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The first reports of patients w ith acalculia m ention a calculation centre in the left angular gyrus (Peritz, 1918) or m ore generally in the left retrorolandic areas (Poppelreuter, 1917 and Sittig, 1917). In line w ith this, Lew andow ski and Stadelm ann (1908) reported that their severe acalculic p atien t h ad an hem atom a

rem oved from his left occipital lobe. The authors pro p o sed the existence of a calculation centre in the left occipital lobe. H enschen (1919) clearly im plicated the left an g u lar gyrus and su rro u n d in g areas in acalculic disorders. Shortly after these first w orks, Berger (1926) d o cu m en ted th at his th ree "pure" acalculic patients h ad intact left angular gyruses though they did present w ith left occipital an d tem p o ral lesions. H ead (1926) m en tio n ed the im p o rtan ce of tem p o ro ­ p arietal stru ctu res w hile K rapf (1937) stressed again th e im p o rtan ce of the occipital and parietal lobes, in particular the angular gyrus, for num erical skills although he did not differentiate betw een the tw o hem ispheres. G oldstein (1948) stated th at disturbances of calculation are m ore frequent for lesions of the left hem isphere, in particular for lesions located in the p arietal an d occipital lobes, an d m ore occasionally for lesions in the frontal lobe. Unlike Krapf, he believed th at the right hem isphere played no role in num eracy.

G rew el (1952, 1962, 1969) an d later Luria (1966; 1970;1973) offered an interesting account of the localization of acalculia. Both authors w ere against the notion of a center for calculation. They clearly stressed the idea th at there is no unitary acalculia that can be specifically lateralized and localized. They suggested that num erical skills can be d isru p ted in a num ber of w ays for w hich there m ay be corresponding specifiable lesions. From their perspective num erical skills are the p ro d u ct of the integrated interactions of various cerebral structures. They suggested that frontal, tem poral, parietal and occipital areas w ere responsible for different forms of acalculia.

G rew el suggested th at frontal acalculia sh o u ld re su lt in "m ore severe disturbances in dealing w ith abstract than w ith concrete num bers" even though the au th o r expressed doubts that any num ber could really be considered abstract. He then added that this type of acalculia is secondary to a general reduction of the level of "p roductive th in k in g a n d of initiative". Tem poral acalculia was associated w ith a specific failu re in u n d e rstan d in g sp o k en n u m b er nam es. Occipital acalculia was suggested to include a variety of disorders d u e to a basic failure in the ability to visualize a n d /o r retain the num bers an d the results of the d ifferen t steps of a calculation. Finally, the a u th o r co n sid ered parietal acalculia as p resen tin g the m ore in terestin g sym ptom s from a psychological point of view. H e stated that for this type of acalculia different com binations of sy m p to m s have been ob serv ed a n d no exact d ifferen tiatio n w as possible. H ow ever, he stressed the im p o rtan ce of different "spatial" a n d "linguistic" disorders as responsible for the patterns observed.

Similarly, Luria stressed the im portance of spatial disorders for acalculia follow ing parieto-occipital and parietal lesions. For exam ple, he observed th at for parieto-occipital lesions p atien ts often p resen ted difficulties in carry in g o u t m u ltid ig it arithm etical problem s req u irin g the carry in g a n d /o r the sp atial alignm ents of arabic num erals. In contrast, linguistic factors w ere suggested to be m ore im p o rta n t for acalculia follow ing temporal lesions. In this case, he suggested that patients presented w ith a loss of the concept of num bers a n d /o r a loss of the concept of arithm etical signs. In addition, patients could also present deficits in the retrieval of arithm etical facts. This im pairm ent w as in terp reted as secondary to a deficit of auditory-verbal m em ory. Luria believed acalculia could also be a consequence also of frontal lesions though n o t frequently. In this case, the calcu latio n d iso rd ers w ere seco n d ary to a m o re g en eral d ifficu lty in p ro g ram m in g , reg u latin g an d verifying cognitive activity. T ypically, these patients failed to conceptualize and link perform ance on subcom ponents of an arithm etical problem in order to achieve the correct solution. An exam ple of one of his frontal lobe patients w ith acalculia (reported by Shallice, 1988) is the following: a patient was presented w ith a problem s such as " there w ere 18 books on tw o shelves and there w ere twice as m any on one than on the other. H ow m any books w ere there on each shelf?". His answ er w as 36 having first carried out the operation 18 : 2 = 9 (corresponding to the first p a rt of the problem ) and secondly having m ultiplied 18 x 2 = 36 (corresponding to the second p a rt of the problem ).

1.62 D ata from the clinical neuropsychological literature

Similar to the conclusions of Grewel and Luria, Critchley (1967) proposed that acalculia m ay occur w ith bilateral or unilateral, left or rig h t sided lesions located anyw here from the frontal to the occipital lobe. The au th o r how ever re p o rte d th a t acalculic sy m p to m s w ere m ore com m on a n d m o re severe follow ing left hem isphere lesions. Cohn (1961) expressed the view that lesions of disparate regions of the brain m ay lead to acalculia.

A num ber of authors suggested that acalculic patients p resented bilateral lesions p rim arily in the parietal-occipital a n d /o r p arietal-tem p o ral regions. H ecaen a n d A ngelergues (1961) rep o rte d th at alexia and agraphia acalculia followed in 84% of patients w ith left tem poro-occipital lesions an d only in 8% of p atien ts w ith rig h t p osterior lesions. Spatial acalculia w as rep o rte d as being

highly associated, b u t not exclusively associated w ith rig h t hem isphere lesions m ainly involving parieto-tem poro-occipital (57% of the cases) o r only parietal regions (16% of the cases). Anarithm etia acalculia w as rep o rted for left (53% to 62% of the cases) or rig h t (6 to 15% of the cases) hem ispheric lesions. In a following w ork (Hecaen, 1962), different percentages w ere reported regarding the frequence of the acalculic disorders (presum ably because he w as now referring to a p o p u la tio n of p atien ts w hose b rain lesions w ere n o t re stric te d on ly to retrorolandic areas) b u t the im portance of the cerebral areas m en tio n ed above was not questioned.

A c o n firm atio n of th e in v o lv e m en t of th e re tro ro la n d ic areas in perform ing calculation comes from a study of W ilkins, Zifkin, A n d erm an n and M cG overn (1982). They described a case of "epilepsy arithm etics". The term d escrib ed a form of reflex epilepsy in w hich m en tal a rith m etic can evoke epileptiform discharges in the scalp electroencephalogram (EEC) a n d /o r clinical seizures. This epileptic patient observed that seizures w ere reliably precipitated by m en tal arithm etic. The e x p erim en tal in v estig atio n sh o w e d th a t tasks in v o lv in g m u ltip licatio n , d ivision and m an ip u latio n of sp a tia l in fo rm atio n w ere significantly associated w ith discharges. Interestingly few, if any, discharges a p p ea re d w hen ad d itio n an d subtraction tasks of eq u iv alen t difficulty w ere perform ed. The authors suggested that the bilateral parietal regions of the patient m ay have been diffusely hyperexcitable.

There have being a num ber of studies suggesting the involvem ent of the right h em isp h ere in calculation. Barbizet an d co-w orkers (1967) ad v an ced the hypothesis th at the spared perform ance on num ber processing tasks of their p a tie n t w ith extrem ely severe left h em isp h ere lesions c o u ld h a v e being m ediated by the right hemisphere. In line w ith this, Leleux et al. (1979) reported a rig h t h a n d e d teacher of m athem atics w ho becam e acalculic fo llo w in g the rem oval of a rig h t fronto parietal hem atom a. Troup, B radshaw an d N ettleton (1983), review ing a series of studies on the lateralization of num erical processing, concluded th at rig h t hem isphere structures are involved in n u m b er processing an d calculation. G rafm an, K am pen, R osenberg, Salazar a n d Boiler (1989b) described a patient (J.S.) w ith a penetrating brain w ound that destroyed alm ost all of his left hem isphere w hile leaving his right hem isphere intact. J.S., w hile being severely aphasie, was still able to read, w rite, and com prehend arabic num erals and arithm etical signs and to perform sim ple addition and subtraction problem s.

There are, however, a large num ber of studies w hich rep o rt th at patients w ith left hem isphere brain dam age perform ed w orse th an th o se w ith rig h t h em isp h ere dam age. C ollignon et al. (1977) rep o rte d th at acalculia is m ore frequently found in left than in right hem isphere dam aged patients. G rafm an et ai. (1982) reported that patients w ith posterior left hem isphere lesions perform ed significantly w orse on both quantitative and qualitative results of an arithm etical test battery than patients w ith right hem isphere lesions. Similarly, D ahm en et al. (1982) fo u n d th a t p atien ts w ith lesions to the left h e m isp h e re p erfo rm ed significantly w orse on num eracy tasks th an p atien ts w ith rig h t h em isp h ere lesions. Jackson an d W arrington (1986) tested the ability of rig h t a n d left hem isphere brain dam aged patients to perform spoken addition an d subtraction problem s at speed (graded difficulty arithm etic test). They found th at patients w ith left hem isphere lesions presented a significantly w orse perform ance than p atien ts w ith right hem isphere lesions. The perform ance of this latter gro u p of p atien ts w as not significantly w orse than the control group. In line w ith this, W arrin g to n , Jam es and M aciejewski (1986) rep o rte d th a t p a tie n ts w ith left hem isphere lesions were significantly m ore im paired on the arithm etical subtest of th e W.A.I.S. than p atien ts w ith rig h t h em isp h ere lesions. In p articu lar, p a tien ts w ith left parietal involvem ent w ere m ore im p aired th an w ere those w ith o u t parietal involvem ent.

Data coming from single cases also seems to confirm the pred o m in an t role of th e left hem isphere in acalculia. In particular, p arietal regions ap p ear to be critically involved in num ber processing and calculation skills. The tw o patients described by Benson and Denckla (1969) had left hem isphere dam age. One of the two patients h ad a gliom a rem oved in the left posterior an d parietal area. The p a tie n t described by Benson and W eir (1972) also show ed a lesion in the left p osterior parietal area. Similarly, the patient reported by Assal an d Descombes (1984) show ed a lesion in the left p osterior p a rt of T l, T2 a n d p a rts of T3 extending into the gyrus angularis and gyrus supram arginalis. O ne of the tw o p atients reported by Ferro and Bothelo had an intracerebral haem atom a located in th e left tem poro-occipital junction w hile the second one h ad a left parietal lesion. The patient reported by G ardner et al. (1975) suffered an occlusion of the left m iddle cerebral artery.

1.63 D ata from the cognitive neuropsychological literature

As m entioned above cognitive neuropsychological studies have reported little data on brain localisation. H ow ever, the evidence collected seem s to m ainly su g g est an invo lv em en t of the p o ste rio r region of the left h em isp h ere. A num ber of studies reported as the left parietal lobe is having a crucial role in n u m b er p ro ce ssin g . The CT scan on p atien t S.T.H. w ho h a d difficulty in the com prehension of num bers in d icated som e low a tten u atio n in th e p osterior tem poral lobe and the occipito-tem poral gyri of the left hem isphere (W arrington, 1982). The patients (H.Y. and J.C.), described by McCloskey et al. (1986) w ho had d ifficu lties in re a d in g a lo u d arab ic n u m e ra ls b o th p re s e n te d w ith left hem isphere lesions. Patient H.Y. show ed an intracerebral hem orrhage in the left tem p o ral a n d p a rie ta l lobes w h ile p a tie n t J.C. p re s e n te d w ith a sm all intracerebral hem atom a in the left tem p o ro p o sterio r region. A n o th er p a tien t (Y.M.) w ith a selective disorder of arabic num eral read in g also show ed a left tem poral tu m o u r extending in to the left parietal lobe and the low er p a rt of the splenium (Cohen and Dehaene, 1991).

Some single case studies seem to suggest th at also patients w ith acquired calculation difficulties m ainly present w ith lesions im plicating dam age to the left h e m isp h ere. P a tie n t D.R.C., w ith a selective d eficit in th e re trie v a l of arithm etical facts show ed an area of increased attenuation in the left posterior parieto-occipital (W arrington 1982). The two patients w ith im paired arithm etical fact retriev al presented by Sokol et al. (1991) also bo th p resen ted w ith left hem isphere lesions. Patient N.A.U., w ho h ad a severe num ber processing and calculation d iso rd er p resen ted at the CT scan w ith a large tem poro-parieto- occipital h y p o d en sity (D ehaene a n d C ohen, 1991). The p resen ce of such a m assive lesion lead the authors to suggest th at his selective p reserv atio n of "approxim ate" calculation abilities w as m ed iate d by th e rig h t hem isphere. R ecently, L uchelli and De Renzi (1993) d escrib ed a p a tie n t w ith a m ild im pairm ent in the retrieval of arithm etical facts and a severe im pairm ent in the execution of calculation procedures w ho presented w ith an u n u su al lesion site. The p a tien t h ad an infarct in the territo ry of the left anterior cerebral artery w hich lesioned the m edial cortex of the frontal lobe.

1.64 A natom ical conclusion

The p resen t review indicates th at num erical skills have a discrete and in d e p e n d e n t b rain substrate. M ost of the d ata rev iew ed so far, sh o w ed that posterior lesions of both hem ispheres are found to lead to greater im pairm ents of m athem atical ability than anterior lesions. In particular, the m ajority of the evidence in d icated the involvem ent of the left p o sterio r areas. The rep o rts available do n o t allow for a conclusive localization of lesion w ith in th e left posterior quadrant, however, it can be suggested that the left parietal lobe plays a crucial role in calculation and num ber processing.

Chapter 2 The selective impairment of number know ledge in a case