain injuries (TBIs) are a major reason formortality and morbidity in trauma [39] [19]. TBIs resultfrom mechanical energy transferred to the brain from phys-ical forces that act directly on the head or are transmittedthrough the head-neck complex [30]. This mechanical en-ergy produces deformations of the brain, its neurons, its sup-porting structures or its vasculature beyond tolerable levelswhich result in the injuries. Various methodologies includ-ing head impact experiments with animals have been usedby researchers to investigate TBIs on impact [27]. The headacceleration modes (rotational vs linear) and its magnitudehave been historically related to the severity of brain injury[27] [2] [32]. While it is expected that the brain regions inthe surroundings of the head impact location will be affected,often there is one or more distal region of the brain that is in-jured (non-coup injury). Localisation of trauma within thebrain has been researched considering the brain matter as aviscoelastic continuum [5], but still remains a problematicissue. The skull motion (acceleration) is also used to pre-dict brain injury using finite element (FE) model [26] [40][7] [44] by assessing strains in the brain tissue. Apart fromthe second derivative of motion, parameters like brain size and shape also affect the risk of injury [43]. Consequently,multivariate regression (learning) techniques have also beenused for vulnerability assessment of injuries [42]. A numberof mathematical functions have been proposed to predict thelikelihood of injuries during impact. One of such functions,is the Head Injury Criteria, has been adopted globally as astandard for motor vehicle and helmet safety evaluation. Thehead injury criterion (HIC) uses linear acceleration param-eters to determine the likelihood of the head injury duringimpact [15]. The HIC is given by a computable expression(equation 1).
HIC = max t , t ( t − t )( t − t (cid:90) t t a ( t ) . dt ) . (1)where, a(t) is the resultant acceleration measured at thecentre of gravity of the head; t and t suggests the onsettime and the end time of impact, respectively. The HIC hasbeen criticized for not accounting for factors known to af-fect head injury, such as the impact direction, shape of thehead, the area of contact or the rotational component of thehead motion. Further, the HIC was developed on the ba-sis of experimental data of severe head injury and it maynot be applicable for milder forms of TBI, especially thosenot including a skull fracture [11] [14]. Head impacts re-sult in a combination of the linear and angular motion of thehead. The probability of injury due to impacts that cause ro-tation free linear acceleration vs those caused by rotationalmotion (acceleration) have been examined [16] through hel-meted head-form impacts and validated through injury out-comes of on-field football impacts. The contribution of an-gular rotation of the head on trauma, particularly on diffuseaxonal injuries, has been demonstrated by experimenting onnon-human primates [12] [1], rats [8] and swine [10]. Com-parable linear acceleration is also known to cause concussionand subarachnoid hematoma in primates [31], rats [28] andswine [10]. Consequently, combinations of the linear and therotational head motion have also been proposed as predictorsof brain injury [34] [21]. Further, regression models that re-late the probability of concussion [35] with a combination1 a r X i v : . [ q - b i o . Q M ] D ec f rotational and linear acceleration components have beendeveloped, as illustrated by equation 2. Probability of Concussion = + e − ( β + β a + β α + β a α ) (2)where β i are regression constants (where i = 0,1,2,3), “a”is the peak linear acceleration and “ α ” is the peak rotationalacceleration. The effect of acceleration-time history of thehead (post impact) [33] has also been studied in relation tobrain injury [17]. Brian strain fields have been also assessedfrom experimental data on acceleration [36]. FE methodsare the only current method of identifying trauma locationsand usually indicates regions in the periphery. Other meth-ods predict existence of brain injury. We note that none ofthese studies incorporate information about the nature (di-rection) of incident impact. An important aspect from thefield data that is under-reported is the nature of impact thathas induced the observed kinematics. The head kinematicsresulting from the impact are related to the magnitude anddirection of the impacting impulse. Specifically, the two arerelated by writing the Newton-Euler equations about the CG,or a point on the body fixed in space, or a point acceleratingtowards the CG. Impact to the brain, when measured exter-nally, is available as a combination of angular accelerationand linear acceleration for the skull-bone with reference to apoint of reference, which is usually the CG of the head. Incontrast, the impact, when represented as a wrench [9] [18],is independent of the choice of the point of reference, as longas the observation is from a Newtonian frame of reference.Screw theory [9] allows study of system dynamics of a rigidbody as the effect of force motors (or wrenches) resulting inacceleration motors. Here the word “motors” is indicative ofa combination of a vector and a moment resulting from thevector. It also allows inferring the direction of impact know-ing the location of impact, and the consequent head kinemat-ics [9] [18]. Protection by design of safety equipment, min-imises the transmission of the impact energy to the brain. Wehypothesize that the energy associated with small finite vol-umes of the brain due to the impact can be a measure of thedosage of the part. The instantaneous work done by the im-pact force at each mass node is representative of the stretch-ing of segments the brain would undergo. To ascertain thecritical regions, it is hence, sufficient to map an energy quan-tifier at different positions when the body is impacted. Theenergy quantifier relates to the amount of energy transmittedto the soft part of the brain from the skull at different loca-tions. The aim of our study is to formulate, leveraging screwtheory, a function that relates a head impact wrench (forceand moment combination) to the affected non-coup regionsof the brain. The function developed utilises the directionof the impact to provide a metric of concussion. We sup-port the theoretical formulations with examples of head im-pact experiments conducted on non-human primates in the1970s, with numerical simulations of the past experimentsconducted and published recently [4] [3]. The numerical val-ues are compared with the standard metrics like HIC and the empirical formulation of probability of concussion. Non-coup injuries are those that occur at the distal lo-cation away from the impact. When the head sustains animpact, the trauma to a region of the brain is posited to beproportional to a quantification of energy that passes throughthe region. This is consistent with the product of strain ε and strain rate at the midbrain region, ε d ε dt max , providing oneof the strongest correlation with the occurrence of mild trau-matic brain injury [25]. In the conventional global head kine-matics approach, the motion of the CG is studied as a repre-sentative point for assessment of injuries. The velocity (ac-celeration) and angular velocity (acceleration) of the CG (orany other point in the rigid body) are represented using sixscalar variables. The theory of kinematic motors [9] allowsrepresentation of the motion of rigid body as a whole usingfive scalar parameters. An inner product of motion variablesto the impacting wrench with units compatible in SI units asJoules, which we call the energy is used to map the traumaticcontour. The theory of kinematic screws is used to determine amap consisting of the instantaneous reciprocal product be-tween the impacting wrench and the resulting screw. Thefunction is defined across the brain volume by integrating theinstantaneous working rate over the duration of the impact, toobtain a measure which we call as the Predictor Map. Max-ima in the Predictor map gives the most probable region ofnon-coup injury, as it is a measure of energy passing throughan affected region. The head has a reasonably rigid shell en-closing compliant matter. The energy associated is a simpli-fication of the mechanics of the head. The different portionsof the rigid part of the head interact with the soft part. Theequivalent energy transmitted at each rigid-soft interface inthe brain is considered to formulate an energy map for thenon-coup injuries. For a given impact, the region with max-imum energy transfer from the rigid to the soft part wouldbe the portion of the brain leading with maximum non coupconcussion.
The brain has an irregular shape. However, geometri-cally simpler models of the brain, (ellipsoid, spheroidal oreven spherical) have been traditionally used to develop theo-ries by researchers [23] [22] [6]. In the current study, we, tooadopt a super-ellipsoid as a simplified geometry representa-tive of a primate brain (Figure 1). This allows the functiondescribed to be calculated analytically.Throughout the paper, we use terminologies introducedin [9] and [18]. The impact force is modelled as a wrenchwith zero moment component and with the axis same as thatof the direction of the force. The brain geometry is assumedto be super ellipsoid as such geometries with variable pa-rameters define surface transition from a cube to a sphere as2 ig. 1. Super-ellipsoid model developed as a simplistic representa-tive of the primate (Macaque) brain plotted in Figure 2. The super ellipsoids give sufficient formadaptability (equation 3), and its first and second momentscan be found in closed form. ( | x / a m | d + | y / b | d ) hd + | z / c | d ≤ a m , b , c are the parameters defining the symmetry of the geometry(for instance, it defines the magnitude of axes in a an ellip-soid) and h and d represent the shape of the system.A parametric formulation of the super ellipsoids is givenin equation 4 to 8. x ( δ , γ ) = ρ a m f ( γ , / h ) f ( γ , / d ) (4) y ( δ , γ ) = ρ b f ( γ , / h ) f ( γ , / d ) (5) z ( δ , γ ) = ρ c f ( γ , / h ) (6) ∨ − π < γ < π and − π < γ < π f ( η , ζ ) = sgn ( cos ( η )) | cos ( η ) | ζ (7) f ( η , ζ ) = sgn ( sin ( η )) | sin ( η ) | ζ (8)where γ and δ are the parameters and sgn ( f ) provides thesign of the output of f . For the super-ellipsoid, the moment of order ‘p,q,s’ with respect to x,y,z axes is defined as a Rie-mann integral [20] (equation 9) and a closed form expression(equations 10-12) allows direct computation of the moments. M pqu = (cid:90) ∞ − ∞ (cid:90) ∞ − ∞ (cid:90) ∞ − ∞ x p y q z u f ( x , y , z ) dxdydz (9) M pqu = p + q + a p + m b q + c u + ε ε β f (( u + ) ε , ( p + q + ) ε + ) β f (( q + ) ε / , ( p + ) ε / + ) (10) ε = h (11) ε = d (12)where β f is the standard beta function, given by β f ( n p , m p ) = (cid:82) t n p − ( − t ) m p − dt .The external surface is discretised into a grid of rectangularpatches. For each grid element the centroid of the rectangu-lar patch is the point of reference for the element. We willsubsequently reduce the motors (combination of vector andmoments) to the points of reference of the elements to com-pute the wrenches needed to calculate the Predictor Map. To quantify the function on a finite volume on the brainsurface, we use the definition of instantaneous work done asthe scalar product of the force and displacement at a pointadded to the product of the applied moment with angularmotion. The reciprocal product [24] of an applied wrenchwith the resultant twist is a compact descriptor of the instan-taneous work done. Conventionally, the reciprocity relationis converted to an inner product by reordering the vectors.The reciprocal product of a wrench and a twist screw is thesum of the vector inner product of the force component tolinear velocity and vector inner product of the moment com-ponent with angular velocity. Following screw theory [9] aset of forces ( F ) and moments ( M ) acting on a rigid bodycan be reduced to a wrench motor at the centroid of eachgrid element (equation 13) where r OJ represents the radiusvector from the centroid of the j th element to the point ofapplication of the impacting wrench and W = ( F M ) T is the3 ig. 2. Super ellipsoids with varying parameters (a) Equidistant diameters with h= 1.8, d = 1 used as the model for the brain. (b) Parameters:h = 1, d = 1.8, the model goes flatter moving away from a cube. (c) Parameters: h = 2, d=2, the output system is a sphere as the diametersalong different axis are equal (d) Parameters: h = 2, d = 2 but with varying diameters along varying axes, forming an Ellipsoid. impacting wrench to the system. R j = (cid:18) Fr OJ × F + M (cid:19) (13)Analogously, the instantaneous motion of a rigid bodycan be reduced to a twist motor of the type (equation 14) V Gi = (cid:18) v Gi r OJ × ω G (cid:19) (14)where v Gi is the velocity vector of the centroid of the i th gridelement, ω G is the angular velocity component and G is theframe fixed to the ground. In impact problems, the net mo-tion of the body over the impact duration is usually smallwhile velocity change can be substantial. It is hence reason-able to assume that the inertia binor (see appendix) of thesystem does not change significantly over the impact dura-tion. For super-ellipsoids which are not excessively prolate,the differential of the binor of inertia with respect to timecan hence be disregarded giving us the following equationrelating wrench and twist (equation 15) T d V dt + V d T dt = W (15)where, T is the inertia binor and W is the impacting wrench.Development of dynamics based on screw theory with def-initions of the parameters of the dynamics’ equation is pre-sented in the Appendix. Let the displacement motor of thecentroid of the grid element J be φ J . The work done, E j , bythe impacting wrench affecting element ‘j’ ( R j ) to displace the considered grid element by φ j is E j = < R j , φ j > which isa scalar quantity. The displacement motor is the componentwise integral of the twist screw and the reciprocal productis the inner product of the element reduced wrench and dis-placement motor. To verify the methodology proposed to predict the majorregions of concussion, we used data from impacts controlledon a group of primates. This methodology is presented insection 2.2.1. The results are also compared with the FEsimulation results.
Series of head impact experiments in controlled environ-ments with non-human primates were conducted at the JapanAutomobile Research Institute (JARI) in the 1970’s [29].The impacts can be grouped into three classes, those in whichthe impact was nearly normal to the frontal region, those inwhich the impact was tangential to the top surface of the skulland impacts that were off normal to impacted surfaces. Thesetests produced injuries of severity that were not consistentwith HIC as a measure and structural damage appeared inunpredictable zones. We shall demonstrate that the approachproposed resolves this anomaly. The impacting wrench, forthe cases where it was not explicitly recorded, was calcu-lated from the acceleration data (equation 15). Thereafter,the experimental data was classified into either the impactbeing near normal or near tangential or intermediate. As theimpacting velocity in all cases was similar, wrench of samenormalized magnitude was used in our analysis. The im-pacting force directions for various test clusters are shown inFigure 3. The dominant normal impacts are realized using apadded impactor. The experimental data has been published4ver the years in 1980s by JARI [29] [38] [31] [37].
To demonstrate the approach developed in section 2.1,a time varying normalised ramp impulse has been applied tovarious locations of the head model and the parts of the brainwith top 10 % of the energy function value are tracked toanalyse the most critical areas under impact. This approachthrows new light on the possible location of non-coup in-juries and its dependence on the nature of the initial impact,parameterised in terms of direction and magnitude. Traumalocations from extensive experiments done in the 1970’s onmonkeys at JARI, Japan, is shown to be consistent with thismethodology. The finite element simulation results pub-lished by the co-author of this paper in [4] are also shown tobe consistent with this methodology. We, hence, show thatby combining with the direction of impact rather than usingonly the overall head kinematics, good injury predictability,in magnitude and location is achieved.
Figure 4 shows the case of an impact on the pole of thesuper ellipsoid. The arrow shows the direction of impact andpole of the arrow is the point of impact. The line of impactpasses through the lower pole as well as the upper pole, mak-ing it along an axis of symmetry. Different values of the pre-dictor map are colour coded and indicated by the bar on theright side of the figure, with darker colors indicating higherlevels. The higher-level elements are concentrated along thefour pseudo edges of the volume. The most traumatic regionfor the normal impact is at the farthest possible point in thevolume from the line of impact. As the impact is along theaxes of symmetry all the four regions are approximately atthe same distance from the line of impact, which are all ap-proximately maximum. Figure 5 shows the predictor mapobtained for an impact at the same location, but tangentialdirection to the surface. This case represents on idealiza-tion of gliding type impacts and illustrates the change of thelocations of peak value elements on modifying the impactdirection. For the super-ellipsoid geometry, two path param-eters (t,s) (latitudinal and longitudinal) are naturally defined(Figure 6). The contours showing the function value of thepredictor function along the path parameters are extracted tostudy the location of the critical regions from the point ofimpact. The contour plots, parameterized by “s” and “t” areconveniently mapped to 2D, which is amenable to a graphi-cal interpretation. For a near-normal impact, the variation ofthe function value with respect to the longitudinal parameter,‘t’, is shown in Figure 7. The bars represent the variationwith ‘s’ and we see a wavelike change with ‘t’. The ‘t-s’ sur-face is shown in Figure 8, which shows an additional periodicvariation of the function in ‘s’. For near normal impact, thetraumatic region is on the sides (central longitudinal param-eter). Generally, for near normal impact, the most criticalregion of injury (apart from just at the point of impact) is atthe point of farthest distance from the line of the impacting force. Near-normal impact means the impact direction pass-ing through a bounded region in the vicinity of the geometriccentre of the geometry. Similarly, near tangential impact isnearly perpendicular to the near normal impact direction.In the figures above, the impact is at the poles. A forceof magnitude 5000 N is applied at the pole for an instance of0.02 seconds. The normal impact leads to bifurcation in thetraumatic zones into four regions that are equally dispersedwith respect to the central axes. However, in case of tangen-tial impact at the pole, the maximum trauma is located at themaximum distance from the point of impact, which is the op-posite side pole. The effect of tangential impact is shown inFigure 9 and Figure 10.An additional observation is that when the predictor mapfor tangential impact as the normal impact of the same mag-nitude, the trauamatic zone is more localised. It would beinteresting to conjecture that this would lead to more traumathan normal impact where the maximum area region is splitin the ‘s’ domain. The normalised maximum predicted en-ergy is of around 27500 J, where as it is around 58000 J fortangential impact (Figure 8 and Figure 10). When the modelis tangentially impacted at the super ellipsoid surface (nearthe side), the most critical regions are again farthest from thepoint of impact (the rear of the head) at a single location. Thetwo views showing the most critical region for this are shownin Figure 11 where the arrows are representative of the im-pact direction. The tail of the arrow is the point of impact.In Figure 11, the impact point is nearly at the centre of theface. The darkest regions are located farthest from the pointof impact. For a general impact, with components, alongboth normal and tangential direction, the resulting contour isbi-normal in the ‘t’ domain, resulting from the combinationof the two modes and is shown in Figure 12. The result-ing contour can be seen as a combination of the two modespresented above. In summary, in the cases of near tangen-tial impact, the critical regions are the areas farthest fromthe point of impact. The critical zones, in this case does notchange rapidly with the direction of impact. Whereas in caseof near normal impacts, the critical regions are farthest fromthe line of impact, and hence dependent on the impact direc-tion. For the general impact force which cannot be classifiedas normally dominated or tangentially dominated impact, theresultant areas of maximum injury are a combination of thetwo canonical modes. The zone with higher Predictor valueor the traumatic region is focused with larger magnitude incase of a tangential impact.
To analyse the experimental data, we have classified thecases based on the direction of impact with respect of thesurface normal. Section 3.2.1 compares the near normal im-pact to the simulation results, section 3.2.2 deals with neartangential impacts whereas section 3.2.3 is based on generalimpacts which are a combination of near normal and neartangential impacts.5 ig. 3. (a) A padded normal dominant impact from front. (b) A padded normal dominant impact from rear (c) A dominant tangential impact(d) A combination of a normal and tangential impact.Fig. 4. For the impact near poles, the maximum effect is split alongthe four pseudo edges of the super ellipsoid. The dark regions sug-gest regions of traumatic injury
The experimental results of the near normal impacts,were reported in [29]. The impact is on the frontal region ofthe skull and nearly normal to the surface. Literature sug-gests that the contra-coup zones [13] are expected on thesurface diametrically opposite to the impacted surface, dueto wave transmission. But several cases in the series report anomalous injuries in other regions as well. We show nextthat they are in zones predicted by the model proposed here.Upon experimentation on primates by JARI, the exper-imental results of frontal impacts on 13 primates were pub-lished [29]. Out of 13 impacts, five of them showed no resid-ual injury and four of them had injuries at regions apart fromthe impacted regions. The macroscopic findings for the non-coup injuries included injuries near the base of the brain ge-ometry. Additionally, in two of the cases, injuries were alsopresent in the “occipital” and “temporal” lobe. For the caseswith no macroscopic findings of the injured region, micro-scopic injuries were notified at the “brain stem” includingthe effect on the brain stem neurons. In all cases the primateexperimental results indicate that the effect of normal impactresults in injuries around the region farthest from the lineof impact. The same is predicted by the mechanics-basedmodel.
The case of primates being impacted near the top of thehead tangentially, were reported in [38]. The report sug-gested that 7 out of 12 cases indicated injuries at the para-sagittal area and base of the frontal and temporal lobe. Apartfrom these, four additional cases were published separately[31] and for which detailed autopsy results were available.6 ig. 5. Response of the function developed under tangential impact at the pole. (a) and (b) shows two perspective views of the response ofthe brain under impact. The arrow shows the direction of the impact.Fig. 6. Response of the function developed under tangential impactat the pole. (a) and (b) shows two perspective views of the responseof the brain under impact. The arrow shows the direction of the im-pact.Fig. 7. Variation of the energy predictor function with respect to thevariation of the longitudinal parameter for near normal impact. Thelength of each segment for a “t” value is an indicator of the variationwith the “s” value
Injuries reported for the four primates includes pulmonaryhemorrhage, transection of medulla, hematoma in the pitu-itary gland, pulmonary congestion, cerebral hemorrhage andeffect on medulla. The significantly injured area in thesecases (apart from the area of impact) was near the base ofthe brain. The region effectively is farthest from the point ofimpact, which is predicted by the mechanics-based model.The super-ellipsoid closest fit to primate brain was utilizedfor pure tangential impact and results are presented in Fig-ure 5, Figure 9 and Figure 10. For the simulated case, the
Fig. 8. Variation of the energy predictor function in the ‘s-t’ domainfor near normal impactFig. 9. Variation of the Predictor function with the longitudinal pa-rameter for tangential impact. region of maximum trauma is farthest from the point of im-pact, as explained for the cases in section 3. It was reportedthat for a similar impacting velocity, the primates survivedduring the near-normal impact but the tangential impact hadfatal outcomes. This finding is consistent with the mechan-ics model, where under near-normal impact, the predictedmaximum energy of impact was around 27500 J whereas forthe same wrench applied tangentially, the maximum energy7 ig. 10. Formation of the energy based function surface for tangen-tial impact.Fig. 11. Most critical region for a tangential impact is farthest fromthe point of impact.Fig. 12. Binormal Function Value region for general impactingwrenchFig. 13. Comparison of the finite element simulation results with themethodology developed. of impact was around 58000 J. The ratio of the numbers forthe peak energy for impacting wrench of the same magnitude was 1:2.1, suggesting higher intensity of energy and higherinjury probability for tangential impact.
The injuries resulting from a general wrench lateral im-pact are reported in [37]. A combination of tangential andnormal impact was applied with the head being impacted lat-erally resulting in a combination of linear and rotational ac-celeration of the head. Apart from the coup injuries (justbelow the point of impact), the injuries caused were pri-marily a combination of the near-normal impact and near-tangential impact. The major regions of injury included basalcistern and injury at the medullo-spinal junction, which canbe explained by the tangential component of the impact-ing wrench, as these regions are farthest from the point ofimpact. Apart from this there were injuries in the parieto-occipital region as well, which can only be explained by thenear-normal component of the impacting wrench, as the sys-tem is farthest from the line of impact.The major injury is due to the tangential component usu-ally but the near normal component also plays a role. Asshown in Figure 12, there are two regions of local maximaof the energy based function, leading to two critical regions.A set of finite element simulations developed based on fea-tures of a Macaque monkey brain by the co-author and pub-lished in [4] [3], report two non-coup injuries on the regions.The details of the FE simulations are published separatelyin [4] [3]. This can be attributed to the tangential componentand normal component, when the general impacting wrenchis discretized into a tangential component and a normal com-ponent. The result is shown schematically in Figure 13.
The methodology is applied to brain of different shapeand size. As, true brain geometries aren’t regular, the instan-taneous centers of curvature changes unlike symmetric superellipsoids. We define near-normal impact as the impact pass-ing closely to the geometric center of the geometry and neartangential as perpendicular to it. The method doesn’t usethe local curvature in defining near-normal or near tangentialimpacts. Also the closed form expression for the momentsfor true brain geometries are unknown, we calculate the inte-grals in equation 9 numerically. Two sets of brain including aMacaque primate monkey brain and a Marmoset brain weresimulated to find the most critical regions.For the case of Macaque primate brain, which is similar tothe human brain in shape various impact cases and subse-quent injuries are presented in Figure 14. The variation ofcolour with the magnitude of injury is same as in figure 5.The top 10% critical region is shown in the dark color. Forall the cases in Figure 14, the arrow tail shows the point ofimpact except in Figure 14 (c) where the arrow head showsthe point of impact. The arrow shows the direction of im-pact. Figure 14 (a) shows a tangential impact on the side ofthe brain, the most critical region being at maximum regionfrom the point of impact. The most critical region occurs on8
Most Critical Region (Occipital)
Fig. 14. Simulation on Primates’ Brain for different impact locations and direction the other side of the brain. Figure 14 (b) shows an impactnear the top of the brain, as a general impact, similar to thatin Figure 13, with critical regions resulting near to the occip-ital region of the brain. Figure 14 (c) and Figure 14 (d) aretwo cases of near normal impact, but with a slight variation in angle (around 20 o ). The most critical regions are primar-ily farthest from the line of impact. When the impact angleis directed more towards the spine, the occipital region startscoming into the critical regime. For a pure tangential impact(Figure 14 (e)), the most critical region occurs at the back9 ig. 15. Simulation on Marmoset Brain for a general impact direction (two views presented) Figure 16.
Comparison of Probability of Concussion with proposed Metric. The value of the metric proposed here was normalised by the peak value (80662 J) for proper comparison
Proposed Metric Probability of Concussion
Fig. 16. Comparison of Probability of Concussion with proposedMetric. The value of the metric proposed here was normalised bythe peak value (80662 J) for proper comparison of the brain (occipital region). The marmoset brain differsfrom the human brain with the medial extent being greaterthan the vertical extent. A general impact to the marmosetbrain was analysed for most critical regions. The point cloudfor marmoset brain was generated using the CT images asin [41]. The analysis results are presented in Figure 15. Thetwo views show the most critical region under a general im-pact which is more near normal than near tangent. The mostcritical area occurs on the temporal region as shown.
This section focuses on comparing the numerical val-ues with existent metrics. The HIC value (equation 1) andthe probability of concussion (equation 2) is compared forthe super-ellipsoid approximation model. A force of 5000 Nwas applied for a duration of 0.02 s. The force was appliedat the frontal surface with varying directions of impact. Theidea is shown in the Figure 16. The direction of impactingforce does not impact the HIC value as it is only affectedby the translational acceleration but the probability of con-cussion shows correlation with the proposed metric. For thecase in Figure 16 the HIC value was constant at 5245 units.To compare the probability of concussion with the method proposed here, the energy terms are normalized by the max-imum value (to compare with the probability of concussion).It can be observes that the method proposed here is coherentwith the empirical relation suggesting probability of concus-sion.
The methodology developed can also be used to predictthe direction of the impact on the human head, knowingthe position of the coup and the non-coup injury. The CTimages of traumatic brain injury cases are shown in Figure17. In both the cases non-coup injury region is not furthestfrom the point of impact, ruling out a near tangential impact.We see that further away from the line of impact is a betterdescriptor than furthest from the point of impact. So, themajor component of the impacting wrench would be normalto the surface. Considering another case of a brain traumacase in JPNATC, AIIMS, Delhi, India, where the impact wasat the frontal lobe and the CT image is presented in Figure18. As the non-coup region is farthest from the point ofimpact, it can be predicted that the impacting wrench wouldbe primarily tangential.
This paper presents a method to predict the location ofregions of brain trauma. The method utilizes the direction of10 (a) (b)
Fig. 17. CT Image of a patient suffering head injury. Two cases of different patients are presented in (a) and (b). In both the cases the injuryregion is farthest from the projected normal to the surface of the brain geometry. The normal drawn are approximately projected normal tothe CT plane
Fig. 18. CT Image of a patient suffering head injury via frontal im-pact the impact force. Analytical estimates using an energy mea-sure with a simplified brain geometry model indicates thathuman brain regions that sustain the highest damage poten-tial are dependent on direction and position of impact. Thereis a large variation of the position of trauma between nearnormal, near tangential or a combination of these. Thesecomputations are consistent with past head impact experi-ments with non-human primates. The experimental observa-tion that tangential impacts can be more injurious than cor-responding normal impacts is consistent with this analysis.Further, the methodology proposed can also be applied toestimate the direction of the impact from the injury pattern,which should be useful to forensics.
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The general motion of a rigid body can be represented usingthe concept of motors. A motor (’mo’ from moment and ’tor’from vector) is a combination of a vector and the moment of avector. A general motor can also be represented as a complexvector [9] with the real part as the vector and the imaginarypart as the moment with respect to a point of observation(equation 16). M = α + i β (16)where α represents the real part and β represents the imagi-nary part. The total momentum of the body is represented bya momentum motor defined as K ( k , K o ) with real and imag-inary vectors k and K o . To analyse the motion of the bodyfollowing an impact, the kinematic motors and the force mo-tors (wrenches) are related following the development in Di-mentberg [9]. Dimentberg in particular, represents a generalscrew as a system of complex numbers. The instantaneousmotion of the brain is related to the wrench W ( F , M ) usingthe inertia binor matrix relating the instantaneous mechani-cal properties of the body. The inertia binor is representedas a rectangular matrix, with two 3 × T = (cid:34) iI S − iD − S − iD m − iS iS − S − iD iI − S − iD iS m − iS S − iD − S − iD iI − iS iS m (cid:35) (17)where ‘i’ as usual represents the imaginary part of acomplex number and S i represents the first moment of in-ertia , D i are the cross product terms of the classical momentof inertia tensor and I i are the leading diagonal terms of theinertia tensor. The elements of the binor are a function of thespatial distribution of mass in the brain with respect to thepoint of observation (equation 18-20). The total mass is mand the other elements of the binor are as per the equations18-20 with ρ i , i = x , y , z being the respective Cartesian coor-dinates of a differential mass element dm. We note that theelements of the binor are a reassuring arrangement of classi-cal mechanics inertia terms. (cid:90) ρ x dm = S ; (cid:90) ρ y dm = S ; (cid:90) ρ z dm = S (18) (cid:90) ρ y ρ z dm = D ; (cid:90) ρ x ρ z dm = D ; (cid:90) ρ y ρ x dm = D (19) (cid:90) ( ρ x + ρ y ) dm = I ; (cid:90) ( ρ x + ρ z ) dm = I ; (cid:90) ( ρ z + ρ y ) dm = I3