Clinical manifestations

Presentation and course

A major distinction can be established between primary acalculia (anarithmetia) and secondary acalculia (aphasic acalculia, alexic acalculia, agraphic acalculia, frontal acalculia, and spatial acalculia).

Anarithmetia. Anarithmetia represents a basic defect in the computational ability. Patients with anarithmetia present with a loss of numerical concepts, inability to understand quantities, defects in using syntactic rules in calculation (eg, "to borrow"), and deficits in understanding numerical signs. They may count and conserve some numerical knowledge but fail in comparing numbers, performing arithmetical operations, and solving numerical problems. The failure in calculation tasks has to be found in both oral and written operations.

Luria emphasized the defects in spatial conceptualization underlying the primary acalculia observed in left-parietal damaged patients and also emphasized the strong association between acalculia and semantic aphasia (41). He proposed that left parietal-temporal-occipital damage could produce components of spatial apraxia, spatial agnosia, semantic aphasia, and acalculia. Luria considered that the same cognitive defects were present in semantic aphasia and acalculia; therefore, acalculia was associated with semantic aphasia.

Ardila proposed that left angular gyrus (Gerstmann) syndrome should be restated to include acalculia, finger agnosia, right left disorientation, and semantic aphasia. A single underlying deficit can account for the simultaneous presentations of these 4 clinical signs (02).

Aphasic acalculia. Dahmen and colleagues studied calculation deficits in patients with Broca and Wernicke types of aphasia (23). Using a factor analysis, they were able to identify 2 different factors: numeric-symbolic and visual-spatial. The milder calculation defects found in Broca aphasia patients are derived from linguistic alterations, whereas with Wernicke aphasia, defects in visual-spatial processing significantly contribute to calculation difficulties. Syntax of calculation is impaired in Broca aphasics. These patients present morphologic errors (eg, 14 is read as 40), and they have difficulties in counting backward and in successive operations. Omission of words is noted when transcoding tasks are given to the patients (that is, converting numbers in their written representation; for example, “1245” to “one thousand two hundred forty-five”; or the inverse task).

Wernicke aphasics present semantic errors in the reading and writing of numbers. Verbal paralexias for numbers are abundant. Verbal memory defect is evident in mathematical problem solving when the patient has to retain different elements of the problem. Lexicalization (eg, 634 is written 600304) and decomposition errors are frequent (eg, 1527 is written 15 27).

In conduction aphasia both oral and written calculation may be impaired. Sequencing operations, solving of mathematical problems, and transcoding tasks are abnormal. Order and hierarchy errors may be observed. As a matter of fact, the neuroanatomical site of damage most commonly noted in conduction aphasia (left parietal lobe) lies close to the area suspected for anarithmetia (angular gyrus). Hence, it is not infrequent to find conduction aphasia associated with primary acalculia.

Calculation disturbances are also observed in other types of aphasia. In extrasylvian (transcortical) motor aphasia the patient may have difficulties in initiating and maintaining numerical sequences. Problem solving ability may be significantly impaired. In extrasylvian (transcortical) sensory aphasia, significant calculation defects are usually found. Temporal-parietal damage results in a variety of language disturbances and significant calculation defects.

Alexic acalculia. The calculation deficits observed in alexia without agraphia (occipital alexia or pure alexia) are limited to reading numbers. Defects in reading numbers parallel the defects found in reading written words and sentences. Digit-by-digit reading may be observed. There is an asymmetrical reading, and the initial digits in a number are easier to read than the last ones. The patient may read the initial digits and omit the remainders (eg, 746 is read 74). Written arithmetical operations may be difficult due to the visual exploration disorders.

Central (parietal-temporal) alexia includes an inability to read written numbers and numerical signs. The ability to perform mental calculation usually is considerably better. Often, central alexia is associated with anarithmetia. Reading and writing difficulties plus computational disturbances indicates severe acalculia.

Agraphic acalculia. Difficulties in writing numbers are correlated with the particular type of agraphia. Broca aphasia patients present a nonfluent agraphia with perseveration and order reversal. Difficulties in transcoding tasks, substitutions, and omissions are evident. Writing number sequences, particularly in reverse order, may be especially difficult.

In Wernicke aphasia, a fluent agraphia is observed. Lexical errors are frequent; abundant numerical verbal paralexias and paragraphias are observed. Language-understanding defects impair the ability to write quantities under dictation.

Some degree of apractic agraphia is frequently found in conduction aphasia. This apractic agraphia will also be found in writing quantities. Self-corrections and approximations are found, and frequently the patients fail in converting the numbers they hear in a correct graphic form.

Frontal acalculia. Patients with damage in the prefrontal areas of the brain may display serious difficulties in mental operations, successive operations (particularly backward operations, such as 100 minus 7), and multistep numerical problems. Written arithmetical operations are notoriously easier than mental operations. Difficulties in calculation tasks in these patients correspond to 3 different types: (1) attention difficulties, (2) perseveration, and (3) impairment of complex mathematical concepts. Attention difficulties result in defects in maintaining the conditions of the tasks and impulsiveness in answers. Perseveration results in the incorrect answer (eg, 100 minus 7: 93, 83, 73, etc.). Impairment in the use of complex mathematical concepts results in an inability to analyze the conditions of numerical problems and to develop an algorithm for its solution. Most significant defects are found in solving numerical problems, whereas elementary arithmetic is usually much better preserved. When arithmetical problems are presented in mixed formats (eg, multiplications and additions), patients with frontal damage may fail, whereas no defects are usually observed when a single operation is used and no switching is required (29).

Spatial acalculia. Spatial acalculia is most frequently observed in right hemisphere pathology (04; 12). Hemineglect, topographic agnosia, constructional apraxia, and general spatial defects are usually correlated with spatial acalculia. Patients with spatial acalculia perform much better in orally presented arithmetical tasks than in written ones. Writing numbers is defective, with exclusive use of the right side of the page. Iterations (eg, 44 is written 444), inability to maintain a straight line, and spatial disorganization are observed. Alignment of the numbers in columns is severely abnormal. In performing written operations there may be confusion about where to locate the "carried" quantity; left neglect may result in failure to complete the operation, or inability to read the numerical sign, usually located at the left. In multiplication procedures, these patients may know the required steps in the operation, that is, when and how to carry, but they do not know where to place the quantities. These patients may have difficulty remembering multiplication tables and frequently mix procedures (eg, when subtracting, they add). Of note, patients with right hemisphere pathology more frequently make transcoding errors involving zeros because transcoding zeros entails visuospatial and integrative processes typically associated with the right hemisphere (11).

Prognosis and complications

The severity of acalculia and the pattern of recovery are variable. Recovery may be related to the size of the lesion and the etiology of the damage. In general, recovery in primary acalculia is limited. In secondary acalculias, recovery correlates with the recovery observed in the primary defect (aphasia, alexia, etc.). Caporali and colleagues suggest that recovery from acalculia is possible in the first months poststroke, that initial severity does not significantly influence recovery, and that initial severity correlates with recovery of auditory comprehension (19). Bernal and colleagues reported a patient with a severe acalculia and compared the fMRI pattern of activation with 4 normal age-matched controls (15). Both patient and controls showed specific task-related activation (especially left posterior temporal-parietal cortex) with some differences that may indicate brain plasticity, even though no significant recovery in calculation abilities was observed in the patient.

Basso and colleagues (09) analyzed the co-occurrence of aphasia and acalculia in 98 left-brain-damaged patients and the spontaneous recovery from acalculia in 92 patients. A significant association between aphasia and acalculia was observed although 19 participants exhibited aphasia with no acalculia and 6 acalculia with no aphasia. The authors concluded that dissociations between language and calculation disorders are not uncommon. A significant improvement was observed between a first examination carried out within the first 5 months post-onset and a second examination carried out on average 5 months later.

Clinical vignette

A 68-year-old woman with a high school level of education was hospitalized due to a sudden loss of sensitivity in her left hemibody. At the incoming neurologic exam, left hypoesthesia in face and arm, left homonymous hemianopia, disorientation in time and place, and left hemineglect were found. CT scans revealed an ischemic lesion involving the temporal and parietal branches of the right middle cerebral artery.

Neuropsychological testing indicated a significant left neglect. Neglect was observed in drawing, reading, writing, and performing spontaneous activities. Severe spatial and constructional defects were also found.

A severe spatial acalculia was noted. The patient could not align numbers in columns. Adding and multiplying were impossible because the patient could not find where to place the "carried" or "borrowed" quantities. She confused "plus" and "multiplication" signs and could not read numbers at the left (eg, 9231 was read as 31). Due to these defects, the patient failed in performing all the written numerical operations that were presented. Nonetheless, her ability to perform arithmetical operations orally and to solve numerical problems was nearly normal. A diagnosis of spatial acalculia associated with spatial alexia, spatial agraphia, hemispatial neglect, and general spatial defects was presented.

During the following weeks neglect mildly improved. Still, she continued to be unable to perform any written arithmetical operation. Defects in spatial orientation continued to be significant. One year later, she began to attend a rehabilitation program that lasted for about 1 year. Improvement was significant. At the end of this program, the patient could perform simple written arithmetical operations.

Biological basis

Etiology and pathogenesis

Acalculia is most often caused by a stroke, tumor, or trauma. Acalculia is usually present in dementia (47). Acalculia is also common in posterior cortical atrophy (39), corticobasal degeneration (46), left posterior peri-insular infarct (16), and left thalamic damage (38).

Primary acalculia is associated with left posterior parietal damage (05). Pure global acalculia has been reported following left subangular lesions (42). It has been suggested that different cerebral pathways may be responsible for processing rote numerical knowledge (eg, multiplication tables) and semantic knowledge of numerical quantities. Dehaene and Cohen have proposed that a left subcortical network contributes to the storage and retrieval of rote verbal arithmetical facts, whereas an inferior parietal network is dedicated to the mental manipulation of numerical quantities (24). Asada and colleagues suggested that acalculia is associated with a defect in manipulating mental images; moreover, mental imagery disturbances can be found in cases of right and left parietal damage, and hence acalculia can be due to both right and left hemisphere pathology (06). “Primary progressive acalculia” as initial manifestation of a degenerative disease has been described (03).

Neuroimaging techniques (eg, fMRI) have been used to analyze the pattern of brain activation during calculation tasks. It has been demonstrated that different brain areas are active during arithmetical tasks, but the specific pattern of brain activity depends on the type of test that is used. It has been found that at least the following brain areas become activate during calculation: upper cortical surface and anterior aspect of the left middle frontal gyrus; supramarginal and angular gyrus (bilaterally) (59); left dorsolateral prefrontal and premotor cortices, Broca area, and inferior parietal cortex; and left parietal and inferior occipitotemporal regions (lingual and fusiform gyri) (51). The diversity of brain areas involved in arithmetical processes support the assumption that calculation ability represents a multifactor skill, including verbal, spatial, memory, body knowledge, and executive function abilities (05). Dehaene and colleagues, however, have proposed that the human ability for arithmetic is associated with activation of specific brain areas (25). Neuroimaging studies with humans have demonstrated that the intraparietal sulcus is systematically activated during diverse number tasks and could be regarded as the most crucial brain region in the understanding and the use of quantities (45). Other brain areas such as the precentral area and the inferior prefrontal cortex are also activated when subjects engage in mental calculations. An unusual case of acalculia following an infarct restricted to the left intraparietal sulcus was reported (07); as expected, the patient presented significant deficits in basic numerical processing tasks. Defects were also observed in numerical comparisons, counting, and subitizing (direct perceptual apprehension of the numerosity of a group). The authors concluded that the underlying deficit referred to an impairment in the perception and manipulation of quantities. Using fMRI, a cluster of neurons selective for visually presented numbers in healthy human adults was found (34). This visual number form area was observed in the inferior temporal gyrus of both hemispheres. In addition to the intraparietal sulcus, a circuit involving an extended set of interconnected areas, including prefrontal, posterior parietal, occipito-temporal, and hippocampal areas, supporting arithmetical abilities, has been proposed (Peters and Smedt 2017).

It has been suggested that right hemisphere acalculia encompasses a wide variety of complex manifestations, affecting even simple calculation that cannot be explained as spatial disorders (60). Benavides-Varela and colleagues studied 30 right-hemisphere damaged patients and 35 normal controls on several numerical tasks (13). The results showed that both groups significantly differ in number comprehension, transcoding, and written operations, particularly subtractions and multiplications. Spatial errors were associated with extensive lesions in fronto-temporo-parietal, whereas pure arithmetical errors appeared with more confined lesions in the right angular gyrus and its proximity. The authors concluded unilateral right hemisphere lesions can directly affect basic numerical processes, and right-hemisphere acalculia is not only ascribable to visuospatial impairments.

Rosca proposed that arithmetic knowledge impairments can be due to either "memory" defects or "monitoring" problems. Patients with "memory" disturbances typically have left parietal lesions, whereas "monitoring" impairments are associated with frontal damage (52). Arithmetical procedural impairments, on the other hand, can be observed in cases of basal ganglia pathology. In consequence, it can be assumed that a fronto-parieto-subcortical circuit is responsible for complex arithmetical processing.

Using functional magnetic resonance imaging (fMRI), Zhou and colleagues analyzed the neural circuits supporting mathematical problem solving and arithmetical computation; results revealed that mathematical problem solving typically had greater activation than arithmetical computation in the 7 regions of the semantic system proposed by Binder and colleagues: posterior inferior parietal lobe (angular gyrus), middle temporal gyrus, fusiform and parahippocampal gyri, dorsomedial prefrontal cortex, inferior frontal gyrus, ventromedial prefrontal cortex, and posterior cingulate gyrus (17; 68). Arithmetical computation typically had greater activation in the supplementary motor area and left precentral gyrus. The authors suggested that the brain semantic system supports mathematical problem solving.

Symptoms of Gerstmann syndrome (acalculia with agraphia, disorders in right-left orientation, and finger agnosia) (31) can be found during direct cortical stimulation in the angular gyrus region (57). Using fMRI it has been observed that the left angular gyrus is not only involved in arithmetic tasks requiring simple fact retrieval, but may show significant activations as a result of relatively short training of complex calculation (26). Ictal Gerstmann syndrome has been reported in a patient with focal seizures in the speech dominant hemisphere (69).

Colvin and colleagues (21) investigated numerical abilities in a split-brain patient using experiments that examined the hemispheres' abilities to make magnitude comparisons. One experiment examined the ability to enumerate sets of stimuli and another 2 experiments required judgments about 2 concurrently presented stimuli that were either identically coded (ie, 2 Arabic numerals, 2 number words, or 2 arrays of dots) or differently coded (eg, an Arabic numeral and a number word). Both hemispheres were equally able to enumerate stimuli and to make comparisons between numerical representations regardless of stimuli coding. However, the left hemisphere was more accurate than the right when the task involved number words. Pu and colleagues, using electrical stimulation during brain surgery, observed that different brain sites in the left parietal lobe were specifically related to multiplication or subtraction processing whereas in the right hemisphere no brain sites associated to numerical processing were found (49).

Cohen and colleagues reported a patient with a lesion in the left perisylvian area who showed a severe impairment in all tasks involving numbers in a verbal format, such as reading aloud, writing to dictation, or responding verbally to questions of numerical knowledge. In contrast, her ability to manipulate nonverbal representation of numbers, ie, Arabic numerals, was comparatively well preserved (20). This observation supports the proposal that language and calculation disorders can be dissociated. When patients with aphasia are tested with a calculation battery adapted to their language difficulties, remarkable improvement in arithmetical skills may be observed (54).

Some authors have assumed that language and numerical concepts are differently organized in the brain and follow distinct developmental patterns in children (30). Other authors have suggested that calculation and language are mediated by partially different and also partially overlapped brain systems (08). Semenza and colleagues have emphasized that calculation and language usually share the same hemisphere (61). Using electrostimulation mapping it has also been observed that language and calculation areas are partially overlapped and partially independent (58).

Differential diagnosis

Acalculia represents a significant component in global cognitive disorders, usually observed in dementia and in cases of traumatic brain injury. It has been proposed that mathematical abilities represent excellent predictors of general intellectual performance in dementia (56; 50). Acalculia correlates with severity of dementia. Although calculation and numeral transcoding deficits are often prominent in early courses of the disease, deficits in semantic processing and basic number processing are less severe. The retrieval of arithmetic facts is generally spared with aging but can be impaired in Alzheimer disease patients. Using several mathematical tests Crutch and Warrington observed different patterns of calculation disorders in Alzheimer disease and frontotemporal dementia (22). The group with frontotemporal dementia was subdivided into those with a semantic dementia and those with prominent frontal features (nonsemantic dementia). On the quantity facts test, the semantic dementia subgroup was more impaired than both the Alzheimer disease patient group and the nonsemantic frontotemporal dementia subgroup. A further analysis revealed several interesting patterns of performance, including a dissociation between impaired performance on the quantity facts and number operations tests and preserved performance on the Graded Difficulty Arithmetic Test. Of note, calculation defects in dementia can be different depending on the characteristics of the writing system; thus, it has been observed that Chinese-speaking patients make significantly more intrusion errors than English-speaking ones, due to the ideographical nature of both Chinese characters and Arabic numbers (63).

Calculation abilities are strongly correlated with educational level. People with lower education may have difficulties in performing some arithmetical tasks.

Diagnostic workup

A few models of testing for calculation abilities have been developed (27; 05; 28; 67). Usually it is considered that testing for acalculia should include the following tasks:

	• Counting forward and backward. Patients are asked to count from 1 to 10 or from 30 to 40 (forward condition), and from 10 to 1 or 40 to 30 (backward condition).
	• Writing and reading numbers with different levels of complexity (eg, 7, 31, 106, 1639, 10002). Different quantities are dictated to the patient: 1-digit numbers, 2-digit numbers, numbers requiring the use of a positional value, etc.
	• Transcoding from a verbal to a numerical code and from a numerical to a verbal code. Numbers using a numerical code are written (eg, 7, 23, 109), and the patient is asked to write them with letters (eg, seven, twenty-three, one hundred nine). Numbers using a verbal code are written (eg, three, forty-nine, three hundred seventy-six) and the patient is asked to write these quantities with numbers (eg, 3, 49, 376).
	• Reading and writing arithmetical signs (+, -, ·/·, x, =), interpreting arithmetical signs (eg, 9 + 5 = _; 9 - 5 = _).
	• Magnitude comparisons: The patient is asked which number is larger and which is smaller (eg, 79 and 103).
	• Mental calculation: adding, subtracting, multiplying, and dividing single-digit simple quantities and complex (2 or 3 digits) quantities.
	• Written calculation: adding, subtracting, multiplying, and dividing single-digit simple quantities, and complex (2 or 3 digits) quantities.
	• Successive arithmetical operations requiring the patient to add or to subtract or to do both (eg, 1, 4, 7....; 100, 93, 86...).
	• Aligning numbers in columns: a series of numbers (eg, 27, 2, 2407) are dictated to the patient. The patient is required to place them in a column, as if for adding.
	• Numerical problems: problems requiring the use of 1 or several numerical operations are presented to the patient (eg, "There are 18 books in 2 shelves; in one of them there is double the number than in the other. How many books are there in each one?").
	• Numerical knowledge (eg, "How many days are in a week?" "How many weeks are there in a year?").

Ardila suggested that due to new technological advances (eg, the extended use of pocket calculators), new assessment procedures more in accordance with current living conditions are to be developed (01).

Management

The recovery from acalculia is variable. In a sample of 59 stroke patients with alexia, agraphia, acalculia, and combinations, 13 (22%) fully recovered 24 months later; the main factors influencing recovery were initial severity of reading, writing and calculation impairment, age, neglect, and level of education (70).

Several case studies have addressed the rehabilitation of calculation defects. Girelli and colleagues reported significant improvement following a rehabilitation program in 2 patients with a selective deficit in multiplication facts (32). Hittmair-Delazer and colleagues reported a patient unable to recall and use "arithmetical facts.” After the application of a rehabilitation program, a selective improvement for each single arithmetical fact was observed (37). Rosselli and Ardila reported the rehabilitation of a 68-year-old patient presenting with spatial acalculia associated with spatial alexia and spatial agraphia (55). The rehabilitation program was based on overcoming the hemispatial neglect and the associated spatial difficulties. A significant improvement was observed.

Tsvetkova proposes a multistep rehabilitation program for primary calculation disturbances associated with brain pathology (65). It includes multiple activities that range from simple to progressively more complex: counting real objects, matching the quantity of object with the written number, dividing the objects in subgroups (equal and nonequal), counting the objects in the subgroups, matching the written number, training on the positional value of the numbers (units, tens, hundreds, etc.), using numbers in columns, training on arithmetical operations using a multistep and piecemeal detailed process, dividing big numbers into smaller ones, analyzing the components of the quantities, and training in the correct sequence of arithmetical operations. Tsvetkova reports significant improvements after long-term training.

A review of the literature published from 1980 to 2007 dealing with the treatment efficacy of calculation disorders found few studies that tackled the issue of efficacy; efficiency of acalculia therapy seems promising, but data are scanty (10). Rosselli and Ardila point out that rehabilitation must start with an analysis of the errors presented by the acalculic patient and of the types of associated disorders (55). Different types of errors can be found in calculation tasks:

	(1) Errors in numbers, such as: errors in understanding the value of the numbers (inability to recognize which number is larger (eg, 96 or 102), lexicalization (127 = 10027), decomposition (137 = 13, 7), inversions (21 = 12), substitutions (hierarchy errors; eg, 500 = 50; or, order, eg, 5 = 6), omissions (eg, 712 = 71), additions (eg, 23 = 233), transcoding errors (7 = 9; 3 = eight).
	(2) "Carrying" errors: carrying is omitted or inappropriately used.
	(3) "Borrowing errors": difficulties with "zero"; "borrowing is omitted, or inappropriately used.
	(4) Errors in the use of basic numerical principles, such as the use of zero and the multiplication tables.
	(5) Errors in algorithms, such as: inability to complete a numerical operation, inappropriate use of the space, mixture of procedures (eg, adding and multiplying), wrong sequence (numerical operation is begun at the left), inappropriate algorithm (eg, when multiplying, numbers are placed as for dividing), and reasoning errors (ie, impossible results).
	(6) Errors in the use of symbols, such as forgetfulness and substitutions of numerical symbols.

Calculation disturbances associated with different brain pathologies are diverse (anarithmetia, spatial acalculia, aphasic acalculia, etc.). Rehabilitation must focus on the specific deficits with which the patient presents. In cases of primary acalculia (anarithmetia) the rehabilitation program must be directed specifically to the fundamental computational defect. In cases of secondary acalculia, the therapy must be directed to the primary defect (spatial neglect, attention deficit, aphasia, etc.).

Patients with aphasic acalculia, when improving from their oral language defect, also significantly and concurrently improve in their calculation disturbance. Also patients with spatial acalculia, when improving from neglect, significantly improve in their written calculation ability.